Shalyt/ASyMOB-Algebraic_Symbolic_Mathematical_Operations_Benchmark huggingface dataset:
Problem: Compute the first 5 nonzero terms of the Maclaurin series of $ e^{\sin(x)} $
Answer (LaTeX): 1+x+\frac{x^2}{2}-\frac{x^4}{8}-\frac{x^5}{15}+\cdots
Answer (SymPy): -x**5/15 - x**4/8 + x**2/2 + x + 1
Problem: Find the radius of convergence of the series:
$ \sum_{n=1}^\infty \left(\frac{ \left((2 \cdot n)!\right) \cdot x^n }{ n^{2 \cdot n} }\right) $
Answer (LaTeX): \frac{e^2}{4}
Answer (SymPy): E**2 / 4
Problem: Compute the first 6 nonzero terms of the Maclaurin series of $f(x) = \sin(x) \cdot \cos\left(\frac{ \pi }{ 4 }\right) + \cos(x) \cdot \sin\left(\frac{ \pi }{ 4 }\right)$
Answer (LaTeX): \frac{1}{34560\cdot\sqrt{2}}\cdot\left(288\cdot x^5+1440\cdot x^4-5760\cdot x^3-17280\cdot x^2+34560\cdot x+34560\right)
Answer (SymPy): sqrt(2)*(x**5 + 5*x**4 - 20*x**3 - 60*x**2 + 120*x + 120)/240
Problem: Compute the first 4 nonzero terms of the Maclaurin series of $f(x) = e^x \cdot \cos(x)$
Answer (LaTeX): 1+x-\frac{x^3}{3}-\frac{x^4}{6}
Answer (SymPy): -x**4/6 - x**3/3 + x + 1
Problem: Compute $\lim_{x \to 0}\left(\frac{ 2 \cdot \cos(x)+4 }{ 5 \cdot x^3 \cdot \sin(x) }-\frac{ 6 }{ 5 \cdot x^4 }\right)$
Answer (LaTeX): \frac{1}{150}
Answer (SymPy): 1/150
Problem: Evaluate $\lim_{x \to 0^{+}}\left(\left(\frac{ \tan\left(\frac{ x }{ 2 }\right) }{ \frac{ x }{ 2 } }\right)^{\frac{ 3 }{ x^2 }}\right)$
Answer (LaTeX): $e^{\frac{1}{4}}$
Answer (SymPy): exp(1/4)
Problem: Evaluate
$ \lim_{x \to 5} \left( \frac{ 3 \cdot x }{ x-5 }-\frac{ 3 }{ \ln\left(\frac{ x }{ 5 }\right) } \right) $
Answer (LaTeX): \frac{3}{2}
Answer (SymPy): 3/2
Problem: Evaluate
$ \lim_{x \to \infty} \left(x - x^2 \cdot \ln\left(1 + \frac{ 1 }{ x }\right)\right) $
Answer (LaTeX): \frac{1}{2}
Answer (SymPy): 1/2
Problem: Evaluate
$ \lim_{x \to 0^{+}} \left( \left( \frac{ \tan(2 \cdot x) }{ 2 \cdot x } \right)^{\frac{ 1 }{ 3 \cdot x^2 }} \right) $
Answer (LaTeX): e^{\frac{4}{9}}
Answer (SymPy): e**(4/9)
Problem: Evaluate
$ \lim_{x \to 0}\left( \left| \frac{ -\sin(x) }{ x } \right| \right)^{\frac{ 1 }{ 4 \cdot x^2 }} $
Answer (LaTeX): e^{\frac{-1}{24}}
Answer (SymPy): e**(-1/24)
Problem: Integrate 
$ \int \sin(x)^4 \cdot \cos(x)^6 dx $
Answer (LaTeX): $C+\frac{1}{320}\cdot\left(\sin(2\cdot x)\right)^5+\frac{1}{128}\cdot\left(\frac{3\cdot x}{2}-\frac{\sin(4\cdot x)}{2}+\frac{\sin(8\cdot x)}{16}\right)$
Answer (SymPy): 3*x/256 + sin(2*x)**5/320 - sin(4*x)/256 + sin(8*x)/2048
Problem: Calculate the integral:
$
\int \frac{ \sqrt[5]{x}+\sqrt[5]{x^4}+x \cdot \sqrt[5]{x} }{ x \cdot \left(1+\sqrt[5]{x^2}\right) }   dx
$
Answer (LaTeX): C+5\cdot\arctan\left(\sqrt[5]{x}\right)+\frac{5}{4}\cdot\sqrt[5]{x}^4
Answer (SymPy): C + 5*x**(4/5)/4 + 5*atan(x**(1/5))
Problem: Solve the integral:
$
\int \frac{ 1 }{ \sin(x)^7 \cdot \cos(x) }   dx
$
Answer (LaTeX): C+\ln\left(\left|\tan(x)\right|\right)-\frac{3}{2\cdot\left(\tan(x)\right)^2}-\frac{3}{4\cdot\left(\tan(x)\right)^4}-\frac{1}{6\cdot\left(\tan(x)\right)^6}
Answer (SymPy): C + log(Abs(tan(x))) - 3 / (2 * tan(x)**2) - 3 / (4 * tan(x)**4) - 1 / (6 * tan(x)**6)
Problem: Compute the integral:
$
-2 \cdot \int x^{-4} \cdot \left(4+x^2\right)^{\frac{ 1 }{ 2 }}   dx
$
Answer (LaTeX): C+\frac{1}{6}\cdot\left(\frac{4}{x^2}+1\right)\cdot\sqrt{\frac{4}{x^2}+1}
Answer (SymPy): (C*x**2 + sqrt((x**2 + 4)/x**2)*(x**2 + 4)/6)/x**2
Problem: Solve the integral:
$
\int \left(\frac{ x+4 }{ x-4 } \right)^{\frac{ 3 }{ 2 }}   dx
$
Answer (LaTeX): C+\sqrt{\frac{x+4}{x-4}}\cdot(x-20)-12\cdot\ln\left(\left|\frac{\sqrt{x-4}-\sqrt{x+4}}{\sqrt{x-4}+\sqrt{x+4}}\right|\right)
Answer (SymPy): C + sqrt((x + 4)/(x - 4)) * (x - 20) - 12 * log(Abs((sqrt(x - 4) - sqrt(x + 4)) / (sqrt(x - 4) + sqrt(x + 4))))
Problem: Compute the integral: $ \int \frac{ -1 }{ x^2 \cdot \left(3+x^3\right)^{\frac{ 5 }{ 3 }} }   dx $
Answer (LaTeX): \frac{1}{9}\cdot\sqrt[3]{1+\frac{3}{x^3}}+\frac{1}{18\cdot\left(1+\frac{3}{x^3}\right)^{\frac{2}{3}}}
Answer (SymPy): (x**3 + 2)/(6*x**3*(1 + 3/x**3)**(2/3))
Problem: Compute the integral:
$ \int \frac{ 4 \cdot x+\sqrt{4 \cdot x-5} }{ 5 \cdot \sqrt[4]{4 \cdot x-5}+\sqrt[4]{(4 \cdot x-5)^3} }   dx $
Answer (LaTeX): C+25\cdot\sqrt[4]{4\cdot x-5}+\frac{1}{5}\cdot\sqrt[4]{4\cdot x-5}^5-\frac{4}{3}\cdot\sqrt[4]{4\cdot x-5}^3-\frac{125}{\sqrt{5}}\cdot\arctan\left(\frac{1}{\sqrt{5}}\cdot\sqrt[4]{4\cdot x-5}\right)
Answer (SymPy): C + (4*x - 5)**(5/4)/5 - 4*(4*x - 5)**(3/4)/3 + 25*(4*x - 5)**(1/4) - 25*sqrt(5)*atan(sqrt(5)*(4*x - 5)**(1/4)/5)
Problem: Solve the integral:
$
\int \frac{ 3 }{ \sin(2 \cdot x)^7 \cdot \cos(-2 \cdot x) }   dx
$
Answer (LaTeX): C+\frac{3}{2}\cdot\left(\ln\left(\left|\tan(2\cdot x)\right|\right)-\frac{3}{2\cdot\left(\tan(2\cdot x)\right)^2}-\frac{3}{4\cdot\left(\tan(2\cdot x)\right)^4}-\frac{1}{6\cdot\left(\tan(2\cdot x)\right)^6}\right)
Answer (SymPy): C + (3/2) * (log(Abs(tan(2*x))) - 3/(2 * tan(2*x)**2) - 3/(4 * tan(2*x)**4) - 1/(6 * tan(2*x)**6)
)
Problem: Solve the integral:
$
\int \frac{ 1 }{ (\sin(8 \cdot x))^5 }   dx
$
Answer (LaTeX): C+\frac{1}{128}\cdot\left(2\cdot\left(\tan(4\cdot x)\right)^2+6\cdot\ln\left(\left|\tan(4\cdot x)\right|\right)+\frac{1}{4}\cdot\left(\tan(4\cdot x)\right)^4-\frac{2}{\left(\tan(4\cdot x)\right)^2}-\frac{1}{4\cdot\left(\tan(4\cdot x)\right)^4}\right)
Answer (SymPy): C + Rational(1, 128) * (2 * tan(4 * x)**2 + 6 * log(Abs(tan(4 * x))) + Rational(1, 4) * tan(4 * x)**4 - 2 / tan(4 * x)**2 - 1 / (4 * tan(4 * x)**4))
Problem: Evaluate the integral:
$ \int \left(x^3 + 3\right) \cdot \cos(2 \cdot x)   dx $
Answer (LaTeX): \frac{1}{256}\cdot\left(384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-256\cdot C-192\cdot x\cdot\sin(2\cdot x)\right)
Answer (SymPy): -C + x**3*sin(2*x)/2 + 3*x**2*cos(2*x)/4 - 3*x*sin(2*x)/4 + 3*sin(2*x)/2 - 3*cos(2*x)/8
Problem: Use factoring to calculate the following limit.
$ \lim_{x \rightarrow K} \frac {{x}^4-K^4} {{x}^5-K^5} $
Answer (LaTeX): \frac{4}{5 K}
Answer (SymPy): 4/(5*K)
Problem: Find the limit. $ \lim_{x \to 0} \frac{1-\cos\!\left(10x\right)}{\cos^{2}\!\left(6x\right)-1}$
Answer (LaTeX): \frac{-25}{18}
Answer (SymPy): -25/18
Problem: Evaluate the limit. $ \lim_{x\to 1} \dfrac{x^2+11x-12}{\ln x}=$
Answer (LaTeX): 13
Answer (SymPy): 13
Problem: Evaluate the limit below, given that $f(t)=\left(\frac{4^t+6^t}{4}\right)^{1/t}$. $\lim\limits_{t\to+\infty} f(t)$
Answer (LaTeX): 6
Answer (SymPy): 6
Problem: Calculate the integral. $\int_{2}^{\infty} 3x^{2}e^{-x^{3}}  dx$
Answer (LaTeX): \frac{1}{e^{8}}
Answer (SymPy): e**(-8)
Problem: Evaluate the indefinite integral. $\int \tan^{3}\!\left(x\right)\sec^{9}\!\left(x\right)   dx$
Answer (LaTeX): \frac{\sec^{11}{\left(x \right)}}{11} - \frac{\sec^{9}{\left(x \right)}}{9}
Answer (SymPy): sec(x)**11/11 - sec(x)**9/9
Problem: Evaluate the indefinite integral.
$\int 208 \cos^4(16x) dx$
Answer (LaTeX): 78 x + \frac{13 \sin{\left(16 x \right)} \cos^{3}{\left(16 x \right)}}{4} + \frac{39 \sin{\left(16 x \right)} \cos{\left(16 x \right)}}{8}
Answer (SymPy): 78*x + 13*sin(16*x)*cos(16*x)**3/4 + 39*sin(16*x)*cos(16*x)/8
Problem: Evaluate the integral.
$ \int \frac{10x^2-48x-38}{x^3-5x^2-8x+48} dx $
Answer (LaTeX): \frac{2 \left(\left(x - 4\right) \left(3 \log{\left(\left|{x - 4}\right| \right)} + 2 \log{\left(\left|{x + 3}\right| \right)}\right) + 5\right)}{x - 4} 
Answer (SymPy): 2*((x - 4)*(3*log(Abs(x - 4)) + 2*log(Abs(x + 3))) + 5)/(x - 4)
Problem: Evaluate the integral. $ \int e^{x}\sqrt{64-e^{2x}} dx$
Answer (LaTeX): \frac{e^{x} \sqrt{64 - e^{2 x}}}{2} + 32 \operatorname{asin}{\left(\frac{e^{x}}{8} \right)}
Answer (SymPy): e**x*sqrt(64 - e**(2*x))/2 + 32*asin(e**x/8)
Problem: Evaluate $\lim_{x \to 0} \frac{e^{-3x^3}-1+3x^3-\frac{9}{2}x^6}{12x^9}$
Answer (LaTeX): \frac{-3}{8}
Answer (SymPy): -3/8
Problem: Solve the following first-order differential equation:
$ \frac{dy}{dx} + 2y = e^{-x}, \quad y(0) = 1. $
Answer (LaTeX): e^{-x}
Answer (SymPy): e**(-x)
Problem: Consider the differential equation $\frac{dy}{dx} = xy$. Find the value of $y(\sqrt{2})$ given that $y(0) = 2$.
Answer (LaTeX): 2e
Answer (SymPy): 2*e
Problem: Evaluate the following limit:
$
\lim_{n \to \infty} \left(\sqrt{n^2+2n-1}-\sqrt{n^2+3}\right).
$
Answer (LaTeX): 1
Answer (SymPy): 1
Problem: Evaluate $\lim\limits_{x\to 4}\frac{x-4}{\sqrt{x}-2}$.
Answer (LaTeX): 4
Answer (SymPy): 4
Problem: Evaluate $ \int_0^4(2x-\sqrt{16-x^2})dx$.
Answer (LaTeX): 16 - 4 \pi
Answer (SymPy): 16 - 4*pi
Problem: Evaluate the series $\sum\limits_{n=1}^\infty\frac{1}{(n+1)(n+3)}$.
Answer (LaTeX): \frac{5}{12}
Answer (SymPy): 5/12
Problem: Evaluate the limit $\lim\limits_{x\to 0}\frac{(1+x)^{\frac{1}{x}}-e}{x}$.
Answer (LaTeX): -\frac{ e}{2}
Answer (SymPy): -e/2
Problem: Evaluate the series $\sum\limits_{n=0}^\infty \frac{1}{2n+1}\left(\frac12\right)^{2n+1}$.
Answer (LaTeX): \ln\sqrt{3}
Answer (SymPy): log(3)/2
Problem: Evaluate the limit $\lim\limits_{n\to\infty}\sum\limits_{k=0}^{n-1}\frac{1}{\sqrt{n^2-k^2}}$.
Answer (LaTeX): \frac{\pi}{2}
Answer (SymPy): pi/2
Problem: Evaluate the iterated integral $\displaystyle{\int_0^1dy\int_y^1(e^{-x^2}+e^x)dx}$.
Answer (LaTeX): \frac{3}{2}-\frac12 e^{-1}
Answer (SymPy): (3*e - 1)/(2*e)
Problem: What is the integral of $ 2x - x^7atan(3) $
Answer (LaTeX): x^2-\frac{1}{8} x^8 \tan ^{-1}(3)
Answer (SymPy): -x**8*atan(3)/8 + x**2
Problem: What is the integral of $ 1 + x + x^3*cosh(2) $
Answer (LaTeX): \frac{1}{4} x^4 \cosh (2)+\frac{x^2}{2}+x
Answer (SymPy): x**4*cosh(2)/4 + x**2/2 + x
Problem: What is the integral of $ 12 + 6cosh(x) $
Answer (LaTeX): 12 x + 6 \sinh{\left(x \right)}
Answer (SymPy): 12*x + 6*sinh(x)
Problem: What is the integral of 4x^7 + sin(1 + x)
Answer (LaTeX): \frac{x^8}{2} - \cos(1+x) 
Answer (SymPy): x**8/2 - cos(x + 1)
Problem: What is the integral of 2x + 2x^2 + x*[(x + x*e^x)^(-1)]
Answer (LaTeX): \frac{2 x^3}{3}+x^2-2 \tanh ^{-1}\left(2 e^x+1\right)
Answer (SymPy): 2*x**3/3 + x**2 + x - log(exp(x) + 1)
Problem: What is the integral of -x + cos[ln(sin(3))] * ln(3x)
Answer (LaTeX): -\frac{1}{2} x (x-2 \log (3 x) \cos (\log (\sin (3)))+2 \cos (\log (\sin (3))))
Answer (SymPy): -1*x*((x - 2*log(3*x, E)*cos(log(sin(3), E))) + 2*cos(log(sin(3), E)))/2
Problem: What is the integral of 3x - 4*[cos(x+3)]*x^2
Answer (LaTeX): \frac{3 x^2}{2}-4 \left(x^2-2\right) \sin (x+3)-8 x \cos (x+3)
Answer (SymPy): -8*x*cos(x + 3) + ((3*x**2)/2 - 4*(x**2 - 2)*sin(x + 3))
Problem: What is the integral of -3 + atan(x) + ln(tanh(3))
Answer (LaTeX): x \arctan(x) - \frac{1}{2} \ln(1 + x^2) + x \ln(\tanh(3)) - 3x + C
Answer (SymPy): x*atan(x) - 3*x + x*log(tanh(3)) - log(x**2 + 1)/2
Problem: What is the integral of e^{x \left(x + 4\right)^{2}} \left(x + 4\right) \left(3 x + 4\right)
Answer (LaTeX): e^{x (x+4)^2}
Answer (SymPy): e**(x*(x + 4)**2)
Problem: What is the integral of -e^{3x} * sin(e^{3x})
Answer (LaTeX): \frac{1}{3} \cos \left(e^{3 x}\right)
Answer (SymPy): cos(e**(3*x))/3
Problem: If $\log _{2} x-2 \log _{2} y=2$, determine $y$, as a function of $x$
Answer (LaTeX): \frac{1}{2} \sqrt{x}
Answer (SymPy): sqrt(x)/2
Problem: If $f(x)=2 x+1$ and $g(f(x))=4 x^{2}+1$, determine an expression for $g(x)$.
Answer (LaTeX): x^2-2 x+2
Answer (SymPy): x**2 - 2*x + 2
Problem: Solve the following integral $\int_0^{\frac{\pi}{2}} \frac{x \sin(2x)}{1 + \cos^2(2x)}   dx$
Answer (LaTeX): Pi^2 / 16
Answer (SymPy): pi**2 / 16
Problem: Solve the following integral:
$\int_{1}^{2} \frac{e^x(x - 1)}{x(x + e^x)}   dx$ 
Answer (LaTeX): \ln\left( \frac{2 + e^2}{2 + 2e} \right)
Answer (SymPy): log((E**2 + 2)/(2*E + 2), E)
Problem: Solve the following integral:
$\int_{0}^{\pi} \log(\sin(x))   dx$
Answer (LaTeX): -\pi \log (2)
Answer (SymPy): -pi*log(2, E)
Problem: Evaluate the following hypergeometric function. Return a closed-form symbolic answer.

$ {}_2F_1\left( \begin{array}{c} 1,1\ \\ 2 \end{array}; -1 \right) $
Answer (LaTeX): \log (2)
Answer (SymPy): log(2, E)
Problem: Evaluate the following hypergeometric function. Return a closed-form symbolic answer.

$ {}_2F_1\left( \begin{array}{c} 1,1 \\ 3 \end{array}; -2 \right) $
Answer (LaTeX): \frac{3 \log (3)}{2}-1
Answer (SymPy): -1 + (3*log(3, E))/2
Problem: Evaluate the following hypergeometric function. Return a closed-form symbolic answer.

$ {}_3F_2\left( \begin{array}{c} 1,1,1 \\ 2,2 \end{array}; -1 \right) $
Answer (LaTeX): \frac{\pi ^2}{12}
Answer (SymPy): pi**2/12
Problem: Evaluate the following hypergeometric function. Return a closed-form symbolic answer.

$ {}_3F_2\left( \begin{array}{c} -1,-1,-1 \\ -1,-1 \end{array}; x \right) $
Answer (LaTeX): 1-x
Answer (SymPy): 1-x
Problem: Solve the following integral. Return a closed-form symbolic answer.

\int \frac{ 1 }{ 1 + x^3 }   dx
Answer (LaTeX): -\frac{1}{6} \log \left(x^2-x+1\right)+\frac{1}{3} \log (x+1)+\frac{\tan ^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{\sqrt{3}}
Answer (SymPy): (log(x + 1, E)/3 - 1*log((x**2 - x) + 1, E)/6) + atan((2*x - 1)/(sqrt(3)))/(sqrt(3))
Problem: Solve the following integral.

\int \frac{(4 + (4 - 1)x^1)x^{2-1}}{2(1 + x^1 + x^{4})\sqrt{1 + x^1}} dx
Answer (LaTeX): \tan ^{-1}\left(\frac{x^2}{\sqrt{x+1}}\right)
Answer (SymPy): atan(x**2/sqrt(x + 1))
Problem: Evaluate the following hypergeometric function. Return a closed-form symbolic answer.

$ {}_2F_1\left( \begin{array}{c} 1, 1 \\ 1 \end{array}; -1 \right) $
Answer (LaTeX): \frac{1}{2}
Answer (SymPy): 1/2
Problem: Evaluate the following hypergeometric function.Return a closed-form symbolic answer.

$ {}_1F_1\left( \begin{array}{c} 1 \\ 1 \end{array}; 1 \right) $
Answer (LaTeX): e
Answer (SymPy): E
Problem: Evaluate the following hypergeometric function. Return a closed-form symbolic answer.

$ {}_2F_1\left( \begin{array}{c} 1, -2 \\ 2+1 \end{array}; 1 \right) $
Answer (LaTeX): \frac{1}{2}
Answer (SymPy): 1/2
Problem: Evaluate the following hypergeometric function. Return a closed-form symbolic answer.

$ {}_3F_1\left( \begin{array}{c} (1 + 1), -2, (1 + 3) \\ (2 + 1) \end{array}; \frac{1}{2} \right) $
Answer (LaTeX): \frac{5}{6}
Answer (SymPy): 5/6
Problem: Solve the following differential equation:
$ x \cdot y' + y = x \cdot \sin(x), y(\pi) = 1 $
Answer (LaTeX): -\cos(x) + \frac{\sin(x)}{x} 
Answer (SymPy): -cos(x) + sin(x)/x
Problem: Solve the following differential equation:
$ y' =  e^{ x } \cdot y , y(1) = 1 $
Answer (LaTeX): e^{e^x-e}
Answer (SymPy): e**(-e + e**x)
Problem: Solve the following differential equation:
$ y' = 2 \cdot x \cdot y^2 - y , y(1) = 1 $
Answer (LaTeX): \frac{e}{2 e x-3 e^x+2 e}
Answer (SymPy): e/(2*e*x + 2*e - 3*e**x)
Problem: Solve the following differential equation:
$ y' = x y^2 - y , y(1) = 1 $
Answer (LaTeX): \frac{e}{e x-e^x+e}
Answer (SymPy): e/(e*x + e - e**x)
Problem: Solve the following differential equation:
$ y' = \frac{y^2 + 2 \cdot x \cdot y }{x^2} , y(1) = 1 $
Answer (LaTeX): -\frac{x^2}{x-2}
Answer (SymPy): -x**2/(x - 2)
Problem: Solve the following differential equation:
$ y' = e^{-x} - 2 \cdot y , y(1) = 1 $
Answer (LaTeX): e^{-2 x} \left(e^x-e+e^2\right)
Answer (SymPy): (e**2 - e + e**x)/e**(2*x)
Problem: Solve the following differential equation:
$ y' = 3 \cdot x^2 \cdot \left( y^2 + 1 \right) , y(1) = 1 $
Answer (LaTeX): \tan \left(x^3+\frac{1}{4} (\pi -4)\right)
Answer (SymPy): tan(pi/4 + x**3 - 1)
Problem: Solve the following differential equation:
$ y' = \frac{2 (y + x)}{x} , y(1) = 1 $
Answer (LaTeX): 3 x^2-2 x
Answer (SymPy): x*(3*x - 2)
Problem: Solve the following differential equation:
$ y' = \frac{ (y + x)}{x} , y(1) = 1 $
Answer (LaTeX): x+x \log (x)
Answer (SymPy): x*(log(x) + 1)
Problem: Solve the following differential equation:
$ y' = x , y(1) = 1 $
Answer (LaTeX): \frac{1}{2} \left(x^2+1\right)
Answer (SymPy): x**2/2 + 1/2
Problem: Solve the following differential equation:
$ y' = x - 2 \cdot x \cdot y , y(1) = 1 $
Answer (LaTeX): \frac{1}{2} e^{-x^2} \left(e^{x^2}+e\right)
Answer (SymPy): e/(2*e**(x**2)) + 1/2
Problem: Solve the following differential equation:
$ y' = -y + 2 \cdot \sin (x) + 5 \cdot \sin (2 \cdot x) , y(0) = 0 $
Answer (LaTeX): -e^{-x} \left(-e^x \sin (x)-e^x \sin (2 x)+e^x \cos (x)+2 e^x \cos (2 x)-3\right)
Answer (SymPy): sin(2*x) - 2*cos(2*x) - sqrt(2)*cos(x + pi/4) + 3/e**x
Problem: Solve the following differential equation:
$ y' = \tan (y) , y(1) = 1 $
Answer (LaTeX): \sin ^{-1}\left(e^{x-1} \sin (1)\right)
Answer (SymPy): asin(e**(x - 1)*sin(1))
Problem: Solve the following differential equation:
$ y' = \sin ^2(y) + 2 \cos ^2(y) , y(1) = 1 $
Answer (LaTeX): -\tan ^{-1}\left(\sqrt{2} \tan \left(-\sqrt{2} x+\sqrt{2}-\tan
   ^{-1}\left(\frac{\tan (1)}{\sqrt{2}}\right)\right)\right)
Answer (SymPy): atan(sqrt(2)*tan(sqrt(2)*x - sqrt(2) + atan(sqrt(2)*tan(1)/2)))
Problem: Solve the following differential equation:
$ y' = 2 \cos ^2(y) - \sin ^2(y) , y(0) = 0 $
Answer (LaTeX): \tan ^{-1}\left(\sqrt{2} \tanh \left(\sqrt{2} x\right)\right)
Answer (SymPy): atan(sqrt(2)*tanh(sqrt(2)*x))
Problem: Solve the following differential equation:
$ y'' + y' + y + x = 0 , y(0) = 1 , y'(0) = 1 $
Answer (LaTeX): -\frac{1}{3} e^{-x/2} \left(3 e^{x/2} x-3 e^{x/2}-4 \sqrt{3} \sin
   \left(\frac{\sqrt{3} x}{2}\right)\right)
Answer (SymPy): -x + 1 + 4*sqrt(3)*sin(sqrt(3)*x/2)/(3*e**(x/2))
Problem: Solve the following differential equation:
$ y'' + y' - 6 \cdot y = 0 , y(0) = 0 , y'(0) = 1 $
Answer (LaTeX): \frac{1}{5} e^{-3 x} \left(e^{5 x}-1\right)
Answer (SymPy): (e**(5*x) - 1)/(5*e**(3*x))
Problem: Solve the following differential equation:
$ y'' + y = 0 , y(1) = 1 , y'(0) = 1 $
Answer (LaTeX): \sin (x)+\cos (x) (\sec (1)-\tan (1))
Answer (SymPy): sin(x) - (-sec(1) + tan(1))*cos(x)
Problem: Solve the following differential equation:
$ y'' - y = 0 , y(1) = 1 , y'(1) = 1 $
Answer (LaTeX): e^{x-1}
Answer (SymPy): e**(x - 1)
Problem: Solve the following differential equation:
$ y'' - 2 \cdot y' - 3 \cdot y = \sin (x) , y(1) = 1 , y'(1) = 1 $
Answer (LaTeX): \frac{1}{40} e^{-x-3} \left(20 e^{4 x}-8 e^{x+3} \sin (x)+3 e^{4 x} \sin (1)+4
   e^{x+3} \cos (x)+e^{4 x} \cos (1)+20 e^4+5 e^4 \sin (1)-5 e^4 \cos (1)\right)
Answer (SymPy): e**(-x - 3)*(-5*sqrt(2)*e**4*cos(pi/4 + 1) + 20*e**4 + e**(4*x)*cos(1) + 3*e**(4*x)*sin(1) + 20*e**(4*x) - 8*e**(x + 3)*sin(x) + 4*e**(x + 3)*cos(x))/40
Problem: Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty 1-\frac{81}{16 (n+1)^4} $
Answer (LaTeX): \frac{64 \sinh \left(\frac{3 \pi }{2}\right)}{585 \pi ^2}
Answer (SymPy): 64*sinh(3*pi/2)/(585*pi**2)
Problem: Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty \frac{\left(1-\frac{1}{(n+1)^2}\right) \left((n+1)^3-1\right)}{(n+1)^3+1} $
Answer (LaTeX): \frac{1}{3}
Answer (SymPy): 1/3
Problem: Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty 1-\frac{1}{4 n^2} $
Answer (LaTeX): \frac{2}{\pi }
Answer (SymPy): 2/pi
Problem: Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty \frac{1}{\left\lfloor \frac{n}{n+1}+\left\lfloor \frac{1}{n^2+1}\right\rfloor
   \right\rfloor +n^2}+1 $
Answer (LaTeX): \frac{\sinh (\pi )}{\pi }
Answer (SymPy): sinh(pi)/pi
Problem: Calculate the following infinite product. Give a finite, closed form answer.
$ \prod_{n=1}^\infty \left(1-\frac{1}{64 \cdot n^6}\right) \left(1-\frac{4}{3} \sin^2\left(3^{-n}\right)\right) $
Answer (LaTeX): \frac{4 \frac{\sin(1)}{ 1} \cosh \left(\frac{\sqrt{3} \pi }{2}\right)}{\pi ^3}
Answer (SymPy): 4*sin(1)*cosh(sqrt(3)*pi/2)/pi**3
Problem: Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \left(\frac{3}{5}\right)^n \left(1-\frac{1}{n}\right) $
Answer (LaTeX): \frac{3}{2}-\log \left(\frac{5}{2}\right)
Answer (SymPy): 3/2 - log(5/2)
Problem: Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{(-1)^n}{2 n+5} $
Answer (LaTeX): \frac{\pi }{4}-\frac{13}{15}
Answer (SymPy): pi/4 - 13/15
Problem: Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty 2^{1-3 n} \cdot 3^{n+1} $
Answer (LaTeX): \frac{18}{5}
Answer (SymPy): 18/5
Problem: Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{1}{n!} $
Answer (LaTeX): e-1
Answer (SymPy): e - 1
Problem: Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{1}{n^2-n-1} $
Answer (LaTeX): -\frac{\left(\sqrt{5}-5\right) \pi  \tan \left(\frac{\sqrt{5} \pi }{2}\right)}{5
   \left(\sqrt{5}-1\right)}
Answer (SymPy): sqrt(5)*pi*tan(sqrt(5)*pi/2)/5
Problem: Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{n^3 + n^2 + n + 1}{n \cdot (n+1)^2 \cdot (n+2)^2} $
Answer (LaTeX): \frac{5}{12} \left(\pi ^2-9\right)
Answer (SymPy): 5*pi**2/12 - 15/4
Problem: Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{n+1}{(n+2) \cdot (n+3) \cdot (n+4)} $
Answer (LaTeX): \frac{5}{24}
Answer (SymPy): 5/24
Problem: Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{(-1)^n \cdot n+1}{(n+1) \cdot (n+2) \cdot (n+3)} $
Answer (LaTeX): \frac{17}{6}-4 \log (2)
Answer (SymPy): 17/6 - log(16)
Problem: Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty (-2)^{-n} \left(n^2 + n + (-1)^n\right) $
Answer (LaTeX): \frac{19}{27}
Answer (SymPy): 19/27
Problem: Calculate the following infinite sum. Give a finite, closed form answer.
$ \sum_{n=1}^\infty \frac{(-1)^{n+1}+1}{n^2+1} $
Answer (LaTeX): \frac{1}{2} \pi  \tanh \left(\frac{\pi }{2}\right)
Answer (SymPy): pi*tanh(pi/2)/2
Problem: Compute up to degree 5 ($x^5$) the terms of the Maclaurin series of $ f(x) =  A \cdot e^{B \cdot \sin(x)} $.
Assume A and B are real and positive.
Answer (LaTeX): \frac{1}{6} A \left(B^3-B\right) x^3+\frac{1}{2} A B^2 x^2+\frac{1}{120} A \left(B^5-10 B^3+B\right) x^5+\frac{1}{24} A
   \left(B^4-4 B^2\right) x^4+A B x+A
Answer (SymPy): A*B**2*x**2/2 + A*B*x + A + x**5*A*(B**5 - 10*B**3 + B)/120 + x**4*A*(B**4 - 4*B**2)/24 + x**3*A*(B**3 - B)/6
Problem: Find the radius of convergence of the following series. Assume A,B,G,H are real and positive.

$ \sum_{n=1}^\infty \frac{B \cdot (2 \cdot n)! \cdot (A \cdot x)^n}{G \cdot (H \cdot n)^{2 \cdot n}} $
Answer (LaTeX): \frac{e^2 \cdot H^2}{4 \cdot A}
Answer (SymPy): (E**2 * H**2) / (4 * A)
Problem: Compute the first 6 nonzero terms of the Maclaurin series of $f(x) = G \cdot \left(\sin(A \cdot x) \cdot \cos\left(\frac{ B \cdot \pi }{ 4 }\right) + \cos(A \cdot x) \cdot \sin\left(\frac{ B \cdot \pi }{ 4 }\right)\right)$
Assume A,B,G are real and positive.
Answer (LaTeX): \frac{1}{120} A^5 G x^5 \cos \left(\frac{\pi  B}{4}\right)+\frac{1}{24} A^4 G x^4 \sin \left(\frac{\pi 
   B}{4}\right)-\frac{1}{6} A^3 G x^3 \cos \left(\frac{\pi  B}{4}\right)-\frac{1}{2} A^2 G x^2 \sin \left(\frac{\pi 
   B}{4}\right)+A G x \cos \left(\frac{\pi  B}{4}\right)+ G \sin \left(\frac{\pi  B}{4}\right)
Answer (SymPy): G*(A**5*x**5*cos(B*pi/4) + 5*A**4*x**4*sin(B*pi/4) - 20*A**3*x**3*cos(B*pi/4) - 60*A**2*x**2*sin(B*pi/4) + 120*A*x*cos(B*pi/4) + 120*sin(B*pi/4))/120
Problem: Compute the terms up to order 4 ($x^4$) of the Maclaurin series of $f(x) = G /cdot e^{A cdot x} \cdot \cos(B cdot x)$.
Assume A,B,G are real and positive.
Answer (LaTeX): G x^3 \left(\frac{A^3}{6}-\frac{A B^2}{2}\right)+\frac{1}{2} G x^2 \left(A^2-B^2\right)+G x^4 \left(\frac{A^4}{24}-\frac{A^2
   B^2}{4}+\frac{B^4}{24}\right)+A G x+G
Answer (SymPy): G*(4*A*x**3*(A**2 - 3*B**2) + 24*A*x + x**4*(A**4 - 6*A**2*B**2 + B**4) + 12*x**2*(A**2 - B**2) + 24)/24
Problem: Compute $\lim_{x \to 0}\frac{(2 \cos (J \cdot x)+4) \csc (J \cdot x)}{5 \cdot A \cdot (J \cdot x)^3}-\frac{6}{5 \cdot A \cdot (J \cdot x)^4}$
Assume A,J are real and positive.
Answer (LaTeX): \frac{1}{A 150}
Answer (SymPy): 1/(150*A)
Problem: Evaluate 
$ \lim_{x \to 0^+} A \cdot \left( \frac{ \tan\left( \frac{B \cdot x}{2} \right) }{ \frac{B \cdot x}{2} } \right)^{ \frac{3 \cdot J}{(B \cdot x)^2} } $
Assume A,B,J are real and positive.
Answer (LaTeX): $A /cdot e^{\frac{J}{4}}$
Answer (SymPy): A*e**(J/4)
Problem: Evaluate
$ \lim_{x \to {5 \cdot A} } \left( \frac{ 3 \cdot B \cdot x }{ x - 5 \cdot A }-\frac{ 3 \cdot B }{ \ln\left(\frac{ x }{ 5 \cdot A }\right) } \right)^{J} $
Assume A,B,J are real and positive.
Answer (LaTeX): \left( \frac{3B}{2} \right)^J
Answer (SymPy): (3*B/2)**J