Shalyt/ASyMOB-Algebraic_Symbolic_Mathematical_Operations_Benchmark huggingface dataset:
Problem: Solve the following differential equation:
$ y' = 2 \cdot A \cdot \sin (G \cdot x) + 5 \cdot B \cdot \sin (2 x) - y , y(H - 1) = J - 1 $
Assume the parameters: A,B,G,H,J are real and positive numbers.
Answer (LaTeX): -\frac{e^{-x-1} \left(2 A e^H \sin (G (H-1))-2 A G e^H \cos (G (H-1))-2 A e^{x+1}
   \sin (G x)+2 A G e^{x+1} \cos (G x)-2 B G^2 e^H \cos (2 (H-1))-2 B G^2 e^H \sin
   (1-H) \cos (1-H)+2 B G^2 e^{x+1} \cos (2 x)-2 B G^2 e^{x+1} \sin (x) \cos (x)-2
   B e^H \cos (2 (H-1))-2 B e^H \sin (1-H) \cos (1-H)+2 B e^{x+1} \cos (2 x)-2 B
   e^{x+1} \sin (x) \cos (x)-G^2 e^H J+G^2 e^H-e^H J+e^H\right)}{G^2+1}
Answer (SymPy): e**(-x - 1)*(2*A*G*e**H*cos(G*(H - 1)) - 2*A*G*e**(x + 1)*cos(G*x) - 2*A*e**H*sin(G*(H - 1)) + 2*A*e**(x + 1)*sin(G*x) - B*G**2*e**H*sin(2*H - 2) + 2*B*G**2*e**H*cos(2*H - 2) + B*G**2*e**(x + 1)*sin(2*x) - 2*B*G**2*e**(x + 1)*cos(2*x) - B*e**H*sin(2*H - 2) + 2*B*e**H*cos(2*H - 2) + B*e**(x + 1)*sin(2*x) - 2*B*e**(x + 1)*cos(2*x) + G**2*J*e**H - G**2*e**H + J*e**H - e**H)/(G**2 + 1)
Problem: Solve the following differential equation:
$ y' = A \cdot \tan (B \cdot y) , y(H) = J $
Assume the parameters: A,B,H,J are real and positive numbers.
Answer (LaTeX): \frac{\sin ^{-1}\left(\sin (B J) e^{A B x-A B H}\right)}{B}
Answer (SymPy): asin(e**(A*B*(-H + x))*sin(B*J))/B
Problem: Solve the following differential equation:
$ A \cdot y' = \sin^2(B \cdot y) + 2 \cdot \cos^2(B \cdot y) , y(H) = J $
Assume the parameters: A,B,H,J are real and positive numbers.
Answer (LaTeX): \frac{\tan ^{-1}\left(\sqrt{2} \tan \left(\frac{\frac{\sqrt{2} A \tan
   ^{-1}\left(\frac{\tan (B J)}{\sqrt{2}}\right)-2 B H}{\sqrt{2}}+\sqrt{2} B
   x}{A}\right)\right)}{B}
Answer (SymPy): atan(sqrt(2)*tan((A*atan(sqrt(2)*tan(B*J)/2) - sqrt(2)*B*H + sqrt(2)*B*x)/A))/B
Problem: Solve the following differential equation:
$ A \cdot y' = 2 \cdot G \cdot \cos ^2(B \cdot y) - \sin ^2(B \cdot y) , y(H - 1) = J - 1 $
Assume the parameters: A,B,G,H,J are real and positive numbers.
Answer (LaTeX): \frac{\tan ^{-1}\left(\sqrt{2} \sqrt{G} \tanh \left(\frac{\frac{\sqrt{2} A \tanh
   ^{-1}\left(\frac{\tan (B (J-1))}{\sqrt{2} \sqrt{G}}\right)-2 B \sqrt{G} H+2 B
   \sqrt{G}}{\sqrt{2}}+\sqrt{2} B \sqrt{G} x}{A}\right)\right)}{B}
Answer (SymPy): atan(sqrt(2)*sqrt(G)*tanh((A*atanh(sqrt(2)*tan(B*(J - 1))/(2*sqrt(G))) - sqrt(2)*B*sqrt(G)*H + sqrt(2)*B*sqrt(G)*x + sqrt(2)*B*sqrt(G))/A))/B
Problem: Solve the following differential equation:
$ G \cdot x + y'' + y' + y = 0 , y(0) = H, y'(0) = J $
Assume the parameters: G,H,J are real and positive numbers.
Answer (LaTeX): -\frac{1}{3} e^{-x/2} \left(-3 G e^{x/2}+3 G e^{x/2} x-\sqrt{3} G \sin
   \left(\frac{\sqrt{3} x}{2}\right)+3 G \cos \left(\frac{\sqrt{3}
   x}{2}\right)-\sqrt{3} H \sin \left(\frac{\sqrt{3} x}{2}\right)-3 H \cos
   \left(\frac{\sqrt{3} x}{2}\right)-2 \sqrt{3} J \sin \left(\frac{\sqrt{3}
   x}{2}\right)\right)
Answer (SymPy): (-3*G*e**(x/2)*x + 3*G*e**(x/2) - 2*sqrt(3)*G*cos(sqrt(3)*x/2 + pi/6) + 2*sqrt(3)*H*sin(sqrt(3)*x/2 + pi/3) + 2*sqrt(3)*J*sin(sqrt(3)*x/2))/(3*e**(x/2))
Problem: Solve the following differential equation:
$ A \cdot y'' + B \cdot y' - 6 \cdot G \cdot y = 0 , y(0) = H - 1, y'(0) = J $
Assume the parameters: A,B,G,H,J are real and positive numbers.
Answer (LaTeX): \frac{e^{-\frac{x \left(\sqrt{24 A G+B^2}+B\right)}{2 A}} \left(e^{\frac{x \sqrt{24
   A G+B^2}}{A}} \left(H \sqrt{24 A G+B^2}-\sqrt{24 A G+B^2}+2 A J\right)+B (H-1)
   \left(e^{\frac{x \sqrt{24 A G+B^2}}{A}}-1\right)+H \sqrt{24 A G+B^2}-\sqrt{24 A
   G+B^2}-2 A J\right)}{2 \sqrt{24 A G+B^2}}
Answer (SymPy): (-2*A*J + H*sqrt(24*A*G + B**2) + e**(x*sqrt(24*A*G + B**2)/A)*(2*A*J + H*sqrt(24*A*G + B**2) - sqrt(24*A*G + B**2)) + (e**(x*sqrt(24*A*G + B**2)/A) - 1)*B*(H - 1) - sqrt(24*A*G + B**2))/(2*e**(x*(B + sqrt(24*A*G + B**2))/(2*A))*sqrt(24*A*G + B**2))
Problem: Solve the following differential equation:
$ G \cdot y + y'' = 0 , y(A) = H, y'(0) = J $
Assume the parameters: A,G,H,J are real and positive numbers.
Answer (LaTeX): \frac{\sec \left(A \sqrt{G}\right) \left(\sqrt{G} H \cos \left(\sqrt{G} x\right)-J
   \sin \left(\sqrt{G} (A-x)\right)\right)}{\sqrt{G}}
Answer (SymPy): (sqrt(G)*H*cos(sqrt(G)*x) - J*sin(sqrt(G)*(A - x)))*sec(A*sqrt(G))/sqrt(G)
Problem: Solve the following differential equation:
$ y'' - G \cdot y = 0 , y(A) = H, y'(B) = J $
Assume the parameters: A,B,G,H,J are real and positive numbers.
Answer (LaTeX): \frac{e^{-\sqrt{G} x} \left(\sqrt{G} H e^{\sqrt{G} (A+2 B)}-J e^{\sqrt{G} (2
   A+B)}+\sqrt{G} H e^{\sqrt{G} (A+2 x)}+J e^{\sqrt{G} (B+2 x)}\right)}{\sqrt{G}
   \left(e^{2 A \sqrt{G}}+e^{2 B \sqrt{G}}\right)}
Answer (SymPy): (sqrt(G)*H*e**(sqrt(G)*(A + 2*B)) + sqrt(G)*H*e**(sqrt(G)*(A + 2*x)) - J*e**(sqrt(G)*(2*A + B)) + J*e**(sqrt(G)*(B + 2*x)))/(sqrt(G)*e**(sqrt(G)*x)*(e**(2*A*sqrt(G)) + e**(2*B*sqrt(G))))
Problem: Solve the following differential equation:
$ y'' - 2 \cdot y' - 3 \cdot y = G \cdot \sin (x) , y(A) = H, y'(B) = J $
Assume the parameters: A,B,G,H,J are real and positive numbers.
Answer (LaTeX): \frac{e^{-x} \left(-e^A G \cos (A) \left(3 e^{4 B}+e^{4 x}\right)-2 e^B G \left(e^{4
   A}-e^{4 x}\right) \cos (B)+6 G e^{A+4 B} \sin (A)-G e^{4 A+B} \sin (B)+30 H
   e^{A+4 B}-10 J e^{4 A+B}+2 G e^{A+4 x} \sin (A)-2 G e^{4 A+x} \sin (x)+G e^{4
   A+x} \cos (x)+10 H e^{A+4 x}+G e^{B+4 x} \sin (B)-6 G e^{4 B+x} \sin (x)+3 G e^{4
   B+x} \cos (x)+10 J e^{B+4 x}\right)}{10 \left(e^{4 A}+3 e^{4 B}\right)}
Answer (SymPy): (-G*e**A*(3*e**(4*B) + e**(4*x))*cos(A) + 6*G*e**(A + 4*B)*sin(A) + 2*G*e**(A + 4*x)*sin(A) - G*e**(4*A + B)*sin(B) - 2*G*e**(4*A + x)*sin(x) + G*e**(4*A + x)*cos(x) + G*e**(B + 4*x)*sin(B) - 6*G*e**(4*B + x)*sin(x) + 3*G*e**(4*B + x)*cos(x) + 30*H*e**(A + 4*B) + 10*H*e**(A + 4*x) - 10*J*e**(4*A + B) + 10*J*e**(B + 4*x) - 2*e**B*G*(e**(4*A) - e**(4*x))*cos(B))/(10*e**x*(e**(4*A) + 3*e**(4*B)))
Problem: Calculate the following infinite product. Give a finite, closed form answer. Assume the parameters: A,B are real and positive.
$ \prod_{n=1}^\infty 1-\frac{\left(\frac{A}{2 B}+1\right)^4}{B^4 (n+1)^4} $
Answer (LaTeX): \frac{4 B^4 \sin \left(\frac{\pi  (A+2 B)}{2 B^2}\right) \sinh \left(\frac{\pi  (A+2
   B)}{2 B^2}\right)}{\pi ^2 (A+2 B)^2 \left(1-\frac{(A+2 B)^4}{16 B^8}\right)}
Answer (SymPy): 64*B**12*sin(pi*(A + 2*B)/(2*B**2))*sinh(pi *(A + 2*B)/(2*B**2))/(pi**2*(A + 2*B)**2*(16*B**8 - (A + 2*B)**4))
Problem: Calculate the following infinite product. Give a finite, closed form answer. Assume the parameters: A,B are real and positive.
$ \prod_{n=1}^\infty \frac{\left((A+n)^3-1\right) \left(1-\frac{1}{(B+n)^2}\right)}{(A+n)^3+1} $
Answer (LaTeX): \frac{A \cdot (A+1) \cdot B}{\left(A^2+A+1\right) (B+1)}
Answer (SymPy): A*B*(A + 1)/((B + 1)*(A**2 + A + 1))
Problem: Calculate the following infinite product. Give a finite, closed form answer. Assume the parameters: A,B are real and positive.
$ \prod_{n=1}^\infty 1-\frac{B^2}{(A+1)^2 n^2} $
Answer (LaTeX): \frac{(A+1) \sin \left(\frac{\pi  B}{A+1}\right)}{\pi  B}
Answer (SymPy): (A + 1)*sin(B*pi/(A + 1))/(B*pi)
Problem: Calculate the following infinite product. Give a finite, closed form answer. Assume the parameters: A,B,G are real and positive.
$ \prod_{n=1}^\infty \frac{1}{A^2 \cdot n^2+\left\lfloor \frac{n}{n+B}+\left\lfloor \frac{1}{n^2 + G}\right\rfloor \right\rfloor }+1$
Answer (LaTeX): \frac{A \sinh \left(\frac{\pi }{A}\right)}{\pi }
Answer (SymPy): A*sinh(pi/A)/pi
Problem: Calculate the following infinite product. Give a finite, closed form answer. Assume the parameters: A,J are real and positive.
$ \prod_{n=1}^\infty \left(1-\frac{J^6}{64 n^6}\right) \left(1-\frac{4}{3} \sin ^2\left(A \cdot 3^{-n}\right)\right) $
Answer (LaTeX): -\frac{4 \frac{\sin( A)}{ A} \sin \left(\frac{\pi  J}{2}\right) \left(\cos
   \left(\frac{\pi  J}{2}\right)-\cosh \left(\frac{1}{2} \sqrt{3} \pi 
   J\right)\right)}{\pi ^3 J^3}
Answer (SymPy): -4*(cos(J*pi/2) - cosh(sqrt(3)*J*pi/2))*sin(A)*sin(J*pi/2)/(A*J**3*pi**3)
Problem: Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G,H,J are real and positive.
$ \sum_{n=1}^\infty H \cdot \left(\frac{3}{5}\right)^n \left(G - \frac{1}{n}\right) \left(\frac{B}{A}\right)^n $
Answer (LaTeX): H \left(\frac{3 B G}{5 A-3 B}+\log \left(1-\frac{3 B}{5 A}\right)\right)
Answer (SymPy): H*((3*B*G + (5*A - 3*B)*log((5*A - 3*B)/(5*A)))/(5*A - 3*B))
Problem: Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B are real and positive.
$ \sum_{n=1}^\infty \frac{(-A)^n}{B \cdot (2 n+5)} $
Answer (LaTeX): \frac{\frac{15 \tan^{-1}\left(\sqrt{A}\right)}{A^{5/2}}-\frac{15}{A^2}+\frac{5}{A}-3}{15 B}
Answer (SymPy): -1/(5*B) + 1/(3*A*B) - 1/(A**2*B) + atan(sqrt(A))/(A**(5/2)*B)
Problem: Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G,H are real and positive.
$ \sum_{n=1}^\infty 3^{A + H \cdot n} \cdot 2^{B - 3 \cdot G \cdot n}  $
Answer (LaTeX): \frac{2^B 3^{A+H}}{8^G-3^H}
Answer (SymPy): -2**B*3**(A + H)/(3**H - 8**G)
Problem: Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G are real and positive.
$ \sum_{n=1}^\infty \frac{B \cdot A^{G \cdot n}}{n!} $
Answer (LaTeX): B \left(e^{A^G}-1\right)
Answer (SymPy): B*(e**(A**G) - 1)
Problem: Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B are real and positive.
$ \sum_{n=1}^\infty \frac{A}{-B + n^2 - n} $
Answer (LaTeX): \frac{\pi  A \tan \left(\frac{1}{2} \pi  \sqrt{4 B+1}\right)}{\sqrt{4 B+1}}
Answer (SymPy): A*pi*tan(pi*sqrt(4*B + 1)/2)/sqrt(4*B + 1)
Problem: Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G,H,J are real and positive.
$ \sum_{n=1}^\infty \frac{A \cdot n^3 + B \cdot n^2 + H \cdot n + J}{G \cdot n \cdot (n+1)^2 \cdot (n+2)^2} $
Answer (LaTeX): \frac{2 \left(5 \pi ^2-48\right) A-6 \left(\pi ^2-10\right) B-39 H+4 \pi ^2 H+30 J-3
   \pi ^2 J}{12 G}
Answer (SymPy): (2*A*(5*pi**2 - 48) - 6*B*(pi**2 - 10) + 4*H*pi**2 - 39*H - 3*J*pi**2 + 30*J)/(12*G)
Problem: Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G are real and positive.
$ \sum_{n=1}^\infty \frac{A \cdot n+B}{G \cdot (n+4) \cdot (n+2) \cdot (n+3)} $
Answer (LaTeX): \frac{4 A+B}{24 G}
Answer (SymPy): (4*A + B)/(24*G)
Problem: Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G are real and positive.
$ \sum_{n=1}^\infty \frac{A \cdot (-1)^n \cdot n + B}{G \cdot (n+1) \cdot (n+2) \cdot (n+3)} $
Answer (LaTeX): \frac{A (33-48 \log (2))+B}{12 G}
Answer (SymPy): (B + A*(33 - 48*log(2)))/(12*G)
Problem: Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G,H,J are real and positive.
$ \sum_{n=1}^\infty \frac{(-2)^{-n} \cdot {J}^{-n} \cdot \left(A \cdot (-1)^n + B \cdot n^2 + H \cdot n \right)}{G} $
Answer (LaTeX): \frac{\frac{A}{2 J - 1}-\frac{2 J \cdot (B \cdot (2 J - 1) + 2 H J + H)}{(2 J+1)^3}}{G}
Answer (SymPy): (A*(2*J + 1)**3 - 2*J*(2*J - 1)*(B*(2*J - 1) + 2*H*J + H))/(G*(2*J - 1)*(2*J + 1)**3)
Problem: Calculate the following infinite sum. Give a finite, closed form answer. Assume the parameters: A,B,G,H,J are real and positive.
$ \sum_{n=1}^\infty \frac{A \cdot (-1)^{B + n} + H}{G \cdot \left(J^2 + n^2\right)} $
Answer (LaTeX): \frac{\pi \cdot A \cdot (-1)^{B} \cdot J \cdot \frac{1}{\sinh(\pi \cdot J)}+A \cdot \left(-(-1)^{B}\right)+\pi  H \cdot J \cdot \coth (\pi \cdot J) - H}{2 \cdot G \cdot J^2}
Answer (SymPy): ((-1)**B*A*J*pi/sinh(J*pi) + (-1)**(B + 1)*A + H*J*pi*coth(J*pi) - H)/(2*G*J**2)