hcju/proofwikiqa huggingface dataset:
{{begin-eqn}} {{eqn | l = z       | r = \sech a x       | c = }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = -a \sech a x \tanh a x       | c = Derivative of $\sech a x$ }} {{eqn | ll= \leadsto       | l = \int \sech^n a x \tanh a x \rd x       | r = \int \frac {-z^{n - 1} \rd z} a       | c = Integration by Substitution }} {{eqn | r = \frac {-1} a \int z^{n - 1} \rd z       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac {-1} a \frac {z^n} n       | c = Primitive of Power }} {{eqn | r = \frac {-\sech^n a x} {n a} + C       | c = substituting for $z$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = z       | r = \csch a x       | c = }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\d x}       | r = -a \csch a x \coth a x       | c = Derivative of $\csch a x$ }} {{eqn | ll= \leadsto       | l = \int \csch^n a x \coth a x \rd x       | r = \int \frac {-z^{n - 1} \rd z} a       | c = Integration by Substitution }} {{eqn | r = \frac {-1} a \int z^{n - 1} \rd z       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac {-1} a \frac {z^n} n       | c = Primitive of Power }} {{eqn | r = \frac {-\csch^n a x} {n a} + C       | c = substituting for $z$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {\sinh^2 a x \cosh^2 a x}       | r = \int \frac {\mathrm d x} {\left({\sinh a x \cosh a x}\right)^2}       | c =  }} {{eqn | r = \int \frac {\mathrm d x} {\left({\dfrac {\sinh^2 2 a x} 2}\right)^2}       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = 4 \int \frac {\mathrm d x} {\sinh^2 2 a x}       | c = Primitive of Constant Multiple of Function }} {{eqn | r = 4 \int \operatorname{csch}^2 2 a x \ \mathrm d x       | c = Definition of Hyperbolic Cosecant }} {{eqn | r = 4 \frac {-\coth 2 a x} {2 a} + C       | c = Primitive of $\operatorname{csch}^2 a x$ }} {{eqn | r = \frac {-2 \coth 2 a x} a + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \frac {\sinh^2 a x \ \mathrm d x} {\cosh a x}       | r = \int \frac {\left({\cosh^2 a x - 1}\right) \ \mathrm d x} {\cosh a x}       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn | r = \int \frac {\left({\cosh^2 a x}\right) \ \mathrm d x} {\cosh a x} - \int \frac {\mathrm d x} {\cosh a x}       | c = Linear Combination of Integrals }} {{eqn | r = \int \cosh a x \ \mathrm d x - \int \operatorname{sech} a x \ \mathrm d x       | c = Definition of Hyperbolic Secant }} {{eqn | r = \frac {\sinh a x} a - \int \operatorname{sech} a x \ \mathrm d x       | c = Primitive of $\cosh a x$ }} {{eqn | r = \frac {\sinh a x} a - \frac 1 a \arctan \left({\sinh a x}\right) + C       | c = Primitive of $\operatorname{sech} a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn 	| l = \int \frac {\rd x} {\cosh a x \paren {1 + \sinh a x} }  	| r = \int \frac {\sech^2 a x} {\sech a x + \tanh a x} \rd x 	| c = {{Defof|Hyperbolic Cosecant}}, {{Defof|Hyperbolic Tangent}} }} {{eqn 	| r = \frac 1 2 \int \frac {2 \sech^2 a x} {\sech a x + \tanh a x} \rd x 	| c = multiplying and dividing by $2$ }} {{eqn 	| r = \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x + \sech^2 a x + \sech a x \tanh a x} {\sech a x + \tanh a x} \rd x 	| c = adding and subtracting $\sech a x \tanh a x$ }} {{eqn 	| r = \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 2 \int \sech ax \paren {\frac {\sech a x + \tanh a x} {\sech a x + \tanh a x} } \rd x 	| c = Linear Combination of Primitives }} {{eqn 	| r = \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 2 \int \sech a x \rd x 	| c = simplifying }} {{eqn 	| r = \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 a \map \arctan {e^{a x} } 	| c = Primitive of $\sech a x$: Arctangent of Exponential Form }} {{eqn 	| r = \frac 1 {2 a} \int \frac {-a \sech a x \tanh a x + a \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 a \map \arctan {e^{a x} } 	| c = multiplying and dividing by $a$ }} {{eqn 	| r = \frac 1 {2 a} \ln \size {\sech a x + \tanh a x} + \frac 1 a \map \arctan {e^{a x} } 	| c = Derivative of $\sech a x$, Derivative of $\tanh a x$, Primitive of Function under its Derivative }} {{eqn 	| r = \frac 1 {2 a} \ln \size {\frac {1 + \sinh a x} {\cosh a x} } + \frac 1 a \map \arctan {e^{a x} } + C 	| c = simplifying and adding integration constant }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn 	| l = \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} } 	| r = \int \frac {\map \sinh {a x} \rd x} {\sinh^2 a x \cosh a x - \sinh^2 a x} 	| c = multiplying through $\dfrac {\sinh a x} {\sinh a x}$ }} {{eqn 	| r = \int \frac {\map \sinh {a x} \rd x} {\cosh a x \paren {\cosh^2 a x - 1} + 1 - \cosh^2 a x} 	| c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn 	| r = \int \frac {\map \sinh {a x} \rd x} {\cosh^3 a x - \cosh^2 a x - \cosh a x + 1}  }} {{eqn 	| r = \frac 1 a \int \frac {\d t} {t^3 - t^2 - t + 1} 	| c = Integration by Substitution: $t \to \map \cosh {a x}$ }} {{eqn 	| r = \frac 1 {4 a} \int \frac 1 {t + 1} \rd t - \frac 1 {4 a} \int \frac 1 {t - 1} \rd t + \frac 1 {2 a} \int \frac 1 {\paren {t - 1}^2} \rd t 	| c = Partial Fractions Expansion }} {{eqn 	| r = \frac 1 {4 a} \map \ln {t + 1} - \frac 1 {4 a} \map \ln {t - 1} - \frac 1 {2 a \paren {t - 1} } + C 	| c = Primitive of Reciprocal }} {{eqn 	| r = \frac 1 {4 a} \map \ln { \frac {\cosh ax + 1} {\cosh ax - 1} } - \frac 1 {2 a \paren {\cosh a x - 1} } + C 	| c = substituting back $t \to \cosh ax$ }} {{eqn 	| r = \frac 1 {4 a} \map \ln {\frac {2 \cosh^2 \frac {a x} 2} {2 \sinh^2 \frac {a x} 2} } - \frac 1 {2 a \paren {\cosh ax - 1} } + C 	| c = Double Angle Formula for Hyperbolic Cosine }} {{eqn 	| r = \frac 1 {4 a} \map \ln {\coth^2 \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C 	| c = {{Defof|Hyperbolic Cotangent}} }} {{eqn 	| r = \frac 1 {2 a} \ln \size {\coth \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C 	| c = Natural Logarithm of Power }} {{eqn 	| r = -\frac 1 {2 a} \ln \size {\tanh \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C 	| c = Logarithm of Reciprocal }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \tanh^3 a x \ \mathrm d x       | r = \int \tanh a x \tanh^2 a x \ \mathrm d x       | c =  }} {{eqn | r = \int \tanh a x \left({1 - \operatorname{sech}^2 a x}\right) \ \mathrm d x       | c = Sum of Squares of Hyperbolic Secant and Tangent }} {{eqn | r = \int \tanh a x \ \mathrm d x - \int \tanh a x \operatorname{sech}^2 a x \ \mathrm d x       | c = Linear Combination of Integrals }} {{eqn | r = \frac {\ln \left\vert{\cosh a x}\right\vert} a - \int \tanh a x \operatorname{sech}^2 a x \ \mathrm d x       | c = Primitive of $\tanh a x$ }} {{eqn | r = \frac {\ln \left\vert{\cosh a x}\right\vert} a - \frac {\tanh^2 a x} {2 a} + C       | c = Primitive of $\tanh^n a x \operatorname{sech}^2 a x$: $n = 1$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \tanh^3 a x \ \mathrm d x       | r = - \frac {\tanh^2 a x} {2 a} + \int \tanh a x \ \mathrm d x       | c = Primitive of Power of $\tanh^n a x$ with $n = 3$ }} {{eqn | r = \frac {\ln \left\vert{\cosh a x}\right\vert} a - \frac {\tanh^2 a x} {2 a} + C       | c = Primitive of $\tanh a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \coth^3 a x \ \mathrm d x       | r = \int \coth a x \coth^2 a x \ \mathrm d x       | c =  }} {{eqn | r = \int \coth a x \left({1 + \operatorname{csch}^2 a x}\right) \ \mathrm d x       | c = Difference of Squares of Hyperbolic Cotangent and Cosecant }} {{eqn | r = \int \coth a x \ \mathrm d x + \int \coth a x \operatorname{csch}^2 a x \ \mathrm d x       | c = Linear Combination of Integrals }} {{eqn | r = \frac {\ln \left\vert{\sinh a x}\right\vert} a + \int \coth a x \operatorname{csch}^2 a x \ \mathrm d x       | c = Primitive of $\tanh a x$ }} {{eqn | r = \frac {\ln \left\vert{\sinh a x}\right\vert} a - \frac {\coth^2 a x} {2 a} + C       | c = Primitive of $\coth^n a x \operatorname{csch}^2 a x$: $n = 1$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \coth^3 a x \ \mathrm d x       | r = - \frac {\coth^2 a x} {2 a} + \int \coth a x \ \mathrm d x       | c = Primitive of Power of $\coth^n a x$ with $n = 3$ }} {{eqn | r = \frac {\ln \left\vert{\sinh a x}\right\vert} a - \frac {\coth^2 a x} {2 a} + C       | c = Primitive of $\coth a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn 	| l = \int x \tanh a x \rd x  	| r = \frac 1 {a^2} \int \theta \tanh \theta \rd \theta  	| c = Substitution of $a x \to \theta$ }} {{eqn 	| r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta 	| c = Power Series Expansion for Hyperbolic Tangent Function }} {{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} \int \theta^{2 n} \rd \theta 	| c = Fubini's Theorem }} {{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!}  + C 	| c = Substituting back $\theta \to ax$ }}  {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn 	| l = \int x \coth a x \rd x  	| r = \frac 1 {a^2} \int \theta \coth \theta \rd \theta  	| c = Substitution of $a x \to \theta$ }} {{eqn 	| r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{2^{2 n} B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta 	| c = Power Series Expansion for Hyperbolic Cotangent Function }} {{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} } {\paren {2 n}!} \int \theta^{2 n} \rd \theta 	| c = Fubini's Theorem }} {{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!}  + C 	| c = Substituting back $\theta \to ax$ }}  {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \frac \d {\d x} \coth a x       | r = -a \csch^2 a x       | c = Derivative of $\coth a x$ }} {{eqn | ll= \leadsto       | l = \int \frac {-a \csch^2 a x \rd x} {\coth a x}       | r = \ln \size {\coth a x} + C       | c = Primitive of Function under its Derivative }} {{eqn | ll= \leadsto       | l = \int \frac {\csch^2 a x \rd x} {\coth a x}       | r = \frac {-1} a \ln \size {\coth a x} + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \sech^3 x \rd x       | r = \frac {\sech a x \tanh a x} {2 a} + \frac 1 2 \int \sech a x \rd x       | c = Primitive of $\sech^n a x$ where $n = 3$ }} {{eqn | r = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C       | c = Primitive of $\sech a x$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \csch^3 x \rd x       | r = \frac {\csch a x \coth a x} {2 a} - \frac 1 2 \int \csch a x \rd x       | c = Primitive of $\csch^n a x$ where $n = 3$ }} {{eqn | r = \frac {\csch a x \coth a x} {2 a} - \ln \size {\tanh a x} + C       | c = Primitive of $\csch a x$ }} {{end-eqn}} {{qed}}
We have that: {{begin-eqn}} {{eqn | l = \int \sech x \rd x       | r = \int \frac {\d x} {\cosh x}       | c = {{Defof|Hyperbolic Secant|index = 2}} }} {{eqn | r = \int \frac {\cosh x \rd x} {\cosh^2 x}       | c = multiplying top and bottom by $\cosh x$ }} {{eqn | r = \int \frac {\cosh x \rd x} {1 + \sinh^2 x}       | c = Difference of Squares of Hyperbolic Cosine and Sine }} {{end-eqn}} Let: {{begin-eqn}} {{eqn | l = u       | r = \sinh x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \cosh x       | c = Derivative of Hyperbolic Sine }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \sech x \rd x       | r = \int \frac {\d u} {u^2 + 1}       | c = Integration by Substitution }} {{eqn | r = \arctan u + C       | c = Primitive of $\dfrac 1 {x^2 + a^2}$: Arctangent Form }} {{eqn | r = \map \arctan {\sinh x} + C       | c = substituting for $u$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \operatorname{sech} x \ \mathrm d x       | r = \arcsin \left({\tanh x}\right)       | c = Primitive of $\operatorname{sech} x$: Arcsine form }} {{eqn | ll= \implies       | l = \int \operatorname{sech} a x \ \mathrm d x       | r = \frac 1 a \left({\arcsin \left({\tanh a x}\right) }\right) + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {\arcsin \left({\tanh a x}\right)} a + C       | c = simplifying }} {{end-eqn}} {{qed}} Category:Primitives involving Hyperbolic Secant Function 286k72unv29baibz0j9l8nr95l5511e
{{begin-eqn}} {{eqn | l = \int \sech x \rd x       | r = 2 \map \arctan {e^x} + C       | c = Primitive of $\sech x$: Arctangent of Exponential Form }} {{eqn | ll= \leadsto       | l = \int \sech a x \rd x       | r = \frac 1 a \paren {2 \map \arctan {e^{a x} } } + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {2 \map \arctan {e^{a x} } } a + C       | c = simplifying }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int \operatorname{sech} x \ \mathrm d x       | r = \arctan \left({\sinh x}\right)       | c = Primitive of $\operatorname{sech} x$: Arctangent of Hyperbolic Sine form }} {{eqn | ll= \implies       | l = \int \operatorname{sech} a x \ \mathrm d x       | r = \frac 1 a \left({\arctan \left({\sinh a x}\right) }\right) + C       | c = Primitive of Function of Constant Multiple }} {{eqn | r = \frac {\arctan \left({\sinh a x}\right)} a + C       | c = simplifying }} {{end-eqn}} {{qed}} Category:Primitives involving Hyperbolic Secant Function b2eqzaz6izdhq0m44q1069zyforgeoq
{{begin-eqn}} {{eqn 	| l = \int x \sech a x \rd x  	| r = \frac 1 {a^2} \int \theta \sech \theta \rd \theta  	| c = Substitution of $a x \to \theta$ }} {{eqn 	| r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac{E_{2 n} \, \theta^{2 n} } {\paren {2 n}!} \rd \theta 	| c = Power Series Expansion for Hyperbolic Secant Function }} {{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} } {\paren {2 n}!} \int \theta^{2 n + 1} \rd \theta 	| c = Fubini's Theorem }} {{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {a x}^{2 n + 2} } {\paren {2 n + 2} \paren {2 n}!}  + C 	| c = Substituting back $\theta \to ax$ }}  {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn 	| l = \int x \csch a x \rd x  	| r = \frac 1 {a^2} \int \theta \csch \theta \rd \theta  	| c = Substitution of $a x \to \theta$ }} {{eqn 	| r = \frac 1 {a^2} \int \theta \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \, \theta^{2 n - 1} } {\paren {2 n}!} \rd \theta 	| c = Power Series Expansion for Hyperbolic Cosecant Function }} {{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} } {\paren {2 n}!} \int \theta^{2 n} \rd \theta 	| c = Fubini's Theorem }} {{eqn 	| r = \frac 1 {a^2} \sum_{n \mathop = 0}^\infty \frac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!}  + C 	| c = Substituting back $\theta \to ax$ }}  {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \frac {\map \d {\map {\sinh^{-1} } {\frac x a} } } {\d x}       | r = \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} + 1}       | c = Derivative of $\sinh^{-1}$ and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac 1 {\sqrt {\frac {x^2 + a^2} {a^2} } }       | c =  }} {{eqn | r = \frac 1 a \frac a {\sqrt {x^2 + a^2} }       | c =  }} {{eqn | r = \frac 1 {\sqrt {x^2 + a^2} }       | c =  }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \sinh^{-1} \frac x a       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \frac 1 {\sqrt {x^2 + a^2} }       | c = Derivative of $\sinh^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = 1       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = x       | c = Primitive of Constant }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \sinh^{-1} \frac x a \rd x       | r = x \sinh^{-1} \frac x a - \int x \paren {\frac 1 {\sqrt {x^2 + a^2} } } \rd x + C       | c = Integration by Parts }} {{eqn | r = x \sinh^{-1} \frac x a - \int \frac {x \rd x} {\sqrt {x^2 + a^2} } + C       | c = simplifying }} {{eqn | r = x \sinh^{-1} \frac x a - \sqrt {x^2 + a^2} + C       | c = Primitive of $\dfrac x {\sqrt {x^2 + a^2} }$ }} {{end-eqn}} {{qed}}
Let $x > a$. Then $\dfrac x a > 1$ and so: {{begin-eqn}} {{eqn | l = \frac {\map \d {\map {\cosh^{-1} } {\frac x a} } } {\d x}       | r = \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} - 1}       | c = Derivative of $\cosh^{-1}$ and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac 1 {\sqrt {\frac {x^2 - a^2} {a^2} } }       | c =  }} {{eqn | r = \frac 1 a \frac a {\sqrt {x^2 - a^2} }       | c =  }} {{eqn | r = \frac 1 {\sqrt {x^2 - a^2} }       | c =  }} {{end-eqn}} $\cosh^{-1} \dfrac x a$ is not defined when $x \le a$. When $x = a$ we have that $\sqrt {x^2 - a^2} = 0$ and so $\dfrac 1 {\sqrt {x^2 - a^2} }$ is not defined. Hence the restriction on the domain. {{qed}}
Let $-a < x < a$. Then $-1 < \dfrac x a < 1$ and so: {{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\map {\tanh^{-1} } {\dfrac x a} }       | r = \frac 1 a \frac 1 {1 - \paren {\frac x a}^2}       | c = Derivative of $\tanh^{-1}$ and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac 1 {\frac {a^2 - x^2} {a^2} }       | c =  }} {{eqn | r = \frac 1 a \frac {a^2} {a^2 - x^2}       | c =  }} {{eqn | r = \frac a {a^2 - x^2}       | c =  }} {{end-eqn}} $\tanh^{-1} \dfrac x a$ is not defined when either $x \le -a$ or $x \ge a$. {{qed}}
Let $x^2 > a^2$. Then either $\dfrac x a < -1$ or $\dfrac x a > 1$ and so: {{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\map {\coth^{-1} } {\dfrac x a} }       | r = \frac 1 a \frac 1 {1 - \paren {\frac x a}^2}       | c = Derivative of $\coth^{-1}$ and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac 1 {\frac {a^2 - x^2} {a^2} }       | c =  }} {{eqn | r = -\frac 1 a \frac {a^2} {x^2 - a^2}       | c =  }} {{eqn | r = \frac {-a} {x^2 - a^2}       | c =  }} {{end-eqn}} $\coth^{-1} \dfrac x a$ is not defined when $x^2 \le a^2$. {{qed}}
Let $0 < x < a$. Then $0 < \dfrac x a < 1$ and so: {{begin-eqn}} {{eqn | l = \frac {\mathrm d \left({\operatorname{sech}^{-1} \frac x a}\right)} {\mathrm d x}       | r = \frac 1 a \frac {-1} {\frac x a \sqrt{1 - \left({\frac x a}\right)^2} }       | c = Derivative of $\operatorname{sech}^{-1}$ and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac {-a} {x \sqrt{\frac {a^2 - x^2} {a^2} } }       | c =  }} {{eqn | r = \frac {-a} {x \sqrt{a^2 - x^2} }       | c =  }} {{end-eqn}} $\operatorname{sech}^{-1} \dfrac x a$ is not defined when $x \le 0$ or $x \ge a$. {{qed}}
Let $0 < x < a$. Then $0 < \dfrac x a < 1$ and so: {{begin-eqn}} {{eqn | l = \frac {\map \d {\csch^{-1} } {\frac x a} } {\d x}       | r = \frac 1 a \dfrac {-1} {\size {\frac x a} \sqrt {1 + \paren {\frac x a}^2} }       | c = Derivative of $\csch^{-1}$ and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac {-a} {\size x \sqrt {\frac {a^2 + x^2} {a^2} } }       | c =  }} {{eqn | r = \frac {-a} {\size x \sqrt{a^2 + x^2} }       | c =  }} {{end-eqn}} $\csch^{-1} \dfrac x a$ is not defined when $x = 0$. {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \cosh^{-1} \frac x a       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = \frac 1 {\sqrt {x^2 - a^2} }       | c = Derivative of $\cosh^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = 1       | c =  }} {{eqn | ll= \implies       | l = v       | r = x       | c = Primitive of Constant }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \cosh^{-1} \frac x a \ \mathrm d x       | r = x \cosh^{-1} \frac x a - \int x \left({\frac 1 {\sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = x \cosh^{-1} \frac x a - \int \frac {x \ \mathrm d x} {\sqrt {x^2 - a^2} } + C       | c = simplifying }} {{eqn | r = x \cosh^{-1} \frac x a - \sqrt {x^2 + a^2} + C       | c = Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$ }} {{end-eqn}} {{finish|Analysis to be done for the negative case. The confusion here is that $\cosh^{-1} \frac x a$ is ''defined'' as being positive, like $\sqrt x$.}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \tanh^{-1} \frac x a       | c =  }} {{eqn | ll= \implies       | l = \frac {\d u} {\d x}       | r = \frac a {a^2 - x^2}       | c = Derivative of $\tanh^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = 1       | c =  }} {{eqn | ll= \implies       | l = v       | r = x       | c = Primitive of Constant }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \tanh^{-1} \frac x a \rd x       | r = x \tanh^{-1} \frac x a - \int x \left({\frac a {a^2 - x^2} }\right) \rd x + C       | c = Integration by Parts }} {{eqn | r = x \tanh^{-1} \frac x a - a \int \frac {x \rd x} {a^2 - x^2} + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = x \tanh^{-1} \frac x a - a \left({- \frac 1 2 \ln \left({a^2 - x^2}\right)}\right) + C       | c = Primitive of $\dfrac x {a^2 - x^2}$ }} {{eqn | r = x \tanh^{-1} \dfrac x a + \frac {a \ln \left({a^2 - x^2}\right)} 2 + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \coth^{-1} \frac x a       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = \frac {-a} {x^2 - a^2}       | c = Derivative of $\coth^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = 1       | c =  }} {{eqn | ll= \implies       | l = v       | r = x       | c = Primitive of Constant }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \coth^{-1} \frac x a \ \mathrm d x       | r = x \tanh^{-1} \frac x a - \int x \left({\frac {-a} {x^2 - a^2} }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = x \coth^{-1} \frac x a + a \int \frac {x \ \mathrm d x} {x^2 - a^2} + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = x \coth^{-1} \frac x a + a \left({\frac 1 2 \ln \left({x^2 - a^2}\right)}\right) + C       | c = Primitive of $\dfrac x {x^2 - a^2}$ }} {{eqn | r = x \coth^{-1} \dfrac x a + \frac {a \ln \left({x^2 - a^2}\right)} 2 + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \sech^{-1} \frac x a       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \frac {-a} {x \sqrt {a^2 - x^2} }       | c = Derivative of $\sech^{-1} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = 1       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = x       | c = Primitive of Constant }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int \sech^{-1} \frac x a \rd x       | r = x \sech^{-1} \frac x a - \int x \paren {\frac {-a} {x \sqrt {a^2 - x^2} } } \rd x + C       | c = Integration by Parts }} {{eqn | r = x \sech^{-1} \frac x a + a \int \frac {\d x} {\sqrt {a^2 - x^2} } + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = x \sech^{-1} \frac x a + \arcsin \frac x a + C       | c = Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$ }} {{end-eqn}} {{finish|Analysis to be done for the negative case. The confusion here is that $\sech^{-1} \frac x a$ is ''defined'' as being positive, like $\sqrt x$.}}