avewright/proofwiki-math huggingface dataset:
Theorem: Primitive of Reciprocal of Hyperbolic Cosine of a x by 1 plus Hyperbolic Sine of a x

:$\ds \int \frac {\rd x} {\cosh a x \paren {1 + \sinh a x} } = \frac 1 {2 a} \ln \size {\frac {1 + \sinh a x} {\cosh a x} } + \frac 1 a \map \arctan {e^{a x} } + C$

Proof:
{{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}}
Theorem: Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x plus 1

:$\ds \int \frac {\d x} {\sinh a x \paren {\cosh a x + 1} } = \frac 1 {2 a} \ln \size {\tanh \frac {a x} 2} + \frac 1 {2 a \paren {\cosh a x + 1} } + C$

Proof:
Let:
{{begin-eqn}}
{{eqn | l = u
      | r = \cosh a x
      | c = 
}}
{{eqn | l = \frac {\d u} {\d x}
      | r = a \sinh a x
      | c = Derivative of $\cosh a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh a x \paren {\cosh a x + 1} }
      | r = \int \frac {\sinh a x \rd x} {\sinh^2 a x \paren {\cosh a x + 1} }
      | c = multiplying top and bottom by $\sinh a x$
}}
{{eqn | r = \int \frac {\sinh a x \rd x} {\paren {\cosh^2 a x - 1} \paren {\cosh a x + 1} }
      | c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | r = \frac 1 a \int \frac {\d u} {\paren {u^2 - 1} \paren {u + 1} }
      | c = Integration by Substitution: $u = \cosh a x$
}}
{{eqn | r = \frac 1 a \int \frac {\d u} {\paren {u + 1}^2 \paren {u - 1} }
      | c = Difference of Two Squares
}}
{{eqn | r = \frac 1 a \paren {\frac 1 2 \paren {\frac 1 {u + 1} + \frac 1 2 \ln \size {\frac {u - 1} {u + 1} } } } + C
      | c = Primitive of $\dfrac 1 {\paren {a x + b}^2 \paren {p x + q} }$
}}
{{eqn | r = \frac 1 {2 a \paren {u + 1} } + \frac 1 {4 a} \ln \size {\frac {u - 1} {u + 1} } + C
      | c = simplifying
}}
{{eqn | r = \frac 1 {2 a \paren {\cosh a x + 1} } + \frac 1 {4 a} \ln \size {\frac {\cosh a x - 1} {\cosh a x + 1} } + C
      | c = substituting for $u$
}}
{{eqn | r = \frac 1 {2 a \paren {\cosh a x + 1} } + \frac 1 {4 a} \ln \size {\frac {\frac 1 2 \sech^2 \dfrac x 2} {\frac 1 2 \csch^2 \frac {a x} 2} } + C
      | c = Reciprocal of $\cosh - 1$ and Reciprocal of $\cosh + 1$
}}
{{eqn | r = \frac 1 {2 a \paren {\cosh a x + 1} } + \frac 1 {4 a} \ln \size {\frac {\sinh^2 \frac {a x} 2} {\cosh^2 \frac {a x} 2} } + C
      | c = {{Defof|Hyperbolic Secant|index = 2}}, {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac 1 {2 a \paren {\cosh a x + 1} } + \frac 1 {4 a} \ln \size {\tanh^2 \frac {a x} 2} + C
      | c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac 1 {2 a \paren {\cosh a x + 1} } + \frac 1 {2 a} \ln \size {\tanh \frac {a x} 2} + C
      | c = Logarithm of Power
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x minus 1

:$\ds \int \frac {\d x} {\sinh a x \paren {\cosh a x - 1} } = \frac {-1} {2 a} \ln \size {\tanh \frac {a x} 2} - \frac 1 {2 a \paren {\cosh a x - 1} } + C$

Proof:
{{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, Primitive of Power
}}
{{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}}
Theorem: Primitive of Cube of Hyperbolic Tangent of a x

:$\ds \int \tanh^3 a x \rd x = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \tanh^3 a x \rd x
      | r = \int \tanh a x \tanh^2 a x \rd x
      | c = 
}}
{{eqn | r = \int \tanh a x \paren {1 - \sech^2 a x} \rd x
      | c = Sum of Squares of Hyperbolic Secant and Tangent
}}
{{eqn | r = \int \tanh a x \rd x - \int \tanh a x \sech^2 a x \rd x
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {\ln \size {\cosh a x} } a - \int \tanh a x \sech^2 a x \rd x
      | c = Primitive of $\tanh a x$
}}
{{eqn | r = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C
      | c = Primitive of $\tanh^n a x \sech^2 a x$: $n = 1$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Cube of Hyperbolic Tangent of a x

:$\ds \int \tanh^3 a x \rd x = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \tanh^3 a x \rd x
      | r = -\frac {\tanh^2 a x} {2 a} + \int \tanh a x \rd x
      | c = Primitive of Power of $\tanh^n a x$ with $n = 3$
}}
{{eqn | r = \frac {\ln \size {\cosh a x} } a - \frac {\tanh^2 a x} {2 a} + C
      | c = Primitive of $\tanh a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Cube of Hyperbolic Cotangent of a x

:$\ds \int \coth^3 a x \rd x = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \coth^3 a x \rd x
      | r = \int \coth a x \coth^2 a x \rd x
      | c = 
}}
{{eqn | r = \int \coth a x \paren {1 + \csch^2 a x} \rd x
      | c = Difference of Squares of Hyperbolic Cotangent and Cosecant
}}
{{eqn | r = \int \coth a x \rd x + \int \coth a x \csch^2 a x \rd x
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {\ln \size {\sinh a x} } a + \int \coth a x \csch^2 a x \rd x
      | c = Primitive of $\tanh a x$
}}
{{eqn | r = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C
      | c = Primitive of $\coth^n a x \csch^2 a x$: $n = 1$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Cube of Hyperbolic Cotangent of a x

:$\ds \int \coth^3 a x \rd x = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \coth^3 a x \rd x
      | r = -\frac {\coth^2 a x} {2 a} + \int \coth a x \rd x
      | c = Primitive of Power of $\coth^n a x$ with $n = 3$
}}
{{eqn | r = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C
      | c = Primitive of $\coth a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x by Hyperbolic Tangent of a x

:$\ds \int x \tanh a x \rd x = \frac 1 {a^2} \paren {\frac {\paren {a x}^3} 3 - \frac {\paren {a x}^5} {15} + \frac {2 \paren {a x}^7} {105} + \cdots + \frac { 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$

Proof:
{{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}}
Theorem: Primitive of x by Hyperbolic Cotangent of a x

:$\ds \int x \coth a x \rd x = \frac 1 {a^2} \paren {a x + \frac {\paren {a x}^3} 9 - \frac {\paren {a x}^5} {225} + \cdots + \frac {2^{2 n} B_{2 n} \paren {a x}^{2 n + 1} } {\paren {2 n + 1}!} + \cdots} + C$

Proof:
{{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}}
Theorem: Primitive of Square of Hyperbolic Cosecant of a x over Hyperbolic Cotangent of a x

:$\ds \int \frac {\csch^2 a x \rd x} {\coth a x} = \frac {-1} a \ln \size {\coth a x} + C$

Proof:
{{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}}
Theorem: Primitive of Cube of Hyperbolic Secant of a x

:$\ds \int \sech^3 a x \rd x = \frac {\sech a x \tanh a x} {2 a} + \frac 1 {2 a} \map \arctan {\sinh a x} + C$

Proof:
{{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}}
Theorem: Primitive of Cube of Hyperbolic Cosecant of a x

:$\ds \int \csch^3 a x \rd x = \frac {-\csch a x \coth a x} {2 a} - \frac 1 {2 a} \ln \size {\tanh a x} + C$

Proof:
{{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}}
Theorem: Primitive of Hyperbolic Secant Function/Arctangent of Hyperbolic Sine Form

:$\ds \int \sech x \rd x =  \map \arctan {\sinh x} + C$

Proof:
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}}
Theorem: Primitive of Hyperbolic Secant of a x/Arcsine Form

:$\ds \int \sech a x \rd x = \frac {\map \arcsin {\tanh a x} } a + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \sech x \rd x
      | r = \map \arcsin {\tanh x}
      | c = Primitive of $\sech x$: Arcsine form
}}
{{eqn | ll= \leadsto
      | l = \int \sech a x \rd x
      | r = \frac 1 a \map \arcsin {\tanh a x} + C
      | c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\map \arcsin {\tanh a x} } a + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Secant of a x/Arctangent of Exponential Form

:$\ds \int \sech a x \rd x = \frac {2 \map \arctan {e^{a x} } } a + C$

Proof:
{{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}}
Theorem: Primitive of Hyperbolic Secant of a x/Arctangent of Hyperbolic Sine Form

:$\ds \int \sech a x \rd x = \frac {\map \arctan {\sinh a x} } a + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \sech x \rd x
      | r = \map \arctan {\sinh x}
      | c = Primitive of $\sech x$: Arctangent of Hyperbolic Sine form
}}
{{eqn | ll= \leadsto
      | l = \int \sech a x \rd x
      | r = \frac 1 a \map \arctan {\sinh a x} + C
      | c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\map \arctan {\sinh a x} } a + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Category:Primitives involving Hyperbolic Secant Function
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Theorem: Primitive of x by Hyperbolic Secant of a x

{{begin-eqn}}
{{eqn | l = \int x \sech a x \rd x
      | 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
}}
{{eqn | r = \frac 1 {a^2} \paren {\frac {\paren {a x}^2} 2 - \frac {\paren {a x}^4} 8 + \frac {5 \paren {a x}^6} {144} - \cdots } + C
}}
{{end-eqn}}

Proof:
{{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}}
Theorem: Primitive of x by Hyperbolic Cosecant of a x

{{begin-eqn}}
{{eqn | l = \int x \csch a x \rd x
      | 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
}}
{{eqn | r = \dfrac 1 {a^2} \paren {a x - \dfrac {\paren {a x}^3} {18} + \dfrac {7 \paren {a x}^5} {1800} - \cdots} + C
}}
{{end-eqn}}
where $B_{2 n}$ denotes the $2 n$th Bernoulli number.

Proof:
{{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}}
Theorem: Derivative of Real Area Hyperbolic Sine of x over a

:$\dfrac {\map \d {\map \arsinh {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 + a^2}}$

Proof:
{{begin-eqn}}
{{eqn | l = \frac {\map \d {\map \arsinh {\frac x a} } } {\d x}
      | r = \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} + 1}
      | c = Derivative of $\arsinh$ 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}}
Theorem: Derivative of Real Area Hyperbolic Cosine of x over a

:$\dfrac {\map \d {\map \arcosh {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 - a^2} }$
where $x > a$.

Proof:
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 $\arcosh$ 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}}
Theorem: Derivative of Real Area Hyperbolic Tangent of x over a

:$\map {\dfrac \d {\d x} } {\map \artanh {\dfrac x a} } = \dfrac a {a^2 - x^2}$
where $-a < x < a$.

Proof:
Let $-a < x < a$.
Then $-1 < \dfrac x a < 1$ and so:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\map \artanh {\dfrac x a} }
      | r = \frac 1 a \frac 1 {1 - \paren {\frac x a}^2}
      | c = Derivative of $\artanh$ 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}}
$\artanh \dfrac x a$ is not defined when either $x \le -a$ or $x \ge a$.
{{qed}}
Theorem: Derivative of Real Area Hyperbolic Cotangent of x over a

:$\map {\dfrac \d {\d x} } {\map \arcoth {\dfrac x a} } = \dfrac {-a} {x^2 - a^2}$
where $x^2 > a^2$.

Proof:
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 \arcoth {\dfrac x a} }
      | r = \frac 1 a \frac 1 {1 - \paren {\frac x a}^2}
      | c = Derivative of $\arcoth$ 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}}
$\arcoth \dfrac x a$ is not defined when $x^2 \le a^2$.
{{qed}}
Theorem: Derivative of Real Area Hyperbolic Secant of x over a

:$\dfrac {\map \d {\map \arsech {\frac x a} } } {\d x} = \dfrac {-a} {x \sqrt{a^2 - x^2} }$
where $0 < x < a$.

Proof:
Let $0 < x < a$.
Then $0 < \dfrac x a < 1$ and so:
{{begin-eqn}}
{{eqn | l = \frac {\map \rd {\map \arsech {\frac x a} } } {\rd x}
      | r = \frac 1 a \frac {-1} {\frac x a \sqrt {1 - \paren {\frac x a}^2} }
      | c = Derivative of $\arsech$ 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}}
$\arsech \dfrac x a$ is not defined when $x \le 0$ or $x \ge a$.
{{qed}}
Theorem: Derivative of Real Area Hyperbolic Cosecant of x over a

:$\dfrac {\map \d {\arcsch \dfrac x a} } {\d x} = \dfrac {-a} {\size x \sqrt {a^2 + x^2} }$
where $x \ne 0$.

Proof:
Let $0 < x < a$.
Then $0 < \dfrac x a < 1$ and so:
{{begin-eqn}}
{{eqn | l = \frac {\map \d {\arcsch \dfrac x a} } {\d x}
      | r = \frac 1 a \dfrac {-1} {\size {\frac x a} \sqrt {1 + \paren {\frac x a}^2} }
      | c = Derivative of $\arcsch$ 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}}
$\arcsch \dfrac x a$ is not defined when $x = 0$.
{{qed}}
Theorem: Primitive of Inverse Hyperbolic Secant of x over a

:$\ds \int \arsech \frac x a \rd x = x \arsech \dfrac x a + a \arcsin \dfrac x a + C$

Proof:
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
      | r = \arsech \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 $\arsech \dfrac x a$
}}
{{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 \arsech \frac x a \rd x
      | r = x \arsech \frac x a - \int x \paren {\frac {-a} {x \sqrt {a^2 - x^2} } } \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = x \arsech \frac x a + a \int \frac {\d x} {\sqrt {a^2 - x^2} } + C
      | c = Primitive of Constant Multiple of Function
}}
{{eqn | r = x \arsech \frac x a + \arcsin \frac x a + C
      | c = Primitive of $\dfrac 1 {\sqrt {a^2 - x^2} }$
}}
{{end-eqn}}
{{qed}}