avewright/proofwiki-math huggingface dataset:
Theorem: Primitive of Hyperbolic Cosecant of a x over x

{{begin-eqn}}
{{eqn | l = \int \frac {\csch a x \rd x} x
      | r = \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1}\paren {2 k}!} + C
}}
{{eqn | r = -\frac 1 {a x} - \frac {a x} 6 + \frac {7 \paren {a x}^3} {1080} + \cdots + C
}}
{{end-eqn}}
where $B_{2 k}$ denotes the $2 k$th Bernoulli number.

Proof:
{{begin-eqn}}
{{eqn | l = \csch x
      | r = \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} x^{2 k - 1} } {\paren {2 k}!}
      | c = Power Series Expansion for Hyperbolic Cosecant Function
}}
{{eqn | ll= \leadsto
      | l = \frac {\csch a x} x
      | r = \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} \paren {a x}^{2 k - 1} } {x \paren {2 k}!}
      | c = 
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\csch a x \rd x} x
      | r = \int \paren {\sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} } \rd x
      | c = 
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \int \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} \rd x
      | c = 
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} a^{2 k - 1} x^{2 k - 1} } {\paren {2 k - 1}\paren {2 k}!}
      | c = Primitive of Power
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 k - 1} } B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1}\paren {2 k}!}
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Hyperbolic Sine of a x

:$\map {D_x} {\sinh a x} = a \cosh a x$

Proof:
{{begin-eqn}}
{{eqn | l = \map {D_x} {\sinh x}
      | r = \cosh x
      | c = Derivative of $\sinh x$
}}
{{eqn | ll= \leadsto
      | l = \map {D_x} {\sinh a x}
      | r = a \cosh a x
      | c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Hyperbolic Cosine of a x

:$\map {\dfrac \d {\d x} } {\cosh a x} = a \sinh a x$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\cosh x}
      | r = \sinh x
      | c = Derivative of $\cosh x$
}}
{{eqn | ll= \leadsto
      | l = \map {\dfrac \d {\d x} } {\cosh a x}
      | r = a \sinh a x
      | c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Sine of a x over x squared

:$\ds \int \frac {\sinh a x \ \d x} {x^2} = -\frac {\sinh a x} x + a \int \frac {\cosh a x \rd x} x$

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 = \sinh a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = a \cosh a x
      | c = Derivative of $\sinh a x$
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \frac 1 {x^2}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = -\frac 1 x
      | c = Primitive of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\sinh a x \rd x} {x^2}
      | r = \int \sinh a x \frac 1 {x^2} \rd x
      | c = 
}}
{{eqn | r = \sinh a x \paren {-\frac 1 x} - \int \paren {-\frac 1 x} a \cosh a x \rd x
      | c = Integration by Parts
}}
{{eqn | r = -\frac {\sinh a x} x + a \int \frac {\cosh a x \rd x} x
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Cosine of a x over x squared

:$\ds \int \frac {\cosh a x \rd x} {x^2} = -\frac {\cosh a x} x + a \int \frac {\sinh a x \rd x} x$

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 = \cosh a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = a \sinh a x
      | c = Derivative of $\cosh a x$
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \frac 1 {x^2}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = -\frac 1 x
      | c = Primitive of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\cosh a x \rd x} {x^2}
      | r = \int \cosh a x \frac 1 {x^2} \rd x
      | c = 
}}
{{eqn | r = \cosh a x \paren {-\frac 1 x} - \int \paren {-\frac 1 x} a \sinh a x \rd x
      | c = Integration by Parts
}}
{{eqn | r = -\frac {\cosh a x} x + a \int \frac {\sinh a x \rd x} x
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Hyperbolic Sine of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh a x}
      | r = \int \csch a x \rd x
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C
      | c = Primitive of $\csch a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Hyperbolic Cosine of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cosh a x}
      | r = \int \sech a x \rd x
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {2 \map \arctan {e^{a x} } } a + C
      | c = Primitive of $\sech a x$: Arctangent of Exponential Form
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Hyperbolic Tangent of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\tanh a x}
      | r = \int \coth a x \rd x
      | c = Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent
}}
{{eqn | r = \frac {\ln \size {\sinh a x} } a + C
      | c = Primitive of $\coth a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent

:$\coth x = \dfrac 1 {\tanh x}$
where $\tanh$ and $\coth$ denote hyperbolic tangent and hyperbolic cotangent respectively.

Proof:
{{begin-eqn}}
{{eqn | l = \coth x
      | r = \frac {\cosh x} {\sinh x}
      | c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac 1 {\sinh x / \cosh x}
      | c = 
}}
{{eqn | r = \frac 1 {\tanh x}
      | c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Hyperbolic Cotangent of a x

:$\ds \int \frac {\d x} {\coth a x} = \frac {\ln \size {\cosh a x} } a + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\coth a x}
      | r = \int \tanh a x \rd x
      | c = Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent
}}
{{eqn | r = \frac {\ln \size {\cosh a x} } a + C
      | c = Primitive of $\tanh a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Hyperbolic Secant of a x

:$\ds \int \frac {\d x} {\sech a x} = \frac {\sinh a x} a + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sech a x}
      | r = \int \cosh a x \rd x
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {\sinh a x} a + C
      | c = Primitive of $\cosh a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Hyperbolic Cosecant of a x

:$\ds \int \frac {\d x} {\csch a x} = \frac {\cosh a x} a + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\csch a x}
      | r = \int \sinh a x \rd x
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {\cosh a x} a + C
      | c = Primitive of $\sinh a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x over Hyperbolic Sine of a x

{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\sinh a 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 \frac {x \rd x} {\sinh a x}
      | r = \int x \csch a x \rd x
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{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 = Primitive of $x \csch a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x over Hyperbolic Cosine of a x

{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\cosh a 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}}
where $E_{2 n}$ denotes the $2 n$th Euler number.

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\cosh a x}
      | r = \int x \sech a x \rd x
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{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 = Primitive of $x \sech a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Sine of a x

:$\ds \int \sinh^2 a x \rd x = \dfrac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \sinh^2 x \rd x
      | r = \frac {\sinh x \cosh x - x} 2 + C
      | c = Primitive of $\sinh^2 x$
}}
{{eqn | ll= \leadsto
      | l = \int \sinh^2 a x \rd x
      | r = \frac 1 a \paren {\frac {\sinh a x \cosh a x - a x} 2} + C
      | c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Cosine of a x

:$\ds \int \cosh^2 a x \rd x = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \cosh^2 x \rd x
      | r = \frac {\sinh x \cosh x + x} 2 + C
      | c = Primitive of $\cosh^2 x$
}}
{{eqn | ll= \leadsto
      | l = \int \cosh^2 a x \rd x
      | r = \frac 1 a \paren {\frac {\sinh a x \cosh a x + a x} 2} + C
      | c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Tangent of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \tanh^2 x \rd x
      | r = x - \tanh x + C
      | c = Primitive of $\tanh^2 x$
}}
{{eqn | ll= \leadsto
      | l = \int \tanh^2 a x \rd x
      | r = \frac 1 a \paren {a x - \tanh a x} + C
      | c = Primitive of Function of Constant Multiple
}}
{{eqn | r = x - \frac {\tanh a x} a + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Cotangent of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \coth^2 x \rd x
      | r = x - \coth x + C
      | c = Primitive of $\coth^2 x$
}}
{{eqn | ll= \leadsto
      | l = \int \coth^2 a x \rd x
      | r = \frac 1 a \paren {a x - \coth a x} + C
      | c = Primitive of Function of Constant Multiple
}}
{{eqn | r = x - \frac {\coth a x} a + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Secant of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \sech^2 x \rd x
      | r = \tanh x + C
      | c = Primitive of $\sech^2 x$
}}
{{eqn | ll= \leadsto
      | l = \int \sech^2 a x \rd x
      | r = \frac 1 a \paren {\tanh a x} + C
      | c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {\tanh a x} a + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Cosecant of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \csch^2 x \rd x
      | r = -\coth x + C
      | c = Primitive of $\csch^2 x$
}}
{{eqn | ll= \leadsto
      | l = \int \csch^2 a x \rd x
      | r = \frac 1 a \paren {-\coth a x} + C
      | c = Primitive of Function of Constant Multiple
}}
{{eqn | r = \frac {-\coth a x} a + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Sine of a x/Corollary

:$\ds \int \sinh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} - \frac x 2 + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \sinh^2 a x \rd x
      | r = \frac {\sinh a x \cosh a x} {2 a} - \frac x 2 + C
      | c = Primitive of $\sinh^2 a x$
}}
{{eqn | r = \frac {\frac {\sinh 2 a x} 2} {2 a} - \frac x 2 + C
      | c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \dfrac {\sinh 2 a x} {4 a} - \frac x 2 + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x by Square of Hyperbolic Sine of a x

:$\ds \int x \sinh^2 a x \rd x = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C$

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 = x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 1
      | c = Derivative of Identity Function
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \sinh^2 a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {\sinh 2 a x} {4 a} - \frac x 2
      | c = Primitive of $\sinh^2 a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \sinh^2 a x \rd x
      | r = x \paren {\frac {\sinh 2 a x} {4 a} - \frac x 2} - \int \paren {\frac {\sinh 2 a x} {4 a} - \frac x 2} \times 1 \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x + \frac 1 2 \int x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x + \frac {x^2} 4 + C
      | c = Primitive of Power
}}
{{eqn | r = \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \frac {\cosh 2 a x} {2 a} + \frac {x^2} 4 + C
      | c = Primitive of $\sinh a x$
}}
{{eqn | r = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Cosine of a x/Corollary

:$\ds \int \cosh^2 a x \rd x = \frac {\sinh 2 a x} {4 a} + \frac x 2 + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \cosh^2 a x \rd x
      | r = \frac {\sinh a x \cosh a x} {2 a} + \frac x 2 + C
      | c = Primitive of $\cosh^2 a x$
}}
{{eqn | r = \frac {\frac {\sinh 2 a x} 2} {2 a} + \frac x 2 + C
      | c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \dfrac {\sinh 2 a x} {4 a} + \frac x 2 + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x by Square of Hyperbolic Cosine of a x

:$\ds \int x \cosh^2 a x \rd x = \frac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} + \frac {x^2} 4 + C$

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 = x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 1
      | c = Derivative of Identity Function
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \cosh^2 a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {\sinh 2 a x} {4 a} + \frac x 2
      | c = Primitive of $\cosh^2 a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \cosh^2 a x \rd x
      | r = x \paren {\frac {\sinh 2 a x} {4 a} + \frac x 2} - \int \paren {\frac {\sinh 2 a x} {4 a} + \frac x 2} \times 1 \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x \sinh 2 a x} {4 a} + \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x - \frac 1 2 \int x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {x \sinh 2 a x} {4 a} + \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x - \frac {x^2} 4 + C
      | c = Primitive of Power
}}
{{eqn | r = \frac {x \sinh 2 a x} {4 a} + \frac {x^2} 2 - \frac 1 {4 a} \frac {\cosh 2 a x} {2 a} - \frac {x^2} 4 + C
      | c = Primitive of $\sinh a x$
}}
{{eqn | r = \frac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} + \frac {x^2} 4 + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x by Square of Hyperbolic Tangent of a x

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

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 = x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 1
      | c = Derivative of Identity Function
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \tanh^2 a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = x - \frac {\tanh a x} a
      | c = Primitive of $\tanh^2 a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \tanh^2 a x \rd x
      | r = x \paren {x - \frac {\tanh a x} a} - \int \paren {x - \frac {\tanh a x} a} \times 1 \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = x^2 - \frac {x \tanh a x} a + \int x \rd x + \frac 1 a \int \tanh a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = x^2 - \frac {x \tanh a x} a + \frac {x^2} 2 + \frac 1 a \int \tanh a x \rd x + C
      | c = Primitive of Power
}}
{{eqn | r = x^2 - \frac {x \tanh a x} a + \frac {x^2} 2 + \frac 1 a \frac {\ln \size {\cosh a x} } a + C
      | c = Primitive of $\tanh a x$
}}
{{eqn | r = \frac {x^2} 2 - \frac {x \tanh a x} a + \frac 1 {a^2} \ln \size {\cosh a x} + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x by Square of Hyperbolic Cotangent of a x

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

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 = x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 1
      | c = Derivative of Identity Function
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \coth^2 a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = x - \frac {\coth a x} a
      | c = Primitive of $\coth^2 a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \coth^2 a x \rd x
      | r = x \paren {x - \frac {\coth a x} a} - \int \paren {x - \frac {\coth a x} a} \times 1 \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = x^2 - \frac {x \coth a x} a + \int x \rd x + \frac 1 a \int \coth a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = x^2 - \frac {x \coth a x} a + \frac {x^2} 2 + \frac 1 a \int \coth a x \rd x + C
      | c = Primitive of Power
}}
{{eqn | r = x^2 - \frac {x \coth a x} a + \frac {x^2} 2 + \frac 1 a \frac {\ln \size {\sinh a x} } a + C
      | c = Primitive of $\coth a x$
}}
{{eqn | r = \frac {x^2} 2 - \frac {x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x by Square of Hyperbolic Secant of a x

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

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 = x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 1
      | c = Derivative of Identity Function
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \sech^2 a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {\tanh a x} a
      | c = Primitive of $\sech^2 a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \sech^2 a x \rd x
      | r = x \paren {\frac {\tanh a x} a} - \int \paren {\frac {\tanh a x} a} \times 1 \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x \tanh a x} a - \frac 1 a \int \tanh a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {x \tanh a x} a - \frac 1 a \frac {\ln \size {\cosh a x} } a + C
      | c = Primitive of $\tanh a x$
}}
{{eqn | r = \frac {x \tanh a x} a - \frac 1 {a^2} \ln \size {\cosh a x} + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x by Square of Hyperbolic Cosecant of a x

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

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 = x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 1
      | c = Derivative of Identity Function
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \csch^2 a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = -\frac {\coth a x} a
      | c = Primitive of $\csch^2 a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x \csch^2 a x \rd x
      | r = x \paren {-\frac {\coth a x} a} - \int \paren {-\frac {\coth a x} a} \times 1 \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {-x \coth a x} a + \frac 1 a \int \coth a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {-x \coth a x} a + \frac 1 a \frac {\ln \size {\sinh a x} } a + C
      | c = Primitive of $\coth a x$
}}
{{eqn | r = \frac {-x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Square of Hyperbolic Sine of a x

:$\ds \int \frac {\d x} {\sinh^2 a x} = -\frac {\coth a x} a + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x}
      | r = \int \csch^2 a x \rd x
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = -\frac {\coth a x} a + C
      | c = Primitive of $\csch^2 a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Square of Hyperbolic Cosine of a x

:$\ds \int \frac {\d x} {\cosh^2 a x} = \frac {\tanh a x} a + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cosh^2 a x}
      | r = \int \sech^2 a x \rd x
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {\tanh a x} a + C
      | c = Primitive of $\sech^2 a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Sine of a x by Hyperbolic Sine of p x

:$\ds \int \sinh a x \sinh p x \rd x = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } - \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \sinh a x \sinh p x \rd x
      | r = \int \paren {\frac {\map \cosh {a x + p x} - \map \cosh {a x - p x} } 2} \rd x
      | c = Werner Formula for Hyperbolic Sine by Hyperbolic Sine
}}
{{eqn | r = \frac 1 2 \int \map \cosh {a + p} x \rd x - \frac 1 2 \int \map \cosh {a - p} x \rd x
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac 1 2 \frac {\map \sinh {a + p} x} {a + p} - \frac 1 2 \frac {\map \sinh {a - p} x} {a - p} + C
      | c = Primitive of $\cosh a x$
}}
{{eqn | r = \frac {\map \sinh {a + p} x} {2 \paren {a + p} } - \frac {\map \sinh {a - p} x} {2 \paren {a - p} } + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}