avewright/proofwiki-math huggingface dataset:
Theorem: Primitive of Reciprocal of p squared minus Square of q by Hyperbolic Sine of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {p^2 - q^2 \sinh^2 a x}
      | r = \int \frac {\sech^2 a x \rd x} {p^2 \sech^2 a x - q^2 \tanh^2 a x}
      | c = multiplying numerator and denominator by $\sech^2 a x$
}}
{{eqn | r = \int \frac {\sech^2 a x \rd x} {p^2 \paren {1 - \tanh^2 a x} - q^2 \tanh^2 a x}
      | c = Sum of Squares of Hyperbolic Secant and Tangent
}}
{{eqn | r = \int \frac {\sech^2 a x \rd x} {p^2 - \paren {p^2 + q^2} \tanh^2 a x}
      | c = 
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = u
      | r = \tanh a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \d u
      | r = a \sech^2 a x \rd x
      | c = Derivative of Hyperbolic Tangent Function
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\sech^2 a x \rd x} {p^2 - \paren {p^2 + q^2} \tanh^2 a x}
      | r = \int \frac {\d u} {a p^2 - a \paren {p^2 + q^2} u^2}
      | c = Integration by Substitution: $u = \tanh a x$
}}
{{eqn | r = \frac 1 {a \paren {p^2 + q^2} } \int \frac {\d u} {\paren {\frac {p^2} {p^2 + q^2} } - u^2}
      | c = 
}}
{{eqn | r = \frac {-1} {a \paren {p^2 + q^2} } \int \frac {\d u} {u^2 - \paren {\frac p {\sqrt {p^2 + q^2} } }^2}
      | c = 
}}
{{eqn | r = \frac {-1} {a \paren {p^2 + q^2} } \frac {\sqrt {p^2 + q^2} } {2 p} \ln \size {\frac {u - \frac p {\sqrt {p^2 + q^2} } } {u + \frac p {\sqrt {p^2 + q^2} } } } + C
      | c = Primitive of $\dfrac 1 {x^2 - a^2}$
}}
{{eqn | r = \frac {-1} {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {\sqrt {p^2 + q^2} u - p} {\sqrt {p^2 + q^2} u + p} } + C
      | c = 
}}
{{eqn | r = \frac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {\sqrt {p^2 + q^2} u + p} {\sqrt {p^2 + q^2} u - p} } + C
      | c = Logarithm of Reciprocal
}}
{{eqn | r = \frac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {\sqrt {p^2 + q^2} \tanh a x + p} {\sqrt {p^2 + q^2} \tanh a x - p} } + C
      | c = substituting $u = \tanh a x$
}}
{{eqn | r = \frac 1 {2 a p \sqrt {p^2 + q^2} } \ln \size {\frac {p + \sqrt {p^2 + q^2} \tanh a x} {p - \sqrt {p^2 + q^2} \tanh a x} } + C
      | c = as $\size {a - b} - \size {b - a}$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of x by Hyperbolic Sine of a x

:$\ds \int x^m \sinh a x \rd x = \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \rd x$

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

:$\ds \int x^m \cosh a x \rd x = \frac {x^m \sinh a x} a - \frac m a \int x^{m - 1} \sinh a x \rd x + C$

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

:$\ds \int \sinh^n a x \rd x = \frac {\sinh^{n - 1} a x \cosh a x} {a n} - \frac {n - 1} n \int \sinh^{n - 2} a x \rd x$
for $n \ne 0$.

Proof:
With a view to expressing the problem 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^{n - 1} a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = \paren {n - 1} a \sinh^{n - 2} a x \cosh a x
      | c = Chain Rule for Derivatives, Derivative of $\sinh a x$, Derivative of Power
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \sinh a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {\cosh a x} a
      | c = Primitive of $\sinh a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \sinh^n a x \rd x
      | r = \int \sinh^{n - 1} a x \sinh a x \rd x
      | c = 
}}
{{eqn | r = \sinh^{n - 1} a x \paren {\frac {\cosh a x} a}
      | c = Integration by Parts
}}
{{eqn | o = 
      | ro= -
      | r = \int \paren {\frac {\cosh a x} a} \paren {\paren {n - 1} a \sinh ^{n - 2} a x \cosh a x} \rd x
      | c = 
}}
{{eqn | r = \frac {\sinh^{n - 1} a x \cosh a x} a
      | c = simplifying
}}
{{eqn | o = 
      | ro= -
      | r = \int \paren {n - 1} \sinh^{n - 2} a x \cosh^2 a x \rd x
      | c = 
}}
{{eqn | r = \frac {\sinh^{n - 1} a x \cosh a x} a
      | c = Difference of Squares of $\cosh$ and $\sinh$
}}
{{eqn | o = 
      | ro= -
      | r = \int \paren {n - 1} \sinh^{n - 2} a x \paren {1 + \sinh^2 a x} \rd x
      | c = 
}}
{{eqn | r = \frac {\sinh^{n - 1} a x \cosh a x} a - \paren {n - 1} \int \sinh^{n - 2} a x \rd x
      | c = Linear Combination of Primitives
}}
{{eqn | o = 
      | ro= -
      | r = \paren {n - 1} \int \sinh^n a x \rd x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = n \int \sinh^n a x \rd x
      | r = \frac {\sinh^{n - 1} a x \cosh a x} a - \paren {n - 1} \int \sinh^{n - 2} a x \rd x
      | c = rearranging
}}
{{eqn | ll= \leadsto
      | l = \int \sinh^n a x \rd x
      | r = \frac {\sinh^{n - 1} a x \cosh a x} {a n} - \frac {n - 1} n \int \sinh^{n - 2} a x \rd x
      | c = dividing both sides by $n$
}}
{{end-eqn}}
We note that if $n = 0$, then $\dfrac {n - 1} n$ is undefined, rendering the derivation invalid.
{{qed}}
Theorem: Primitive of Power of Hyperbolic Cosine of a x

:$\ds \int \cosh^n a x \rd x = \frac {\cosh^{n - 1} a x \sinh a x} {a n} + \frac {n - 1} n \int \cosh^{n - 2} a x \rd x$
for $n \ne 0$.

Proof:
With a view to expressing the problem 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^{n - 1} a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = \paren {n - 1} a \cosh^{n - 2} a x \sinh a x
      | c = Chain Rule for Derivatives, Derivative of $\cosh a x$, Derivative of Power
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = \cosh a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {\sinh a x} a
      | c = Primitive of $\cosh a x$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \cosh^n a x \rd x
      | r = \int \cosh^{n - 1} a x \cosh a x \rd x
      | c = 
}}
{{eqn | r = \cosh^{n - 1} a x \paren {\frac {\sinh a x} a}
      | c = Integration by Parts
}}
{{eqn | o = 
      | ro= -
      | r = \int \paren {\frac {\sinh a x} a} \paren {\paren {n - 1} a \cosh^{n - 2} a x \sinh a x} \rd x
      | c = 
}}
{{eqn | r = \frac {\cosh^{n - 1} a x \sinh a x} a
      | c = simplifying
}}
{{eqn | o = 
      | ro= -
      | r = \int \paren {n - 1} \cosh^{n - 2} a x \sinh^2 a x \rd x
      | c = 
}}
{{eqn | r = \frac {\cosh^{n - 1} a x \sinh a x} a
      | c = Difference of Squares of $\cosh$ and $\sinh$
}}
{{eqn | o = 
      | ro= -
      | r = \int \paren {n - 1} \cosh^{n - 2} a x \paren {\cosh^2 a x - 1} \rd x
      | c = 
}}
{{eqn | r = \frac {\cosh^{n - 1} a x \sinh a x} a + \paren {n - 1} \int \cosh^{n - 2} a x \rd x
      | c = Linear Combination of Primitives
}}
{{eqn | o = 
      | ro= -
      | r = \paren {n - 1} \int \cosh^n a x \rd x 
      | c = 
}}
{{eqn | ll= \leadsto
      | l = n \int \cosh^n a x \rd x
      | r = \frac {\cosh^{n - 1} a x \sinh a x} a + \paren {n - 1} \int \cosh^{n - 2} a x \rd x 
      | c = rearranging
}}
{{eqn | ll= \leadsto
      | l = \int \cosh^n a x \rd x
      | r = \frac {\cosh^{n - 1} a x \sinh a x} {a n} + \frac {n - 1} n \int \cosh^{n - 2} a x \rd x
      | c = dividing both sides by $n$
}}
{{end-eqn}}
We note that if $n = 0$, then $\dfrac {n - 1} n$ is undefined, rendering the derivation invalid.
{{qed}}
Theorem: Primitive of Power of Hyperbolic Tangent of a x

:$\ds \int \tanh^n a x \rd x = \frac {-\tanh^{n - 1} a x} {a \paren {n - 1} } + \int \tanh^{n - 2} a x \rd x + C$

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

:$\ds \int \tanh^n a x \sech^2 a x \rd x = \frac {\tanh^{n + 1} a x} {\paren {n + 1} a} + C$

Proof:
{{begin-eqn}}
{{eqn | l = z
      | r = \tanh a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d z} {\d x}
      | r = a \sech^2 a x
      | c = Derivative of $\tanh a x$
}}
{{eqn | ll= \leadsto
      | l = \int \tanh^n a x \sech^2 a x \rd x
      | r = \int \frac 1 a z^n \rd z
      | c = Integration by Substitution
}}
{{eqn | r = \frac 1 a \frac {z^{n + 1} } {n + 1}
      | c = Primitive of Power
}}
{{eqn | r = \frac {\tanh^{n + 1} a x} {\paren {n + 1} a} + C
      | c = substituting for $z$ and simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Hyperbolic Tangent of a x

:$\map {\dfrac \d {\d x} } {\tanh a x} = a \sech^2 a x$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\tanh x}
      | r = \sech^2 x
      | c = Derivative of $\tanh x$
}}
{{eqn | ll= \leadsto
      | l = \map {\dfrac \d {\d x} } {\tanh a x}
      | r = a \sech^2 a x
      | c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Hyperbolic Cotangent of a x

:$\map {\dfrac \d {\d x} } {\coth a x} = -a \csch^2 a x$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\coth x}
      | r = -\csch^2 x
      | c = Derivative of $\coth x$
}}
{{eqn | ll= \leadsto
      | l = \map {\dfrac \d {\d x} } {\coth a x}
      | r = -a \csch^2 a x
      | c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of Hyperbolic Cotangent of a x by Square of Hyperbolic Cosecant of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = z
      | r = \coth a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d z} {\d x}
      | r = -a \csch^2 a x
      | c = Derivative of $\coth a x$
}}
{{eqn | ll= \leadsto
      | l = \int \coth^n a x \csch^2 a x \rd x
      | r = \int \frac {-1} a z^n \rd z
      | c = Integration by Substitution
}}
{{eqn | r = \frac {-1} a \frac {z^{n + 1} } {n + 1}
      | c = Primitive of Power
}}
{{eqn | r = \frac {-\coth^{n + 1} a x} {\paren {n + 1} a} + C
      | c = substituting for $z$ and simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of Hyperbolic Cotangent of a x

:$\ds \int \coth^n a x \rd x = \frac {-\coth^{n - 1} a x} {a \paren {n - 1} } + \int \coth^{n - 2} a x \rd x + C$

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

:$\map {\dfrac \d {\d x} } {\sech a x} = -a \sech a x \tanh a x$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\sech x}
      | r = -\sech x \tanh x
      | c = Derivative of $\sech x$
}}
{{eqn | ll= \leadsto
      | l = \map {\dfrac \d {\d x} } {\sech a x}
      | r = -a \sech a x \tanh a x
      | c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}}
Theorem: Derivative of Hyperbolic Cosecant of a x

:$\map {\dfrac \d {\d x} } {\csch a x} = -a \csch a x \coth a x$

Proof:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\csch x}
      | r = -\csch x \coth x
      | c = Derivative of $\csch x$
}}
{{eqn | ll= \leadsto
      | l = \map {\dfrac \d {\d x} } {\csch a x}
      | r = -a \csch a x \coth a x
      | c = Derivative of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of Hyperbolic Secant of a x

:$\ds \int \sech^n a x \rd x = \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C$
for $n \ne -1$.

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 = \sech^{n - 2} a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = -a \paren {n - 2} \sech^{n - 3} a x \sech a x \tanh a x
      | c = Derivative of Power, Derivative of $\sech a x$, Chain Rule for Derivatives
}}
{{eqn | r = -a \paren {n - 2} \sech^{n - 2} a x \tanh a x
      | c = 
}}
{{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 \sech^n a x \rd x
      | r = \int \sech^{n - 2} a x \sech^2 a x \rd x
      | c = 
}}
{{eqn | r = \sech^{n - 2} a x \paren {\frac {\tanh a x} a}
      | c = Integration by Parts
}}
{{eqn | o = 
      | ro= -
      | r = \int \paren {\frac {\tanh a x} a} \paren {-a \paren {n - 2} \sech^{n - 2} a x \tanh a x} \rd x
      | c = 
}}
{{eqn | r = \frac {\sech^{n - 2} a x \tanh a x} a + \paren {n - 2} \int \tanh^2 a x \sech^{n - 2} a x \rd x
      | c = simplifying
}}
{{eqn | r = \frac {\sech^{n - 2} a x \tanh a x} a + \paren {n - 2} \int \paren {1 - \sech^2 a x} \sech^{n - 2} a x \rd x
      | c = Sum of $\sech^2$ and $\tanh^2$
}}
{{eqn | r = \frac {\sech^{n - 2} a x \tanh a x} a - \paren {n - 2} \int \sech^n a x \rd x
      | c = Linear Combination of Primitives
}}
{{eqn | o = 
      | ro= +
      | r = \paren {n - 2} \int \sech^{n - 2} a x \rd x
      | c = 
}}
{{eqn | l = \paren {n - 1} \int \sech^n a x \rd x
      | r = \frac {\sech^{n - 2} a x \tanh a x} a + \paren {n - 2} \int \sech^{n - 2} a x \rd x
      | c = gathering terms
}}
{{eqn | l = \int \sech^n a x \rd x
      | r = \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x
      | c = dividing by $n - 1$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of Hyperbolic Cosecant of a x

:$\ds \int \csch^n a x \rd x = \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x + C$
for $n \ne -1$.

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 = \csch^{n - 2} a x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = -a \paren {n - 2} \csch^{n - 3} a x \csch a x \coth a x
      | c = Derivative of Power, Derivative of $\csch a x$, Chain Rule for Derivatives
}}
{{eqn | r = -a \paren {n - 2} \csch^{n - 2} a x \coth a x
      | c = 
}}
{{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 \csch^n a x \rd x
      | r = \int \csch^{n - 2} a x \csch^2 a x \rd x
      | c = 
}}
{{eqn | r = \csch^{n - 2} a x \paren {\frac {-\coth a x} a}
      | c = Integration by Parts
}}
{{eqn | o = 
      | ro= -
      | r = \int \paren {\frac {-\coth a x} a} \paren {-a \paren {n - 2} \csch^{n - 2} a x \coth a x} \rd x
      | c = 
}}
{{eqn | r = \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \coth^2 a x \csch^{n - 2} a x \rd x
      | c = simplifying
}}
{{eqn | r = \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \paren {1 + \csch^2 a x} \csch^{n - 2} a x \rd x
      | c = Difference of $\coth^2$ and $\csch^2$
}}
{{eqn | r = \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \csch^n a x \rd x
      | c = Linear Combination of Primitives
}}
{{eqn | o = 
      | ro= -
      | r = \paren {n - 2} \int \csch^{n - 2} a x \rd x
      | c = 
}}
{{eqn | l = \paren {n - 1} \int \csch^n a x \rd x
      | r = \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \csch^{n - 2} a x \rd x
      | c = gathering terms
}}
{{eqn | l = \int \csch^n a x \rd x
      | r = \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x
      | c = dividing by $n - 1$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Sine of a x over Power of x

:$\ds \int \frac {\sinh a x \rd x} {x^n} = \frac {-\sinh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cosh a x \rd x} {x^{n - 1} }$

Proof:
With a view to expressing the problem 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^n}
      | c = 
}}
{{eqn | r = x^{-n}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {x^{-n + 1} } {- n + 1}
      | c = Primitive of Power
}}
{{eqn | r = \frac {-1} {\paren {n - 1} x^{n - 1} }
      | c = simplifying
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\sinh a x \rd x} {x^n}
      | r = \sinh a x \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } - \int \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } \paren {a \cosh a x} \rd x
      | c = Integration by Parts
}}
{{eqn | r = \frac {-\sinh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cosh a x \rd x} {x^{n - 1} }
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Cosine of a x over Power of x

:$\ds \int \frac {\cosh a x \rd x} {x^n} = \frac {-\cosh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\sinh a x \rd x} {x^{n - 1} } + C$

Proof:
With a view to expressing the problem 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^n}
      | c = 
}}
{{eqn | r = x^{-n}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {x^{-n + 1} } {- n + 1}
      | c = Primitive of Power
}}
{{eqn | r = \frac {-1} {\paren {n - 1} x^{n - 1} }
      | c = simplifying
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\cosh a x \rd x} {x^n}
      | r = \cosh a x \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } - \int \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } \paren {a \sinh a x} \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {-\cosh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\sinh a x \rd x} {x^{n - 1} } + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Power of Hyperbolic Sine of a x

:$\ds \int \frac {\d x} {\sinh^n a x} = \frac {-\cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac {n - 2} {n - 1} \int \frac {\d x} {\sinh^{n - 2} a x}$
for $n \ne 1$.

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^n a x}
      | r = \int \csch^n a x \rd x
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x
      | c = Primitive of $\csch^n a x$
}}
{{eqn | r = \frac {-\coth a x} {a \paren {n - 1} \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {\d x} {\sinh^{n - 2} a x}
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {-\cosh a x} {a \sinh a x \left({n - 1}\right) \sinh^{n - 2} a x} - \frac {n - 2} {n - 1} \int \frac {\d x} {\sinh^{n - 2} a x}
      | c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \frac {-\cosh a x} {a \paren {n - 1} \sinh^{n - 1} a x} - \frac {n - 2} {n - 1} \int \frac {\d x} {\sinh^{n - 2} a x}
      | c = simplifying
}}
{{end-eqn}}
We note that when $n = 1$, $\dfrac {n - 2} {n - 1}$ is undefined, rendering this derivation invalid.
{{qed}}
Theorem: Primitive of Reciprocal of Power of Hyperbolic Cosine of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\cosh^n a x}
      | r = \int \sech^n a x \rd x
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {\sech^{n - 2} a x \tanh a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sech^{n - 2} a x \rd x + C
      | c = Primitive of $\sech^n a x$
}}
{{eqn | r = \frac {\tanh a x} {a \paren {n - 1} \cosh^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {\sinh a x} {a \cosh a x \paren {n - 1} \cosh^{n - 2} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C
      | c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac {\sinh a x} {a \paren {n - 1} \cosh^{n - 1} a x} + \frac {n - 2} {n - 1} \int \frac {\d x} {\cosh^{n - 2} a x} + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Hyperbolic Cosine of a x plus 1

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

Proof:
{{begin-eqn}}
{{eqn | l = u
      | r = \tanh \frac x 2
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\d x} {1 + \cosh x}
      | r = \int \frac {\dfrac {2 \rd u} {1 - u^2} } {\dfrac {1 + u^2} {1 - u^2} + 1}
      | c = Hyperbolic Tangent Half-Angle Substitution
}}
{{eqn | r = \int \frac {2 \rd u} {1 + u^2 + \paren {1 - u^2} }
      | c = multiplying top and bottom by $1 - u^2$
}}
{{eqn | r = \int \rd u
      | c = simplifying
}}
{{eqn | r = u + C
      | c = Primitive of Constant
}}
{{eqn | r = \tanh \frac x 2 + C
      | c = substituting for $u$
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\d x} {1 + \cosh a x}
      | r = \frac 1 a \tanh \frac {a x} 2 + C
      | c = Primitive of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}}