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

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

Proof:
{{begin-eqn}}
{{eqn | l = u
      | r = \tanh \frac x 2
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\d x} {\cosh x - 1}
      | 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 \frac {\d u} {u^2}
      | c = simplifying
}}
{{eqn | r = \frac {-1} u + C
      | c = Primitive of Power
}}
{{eqn | r = \frac {-1} {\tanh \dfrac x 2} + C
      | c = substituting for $u$
}}
{{eqn | r = -\coth \frac x 2 + C
      | c = Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\d x} {\cosh a x - 1}
      | r = \frac {-1} a \coth \frac {a x} 2 + C
      | c = Primitive of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x over Hyperbolic Cosine of a x plus 1

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

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

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

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 = \frac 1 {\cosh a x + 1}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = -\frac 1 a \coth \frac {a x} 2
      | c = Primitive of $\dfrac 1 {\cosh a x - 1}$
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {x \rd x} {\cosh a x - 1}
      | r = x \paren {-\frac 1 a \coth \frac {a x} 2} - \int \paren {-\frac 1 a \coth \frac {a x} 2} \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = -\frac x a \coth \frac {a x} 2 + \frac 1 a \int \coth {a x} 2 \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = -\frac x a \tanh \frac {a x} 2 + \frac 1 a \paren {\frac 2 a \ln \size {\sinh \frac {a x} 2} } + C
      | c = Primitive of $\coth a x$
}}
{{eqn | r = -\frac x a \coth \frac {a x} 2 + \frac 2 {a^2} \ln \size {\sinh \frac {a x} 2} + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Square of Hyperbolic Cosine of a x plus 1

:$\ds \int \frac {\d x} {\paren {\cosh a x + 1}^2} = \frac 1 {2 a} \tanh \frac {a x} 2 - \frac 1 {6 a} \tanh^3 \frac {a x} 2 + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {\cosh a x + 1}^2}
      | r = \int \paren {\frac 1 2 \sech^2 \frac {a x} 2}^2 \rd x
      | c = Reciprocal of Hyperbolic Cosine Plus One
}}
{{eqn | r = \frac 1 4 \int \sech^4 \frac {a x} 2 \rd x
      | c = simplifying
}}
{{eqn | r = \frac 1 4 \paren {\frac{\sech^2 \dfrac {a x} 2 \tanh \dfrac {a x} 2} {\dfrac {3 a} 2} + \frac 2 3 \int \sech^2 \frac {a x} 2 \rd x} + C
      | c = Primitive of $\sech^n a x$
}}
{{eqn | r = \frac 1 {6 a} \sech^2 \frac {a x} 2 \tanh \dfrac {a x} 2 + \frac 1 6 \int \sech^2 \frac {a x} 2 \rd x + C
      | c = simplifying
}}
{{eqn | r = \frac 1 {6 a} \sech^2 \frac {a x} 2 \tanh \dfrac {a x} 2 + \frac 1 6 \paren {\frac 2 a \tanh \frac {a x} 2} + C
      | c = Primitive of $\sech^2 a x$
}}
{{eqn | r = \frac 1 {6 a} \paren {1 - \tanh^2 \frac {a x} 2} \tan \dfrac {a x} 2 + \frac 2 {6 a} \tanh \frac {a x} 2 + C
      | c = Sum of Squares of Hyperbolic Secant and Tangent
}}
{{eqn | r = \frac 1 {2 a} \tanh \frac {a x} 2 - \frac 1 {6 a} \tanh^3 \frac {a x} 2 + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Square of Hyperbolic Cosine of a x minus 1

:$\ds \int \frac {\d x} {\paren {\cosh a x - 1}^2} = \frac 1 {2 a} \coth \frac {a x} 2 - \frac 1 {6 a} \coth^3 \frac {a x} 2 + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\paren {\cosh a x - 1}^2}
      | r = \int \paren {\frac 1 2 \csch^2 \frac {a x} 2}^2 \rd x
      | c = Reciprocal of Hyperbolic Cosine Minus One
}}
{{eqn | r = \frac 1 4 \int \csch^4 \frac {a x} 2 \rd x
      | c = simplifying
}}
{{eqn | r = \frac 1 4 \paren {\frac {-\csch^2 \dfrac {a x} 2 \coth \dfrac {a x} 2} {\dfrac {3 a} 2} - \frac 2 3 \int \csch^2 \frac {a x} 2 \rd x} + C
      | c = Primitive of $\csch^n a x$
}}
{{eqn | r = \frac {-1} {6 a} \csch^2 \frac {a x} 2 \coth \dfrac {a x} 2 - \frac 1 6 \int \csch^2 \frac {a x} 2 \rd x + C
      | c = simplifying
}}
{{eqn | r = \frac {-1} {6 a} \csch^2 \frac {a x} 2 \coth \dfrac {a x} 2 - \frac 1 6 \paren {\frac {-2} a \coth \frac {a x} 2} + C
      | c = Primitive of $\csch^2 a x$
}}
{{eqn | r = \frac {-1} {6 a} \paren {\coth^2 \frac {a x} 2 - 1} \coth \dfrac {a x} 2 + \frac 2 {6 a} \coth \frac {a x} 2 + C
      | c = Difference of Squares of Hyperbolic Cotangent and Cosecant
}}
{{eqn | r = \frac 1 {2 a} \coth \frac {a x} 2 - \frac 1 {6 a} \coth^3 \frac {a x} 2 + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \sinh a x \cosh a x \rd x
      | r = \int \frac {\sinh 2 a x} 2 \rd x
      | c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \frac 1 2 \int \sinh 2 a x \rd x
      | c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 2 \paren {\frac {\cosh 2 a x} {2 a} } + C
      | c = Primitive of $\sinh a x$
}}
{{eqn | r = \frac 1 2 \paren {\frac {1 + 2 \sinh^2 a x} {2 a} } + C
      | c = {{Corollary|Double Angle Formula for Hyperbolic Cosine|2}}
}}
{{eqn | r = \frac {\sinh^2 a x} {2 a} + \frac 1 {4 a} + C
      | c = simplifying
}}
{{eqn | r = \frac {\sinh^2 a x} {2 a} + C
      | c = subsuming $\dfrac 1 {4 a}$ into arbitrary constant
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \sinh a x \cosh a x \rd x
      | r = \int \cosh a x \sinh a x \rd x
      | c = 
}}
{{eqn | r = \frac {\cosh^2 a x} {2 a} + C
      | c = Primitive of $\cosh^n a x \sinh a x$ using $n = 1$
}}
{{eqn | r = \frac {1 + \sinh^2 a x} {2 a} + C
      | c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | r = \frac 1 {2 a} + \frac {\sinh^2 a x} {2 a} + C
      | c = simplifying
}}
{{eqn | r = \frac {\sinh^2 a x} {2 a} + C
      | c = subsuming $\dfrac 1 {2 a}$ into arbitrary constant
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

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

Proof:
{{begin-eqn}}
{{eqn | n = 1
      | l = \int \sinh^n a x \cosh a x \rd x
      | r = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C
      | c = Primitive of $\sinh^n a x \cosh a x$
}}
{{eqn | ll= \leadsto
      | l = \int \sinh a x \cosh a x \rd x
      | r = \frac {\sinh^2 a x} {2 a} + C
      | c = setting $n = 1$ in $(1)$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

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

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 = \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 = \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 \sinh a x \cosh a x \rd x
      | r = \paren {\sinh a x} \paren {\frac {\sinh a x} a} - \int \paren {\frac {\sinh a x} a} \paren {a \cosh a x} \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {\sinh^2 a x} a - \int \sinh a x \cosh a x \rd x + C
      | c = simplifying
}}
{{eqn | ll= \leadsto
      | l = 2 \int \sinh a x \cosh a x \rd x
      | r = \frac {\sinh^2 a x} a + C
      | c = gathering terms
}}
{{eqn | ll= \leadsto
      | l = \int \sinh a x \cosh a x \rd x
      | r = \frac {\sinh^2 a x} {2 a} + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

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

Proof:
{{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 $\sinh a x$
}}
{{eqn | ll= \leadsto
      | l = \int \sinh a x \cosh a x \rd x
      | r = \int \frac u a \rd u
      | c = Integration by Substitution
}}
{{eqn | r = \frac 1 a \int u \rd u
      | c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 a \frac {u^2} 2 + C
      | c = Primitive of Power
}}
{{eqn | r = \frac {\sinh^2 a x} {2 a} + C
      | c = substituting for $u$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Sine of p x by Hyperbolic Cosine of q x

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

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

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

Proof:
{{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 $\sinh a x$
}}
{{eqn | ll= \leadsto
      | l = \int \sinh^n a x \cosh a x \rd x
      | r = \int \frac {u^n} a \rd u
      | c = Integration by Substitution
}}
{{eqn | r = \frac 1 a \int u^n \rd u
      | c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 a \frac {u^{n + 1} } {n + 1} + C
      | c = Primitive of Power
}}
{{eqn | r = \frac {\sinh^{n + 1} a x} {\paren {n + 1} a} + C
      | c = substituting for $u$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x

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

Proof:
{{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 $\cosh a x$
}}
{{eqn | ll= \leadsto
      | l = \int \cosh^n a x \sinh a x \rd x
      | r = \int \frac {u^n} a \rd u
      | c = Integration by Substitution
}}
{{eqn | r = \frac 1 a \int u^n \rd u
      | c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 a \frac {u^{n + 1} } {n + 1} + C
      | c = Primitive of Power
}}
{{eqn | r = \frac {\cosh^{n + 1} a x} {\paren {n + 1} a} + C
      | c = substituting for $u$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x

:$\ds \int \sinh^2 a x \cosh^2 a x \rd x = \frac {\sinh 4 a x} {32 a} - \frac x 8 + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \sinh^2 a x \cosh^2 a x \rd x
      | r = \int \paren {\sinh a x \cosh a x}^2 \rd x
      | c = 
}}
{{eqn | r = \int \paren {\frac {\sinh 2 a x} 2}^2 \rd x
      | c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = \frac 1 4 \int \sinh^2 2 a x \rd x
      | c = Primitive of Constant Multiple of Function
}}
{{eqn | r = \frac 1 4 \paren {\frac {\sinh 2 \paren {2 a x} } {4 \paren {2 a} } - \frac x 2} + C
      | c = Primitive of $\sinh^2 a x$
}}
{{eqn | r = \frac {\sinh 4 a x} {32 a} - \frac x 8 + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh a x \cosh a x}
      | r = \int \frac {\sech a x \rd x} {\sinh a x}
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \int \frac {\sech^2 a x \rd x} {\sinh a x \sech a x}
      | c = multiplying top and bottom by $\sech a x$
}}
{{eqn | r = \int \frac {\sech^2 a x \rd x} {\frac {\sinh a x} {\cosh a x} }
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \int \frac {\sech^2 a x \rd x} {\tanh a x}
      | c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \frac 1 a \ln \size {\tanh a x} + C
      | c = Primitive of $\dfrac {\sech^2 a x} {\tanh a x}$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Secant of a x over Hyperbolic Tangent of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \frac {\d} {\d x} \tanh a x
      | r = a \sech^2 a x
      | c = Derivative of $\tanh a x$
}}
{{eqn | ll= \leadsto
      | l = \int \frac {a \sech^2 a x \rd x} {\tanh a x}
      | r = \ln \size {\tanh a x} + C
      | c = Primitive of Function under its Derivative
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\sech^2 a x \rd x} {\tanh a x}
      | r = \frac 1 a \ln \size {\tanh a x} + C
      | c = Primitive of Constant Multiple of Function
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Square of Hyperbolic Sine of a x by Hyperbolic Cosine of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x \cosh a x}
      | r = \int \frac {\paren {\cosh^2 a x - \sinh^2 a x} \rd x} {\sinh^2 a x \cosh a x}
      | c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | r = \int \frac {\cosh^2 a x \rd x} {\sinh^2 a x \cosh a x} - \int \frac {\sinh^2 a x \rd x} {\sinh^2 a x \cosh a x}
      | c = Linear Combination of Primitives
}}
{{eqn | r = \int \frac {\cosh a x \rd x} {\sinh^2 a x} - \int \frac {\d x} {\cosh a x}
      | c = simplifying
}}
{{eqn | r = \int \frac {\cosh a x \rd x} {\sinh^2 a x} - \int \sech a x \rd x
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \int \frac {\coth a x \rd x} {\sinh a x} - \int \sech a x \rd x
      | c = {{Defof|Hyperbolic Cotangent|index = 2}}
}}
{{eqn | r = \int \csch a x \coth a x \rd x - \int \sech a x \rd x
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {-\csch a x} a - \int \sech a x \rd x
      | c = Primitive of $\csch^n a x \coth a x$ for $n = 1$
}}
{{eqn | r = \frac {-\csch a x} a - \frac 1 a \map \arctan {\sinh a x}
      | c = Primitive of $\sech a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Reciprocal of Hyperbolic Sine of a x by Square of Hyperbolic Cosine of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x \cosh a x}
      | r = \int \frac {\left({\cosh^2 a x - \sinh^2 a x}\right) \rd x} {\sinh a x \cosh^2 a x}
      | c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | r = \int \frac {\cosh^2 a x \rd x} {\sinh a x \cosh^2 a x} - \int \frac {\sinh^2 a x \rd x} {\sinh a x \cosh^2 a x}
      | c = Linear Combination of Primitives
}}
{{eqn | r = \int \frac {\d x} {\sinh a x} - \int \frac {\sinh a x \rd x} {\cosh^2 a x}
      | c = simplifying
}}
{{eqn | r = \int \csch a x \rd x - \int \frac {\sinh a x \rd x} {\cosh^2 a x}
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \int \csch a x \rd x - \int \frac {\tanh a x \rd x} {\cosh a x}
      | c = {{Defof|Hyperbolic Tangent|index = 2}}
}}
{{eqn | r = \int \csch a x \rd x - \int \sech a x \tanh a x \rd x
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \int \csch a x \rd x - \frac {-\sech a x} a + C
      | c = Primitive of $\sech^n a x \tanh a x$ for $n = 1$
}}
{{eqn | r = \frac 1 a \ln \left\vert {\tanh \frac {a x} 2} \right\vert + \frac {\sech a x} a + C
      | c = Primitive of $\csch a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of Hyperbolic Secant of a x by Hyperbolic Tangent of a x

:$\ds \int \sech^n a x \tanh a x \rd x = \frac {-\sech^n a x} {n a} + C$
for $n \ne 0$.

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

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

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

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sinh^2 a x \cosh^2 a x}
      | r = \int \frac {\d x} {\paren {\sinh a x \cosh a x}^2}
      | c = 
}}
{{eqn | r = \int \frac {\d x} {\paren {\dfrac {\sinh^2 2 a x} 2}^2}
      | c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | r = 4 \int \frac {\d x} {\sinh^2 2 a x}
      | c = Primitive of Constant Multiple of Function
}}
{{eqn | r = 4 \int \csch^2 2 a x \rd x
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = 4 \frac {-\coth 2 a x} {2 a} + C
      | c = Primitive of $\csch^2 a x$
}}
{{eqn | r = \frac {-2 \coth 2 a x} a + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Sine of a x over Hyperbolic Cosine of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\sinh^2 a x \rd x} {\cosh a x}
      | r = \int \frac {\paren {\cosh^2 a x - 1} \rd x} {\cosh a x}
      | c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | r = \int \frac {\paren {\cosh^2 a x} \rd x} {\cosh a x} - \int \frac {\d x} {\cosh a x}
      | c = Linear Combination of Primitives
}}
{{eqn | r = \int \cosh a x \rd x - \int \sech a x \rd x
      | c = {{Defof|Hyperbolic Secant|index = 2}}
}}
{{eqn | r = \frac {\sinh a x} a - \int \sech a x \rd x
      | c = Primitive of $\cosh a x$
}}
{{eqn | r = \frac {\sinh a x} a - \frac 1 a \map \arctan {\sinh a x} + C
      | c = Primitive of $\sech a x$
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Square of Hyperbolic Cosine of a x over Hyperbolic Sine of a x

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\cosh^2 a x \rd x} {\sinh a x}
      | r = \int \frac {\paren {\sinh^2 a x + 1} \rd x} {\sinh a x}
      | c = Difference of Squares of Hyperbolic Cosine and Sine
}}
{{eqn | r = \int \frac {\sinh^2 a x \rd x} {\sinh a x} + \int \frac {\d x} {\sinh a x}
      | c = Linear Combination of Primitives
}}
{{eqn | r = \int \sinh a x \rd x + \int \csch a x \rd x
      | c = {{Defof|Hyperbolic Cosecant|index = 2}}
}}
{{eqn | r = \frac {\cosh a x} a + \int \csch a x \rd x
      | c = Primitive of $\sinh a x$
}}
{{eqn | r = \frac {\cosh a x} a + \frac 1 a \ln \size {\tanh \frac {a x} 2} + C
      | c = Primitive of $\csch a x$
}}
{{end-eqn}}
{{qed}}