avewright/proofwiki-math huggingface dataset:
Theorem: Primitive of Power of x by Power of Logarithm of x

:$\ds \int x^m \ln^n x \rd x = \frac {x^{m + 1} \ln^n x} {m + 1} - \frac n {m + 1} \int x^m \ln^{n - 1} x \rd x + C$
where $m \ne -1$.

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 = \ln^n x
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = n \ln^{n - 1} x \frac 1 x
      | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = x^m
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {x^{m + 1} } {m + 1}
      | c = Primitive of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \ln^n x \rd x
      | r = \frac {x^{m + 1} } {m + 1} \ln^n x - \int \frac {x^{m + 1} } {m + 1} \paren {n \ln^{n - 1} x \frac 1 x} \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x^{m + 1} \ln^n x} {m + 1} - \frac n {m + 1} \int x^m \ln^{n - 1} x \rd x + C
      | c = Primitive of Constant Multiple of Function
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Logarithm of x squared plus a squared

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

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 = \map \ln {x^2 + a^2}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = \frac {2 x} {x^2 + a^2}
      | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives
}}
{{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 Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \map \ln {x^2 + a^2} \rd x
      | r = x \map \ln {x^2 + a^2} - \int \frac {2 x^2 \rd x} {x^2 + a^2} + C
      | c = Integration by Parts
}}
{{eqn | r = x \map \ln {x^2 + a^2} - 2 \int \frac {x^2 \rd x} {x^2 + a^2} + C
      | c = Primitive of Constant Multiple of Function
}}
{{eqn | r = x \map \ln {x^2 + a^2} - 2 \paren {x - a \arctan {\frac x a} } + C
      | c = Primitive of $\dfrac {x^2} {x^2 + a^2}$
}}
{{eqn | r = x \map \ln {x^2 + a^2} - 2 x + 2 a \arctan \frac x a + C
      | c = simplifying
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Logarithm of x squared minus a squared

:$\ds \int \map \ln {x^2 - a^2} \rd x = x \map \ln {x^2 - a^2} - 2 x + a \map \ln {\frac {x + a} {x - a} } + C$
for $x^2 > a^2$.

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 = \map \ln {x^2 - a^2}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = \frac {2 x} {x^2 - a^2}
      | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives
}}
{{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 Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int \map \ln {x^2 - a^2} \rd x
      | r = x \map \ln {x^2 - a^2} - \int \frac {2 x^2 \rd x} {x^2 - a^2} + C
      | c = Integration by Parts
}}
{{eqn | r = x \map \ln {x^2 - a^2} - 2 \int \frac {x^2 \rd x} {x^2 - a^2} + C
      | c = Primitive of Constant Multiple of Function
}}
{{eqn | r = x \map \ln {x^2 - a^2} - 2 \paren {x + \frac a 2 \map \ln {\frac {x - a} {x + a} } } + C
      | c = Primitive of $\dfrac {x^2} {x^2 - a^2}$
}}
{{eqn | r = x \map \ln {x^2 - a^2} - 2 x - a \map \ln {\frac {x + a} {x - a} } + C
      | c = simplifying
}}
{{eqn | r = x \map \ln {x^2 - a^2} - 2 x + a \map \ln {\frac {x + a} {x - a} } + C
      | c = Logarithm of Reciprocal
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of x by Logarithm of x squared plus a squared

:$\ds \int x^m \map \ln {x^2 + a^2} \rd x = \frac {x^{m + 1} \map \ln {x^2 + a^2} } {m + 1} - \frac 2 {m + 1} \int \frac {x^{m + 2} } {x^2 + a^2} \rd x$

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 = \map \ln {x^2 + a^2}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = \frac {2 x} {x^2 + a^2}
      | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = x^m
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {x^{m + 1} } {m + 1}
      | c = Primitive of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \map \ln {x^2 + a^2} \rd x
      | r = \frac {x^{m + 1} } {m + 1} \map \ln {x^2 + a^2} - \int \frac {x^{m + 1} } {m + 1} \frac {2 x \rd x} {x^2 + a^2} + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x^{m + 1} \map \ln {x^2 + a^2} } {m + 1} - \frac 2 {m + 1} \int \frac {x^{m + 2} } {x^2 + a^2} \d x
      | c = Primitive of Constant Multiple of Function
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Power of x by Logarithm of x squared minus a squared

:$\ds \int x^m \, \map \ln {x^2 - a^2} \rd x = \frac {x^{m + 1} \map \ln {x^2 - a^2} } {m + 1} - \frac 2 {m + 1} \int \frac {x^{m + 2} } {x^2 - a^2} \rd x$

Proof:
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
      | r = \map \ln {x^2 - a^2}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = \frac {2 x} {x^2 - a^2}
      | c = Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives
}}
{{end-eqn}}
and let:
{{begin-eqn}}
{{eqn | l = \frac {\d v} {\d x}
      | r = x^m
      | c = 
}}
{{eqn | ll= \leadsto
      | l = v
      | r = \frac {x^{m + 1} } {m + 1}
      | c = Primitive of Power
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \int x^m \, \map \ln {x^2 - a^2} \rd x
      | r = \frac {x^{m + 1} } {m + 1} \map \ln {x^2 - a^2} - \int \frac {x^{m + 1} } {m + 1} \frac {2 x \rd x} {x^2 - a^2} + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x^{m + 1} \map \ln {x^2 - a^2} } {m + 1} - \frac 2 {m + 1} \int \frac {x^{m + 2} } {x^2 - a^2} \rd x
      | c = Primitive of Constant Multiple of Function
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Sine of a x

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

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

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

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

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

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

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

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

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

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 = \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

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

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 Cosine of a x

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

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

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

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

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

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

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \csch x \rd x
      | r = \ln \size {\tanh \frac x 2} + C
      | c = Primitive of $\csch x$
}}
{{eqn | ll= \leadsto
      | l = \int \csch a x \rd x
      | r = \frac 1 a \ln \size {\tanh \frac {a x} 2} + C
      | c = Primitive of Function of Constant Multiple
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x by Hyperbolic Sine of a x

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

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 = \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 \sinh a x \rd x
      | r = x \paren {\frac {\cosh a x} a} - \int \paren {\frac {\cosh a x} a} \times 1 \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x \cosh a x} a - \frac 1 a \int \cosh a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {x \cosh a x} a - \frac 1 a \paren {\frac {\sinh a x} a} + C
      | c = Primitive of $\cosh a x$
}}
{{eqn | r = \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C
      | c = simplification
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x by Hyperbolic Cosine of a x

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

Proof:
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\rd u} {\rd x} \rd x$
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 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 \cosh a x \rd x
      | r = x \paren {\frac {\sinh a x} a} - \int \paren {\frac {\sinh a x} a} \times 1 \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x \sinh a x} a - \frac 1 a \int \sinh a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {x \sinh a x} a - \frac 1 a \paren {\frac {\cosh a x} a} + C
      | c = Primitive of $\sinh a x$
}}
{{eqn | r = \frac {x \sinh a x} a - \frac {\cosh a x} {a^2} + C
      | c = simplification
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x squared by Hyperbolic Sine of a x

:$\ds \int x^2 \sinh a x \rd x = \paren {\frac {x^2} a + \frac 2 {a^3} } \cosh a x - \frac {2 x \sinh a x} {a^2} + C$

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^2
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 2 x
      | 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^2 \sinh a x \rd x
      | r = x^2 \paren {\frac {\cosh a x} a} - \int 2 x \paren {\frac {\cosh a x} a} \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x^2 \cosh a x} a - \frac 2 a \int x \cosh a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {x^2 \cosh a x} a - \frac 2 a \paren {\frac {x \sinh a x} a - \frac {\cosh a x} {a^2} } + C
      | c = Primitive of $x \cosh a x$
}}
{{eqn | r = \paren {\frac {x^2} a + \frac 2 {a^3} } \cosh a x - \frac {2 x \sinh a x} {a^2} + C
      | c = simplification
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of x squared by Hyperbolic Cosine of a x

:$\ds \int x^2 \cosh a x \rd x = \paren {\frac {x^2} a + \frac 2 {a^3} } \sinh a x - \frac {2 x \cosh a x} {a^2} + C$

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^2
      | c = 
}}
{{eqn | ll= \leadsto
      | l = \frac {\d u} {\d x}
      | r = 2 x
      | 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^2 \sinh a x \rd x
      | r = x^2 \paren {\frac {\sinh a x} a} - \int 2 x \paren {\frac {\sinh a x} a} \rd x + C
      | c = Integration by Parts
}}
{{eqn | r = \frac {x^2 \sinh a x} a - \frac 2 a \int x \sinh a x \rd x + C
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac {x^2 \sinh a x} a - \frac 2 a \paren {\frac {x \cosh a x} a - \frac {\sinh a x} {a^2} } + C
      | c = Primitive of $x \cosh a x$
}}
{{eqn | r = \paren {\frac {x^2} a + \frac 2 {a^3} } \sinh a x - \frac {2 x \cosh a x} {a^2} + C
      | c = simplification
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Tangent of a x over x

{{begin-eqn}}
{{eqn | l = \int \frac {\tanh a x \rd x} x
      | r = \sum_{k \mathop \ge 1} \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1} \paren {2 k}!} + C
}}
{{eqn | r = a x - \frac {\paren {a x}^3} 9 + \frac {2 \paren {a x}^5} {75} - \cdots + C
}}
{{end-eqn}}
where $B_k$ denotes the $k$th Bernoulli number.

Proof:
{{begin-eqn}}
{{eqn | l = \tanh x
      | r = \sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}
      | c = Power Series Expansion for Hyperbolic Tangent Function
}}
{{eqn | ll= \leadsto
      | l = \frac {\tanh a x} x
      | r = \sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \paren {a x}^{2 k - 1} } {x \paren {2 k}!}
      | c = 
}}
{{eqn | r = \sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {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 {\tanh a x \rd x} x
      | r = \int \paren {\sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {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 = 1}^\infty \int \frac {2^{2 k} \paren {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 = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} a^{2 k - 1} x^{2 k - 1} } {\paren {2 k - 1} \paren {2 k}!} + C
      | c = Primitive of Power
}}
{{eqn | r = \sum_{k \mathop = 1}^\infty \frac {2^{2 k} \paren {2^{2 k} - 1} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k - 1} \paren {2 k}!} + C
      | c = 
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Cotangent of a x over x

{{begin-eqn}}
{{eqn | l = \int \frac {\coth a x \rd x} x
      | r = \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k}!} + C
}}
{{eqn | r = -\frac 1 {a x} + \frac {a x} 3 - \frac {\paren {a x}^3} {135} + \cdots + C
}}
{{end-eqn}}
where $B_k$ denotes the $k$th Bernoulli number.

Proof:
{{begin-eqn}}
{{eqn | l = \coth x
      | r = \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}
      | c = Power Series Expansion for Hyperbolic Cotangent Function
}}
{{eqn | r = \dfrac 1 x + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}
      | c = extracting the first term, just in case
}}
{{eqn | ll= \leadsto
      | l = \frac {\coth a x} x
      | r = \dfrac 1 {a x^2} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} \, \paren {a x}^{2 k - 1} } {x \paren {2 k}!}
      | c = 
}}
{{eqn | r = \dfrac 1 {a x^2} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!}
      | c = 
}}
{{eqn-intertext|It is noted that the exponent of $x$ is always even, so there is no need to consider the special case where $x^{-1}$.}}
{{eqn | ll= \leadsto
      | l = \int \frac {\coth a x \rd x} x
      | r = \int \paren {\dfrac 1 {a x^2} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} } \rd x
      | c = 
}}
{{eqn | r = \int \dfrac 1 {a x^2} \rd x + \sum_{k \mathop = 1}^\infty \int \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} \rd x
      | c = 
}}
{{eqn | r = -\dfrac 1 {a x} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 1} } {\paren {2 k - 1} \paren {2 k}!} + C
      | c = Primitive of Power
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k}!} + C
      | c = bringing the first term back inside the summation
}}
{{end-eqn}}
{{qed}}
Theorem: Primitive of Hyperbolic Secant of a x over x

{{begin-eqn}}
{{eqn | l = \int \frac {\sech a x \rd x} x
      | r = \ln \size x + \sum_{k \mathop \ge 1} \frac {E_{2 k} \paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!} + C
}}
{{eqn | r = \ln \size x - \frac {\paren {a x}^2} 4 + \frac {\paren {a x}^4} {96} - \frac {\paren {a x}^6} {4320} + \cdots + C
}}
{{end-eqn}}
where $E_k$ denotes the $k$th Euler number.

Proof:
{{begin-eqn}}
{{eqn | l = \sech x
      | r = \sum_{n \mathop = 0}^\infty \frac {E_{2 k} x^{2 k} } {\paren {2 k}!}
      | c = Power Series Expansion for Hyperbolic Secant Function
}}
{{eqn | ll= \leadsto
      | l = \frac {\sech a x} x
      | r = \sum_{n \mathop = 0}^\infty \frac {E_{2 k} \paren {a x}^{2 k} } {x \paren {2 k}!}
      | c = 
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {E_{2 k} a^{2 k} x^{2 k - 1} } {\paren {2 k}!}
      | c = 
}}
{{eqn | r = \frac 1 x + \sum_{n \mathop = 1}^\infty \frac {E_{2 k} a^{2 k} x^{2 k - 1} } {\paren {2 k}!}
      | c = extracting the first term, which needs to be handled separately
}}
{{eqn | ll= \leadsto
      | l = \int \frac {\sech a x \rd x} x
      | r = \int \paren {\frac 1 x + \sum_{n \mathop = 1}^\infty \frac {E_{2 k} a^{2 k} x^{2 k - 1} } {\paren {2 k}!} } \rd x
      | c = 
}}
{{eqn | r = \int \frac 1 x \rd x + \sum_{k \mathop = 1}^\infty \int \frac {E_{2 k} a^{2 k} x^{2 k - 1} } {\paren {2 k}!} \rd x
      | c = 
}}
{{eqn | r = \ln \size x + \sum_{k \mathop = 1}^\infty \int \frac {E_{2 k} a^{2 k} x^{2 k - 1} } {\paren {2 k}!} \rd x
      | c = Primitive of Reciprocal
}}
{{eqn | r = \ln \size x + \sum_{k \mathop = 1}^\infty \frac {E_{2 k} a^{2 k} x^{2 k} } {\paren {2 k} \paren {2 k}!}
      | c = Primitive of Power
}}
{{eqn | r = \ln \size x + \sum_{k \mathop = 1}^\infty \frac {E_{2 k} \paren {a x}^{2 k} } {\paren {2 k} \paren {2 k}!}
      | c = 
}}
{{end-eqn}}
{{qed}}