# Primitive of Reciprocal of Hyperbolic Cosine of a x by 1 plus Hyperbolic Sine of a x

## Theorem

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

## Proof

 $\displaystyle \int \frac {\rd x} {\cosh a x \paren {1 + \sinh a x} }$ $=$ $\displaystyle \int \frac {\sech^2 a x} {\sech a x + \tanh a x} \rd x$ Definition of Hyperbolic Cosecant, Definition of Hyperbolic Tangent $\displaystyle$ $=$ $\displaystyle \frac 1 2 \int \frac {2 \sech^2 a x} {\sech a x + \tanh a x} \rd x$ multiplying and dividing by $2$ $\displaystyle$ $=$ $\displaystyle \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x + \sech^2 a x + \sech a x \tanh a x} {\sech a x + \tanh a x} \rd x$ adding and subtracting $\sech a x \tanh a x$ $\displaystyle$ $=$ $\displaystyle \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 2 \int \sech ax \paren {\frac {\sech a x + \tanh a x} {\sech a x + \tanh a x} } \rd x$ Linear Combination of Primitives $\displaystyle$ $=$ $\displaystyle \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 2 \int \sech a x \rd x$ simplifying $\displaystyle$ $=$ $\displaystyle \frac 1 2 \int \frac {-\sech a x \tanh a x + \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 a \map \arctan {e^{a x} }$ Primitive of $\sech a x$ $\displaystyle$ $=$ $\displaystyle \frac 1 {2 a} \int \frac {-a \sech a x \tanh a x + a \sech^2 a x} {\sech a x + \tanh a x} \rd x + \frac 1 a \map \arctan {e^{a x} }$ multiplying and dividing by $a$ $\displaystyle$ $=$ $\displaystyle \frac 1 {2 a} \ln \size {\sech a x + \tanh a x} + \frac 1 a \map \arctan {e^{a x} }$ Derivative of $\sech a x$, Derivative of $\tanh a x$, Primitive of Function under its Derivative $\displaystyle$ $=$ $\displaystyle \frac 1 {2 a} \ln \size {\frac {1 + \sinh a x} {\cosh a x} } + \frac 1 a \map \arctan {e^{a x} } + C$ simplifying and adding integration constant

$\blacksquare$