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

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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

$\blacksquare$


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