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

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Theorem

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


Proof

Let:

\(\ds u\) \(=\) \(\ds \cosh a x\)
\(\ds \frac {\d u} {\d x}\) \(=\) \(\ds a \sinh a x\) Derivative of $\cosh a x$


Then:

\(\ds \int \frac {\d x} {\sinh a x \paren {\cosh a x + 1} }\) \(=\) \(\ds \int \frac {\sinh a x \rd x} {\sinh^2 a x \paren {\cosh a x + 1} }\) multiplying top and bottom by $\sinh a x$
\(\ds \) \(=\) \(\ds \int \frac {\sinh a x \rd x} {\paren {\cosh^2 a x - 1} \paren {\cosh a x + 1} }\) Difference of Squares of Hyperbolic Cosine and Sine
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {\d u} {\paren {u^2 - 1} \paren {u + 1} }\) Integration by Substitution: $u = \cosh a x$
\(\ds \) \(=\) \(\ds \frac 1 a \int \frac {\d u} {\paren {u + 1}^2 \paren {u - 1} }\) Difference of Two Squares
\(\ds \) \(=\) \(\ds \frac 1 a \paren {\frac 1 2 \paren {\frac 1 {u + 1} + \frac 1 2 \ln \size {\frac {u - 1} {u + 1} } } } + C\) Primitive of $\dfrac 1 {\paren {a x + b}^2 \paren {p x + q} }$
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {u + 1} } + \frac 1 {4 a} \ln \size {\frac {u - 1} {u + 1} } + C\) simplifying
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {\cosh a x + 1} } + \frac 1 {4 a} \ln \size {\frac {\cosh a x - 1} {\cosh a x + 1} } + C\) substituting for $u$
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {\cosh a x + 1} } + \frac 1 {4 a} \ln \size {\frac {\frac 1 2 \sech^2 \dfrac x 2} {\frac 1 2 \csch^2 \frac {a x} 2} } + C\) Reciprocal of $\cosh - 1$ and Reciprocal of $\cosh + 1$
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {\cosh a x + 1} } + \frac 1 {4 a} \ln \size {\frac {\sinh^2 \frac {a x} 2} {\cosh^2 \frac {a x} 2} } + C\) Definition 2 of Hyperbolic Secant, Definition 2 of Hyperbolic Cosecant
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {\cosh a x + 1} } + \frac 1 {4 a} \ln \size {\tanh^2 \frac {a x} 2} + C\) Definition 2 of Hyperbolic Tangent
\(\ds \) \(=\) \(\ds \frac 1 {2 a \paren {\cosh a x + 1} } + \frac 1 {2 a} \ln \size {\tanh \frac {a x} 2} + C\) Logarithm of Power

$\blacksquare$


Sources

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