Optimal. Leaf size=54 \[ \frac{\tan ^{-1}(\sinh (x))}{a}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a \sqrt{a-b} \sqrt{a+b}} \]
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Rubi [A] time = 0.067154, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2747, 3770, 2659, 208} \[ \frac{\tan ^{-1}(\sinh (x))}{a}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a \sqrt{a-b} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 2747
Rule 3770
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{\text{sech}(x)}{a+b \cosh (x)} \, dx &=\frac{\int \text{sech}(x) \, dx}{a}-\frac{b \int \frac{1}{a+b \cosh (x)} \, dx}{a}\\ &=\frac{\tan ^{-1}(\sinh (x))}{a}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a}\\ &=\frac{\tan ^{-1}(\sinh (x))}{a}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a \sqrt{a-b} \sqrt{a+b}}\\ \end{align*}
Mathematica [A] time = 0.0507831, size = 54, normalized size = 1. \[ \frac{2 \left (\frac{b \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}+\tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 51, normalized size = 0.9 \begin{align*} -2\,{\frac{b}{a\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tanh \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+2\,{\frac{\arctan \left ( \tanh \left ( x/2 \right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51079, size = 602, normalized size = 11.15 \begin{align*} \left [\frac{\sqrt{a^{2} - b^{2}} b \log \left (\frac{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} - b^{2} + 2 \,{\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt{a^{2} - b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b}\right ) + 2 \,{\left (a^{2} - b^{2}\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}{a^{3} - a b^{2}}, \frac{2 \,{\left (\sqrt{-a^{2} + b^{2}} b \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{a^{2} - b^{2}}\right ) +{\left (a^{2} - b^{2}\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )\right )}}{a^{3} - a b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}{\left (x \right )}}{a + b \cosh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18787, size = 61, normalized size = 1.13 \begin{align*} -\frac{2 \, b \arctan \left (\frac{b e^{x} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}} a} + \frac{2 \, \arctan \left (e^{x}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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