Optimal. Leaf size=50 \[ \frac{2 a \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b \sqrt{a^2+b^2}}-\frac{\tanh ^{-1}(\cosh (x))}{b} \]
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Rubi [A] time = 0.114566, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {3789, 3770, 3831, 2660, 618, 206} \[ \frac{2 a \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b \sqrt{a^2+b^2}}-\frac{\tanh ^{-1}(\cosh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 3789
Rule 3770
Rule 3831
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(x)}{a+b \text{csch}(x)} \, dx &=\frac{\int \text{csch}(x) \, dx}{b}-\frac{a \int \frac{\text{csch}(x)}{a+b \text{csch}(x)} \, dx}{b}\\ &=-\frac{\tanh ^{-1}(\cosh (x))}{b}-\frac{a \int \frac{1}{1+\frac{a \sinh (x)}{b}} \, dx}{b^2}\\ &=-\frac{\tanh ^{-1}(\cosh (x))}{b}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{1+\frac{2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b^2}\\ &=-\frac{\tanh ^{-1}(\cosh (x))}{b}+\frac{(4 a) \operatorname{Subst}\left (\int \frac{1}{4 \left (1+\frac{a^2}{b^2}\right )-x^2} \, dx,x,\frac{2 a}{b}-2 \tanh \left (\frac{x}{2}\right )\right )}{b^2}\\ &=-\frac{\tanh ^{-1}(\cosh (x))}{b}+\frac{2 a \tanh ^{-1}\left (\frac{b \left (\frac{a}{b}-\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{a^2+b^2}}\right )}{b \sqrt{a^2+b^2}}\\ \end{align*}
Mathematica [A] time = 0.0488853, size = 58, normalized size = 1.16 \[ \frac{\log \left (\tanh \left (\frac{x}{2}\right )\right )-\frac{2 a \tan ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 49, normalized size = 1. \begin{align*} -2\,{\frac{a}{b\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,\tanh \left ( x/2 \right ) b-2\,a}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }+{\frac{1}{b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \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 [B] time = 2.04562, size = 456, normalized size = 9.12 \begin{align*} \frac{\sqrt{a^{2} + b^{2}} a \log \left (\frac{a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \,{\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt{a^{2} + b^{2}}{\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \,{\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right ) -{\left (a^{2} + b^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) +{\left (a^{2} + b^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{2} b + b^{3}} \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{csch}^{2}{\left (x \right )}}{a + b \operatorname{csch}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18426, size = 111, normalized size = 2.22 \begin{align*} -\frac{a \log \left (\frac{{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} b} - \frac{\log \left (e^{x} + 1\right )}{b} + \frac{\log \left ({\left | e^{x} - 1 \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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