Optimal. Leaf size=56 \[ -\frac{2 \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^2}+\frac{b \tanh ^{-1}(\cosh (x))}{a^2}-\frac{\coth (x)}{a} \]
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Rubi [A] time = 0.234705, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {2723, 3056, 3001, 3770, 2660, 618, 206} \[ -\frac{2 \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^2}+\frac{b \tanh ^{-1}(\cosh (x))}{a^2}-\frac{\coth (x)}{a} \]
Antiderivative was successfully verified.
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Rule 2723
Rule 3056
Rule 3001
Rule 3770
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{\coth ^2(x)}{a+b \sinh (x)} \, dx &=\int \frac{\text{csch}^2(x) \left (1+\sinh ^2(x)\right )}{a+b \sinh (x)} \, dx\\ &=-\frac{\coth (x)}{a}+\frac{i \int \frac{\text{csch}(x) (i b-i a \sinh (x))}{a+b \sinh (x)} \, dx}{a}\\ &=-\frac{\coth (x)}{a}-\frac{b \int \text{csch}(x) \, dx}{a^2}+\frac{\left (a^2+b^2\right ) \int \frac{1}{a+b \sinh (x)} \, dx}{a^2}\\ &=\frac{b \tanh ^{-1}(\cosh (x))}{a^2}-\frac{\coth (x)}{a}+\frac{\left (2 \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{a^2}\\ &=\frac{b \tanh ^{-1}(\cosh (x))}{a^2}-\frac{\coth (x)}{a}-\frac{\left (4 \left (a^2+b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac{x}{2}\right )\right )}{a^2}\\ &=\frac{b \tanh ^{-1}(\cosh (x))}{a^2}-\frac{2 \sqrt{a^2+b^2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^2}-\frac{\coth (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.175744, size = 82, normalized size = 1.46 \[ -\frac{\text{csch}\left (\frac{x}{2}\right ) \text{sech}\left (\frac{x}{2}\right ) \left (\sinh (x) \left (2 \sqrt{-a^2-b^2} \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )+b \log \left (\tanh \left (\frac{x}{2}\right )\right )\right )+a \cosh (x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 107, normalized size = 1.9 \begin{align*} -{\frac{1}{2\,a}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{b}{{a}^{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+2\,{\frac{1}{\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }+2\,{\frac{{b}^{2}}{{a}^{2}\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \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 = 2.23027, size = 705, normalized size = 12.59 \begin{align*} \frac{\sqrt{a^{2} + b^{2}}{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \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 ) +{\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} - b\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) -{\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} - b\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 2 \, a}{a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} - a^{2}} \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{\coth ^{2}{\left (x \right )}}{a + b \sinh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14298, size = 128, normalized size = 2.29 \begin{align*} \frac{b \log \left (e^{x} + 1\right )}{a^{2}} - \frac{b \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{2}} + \frac{\sqrt{a^{2} + b^{2}} \log \left (\frac{{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{a^{2}} - \frac{2}{a{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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