Optimal. Leaf size=59 \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b^2 \sqrt{a^2+b^2}}+\frac{a \tanh ^{-1}(\cosh (x))}{b^2}-\frac{\coth (x)}{b} \]
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Rubi [A] time = 0.159803, antiderivative size = 59, 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 = {3790, 3789, 3770, 3831, 2660, 618, 206} \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b^2 \sqrt{a^2+b^2}}+\frac{a \tanh ^{-1}(\cosh (x))}{b^2}-\frac{\coth (x)}{b} \]
Antiderivative was successfully verified.
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Rule 3790
Rule 3789
Rule 3770
Rule 3831
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{csch}^3(x)}{a+b \text{csch}(x)} \, dx &=-\frac{\coth (x)}{b}-\frac{a \int \frac{\text{csch}^2(x)}{a+b \text{csch}(x)} \, dx}{b}\\ &=-\frac{\coth (x)}{b}-\frac{a \int \text{csch}(x) \, dx}{b^2}+\frac{a^2 \int \frac{\text{csch}(x)}{a+b \text{csch}(x)} \, dx}{b^2}\\ &=\frac{a \tanh ^{-1}(\cosh (x))}{b^2}-\frac{\coth (x)}{b}+\frac{a^2 \int \frac{1}{1+\frac{a \sinh (x)}{b}} \, dx}{b^3}\\ &=\frac{a \tanh ^{-1}(\cosh (x))}{b^2}-\frac{\coth (x)}{b}+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b^3}\\ &=\frac{a \tanh ^{-1}(\cosh (x))}{b^2}-\frac{\coth (x)}{b}-\frac{\left (4 a^2\right ) \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^3}\\ &=\frac{a \tanh ^{-1}(\cosh (x))}{b^2}-\frac{2 a^2 \tanh ^{-1}\left (\frac{b \left (\frac{a}{b}-\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{a^2+b^2}}\right )}{b^2 \sqrt{a^2+b^2}}-\frac{\coth (x)}{b}\\ \end{align*}
Mathematica [A] time = 0.278098, size = 71, normalized size = 1.2 \[ \frac{\frac{4 a^2 \tan ^{-1}\left (\frac{a-b \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}-2 a \log \left (\tanh \left (\frac{x}{2}\right )\right )-2 b \coth (x)}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 73, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,b}\tanh \left ({\frac{x}{2}} \right ) }+2\,{\frac{{a}^{2}}{{b}^{2}\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}{2\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{a}{{b}^{2}}\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 = 2.19974, size = 926, normalized size = 15.69 \begin{align*} \frac{2 \, a^{2} b + 2 \, b^{3} -{\left (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}\right )} \sqrt{a^{2} + b^{2}} \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^{3} + a b^{2} -{\left (a^{3} + a b^{2}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (a^{3} + a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (a^{3} + a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) -{\left (a^{3} + a b^{2} -{\left (a^{3} + a b^{2}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (a^{3} + a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (a^{3} + a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{2} b^{2} + b^{4} -{\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} - 2 \,{\left (a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) -{\left (a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{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{\operatorname{csch}^{3}{\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 = 1.20938, size = 132, normalized size = 2.24 \begin{align*} \frac{a^{2} \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^{2}} + \frac{a \log \left (e^{x} + 1\right )}{b^{2}} - \frac{a \log \left ({\left | e^{x} - 1 \right |}\right )}{b^{2}} - \frac{2}{b{\left (e^{\left (2 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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