Optimal. Leaf size=32 \[ \left (\frac{a}{b^2}+\frac{1}{a}\right ) \log (a+b \text{csch}(x))+\frac{\log (\sinh (x))}{a}-\frac{\text{csch}(x)}{b} \]
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Rubi [A] time = 0.0721172, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3885, 894} \[ \left (\frac{a}{b^2}+\frac{1}{a}\right ) \log (a+b \text{csch}(x))+\frac{\log (\sinh (x))}{a}-\frac{\text{csch}(x)}{b} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 894
Rubi steps
\begin{align*} \int \frac{\coth ^3(x)}{a+b \text{csch}(x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-b^2-x^2}{x (a+x)} \, dx,x,b \text{csch}(x)\right )}{b^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1-\frac{b^2}{a x}+\frac{a^2+b^2}{a (a+x)}\right ) \, dx,x,b \text{csch}(x)\right )}{b^2}\\ &=-\frac{\text{csch}(x)}{b}+\left (\frac{1}{a}+\frac{a}{b^2}\right ) \log (a+b \text{csch}(x))+\frac{\log (\sinh (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0502377, size = 37, normalized size = 1.16 \[ \frac{\left (a^2+b^2\right ) \log (a \sinh (x)+b)+a^2 (-\log (\sinh (x)))-a b \text{csch}(x)}{a b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 106, normalized size = 3.3 \begin{align*}{\frac{1}{2\,b}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{a}{{b}^{2}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-2\,a\tanh \left ( x/2 \right ) -b \right ) }+{\frac{1}{a}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-2\,a\tanh \left ( x/2 \right ) -b \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 ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00749, size = 111, normalized size = 3.47 \begin{align*} \frac{x}{a} + \frac{2 \, e^{\left (-x\right )}}{b e^{\left (-2 \, x\right )} - b} - \frac{a \log \left (e^{\left (-x\right )} + 1\right )}{b^{2}} - \frac{a \log \left (e^{\left (-x\right )} - 1\right )}{b^{2}} + \frac{{\left (a^{2} + b^{2}\right )} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.66492, size = 543, normalized size = 16.97 \begin{align*} -\frac{b^{2} x \cosh \left (x\right )^{2} + b^{2} x \sinh \left (x\right )^{2} - b^{2} x + 2 \, a b \cosh \left (x\right ) -{\left ({\left (a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{2} + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{2} + b^{2}\right )} \sinh \left (x\right )^{2} - a^{2} - b^{2}\right )} \log \left (\frac{2 \,{\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) +{\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 )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (b^{2} x \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )}{a b^{2} \cosh \left (x\right )^{2} + 2 \, a b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a b^{2} \sinh \left (x\right )^{2} - a b^{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 ^{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 [B] time = 1.19312, size = 108, normalized size = 3.38 \begin{align*} -\frac{a \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{b^{2}} + \frac{{\left (a^{2} + b^{2}\right )} \log \left ({\left | -a{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a b^{2}} + \frac{a{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b}{b^{2}{\left (e^{\left (-x\right )} - e^{x}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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