Optimal. Leaf size=37 \[ -\frac{3 x}{2}-\coth ^2(x)+\frac{3 \coth (x)}{2}+2 \log (\sinh (x))+\frac{\coth ^2(x)}{2 (\tanh (x)+1)} \]
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Rubi [A] time = 0.0888637, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3552, 3529, 3531, 3475} \[ -\frac{3 x}{2}-\coth ^2(x)+\frac{3 \coth (x)}{2}+2 \log (\sinh (x))+\frac{\coth ^2(x)}{2 (\tanh (x)+1)} \]
Antiderivative was successfully verified.
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Rule 3552
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \frac{\coth ^3(x)}{1+\tanh (x)} \, dx &=\frac{\coth ^2(x)}{2 (1+\tanh (x))}-\frac{1}{2} \int \coth ^3(x) (-4+3 \tanh (x)) \, dx\\ &=-\coth ^2(x)+\frac{\coth ^2(x)}{2 (1+\tanh (x))}-\frac{1}{2} i \int \coth ^2(x) (-3 i+4 i \tanh (x)) \, dx\\ &=\frac{3 \coth (x)}{2}-\coth ^2(x)+\frac{\coth ^2(x)}{2 (1+\tanh (x))}+\frac{1}{2} \int \coth (x) (4-3 \tanh (x)) \, dx\\ &=-\frac{3 x}{2}+\frac{3 \coth (x)}{2}-\coth ^2(x)+\frac{\coth ^2(x)}{2 (1+\tanh (x))}+2 \int \coth (x) \, dx\\ &=-\frac{3 x}{2}+\frac{3 \coth (x)}{2}-\coth ^2(x)+2 \log (\sinh (x))+\frac{\coth ^2(x)}{2 (1+\tanh (x))}\\ \end{align*}
Mathematica [A] time = 0.0589725, size = 33, normalized size = 0.89 \[ \frac{1}{4} \left (-6 x-\sinh (2 x)+\cosh (2 x)+4 \coth (x)-2 \text{csch}^2(x)+8 \log (\sinh (x))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.032, size = 75, normalized size = 2. \begin{align*} -{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) }+ \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}- \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}-{\frac{7}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+2\,\ln \left ( \tanh \left ( x/2 \right ) \right ) -{\frac{1}{2}\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 [A] time = 1.39836, size = 73, normalized size = 1.97 \begin{align*} \frac{1}{2} \, x + \frac{2 \,{\left (2 \, e^{\left (-2 \, x\right )} - 1\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac{1}{4} \, e^{\left (-2 \, x\right )} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) + 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.21779, size = 1168, normalized size = 31.57 \begin{align*} \text{result too large to display} \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 )}}{\tanh{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18321, size = 54, normalized size = 1.46 \begin{align*} -\frac{7}{2} \, x + \frac{{\left (e^{\left (4 \, x\right )} - 10 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}}{4 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + 2 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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