Optimal. Leaf size=31 \[ \frac{3 x}{2}+\frac{\coth ^2(x)}{2 (\coth (x)+1)}-\frac{3 \coth (x)}{2}-\log (\sinh (x)) \]
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Rubi [A] time = 0.0553428, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3550, 3525, 3475} \[ \frac{3 x}{2}+\frac{\coth ^2(x)}{2 (\coth (x)+1)}-\frac{3 \coth (x)}{2}-\log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3550
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \frac{\coth ^3(x)}{1+\coth (x)} \, dx &=\frac{\coth ^2(x)}{2 (1+\coth (x))}-\frac{1}{2} \int (2-3 \coth (x)) \coth (x) \, dx\\ &=\frac{3 x}{2}-\frac{3 \coth (x)}{2}+\frac{\coth ^2(x)}{2 (1+\coth (x))}-\int \coth (x) \, dx\\ &=\frac{3 x}{2}-\frac{3 \coth (x)}{2}+\frac{\coth ^2(x)}{2 (1+\coth (x))}-\log (\sinh (x))\\ \end{align*}
Mathematica [A] time = 0.0413891, size = 27, normalized size = 0.87 \[ \frac{1}{4} (6 x+\sinh (2 x)-\cosh (2 x)-4 \coth (x)-4 \log (\sinh (x))) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 28, normalized size = 0.9 \begin{align*} -{\rm coth} \left (x\right )+{\frac{1}{2+2\,{\rm coth} \left (x\right )}}+{\frac{5\,\ln \left ( 1+{\rm coth} \left (x\right ) \right ) }{4}}-{\frac{\ln \left ({\rm coth} \left (x\right )-1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09329, size = 51, normalized size = 1.65 \begin{align*} \frac{1}{2} \, x + \frac{2}{e^{\left (-2 \, x\right )} - 1} - \frac{1}{4} \, e^{\left (-2 \, x\right )} - \log \left (e^{\left (-x\right )} + 1\right ) - \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 = 3.01997, size = 632, normalized size = 20.39 \begin{align*} \frac{10 \, x \cosh \left (x\right )^{4} + 40 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + 10 \, x \sinh \left (x\right )^{4} -{\left (10 \, x + 9\right )} \cosh \left (x\right )^{2} +{\left (60 \, x \cosh \left (x\right )^{2} - 10 \, x - 9\right )} \sinh \left (x\right )^{2} - 4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (20 \, x \cosh \left (x\right )^{3} -{\left (10 \, x + 9\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.3806, size = 162, normalized size = 5.23 \begin{align*} \frac{x \tanh ^{2}{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} + \frac{x \tanh{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} + \frac{2 \log{\left (\tanh{\left (x \right )} + 1 \right )} \tanh ^{2}{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} + \frac{2 \log{\left (\tanh{\left (x \right )} + 1 \right )} \tanh{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} - \frac{2 \log{\left (\tanh{\left (x \right )} \right )} \tanh ^{2}{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} - \frac{2 \log{\left (\tanh{\left (x \right )} \right )} \tanh{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} + \frac{3 \tanh ^{2}{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} - \frac{2}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13294, size = 49, normalized size = 1.58 \begin{align*} \frac{5}{2} \, x - \frac{{\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}}{4 \,{\left (e^{\left (2 \, x\right )} - 1\right )}} - \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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