Optimal. Leaf size=19 \[ -\frac{x}{2}+\frac{1}{2 (\tanh (x)+1)}+\log (\sinh (x)) \]
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Rubi [A] time = 0.0404755, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {3551, 3479, 8, 3475} \[ -\frac{x}{2}+\frac{1}{2 (\tanh (x)+1)}+\log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3551
Rule 3479
Rule 8
Rule 3475
Rubi steps
\begin{align*} \int \frac{\coth (x)}{1+\tanh (x)} \, dx &=\int \coth (x) \, dx-\int \frac{1}{1+\tanh (x)} \, dx\\ &=\log (\sinh (x))+\frac{1}{2 (1+\tanh (x))}-\frac{\int 1 \, dx}{2}\\ &=-\frac{x}{2}+\log (\sinh (x))+\frac{1}{2 (1+\tanh (x))}\\ \end{align*}
Mathematica [A] time = 0.0281768, size = 23, normalized size = 1.21 \[ \frac{1}{4} (-2 x-\sinh (2 x)+\cosh (2 x)+4 \log (\sinh (x))) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 43, normalized size = 2.3 \begin{align*} - \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}+ \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}-{\frac{3}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+\ln \left ( \tanh \left ({\frac{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 = 1.27018, size = 32, normalized size = 1.68 \begin{align*} \frac{1}{2} \, x + \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 = 2.37501, size = 259, normalized size = 13.63 \begin{align*} -\frac{6 \, x \cosh \left (x\right )^{2} + 12 \, x \cosh \left (x\right ) \sinh \left (x\right ) + 6 \, x \sinh \left (x\right )^{2} - 4 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 1}{4 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \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{\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 = 1.20533, size = 24, normalized size = 1.26 \begin{align*} -\frac{3}{2} \, x + \frac{1}{4} \, e^{\left (-2 \, x\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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