Optimal. Leaf size=29 \[ \frac{3 x}{2}-\frac{3 \tanh (x)}{2}-\log (\cosh (x))+\frac{\tanh (x)}{2 (\coth (x)+1)} \]
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Rubi [A] time = 0.0742931, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3552, 3529, 3531, 3475} \[ \frac{3 x}{2}-\frac{3 \tanh (x)}{2}-\log (\cosh (x))+\frac{\tanh (x)}{2 (\coth (x)+1)} \]
Antiderivative was successfully verified.
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Rule 3552
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tanh ^2(x)}{1+\coth (x)} \, dx &=\frac{\tanh (x)}{2 (1+\coth (x))}-\frac{1}{2} \int (-3+2 \coth (x)) \tanh ^2(x) \, dx\\ &=-\frac{3 \tanh (x)}{2}+\frac{\tanh (x)}{2 (1+\coth (x))}-\frac{1}{2} i \int (-2 i+3 i \coth (x)) \tanh (x) \, dx\\ &=\frac{3 x}{2}-\frac{3 \tanh (x)}{2}+\frac{\tanh (x)}{2 (1+\coth (x))}-\int \tanh (x) \, dx\\ &=\frac{3 x}{2}-\log (\cosh (x))-\frac{3 \tanh (x)}{2}+\frac{\tanh (x)}{2 (1+\coth (x))}\\ \end{align*}
Mathematica [A] time = 0.0456504, size = 27, normalized size = 0.93 \[ \frac{1}{4} (6 x-\sinh (2 x)+\cosh (2 x)-4 \tanh (x)-4 \log (\cosh (x))) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 65, normalized size = 2.2 \begin{align*} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}- \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}+{\frac{5}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }-2\,{\frac{\tanh \left ( x/2 \right ) }{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1}}-\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67494, size = 39, normalized size = 1.34 \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 (-2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.69727, size = 632, normalized size = 21.79 \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 \, \cosh \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{2}{\left (x \right )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14485, size = 47, normalized size = 1.62 \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 (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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