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 = 0.07, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 3475
Rule 3529
Rule 3531
Rule 3552
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*}
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Mathematica [A] time = 0.05, 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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fricas [B] time = 0.44, size = 186, normalized size = 6.41 \[ \frac {10 \, x \cosh \relax (x)^{4} + 40 \, x \cosh \relax (x) \sinh \relax (x)^{3} + 10 \, x \sinh \relax (x)^{4} + {\left (10 \, x + 9\right )} \cosh \relax (x)^{2} + {\left (60 \, x \cosh \relax (x)^{2} + 10 \, x + 9\right )} \sinh \relax (x)^{2} - 4 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + {\left (6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + 2 \, {\left (2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (20 \, x \cosh \relax (x)^{3} + {\left (10 \, x + 9\right )} \cosh \relax (x)\right )} \sinh \relax (x) + 1}{4 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + {\left (6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + 2 \, {\left (2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 35, normalized size = 1.21 \[ \frac {5}{2} \, x + \frac {{\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )}}{4 \, {\left (e^{\left (2 \, x\right )} + 1\right )}} - \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 65, normalized size = 2.24 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}-\frac {2 \tanh \left (\frac {x}{2}\right )}{\tanh ^{2}\left (\frac {x}{2}\right )+1}-\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 29, normalized size = 1.00 \[ \frac {1}{2} \, x - \frac {2}{e^{\left (-2 \, x\right )} + 1} + \frac {1}{4} \, e^{\left (-2 \, x\right )} - \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 29, normalized size = 1.00 \[ \frac {5\,x}{2}-\ln \left ({\mathrm {e}}^{2\,x}+1\right )+\frac {{\mathrm {e}}^{-2\,x}}{4}+\frac {2}{{\mathrm {e}}^{2\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{2}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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