Optimal. Leaf size=37 \[ -\frac {3 x}{2}+2 \log (\cosh (x))+\frac {3 \tanh (x)}{2}-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 = {3633, 3610,
3612, 3556} \begin {gather*} -\frac {3 x}{2}-\tanh ^2(x)+\frac {3 \tanh (x)}{2}+2 \log (\cosh (x))+\frac {\tanh ^2(x)}{2 (\coth (x)+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rule 3633
Rubi steps
\begin {align*} \int \frac {\tanh ^3(x)}{1+\coth (x)} \, dx &=\frac {\tanh ^2(x)}{2 (1+\coth (x))}-\frac {1}{2} \int (-4+3 \coth (x)) \tanh ^3(x) \, dx\\ &=-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))}-\frac {1}{2} i \int (-3 i+4 i \coth (x)) \tanh ^2(x) \, dx\\ &=\frac {3 \tanh (x)}{2}-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))}+\frac {1}{2} \int (4-3 \coth (x)) \tanh (x) \, dx\\ &=-\frac {3 x}{2}+\frac {3 \tanh (x)}{2}-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))}+2 \int \tanh (x) \, dx\\ &=-\frac {3 x}{2}+2 \log (\cosh (x))+\frac {3 \tanh (x)}{2}-\tanh ^2(x)+\frac {\tanh ^2(x)}{2 (1+\coth (x))}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 33, normalized size = 0.89 \begin {gather*} \frac {1}{4} \left (-6 x-\cosh (2 x)+8 \log (\cosh (x))+2 \text {sech}^2(x)+\sinh (2 x)+4 \tanh (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs.
\(2(31)=62\).
time = 0.58, size = 80, normalized size = 2.16
method | result | size |
risch | \(-\frac {7 x}{2}-\frac {{\mathrm e}^{-2 x}}{4}-\frac {2}{\left (1+{\mathrm e}^{2 x}\right )^{2}}+2 \ln \left (1+{\mathrm e}^{2 x}\right )\) | \(30\) |
default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {2 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )-2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+2 \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {7 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 43, normalized size = 1.16 \begin {gather*} \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 (-2 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 354 vs.
\(2 (31) = 62\).
time = 0.36, size = 354, normalized size = 9.57 \begin {gather*} -\frac {14 \, x \cosh \left (x\right )^{6} + 84 \, x \cosh \left (x\right ) \sinh \left (x\right )^{5} + 14 \, x \sinh \left (x\right )^{6} + {\left (28 \, x + 1\right )} \cosh \left (x\right )^{4} + {\left (210 \, x \cosh \left (x\right )^{2} + 28 \, x + 1\right )} \sinh \left (x\right )^{4} + 4 \, {\left (70 \, x \cosh \left (x\right )^{3} + {\left (28 \, x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (7 \, x + 5\right )} \cosh \left (x\right )^{2} + 2 \, {\left (105 \, x \cosh \left (x\right )^{4} + 3 \, {\left (28 \, x + 1\right )} \cosh \left (x\right )^{2} + 7 \, x + 5\right )} \sinh \left (x\right )^{2} - 8 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + {\left (15 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} + 4 \, \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 ) + 4 \, {\left (21 \, x \cosh \left (x\right )^{5} + {\left (28 \, x + 1\right )} \cosh \left (x\right )^{3} + {\left (7 \, x + 5\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{4 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + {\left (15 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (15 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{5} + 4 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\tanh ^{3}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 39, normalized size = 1.05 \begin {gather*} -\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 (e^{\left (2 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 35, normalized size = 0.95 \begin {gather*} 2\,\ln \left ({\mathrm {e}}^{2\,x}+1\right )-\frac {7\,x}{2}-\frac {{\mathrm {e}}^{-2\,x}}{4}-\frac {2}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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