Optimal. Leaf size=43 \[ \frac {5 x}{2}-\frac {5 \tanh ^3(x)}{6}+\tanh ^2(x)-\frac {5 \tanh (x)}{2}-2 \log (\cosh (x))+\frac {\tanh ^3(x)}{2 (\coth (x)+1)} \]
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Rubi [A] time = 0.11, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3552, 3529, 3531, 3475} \[ \frac {5 x}{2}-\frac {5 \tanh ^3(x)}{6}+\tanh ^2(x)-\frac {5 \tanh (x)}{2}-2 \log (\cosh (x))+\frac {\tanh ^3(x)}{2 (\coth (x)+1)} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rule 3552
Rubi steps
\begin {align*} \int \frac {\tanh ^4(x)}{1+\coth (x)} \, dx &=\frac {\tanh ^3(x)}{2 (1+\coth (x))}-\frac {1}{2} \int (-5+4 \coth (x)) \tanh ^4(x) \, dx\\ &=-\frac {5}{6} \tanh ^3(x)+\frac {\tanh ^3(x)}{2 (1+\coth (x))}-\frac {1}{2} i \int (-4 i+5 i \coth (x)) \tanh ^3(x) \, dx\\ &=\tanh ^2(x)-\frac {5 \tanh ^3(x)}{6}+\frac {\tanh ^3(x)}{2 (1+\coth (x))}+\frac {1}{2} \int (5-4 \coth (x)) \tanh ^2(x) \, dx\\ &=-\frac {5 \tanh (x)}{2}+\tanh ^2(x)-\frac {5 \tanh ^3(x)}{6}+\frac {\tanh ^3(x)}{2 (1+\coth (x))}+\frac {1}{2} i \int (4 i-5 i \coth (x)) \tanh (x) \, dx\\ &=\frac {5 x}{2}-\frac {5 \tanh (x)}{2}+\tanh ^2(x)-\frac {5 \tanh ^3(x)}{6}+\frac {\tanh ^3(x)}{2 (1+\coth (x))}-2 \int \tanh (x) \, dx\\ &=\frac {5 x}{2}-2 \log (\cosh (x))-\frac {5 \tanh (x)}{2}+\tanh ^2(x)-\frac {5 \tanh ^3(x)}{6}+\frac {\tanh ^3(x)}{2 (1+\coth (x))}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 40, normalized size = 0.93 \[ \frac {1}{12} \left (30 x-3 \sinh (2 x)+3 \cosh (2 x)-28 \tanh (x)-24 \log (\cosh (x))+(4 \tanh (x)-6) \text {sech}^2(x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 571, normalized size = 13.28 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 47, normalized size = 1.09 \[ \frac {9}{2} \, x + \frac {{\left (51 \, e^{\left (6 \, x\right )} + 81 \, e^{\left (4 \, x\right )} + 65 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-2 \, x\right )}}{12 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} - 2 \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 96, normalized size = 2.23 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+\frac {9 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}-\frac {4 \left (\tanh ^{5}\left (\frac {x}{2}\right )-\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{2}+\frac {8 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2}+\tanh \left (\frac {x}{2}\right )\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-2 \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 55, normalized size = 1.28 \[ \frac {1}{2} \, x - \frac {2 \, {\left (15 \, e^{\left (-2 \, x\right )} + 12 \, e^{\left (-4 \, x\right )} + 7\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} + \frac {1}{4} \, e^{\left (-2 \, x\right )} - 2 \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 69, normalized size = 1.60 \[ \frac {9\,x}{2}-2\,\ln \left ({\mathrm {e}}^{2\,x}+1\right )+\frac {{\mathrm {e}}^{-2\,x}}{4}+\frac {8}{3\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}-\frac {2}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {4}{{\mathrm {e}}^{2\,x}+1} \]
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 \[ \int \frac {\tanh ^{4}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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