Optimal. Leaf size=43 \[ \frac {5 x}{2}+\frac {\tanh ^4(x)}{2 (\tanh (x)+1)}-\frac {5 \tanh ^3(x)}{6}+\tanh ^2(x)-\frac {5 \tanh (x)}{2}-2 \log (\cosh (x)) \]
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Rubi [A] time = 0.09, antiderivative size = 43, 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 = {3550, 3528, 3525, 3475} \[ \frac {5 x}{2}+\frac {\tanh ^4(x)}{2 (\tanh (x)+1)}-\frac {5 \tanh ^3(x)}{6}+\tanh ^2(x)-\frac {5 \tanh (x)}{2}-2 \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
Rule 3550
Rubi steps
\begin {align*} \int \frac {\tanh ^5(x)}{1+\tanh (x)} \, dx &=\frac {\tanh ^4(x)}{2 (1+\tanh (x))}-\frac {1}{2} \int (4-5 \tanh (x)) \tanh ^3(x) \, dx\\ &=-\frac {5}{6} \tanh ^3(x)+\frac {\tanh ^4(x)}{2 (1+\tanh (x))}+\frac {1}{2} i \int (-5 i+4 i \tanh (x)) \tanh ^2(x) \, dx\\ &=\tanh ^2(x)-\frac {5 \tanh ^3(x)}{6}+\frac {\tanh ^4(x)}{2 (1+\tanh (x))}+\frac {1}{2} \int \tanh (x) (-4+5 \tanh (x)) \, dx\\ &=\frac {5 x}{2}-\frac {5 \tanh (x)}{2}+\tanh ^2(x)-\frac {5 \tanh ^3(x)}{6}+\frac {\tanh ^4(x)}{2 (1+\tanh (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 ^4(x)}{2 (1+\tanh (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 = 0.80, 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 = 0.12, 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 [A] time = 0.08, size = 40, normalized size = 0.93 \[ -\frac {\left (\tanh ^{3}\relax (x )\right )}{3}+\frac {\left (\tanh ^{2}\relax (x )\right )}{2}-2 \tanh \relax (x )-\frac {\ln \left (\tanh \relax (x )-1\right )}{4}+\frac {1}{2+2 \tanh \relax (x )}+\frac {9 \ln \left (1+\tanh \relax (x )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, 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 = 0.10, size = 35, normalized size = 0.81 \[ \frac {x}{2}+2\,\ln \left (\mathrm {tanh}\relax (x)+1\right )-2\,\mathrm {tanh}\relax (x)+\frac {{\mathrm {tanh}\relax (x)}^2}{2}-\frac {{\mathrm {tanh}\relax (x)}^3}{3}+\frac {1}{2\,\left (\mathrm {tanh}\relax (x)+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.47, size = 104, normalized size = 2.42 \[ \frac {3 x \tanh {\relax (x )}}{6 \tanh {\relax (x )} + 6} + \frac {3 x}{6 \tanh {\relax (x )} + 6} + \frac {12 \log {\left (\tanh {\relax (x )} + 1 \right )} \tanh {\relax (x )}}{6 \tanh {\relax (x )} + 6} + \frac {12 \log {\left (\tanh {\relax (x )} + 1 \right )}}{6 \tanh {\relax (x )} + 6} - \frac {2 \tanh ^{4}{\relax (x )}}{6 \tanh {\relax (x )} + 6} + \frac {\tanh ^{3}{\relax (x )}}{6 \tanh {\relax (x )} + 6} - \frac {9 \tanh ^{2}{\relax (x )}}{6 \tanh {\relax (x )} + 6} + \frac {15}{6 \tanh {\relax (x )} + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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