Optimal. Leaf size=43 \[ \frac {5 x}{2}-\frac {5 \coth ^3(x)}{6}+\coth ^2(x)-\frac {5 \coth (x)}{2}-2 \log (\sinh (x))+\frac {\coth ^3(x)}{2 (\tanh (x)+1)} \]
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Rubi [A] time = 0.10, 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} \[ \frac {5 x}{2}-\frac {5 \coth ^3(x)}{6}+\coth ^2(x)-\frac {5 \coth (x)}{2}-2 \log (\sinh (x))+\frac {\coth ^3(x)}{2 (\tanh (x)+1)} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rule 3552
Rubi steps
\begin {align*} \int \frac {\coth ^4(x)}{1+\tanh (x)} \, dx &=\frac {\coth ^3(x)}{2 (1+\tanh (x))}-\frac {1}{2} \int \coth ^4(x) (-5+4 \tanh (x)) \, dx\\ &=-\frac {5}{6} \coth ^3(x)+\frac {\coth ^3(x)}{2 (1+\tanh (x))}-\frac {1}{2} i \int \coth ^3(x) (-4 i+5 i \tanh (x)) \, dx\\ &=\coth ^2(x)-\frac {5 \coth ^3(x)}{6}+\frac {\coth ^3(x)}{2 (1+\tanh (x))}+\frac {1}{2} \int \coth ^2(x) (5-4 \tanh (x)) \, dx\\ &=-\frac {5 \coth (x)}{2}+\coth ^2(x)-\frac {5 \coth ^3(x)}{6}+\frac {\coth ^3(x)}{2 (1+\tanh (x))}+\frac {1}{2} i \int \coth (x) (4 i-5 i \tanh (x)) \, dx\\ &=\frac {5 x}{2}-\frac {5 \coth (x)}{2}+\coth ^2(x)-\frac {5 \coth ^3(x)}{6}+\frac {\coth ^3(x)}{2 (1+\tanh (x))}-2 \int \coth (x) \, dx\\ &=\frac {5 x}{2}-\frac {5 \coth (x)}{2}+\coth ^2(x)-\frac {5 \coth ^3(x)}{6}-2 \log (\sinh (x))+\frac {\coth ^3(x)}{2 (1+\tanh (x))}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 42, normalized size = 0.98 \[ \frac {1}{12} \left (-3 \cosh (2 x)-4 \coth (x) \left (\text {csch}^2(x)+7\right )+3 \left (10 x+\sinh (2 x)+2 \text {csch}^2(x)-8 \log (\sinh (x))\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 582, normalized size = 13.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 48, normalized size = 1.12 \[ \frac {9}{2} \, x - \frac {{\left (51 \, e^{\left (6 \, x\right )} - 81 \, e^{\left (4 \, x\right )} + 65 \, e^{\left (2 \, x\right )} - 3\right )} e^{\left (-2 \, x\right )}}{12 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} - 2 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 91, normalized size = 2.12 \[ -\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}-\frac {9 \tanh \left (\frac {x}{2}\right )}{8}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+\frac {9 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}-\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}+\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}-\frac {9}{8 \tanh \left (\frac {x}{2}\right )}-2 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 64, normalized size = 1.49 \[ \frac {1}{2} \, x - \frac {2 \, {\left (15 \, e^{\left (-2 \, x\right )} - 12 \, e^{\left (-4 \, x\right )} - 7\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac {1}{4} \, e^{\left (-2 \, x\right )} - 2 \, \log \left (e^{\left (-x\right )} + 1\right ) - 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 69, normalized size = 1.60 \[ \frac {9\,x}{2}-2\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\frac {{\mathrm {e}}^{-2\,x}}{4}-\frac {8}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {2}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {4}{{\mathrm {e}}^{2\,x}-1} \]
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 \[ \int \frac {\coth ^{4}{\relax (x )}}{\tanh {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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