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 = 0.114614, antiderivative size = 43, normalized size of antiderivative = 1., 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 3552
Rule 3529
Rule 3531
Rule 3475
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*}
Mathematica [A] time = 0.095612, 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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Maple [B] time = 0.043, size = 96, normalized size = 2.2 \begin{align*} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}- \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}+{\frac{9}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }-4\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{5}-1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+8/3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-1/2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+\tanh \left ( x/2 \right ) }{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-2\,\ln \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70061, size = 74, normalized size = 1.72 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.6535, size = 1897, normalized size = 44.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{4}{\left (x \right )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11034, size = 63, normalized size = 1.47 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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