Optimal. Leaf size=23 \[ 4 x-\frac{1}{2} (\coth (x)+1)^2-2 \coth (x)+4 \log (\sinh (x)) \]
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Rubi [A] time = 0.0214076, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3478, 3477, 3475} \[ 4 x-\frac{1}{2} (\coth (x)+1)^2-2 \coth (x)+4 \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3478
Rule 3477
Rule 3475
Rubi steps
\begin{align*} \int (1+\coth (x))^3 \, dx &=-\frac{1}{2} (1+\coth (x))^2+2 \int (1+\coth (x))^2 \, dx\\ &=4 x-2 \coth (x)-\frac{1}{2} (1+\coth (x))^2+4 \int \coth (x) \, dx\\ &=4 x-2 \coth (x)-\frac{1}{2} (1+\coth (x))^2+4 \log (\sinh (x))\\ \end{align*}
Mathematica [C] time = 0.149008, size = 61, normalized size = 2.65 \[ \frac{1}{4} \text{csch}^2(x) \left (-6 \sinh (2 x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\tanh ^2(x)\right )-2 x-8 \log (\tanh (x))-8 \log (\cosh (x))+\cosh (2 x) (2 x+8 \log (\tanh (x))+8 \log (\cosh (x))-1)-1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 19, normalized size = 0.8 \begin{align*} -{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{2}}{2}}-3\,{\rm coth} \left (x\right )-4\,\ln \left ({\rm coth} \left (x\right )-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03956, size = 74, normalized size = 3.22 \begin{align*} 5 \, x + \frac{2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac{6}{e^{\left (-2 \, x\right )} - 1} + \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) + 3 \, \log \left (\sinh \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03912, size = 479, normalized size = 20.83 \begin{align*} -\frac{2 \,{\left (4 \, \cosh \left (x\right )^{2} - 2 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 8 \, \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sinh \left (x\right )^{2} - 3\right )}}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.17971, size = 31, normalized size = 1.35 \begin{align*} 8 x - 4 \log{\left (\tanh{\left (x \right )} + 1 \right )} + 4 \log{\left (\tanh{\left (x \right )} \right )} - \frac{3}{\tanh{\left (x \right )}} - \frac{1}{2 \tanh ^{2}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1699, size = 39, normalized size = 1.7 \begin{align*} -\frac{2 \,{\left (4 \, e^{\left (2 \, x\right )} - 3\right )}}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + 4 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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