Optimal. Leaf size=45 \[ -\frac{2}{3} (\coth (x)+1)^{3/2}-4 \sqrt{\coth (x)+1}+4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.031965, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3478, 3480, 206} \[ -\frac{2}{3} (\coth (x)+1)^{3/2}-4 \sqrt{\coth (x)+1}+4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 3478
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int (1+\coth (x))^{5/2} \, dx &=-\frac{2}{3} (1+\coth (x))^{3/2}+2 \int (1+\coth (x))^{3/2} \, dx\\ &=-4 \sqrt{1+\coth (x)}-\frac{2}{3} (1+\coth (x))^{3/2}+4 \int \sqrt{1+\coth (x)} \, dx\\ &=-4 \sqrt{1+\coth (x)}-\frac{2}{3} (1+\coth (x))^{3/2}+8 \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\coth (x)}\right )\\ &=4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1+\coth (x)}}{\sqrt{2}}\right )-4 \sqrt{1+\coth (x)}-\frac{2}{3} (1+\coth (x))^{3/2}\\ \end{align*}
Mathematica [C] time = 0.153248, size = 92, normalized size = 2.04 \[ -\frac{2 \sinh (x) (\coth (x)+1)^{5/2} \left (\cosh (x) \sqrt{i (\coth (x)+1)}+\sinh (x) \left (7 \sqrt{i (\coth (x)+1)}-(6-6 i) \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{i (\coth (x)+1)}\right )\right )\right )}{3 \sqrt{i (\coth (x)+1)} (\sinh (x)+\cosh (x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 35, normalized size = 0.8 \begin{align*} -{\frac{2}{3} \left ( 1+{\rm coth} \left (x\right ) \right ) ^{{\frac{3}{2}}}}+4\,{\it Artanh} \left ( 1/2\,\sqrt{1+{\rm coth} \left (x\right )}\sqrt{2} \right ) \sqrt{2}-4\,\sqrt{1+{\rm coth} \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\coth \left (x\right ) + 1\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.39008, size = 869, normalized size = 19.31 \begin{align*} -\frac{2 \,{\left (2 \, \sqrt{2}{\left (4 \, \sqrt{2} \cosh \left (x\right )^{3} + 12 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + 4 \, \sqrt{2} \sinh \left (x\right )^{3} + 3 \,{\left (4 \, \sqrt{2} \cosh \left (x\right )^{2} - \sqrt{2}\right )} \sinh \left (x\right ) - 3 \, \sqrt{2} \cosh \left (x\right )\right )} \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \,{\left (\sqrt{2} \cosh \left (x\right )^{4} + 4 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt{2} \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \sqrt{2} \cosh \left (x\right )^{2} - \sqrt{2}\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt{2} \cosh \left (x\right )^{2} + 4 \,{\left (\sqrt{2} \cosh \left (x\right )^{3} - \sqrt{2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt{2}\right )} \log \left (2 \, \sqrt{2} \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - 1\right )\right )}}{3 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17387, size = 151, normalized size = 3.36 \begin{align*} -\frac{2}{3} \, \sqrt{2}{\left (\frac{2 \,{\left (6 \,{\left (\sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} + 9 \, \sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 9 \, e^{\left (2 \, x\right )} + 4\right )}}{{\left (\sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1\right )}^{3}} + 3 \, \log \left ({\left | 2 \, \sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )} \mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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