Optimal. Leaf size=61 \[ -\frac{1}{4 \sqrt{\coth (x)+1}}-\frac{1}{6 (\coth (x)+1)^{3/2}}-\frac{1}{5 (\coth (x)+1)^{5/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
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Rubi [A] time = 0.0419238, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3479, 3480, 206} \[ -\frac{1}{4 \sqrt{\coth (x)+1}}-\frac{1}{6 (\coth (x)+1)^{3/2}}-\frac{1}{5 (\coth (x)+1)^{5/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 3479
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(1+\coth (x))^{5/2}} \, dx &=-\frac{1}{5 (1+\coth (x))^{5/2}}+\frac{1}{2} \int \frac{1}{(1+\coth (x))^{3/2}} \, dx\\ &=-\frac{1}{5 (1+\coth (x))^{5/2}}-\frac{1}{6 (1+\coth (x))^{3/2}}+\frac{1}{4} \int \frac{1}{\sqrt{1+\coth (x)}} \, dx\\ &=-\frac{1}{5 (1+\coth (x))^{5/2}}-\frac{1}{6 (1+\coth (x))^{3/2}}-\frac{1}{4 \sqrt{1+\coth (x)}}+\frac{1}{8} \int \sqrt{1+\coth (x)} \, dx\\ &=-\frac{1}{5 (1+\coth (x))^{5/2}}-\frac{1}{6 (1+\coth (x))^{3/2}}-\frac{1}{4 \sqrt{1+\coth (x)}}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\coth (x)}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{1+\coth (x)}}{\sqrt{2}}\right )}{4 \sqrt{2}}-\frac{1}{5 (1+\coth (x))^{5/2}}-\frac{1}{6 (1+\coth (x))^{3/2}}-\frac{1}{4 \sqrt{1+\coth (x)}}\\ \end{align*}
Mathematica [C] time = 0.757937, size = 94, normalized size = 1.54 \[ -\frac{1}{60} \sqrt{\coth (x)+1} (\cosh (3 x)-\sinh (3 x)) (-24 \sinh (x)+13 \sinh (3 x)-10 \cosh (x)+10 \cosh (3 x))+\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) (\coth (x)+1)^{3/2} \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{i (\coth (x)+1)}\right )}{(i (\coth (x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 43, normalized size = 0.7 \begin{align*} -{\frac{1}{5} \left ( 1+{\rm coth} \left (x\right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{1}{6} \left ( 1+{\rm coth} \left (x\right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{\sqrt{2}}{8}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{1+{\rm coth} \left (x\right )}} \right ) }-{\frac{1}{4}{\frac{1}{\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 \frac{1}{{\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 = 2.45624, size = 919, normalized size = 15.07 \begin{align*} -\frac{2 \, \sqrt{2}{\left (23 \, \sqrt{2} \cosh \left (x\right )^{4} + 92 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + 23 \, \sqrt{2} \sinh \left (x\right )^{4} +{\left (138 \, \sqrt{2} \cosh \left (x\right )^{2} - 11 \, \sqrt{2}\right )} \sinh \left (x\right )^{2} - 11 \, \sqrt{2} \cosh \left (x\right )^{2} + 2 \,{\left (46 \, \sqrt{2} \cosh \left (x\right )^{3} - 11 \, \sqrt{2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3 \, \sqrt{2}\right )} \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 15 \,{\left (\sqrt{2} \cosh \left (x\right )^{5} + 5 \, \sqrt{2} \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \sqrt{2} \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \sqrt{2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sqrt{2} \sinh \left (x\right )^{5}\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 )}{240 \,{\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\right )}} \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{1}{\left (\coth{\left (x \right )} + 1\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22279, size = 242, normalized size = 3.97 \begin{align*} -\frac{1}{240} \, \sqrt{2}{\left (\frac{15 \, \log \left ({\left | 2 \, \sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )}{\mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \frac{2 \,{\left (45 \,{\left (\sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{4} + 45 \,{\left (\sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} + 35 \,{\left (\sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} + 15 \, \sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 15 \, e^{\left (2 \, x\right )} + 3\right )}}{{\left (\sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{5} \mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - 46 \, \mathrm{sgn}\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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