Optimal. Leaf size=61 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )}{4 \sqrt{2}}-\frac{1}{4 \sqrt{\tanh (x)+1}}-\frac{1}{6 (\tanh (x)+1)^{3/2}}-\frac{1}{5 (\tanh (x)+1)^{5/2}} \]
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Rubi [A] time = 0.0414043, 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} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )}{4 \sqrt{2}}-\frac{1}{4 \sqrt{\tanh (x)+1}}-\frac{1}{6 (\tanh (x)+1)^{3/2}}-\frac{1}{5 (\tanh (x)+1)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3479
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(1+\tanh (x))^{5/2}} \, dx &=-\frac{1}{5 (1+\tanh (x))^{5/2}}+\frac{1}{2} \int \frac{1}{(1+\tanh (x))^{3/2}} \, dx\\ &=-\frac{1}{5 (1+\tanh (x))^{5/2}}-\frac{1}{6 (1+\tanh (x))^{3/2}}+\frac{1}{4} \int \frac{1}{\sqrt{1+\tanh (x)}} \, dx\\ &=-\frac{1}{5 (1+\tanh (x))^{5/2}}-\frac{1}{6 (1+\tanh (x))^{3/2}}-\frac{1}{4 \sqrt{1+\tanh (x)}}+\frac{1}{8} \int \sqrt{1+\tanh (x)} \, dx\\ &=-\frac{1}{5 (1+\tanh (x))^{5/2}}-\frac{1}{6 (1+\tanh (x))^{3/2}}-\frac{1}{4 \sqrt{1+\tanh (x)}}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\tanh (x)}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{1+\tanh (x)}}{\sqrt{2}}\right )}{4 \sqrt{2}}-\frac{1}{5 (1+\tanh (x))^{5/2}}-\frac{1}{6 (1+\tanh (x))^{3/2}}-\frac{1}{4 \sqrt{1+\tanh (x)}}\\ \end{align*}
Mathematica [A] time = 0.229374, size = 62, normalized size = 1.02 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )}{4 \sqrt{2}}+\frac{(\sinh (2 x)-\cosh (2 x)) (20 \sinh (2 x)+26 \cosh (2 x)+11)}{60 \sqrt{\tanh (x)+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 43, normalized size = 0.7 \begin{align*}{\frac{\sqrt{2}}{8}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{1+\tanh \left ( x \right ) }} \right ) }-{\frac{1}{4}{\frac{1}{\sqrt{1+\tanh \left ( x \right ) }}}}-{\frac{1}{5} \left ( 1+\tanh \left ( x \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{1}{6} \left ( 1+\tanh \left ( x \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64044, size = 107, normalized size = 1.75 \begin{align*} -\frac{1}{120} \, \sqrt{2}{\left (\frac{5}{e^{\left (-2 \, x\right )} + 1} + \frac{15}{{\left (e^{\left (-2 \, x\right )} + 1\right )}^{2}} + 3\right )}{\left (e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}} - \frac{1}{16} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{\sqrt{2}}{\sqrt{e^{\left (-2 \, x\right )} + 1}}}{\sqrt{2} + \frac{\sqrt{2}}{\sqrt{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 = 2.22798, size = 921, normalized size = 15.1 \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{\cosh \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{\cosh \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 (\tanh{\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 = 1.24633, size = 189, normalized size = 3.1 \begin{align*} -\frac{1}{240} \, \sqrt{2}{\left (\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}} + 15 \, \log \left (-2 \, \sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right ) - 46\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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