Optimal. Leaf size=90 \[ \frac{2 \cosh (c+d x)}{5 b d (b \text{csch}(c+d x))^{3/2}}+\frac{6 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{5 b^2 d \sqrt{i \sinh (c+d x)} \sqrt{b \text{csch}(c+d x)}} \]
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Rubi [A] time = 0.0382342, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3769, 3771, 2639} \[ \frac{2 \cosh (c+d x)}{5 b d (b \text{csch}(c+d x))^{3/2}}+\frac{6 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{5 b^2 d \sqrt{i \sinh (c+d x)} \sqrt{b \text{csch}(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(b \text{csch}(c+d x))^{5/2}} \, dx &=\frac{2 \cosh (c+d x)}{5 b d (b \text{csch}(c+d x))^{3/2}}-\frac{3 \int \frac{1}{\sqrt{b \text{csch}(c+d x)}} \, dx}{5 b^2}\\ &=\frac{2 \cosh (c+d x)}{5 b d (b \text{csch}(c+d x))^{3/2}}-\frac{3 \int \sqrt{i \sinh (c+d x)} \, dx}{5 b^2 \sqrt{b \text{csch}(c+d x)} \sqrt{i \sinh (c+d x)}}\\ &=\frac{2 \cosh (c+d x)}{5 b d (b \text{csch}(c+d x))^{3/2}}+\frac{6 i E\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right )}{5 b^2 d \sqrt{b \text{csch}(c+d x)} \sqrt{i \sinh (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.13045, size = 68, normalized size = 0.76 \[ \frac{\sinh (2 (c+d x))-\frac{6 i E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right )}{\sqrt{i \sinh (c+d x)}}}{5 b^2 d \sqrt{b \text{csch}(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int \left ( b{\rm csch} \left (dx+c\right ) \right ) ^{-{\frac{5}{2}}}\, dx \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 (b \operatorname{csch}\left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \operatorname{csch}\left (d x + c\right )}}{b^{3} \operatorname{csch}\left (d x + c\right )^{3}}, x\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 (b \operatorname{csch}{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \operatorname{csch}\left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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