Optimal. Leaf size=88 \[ -\frac{2 b \cosh (c+d x) (b \text{csch}(c+d x))^{3/2}}{3 d}+\frac{2 i b^2 \sqrt{i \sinh (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right ),2\right ) \sqrt{b \text{csch}(c+d x)}}{3 d} \]
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Rubi [A] time = 0.038811, antiderivative size = 88, 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 = {3768, 3771, 2641} \[ -\frac{2 b \cosh (c+d x) (b \text{csch}(c+d x))^{3/2}}{3 d}+\frac{2 i b^2 \sqrt{i \sinh (c+d x)} F\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \text{csch}(c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int (b \text{csch}(c+d x))^{5/2} \, dx &=-\frac{2 b \cosh (c+d x) (b \text{csch}(c+d x))^{3/2}}{3 d}-\frac{1}{3} b^2 \int \sqrt{b \text{csch}(c+d x)} \, dx\\ &=-\frac{2 b \cosh (c+d x) (b \text{csch}(c+d x))^{3/2}}{3 d}-\frac{1}{3} \left (b^2 \sqrt{b \text{csch}(c+d x)} \sqrt{i \sinh (c+d x)}\right ) \int \frac{1}{\sqrt{i \sinh (c+d x)}} \, dx\\ &=-\frac{2 b \cosh (c+d x) (b \text{csch}(c+d x))^{3/2}}{3 d}+\frac{2 i b^2 \sqrt{b \text{csch}(c+d x)} F\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right ) \sqrt{i \sinh (c+d x)}}{3 d}\\ \end{align*}
Mathematica [A] time = 0.100849, size = 66, normalized size = 0.75 \[ -\frac{2 b^2 \sqrt{b \text{csch}(c+d x)} \left (\coth (c+d x)+i \sqrt{i \sinh (c+d x)} \text{EllipticF}\left (\frac{1}{4} (-2 i c-2 i d x+\pi ),2\right )\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.072, 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 \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 (\sqrt{b \operatorname{csch}\left (d x + c\right )} b^{2} \operatorname{csch}\left (d x + c\right )^{2}, 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 \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 \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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