Optimal. Leaf size=89 \[ -\frac{14 \cosh (x)}{45 a \sqrt{a \text{csch}^3(x)}}+\frac{2 \sinh ^2(x) \cosh (x)}{9 a \sqrt{a \text{csch}^3(x)}}+\frac{14 i \text{csch}(x) E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right )}{15 a \sqrt{i \sinh (x)} \sqrt{a \text{csch}^3(x)}} \]
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Rubi [A] time = 0.0479715, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3769, 3771, 2639} \[ -\frac{14 \cosh (x)}{45 a \sqrt{a \text{csch}^3(x)}}+\frac{2 \sinh ^2(x) \cosh (x)}{9 a \sqrt{a \text{csch}^3(x)}}+\frac{14 i \text{csch}(x) E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right )}{15 a \sqrt{i \sinh (x)} \sqrt{a \text{csch}^3(x)}} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\left (a \text{csch}^3(x)\right )^{3/2}} \, dx &=-\frac{\left (i (i \text{csch}(x))^{3/2}\right ) \int \frac{1}{(i \text{csch}(x))^{9/2}} \, dx}{a \sqrt{a \text{csch}^3(x)}}\\ &=\frac{2 \cosh (x) \sinh ^2(x)}{9 a \sqrt{a \text{csch}^3(x)}}-\frac{\left (7 i (i \text{csch}(x))^{3/2}\right ) \int \frac{1}{(i \text{csch}(x))^{5/2}} \, dx}{9 a \sqrt{a \text{csch}^3(x)}}\\ &=-\frac{14 \cosh (x)}{45 a \sqrt{a \text{csch}^3(x)}}+\frac{2 \cosh (x) \sinh ^2(x)}{9 a \sqrt{a \text{csch}^3(x)}}-\frac{\left (7 i (i \text{csch}(x))^{3/2}\right ) \int \frac{1}{\sqrt{i \text{csch}(x)}} \, dx}{15 a \sqrt{a \text{csch}^3(x)}}\\ &=-\frac{14 \cosh (x)}{45 a \sqrt{a \text{csch}^3(x)}}+\frac{2 \cosh (x) \sinh ^2(x)}{9 a \sqrt{a \text{csch}^3(x)}}+\frac{(7 \text{csch}(x)) \int \sqrt{i \sinh (x)} \, dx}{15 a \sqrt{a \text{csch}^3(x)} \sqrt{i \sinh (x)}}\\ &=-\frac{14 \cosh (x)}{45 a \sqrt{a \text{csch}^3(x)}}+\frac{14 i \text{csch}(x) E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right )}{15 a \sqrt{a \text{csch}^3(x)} \sqrt{i \sinh (x)}}+\frac{2 \cosh (x) \sinh ^2(x)}{9 a \sqrt{a \text{csch}^3(x)}}\\ \end{align*}
Mathematica [A] time = 0.0750177, size = 57, normalized size = 0.64 \[ \frac{-33 \cosh (x)+5 \cosh (3 x)+84 \sqrt{i \sinh (x)} \text{csch}^2(x) E\left (\left .\frac{1}{4} (\pi -2 i x)\right |2\right )}{90 a \sqrt{a \text{csch}^3(x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ({\rm csch} \left (x\right ) \right ) ^{3} \right ) ^{-{\frac{3}{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 (a \operatorname{csch}\left (x\right )^{3}\right )^{\frac{3}{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{a \operatorname{csch}\left (x\right )^{3}}}{a^{2} \operatorname{csch}\left (x\right )^{6}}, 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 (a \operatorname{csch}^{3}{\left (x \right )}\right )^{\frac{3}{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 (a \operatorname{csch}\left (x\right )^{3}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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