Optimal. Leaf size=135 \[ -\frac{2}{13} a^2 \coth (x) \text{csch}^4(x) \sqrt{a \text{csch}^3(x)}+\frac{22}{117} a^2 \coth (x) \text{csch}^2(x) \sqrt{a \text{csch}^3(x)}-\frac{154}{585} a^2 \coth (x) \sqrt{a \text{csch}^3(x)}+\frac{154}{195} a^2 \sinh (x) \cosh (x) \sqrt{a \text{csch}^3(x)}-\frac{154 i a^2 \sinh ^2(x) E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{a \text{csch}^3(x)}}{195 \sqrt{i \sinh (x)}} \]
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Rubi [A] time = 0.0698942, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3768, 3771, 2639} \[ -\frac{2}{13} a^2 \coth (x) \text{csch}^4(x) \sqrt{a \text{csch}^3(x)}+\frac{22}{117} a^2 \coth (x) \text{csch}^2(x) \sqrt{a \text{csch}^3(x)}-\frac{154}{585} a^2 \coth (x) \sqrt{a \text{csch}^3(x)}+\frac{154}{195} a^2 \sinh (x) \cosh (x) \sqrt{a \text{csch}^3(x)}-\frac{154 i a^2 \sinh ^2(x) E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{a \text{csch}^3(x)}}{195 \sqrt{i \sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \left (a \text{csch}^3(x)\right )^{5/2} \, dx &=-\frac{\left (a^2 \sqrt{a \text{csch}^3(x)}\right ) \int (i \text{csch}(x))^{15/2} \, dx}{(i \text{csch}(x))^{3/2}}\\ &=-\frac{2}{13} a^2 \coth (x) \text{csch}^4(x) \sqrt{a \text{csch}^3(x)}-\frac{\left (11 a^2 \sqrt{a \text{csch}^3(x)}\right ) \int (i \text{csch}(x))^{11/2} \, dx}{13 (i \text{csch}(x))^{3/2}}\\ &=\frac{22}{117} a^2 \coth (x) \text{csch}^2(x) \sqrt{a \text{csch}^3(x)}-\frac{2}{13} a^2 \coth (x) \text{csch}^4(x) \sqrt{a \text{csch}^3(x)}-\frac{\left (77 a^2 \sqrt{a \text{csch}^3(x)}\right ) \int (i \text{csch}(x))^{7/2} \, dx}{117 (i \text{csch}(x))^{3/2}}\\ &=-\frac{154}{585} a^2 \coth (x) \sqrt{a \text{csch}^3(x)}+\frac{22}{117} a^2 \coth (x) \text{csch}^2(x) \sqrt{a \text{csch}^3(x)}-\frac{2}{13} a^2 \coth (x) \text{csch}^4(x) \sqrt{a \text{csch}^3(x)}-\frac{\left (77 a^2 \sqrt{a \text{csch}^3(x)}\right ) \int (i \text{csch}(x))^{3/2} \, dx}{195 (i \text{csch}(x))^{3/2}}\\ &=-\frac{154}{585} a^2 \coth (x) \sqrt{a \text{csch}^3(x)}+\frac{22}{117} a^2 \coth (x) \text{csch}^2(x) \sqrt{a \text{csch}^3(x)}-\frac{2}{13} a^2 \coth (x) \text{csch}^4(x) \sqrt{a \text{csch}^3(x)}+\frac{154}{195} a^2 \cosh (x) \sqrt{a \text{csch}^3(x)} \sinh (x)+\frac{\left (77 a^2 \sqrt{a \text{csch}^3(x)}\right ) \int \frac{1}{\sqrt{i \text{csch}(x)}} \, dx}{195 (i \text{csch}(x))^{3/2}}\\ &=-\frac{154}{585} a^2 \coth (x) \sqrt{a \text{csch}^3(x)}+\frac{22}{117} a^2 \coth (x) \text{csch}^2(x) \sqrt{a \text{csch}^3(x)}-\frac{2}{13} a^2 \coth (x) \text{csch}^4(x) \sqrt{a \text{csch}^3(x)}+\frac{154}{195} a^2 \cosh (x) \sqrt{a \text{csch}^3(x)} \sinh (x)-\frac{\left (77 a^2 \sqrt{a \text{csch}^3(x)} \sinh ^2(x)\right ) \int \sqrt{i \sinh (x)} \, dx}{195 \sqrt{i \sinh (x)}}\\ &=-\frac{154}{585} a^2 \coth (x) \sqrt{a \text{csch}^3(x)}+\frac{22}{117} a^2 \coth (x) \text{csch}^2(x) \sqrt{a \text{csch}^3(x)}-\frac{2}{13} a^2 \coth (x) \text{csch}^4(x) \sqrt{a \text{csch}^3(x)}+\frac{154}{195} a^2 \cosh (x) \sqrt{a \text{csch}^3(x)} \sinh (x)-\frac{154 i a^2 \sqrt{a \text{csch}^3(x)} E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sinh ^2(x)}{195 \sqrt{i \sinh (x)}}\\ \end{align*}
Mathematica [A] time = 0.167947, size = 68, normalized size = 0.5 \[ -\frac{2}{585} a^2 \sinh (x) \sqrt{a \text{csch}^3(x)} \left (-231 \cosh (x)+\coth (x) \text{csch}(x) \left (45 \text{csch}^4(x)-55 \text{csch}^2(x)+77\right )+231 \sqrt{i \sinh (x)} E\left (\left .\frac{1}{4} (\pi -2 i x)\right |2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ({\rm csch} \left (x\right ) \right ) ^{3} \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 (a \operatorname{csch}\left (x\right )^{3}\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{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 \left (a \operatorname{csch}^{3}{\left (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 (a \operatorname{csch}\left (x\right )^{3}\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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