Optimal. Leaf size=135 \[ -\frac {154}{585} a^2 \coth (x) \sqrt {a \text {csch}^3(x)}-\frac {2}{13} a^2 \coth (x) \text {csch}^4(x) \sqrt {a \text {csch}^3(x)}+\frac {22}{117} a^2 \coth (x) \text {csch}^2(x) \sqrt {a \text {csch}^3(x)}+\frac {154}{195} a^2 \sinh (x) \cosh (x) \sqrt {a \text {csch}^3(x)}-\frac {154 i a^2 \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \text {csch}^3(x)}}{195 \sqrt {i \sinh (x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 2639
Rule 3768
Rule 3771
Rule 4123
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*}
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Mathematica [A] time = 0.18, size = 68, normalized size = 0.50 \[ -\frac {2}{585} a^2 \sinh (x) \sqrt {a \text {csch}^3(x)} \left (-231 \cosh (x)+\coth (x) \text {csch}(x) \left (45 \text {csch}^4(x)-55 \text {csch}^2(x)+77\right )+231 \sqrt {i \sinh (x)} E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \operatorname {csch}\relax (x)^{3}} a^{2} \operatorname {csch}\relax (x)^{6}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \operatorname {csch}\relax (x)^{3}\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (a \mathrm {csch}\relax (x )^{3}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \operatorname {csch}\relax (x)^{3}\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {a}{{\mathrm {sinh}\relax (x)}^3}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \operatorname {csch}^{3}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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