Optimal. Leaf size=56 \[ -2 \sinh (x) \cosh (x) \sqrt {a \text {csch}^3(x)}-2 i (i \sinh (x))^{3/2} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \text {csch}^3(x)} \]
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Rubi [A] time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3768, 3771, 2639} \[ -2 \sinh (x) \cosh (x) \sqrt {a \text {csch}^3(x)}+\frac {2 i \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \text {csch}^3(x)}}{\sqrt {i \sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3768
Rule 3771
Rule 4123
Rubi steps
\begin {align*} \int \sqrt {a \text {csch}^3(x)} \, dx &=\frac {\sqrt {a \text {csch}^3(x)} \int (i \text {csch}(x))^{3/2} \, dx}{(i \text {csch}(x))^{3/2}}\\ &=-2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x)-\frac {\sqrt {a \text {csch}^3(x)} \int \frac {1}{\sqrt {i \text {csch}(x)}} \, dx}{(i \text {csch}(x))^{3/2}}\\ &=-2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x)+\frac {\left (\sqrt {a \text {csch}^3(x)} \sinh ^2(x)\right ) \int \sqrt {i \sinh (x)} \, dx}{\sqrt {i \sinh (x)}}\\ &=-2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x)+\frac {2 i \sqrt {a \text {csch}^3(x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{\sqrt {i \sinh (x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.75 \[ -2 \sinh (x) \sqrt {a \text {csch}^3(x)} \left (\cosh (x)-\sqrt {i \sinh (x)} E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \operatorname {csch}\relax (x)^{3}}, x\right ) \]
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 \[ \int \sqrt {a \operatorname {csch}\relax (x)^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \sqrt {a \mathrm {csch}\relax (x )^{3}}\, dx \]
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 \[ \int \sqrt {a \operatorname {csch}\relax (x)^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\frac {a}{{\mathrm {sinh}\relax (x)}^3}} \,d x \]
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 \[ \int \sqrt {a \operatorname {csch}^{3}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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