Integrand size = 10, antiderivative size = 56 \[ \int \sqrt {a \text {csch}^3(x)} \, dx=-2 i \sqrt {a \text {csch}^3(x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) (i \sinh (x))^{3/2}-2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x) \]
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Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2719} \[ \int \sqrt {a \text {csch}^3(x)} \, dx=-2 \sinh (x) \cosh (x) \sqrt {a \text {csch}^3(x)}+\frac {2 i \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {a \text {csch}^3(x)}}{\sqrt {i \sinh (x)}} \]
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Rule 2719
Rule 3853
Rule 3856
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.75 \[ \int \sqrt {a \text {csch}^3(x)} \, dx=-2 \sqrt {a \text {csch}^3(x)} \left (\cosh (x)-E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) \sqrt {i \sinh (x)}\right ) \sinh (x) \]
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\[\int \sqrt {a \operatorname {csch}\left (x \right )^{3}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07 \[ \int \sqrt {a \text {csch}^3(x)} \, dx=-2 \, \sqrt {2} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \sqrt {2} \sqrt {a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) \]
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\[ \int \sqrt {a \text {csch}^3(x)} \, dx=\int \sqrt {a \operatorname {csch}^{3}{\left (x \right )}}\, dx \]
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\[ \int \sqrt {a \text {csch}^3(x)} \, dx=\int { \sqrt {a \operatorname {csch}\left (x\right )^{3}} \,d x } \]
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\[ \int \sqrt {a \text {csch}^3(x)} \, dx=\int { \sqrt {a \operatorname {csch}\left (x\right )^{3}} \,d x } \]
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Timed out. \[ \int \sqrt {a \text {csch}^3(x)} \, dx=\int \sqrt {\frac {a}{{\mathrm {sinh}\left (x\right )}^3}} \,d x \]
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