Integrand size = 10, antiderivative size = 135 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=-\frac {26 \coth (x)}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {26 i \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)}}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt {a \text {csch}^3(x)}} \]
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Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3854, 3856, 2720} \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=-\frac {26 \coth (x)}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \sinh ^5(x) \cosh (x)}{15 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {26 \sinh ^3(x) \cosh (x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {78 \sinh (x) \cosh (x)}{385 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {26 i \sqrt {i \sinh (x)} \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right )}{77 a^2 \sqrt {a \text {csch}^3(x)}} \]
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Rule 2720
Rule 3854
Rule 3856
Rule 4208
Rubi steps \begin{align*} \text {integral}& = -\frac {(i \text {csch}(x))^{3/2} \int \frac {1}{(i \text {csch}(x))^{15/2}} \, dx}{a^2 \sqrt {a \text {csch}^3(x)}} \\ & = \frac {2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {\left (13 (i \text {csch}(x))^{3/2}\right ) \int \frac {1}{(i \text {csch}(x))^{11/2}} \, dx}{15 a^2 \sqrt {a \text {csch}^3(x)}} \\ & = -\frac {26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {\left (39 (i \text {csch}(x))^{3/2}\right ) \int \frac {1}{(i \text {csch}(x))^{7/2}} \, dx}{55 a^2 \sqrt {a \text {csch}^3(x)}} \\ & = \frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {\left (39 (i \text {csch}(x))^{3/2}\right ) \int \frac {1}{(i \text {csch}(x))^{3/2}} \, dx}{77 a^2 \sqrt {a \text {csch}^3(x)}} \\ & = -\frac {26 \coth (x)}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {\left (13 (i \text {csch}(x))^{3/2}\right ) \int \sqrt {i \text {csch}(x)} \, dx}{77 a^2 \sqrt {a \text {csch}^3(x)}} \\ & = -\frac {26 \coth (x)}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {\left (13 \text {csch}^2(x) \sqrt {i \sinh (x)}\right ) \int \frac {1}{\sqrt {i \sinh (x)}} \, dx}{77 a^2 \sqrt {a \text {csch}^3(x)}} \\ & = -\frac {26 \coth (x)}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {26 i \text {csch}^2(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},2\right ) \sqrt {i \sinh (x)}}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {26 \cosh (x) \sinh ^3(x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \cosh (x) \sinh ^5(x)}{15 a^2 \sqrt {a \text {csch}^3(x)}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\frac {\sqrt {a \text {csch}^3(x)} \sinh (x) \left (24960 i \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),2\right ) \sqrt {i \sinh (x)}-19122 \sinh (2 x)+4406 \sinh (4 x)-826 \sinh (6 x)+77 \sinh (8 x)\right )}{73920 a^3} \]
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\[\int \frac {1}{\left (a \operatorname {csch}\left (x \right )^{3}\right )^{\frac {5}{2}}}d x\]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 718, normalized size of antiderivative = 5.32 \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \operatorname {csch}^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \operatorname {csch}\left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \operatorname {csch}\left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\left (x\right )}^3}\right )}^{5/2}} \,d x \]
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