Optimal. Leaf size=135 \[ -\frac {26 \coth (x)}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \sinh ^5(x) \cosh (x)}{15 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {26 \sinh ^3(x) \cosh (x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {78 \sinh (x) \cosh (x)}{385 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {26 i \sqrt {i \sinh (x)} \text {csch}^2(x) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{77 a^2 \sqrt {a \text {csch}^3(x)}} \]
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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, 3769, 3771, 2641} \[ -\frac {26 \coth (x)}{77 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {2 \sinh ^5(x) \cosh (x)}{15 a^2 \sqrt {a \text {csch}^3(x)}}-\frac {26 \sinh ^3(x) \cosh (x)}{165 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {78 \sinh (x) \cosh (x)}{385 a^2 \sqrt {a \text {csch}^3(x)}}+\frac {26 i \sqrt {i \sinh (x)} \text {csch}^2(x) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{77 a^2 \sqrt {a \text {csch}^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rule 4123
Rubi steps
\begin {align*} \int \frac {1}{\left (a \text {csch}^3(x)\right )^{5/2}} \, dx &=-\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) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |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*}
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Mathematica [A] time = 0.13, size = 71, normalized size = 0.53 \[ \frac {\sinh (x) \sqrt {a \text {csch}^3(x)} \left (-19122 \sinh (2 x)+4406 \sinh (4 x)-826 \sinh (6 x)+77 \sinh (8 x)+24960 i \sqrt {i \sinh (x)} F\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )\right )}{73920 a^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \operatorname {csch}\relax (x)^{3}}}{a^{3} \operatorname {csch}\relax (x)^{9}}, x\right ) \]
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 \[ \int \frac {1}{\left (a \operatorname {csch}\relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \mathrm {csch}\relax (x )^{3}\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {1}{\left (a \operatorname {csch}\relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\relax (x)}^3}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \operatorname {csch}^{3}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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