Optimal. Leaf size=87 \[ \frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {10 i F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {i \sinh (x)} \sinh (x)}{21 a \sqrt {a \sinh ^3(x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3286, 2716,
2721, 2720} \begin {gather*} \frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}+\frac {10 i \sqrt {i \sinh (x)} \sinh (x) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2720
Rule 2721
Rule 3286
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx &=\frac {\sinh ^{\frac {3}{2}}(x) \int \frac {1}{\sinh ^{\frac {9}{2}}(x)} \, dx}{a \sqrt {a \sinh ^3(x)}}\\ &=-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}-\frac {\left (5 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sinh ^{\frac {5}{2}}(x)} \, dx}{7 a \sqrt {a \sinh ^3(x)}}\\ &=\frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {\left (5 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sqrt {\sinh (x)}} \, dx}{21 a \sqrt {a \sinh ^3(x)}}\\ &=\frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {\left (5 \sqrt {i \sinh (x)} \sinh (x)\right ) \int \frac {1}{\sqrt {i \sinh (x)}} \, dx}{21 a \sqrt {a \sinh ^3(x)}}\\ &=\frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {10 i F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {i \sinh (x)} \sinh (x)}{21 a \sqrt {a \sinh ^3(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 53, normalized size = 0.61 \begin {gather*} \frac {2 \left (5 \cosh (x)-3 \coth (x) \text {csch}(x)+5 F\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) (i \sinh (x))^{3/2}\right )}{21 a \sqrt {a \sinh ^3(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.78, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \left (\sinh ^{3}\left (x \right )\right )\right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 639, normalized size = 7.34 \begin {gather*} \frac {2 \, {\left (5 \, {\left (\sqrt {2} \cosh \left (x\right )^{8} + 8 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sqrt {2} \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right )^{6} - 4 \, \sqrt {2} \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{3} - 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \sqrt {2} \cosh \left (x\right )^{4} - 30 \, \sqrt {2} \cosh \left (x\right )^{2} + 3 \, \sqrt {2}\right )} \sinh \left (x\right )^{4} + 6 \, \sqrt {2} \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{5} - 10 \, \sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{6} - 15 \, \sqrt {2} \cosh \left (x\right )^{4} + 9 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right )^{2} - 4 \, \sqrt {2} \cosh \left (x\right )^{2} + 8 \, {\left (\sqrt {2} \cosh \left (x\right )^{7} - 3 \, \sqrt {2} \cosh \left (x\right )^{5} + 3 \, \sqrt {2} \cosh \left (x\right )^{3} - \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, {\left (5 \, \cosh \left (x\right )^{7} + 35 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + 5 \, \sinh \left (x\right )^{7} + {\left (105 \, \cosh \left (x\right )^{2} - 17\right )} \sinh \left (x\right )^{5} - 17 \, \cosh \left (x\right )^{5} + 5 \, {\left (35 \, \cosh \left (x\right )^{3} - 17 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (175 \, \cosh \left (x\right )^{4} - 170 \, \cosh \left (x\right )^{2} - 17\right )} \sinh \left (x\right )^{3} - 17 \, \cosh \left (x\right )^{3} + {\left (105 \, \cosh \left (x\right )^{5} - 170 \, \cosh \left (x\right )^{3} - 51 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (35 \, \cosh \left (x\right )^{6} - 85 \, \cosh \left (x\right )^{4} - 51 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right ) + 5 \, \cosh \left (x\right )\right )} \sqrt {a \sinh \left (x\right )}\right )}}{21 \, {\left (a^{2} \cosh \left (x\right )^{8} + 8 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{7} + a^{2} \sinh \left (x\right )^{8} - 4 \, a^{2} \cosh \left (x\right )^{6} + 4 \, {\left (7 \, a^{2} \cosh \left (x\right )^{2} - a^{2}\right )} \sinh \left (x\right )^{6} + 6 \, a^{2} \cosh \left (x\right )^{4} + 8 \, {\left (7 \, a^{2} \cosh \left (x\right )^{3} - 3 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, a^{2} \cosh \left (x\right )^{4} - 30 \, a^{2} \cosh \left (x\right )^{2} + 3 \, a^{2}\right )} \sinh \left (x\right )^{4} - 4 \, a^{2} \cosh \left (x\right )^{2} + 8 \, {\left (7 \, a^{2} \cosh \left (x\right )^{5} - 10 \, a^{2} \cosh \left (x\right )^{3} + 3 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, a^{2} \cosh \left (x\right )^{6} - 15 \, a^{2} \cosh \left (x\right )^{4} + 9 \, a^{2} \cosh \left (x\right )^{2} - a^{2}\right )} \sinh \left (x\right )^{2} + a^{2} + 8 \, {\left (a^{2} \cosh \left (x\right )^{7} - 3 \, a^{2} \cosh \left (x\right )^{5} + 3 \, a^{2} \cosh \left (x\right )^{3} - a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (a \sinh ^{3}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (a\,{\mathrm {sinh}\left (x\right )}^3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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