Optimal. Leaf size=123 \[ -\frac {26}{77} a^2 \cot (x) \sqrt {a \sin ^3(x)}-\frac {26 a^2 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{77 \sin ^{\frac {3}{2}}(x)}-\frac {78}{385} a^2 \cos (x) \sin (x) \sqrt {a \sin ^3(x)}-\frac {26}{165} a^2 \cos (x) \sin ^3(x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715,
2720} \begin {gather*} -\frac {26}{165} a^2 \sin ^3(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {78}{385} a^2 \sin (x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \sin ^5(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {26}{77} a^2 \cot (x) \sqrt {a \sin ^3(x)}-\frac {26 a^2 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{77 \sin ^{\frac {3}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 3286
Rubi steps
\begin {align*} \int \left (a \sin ^3(x)\right )^{5/2} \, dx &=\frac {\left (a^2 \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {15}{2}}(x) \, dx}{\sin ^{\frac {3}{2}}(x)}\\ &=-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)}+\frac {\left (13 a^2 \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {11}{2}}(x) \, dx}{15 \sin ^{\frac {3}{2}}(x)}\\ &=-\frac {26}{165} a^2 \cos (x) \sin ^3(x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)}+\frac {\left (39 a^2 \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {7}{2}}(x) \, dx}{55 \sin ^{\frac {3}{2}}(x)}\\ &=-\frac {78}{385} a^2 \cos (x) \sin (x) \sqrt {a \sin ^3(x)}-\frac {26}{165} a^2 \cos (x) \sin ^3(x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)}+\frac {\left (39 a^2 \sqrt {a \sin ^3(x)}\right ) \int \sin ^{\frac {3}{2}}(x) \, dx}{77 \sin ^{\frac {3}{2}}(x)}\\ &=-\frac {26}{77} a^2 \cot (x) \sqrt {a \sin ^3(x)}-\frac {78}{385} a^2 \cos (x) \sin (x) \sqrt {a \sin ^3(x)}-\frac {26}{165} a^2 \cos (x) \sin ^3(x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)}+\frac {\left (13 a^2 \sqrt {a \sin ^3(x)}\right ) \int \frac {1}{\sqrt {\sin (x)}} \, dx}{77 \sin ^{\frac {3}{2}}(x)}\\ &=-\frac {26}{77} a^2 \cot (x) \sqrt {a \sin ^3(x)}-\frac {26 a^2 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{77 \sin ^{\frac {3}{2}}(x)}-\frac {78}{385} a^2 \cos (x) \sin (x) \sqrt {a \sin ^3(x)}-\frac {26}{165} a^2 \cos (x) \sin ^3(x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \cos (x) \sin ^5(x) \sqrt {a \sin ^3(x)}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 65, normalized size = 0.53 \begin {gather*} \frac {a \left (-12480 F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )+(-15465 \cos (x)+3657 \cos (3 x)-749 \cos (5 x)+77 \cos (7 x)) \sqrt {\sin (x)}\right ) \left (a \sin ^3(x)\right )^{3/2}}{36960 \sin ^{\frac {9}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.54, size = 152, normalized size = 1.24
method | result | size |
default | \(-\frac {\left (-154 \left (\cos ^{8}\left (x \right )\right )+195 i \sqrt {2}\, \sin \left (x \right ) \sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}}\, \sqrt {\frac {-i \cos \left (x \right )+\sin \left (x \right )+i}{\sin \left (x \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (x \right )+\sin \left (x \right )-i}{\sin \left (x \right )}}, \frac {\sqrt {2}}{2}\right )+154 \left (\cos ^{7}\left (x \right )\right )+644 \left (\cos ^{6}\left (x \right )\right )-644 \left (\cos ^{5}\left (x \right )\right )-1060 \left (\cos ^{4}\left (x \right )\right )+1060 \left (\cos ^{3}\left (x \right )\right )+960 \left (\cos ^{2}\left (x \right )\right )-960 \cos \left (x \right )\right ) \left (a \left (1-\left (\cos ^{2}\left (x \right )\right )\right ) \sin \left (x \right )\right )^{\frac {5}{2}}}{1155 \sin \left (x \right )^{7} \left (-1+\cos \left (x \right )\right )}\) | \(152\) |
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 = 0.12, size = 110, normalized size = 0.89 \begin {gather*} \frac {195 \, \sqrt {2} \sqrt {-i \, a} a^{2} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 195 \, \sqrt {2} \sqrt {i \, a} a^{2} \sin \left (x\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (77 \, a^{2} \cos \left (x\right )^{7} - 322 \, a^{2} \cos \left (x\right )^{5} + 530 \, a^{2} \cos \left (x\right )^{3} - 480 \, a^{2} \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{1155 \, \sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\left (a\,{\sin \left (x\right )}^3\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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