Optimal. Leaf size=123 \[ -\frac {26}{165} a^2 \sin ^3(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {78}{385} a^2 \sin (x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {2}{15} a^2 \sin ^5(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {26}{77} a^2 \cot (x) \sqrt {a \sin ^3(x)}-\frac {26 a^2 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{77 \sin ^{\frac {3}{2}}(x)} \]
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Rubi [A] time = 0.04, 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 = {3207, 2635, 2641} \[ -\frac {2}{15} a^2 \sin ^5(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {26}{165} a^2 \sin ^3(x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {78}{385} a^2 \sin (x) \cos (x) \sqrt {a \sin ^3(x)}-\frac {26}{77} a^2 \cot (x) \sqrt {a \sin ^3(x)}-\frac {26 a^2 F\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \sin ^3(x)}}{77 \sin ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 3207
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 = 0.17, size = 65, normalized size = 0.53 \[ \frac {a \left (a \sin ^3(x)\right )^{3/2} \left (\sqrt {\sin (x)} (-15465 \cos (x)+3657 \cos (3 x)-749 \cos (5 x)+77 \cos (7 x))-12480 F\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )\right )}{36960 \sin ^{\frac {9}{2}}(x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} \cos \relax (x)^{6} - 3 \, a^{2} \cos \relax (x)^{4} + 3 \, a^{2} \cos \relax (x)^{2} - a^{2}\right )} \sqrt {-{\left (a \cos \relax (x)^{2} - a\right )} \sin \relax (x)}, x\right ) \]
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 \[ \int \left (a \sin \relax (x)^{3}\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.43, size = 155, normalized size = 1.26 \[ -\frac {\left (-154 \left (\cos ^{8}\relax (x )\right )+195 i \sqrt {2}\, \sin \relax (x ) \sqrt {-\frac {i \cos \relax (x )-\sin \relax (x )-i}{\sin \relax (x )}}\, \EllipticF \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}+154 \left (\cos ^{7}\relax (x )\right )+644 \left (\cos ^{6}\relax (x )\right )-644 \left (\cos ^{5}\relax (x )\right )-1060 \left (\cos ^{4}\relax (x )\right )+1060 \left (\cos ^{3}\relax (x )\right )+960 \left (\cos ^{2}\relax (x )\right )-960 \cos \relax (x )\right ) \left (a \left (1-\left (\cos ^{2}\relax (x )\right )\right ) \sin \relax (x )\right )^{\frac {5}{2}}}{1155 \sin \relax (x )^{7} \left (-1+\cos \relax (x )\right )} \]
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 \[ \int \left (a \sin \relax (x)^{3}\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a\,{\sin \relax (x)}^3\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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