Optimal. Leaf size=123 \[ -\frac{2}{15} a^2 \sin ^5(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{26}{165} a^2 \sin ^3(x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{78}{385} a^2 \sin (x) \cos (x) \sqrt{a \sin ^3(x)}-\frac{26}{77} a^2 \cot (x) \sqrt{a \sin ^3(x)}-\frac{26 a^2 F\left (\left .\frac{\pi }{4}-\frac{x}{2}\right |2\right ) \sqrt{a \sin ^3(x)}}{77 \sin ^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.0407798, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 3207
Rule 2635
Rule 2641
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*}
Mathematica [A] time = 0.188814, 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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Maple [C] time = 0.224, size = 149, normalized size = 1.2 \begin{align*} -{\frac{1}{1155\, \left ( \sin \left ( x \right ) \right ) ^{7} \left ( -1+\cos \left ( x \right ) \right ) } \left ( -154\, \left ( \cos \left ( x \right ) \right ) ^{8}+195\,i\sqrt{2}\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 ) }}}\sqrt{-{\frac{i\cos \left ( x \right ) -\sin \left ( x \right ) -i}{\sin \left ( x \right ) }}}{\it 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 \left ( x \right ) \right ) ^{7}+644\, \left ( \cos \left ( x \right ) \right ) ^{6}-644\, \left ( \cos \left ( x \right ) \right ) ^{5}-1060\, \left ( \cos \left ( x \right ) \right ) ^{4}+1060\, \left ( \cos \left ( x \right ) \right ) ^{3}+960\, \left ( \cos \left ( x \right ) \right ) ^{2}-960\,\cos \left ( x \right ) \right ) \left ( a \left ( \sin \left ( x \right ) \right ) ^{3} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin \left (x\right )^{3}\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (a^{2} \cos \left (x\right )^{6} - 3 \, a^{2} \cos \left (x\right )^{4} + 3 \, a^{2} \cos \left (x\right )^{2} - a^{2}\right )} \sqrt{-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sin \left (x\right )^{3}\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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