Optimal. Leaf size=117 \[ \frac{2}{15} a^2 \sin (x) \cos ^5(x) \sqrt{a \cos ^3(x)}+\frac{26}{165} a^2 \sin (x) \cos ^3(x) \sqrt{a \cos ^3(x)}+\frac{78}{385} a^2 \sin (x) \cos (x) \sqrt{a \cos ^3(x)}+\frac{26}{77} a^2 \tan (x) \sqrt{a \cos ^3(x)}+\frac{26 a^2 F\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{77 \cos ^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.065901, antiderivative size = 117, 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} \[ \frac{2}{15} a^2 \sin (x) \cos ^5(x) \sqrt{a \cos ^3(x)}+\frac{26}{165} a^2 \sin (x) \cos ^3(x) \sqrt{a \cos ^3(x)}+\frac{78}{385} a^2 \sin (x) \cos (x) \sqrt{a \cos ^3(x)}+\frac{26}{77} a^2 \tan (x) \sqrt{a \cos ^3(x)}+\frac{26 a^2 F\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{77 \cos ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \left (a \cos ^3(x)\right )^{5/2} \, dx &=\frac{\left (a^2 \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{15}{2}}(x) \, dx}{\cos ^{\frac{3}{2}}(x)}\\ &=\frac{2}{15} a^2 \cos ^5(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{\left (13 a^2 \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{11}{2}}(x) \, dx}{15 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{26}{165} a^2 \cos ^3(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{15} a^2 \cos ^5(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{\left (39 a^2 \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{7}{2}}(x) \, dx}{55 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{78}{385} a^2 \cos (x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{26}{165} a^2 \cos ^3(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{15} a^2 \cos ^5(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{\left (39 a^2 \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{3}{2}}(x) \, dx}{77 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{78}{385} a^2 \cos (x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{26}{165} a^2 \cos ^3(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{15} a^2 \cos ^5(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{26}{77} a^2 \sqrt{a \cos ^3(x)} \tan (x)+\frac{\left (13 a^2 \sqrt{a \cos ^3(x)}\right ) \int \frac{1}{\sqrt{\cos (x)}} \, dx}{77 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{26 a^2 \sqrt{a \cos ^3(x)} F\left (\left .\frac{x}{2}\right |2\right )}{77 \cos ^{\frac{3}{2}}(x)}+\frac{78}{385} a^2 \cos (x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{26}{165} a^2 \cos ^3(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{15} a^2 \cos ^5(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{26}{77} a^2 \sqrt{a \cos ^3(x)} \tan (x)\\ \end{align*}
Mathematica [A] time = 0.1198, size = 61, normalized size = 0.52 \[ \frac{a \left (a \cos ^3(x)\right )^{3/2} \left (12480 F\left (\left .\frac{x}{2}\right |2\right )+(15465 \sin (x)+3657 \sin (3 x)+749 \sin (5 x)+77 \sin (7 x)) \sqrt{\cos (x)}\right )}{36960 \cos ^{\frac{9}{2}}(x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.301, size = 114, normalized size = 1. \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( x \right ) \right ) \left ( \cos \left ( x \right ) +1 \right ) ^{2}}{1155\, \left ( \sin \left ( x \right ) \right ) ^{3} \left ( \cos \left ( x \right ) \right ) ^{8}} \left ( -77\, \left ( \cos \left ( x \right ) \right ) ^{8}+77\, \left ( \cos \left ( x \right ) \right ) ^{7}-91\, \left ( \cos \left ( x \right ) \right ) ^{6}+195\,i\sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sin \left ( x \right ) +91\, \left ( \cos \left ( x \right ) \right ) ^{5}-117\, \left ( \cos \left ( x \right ) \right ) ^{4}+117\, \left ( \cos \left ( x \right ) \right ) ^{3}-195\, \left ( \cos \left ( x \right ) \right ) ^{2}+195\,\cos \left ( x \right ) \right ) \left ( a \left ( \cos \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 \cos \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 (\sqrt{a \cos \left (x\right )^{3}} a^{2} \cos \left (x\right )^{6}, 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 \cos \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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