Optimal. Leaf size=67 \[ \frac{2}{9} a \sin (x) \cos ^2(x) \sqrt{a \cos ^3(x)}+\frac{14}{45} a \sin (x) \sqrt{a \cos ^3(x)}+\frac{14 a E\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{15 \cos ^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.0408263, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 2639} \[ \frac{2}{9} a \sin (x) \cos ^2(x) \sqrt{a \cos ^3(x)}+\frac{14}{45} a \sin (x) \sqrt{a \cos ^3(x)}+\frac{14 a E\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{15 \cos ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 2639
Rubi steps
\begin{align*} \int \left (a \cos ^3(x)\right )^{3/2} \, dx &=\frac{\left (a \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{9}{2}}(x) \, dx}{\cos ^{\frac{3}{2}}(x)}\\ &=\frac{2}{9} a \cos ^2(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{\left (7 a \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{5}{2}}(x) \, dx}{9 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{14}{45} a \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{9} a \cos ^2(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{\left (7 a \sqrt{a \cos ^3(x)}\right ) \int \sqrt{\cos (x)} \, dx}{15 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{14 a \sqrt{a \cos ^3(x)} E\left (\left .\frac{x}{2}\right |2\right )}{15 \cos ^{\frac{3}{2}}(x)}+\frac{14}{45} a \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{9} a \cos ^2(x) \sqrt{a \cos ^3(x)} \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0645819, size = 50, normalized size = 0.75 \[ \frac{\left (a \cos ^3(x)\right )^{3/2} \left (168 E\left (\left .\frac{x}{2}\right |2\right )+(38 \sin (2 x)+5 \sin (4 x)) \sqrt{\cos (x)}\right )}{180 \cos ^{\frac{9}{2}}(x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.254, size = 198, normalized size = 3. \begin{align*} -{\frac{2}{45\, \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) } \left ( 5\, \left ( \cos \left ( x \right ) \right ) ^{6}+21\,i\cos \left ( x \right ) \sin \left ( x \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}-21\,i\cos \left ( x \right ) \sin \left ( x \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}+21\,i\sin \left ( x \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}-21\,i\sin \left ( x \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}+2\, \left ( \cos \left ( x \right ) \right ) ^{4}+14\, \left ( \cos \left ( x \right ) \right ) ^{2}-21\,\cos \left ( x \right ) \right ) \left ( a \left ( \cos \left ( x \right ) \right ) ^{3} \right ) ^{{\frac{3}{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{3}{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 \cos \left (x\right )^{3}, 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{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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