Optimal. Leaf size=44 \[ \frac{2}{3} \tan (x) \sqrt{a \cos ^3(x)}+\frac{2 F\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{3 \cos ^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.0283456, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 2641} \[ \frac{2}{3} \tan (x) \sqrt{a \cos ^3(x)}+\frac{2 F\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{3 \cos ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{a \cos ^3(x)} \, dx &=\frac{\sqrt{a \cos ^3(x)} \int \cos ^{\frac{3}{2}}(x) \, dx}{\cos ^{\frac{3}{2}}(x)}\\ &=\frac{2}{3} \sqrt{a \cos ^3(x)} \tan (x)+\frac{\sqrt{a \cos ^3(x)} \int \frac{1}{\sqrt{\cos (x)}} \, dx}{3 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{2 \sqrt{a \cos ^3(x)} F\left (\left .\frac{x}{2}\right |2\right )}{3 \cos ^{\frac{3}{2}}(x)}+\frac{2}{3} \sqrt{a \cos ^3(x)} \tan (x)\\ \end{align*}
Mathematica [A] time = 0.0254141, size = 37, normalized size = 0.84 \[ \frac{2 \sqrt{a \cos ^3(x)} \left (F\left (\left .\frac{x}{2}\right |2\right )+\sin (x) \sqrt{\cos (x)}\right )}{3 \cos ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.262, size = 76, normalized size = 1.7 \begin{align*}{\frac{ \left ( -2+2\,\cos \left ( x \right ) \right ) \left ( \cos \left ( x \right ) +1 \right ) ^{2}}{3\, \left ( \cos \left ( x \right ) \right ) ^{2} \left ( \sin \left ( x \right ) \right ) ^{3}} \left ( -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 ) + \left ( \cos \left ( x \right ) \right ) ^{2}-\cos \left ( x \right ) \right ) \sqrt{a \left ( \cos \left ( x \right ) \right ) ^{3}}} \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 \sqrt{a \cos \left (x\right )^{3}}\,{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}}, 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 \sqrt{a \cos \left (x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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