Optimal. Leaf size=42 \[ \frac{2 \sin (x) \cos (x)}{\sqrt{a \cos ^3(x)}}-\frac{2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{\sqrt{a \cos ^3(x)}} \]
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Rubi [A] time = 0.0234349, antiderivative size = 42, 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, 2636, 2639} \[ \frac{2 \sin (x) \cos (x)}{\sqrt{a \cos ^3(x)}}-\frac{2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{\sqrt{a \cos ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \cos ^3(x)}} \, dx &=\frac{\cos ^{\frac{3}{2}}(x) \int \frac{1}{\cos ^{\frac{3}{2}}(x)} \, dx}{\sqrt{a \cos ^3(x)}}\\ &=\frac{2 \cos (x) \sin (x)}{\sqrt{a \cos ^3(x)}}-\frac{\cos ^{\frac{3}{2}}(x) \int \sqrt{\cos (x)} \, dx}{\sqrt{a \cos ^3(x)}}\\ &=-\frac{2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{\sqrt{a \cos ^3(x)}}+\frac{2 \cos (x) \sin (x)}{\sqrt{a \cos ^3(x)}}\\ \end{align*}
Mathematica [A] time = 0.0210661, size = 31, normalized size = 0.74 \[ \frac{\sin (2 x)-2 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{\sqrt{a \cos ^3(x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.333, size = 191, normalized size = 4.6 \begin{align*} 2\,{\frac{ \left ( \cos \left ( x \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( x \right ) \right ) ^{2}\cos \left ( x \right ) }{\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{3}} \left ( \sin \left ( x \right ) \right ) ^{5}} \left ( i\cos \left ( x \right ) \sin \left ( x \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) -i\cos \left ( x \right ) \sin \left ( x \right ) \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 ) +i\sin \left ( x \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) -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 ) -\cos \left ( x \right ) +1 \right ) } \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 \frac{1}{\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 (\frac{\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 \frac{1}{\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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