Optimal. Leaf size=117 \[ \frac{154 \sin (x) \cos (x)}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{154 \tan (x)}{585 a^2 \sqrt{a \cos ^3(x)}}-\frac{154 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \tan (x) \sec ^4(x)}{13 a^2 \sqrt{a \cos ^3(x)}}+\frac{22 \tan (x) \sec ^2(x)}{117 a^2 \sqrt{a \cos ^3(x)}} \]
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Rubi [A] time = 0.0509484, 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, 2636, 2639} \[ \frac{154 \sin (x) \cos (x)}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{154 \tan (x)}{585 a^2 \sqrt{a \cos ^3(x)}}-\frac{154 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \tan (x) \sec ^4(x)}{13 a^2 \sqrt{a \cos ^3(x)}}+\frac{22 \tan (x) \sec ^2(x)}{117 a^2 \sqrt{a \cos ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\left (a \cos ^3(x)\right )^{5/2}} \, dx &=\frac{\cos ^{\frac{3}{2}}(x) \int \frac{1}{\cos ^{\frac{15}{2}}(x)} \, dx}{a^2 \sqrt{a \cos ^3(x)}}\\ &=\frac{2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt{a \cos ^3(x)}}+\frac{\left (11 \cos ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cos ^{\frac{11}{2}}(x)} \, dx}{13 a^2 \sqrt{a \cos ^3(x)}}\\ &=\frac{22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt{a \cos ^3(x)}}+\frac{\left (77 \cos ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cos ^{\frac{7}{2}}(x)} \, dx}{117 a^2 \sqrt{a \cos ^3(x)}}\\ &=\frac{154 \tan (x)}{585 a^2 \sqrt{a \cos ^3(x)}}+\frac{22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt{a \cos ^3(x)}}+\frac{\left (77 \cos ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(x)} \, dx}{195 a^2 \sqrt{a \cos ^3(x)}}\\ &=\frac{154 \cos (x) \sin (x)}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{154 \tan (x)}{585 a^2 \sqrt{a \cos ^3(x)}}+\frac{22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt{a \cos ^3(x)}}-\frac{\left (77 \cos ^{\frac{3}{2}}(x)\right ) \int \sqrt{\cos (x)} \, dx}{195 a^2 \sqrt{a \cos ^3(x)}}\\ &=-\frac{154 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{154 \cos (x) \sin (x)}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{154 \tan (x)}{585 a^2 \sqrt{a \cos ^3(x)}}+\frac{22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt{a \cos ^3(x)}}\\ \end{align*}
Mathematica [A] time = 0.104767, size = 57, normalized size = 0.49 \[ \frac{-462 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )+462 \sin (x) \cos (x)+2 \tan (x) \left (45 \sec ^4(x)+55 \sec ^2(x)+77\right )}{585 a^2 \sqrt{a \cos ^3(x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.423, size = 223, normalized size = 1.9 \begin{align*} -{\frac{2\, \left ( \cos \left ( x \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( x \right ) \right ) ^{2}\cos \left ( x \right ) }{585\, \left ( \sin \left ( x \right ) \right ) ^{5}} \left ( 231\,i \left ( \cos \left ( x \right ) \right ) ^{7}\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 ) -231\,i \left ( \cos \left ( x \right ) \right ) ^{7}\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 ) +231\,i \left ( \cos \left ( x \right ) \right ) ^{6}\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 ) -231\,i \left ( \cos \left ( x \right ) \right ) ^{6}\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 ) +231\, \left ( \cos \left ( x \right ) \right ) ^{7}-154\, \left ( \cos \left ( x \right ) \right ) ^{6}-22\, \left ( \cos \left ( x \right ) \right ) ^{4}-10\, \left ( \cos \left ( x \right ) \right ) ^{2}-45 \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 \frac{1}{\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 (\frac{\sqrt{a \cos \left (x\right )^{3}}}{a^{3} \cos \left (x\right )^{9}}, 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}{\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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