Optimal. Leaf size=117 \[ \frac{\sin (x) \cos (x)}{a^2 \sqrt{a \cos ^4(x)}}+\frac{\sin ^2(x) \tan ^7(x)}{9 a^2 \sqrt{a \cos ^4(x)}}+\frac{4 \sin ^2(x) \tan ^5(x)}{7 a^2 \sqrt{a \cos ^4(x)}}+\frac{6 \sin ^2(x) \tan ^3(x)}{5 a^2 \sqrt{a \cos ^4(x)}}+\frac{4 \sin ^2(x) \tan (x)}{3 a^2 \sqrt{a \cos ^4(x)}} \]
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Rubi [A] time = 0.0321276, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 3767} \[ \frac{\sin (x) \cos (x)}{a^2 \sqrt{a \cos ^4(x)}}+\frac{\sin ^2(x) \tan ^7(x)}{9 a^2 \sqrt{a \cos ^4(x)}}+\frac{4 \sin ^2(x) \tan ^5(x)}{7 a^2 \sqrt{a \cos ^4(x)}}+\frac{6 \sin ^2(x) \tan ^3(x)}{5 a^2 \sqrt{a \cos ^4(x)}}+\frac{4 \sin ^2(x) \tan (x)}{3 a^2 \sqrt{a \cos ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3767
Rubi steps
\begin{align*} \int \frac{1}{\left (a \cos ^4(x)\right )^{5/2}} \, dx &=\frac{\cos ^2(x) \int \sec ^{10}(x) \, dx}{a^2 \sqrt{a \cos ^4(x)}}\\ &=-\frac{\cos ^2(x) \operatorname{Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,-\tan (x)\right )}{a^2 \sqrt{a \cos ^4(x)}}\\ &=\frac{\cos (x) \sin (x)}{a^2 \sqrt{a \cos ^4(x)}}+\frac{4 \sin ^2(x) \tan (x)}{3 a^2 \sqrt{a \cos ^4(x)}}+\frac{6 \sin ^2(x) \tan ^3(x)}{5 a^2 \sqrt{a \cos ^4(x)}}+\frac{4 \sin ^2(x) \tan ^5(x)}{7 a^2 \sqrt{a \cos ^4(x)}}+\frac{\sin ^2(x) \tan ^7(x)}{9 a^2 \sqrt{a \cos ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.051605, size = 47, normalized size = 0.4 \[ \frac{(130 \cos (2 x)+46 \cos (4 x)+10 \cos (6 x)+\cos (8 x)+128) \tan (x) \sec ^6(x)}{315 a^2 \sqrt{a \cos ^4(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.165, size = 41, normalized size = 0.4 \begin{align*}{\frac{\sin \left ( x \right ) \left ( 128\, \left ( \cos \left ( x \right ) \right ) ^{8}+64\, \left ( \cos \left ( x \right ) \right ) ^{6}+48\, \left ( \cos \left ( x \right ) \right ) ^{4}+40\, \left ( \cos \left ( x \right ) \right ) ^{2}+35 \right ) \cos \left ( x \right ) }{315} \left ( a \left ( \cos \left ( x \right ) \right ) ^{4} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.88253, size = 46, normalized size = 0.39 \begin{align*} \frac{35 \, \tan \left (x\right )^{9} + 180 \, \tan \left (x\right )^{7} + 378 \, \tan \left (x\right )^{5} + 420 \, \tan \left (x\right )^{3} + 315 \, \tan \left (x\right )}{315 \, a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08485, size = 147, normalized size = 1.26 \begin{align*} \frac{{\left (128 \, \cos \left (x\right )^{8} + 64 \, \cos \left (x\right )^{6} + 48 \, \cos \left (x\right )^{4} + 40 \, \cos \left (x\right )^{2} + 35\right )} \sqrt{a \cos \left (x\right )^{4}} \sin \left (x\right )}{315 \, a^{3} \cos \left (x\right )^{11}} \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 [A] time = 1.30331, size = 46, normalized size = 0.39 \begin{align*} \frac{35 \, \tan \left (x\right )^{9} + 180 \, \tan \left (x\right )^{7} + 378 \, \tan \left (x\right )^{5} + 420 \, \tan \left (x\right )^{3} + 315 \, \tan \left (x\right )}{315 \, a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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