Optimal. Leaf size=51 \[ \frac{\tan ^9(x)}{9 a^5}+\frac{4 \tan ^7(x)}{7 a^5}+\frac{6 \tan ^5(x)}{5 a^5}+\frac{4 \tan ^3(x)}{3 a^5}+\frac{\tan (x)}{a^5} \]
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Rubi [A] time = 0.0263289, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3175, 3767} \[ \frac{\tan ^9(x)}{9 a^5}+\frac{4 \tan ^7(x)}{7 a^5}+\frac{6 \tan ^5(x)}{5 a^5}+\frac{4 \tan ^3(x)}{3 a^5}+\frac{\tan (x)}{a^5} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rubi steps
\begin{align*} \int \frac{1}{\left (a-a \sin ^2(x)\right )^5} \, dx &=\frac{\int \sec ^{10}(x) \, dx}{a^5}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,-\tan (x)\right )}{a^5}\\ &=\frac{\tan (x)}{a^5}+\frac{4 \tan ^3(x)}{3 a^5}+\frac{6 \tan ^5(x)}{5 a^5}+\frac{4 \tan ^7(x)}{7 a^5}+\frac{\tan ^9(x)}{9 a^5}\\ \end{align*}
Mathematica [A] time = 0.0059237, size = 51, normalized size = 1. \[ \frac{\frac{128 \tan (x)}{315}+\frac{1}{9} \tan (x) \sec ^8(x)+\frac{8}{63} \tan (x) \sec ^6(x)+\frac{16}{105} \tan (x) \sec ^4(x)+\frac{64}{315} \tan (x) \sec ^2(x)}{a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 32, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{ \left ( \tan \left ( x \right ) \right ) ^{9}}{9}}+{\frac{4\, \left ( \tan \left ( x \right ) \right ) ^{7}}{7}}+{\frac{6\, \left ( \tan \left ( x \right ) \right ) ^{5}}{5}}+{\frac{4\, \left ( \tan \left ( x \right ) \right ) ^{3}}{3}}+\tan \left ( x \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.958202, size = 46, normalized size = 0.9 \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^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86624, size = 123, normalized size = 2.41 \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 )} \sin \left (x\right )}{315 \, a^{5} \cos \left (x\right )^{9}} \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 = 1.15653, size = 46, normalized size = 0.9 \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^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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