Optimal. Leaf size=37 \[ \frac{\tan ^7(x)}{7 a^2}+\frac{3 \tan ^5(x)}{5 a^2}+\frac{\tan ^3(x)}{a^2}+\frac{\tan (x)}{a^2} \]
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Rubi [A] time = 0.0481523, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 3767} \[ \frac{\tan ^7(x)}{7 a^2}+\frac{3 \tan ^5(x)}{5 a^2}+\frac{\tan ^3(x)}{a^2}+\frac{\tan (x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rubi steps
\begin{align*} \int \frac{\sec ^4(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac{\int \sec ^8(x) \, dx}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (x)\right )}{a^2}\\ &=\frac{\tan (x)}{a^2}+\frac{\tan ^3(x)}{a^2}+\frac{3 \tan ^5(x)}{5 a^2}+\frac{\tan ^7(x)}{7 a^2}\\ \end{align*}
Mathematica [A] time = 0.0044555, size = 41, normalized size = 1.11 \[ \frac{\frac{16 \tan (x)}{35}+\frac{1}{7} \tan (x) \sec ^6(x)+\frac{6}{35} \tan (x) \sec ^4(x)+\frac{8}{35} \tan (x) \sec ^2(x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 24, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{ \left ( \tan \left ( x \right ) \right ) ^{7}}{7}}+{\frac{3\, \left ( \tan \left ( x \right ) \right ) ^{5}}{5}}+ \left ( \tan \left ( x \right ) \right ) ^{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 = 0.980879, size = 38, normalized size = 1.03 \begin{align*} \frac{5 \, \tan \left (x\right )^{7} + 21 \, \tan \left (x\right )^{5} + 35 \, \tan \left (x\right )^{3} + 35 \, \tan \left (x\right )}{35 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87242, size = 97, normalized size = 2.62 \begin{align*} \frac{{\left (16 \, \cos \left (x\right )^{6} + 8 \, \cos \left (x\right )^{4} + 6 \, \cos \left (x\right )^{2} + 5\right )} \sin \left (x\right )}{35 \, a^{2} \cos \left (x\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sec ^{4}{\left (x \right )}}{\sin ^{4}{\left (x \right )} - 2 \sin ^{2}{\left (x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1291, size = 38, normalized size = 1.03 \begin{align*} \frac{5 \, \tan \left (x\right )^{7} + 21 \, \tan \left (x\right )^{5} + 35 \, \tan \left (x\right )^{3} + 35 \, \tan \left (x\right )}{35 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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