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 = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, 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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fricas [A] time = 0.41, size = 31, normalized size = 0.84 \[ \frac {{\left (16 \, \cos \relax (x)^{6} + 8 \, \cos \relax (x)^{4} + 6 \, \cos \relax (x)^{2} + 5\right )} \sin \relax (x)}{35 \, a^{2} \cos \relax (x)^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 28, normalized size = 0.76 \[ \frac {5 \, \tan \relax (x)^{7} + 21 \, \tan \relax (x)^{5} + 35 \, \tan \relax (x)^{3} + 35 \, \tan \relax (x)}{35 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 24, normalized size = 0.65 \[ \frac {\frac {\left (\tan ^{7}\relax (x )\right )}{7}+\frac {3 \left (\tan ^{5}\relax (x )\right )}{5}+\tan ^{3}\relax (x )+\tan \relax (x )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 28, normalized size = 0.76 \[ \frac {5 \, \tan \relax (x)^{7} + 21 \, \tan \relax (x)^{5} + 35 \, \tan \relax (x)^{3} + 35 \, \tan \relax (x)}{35 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.84, size = 33, normalized size = 0.89 \[ \frac {\mathrm {tan}\relax (x)}{a^2}+\frac {{\mathrm {tan}\relax (x)}^3}{a^2}+\frac {3\,{\mathrm {tan}\relax (x)}^5}{5\,a^2}+\frac {{\mathrm {tan}\relax (x)}^7}{7\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec ^{4}{\relax (x )}}{\sin ^{4}{\relax (x )} - 2 \sin ^{2}{\relax (x )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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