Optimal. Leaf size=29 \[ \frac {\tan ^5(x)}{5 a}+\frac {2 \tan ^3(x)}{3 a}+\frac {\tan (x)}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 29, 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} \[ \frac {\tan ^5(x)}{5 a}+\frac {2 \tan ^3(x)}{3 a}+\frac {\tan (x)}{a} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rubi steps
\begin {align*} \int \frac {\sec ^4(x)}{a-a \sin ^2(x)} \, dx &=\frac {\int \sec ^6(x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (x)\right )}{a}\\ &=\frac {\tan (x)}{a}+\frac {2 \tan ^3(x)}{3 a}+\frac {\tan ^5(x)}{5 a}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 31, normalized size = 1.07 \[ \frac {\frac {8 \tan (x)}{15}+\frac {1}{5} \tan (x) \sec ^4(x)+\frac {4}{15} \tan (x) \sec ^2(x)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 25, normalized size = 0.86 \[ \frac {{\left (8 \, \cos \relax (x)^{4} + 4 \, \cos \relax (x)^{2} + 3\right )} \sin \relax (x)}{15 \, a \cos \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 22, normalized size = 0.76 \[ \frac {3 \, \tan \relax (x)^{5} + 10 \, \tan \relax (x)^{3} + 15 \, \tan \relax (x)}{15 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 20, normalized size = 0.69 \[ \frac {\frac {\left (\tan ^{5}\relax (x )\right )}{5}+\frac {2 \left (\tan ^{3}\relax (x )\right )}{3}+\tan \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 22, normalized size = 0.76 \[ \frac {3 \, \tan \relax (x)^{5} + 10 \, \tan \relax (x)^{3} + 15 \, \tan \relax (x)}{15 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.92, size = 21, normalized size = 0.72 \[ \frac {\mathrm {tan}\relax (x)\,\left (3\,{\mathrm {tan}\relax (x)}^4+10\,{\mathrm {tan}\relax (x)}^2+15\right )}{15\,a} \]
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 \[ - \frac {\int \frac {\sec ^{4}{\relax (x )}}{\sin ^{2}{\relax (x )} - 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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