Optimal. Leaf size=6 \[ \frac {\tan (x)}{a^2} \]
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Rubi [A] time = 0.04, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3175, 3767, 8} \[ \frac {\tan (x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3175
Rule 3767
Rubi steps
\begin {align*} \int \frac {\cos ^2(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac {\int \sec ^2(x) \, dx}{a^2}\\ &=-\frac {\operatorname {Subst}(\int 1 \, dx,x,-\tan (x))}{a^2}\\ &=\frac {\tan (x)}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 6, normalized size = 1.00 \[ \frac {\tan (x)}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 10, normalized size = 1.67 \[ \frac {\sin \relax (x)}{a^{2} \cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 6, normalized size = 1.00 \[ \frac {\tan \relax (x)}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 7, normalized size = 1.17 \[ \frac {\tan \relax (x )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 6, normalized size = 1.00 \[ \frac {\tan \relax (x)}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.78, size = 6, normalized size = 1.00 \[ \frac {\mathrm {tan}\relax (x)}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.64, size = 20, normalized size = 3.33 \[ - \frac {2 \tan {\left (\frac {x}{2} \right )}}{a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} - a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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