Optimal. Leaf size=6 \[ \frac{\tan (x)}{a^2} \]
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Rubi [A] time = 0.0430509, antiderivative size = 6, normalized size of antiderivative = 1., 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 3175
Rule 3767
Rule 8
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*}
Mathematica [A] time = 0.0022568, size = 6, normalized size = 1. \[ \frac{\tan (x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 7, normalized size = 1.2 \begin{align*}{\frac{\tan \left ( x \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989701, size = 8, normalized size = 1.33 \begin{align*} \frac{\tan \left (x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85202, size = 28, normalized size = 4.67 \begin{align*} \frac{\sin \left (x\right )}{a^{2} \cos \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.59203, size = 20, normalized size = 3.33 \begin{align*} - \frac{2 \tan{\left (\frac{x}{2} \right )}}{a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} - a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0964, size = 8, normalized size = 1.33 \begin{align*} \frac{\tan \left (x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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