Optimal. Leaf size=20 \[ \frac{x}{2 a^2}+\frac{\sin (x) \cos (x)}{2 a^2} \]
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Rubi [A] time = 0.0434516, antiderivative size = 20, 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, 2635, 8} \[ \frac{x}{2 a^2}+\frac{\sin (x) \cos (x)}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^6(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac{\int \cos ^2(x) \, dx}{a^2}\\ &=\frac{\cos (x) \sin (x)}{2 a^2}+\frac{\int 1 \, dx}{2 a^2}\\ &=\frac{x}{2 a^2}+\frac{\cos (x) \sin (x)}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0026244, size = 18, normalized size = 0.9 \[ \frac{\frac{x}{2}+\frac{1}{4} \sin (2 x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 25, normalized size = 1.3 \begin{align*}{\frac{\tan \left ( x \right ) }{2\,{a}^{2} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}+{\frac{\arctan \left ( \tan \left ( x \right ) \right ) }{2\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48092, size = 34, normalized size = 1.7 \begin{align*} \frac{\tan \left (x\right )}{2 \,{\left (a^{2} \tan \left (x\right )^{2} + a^{2}\right )}} + \frac{x}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82461, size = 39, normalized size = 1.95 \begin{align*} \frac{\cos \left (x\right ) \sin \left (x\right ) + x}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 47.708, size = 178, normalized size = 8.9 \begin{align*} \frac{x \tan ^{4}{\left (\frac{x}{2} \right )}}{2 a^{2} \tan ^{4}{\left (\frac{x}{2} \right )} + 4 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 2 a^{2}} + \frac{2 x \tan ^{2}{\left (\frac{x}{2} \right )}}{2 a^{2} \tan ^{4}{\left (\frac{x}{2} \right )} + 4 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 2 a^{2}} + \frac{x}{2 a^{2} \tan ^{4}{\left (\frac{x}{2} \right )} + 4 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 2 a^{2}} - \frac{2 \tan ^{3}{\left (\frac{x}{2} \right )}}{2 a^{2} \tan ^{4}{\left (\frac{x}{2} \right )} + 4 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 2 a^{2}} + \frac{2 \tan{\left (\frac{x}{2} \right )}}{2 a^{2} \tan ^{4}{\left (\frac{x}{2} \right )} + 4 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 2 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16203, size = 30, normalized size = 1.5 \begin{align*} \frac{x}{2 \, a^{2}} + \frac{\tan \left (x\right )}{2 \,{\left (\tan \left (x\right )^{2} + 1\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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