Optimal. Leaf size=18 \[ \frac{\sin (x)}{a^2}-\frac{\sin ^3(x)}{3 a^2} \]
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Rubi [A] time = 0.044215, antiderivative size = 18, normalized size of antiderivative = 1., 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, 2633} \[ \frac{\sin (x)}{a^2}-\frac{\sin ^3(x)}{3 a^2} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^7(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac{\int \cos ^3(x) \, dx}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (x)\right )}{a^2}\\ &=\frac{\sin (x)}{a^2}-\frac{\sin ^3(x)}{3 a^2}\\ \end{align*}
Mathematica [A] time = 0.0027456, size = 19, normalized size = 1.06 \[ \frac{\frac{3 \sin (x)}{4}+\frac{1}{12} \sin (3 x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 14, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{3}}+\sin \left ( x \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985575, size = 19, normalized size = 1.06 \begin{align*} -\frac{\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )}{3 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82188, size = 42, normalized size = 2.33 \begin{align*} \frac{{\left (\cos \left (x\right )^{2} + 2\right )} \sin \left (x\right )}{3 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 74.1649, size = 144, normalized size = 8. \begin{align*} \frac{6 \tan ^{5}{\left (\frac{x}{2} \right )}}{3 a^{2} \tan ^{6}{\left (\frac{x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac{x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a^{2}} + \frac{4 \tan ^{3}{\left (\frac{x}{2} \right )}}{3 a^{2} \tan ^{6}{\left (\frac{x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac{x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a^{2}} + \frac{6 \tan{\left (\frac{x}{2} \right )}}{3 a^{2} \tan ^{6}{\left (\frac{x}{2} \right )} + 9 a^{2} \tan ^{4}{\left (\frac{x}{2} \right )} + 9 a^{2} \tan ^{2}{\left (\frac{x}{2} \right )} + 3 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09902, size = 19, normalized size = 1.06 \begin{align*} -\frac{\sin \left (x\right )^{3} - 3 \, \sin \left (x\right )}{3 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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