Optimal. Leaf size=29 \[ \frac{\sin ^5(x)}{5 a^2}-\frac{2 \sin ^3(x)}{3 a^2}+\frac{\sin (x)}{a^2} \]
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Rubi [A] time = 0.0461488, antiderivative size = 29, 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} \[ \frac{\sin ^5(x)}{5 a^2}-\frac{2 \sin ^3(x)}{3 a^2}+\frac{\sin (x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^9(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac{\int \cos ^5(x) \, dx}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (x)\right )}{a^2}\\ &=\frac{\sin (x)}{a^2}-\frac{2 \sin ^3(x)}{3 a^2}+\frac{\sin ^5(x)}{5 a^2}\\ \end{align*}
Mathematica [A] time = 0.0037578, size = 27, normalized size = 0.93 \[ \frac{\frac{5 \sin (x)}{8}+\frac{5}{48} \sin (3 x)+\frac{1}{80} \sin (5 x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 20, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{ \left ( \sin \left ( x \right ) \right ) ^{5}}{5}}-{\frac{2\, \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 = 0.961846, size = 30, normalized size = 1.03 \begin{align*} \frac{3 \, \sin \left (x\right )^{5} - 10 \, \sin \left (x\right )^{3} + 15 \, \sin \left (x\right )}{15 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83329, size = 63, normalized size = 2.17 \begin{align*} \frac{{\left (3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right )}{15 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11176, size = 30, normalized size = 1.03 \begin{align*} \frac{3 \, \sin \left (x\right )^{5} - 10 \, \sin \left (x\right )^{3} + 15 \, \sin \left (x\right )}{15 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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