Optimal. Leaf size=38 \[ -\frac{\sin ^7(x)}{7 a}+\frac{3 \sin ^5(x)}{5 a}-\frac{\sin ^3(x)}{a}+\frac{\sin (x)}{a} \]
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Rubi [A] time = 0.0538224, antiderivative size = 38, 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} \[ -\frac{\sin ^7(x)}{7 a}+\frac{3 \sin ^5(x)}{5 a}-\frac{\sin ^3(x)}{a}+\frac{\sin (x)}{a} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^9(x)}{a-a \sin ^2(x)} \, dx &=\frac{\int \cos ^7(x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (x)\right )}{a}\\ &=\frac{\sin (x)}{a}-\frac{\sin ^3(x)}{a}+\frac{3 \sin ^5(x)}{5 a}-\frac{\sin ^7(x)}{7 a}\\ \end{align*}
Mathematica [A] time = 0.0044287, size = 35, normalized size = 0.92 \[ \frac{\frac{35 \sin (x)}{64}+\frac{7}{64} \sin (3 x)+\frac{7}{320} \sin (5 x)+\frac{1}{448} \sin (7 x)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 26, normalized size = 0.7 \begin{align*}{\frac{1}{a} \left ( -{\frac{ \left ( \sin \left ( x \right ) \right ) ^{7}}{7}}+{\frac{3\, \left ( \sin \left ( x \right ) \right ) ^{5}}{5}}- \left ( \sin \left ( x \right ) \right ) ^{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 = 0.990582, size = 38, normalized size = 1. \begin{align*} -\frac{5 \, \sin \left (x\right )^{7} - 21 \, \sin \left (x\right )^{5} + 35 \, \sin \left (x\right )^{3} - 35 \, \sin \left (x\right )}{35 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.45918, size = 80, normalized size = 2.11 \begin{align*} \frac{{\left (5 \, \cos \left (x\right )^{6} + 6 \, \cos \left (x\right )^{4} + 8 \, \cos \left (x\right )^{2} + 16\right )} \sin \left (x\right )}{35 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 83.0451, size = 580, normalized size = 15.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10332, size = 38, normalized size = 1. \begin{align*} -\frac{5 \, \sin \left (x\right )^{7} - 21 \, \sin \left (x\right )^{5} + 35 \, \sin \left (x\right )^{3} - 35 \, \sin \left (x\right )}{35 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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