Optimal. Leaf size=19 \[ \frac{\cos ^3(x)}{3 a}-\frac{\cos (x)}{a} \]
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Rubi [A] time = 0.0451872, antiderivative size = 19, 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} \[ \frac{\cos ^3(x)}{3 a}-\frac{\cos (x)}{a} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2633
Rubi steps
\begin{align*} \int \frac{\sin ^5(x)}{a-a \cos ^2(x)} \, dx &=\frac{\int \sin ^3(x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )}{a}\\ &=-\frac{\cos (x)}{a}+\frac{\cos ^3(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0031018, size = 19, normalized size = 1. \[ \frac{\frac{1}{12} \cos (3 x)-\frac{3 \cos (x)}{4}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 16, normalized size = 0.8 \begin{align*}{\frac{1}{a} \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{3}}-\cos \left ( x \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946388, size = 19, normalized size = 1. \begin{align*} \frac{\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73446, size = 39, normalized size = 2.05 \begin{align*} \frac{\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 18.3349, size = 162, normalized size = 8.53 \begin{align*} \frac{14 \tan ^{6}{\left (\frac{x}{2} \right )}}{15 a \tan ^{6}{\left (\frac{x}{2} \right )} + 45 a \tan ^{4}{\left (\frac{x}{2} \right )} + 45 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} + \frac{42 \tan ^{4}{\left (\frac{x}{2} \right )}}{15 a \tan ^{6}{\left (\frac{x}{2} \right )} + 45 a \tan ^{4}{\left (\frac{x}{2} \right )} + 45 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} - \frac{18 \tan ^{2}{\left (\frac{x}{2} \right )}}{15 a \tan ^{6}{\left (\frac{x}{2} \right )} + 45 a \tan ^{4}{\left (\frac{x}{2} \right )} + 45 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} - \frac{6}{15 a \tan ^{6}{\left (\frac{x}{2} \right )} + 45 a \tan ^{4}{\left (\frac{x}{2} \right )} + 45 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11841, size = 19, normalized size = 1. \begin{align*} \frac{\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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