Optimal. Leaf size=19 \[ \frac {\cos ^3(x)}{3 a}-\frac {\cos (x)}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 2633
Rule 3175
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*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.00 \[ \frac {\frac {1}{12} \cos (3 x)-\frac {3 \cos (x)}{4}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 14, normalized size = 0.74 \[ \frac {\cos \relax (x)^{3} - 3 \, \cos \relax (x)}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 14, normalized size = 0.74 \[ \frac {\cos \relax (x)^{3} - 3 \, \cos \relax (x)}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 16, normalized size = 0.84 \[ \frac {\frac {\left (\cos ^{3}\relax (x )\right )}{3}-\cos \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 14, normalized size = 0.74 \[ \frac {\cos \relax (x)^{3} - 3 \, \cos \relax (x)}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 16, normalized size = 0.84 \[ -\frac {3\,\cos \relax (x)-{\cos \relax (x)}^3}{3\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.74, size = 78, normalized size = 4.11 \[ - \frac {12 \tan ^{2}{\left (\frac {x}{2} \right )}}{3 a \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a} - \frac {4}{3 a \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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