Optimal. Leaf size=18 \[ \frac {\sin (x)}{a}-\frac {\sin ^3(x)}{3 a} \]
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Rubi [A] time = 0.05, antiderivative size = 18, 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} \[ \frac {\sin (x)}{a}-\frac {\sin ^3(x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 3175
Rubi steps
\begin {align*} \int \frac {\cos ^5(x)}{a-a \sin ^2(x)} \, dx &=\frac {\int \cos ^3(x) \, dx}{a}\\ &=-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (x)\right )}{a}\\ &=\frac {\sin (x)}{a}-\frac {\sin ^3(x)}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.06 \[ \frac {\frac {3 \sin (x)}{4}+\frac {1}{12} \sin (3 x)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 13, normalized size = 0.72 \[ \frac {{\left (\cos \relax (x)^{2} + 2\right )} \sin \relax (x)}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 14, normalized size = 0.78 \[ -\frac {\sin \relax (x)^{3} - 3 \, \sin \relax (x)}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 14, normalized size = 0.78 \[ \frac {-\frac {\left (\sin ^{3}\relax (x )\right )}{3}+\sin \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 14, normalized size = 0.78 \[ -\frac {\sin \relax (x)^{3} - 3 \, \sin \relax (x)}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 16, normalized size = 0.89 \[ \frac {3\,\sin \relax (x)-{\sin \relax (x)}^3}{3\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.95, size = 124, normalized size = 6.89 \[ \frac {6 \tan ^{5}{\left (\frac {x}{2} \right )}}{3 a \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a} + \frac {4 \tan ^{3}{\left (\frac {x}{2} \right )}}{3 a \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a} + \frac {6 \tan {\left (\frac {x}{2} \right )}}{3 a \tan ^{6}{\left (\frac {x}{2} \right )} + 9 a \tan ^{4}{\left (\frac {x}{2} \right )} + 9 a \tan ^{2}{\left (\frac {x}{2} \right )} + 3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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