Optimal. Leaf size=34 \[ \frac {1}{3} \tan (x) \left (a \cos ^2(x)\right )^{3/2}+\frac {2}{3} a \tan (x) \sqrt {a \cos ^2(x)} \]
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Rubi [A] time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3176, 3203, 3207, 2637} \[ \frac {1}{3} \tan (x) \left (a \cos ^2(x)\right )^{3/2}+\frac {2}{3} a \tan (x) \sqrt {a \cos ^2(x)} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3176
Rule 3203
Rule 3207
Rubi steps
\begin {align*} \int \left (a-a \sin ^2(x)\right )^{3/2} \, dx &=\int \left (a \cos ^2(x)\right )^{3/2} \, dx\\ &=\frac {1}{3} \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{3} (2 a) \int \sqrt {a \cos ^2(x)} \, dx\\ &=\frac {1}{3} \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{3} \left (2 a \sqrt {a \cos ^2(x)} \sec (x)\right ) \int \cos (x) \, dx\\ &=\frac {2}{3} a \sqrt {a \cos ^2(x)} \tan (x)+\frac {1}{3} \left (a \cos ^2(x)\right )^{3/2} \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.76 \[ \frac {1}{12} a (9 \sin (x)+\sin (3 x)) \sec (x) \sqrt {a \cos ^2(x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 26, normalized size = 0.76 \[ \frac {{\left (a \cos \relax (x)^{2} + 2 \, a\right )} \sqrt {a \cos \relax (x)^{2}} \sin \relax (x)}{3 \, \cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 57, normalized size = 1.68 \[ -\frac {2 \, {\left (3 \, a^{\frac {3}{2}} {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right ) - 4 \, a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )\right )}}{3 \, {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.85, size = 24, normalized size = 0.71 \[ \frac {a^{2} \cos \relax (x ) \sin \relax (x ) \left (\cos ^{2}\relax (x )+2\right )}{3 \sqrt {a \left (\cos ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 17, normalized size = 0.50 \[ \frac {1}{12} \, {\left (a \sin \left (3 \, x\right ) + 9 \, a \sin \relax (x)\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (a-a\,{\sin \relax (x)}^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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