Optimal. Leaf size=42 \[ \frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\cos (x) \tanh ^{-1}(\sin (x))}{2 a \sqrt {a \cos ^2(x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 42, 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, 3204, 3207, 3770} \[ \frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\cos (x) \tanh ^{-1}(\sin (x))}{2 a \sqrt {a \cos ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3204
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{\left (a-a \sin ^2(x)\right )^{3/2}} \, dx &=\int \frac {1}{\left (a \cos ^2(x)\right )^{3/2}} \, dx\\ &=\frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\int \frac {1}{\sqrt {a \cos ^2(x)}} \, dx}{2 a}\\ &=\frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\cos (x) \int \sec (x) \, dx}{2 a \sqrt {a \cos ^2(x)}}\\ &=\frac {\tanh ^{-1}(\sin (x)) \cos (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}\\ \end {align*}
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Mathematica [B] time = 0.07, size = 91, normalized size = 2.17 \[ -\frac {\cos (x) \left (-2 \sin (x)+\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\cos (2 x) \left (\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )}{4 \left (a \cos ^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 40, normalized size = 0.95 \[ -\frac {\sqrt {a \cos \relax (x)^{2}} {\left (\cos \relax (x)^{2} \log \left (-\frac {\sin \relax (x) - 1}{\sin \relax (x) + 1}\right ) - 2 \, \sin \relax (x)\right )}}{4 \, a^{2} \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 44, normalized size = 1.05 \[ -\frac {\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )}{{\left ({\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{2} - 4\right )} a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.34, size = 70, normalized size = 1.67 \[ \frac {\sqrt {a \left (\sin ^{2}\relax (x )\right )}\, \left (\ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\relax (x )\right )}+2 a}{\cos \relax (x )}\right ) \left (\cos ^{2}\relax (x )\right ) a +\sqrt {a}\, \sqrt {a \left (\sin ^{2}\relax (x )\right )}\right )}{2 a^{\frac {5}{2}} \cos \relax (x ) \sin \relax (x ) \sqrt {a \left (\cos ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 304, normalized size = 7.24 \[ \frac {4 \, {\left (\sin \left (3 \, x\right ) - \sin \relax (x)\right )} \cos \left (4 \, x\right ) + {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) - 4 \, {\left (\cos \left (3 \, x\right ) - \cos \relax (x)\right )} \sin \left (4 \, x\right ) + 4 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (3 \, x\right ) - 8 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 8 \, \cos \relax (x) \sin \left (2 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \sin \relax (x) - 4 \, \sin \relax (x)}{4 \, {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} + 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} + 2 \, {\left (2 \, a \cos \left (2 \, x\right ) + a\right )} \cos \left (4 \, x\right ) + 4 \, a \cos \left (2 \, x\right ) + a\right )} \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a-a\,{\sin \relax (x)}^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- a \sin ^{2}{\relax (x )} + a\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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