Optimal. Leaf size=42 \[ -\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}-\frac {\sin (x) \tanh ^{-1}(\cos (x))}{2 a \sqrt {a \sin ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3204, 3207, 3770} \[ -\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}-\frac {\sin (x) \tanh ^{-1}(\cos (x))}{2 a \sqrt {a \sin ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3204
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx &=-\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}+\frac {\int \frac {1}{\sqrt {a \sin ^2(x)}} \, dx}{2 a}\\ &=-\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}+\frac {\sin (x) \int \csc (x) \, dx}{2 a \sqrt {a \sin ^2(x)}}\\ &=-\frac {\cot (x)}{2 a \sqrt {a \sin ^2(x)}}-\frac {\tanh ^{-1}(\cos (x)) \sin (x)}{2 a \sqrt {a \sin ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 55, normalized size = 1.31 \[ -\frac {\sin ^3(x) \left (\csc ^2\left (\frac {x}{2}\right )-\sec ^2\left (\frac {x}{2}\right )-4 \log \left (\sin \left (\frac {x}{2}\right )\right )+4 \log \left (\cos \left (\frac {x}{2}\right )\right )\right )}{8 \left (a \sin ^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 58, normalized size = 1.38 \[ \frac {\sqrt {-a \cos \relax (x)^{2} + a} {\left ({\left (\cos \relax (x)^{2} - 1\right )} \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right ) + 2 \, \cos \relax (x)\right )}}{4 \, {\left (a^{2} \cos \relax (x)^{2} - a^{2}\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 61, normalized size = 1.45 \[ \frac {\frac {\tan \left (\frac {1}{2} \, x\right )^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right )} + \frac {2 \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right )} - \frac {2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{2}}}{8 \, a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.56, size = 70, normalized size = 1.67 \[ -\frac {\sqrt {a \left (\cos ^{2}\relax (x )\right )}\, \left (\ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\relax (x )\right )}+2 a}{\sin \relax (x )}\right ) \left (\sin ^{2}\relax (x )\right ) a +\sqrt {a}\, \sqrt {a \left (\cos ^{2}\relax (x )\right )}\right )}{2 a^{\frac {5}{2}} \sin \relax (x ) \cos \relax (x ) \sqrt {a \left (\sin ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.69, size = 314, normalized size = 7.48 \[ -\frac {{\left ({\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right ) + 2 \, {\left (\sin \left (3 \, x\right ) + \sin \relax (x)\right )} \cos \left (4 \, x\right ) - 2 \, {\left (\cos \left (3 \, x\right ) + \cos \relax (x)\right )} \sin \left (4 \, x\right ) - 2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + 4 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \cos \relax (x) \sin \left (2 \, x\right ) - 4 \, \cos \left (2 \, x\right ) \sin \relax (x) + 2 \, \sin \relax (x)\right )} \sqrt {-a}}{2 \, {\left (a^{2} \cos \left (4 \, x\right )^{2} + 4 \, a^{2} \cos \left (2 \, x\right )^{2} + a^{2} \sin \left (4 \, x\right )^{2} - 4 \, a^{2} \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a^{2} \sin \left (2 \, x\right )^{2} - 4 \, a^{2} \cos \left (2 \, x\right ) + a^{2} - 2 \, {\left (2 \, a^{2} \cos \left (2 \, x\right ) - a^{2}\right )} \cos \left (4 \, x\right )\right )}} \]
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 \[ \int \frac {1}{{\left (a\,{\sin \relax (x)}^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \sin ^{2}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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