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 = 0.0242362, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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*}
Mathematica [A] time = 0.058478, 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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Maple [B] time = 1.116, size = 70, normalized size = 1.7 \begin{align*} -{\frac{1}{2\,\sin \left ( x \right ) \cos \left ( x \right ) }\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( \ln \left ( 2\,{\frac{\sqrt{a}\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}+a}{\sin \left ( x \right ) }} \right ) \left ( \sin \left ( x \right ) \right ) ^{2}a+\sqrt{a}\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}} \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.64417, size = 424, normalized size = 10.1 \begin{align*} -\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 \left (x\right ), \cos \left (x\right ) + 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 \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \,{\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \,{\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\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 \left (x\right ) \sin \left (2 \, x\right ) - 4 \, \cos \left (2 \, x\right ) \sin \left (x\right ) + 2 \, \sin \left (x\right )\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6649, size = 158, normalized size = 3.76 \begin{align*} \frac{\sqrt{-a \cos \left (x\right )^{2} + a}{\left ({\left (\cos \left (x\right )^{2} - 1\right )} \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) + 2 \, \cos \left (x\right )\right )}}{4 \,{\left (a^{2} \cos \left (x\right )^{2} - a^{2}\right )} \sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sin ^{2}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27683, size = 82, normalized size = 1.95 \begin{align*} \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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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