Optimal. Leaf size=36 \[ \frac {\cot (x)}{2 \sqrt {-\sin ^2(x)}}+\frac {\sin (x) \tanh ^{-1}(\cos (x))}{2 \sqrt {-\sin ^2(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3176, 3204, 3207, 3770} \[ \frac {\cot (x)}{2 \sqrt {-\sin ^2(x)}}+\frac {\sin (x) \tanh ^{-1}(\cos (x))}{2 \sqrt {-\sin ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3204
Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{\left (-1+\cos ^2(x)\right )^{3/2}} \, dx &=\int \frac {1}{\left (-\sin ^2(x)\right )^{3/2}} \, dx\\ &=\frac {\cot (x)}{2 \sqrt {-\sin ^2(x)}}-\frac {1}{2} \int \frac {1}{\sqrt {-\sin ^2(x)}} \, dx\\ &=\frac {\cot (x)}{2 \sqrt {-\sin ^2(x)}}-\frac {\sin (x) \int \csc (x) \, dx}{2 \sqrt {-\sin ^2(x)}}\\ &=\frac {\cot (x)}{2 \sqrt {-\sin ^2(x)}}+\frac {\tanh ^{-1}(\cos (x)) \sin (x)}{2 \sqrt {-\sin ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 1.47 \[ \frac {\sin (x) \left (\csc ^2\left (\frac {x}{2}\right )-\sec ^2\left (\frac {x}{2}\right )-4 \log \left (\sin \left (\frac {x}{2}\right )\right )+4 \log \left (\cos \left (\frac {x}{2}\right )\right )\right )}{8 \sqrt {-\sin ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.69, size = 90, normalized size = 2.50 \[ -\frac {i \, \tan \left (\frac {1}{2} \, x\right )^{2}}{8 \, \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{3} - \tan \left (\frac {1}{2} \, x\right )\right )} - \frac {i \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right )}{4 \, \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{3} - \tan \left (\frac {1}{2} \, x\right )\right )} + \frac {2 i \, \tan \left (\frac {1}{2} \, x\right )^{2} + i}{8 \, \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{3} - \tan \left (\frac {1}{2} \, x\right )\right ) \tan \left (\frac {1}{2} \, x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.18, size = 51, normalized size = 1.42 \[ -\frac {\sqrt {-\left (\cos ^{2}\relax (x )\right )}\, \left (-\arctan \left (\frac {1}{\sqrt {-\left (\cos ^{2}\relax (x )\right )}}\right ) \left (\sin ^{2}\relax (x )\right )+\sqrt {-\left (\cos ^{2}\relax (x )\right )}\right )}{2 \sin \relax (x ) \cos \relax (x ) \sqrt {-\left (\sin ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.05, size = 284, normalized size = 7.89 \[ \frac {{\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right ) + 2 \, {\left (\sin \left (3 \, x\right ) + \sin \relax (x)\right )} \cos \left (4 \, x\right ) - 2 \, {\left (\cos \left (3 \, x\right ) + \cos \relax (x)\right )} \sin \left (4 \, x\right ) - 2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + 4 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \cos \relax (x) \sin \left (2 \, x\right ) - 4 \, \cos \left (2 \, x\right ) \sin \relax (x) + 2 \, \sin \relax (x)}{2 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \]
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 \[ \int \frac {1}{{\left ({\cos \relax (x)}^2-1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (\cos ^{2}{\relax (x )} - 1\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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