Optimal. Leaf size=36 \[ \frac {\cot (x)}{2 \sqrt {-\sin ^2(x)}}+\frac {\tanh ^{-1}(\cos (x)) \sin (x)}{2 \sqrt {-\sin ^2(x)}} \]
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Rubi [A]
time = 0.02, 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 = {3255, 3283,
3286, 3855} \begin {gather*} \frac {\cot (x)}{2 \sqrt {-\sin ^2(x)}}+\frac {\sin (x) \tanh ^{-1}(\cos (x))}{2 \sqrt {-\sin ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3255
Rule 3283
Rule 3286
Rule 3855
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 = 0.04, size = 53, normalized size = 1.47 \begin {gather*} \frac {\left (\csc ^2\left (\frac {x}{2}\right )+4 \log \left (\cos \left (\frac {x}{2}\right )\right )-4 \log \left (\sin \left (\frac {x}{2}\right )\right )-\sec ^2\left (\frac {x}{2}\right )\right ) \sin (x)}{8 \sqrt {-\sin ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 52, normalized size = 1.44
method | result | size |
default | \(\frac {\sqrt {-\left (\cos ^{2}\left (x \right )\right )}\, \left (\arctan \left (\frac {1}{\sqrt {-\left (\cos ^{2}\left (x \right )\right )}}\right ) \left (\sin ^{2}\left (x \right )\right )-\sqrt {-\left (\cos ^{2}\left (x \right )\right )}\right )}{2 \sin \left (x \right ) \cos \left (x \right ) \sqrt {-\left (\sin ^{2}\left (x \right )\right )}}\) | \(52\) |
risch | \(\frac {i \left ({\mathrm e}^{2 i x}+1\right )}{\left ({\mathrm e}^{2 i x}-1\right ) \sqrt {\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}-\frac {\ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{\sqrt {\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}+\frac {\ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )}{\sqrt {\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}\) | \(95\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 284 vs.
\(2 (28) = 56\).
time = 0.51, size = 284, normalized size = 7.89 \begin {gather*} \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 \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 )}{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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (\cos ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.44, size = 90, normalized size = 2.50 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left ({\cos \left (x\right )}^2-1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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