Optimal. Leaf size=22 \[ -\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{2} \sinh ^{-1}(\cot (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4122, 195, 215} \[ -\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{2} \sinh ^{-1}(\cot (x)) \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 4122
Rubi steps
\begin {align*} \int \csc ^2(x)^{3/2} \, dx &=-\operatorname {Subst}\left (\int \sqrt {1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac {1}{2} \sinh ^{-1}(\cot (x))-\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)}\\ \end {align*}
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Mathematica [B] time = 0.09, size = 51, normalized size = 2.32 \[ \frac {1}{8} \sin (x) \sqrt {\csc ^2(x)} \left (-\csc ^2\left (\frac {x}{2}\right )+\sec ^2\left (\frac {x}{2}\right )+4 \log \left (\sin \left (\frac {x}{2}\right )\right )-4 \log \left (\cos \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 44, normalized size = 2.00 \[ -\frac {{\left (\cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - {\left (\cos \relax (x)^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 2 \, \cos \relax (x)}{4 \, {\left (\cos \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.49, size = 69, normalized size = 3.14 \[ -\frac {{\left (\frac {2 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} - 1\right )} {\left (\cos \relax (x) + 1\right )}}{8 \, {\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )} + \frac {\log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right )}{4 \, \mathrm {sgn}\left (\sin \relax (x)\right )} - \frac {\cos \relax (x) - 1}{8 \, {\left (\cos \relax (x) + 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 52, normalized size = 2.36 \[ -\frac {\left (\left (\cos ^{2}\relax (x )\right ) \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )-\ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )+\cos \relax (x )\right ) \sin \relax (x ) \left (-\frac {1}{-1+\cos ^{2}\relax (x )}\right )^{\frac {3}{2}} \sqrt {4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 300, normalized size = 13.64 \[ -\frac {4 \, {\left (\cos \left (3 \, x\right ) + \cos \relax (x)\right )} \cos \left (4 \, x\right ) - 4 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (3 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \cos \relax (x) + {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) + 4 \, {\left (\sin \left (3 \, x\right ) + \sin \relax (x)\right )} \sin \left (4 \, x\right ) - 8 \, \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, \sin \left (2 \, x\right ) \sin \relax (x) + 4 \, \cos \relax (x)}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int {\left (\frac {1}{{\sin \relax (x)}^2}\right )}^{3/2} \,d x \]
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 \[ \int \left (\csc ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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