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 = 0.0094297, antiderivative size = 22, normalized size of antiderivative = 1., 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 4122
Rule 195
Rule 215
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*}
Mathematica [B] time = 0.0874973, 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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Maple [B] time = 0.086, size = 52, normalized size = 2.4 \begin{align*} -{\frac{\sqrt{4}\sin \left ( x \right ) }{4} \left ( \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +\cos \left ( x \right ) -\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \right ) \left ( - \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-1} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.67446, size = 405, normalized size = 18.41 \begin{align*} -\frac{4 \,{\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\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 \left (x\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 \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \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 )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + 4 \,{\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \sin \left (4 \, x\right ) - 8 \, \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \cos \left (x\right )}{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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.472836, size = 150, normalized size = 6.82 \begin{align*} -\frac{{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2 \, \cos \left (x\right )}{4 \,{\left (\cos \left (x\right )^{2} - 1\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 \left (\csc ^{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 [B] time = 1.30788, size = 93, normalized size = 4.23 \begin{align*} -\frac{{\left (\frac{2 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 1\right )}{\left (\cos \left (x\right ) + 1\right )}}{8 \,{\left (\cos \left (x\right ) - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )} + \frac{\log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )}{4 \, \mathrm{sgn}\left (\sin \left (x\right )\right )} - \frac{\cos \left (x\right ) - 1}{8 \,{\left (\cos \left (x\right ) + 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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