Optimal. Leaf size=36 \[ -\frac{1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac{3}{8} \cot (x) \sqrt{\csc ^2(x)}-\frac{3}{8} \sinh ^{-1}(\cot (x)) \]
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Rubi [A] time = 0.0123468, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4122, 195, 215} \[ -\frac{1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac{3}{8} \cot (x) \sqrt{\csc ^2(x)}-\frac{3}{8} \sinh ^{-1}(\cot (x)) \]
Antiderivative was successfully verified.
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Rule 4122
Rule 195
Rule 215
Rubi steps
\begin{align*} \int \csc ^2(x)^{5/2} \, dx &=-\operatorname{Subst}\left (\int \left (1+x^2\right )^{3/2} \, dx,x,\cot (x)\right )\\ &=-\frac{1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac{3}{4} \operatorname{Subst}\left (\int \sqrt{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac{3}{8} \cot (x) \sqrt{\csc ^2(x)}-\frac{1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac{3}{8} \sinh ^{-1}(\cot (x))-\frac{3}{8} \cot (x) \sqrt{\csc ^2(x)}-\frac{1}{4} \cot (x) \csc ^2(x)^{3/2}\\ \end{align*}
Mathematica [A] time = 0.243038, size = 72, normalized size = 2. \[ \frac{1}{64} \sin (x) \sqrt{\csc ^2(x)} \left (-\csc ^4\left (\frac{x}{2}\right )-6 \csc ^2\left (\frac{x}{2}\right )+\sec ^4\left (\frac{x}{2}\right )+6 \sec ^2\left (\frac{x}{2}\right )+24 \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.17, size = 78, normalized size = 2.2 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{16} \left ( 3\, \left ( \cos \left ( x \right ) \right ) ^{4}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +3\, \left ( \cos \left ( x \right ) \right ) ^{3}-6\, \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) -5\,\cos \left ( x \right ) +3\,\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{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.76746, size = 1173, normalized size = 32.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.481315, size = 227, normalized size = 6.31 \begin{align*} \frac{6 \, \cos \left (x\right )^{3} - 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 10 \, \cos \left (x\right )}{16 \,{\left (\cos \left (x\right )^{4} - 2 \, \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23869, size = 135, normalized size = 3.75 \begin{align*} -\frac{{\left (\cos \left (x\right ) - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{8 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{{\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm{sgn}\left (\sin \left (x\right )\right )}{64 \,{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{{\left (\frac{8 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac{18 \,{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )}{\left (\cos \left (x\right ) + 1\right )}^{2}}{64 \,{\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm{sgn}\left (\sin \left (x\right )\right )} + \frac{3 \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )}{16 \, \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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