Integrand size = 10, antiderivative size = 30 \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \cot ^2(x)^{3/2} \tan (x)-\sqrt {\cot ^2(x)} \log (\sin (x)) \tan (x) \]
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Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4206, 3739, 3554, 3556} \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \tan (x) \cot ^2(x)^{3/2}-\tan (x) \sqrt {\cot ^2(x)} \log (\sin (x)) \]
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Rule 3554
Rule 3556
Rule 3739
Rule 4206
Rubi steps \begin{align*} \text {integral}& = \int \cot ^2(x)^{3/2} \, dx \\ & = \left (\sqrt {\cot ^2(x)} \tan (x)\right ) \int \cot ^3(x) \, dx \\ & = -\frac {1}{2} \cot ^2(x)^{3/2} \tan (x)-\left (\sqrt {\cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx \\ & = -\frac {1}{2} \cot ^2(x)^{3/2} \tan (x)-\sqrt {\cot ^2(x)} \log (\sin (x)) \tan (x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \sqrt {\cot ^2(x)} \left (\cot ^2(x)+2 (\log (\cos (x))+\log (\tan (x)))\right ) \tan (x) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.55 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (\cot \left (x \right )\right ) \left (4 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )-4 \ln \left (\frac {2}{\cos \left (x \right )+1}\right )+\cot \left (x \right )^{2}+\csc \left (x \right )^{2}\right ) \sqrt {4}}{8}\) | \(39\) |
risch | \(\frac {\sqrt {-\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left (i {\mathrm e}^{4 i x} \ln \left ({\mathrm e}^{2 i x}-1\right )-2 i {\mathrm e}^{2 i x} \ln \left ({\mathrm e}^{2 i x}-1\right )+{\mathrm e}^{4 i x} x -2 i {\mathrm e}^{2 i x}+i \ln \left ({\mathrm e}^{2 i x}-1\right )-2 \,{\mathrm e}^{2 i x} x +x \right )}{\left ({\mathrm e}^{2 i x}+1\right ) \left ({\mathrm e}^{2 i x}-1\right )}\) | \(110\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=\frac {2 \, {\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) - 1}{2 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \]
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\[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=\int \left (\csc ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=-\frac {1}{2 \, \tan \left (x\right )^{2}} + \frac {1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) - \log \left (\tan \left (x\right )\right ) \]
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Time = 0.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=\frac {1}{8} \, \tan \left (\frac {1}{2} \, x\right )^{2} - \frac {4 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}{8 \, \tan \left (\frac {1}{2} \, x\right )^{2}} - \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + \frac {1}{2} \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) \]
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Timed out. \[ \int \left (-1+\csc ^2(x)\right )^{3/2} \, dx=\int {\left (\frac {1}{{\sin \left (x\right )}^2}-1\right )}^{3/2} \,d x \]
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