Integrand size = 12, antiderivative size = 54 \[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\frac {5}{2} \arcsin (\cot (x))-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )+\frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)} \]
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Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 427, 537, 222, 385, 209} \[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\frac {5}{2} \arcsin (\cot (x))-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )+\frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)} \]
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Rule 209
Rule 222
Rule 385
Rule 427
Rule 537
Rule 3742
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2}}{1+x^2} \, dx,x,\cot (x)\right ) \\ & = \frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {3-5 x^2}{\sqrt {1-x^2} \left (1+x^2\right )} \, dx,x,\cot (x)\right ) \\ & = \frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)}+\frac {5}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\cot (x)\right )-4 \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (1+x^2\right )} \, dx,x,\cot (x)\right ) \\ & = \frac {5}{2} \arcsin (\cot (x))+\frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)}-4 \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {\cot (x)}{\sqrt {1-\cot ^2(x)}}\right ) \\ & = \frac {5}{2} \arcsin (\cot (x))-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} \cot (x)}{\sqrt {1-\cot ^2(x)}}\right )+\frac {1}{2} \cot (x) \sqrt {1-\cot ^2(x)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(123\) vs. \(2(54)=108\).
Time = 0.51 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.28 \[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\frac {1}{2} \left (1-\cot ^2(x)\right )^{3/2} \sec ^2(2 x) \left (\arctan \left (\frac {\cos (x)}{\sqrt {-\cos (2 x)}}\right ) \sqrt {-\cos (2 x)} \sin ^3(x)+4 \text {arctanh}\left (\frac {\cos (x)}{\sqrt {\cos (2 x)}}\right ) \sqrt {\cos (2 x)} \sin ^3(x)-4 \sqrt {2} \sqrt {\cos (2 x)} \log \left (\sqrt {2} \cos (x)+\sqrt {\cos (2 x)}\right ) \sin ^3(x)-\frac {1}{4} \sin (4 x)\right ) \]
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Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {5 \arcsin \left (\cot \left (x \right )\right )}{2}+\frac {\cot \left (x \right ) \sqrt {1-\cot \left (x \right )^{2}}}{2}+2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {1-\cot \left (x \right )^{2}}\, \cot \left (x \right )}{-1+\cot \left (x \right )^{2}}\right )\) | \(51\) |
default | \(\frac {5 \arcsin \left (\cot \left (x \right )\right )}{2}+\frac {\cot \left (x \right ) \sqrt {1-\cot \left (x \right )^{2}}}{2}+2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {1-\cot \left (x \right )^{2}}\, \cot \left (x \right )}{-1+\cot \left (x \right )^{2}}\right )\) | \(51\) |
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (42) = 84\).
Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.04 \[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\frac {4 \, \sqrt {2} \arctan \left (\frac {\sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}\right ) \sin \left (2 \, x\right ) + \sqrt {2} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) + 1\right )} - 5 \, \arctan \left (\frac {\sqrt {2} \sqrt {\frac {\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{\cos \left (2 \, x\right ) + 1}\right ) \sin \left (2 \, x\right )}{2 \, \sin \left (2 \, x\right )} \]
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\[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\int \left (1 - \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\int { {\left (-\cot \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (42) = 84\).
Time = 0.33 (sec) , antiderivative size = 257, normalized size of antiderivative = 4.76 \[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\frac {1}{4} \, {\left (5 \, \pi \mathrm {sgn}\left (\cos \left (x\right )\right ) - 4 \, \sqrt {2} {\left (\pi \mathrm {sgn}\left (\cos \left (x\right )\right ) + 2 \, \arctan \left (-\frac {{\left (\frac {{\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}^{2}}{\cos \left (x\right )^{2}} - 4\right )} \cos \left (x\right )}{4 \, {\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}}\right )\right )} + \frac {4 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}{\cos \left (x\right )} - \frac {4 \, \cos \left (x\right )}{\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}\right )}}{{\left (\frac {\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}{\cos \left (x\right )} - \frac {4 \, \cos \left (x\right )}{\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}}\right )}^{2} + 8} + 10 \, \arctan \left (-\frac {\sqrt {2} {\left (\frac {{\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}^{2}}{\cos \left (x\right )^{2}} - 4\right )} \cos \left (x\right )}{4 \, {\left (\sqrt {2} \sqrt {-2 \, \cos \left (x\right )^{2} + 1} - \sqrt {2}\right )}}\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \]
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Time = 14.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.93 \[ \int \left (1-\cot ^2(x)\right )^{3/2} \, dx=\frac {5\,\mathrm {asin}\left (\mathrm {cot}\left (x\right )\right )}{2}+\frac {\mathrm {cot}\left (x\right )\,\sqrt {1-{\mathrm {cot}\left (x\right )}^2}}{2}-\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (-1+\mathrm {cot}\left (x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\sqrt {1-{\mathrm {cot}\left (x\right )}^2}\,1{}\mathrm {i}}{\mathrm {cot}\left (x\right )-\mathrm {i}}\right )\,1{}\mathrm {i}+\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (1+\mathrm {cot}\left (x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\sqrt {1-{\mathrm {cot}\left (x\right )}^2}\,1{}\mathrm {i}}{\mathrm {cot}\left (x\right )+1{}\mathrm {i}}\right )\,1{}\mathrm {i} \]
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