Integrand size = 10, antiderivative size = 200 \[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\frac {2}{3} a \sqrt {a \cot ^3(x)}+\frac {a \arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {a \arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {a \sqrt {a \cot ^3(x)} \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {a \sqrt {a \cot ^3(x)} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)} \]
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Time = 0.11 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3739, 3554, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\frac {a \sqrt {a \cot ^3(x)} \arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {a \sqrt {a \cot ^3(x)} \arctan \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {a \sqrt {a \cot ^3(x)} \log \left (\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {a \sqrt {a \cot ^3(x)} \log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)} \]
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3554
Rule 3557
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt {a \cot ^3(x)}\right ) \int \cot ^{\frac {9}{2}}(x) \, dx}{\cot ^{\frac {3}{2}}(x)} \\ & = -\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \int \cot ^{\frac {5}{2}}(x) \, dx}{\cot ^{\frac {3}{2}}(x)} \\ & = \frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}+\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \int \sqrt {\cot (x)} \, dx}{\cot ^{\frac {3}{2}}(x)} \\ & = \frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\cot (x)\right )}{\cot ^{\frac {3}{2}}(x)} \\ & = \frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {\left (2 a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)} \\ & = \frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}+\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)} \\ & = \frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 \cot ^{\frac {3}{2}}(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 \cot ^{\frac {3}{2}}(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)} \\ & = \frac {2}{3} a \sqrt {a \cot ^3(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {a \sqrt {a \cot ^3(x)} \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {a \sqrt {a \cot ^3(x)} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\left (a \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)} \\ & = \frac {2}{3} a \sqrt {a \cot ^3(x)}+\frac {a \arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {a \arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {2}{7} a \cot ^2(x) \sqrt {a \cot ^3(x)}-\frac {a \sqrt {a \cot ^3(x)} \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {a \sqrt {a \cot ^3(x)} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.40 \[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\frac {a \sqrt {a \cot ^3(x)} \left (-21 \arctan \left (\sqrt [4]{-\cot ^2(x)}\right ) \sqrt [4]{-\cot (x)}+21 \text {arctanh}\left (\sqrt [4]{-\cot ^2(x)}\right ) \sqrt [4]{-\cot (x)}+2 \cot ^{\frac {7}{4}}(x) \left (7-3 \cot ^2(x)\right )\right )}{21 \cot ^{\frac {7}{4}}(x)} \]
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Time = 0.09 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(-\frac {\left (a \cot \left (x \right )^{3}\right )^{\frac {3}{2}} \left (24 \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \left (a^{2}\right )^{\frac {1}{4}}+21 a^{4} \sqrt {2}\, \ln \left (-\frac {\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}{a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}\right )+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )-56 a^{2} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} \left (a^{2}\right )^{\frac {1}{4}}\right )}{84 \cot \left (x \right )^{3} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} a^{2} \left (a^{2}\right )^{\frac {1}{4}}}\) | \(189\) |
default | \(-\frac {\left (a \cot \left (x \right )^{3}\right )^{\frac {3}{2}} \left (24 \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \left (a^{2}\right )^{\frac {1}{4}}+21 a^{4} \sqrt {2}\, \ln \left (-\frac {\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}{a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}\right )+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+42 a^{4} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )-56 a^{2} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} \left (a^{2}\right )^{\frac {1}{4}}\right )}{84 \cot \left (x \right )^{3} \left (a \cot \left (x \right )\right )^{\frac {3}{2}} a^{2} \left (a^{2}\right )^{\frac {1}{4}}}\) | \(189\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.00 \[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=-\frac {21 \, \left (-a^{6}\right )^{\frac {1}{4}} {\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (\frac {a^{4} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) + \left (-a^{6}\right )^{\frac {3}{4}} {\left (\cos \left (2 \, x\right ) + 1\right )}}{\cos \left (2 \, x\right ) + 1}\right ) - 21 \, \left (-a^{6}\right )^{\frac {1}{4}} {\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (\frac {a^{4} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) - \left (-a^{6}\right )^{\frac {3}{4}} {\left (\cos \left (2 \, x\right ) + 1\right )}}{\cos \left (2 \, x\right ) + 1}\right ) + 21 \, \left (-a^{6}\right )^{\frac {1}{4}} {\left (-i \, \cos \left (2 \, x\right ) + i\right )} \log \left (\frac {a^{4} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) + \left (-a^{6}\right )^{\frac {3}{4}} {\left (i \, \cos \left (2 \, x\right ) + i\right )}}{\cos \left (2 \, x\right ) + 1}\right ) + 21 \, \left (-a^{6}\right )^{\frac {1}{4}} {\left (i \, \cos \left (2 \, x\right ) - i\right )} \log \left (\frac {a^{4} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) + \left (-a^{6}\right )^{\frac {3}{4}} {\left (-i \, \cos \left (2 \, x\right ) - i\right )}}{\cos \left (2 \, x\right ) + 1}\right ) - 8 \, {\left (5 \, a \cos \left (2 \, x\right ) - 2 \, a\right )} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}}}{42 \, {\left (\cos \left (2 \, x\right ) - 1\right )}} \]
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\[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\int \left (a \cot ^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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none
Time = 0.42 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.56 \[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\frac {1}{4} \, {\left (2 \, \sqrt {2} \sqrt {a} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + 2 \, \sqrt {2} \sqrt {a} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + \sqrt {2} \sqrt {a} \log \left (\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) - \sqrt {2} \sqrt {a} \log \left (-\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right )\right )} a + \frac {2 \, a^{\frac {3}{2}}}{3 \, \tan \left (x\right )^{\frac {3}{2}}} - \frac {2 \, a^{\frac {3}{2}}}{7 \, \tan \left (x\right )^{\frac {7}{2}}} \]
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\[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\int { \left (a \cot \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \left (a \cot ^3(x)\right )^{3/2} \, dx=\int {\left (a\,{\mathrm {cot}\left (x\right )}^3\right )}^{3/2} \,d x \]
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