Integrand size = 10, antiderivative size = 176 \[ \int \sqrt {a \cot ^3(x)} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {\sqrt {a \cot ^3(x)} \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}-2 \sqrt {a \cot ^3(x)} \tan (x) \]
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Time = 0.10 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3739, 3554, 3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \sqrt {a \cot ^3(x)} \, dx=-\frac {\sqrt {a \cot ^3(x)} \arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \arctan \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-2 \tan (x) \sqrt {a \cot ^3(x)}-\frac {\sqrt {a \cot ^3(x)} \log \left (\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)} \]
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3554
Rule 3557
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a \cot ^3(x)} \int \cot ^{\frac {3}{2}}(x) \, dx}{\cot ^{\frac {3}{2}}(x)} \\ & = -2 \sqrt {a \cot ^3(x)} \tan (x)-\frac {\sqrt {a \cot ^3(x)} \int \frac {1}{\sqrt {\cot (x)}} \, dx}{\cot ^{\frac {3}{2}}(x)} \\ & = -2 \sqrt {a \cot ^3(x)} \tan (x)+\frac {\sqrt {a \cot ^3(x)} \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\cot (x)\right )}{\cot ^{\frac {3}{2}}(x)} \\ & = -2 \sqrt {a \cot ^3(x)} \tan (x)+\frac {\left (2 \sqrt {a \cot ^3(x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)} \\ & = -2 \sqrt {a \cot ^3(x)} \tan (x)+\frac {\sqrt {a \cot ^3(x)} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)}+\frac {\sqrt {a \cot ^3(x)} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)} \\ & = -2 \sqrt {a \cot ^3(x)} \tan (x)+\frac {\sqrt {a \cot ^3(x)} \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 {\sqrt {a \cot ^3(x)} \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 {\sqrt {a \cot ^3(x)} \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 {\sqrt {a \cot ^3(x)} \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 {\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 {\sqrt {a \cot ^3(x)} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}-2 \sqrt {a \cot ^3(x)} \tan (x)+\frac {\sqrt {a \cot ^3(x)} \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 {\sqrt {a \cot ^3(x)} \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 {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-\frac {\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 {\sqrt {a \cot ^3(x)} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \cot ^{\frac {3}{2}}(x)}-2 \sqrt {a \cot ^3(x)} \tan (x) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.69 \[ \int \sqrt {a \cot ^3(x)} \, dx=-\frac {\sqrt {a \cot ^3(x)} \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right )+8 \sqrt {\cot (x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )\right )}{4 \cot ^{\frac {3}{2}}(x)} \]
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Time = 0.04 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\sqrt {a \cot \left (x \right )^{3}}\, \left (\left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}{\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}\right )+2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )-8 \sqrt {a \cot \left (x \right )}\right )}{4 \cot \left (x \right ) \sqrt {a \cot \left (x \right )}}\) | \(165\) |
default | \(\frac {\sqrt {a \cot \left (x \right )^{3}}\, \left (\left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}{\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}\right )+2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )-8 \sqrt {a \cot \left (x \right )}\right )}{4 \cot \left (x \right ) \sqrt {a \cot \left (x \right )}}\) | \(165\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.15 \[ \int \sqrt {a \cot ^3(x)} \, dx=\frac {\left (-a^{2}\right )^{\frac {1}{4}} {\left (\cos \left (2 \, x\right ) + 1\right )} \log \left (\frac {\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^{2}\right )^{\frac {1}{4}} {\left (\cos \left (2 \, x\right ) + 1\right )}}{\cos \left (2 \, x\right ) + 1}\right ) - \left (-a^{2}\right )^{\frac {1}{4}} {\left (\cos \left (2 \, x\right ) + 1\right )} \log \left (\frac {\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^{2}\right )^{\frac {1}{4}} {\left (\cos \left (2 \, x\right ) + 1\right )}}{\cos \left (2 \, x\right ) + 1}\right ) + \left (-a^{2}\right )^{\frac {1}{4}} {\left (i \, \cos \left (2 \, x\right ) + i\right )} \log \left (\frac {\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^{2}\right )^{\frac {1}{4}} {\left (i \, \cos \left (2 \, x\right ) + i\right )}}{\cos \left (2 \, x\right ) + 1}\right ) + \left (-a^{2}\right )^{\frac {1}{4}} {\left (-i \, \cos \left (2 \, x\right ) - i\right )} \log \left (\frac {\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^{2}\right )^{\frac {1}{4}} {\left (-i \, \cos \left (2 \, x\right ) - i\right )}}{\cos \left (2 \, x\right ) + 1}\right ) - 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 )}{2 \, {\left (\cos \left (2 \, x\right ) + 1\right )}} \]
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\[ \int \sqrt {a \cot ^3(x)} \, dx=\int \sqrt {a \cot ^{3}{\left (x \right )}}\, dx \]
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none
Time = 0.35 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.53 \[ \int \sqrt {a \cot ^3(x)} \, dx=-\frac {1}{4} \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right )\right )} \sqrt {a} - \frac {2 \, \sqrt {a}}{\sqrt {\tan \left (x\right )}} \]
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\[ \int \sqrt {a \cot ^3(x)} \, dx=\int { \sqrt {a \cot \left (x\right )^{3}} \,d x } \]
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Timed out. \[ \int \sqrt {a \cot ^3(x)} \, dx=\int \sqrt {a\,{\mathrm {cot}\left (x\right )}^3} \,d x \]
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