Integrand size = 10, antiderivative size = 212 \[ \int \frac {1}{\left (a \cot ^3(x)\right )^{3/2}} \, dx=-\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \cot ^{\frac {3}{2}}(x)}{\sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \cot ^{\frac {3}{2}}(x)}{\sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}} \]
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Time = 0.12 (sec) , antiderivative size = 212, 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, 3555, 3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{\left (a \cot ^3(x)\right )^{3/2}} \, dx=\frac {\cot ^{\frac {3}{2}}(x) \arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \arctan \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \log \left (\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}} \]
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\cot ^{\frac {3}{2}}(x) \int \frac {1}{\cot ^{\frac {9}{2}}(x)} \, dx}{a \sqrt {a \cot ^3(x)}} \\ & = \frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \int \frac {1}{\cot ^{\frac {5}{2}}(x)} \, dx}{a \sqrt {a \cot ^3(x)}} \\ & = -\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \int \frac {1}{\sqrt {\cot (x)}} \, dx}{a \sqrt {a \cot ^3(x)}} \\ & = -\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\cot (x)\right )}{a \sqrt {a \cot ^3(x)}} \\ & = -\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\left (2 \cot ^{\frac {3}{2}}(x)\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{a \sqrt {a \cot ^3(x)}} \\ & = -\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{a \sqrt {a \cot ^3(x)}} \\ & = -\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (x)}\right )}{2 a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(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} a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(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} a \sqrt {a \cot ^3(x)}} \\ & = -\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} a \sqrt {a \cot ^3(x)}} \\ & = -\frac {2}{3 a \sqrt {a \cot ^3(x)}}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \cot ^{\frac {3}{2}}(x)}{\sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \cot ^{\frac {3}{2}}(x)}{\sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} a \sqrt {a \cot ^3(x)}}+\frac {2 \tan ^2(x)}{7 a \sqrt {a \cot ^3(x)}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\left (a \cot ^3(x)\right )^{3/2}} \, dx=\frac {-14+21 \arctan \left (\sqrt [4]{-\cot ^2(x)}\right ) \left (-\cot ^2(x)\right )^{3/4}+21 \text {arctanh}\left (\sqrt [4]{-\cot ^2(x)}\right ) \left (-\cot ^2(x)\right )^{3/4}+6 \tan ^2(x)}{21 a \sqrt {a \cot ^3(x)}} \]
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Time = 0.03 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(-\frac {\cot \left (x \right ) \left (21 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \ln \left (-\frac {a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}{\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}\right )+42 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+42 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+56 a^{4} \cot \left (x \right )^{2}-24 a^{4}\right )}{84 a^{4} \left (a \cot \left (x \right )^{3}\right )^{\frac {3}{2}}}\) | \(185\) |
default | \(-\frac {\cot \left (x \right ) \left (21 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \ln \left (-\frac {a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}{\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}\right )+42 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+42 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (a \cot \left (x \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+56 a^{4} \cot \left (x \right )^{2}-24 a^{4}\right )}{84 a^{4} \left (a \cot \left (x \right )^{3}\right )^{\frac {3}{2}}}\) | \(185\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.67 \[ \int \frac {1}{\left (a \cot ^3(x)\right )^{3/2}} \, dx=\frac {8 \, {\left (5 \, \cos \left (2 \, x\right )^{2} - 3 \, \cos \left (2 \, x\right ) - 2\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 )}} \sin \left (2 \, x\right ) - 21 \, {\left (a^{2} \cos \left (2 \, x\right )^{3} + 3 \, a^{2} \cos \left (2 \, x\right )^{2} + 3 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )} \left (-\frac {1}{a^{6}}\right )^{\frac {1}{4}} \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} \cos \left (2 \, x\right ) + a^{2}\right )} \left (-\frac {1}{a^{6}}\right )^{\frac {1}{4}}}{\cos \left (2 \, x\right ) + 1}\right ) + 21 \, {\left (a^{2} \cos \left (2 \, x\right )^{3} + 3 \, a^{2} \cos \left (2 \, x\right )^{2} + 3 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )} \left (-\frac {1}{a^{6}}\right )^{\frac {1}{4}} \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} \cos \left (2 \, x\right ) + a^{2}\right )} \left (-\frac {1}{a^{6}}\right )^{\frac {1}{4}}}{\cos \left (2 \, x\right ) + 1}\right ) - 21 \, {\left (-i \, a^{2} \cos \left (2 \, x\right )^{3} - 3 i \, a^{2} \cos \left (2 \, x\right )^{2} - 3 i \, a^{2} \cos \left (2 \, x\right ) - i \, a^{2}\right )} \left (-\frac {1}{a^{6}}\right )^{\frac {1}{4}} \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 (i \, a^{2} \cos \left (2 \, x\right ) + i \, a^{2}\right )} \left (-\frac {1}{a^{6}}\right )^{\frac {1}{4}}}{\cos \left (2 \, x\right ) + 1}\right ) - 21 \, {\left (i \, a^{2} \cos \left (2 \, x\right )^{3} + 3 i \, a^{2} \cos \left (2 \, x\right )^{2} + 3 i \, a^{2} \cos \left (2 \, x\right ) + i \, a^{2}\right )} \left (-\frac {1}{a^{6}}\right )^{\frac {1}{4}} \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 (-i \, a^{2} \cos \left (2 \, x\right ) - i \, a^{2}\right )} \left (-\frac {1}{a^{6}}\right )^{\frac {1}{4}}}{\cos \left (2 \, x\right ) + 1}\right )}{42 \, {\left (a^{2} \cos \left (2 \, x\right )^{3} + 3 \, a^{2} \cos \left (2 \, x\right )^{2} + 3 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )}} \]
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\[ \int \frac {1}{\left (a \cot ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \cot ^{3}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.37 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.51 \[ \int \frac {1}{\left (a \cot ^3(x)\right )^{3/2}} \, dx=\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right )}{4 \, a^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, \sqrt {a} \tan \left (x\right )^{\frac {7}{2}} - 7 \, \sqrt {a} \tan \left (x\right )^{\frac {3}{2}}\right )}}{21 \, a^{2}} \]
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\[ \int \frac {1}{\left (a \cot ^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \cot \left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a \cot ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a\,{\mathrm {cot}\left (x\right )}^3\right )}^{3/2}} \,d x \]
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