Integrand size = 10, antiderivative size = 176 \[ \int \frac {1}{\sqrt {a \cot ^3(x)}} \, dx=\frac {2 \cot (x)}{\sqrt {a \cot ^3(x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \cot ^{\frac {3}{2}}(x)}{\sqrt {2} \sqrt {a \cot ^3(x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \cot ^{\frac {3}{2}}(x)}{\sqrt {2} \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )}{2 \sqrt {2} \sqrt {a \cot ^3(x)}} \]
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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, 3555, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{\sqrt {a \cot ^3(x)}} \, dx=-\frac {\cot ^{\frac {3}{2}}(x) \arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2} \sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \arctan \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2} \sqrt {a \cot ^3(x)}}+\frac {2 \cot (x)}{\sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \log \left (\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2} \sqrt {a \cot ^3(x)}} \]
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Rule 210
Rule 303
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 {3}{2}}(x)} \, dx}{\sqrt {a \cot ^3(x)}} \\ & = \frac {2 \cot (x)}{\sqrt {a \cot ^3(x)}}-\frac {\cot ^{\frac {3}{2}}(x) \int \sqrt {\cot (x)} \, dx}{\sqrt {a \cot ^3(x)}} \\ & = \frac {2 \cot (x)}{\sqrt {a \cot ^3(x)}}+\frac {\cot ^{\frac {3}{2}}(x) \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\cot (x)\right )}{\sqrt {a \cot ^3(x)}} \\ & = \frac {2 \cot (x)}{\sqrt {a \cot ^3(x)}}+\frac {\left (2 \cot ^{\frac {3}{2}}(x)\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\cot (x)}\right )}{\sqrt {a \cot ^3(x)}} \\ & = \frac {2 \cot (x)}{\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 )}{\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 )}{\sqrt {a \cot ^3(x)}} \\ & = \frac {2 \cot (x)}{\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 \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 \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} \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} \sqrt {a \cot ^3(x)}} \\ & = \frac {2 \cot (x)}{\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} \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} \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} \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} \sqrt {a \cot ^3(x)}} \\ & = \frac {2 \cot (x)}{\sqrt {a \cot ^3(x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \cot ^{\frac {3}{2}}(x)}{\sqrt {2} \sqrt {a \cot ^3(x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \cot ^{\frac {3}{2}}(x)}{\sqrt {2} \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} \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} \sqrt {a \cot ^3(x)}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.34 \[ \int \frac {1}{\sqrt {a \cot ^3(x)}} \, dx=\frac {\cot (x) \left (2+\arctan \left (\sqrt [4]{-\cot ^2(x)}\right ) \sqrt [4]{-\cot ^2(x)}-\text {arctanh}\left (\sqrt [4]{-\cot ^2(x)}\right ) \sqrt [4]{-\cot ^2(x)}\right )}{\sqrt {a \cot ^3(x)}} \]
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Time = 0.04 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {\cot \left (x \right ) \left (\sqrt {2}\, \sqrt {a \cot \left (x \right )}\, \ln \left (-\frac {\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}{a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}\right )+2 \sqrt {2}\, \sqrt {a \cot \left (x \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+2 \sqrt {2}\, \sqrt {a \cot \left (x \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+8 \left (a^{2}\right )^{\frac {1}{4}}\right )}{4 \sqrt {a \cot \left (x \right )^{3}}\, \left (a^{2}\right )^{\frac {1}{4}}}\) | \(164\) |
default | \(\frac {\cot \left (x \right ) \left (\sqrt {2}\, \sqrt {a \cot \left (x \right )}\, \ln \left (-\frac {\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}{a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}\right )+2 \sqrt {2}\, \sqrt {a \cot \left (x \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+2 \sqrt {2}\, \sqrt {a \cot \left (x \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+8 \left (a^{2}\right )^{\frac {1}{4}}\right )}{4 \sqrt {a \cot \left (x \right )^{3}}\, \left (a^{2}\right )^{\frac {1}{4}}}\) | \(164\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.40 \[ \int \frac {1}{\sqrt {a \cot ^3(x)}} \, dx=\frac {{\left (a \cos \left (2 \, x\right ) + a\right )} \left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (a^{2} \cos \left (2 \, x\right ) + a^{2}\right )} \left (-\frac {1}{a^{2}}\right )^{\frac {3}{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 )}{\cos \left (2 \, x\right ) + 1}\right ) - {\left (a \cos \left (2 \, x\right ) + a\right )} \left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (a^{2} \cos \left (2 \, x\right ) + a^{2}\right )} \left (-\frac {1}{a^{2}}\right )^{\frac {3}{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 )}{\cos \left (2 \, x\right ) + 1}\right ) + {\left (i \, a \cos \left (2 \, x\right ) + i \, a\right )} \left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (i \, a^{2} \cos \left (2 \, x\right ) + i \, a^{2}\right )} \left (-\frac {1}{a^{2}}\right )^{\frac {3}{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 )}{\cos \left (2 \, x\right ) + 1}\right ) + {\left (-i \, a \cos \left (2 \, x\right ) - i \, a\right )} \left (-\frac {1}{a^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {{\left (-i \, a^{2} \cos \left (2 \, x\right ) - i \, a^{2}\right )} \left (-\frac {1}{a^{2}}\right )^{\frac {3}{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 )}{\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 )}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{2 \, {\left (a \cos \left (2 \, x\right ) + a\right )}} \]
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\[ \int \frac {1}{\sqrt {a \cot ^3(x)}} \, dx=\int \frac {1}{\sqrt {a \cot ^{3}{\left (x \right )}}}\, dx \]
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none
Time = 0.41 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\sqrt {a \cot ^3(x)}} \, dx=-\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) - \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right )}{4 \, \sqrt {a}} + \frac {2 \, \sqrt {\tan \left (x\right )}}{\sqrt {a}} \]
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\[ \int \frac {1}{\sqrt {a \cot ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \cot \left (x\right )^{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a \cot ^3(x)}} \, dx=\int \frac {1}{\sqrt {a\,{\mathrm {cot}\left (x\right )}^3}} \,d x \]
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