Integrand size = 19, antiderivative size = 199 \[ \int \frac {\cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n} \]
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Time = 0.17 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3554, 3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\arctan \left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2} b n}-\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}+\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}-\frac {2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{b n} \]
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3554
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \cot ^{\frac {3}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\cot (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\cot \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = -\frac {2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {2 \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = -\frac {2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {\text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = -\frac {2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n}+\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n}-\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 \sqrt {2} b n}-\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{2 \sqrt {2} b n} \\ & = -\frac {2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n} \\ & = -\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2}}+2 \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2}}}{b n} \]
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Time = 0.93 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {-2 \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\) | \(139\) |
default | \(\frac {-2 \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}}{n b}\) | \(139\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 627, normalized size of antiderivative = 3.15 \[ \int \frac {\cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}{\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}} \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}\right ) - b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} - \sqrt {\frac {\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}{\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}} \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}\right ) - i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + i \, b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}{\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}} \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}\right ) + i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) - i \, b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}{\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}} \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}{\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}\right ) + 4 \, \sqrt {\frac {\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1}{\sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )}}}{2 \, b n} \]
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\[ \int \frac {\cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\cot ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
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\[ \int \frac {\cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\cot \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
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Time = 28.34 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.40 \[ \int \frac {\cot ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}}{b\,n}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n} \]
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