Integrand size = 19, antiderivative size = 176 \[ \int \frac {\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\log \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {\log \left (1+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+\cot \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n} \]
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Time = 0.13 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\arctan \left (\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2} b n}-\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}+\frac {\log \left (\cot \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n} \]
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {\cot (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\cot \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = -\frac {2 \text {Subst}\left (\int \frac {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 {\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}{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 {\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 {\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 = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {\left (-\arctan \left (\sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}\right )+\text {arctanh}\left (\sqrt [4]{-\cot ^2\left (a+b \log \left (c x^n\right )\right )}\right )\right ) \sqrt [4]{-\cot \left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt [4]{\cot \left (a+b \log \left (c x^n\right )\right )}} \]
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Time = 1.03 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4 n b}\) | \(122\) |
default | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )-\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}{1+\cot \left (a +b \ln \left (c \,x^{n}\right )\right )+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4 n b}\) | \(122\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 539, normalized size of antiderivative = 3.06 \[ \int \frac {\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {1}{2} \, \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{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 ) - \frac {1}{2} \, \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{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 ) + \frac {1}{2} i \, \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{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 ) - \frac {1}{2} i \, \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) - i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{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 ) \]
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\[ \int \frac {\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int \frac {\sqrt {\cot {\left (a + b \log {\left (c x^{n} \right )} \right )}}}{x}\, dx \]
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\[ \int \frac {\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int { \frac {\sqrt {\cot \left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\text {Timed out} \]
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Time = 26.40 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {\cot \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {cot}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n} \]
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