Integrand size = 19, antiderivative size = 199 \[ \int \frac {1}{x \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {2}{b n \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \]
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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 = {3555, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{x \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2} b n}-\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}+\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}-\frac {2}{b n \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \]
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\tan ^{\frac {3}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2}{b n \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}-\frac {\text {Subst}\left (\int \sqrt {\tan (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2}{b n \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}-\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = -\frac {2}{b n \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}-\frac {2 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = -\frac {2}{b n \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}+\frac {\text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan \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 {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = -\frac {2}{b n \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}-\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan \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 {\tan \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 {\tan \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 {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{2 \sqrt {2} b n} \\ & = -\frac {\log \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {2}{b n \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan \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 {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n} \\ & = \frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {2}{b n \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {-2-\arctan \left (\sqrt [4]{-\tan ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt [4]{-\tan ^2\left (a+b \log \left (c x^n\right )\right )}+\text {arctanh}\left (\sqrt [4]{-\tan ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt [4]{-\tan ^2\left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}} \]
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Time = 0.84 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}-\frac {2}{\sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}}}{n b}\) | \(139\) |
default | \(\frac {-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}-\frac {2}{\sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}}}{n b}\) | \(139\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.21 \[ \int \frac {1}{x \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{4}} \log \left (b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {\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 ) \sin \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}} \log \left (i \, b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {\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 ) \sin \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}} \log \left (-i \, b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {\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 ) \sin \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}} \log \left (-b^{3} n^{3} \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {\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 ) \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 4 \, \sqrt {\frac {\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}} {\left (\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1\right )}}{2 \, b n \sin \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )} \]
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\[ \int \frac {1}{x \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \tan ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
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\[ \int \frac {1}{x \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \tan \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{x \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]
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Time = 29.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.40 \[ \int \frac {1}{x \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {2}{b\,n\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}} \]
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