Integrand size = 19, antiderivative size = 201 \[ \int \frac {1}{x \tan ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 \sqrt {2} b n}-\frac {2}{3 b n \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Time = 0.16 (sec) , antiderivative size = 201, 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, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{x \tan ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {2} b n}-\frac {\arctan \left (\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{\sqrt {2} b n}-\frac {2}{3 b n \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )-\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n}-\frac {\log \left (\tan \left (a+b \log \left (c x^n\right )\right )+\sqrt {2} \sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}+1\right )}{2 \sqrt {2} b n} \]
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Rule 210
Rule 217
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 {5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2}{3 b n \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\tan (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2}{3 b n \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \\ & = -\frac {2}{3 b n \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {2 \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \\ & = -\frac {2}{3 b n \tan ^{\frac {3}{2}}\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}{3 b n \tan ^{\frac {3}{2}}\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}{3 b n \tan ^{\frac {3}{2}}\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}{3 b n \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x \tan ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {-2+3 \arctan \left (\sqrt [4]{-\tan ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \left (-\tan ^2\left (a+b \log \left (c x^n\right )\right )\right )^{3/4}+3 \text {arctanh}\left (\sqrt [4]{-\tan ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \left (-\tan ^2\left (a+b \log \left (c x^n\right )\right )\right )^{3/4}}{3 b n \tan ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Time = 0.87 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{1-\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}-\frac {2}{3 {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}}{n b}\) | \(139\) |
default | \(\frac {-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{1-\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )\right )}{4}-\frac {2}{3 {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}}{n b}\) | \(139\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.45 \[ \int \frac {1}{x \tan ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {3 \, {\left (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}}\right )} \log \left (b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{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 ) + 3 \, {\left (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}}\right )} \log \left (i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{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 ) + 3 \, {\left (-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}}\right )} \log \left (-i \, b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{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 ) - 3 \, {\left (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}}\right )} \log \left (-b n \left (-\frac {1}{b^{4} n^{4}}\right )^{\frac {1}{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 ) - 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 )}}{6 \, {\left (b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) - b n\right )}} \]
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\[ \int \frac {1}{x \tan ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \tan ^{\frac {5}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
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\[ \int \frac {1}{x \tan ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \tan \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{x \tan ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]
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Time = 31.70 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.39 \[ \int \frac {1}{x \tan ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2}{3\,b\,n\,{\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )\,1{}\mathrm {i}}{b\,n} \]
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