Integrand size = 19, antiderivative size = 156 \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {1+\frac {2 i}{b d n}}{2 x^2}+\frac {i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2 \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {2 i \operatorname {Hypergeometric2F1}\left (1,\frac {i}{b d n},1+\frac {i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2} \]
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Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4593, 4591, 516, 470, 371} \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {2 i \operatorname {Hypergeometric2F1}\left (1,\frac {i}{b d n},1+\frac {i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2}+\frac {i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2 \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {1+\frac {2 i}{b d n}}{2 x^2} \]
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Rule 371
Rule 470
Rule 516
Rule 4591
Rule 4593
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int x^{-1-\frac {2}{n}} \tan ^2(d (a+b \log (x))) \, dx,x,c x^n\right )}{n x^2} \\ & = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int \frac {x^{-1-\frac {2}{n}} \left (i-i e^{2 i a d} x^{2 i b d}\right )^2}{\left (1+e^{2 i a d} x^{2 i b d}\right )^2} \, dx,x,c x^n\right )}{n x^2} \\ & = \frac {i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2 \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {\left (i e^{-2 i a d} \left (c x^n\right )^{2/n}\right ) \text {Subst}\left (\int \frac {x^{-1-\frac {2}{n}} \left (\frac {2 e^{2 i a d} (2+i b d n)}{n}-\frac {2 e^{4 i a d} (2-i b d n) x^{2 i b d}}{n}\right )}{1+e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{2 b d n x^2} \\ & = \frac {1+\frac {2 i}{b d n}}{2 x^2}+\frac {i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2 \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {\left (4 i \left (c x^n\right )^{2/n}\right ) \text {Subst}\left (\int \frac {x^{-1-\frac {2}{n}}}{1+e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{b d n^2 x^2} \\ & = \frac {1+\frac {2 i}{b d n}}{2 x^2}+\frac {i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2 \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {2 i \operatorname {Hypergeometric2F1}\left (1,\frac {i}{b d n},1+\frac {i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n x^2} \\ \end{align*}
Time = 3.09 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.15 \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {-2 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1+\frac {i}{b d n},2+\frac {i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(i+b d n) \left (b d n-2 i \operatorname {Hypergeometric2F1}\left (1,\frac {i}{b d n},1+\frac {i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+2 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b d n (i+b d n) x^2} \]
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\[\int \frac {{\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}}{x^{3}}d x\]
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\[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{3}} \,d x } \]
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\[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\tan ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{3}} \,d x } \]
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\[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {{\mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2}{x^3} \,d x \]
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