Integrand size = 19, antiderivative size = 159 \[ \int x^3 \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(4 i-b d n) x^4}{4 b d n}+\frac {i x^4 \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {2 i x^4 \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i}{b d n},1-\frac {2 i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n} \]
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Time = 0.22 (sec) , antiderivative size = 159, 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} \[ \int x^3 \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {2 i x^4 \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i}{b d n},1-\frac {2 i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n}+\frac {i x^4 \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}+\frac {x^4 (-b d n+4 i)}{4 b d n} \]
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Rule 371
Rule 470
Rule 516
Rule 4591
Rule 4593
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int x^{-1+\frac {4}{n}} \tan ^2(d (a+b \log (x))) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {4}{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} \\ & = \frac {i x^4 \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \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} x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {4}{n}} \left (-\frac {2 e^{2 i a d} (4-i b d n)}{n}+\frac {2 e^{4 i a d} (4+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} \\ & = -\frac {1}{4} \left (1-\frac {4 i}{b d n}\right ) x^4+\frac {i x^4 \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {\left (8 i x^4 \left (c x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {4}{n}}}{1+e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{b d n^2} \\ & = -\frac {1}{4} \left (1-\frac {4 i}{b d n}\right ) x^4+\frac {i x^4 \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}-\frac {2 i x^4 \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i}{b d n},1-\frac {2 i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{b d n} \\ \end{align*}
Time = 5.60 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.13 \[ \int x^3 \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {x^4 \left (-8 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {2 i}{b d n},2-\frac {2 i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-2 i+b d n) \left (b d n+4 i \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i}{b d n},1-\frac {2 i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )-4 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )\right )}{4 b d n (-2 i+b d n)} \]
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\[\int x^{3} {\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}d x\]
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\[ \int x^3 \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{3} \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
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Timed out. \[ \int x^3 \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \]
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\[ \int x^3 \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{3} \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
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Timed out. \[ \int x^3 \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int x^3 \tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^3\,{\mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2 \,d x \]
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