Integrand size = 17, antiderivative size = 71 \[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {i x^4}{4}+\frac {1}{2} i x^4 \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i}{b d n},1-\frac {2 i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {4593, 4591, 470, 371} \[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} i x^4 \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i}{b d n},1-\frac {2 i}{b d n},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-\frac {i x^4}{4} \]
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Rule 371
Rule 470
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 (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 )}{1+e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{n} \\ & = -\frac {i x^4}{4}+\frac {\left (2 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 )}{n} \\ & = -\frac {i x^4}{4}+\frac {1}{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 ) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(71)=142\).
Time = 5.64 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.06 \[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x^4 \left (2 i e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {2 i}{b d n},2-\frac {2 i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-2 i+b d n) \operatorname {Hypergeometric2F1}\left (1,-\frac {2 i}{b d n},1-\frac {2 i}{b d n},-e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{-8-4 i b d n} \]
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\[\int x^{3} \tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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\[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{3} \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^{3} \tan {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{3} \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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Timed out. \[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int x^3 \tan \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^3\,\mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
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