Integrand size = 15, antiderivative size = 190 \[ \int \tan ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (\frac {i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b d n},-p,p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4589, 4591, 1986, 525, 524} \[ \int \tan ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (\frac {i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b d n},-p,p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
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Rule 524
Rule 525
Rule 1986
Rule 4589
Rule 4591
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \tan ^p(d (a+b \log (x))) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \left (\frac {i-i e^{2 i a d} x^{2 i b d}}{1+e^{2 i a d} x^{2 i b d}}\right )^p \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-1/n} \left (i-i e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (\frac {i-i e^{2 i a d} \left (c x^n\right )^{2 i b d}}{1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \left (i-i e^{2 i a d} x^{2 i b d}\right )^p \left (1+e^{2 i a d} x^{2 i b d}\right )^{-p} \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-1/n} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (\frac {i-i e^{2 i a d} \left (c x^n\right )^{2 i b d}}{1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \left (1-e^{2 i a d} x^{2 i b d}\right )^p \left (1+e^{2 i a d} x^{2 i b d}\right )^{-p} \, dx,x,c x^n\right )}{n} \\ & = x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (\frac {i \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b d n},-p,p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-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. \(458\) vs. \(2(190)=380\).
Time = 1.06 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.41 \[ \int \tan ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(-i+2 b d n) x \left (-\frac {i \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b d n},-p,p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{-2 b d e^{2 i a d} n p \left (c x^n\right )^{2 i b d} \operatorname {AppellF1}\left (1-\frac {i}{2 b d n},1-p,p,2-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-2 b d e^{2 i a d} n p \left (c x^n\right )^{2 i b d} \operatorname {AppellF1}\left (1-\frac {i}{2 b d n},-p,1+p,2-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+(-i+2 b d n) \operatorname {AppellF1}\left (-\frac {i}{2 b d n},-p,p,1-\frac {i}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \]
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\[\int {\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{p}d x\]
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\[ \int \tan ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p} \,d x } \]
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\[ \int \tan ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \tan ^{p}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \]
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\[ \int \tan ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \tan \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p} \,d x } \]
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Timed out. \[ \int \tan ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \tan ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int {\mathrm {tan}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^p \,d x \]
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