Integrand size = 9, antiderivative size = 142 \[ \int \tan ^p(a+b \log (x)) \, dx=x \left (1-e^{2 i a} x^{2 i b}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{2 i b}\right )}{1+e^{2 i a} x^{2 i b}}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b},-p,p,1-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4587, 1986, 441, 440} \[ \int \tan ^p(a+b \log (x)) \, dx=x \left (1-e^{2 i a} x^{2 i b}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{2 i b}\right )}{1+e^{2 i a} x^{2 i b}}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b},-p,p,1-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right ) \]
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Rule 440
Rule 441
Rule 1986
Rule 4587
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {i-i e^{2 i a} x^{2 i b}}{1+e^{2 i a} x^{2 i b}}\right )^p \, dx \\ & = \left (\left (i-i e^{2 i a} x^{2 i b}\right )^{-p} \left (\frac {i-i e^{2 i a} x^{2 i b}}{1+e^{2 i a} x^{2 i b}}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^p\right ) \int \left (i-i e^{2 i a} x^{2 i b}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^{-p} \, dx \\ & = \left (\left (1-e^{2 i a} x^{2 i b}\right )^{-p} \left (\frac {i-i e^{2 i a} x^{2 i b}}{1+e^{2 i a} x^{2 i b}}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^p\right ) \int \left (1-e^{2 i a} x^{2 i b}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^{-p} \, dx \\ & = x \left (1-e^{2 i a} x^{2 i b}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{2 i b}\right )}{1+e^{2 i a} x^{2 i b}}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b},-p,p,1-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right ) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(330\) vs. \(2(142)=284\).
Time = 0.59 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.32 \[ \int \tan ^p(a+b \log (x)) \, dx=\frac {(-i+2 b) x \left (-\frac {i \left (-1+e^{2 i a} x^{2 i b}\right )}{1+e^{2 i a} x^{2 i b}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2 b},-p,p,1-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )}{-2 b e^{2 i a} p x^{2 i b} \operatorname {AppellF1}\left (1-\frac {i}{2 b},1-p,p,2-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )-2 b e^{2 i a} p x^{2 i b} \operatorname {AppellF1}\left (1-\frac {i}{2 b},-p,1+p,2-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )+(-i+2 b) \operatorname {AppellF1}\left (-\frac {i}{2 b},-p,p,1-\frac {i}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )} \]
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\[\int \tan \left (a +b \ln \left (x \right )\right )^{p}d x\]
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\[ \int \tan ^p(a+b \log (x)) \, dx=\int { \tan \left (b \log \left (x\right ) + a\right )^{p} \,d x } \]
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\[ \int \tan ^p(a+b \log (x)) \, dx=\int \tan ^{p}{\left (a + b \log {\left (x \right )} \right )}\, dx \]
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\[ \int \tan ^p(a+b \log (x)) \, dx=\int { \tan \left (b \log \left (x\right ) + a\right )^{p} \,d x } \]
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\[ \int \tan ^p(a+b \log (x)) \, dx=\int { \tan \left (b \log \left (x\right ) + a\right )^{p} \,d x } \]
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Timed out. \[ \int \tan ^p(a+b \log (x)) \, dx=\int {\mathrm {tan}\left (a+b\,\ln \left (x\right )\right )}^p \,d x \]
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