Integrand size = 7, antiderivative size = 120 \[ \int \cot ^p(a+\log (x)) \, dx=\left (1-e^{2 i a} x^{2 i}\right )^p \left (1+e^{2 i a} x^{2 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{1-e^{2 i a} x^{2 i}}\right )^p x \operatorname {AppellF1}\left (-\frac {i}{2},p,-p,1-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4588, 1986, 441, 440} \[ \int \cot ^p(a+\log (x)) \, dx=x \left (1-e^{2 i a} x^{2 i}\right )^p \left (1+e^{2 i a} x^{2 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{1-e^{2 i a} x^{2 i}}\right )^p \operatorname {AppellF1}\left (-\frac {i}{2},p,-p,1-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right ) \]
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Rule 440
Rule 441
Rule 1986
Rule 4588
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-i-i e^{2 i a} x^{2 i}}{1-e^{2 i a} x^{2 i}}\right )^p \, dx \\ & = \left (\left (1-e^{2 i a} x^{2 i}\right )^p \left (-i-i e^{2 i a} x^{2 i}\right )^{-p} \left (\frac {-i-i e^{2 i a} x^{2 i}}{1-e^{2 i a} x^{2 i}}\right )^p\right ) \int \left (1-e^{2 i a} x^{2 i}\right )^{-p} \left (-i-i e^{2 i a} x^{2 i}\right )^p \, dx \\ & = \left (\left (1-e^{2 i a} x^{2 i}\right )^p \left (\frac {-i-i e^{2 i a} x^{2 i}}{1-e^{2 i a} x^{2 i}}\right )^p \left (1+e^{2 i a} x^{2 i}\right )^{-p}\right ) \int \left (1-e^{2 i a} x^{2 i}\right )^{-p} \left (1+e^{2 i a} x^{2 i}\right )^p \, dx \\ & = \left (1-e^{2 i a} x^{2 i}\right )^p \left (1+e^{2 i a} x^{2 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{1-e^{2 i a} x^{2 i}}\right )^p x \operatorname {AppellF1}\left (-\frac {i}{2},p,-p,1-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right ) \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.98 \[ \int \cot ^p(a+\log (x)) \, dx=\frac {(2-i) \left (\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{-1+e^{2 i a} x^{2 i}}\right )^p x \operatorname {AppellF1}\left (-\frac {i}{2},p,-p,1-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )}{(2-i) \operatorname {AppellF1}\left (-\frac {i}{2},p,-p,1-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )+2 e^{2 i a} p x^{2 i} \left (\operatorname {AppellF1}\left (1-\frac {i}{2},p,1-p,2-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )+\operatorname {AppellF1}\left (1-\frac {i}{2},1+p,-p,2-\frac {i}{2},e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )\right )} \]
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\[\int \cot \left (a +\ln \left (x \right )\right )^{p}d x\]
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\[ \int \cot ^p(a+\log (x)) \, dx=\int { \cot \left (a + \log \left (x\right )\right )^{p} \,d x } \]
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\[ \int \cot ^p(a+\log (x)) \, dx=\int \cot ^{p}{\left (a + \log {\left (x \right )} \right )}\, dx \]
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\[ \int \cot ^p(a+\log (x)) \, dx=\int { \cot \left (a + \log \left (x\right )\right )^{p} \,d x } \]
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\[ \int \cot ^p(a+\log (x)) \, dx=\int { \cot \left (a + \log \left (x\right )\right )^{p} \,d x } \]
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Timed out. \[ \int \cot ^p(a+\log (x)) \, dx=\int {\mathrm {cot}\left (a+\ln \left (x\right )\right )}^p \,d x \]
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