Integrand size = 7, antiderivative size = 61 \[ \int \coth ^p(a+\log (x)) \, dx=x \left (-1-e^{2 a} x^2\right )^p \left (1+e^{2 a} x^2\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},p,-p,\frac {3}{2},e^{2 a} x^2,-e^{2 a} x^2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5653, 441, 440} \[ \int \coth ^p(a+\log (x)) \, dx=x \left (-e^{2 a} x^2-1\right )^p \left (e^{2 a} x^2+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},p,-p,\frac {3}{2},e^{2 a} x^2,-e^{2 a} x^2\right ) \]
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Rule 440
Rule 441
Rule 5653
Rubi steps \begin{align*} \text {integral}& = \int \left (-1-e^{2 a} x^2\right )^p \left (1-e^{2 a} x^2\right )^{-p} \, dx \\ & = \left (\left (-1-e^{2 a} x^2\right )^p \left (1+e^{2 a} x^2\right )^{-p}\right ) \int \left (1-e^{2 a} x^2\right )^{-p} \left (1+e^{2 a} x^2\right )^p \, dx \\ & = x \left (-1-e^{2 a} x^2\right )^p \left (1+e^{2 a} x^2\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},p,-p,\frac {3}{2},e^{2 a} x^2,-e^{2 a} x^2\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(171\) vs. \(2(61)=122\).
Time = 0.63 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.80 \[ \int \coth ^p(a+\log (x)) \, dx=\frac {3 x \left (\frac {1+e^{2 a} x^2}{-1+e^{2 a} x^2}\right )^p \operatorname {AppellF1}\left (\frac {1}{2},p,-p,\frac {3}{2},e^{2 a} x^2,-e^{2 a} x^2\right )}{3 \operatorname {AppellF1}\left (\frac {1}{2},p,-p,\frac {3}{2},e^{2 a} x^2,-e^{2 a} x^2\right )+2 e^{2 a} p x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},p,1-p,\frac {5}{2},e^{2 a} x^2,-e^{2 a} x^2\right )+\operatorname {AppellF1}\left (\frac {3}{2},1+p,-p,\frac {5}{2},e^{2 a} x^2,-e^{2 a} x^2\right )\right )} \]
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\[\int \coth \left (a +\ln \left (x \right )\right )^{p}d x\]
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\[ \int \coth ^p(a+\log (x)) \, dx=\int { \coth \left (a + \log \left (x\right )\right )^{p} \,d x } \]
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\[ \int \coth ^p(a+\log (x)) \, dx=\int \coth ^{p}{\left (a + \log {\left (x \right )} \right )}\, dx \]
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\[ \int \coth ^p(a+\log (x)) \, dx=\int { \coth \left (a + \log \left (x\right )\right )^{p} \,d x } \]
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\[ \int \coth ^p(a+\log (x)) \, dx=\int { \coth \left (a + \log \left (x\right )\right )^{p} \,d x } \]
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Timed out. \[ \int \coth ^p(a+\log (x)) \, dx=\int {\mathrm {coth}\left (a+\ln \left (x\right )\right )}^p \,d x \]
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