Integrand size = 9, antiderivative size = 61 \[ \int \tanh ^p(a+2 \log (x)) \, dx=x \left (1-e^{2 a} x^4\right )^{-p} \left (-1+e^{2 a} x^4\right )^p \operatorname {AppellF1}\left (\frac {1}{4},-p,p,\frac {5}{4},e^{2 a} x^4,-e^{2 a} x^4\right ) \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5652, 441, 440} \[ \int \tanh ^p(a+2 \log (x)) \, dx=x \left (1-e^{2 a} x^4\right )^{-p} \left (e^{2 a} x^4-1\right )^p \operatorname {AppellF1}\left (\frac {1}{4},-p,p,\frac {5}{4},e^{2 a} x^4,-e^{2 a} x^4\right ) \]
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Rule 440
Rule 441
Rule 5652
Rubi steps \begin{align*} \text {integral}& = \int \left (-1+e^{2 a} x^4\right )^p \left (1+e^{2 a} x^4\right )^{-p} \, dx \\ & = \left (\left (1-e^{2 a} x^4\right )^{-p} \left (-1+e^{2 a} x^4\right )^p\right ) \int \left (1-e^{2 a} x^4\right )^p \left (1+e^{2 a} x^4\right )^{-p} \, dx \\ & = x \left (1-e^{2 a} x^4\right )^{-p} \left (-1+e^{2 a} x^4\right )^p \operatorname {AppellF1}\left (\frac {1}{4},-p,p,\frac {5}{4},e^{2 a} x^4,-e^{2 a} x^4\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(171\) vs. \(2(61)=122\).
Time = 0.72 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.80 \[ \int \tanh ^p(a+2 \log (x)) \, dx=\frac {5 x \left (\frac {-1+e^{2 a} x^4}{1+e^{2 a} x^4}\right )^p \operatorname {AppellF1}\left (\frac {1}{4},-p,p,\frac {5}{4},e^{2 a} x^4,-e^{2 a} x^4\right )}{5 \operatorname {AppellF1}\left (\frac {1}{4},-p,p,\frac {5}{4},e^{2 a} x^4,-e^{2 a} x^4\right )-4 e^{2 a} p x^4 \left (\operatorname {AppellF1}\left (\frac {5}{4},1-p,p,\frac {9}{4},e^{2 a} x^4,-e^{2 a} x^4\right )+\operatorname {AppellF1}\left (\frac {5}{4},-p,1+p,\frac {9}{4},e^{2 a} x^4,-e^{2 a} x^4\right )\right )} \]
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\[\int \tanh \left (a +2 \ln \left (x \right )\right )^{p}d x\]
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\[ \int \tanh ^p(a+2 \log (x)) \, dx=\int { \tanh \left (a + 2 \, \log \left (x\right )\right )^{p} \,d x } \]
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\[ \int \tanh ^p(a+2 \log (x)) \, dx=\int \tanh ^{p}{\left (a + 2 \log {\left (x \right )} \right )}\, dx \]
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\[ \int \tanh ^p(a+2 \log (x)) \, dx=\int { \tanh \left (a + 2 \, \log \left (x\right )\right )^{p} \,d x } \]
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\[ \int \tanh ^p(a+2 \log (x)) \, dx=\int { \tanh \left (a + 2 \, \log \left (x\right )\right )^{p} \,d x } \]
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Timed out. \[ \int \tanh ^p(a+2 \log (x)) \, dx=\int {\mathrm {tanh}\left (a+2\,\ln \left (x\right )\right )}^p \,d x \]
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