Integrand size = 11, antiderivative size = 194 \[ \int \coth ^p\left (a+\frac {\log (x)}{8}\right ) \, dx=\frac {1}{3} e^{-12 a} \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \left (e^{4 a} \left (3+2 p^2\right )+2 e^{6 a} p \sqrt [4]{x}\right )+e^{-4 a} \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \sqrt {x}-\frac {2^{2-p} e^{-8 a} p \left (2+p^2\right ) \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {1}{2} \left (1+e^{2 a} \sqrt [4]{x}\right )\right )}{3 (1+p)} \]
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Time = 0.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5653, 383, 102, 152, 71} \[ \int \coth ^p\left (a+\frac {\log (x)}{8}\right ) \, dx=-\frac {e^{-8 a} 2^{2-p} p \left (p^2+2\right ) \left (-e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \operatorname {Hypergeometric2F1}\left (p,p+1,p+2,\frac {1}{2} \left (e^{2 a} \sqrt [4]{x}+1\right )\right )}{3 (p+1)}+\frac {1}{3} e^{-12 a} \left (-e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \left (e^{4 a} \left (2 p^2+3\right )+2 e^{6 a} p \sqrt [4]{x}\right ) \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p}+e^{-4 a} \sqrt {x} \left (-e^{2 a} \sqrt [4]{x}-1\right )^{p+1} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \]
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Rule 71
Rule 102
Rule 152
Rule 383
Rule 5653
Rubi steps \begin{align*} \text {integral}& = \int \left (-1-e^{2 a} \sqrt [4]{x}\right )^p \left (1-e^{2 a} \sqrt [4]{x}\right )^{-p} \, dx \\ & = 4 \text {Subst}\left (\int x^3 \left (-1-e^{2 a} x\right )^p \left (1-e^{2 a} x\right )^{-p} \, dx,x,\sqrt [4]{x}\right ) \\ & = e^{-4 a} \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \sqrt {x}+e^{-4 a} \text {Subst}\left (\int x \left (-1-e^{2 a} x\right )^p \left (1-e^{2 a} x\right )^{-p} \left (2+2 e^{2 a} p x\right ) \, dx,x,\sqrt [4]{x}\right ) \\ & = \frac {1}{3} e^{-12 a} \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \left (e^{4 a} \left (3+2 p^2\right )+2 e^{6 a} p \sqrt [4]{x}\right )+e^{-4 a} \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \sqrt {x}+\frac {1}{3} \left (4 e^{-6 a} p \left (2+p^2\right )\right ) \text {Subst}\left (\int \left (-1-e^{2 a} x\right )^p \left (1-e^{2 a} x\right )^{-p} \, dx,x,\sqrt [4]{x}\right ) \\ & = \frac {1}{3} e^{-12 a} \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \left (e^{4 a} \left (3+2 p^2\right )+2 e^{6 a} p \sqrt [4]{x}\right )+e^{-4 a} \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \left (1-e^{2 a} \sqrt [4]{x}\right )^{1-p} \sqrt {x}-\frac {2^{2-p} e^{-8 a} p \left (2+p^2\right ) \left (-1-e^{2 a} \sqrt [4]{x}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {1}{2} \left (1+e^{2 a} \sqrt [4]{x}\right )\right )}{3 (1+p)} \\ \end{align*}
Time = 1.26 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.15 \[ \int \coth ^p\left (a+\frac {\log (x)}{8}\right ) \, dx=\frac {e^{-8 a} \left (1+e^{2 a} \sqrt [4]{x}\right )^{1-p} \left (\frac {1+e^{2 a} \sqrt [4]{x}}{-1+e^{2 a} \sqrt [4]{x}}\right )^{-1+p} \left (-2^{3+p} p \operatorname {Hypergeometric2F1}\left (-2-p,1-p,2-p,\frac {1}{2}-\frac {1}{2} e^{2 a} \sqrt [4]{x}\right )+2^{2+p} (-1+2 p) \operatorname {Hypergeometric2F1}\left (-1-p,1-p,2-p,\frac {1}{2}-\frac {1}{2} e^{2 a} \sqrt [4]{x}\right )+(-1+p) \left (e^{4 a} \left (1+e^{2 a} \sqrt [4]{x}\right )^{1+p} \sqrt {x}-2^{1+p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,\frac {1}{2}-\frac {1}{2} e^{2 a} \sqrt [4]{x}\right )\right )\right )}{-1+p} \]
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\[\int \coth \left (a +\frac {\ln \left (x \right )}{8}\right )^{p}d x\]
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\[ \int \coth ^p\left (a+\frac {\log (x)}{8}\right ) \, dx=\int { \coth \left (a + \frac {1}{8} \, \log \left (x\right )\right )^{p} \,d x } \]
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\[ \int \coth ^p\left (a+\frac {\log (x)}{8}\right ) \, dx=\int \coth ^{p}{\left (a + \frac {\log {\left (x \right )}}{8} \right )}\, dx \]
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\[ \int \coth ^p\left (a+\frac {\log (x)}{8}\right ) \, dx=\int { \coth \left (a + \frac {1}{8} \, \log \left (x\right )\right )^{p} \,d x } \]
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\[ \int \coth ^p\left (a+\frac {\log (x)}{8}\right ) \, dx=\int { \coth \left (a + \frac {1}{8} \, \log \left (x\right )\right )^{p} \,d x } \]
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Timed out. \[ \int \coth ^p\left (a+\frac {\log (x)}{8}\right ) \, dx=\int {\mathrm {coth}\left (a+\frac {\ln \left (x\right )}{8}\right )}^p \,d x \]
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