Integrand size = 19, antiderivative size = 134 \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {1-\frac {1}{b d n}}{x}+\frac {1+e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {2 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 b d n},1-\frac {1}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x} \]
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Time = 0.13 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5659, 5657, 516, 470, 371} \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {2 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 b d n},1-\frac {1}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x}+\frac {e^{2 a d} \left (c x^n\right )^{2 b d}+1}{b d n x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {1-\frac {1}{b d n}}{x} \]
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Rule 371
Rule 470
Rule 516
Rule 5657
Rule 5659
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int x^{-1-\frac {1}{n}} \coth ^2(d (a+b \log (x))) \, dx,x,c x^n\right )}{n x} \\ & = \frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int \frac {x^{-1-\frac {1}{n}} \left (-1-e^{2 a d} x^{2 b d}\right )^2}{\left (1-e^{2 a d} x^{2 b d}\right )^2} \, dx,x,c x^n\right )}{n x} \\ & = \frac {1+e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac {\left (e^{-2 a d} \left (c x^n\right )^{\frac {1}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1-\frac {1}{n}} \left (\frac {2 e^{2 a d} (1+b d n)}{n}+\frac {2 e^{4 a d} (1-b d n) x^{2 b d}}{n}\right )}{1-e^{2 a d} x^{2 b d}} \, dx,x,c x^n\right )}{2 b d n x} \\ & = -\frac {1-\frac {1}{b d n}}{x}+\frac {1+e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac {\left (2 \left (c x^n\right )^{\frac {1}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1-\frac {1}{n}}}{1-e^{2 a d} x^{2 b d}} \, dx,x,c x^n\right )}{b d n^2 x} \\ & = -\frac {1-\frac {1}{b d n}}{x}+\frac {1+e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {2 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 b d n},1-\frac {1}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x} \\ \end{align*}
Time = 3.92 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.18 \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {1}{2 b d n},2-\frac {1}{2 b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(-1+2 b d n) \left (b d n+\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )+\operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 b d n},1-\frac {1}{2 b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b d n (-1+2 b d n) x} \]
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\[\int \frac {{\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}}{x^{2}}d x\]
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\[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {{\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2}{x^2} \,d x \]
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