Integrand size = 19, antiderivative size = 136 \[ \int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{3} \left (1+\frac {3}{b d n}\right ) x^3+\frac {x^3 \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {2 x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 b d n},1+\frac {3}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n} \]
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Time = 0.12 (sec) , antiderivative size = 136, 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} \[ \int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {2 x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 b d n},1+\frac {3}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n}+\frac {x^3 \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac {1}{3} x^3 \left (\frac {3}{b d n}+1\right ) \]
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Rule 371
Rule 470
Rule 516
Rule 5657
Rule 5659
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int x^{-1+\frac {3}{n}} \coth ^2(d (a+b \log (x))) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {3}{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} \\ & = \frac {x^3 \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac {\left (e^{-2 a d} x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {3}{n}} \left (-\frac {2 e^{2 a d} (3-b d n)}{n}-\frac {2 e^{4 a d} (3+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} \\ & = \frac {1}{3} \left (1+\frac {3}{b d n}\right ) x^3+\frac {x^3 \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {\left (6 x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {3}{n}}}{1-e^{2 a d} x^{2 b d}} \, dx,x,c x^n\right )}{b d n^2} \\ & = \frac {1}{3} \left (1+\frac {3}{b d n}\right ) x^3+\frac {x^3 \left (1+e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}-\frac {2 x^3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 b d n},1+\frac {3}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n} \\ \end{align*}
Time = 4.62 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.21 \[ \int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x^3 \left (-9 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1+\frac {3}{2 b d n},2+\frac {3}{2 b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(3+2 b d n) \left (b d n-3 \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-3 \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2 b d n},1+\frac {3}{2 b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )\right )}{3 b d n (3+2 b d n)} \]
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\[\int x^{2} {\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{2}d x\]
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\[ \int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
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\[ \int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^{2} \coth ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
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\[ \int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { x^{2} \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2} \,d x } \]
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Timed out. \[ \int x^2 \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int x^2\,{\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2 \,d x \]
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