Integrand size = 17, antiderivative size = 55 \[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {1}{2 x^2}+\frac {\operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{x^2} \]
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Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5659, 5657, 470, 371} \[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{x^2}-\frac {1}{2 x^2} \]
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Rule 371
Rule 470
Rule 5657
Rule 5659
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int x^{-1-\frac {2}{n}} \coth (d (a+b \log (x))) \, dx,x,c x^n\right )}{n x^2} \\ & = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int \frac {x^{-1-\frac {2}{n}} \left (-1-e^{2 a d} x^{2 b d}\right )}{1-e^{2 a d} x^{2 b d}} \, dx,x,c x^n\right )}{n x^2} \\ & = -\frac {1}{2 x^2}-\frac {\left (2 \left (c x^n\right )^{2/n}\right ) \text {Subst}\left (\int \frac {x^{-1-\frac {2}{n}}}{1-e^{2 a d} x^{2 b d}} \, dx,x,c x^n\right )}{n x^2} \\ & = -\frac {1}{2 x^2}+\frac {\operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{x^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(191\) vs. \(2(55)=110\).
Time = 4.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 3.47 \[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\coth \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )-\frac {e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {1}{b d n},2-\frac {1}{b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{-1+b d n}+\operatorname {Hypergeometric2F1}\left (1,-\frac {1}{b d n},1-\frac {1}{b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+\text {csch}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \text {csch}\left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right ) \sinh (b d n \log (x))}{2 x^2} \]
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\[\int \frac {\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}d x\]
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\[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\coth {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \]
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