Integrand size = 13, antiderivative size = 68 \[ \int \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {4 e^{2 a} x \left (c x^n\right )^{2 b} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2} \left (2+\frac {1}{b n}\right ),\frac {1}{2} \left (4+\frac {1}{b n}\right ),e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+2 b n} \]
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Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5665, 5667, 269, 371} \[ \int \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {4 e^{2 a} x \left (c x^n\right )^{2 b} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2} \left (2+\frac {1}{b n}\right ),\frac {1}{2} \left (4+\frac {1}{b n}\right ),e^{2 a} \left (c x^n\right )^{2 b}\right )}{2 b n+1} \]
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Rule 269
Rule 371
Rule 5665
Rule 5667
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \text {csch}^2(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (4 e^{-2 a} x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1-2 b+\frac {1}{n}}}{\left (1-e^{-2 a} x^{-2 b}\right )^2} \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (4 e^{-2 a} x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+2 b+\frac {1}{n}}}{\left (-e^{-2 a}+x^{2 b}\right )^2} \, dx,x,c x^n\right )}{n} \\ & = \frac {4 e^{2 a} x \left (c x^n\right )^{2 b} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2} \left (2+\frac {1}{b n}\right ),\frac {1}{2} \left (4+\frac {1}{b n}\right ),e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+2 b n} \\ \end{align*}
Time = 2.86 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.85 \[ \int \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (-\coth \left (a+b \log \left (c x^n\right )\right )-\frac {e^{2 a} \left (c x^n\right )^{2 b} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{2 b n},2+\frac {1}{2 b n},e^{2 \left (a+b \log \left (c x^n\right )\right )}\right )}{1+2 b n}-\operatorname {Hypergeometric2F1}\left (1,\frac {1}{2 b n},1+\frac {1}{2 b n},e^{2 \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b n} \]
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\[\int {\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}d x\]
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\[ \int \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{2} \,d x } \]
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\[ \int \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \operatorname {csch}^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{2} \,d x } \]
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\[ \int \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{2} \,d x } \]
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Timed out. \[ \int \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {1}{{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
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