Integrand size = 20, antiderivative size = 90 \[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=-\frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {2}{n (2-p)}} \left (1-e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right ) \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
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Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5665, 5669, 267} \[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=-\frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {2}{n (2-p)}} \left (1-e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right ) \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
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Rule 267
Rule 5665
Rule 5669
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}^p\left (a+\frac {\log (x)}{n (-2+p)}\right ) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-\frac {1}{n}+\frac {p}{n (-2+p)}} \left (1-e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (-2+p)}}\right )^p \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right )\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}-\frac {p}{n (-2+p)}} \left (1-e^{-2 a} x^{-\frac {2}{n (-2+p)}}\right )^{-p} \, dx,x,c x^n\right )}{n} \\ & = -\frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {2}{n (2-p)}} \left (1-e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right ) \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \\ \end{align*}
Time = 1.54 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.28 \[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\frac {2^{-1+p} (-2+p) x \left (\frac {e^a \left (c x^n\right )^{\frac {1}{n (-2+p)}}}{-1+e^{2 a} \left (c x^n\right )^{\frac {2}{n (-2+p)}}}\right )^p \left (1+e^{2 a} \left (c x^n\right )^{\frac {2}{n (-2+p)}} \left (-1+\left (1-e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (-2+p)}}\right )^p\right )\right )}{-1+p} \]
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\[\int {\operatorname {csch}\left (a +\frac {\ln \left (c \,x^{n}\right )}{n \left (-2+p \right )}\right )}^{p}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (76) = 152\).
Time = 0.27 (sec) , antiderivative size = 475, normalized size of antiderivative = 5.28 \[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=-\frac {{\left (p - 2\right )} x \cosh \left (p \log \left (\frac {2 \, {\left (\cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )\right )}}{\cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} - 1}\right )\right ) \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + {\left (p - 2\right )} x \sinh \left (p \log \left (\frac {2 \, {\left (\cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )\right )}}{\cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )^{2} - 1}\right )\right ) \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )}{{\left (p - 1\right )} \cosh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right ) - {\left (p - 1\right )} \sinh \left (\frac {a n p - 2 \, a n + n \log \left (x\right ) + \log \left (c\right )}{n p - 2 \, n}\right )} \]
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\[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\int \operatorname {csch}^{p}{\left (a + \frac {\log {\left (c x^{n} \right )}}{n \left (p - 2\right )} \right )}\, dx \]
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\[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\int { \operatorname {csch}\left (a + \frac {\log \left (c x^{n}\right )}{n {\left (p - 2\right )}}\right )^{p} \,d x } \]
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Timed out. \[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\int {\left (\frac {1}{\mathrm {sinh}\left (a+\frac {\ln \left (c\,x^n\right )}{n\,\left (p-2\right )}\right )}\right )}^p \,d x \]
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