Integrand size = 21, antiderivative size = 65 \[ \int \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\frac {(2-p) x \left (1+e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}\right ) \text {sech}^p\left (a+\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
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Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5664, 5668, 270} \[ \int \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\frac {(2-p) x \left (e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}+1\right ) \text {sech}^p\left (a+\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
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Rule 270
Rule 5664
Rule 5668
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 {sech}^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 {sech}^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 {(2-p) x \left (1+e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}\right ) \text {sech}^p\left (a+\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \\ \end{align*}
Time = 1.42 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.66 \[ \int \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\frac {2^{-1+p} e^{-a} (-2+p) x \left (c x^n\right )^{\frac {1}{n (-2+p)}} \left (\frac {e^{\frac {a (2+p)}{-2+p}} \left (c x^n\right )^{\frac {1}{n (-2+p)}}}{e^{\frac {2 a p}{-2+p}}+e^{\frac {4 a}{-2+p}} \left (c x^n\right )^{\frac {2}{n (-2+p)}}}\right )^{-1+p}}{-1+p} \]
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\[\int {\operatorname {sech}\left (a -\frac {\ln \left (c \,x^{n}\right )}{n \left (-2+p \right )}\right )}^{p}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 538 vs. \(2 (55) = 110\).
Time = 0.27 (sec) , antiderivative size = 538, normalized size of antiderivative = 8.28 \[ \int \text {sech}^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 ) \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 - 2\right )} x \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 (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 )}{{\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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\[ \int \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\int \operatorname {sech}^{p}{\left (a - \frac {\log {\left (c x^{n} \right )}}{n \left (p - 2\right )} \right )}\, dx \]
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\[ \int \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\int { \operatorname {sech}\left (a - \frac {\log \left (c x^{n}\right )}{n {\left (p - 2\right )}}\right )^{p} \,d x } \]
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\[ \int \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\int { \operatorname {sech}\left (a - \frac {\log \left (c x^{n}\right )}{n {\left (p - 2\right )}}\right )^{p} \,d x } \]
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Timed out. \[ \int \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\int {\left (\frac {1}{\mathrm {cosh}\left (a-\frac {\ln \left (c\,x^n\right )}{n\,\left (p-2\right )}\right )}\right )}^p \,d x \]
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