Optimal. Leaf size=65 \[ \frac {(2-p) x \left (e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}+1\right ) \text {sech}^p\left (a+\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
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Rubi [A] time = 0.08, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5545, 5549, 264} \[ \frac {(2-p) x \left (e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}+1\right ) \text {sech}^p\left (a+\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
Antiderivative was successfully verified.
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Rule 264
Rule 5545
Rule 5549
Rubi steps
\begin {align*} \int \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {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 ) \operatorname {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*}
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Mathematica [A] time = 0.88, size = 62, normalized size = 0.95 \[ \frac {e^{-2 a} (p-2) x \left (e^{2 a}+\left (c x^n\right )^{\frac {2}{n (p-2)}}\right ) \text {sech}^p\left (a+\frac {\log \left (c x^n\right )}{2 n-n p}\right )}{2 (p-1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 538, normalized size = 8.28 \[ \frac {{\left (p - 2\right )} x \cosh \left (p \log \left (\frac {2 \, {\left (\cosh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right ) + \sinh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right )\right )}}{\cosh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right ) \sinh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right ) + \sinh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right )^{2} + 1}\right )\right ) \cosh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right ) + {\left (p - 2\right )} x \cosh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right ) \sinh \left (p \log \left (\frac {2 \, {\left (\cosh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right ) + \sinh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right )\right )}}{\cosh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right ) \sinh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right ) + \sinh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right )^{2} + 1}\right )\right )}{{\left (p - 1\right )} \cosh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right ) - {\left (p - 1\right )} \sinh \left (-\frac {a n p - 2 \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {sech}\left (a - \frac {\log \left (c x^{n}\right )}{n {\left (p - 2\right )}}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\left (a -\frac {\ln \left (c \,x^{n}\right )}{n \left (-2+p \right )}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {sech}\left (-a + \frac {\log \left (c x^{n}\right )}{n {\left (p - 2\right )}}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {1}{\mathrm {cosh}\left (a-\frac {\ln \left (c\,x^n\right )}{n\,\left (p-2\right )}\right )}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {sech}^{p}{\left (a - \frac {\log {\left (c x^{n} \right )}}{n \left (p - 2\right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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