Optimal. Leaf size=63 \[ \frac{2 e^a x \left (c x^n\right )^b \, _2F_1\left (1,\frac{b+\frac{1}{n}}{2 b};\frac{1}{2} \left (3+\frac{1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{b n+1} \]
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Rubi [A] time = 0.0586494, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {5545, 5547, 263, 364} \[ \frac{2 e^a x \left (c x^n\right )^b \, _2F_1\left (1,\frac{b+\frac{1}{n}}{2 b};\frac{1}{2} \left (3+\frac{1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{b n+1} \]
Antiderivative was successfully verified.
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Rule 5545
Rule 5547
Rule 263
Rule 364
Rubi steps
\begin{align*} \int \text{sech}\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \text{sech}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (2 e^{-a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1-b+\frac{1}{n}}}{1+e^{-2 a} x^{-2 b}} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (2 e^{-a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+b+\frac{1}{n}}}{e^{-2 a}+x^{2 b}} \, dx,x,c x^n\right )}{n}\\ &=\frac{2 e^a x \left (c x^n\right )^b \, _2F_1\left (1,\frac{b+\frac{1}{n}}{2 b};\frac{1}{2} \left (3+\frac{1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+b n}\\ \end{align*}
Mathematica [A] time = 0.1466, size = 64, normalized size = 1.02 \[ \frac{2 e^a x \left (c x^n\right )^b \, _2F_1\left (1,\frac{1}{2} \left (1+\frac{1}{b n}\right );\frac{1}{2} \left (3+\frac{1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{b n+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.084, size = 0, normalized size = 0. \begin{align*} \int{\rm sech} \left (a+b\ln \left ( c{x}^{n} \right ) \right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}\left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{sech}\left (b \log \left (c x^{n}\right ) + a\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}{\left (a + b \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}\left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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