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 = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 263
Rule 364
Rule 5545
Rule 5547
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*}
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Mathematica [A] time = 0.15, 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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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right ), x\right ) \]
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 \[ \int \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]
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 \[ \int \operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \]
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 \[ \int \frac {1}{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
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 \[ \int \operatorname {sech}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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