Optimal. Leaf size=69 \[ \frac{4 e^{2 a} x \left (c x^n\right )^{2 b} \, _2F_1\left (2,\frac{1}{2} \left (2+\frac{1}{b n}\right );\frac{1}{2} \left (4+\frac{1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{2 b n+1} \]
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Rubi [A] time = 0.0698002, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {5545, 5547, 263, 364} \[ \frac{4 e^{2 a} x \left (c x^n\right )^{2 b} \, _2F_1\left (2,\frac{1}{2} \left (2+\frac{1}{b n}\right );\frac{1}{2} \left (4+\frac{1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{2 b n+1} \]
Antiderivative was successfully verified.
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Rule 5545
Rule 5547
Rule 263
Rule 364
Rubi steps
\begin{align*} \int \text{sech}^2\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}^2(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (4 e^{-2 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1-2 b+\frac{1}{n}}}{\left (1+e^{-2 a} x^{-2 b}\right )^2} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (4 e^{-2 a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+2 b+\frac{1}{n}}}{\left (e^{-2 a}+x^{2 b}\right )^2} \, dx,x,c x^n\right )}{n}\\ &=\frac{4 e^{2 a} x \left (c x^n\right )^{2 b} \, _2F_1\left (2,\frac{1}{2} \left (2+\frac{1}{b n}\right );\frac{1}{2} \left (4+\frac{1}{b n}\right );-e^{2 a} \left (c x^n\right )^{2 b}\right )}{1+2 b n}\\ \end{align*}
Mathematica [A] time = 5.41015, size = 126, normalized size = 1.83 \[ \frac{x \left (-\frac{e^{2 a} \left (c x^n\right )^{2 b} \, _2F_1\left (1,1+\frac{1}{2 b n};2+\frac{1}{2 b n};-e^{2 a} \left (c x^n\right )^{2 b}\right )}{2 b n+1}+\, _2F_1\left (1,\frac{1}{2 b n};1+\frac{1}{2 b n};-e^{2 a} \left (c x^n\right )^{2 b}\right )+\tanh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.099, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm sech} \left (a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}\, 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*} -\frac{2 \, x}{b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n} + 4 \, \int \frac{1}{2 \,{\left (b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\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 )^{2}, 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}^{2}{\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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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