Optimal. Leaf size=87 \[ \frac{2 e^{i a} x^3 \left (c x^n\right )^{i b} \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (1-\frac{3 i}{b n}\right ),\frac{3}{2} \left (1-\frac{i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3+i b n} \]
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Rubi [A] time = 0.0594096, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4509, 4505, 364} \[ \frac{2 e^{i a} x^3 \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac{1}{2} \left (1-\frac{3 i}{b n}\right );\frac{3}{2} \left (1-\frac{i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3+i b n} \]
Antiderivative was successfully verified.
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Rule 4509
Rule 4505
Rule 364
Rubi steps
\begin{align*} \int x^2 \sec \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{3}{n}} \sec (a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (2 e^{i a} x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+i b+\frac{3}{n}}}{1+e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n}\\ &=\frac{2 e^{i a} x^3 \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac{1}{2} \left (1-\frac{3 i}{b n}\right );\frac{3}{2} \left (1-\frac{i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3+i b n}\\ \end{align*}
Mathematica [A] time = 0.167949, size = 86, normalized size = 0.99 \[ -\frac{2 i e^{i a} x^3 \left (c x^n\right )^{i b} \text{Hypergeometric2F1}\left (1,\frac{1}{2}-\frac{3 i}{2 b n},\frac{3}{2}-\frac{3 i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{b n-3 i} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.275, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\sec \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 x^{2} \sec \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 (x^{2} \sec \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 x^{2} \sec{\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 x^{2} \sec \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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