Optimal. Leaf size=106 \[ \frac{x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \text{Hypergeometric2F1}\left (p,\frac{1}{2} \left (p-\frac{2 i}{b n}\right ),\frac{1}{2} \left (-\frac{2 i}{b n}+p+2\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^p\left (a+b \log \left (c x^n\right )\right )}{2+i b n p} \]
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Rubi [A] time = 0.0827361, antiderivative size = 106, 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, 4507, 364} \[ \frac{x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \, _2F_1\left (p,\frac{1}{2} \left (p-\frac{2 i}{b n}\right );\frac{1}{2} \left (p-\frac{2 i}{b n}+2\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^p\left (a+b \log \left (c x^n\right )\right )}{2+i b n p} \]
Antiderivative was successfully verified.
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Rule 4509
Rule 4507
Rule 364
Rubi steps
\begin{align*} \int x \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{2}{n}} \sec ^p(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x^2 \left (c x^n\right )^{-\frac{2}{n}-i b p} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \sec ^p\left (a+b \log \left (c x^n\right )\right )\right ) \operatorname{Subst}\left (\int x^{-1+\frac{2}{n}+i b p} \left (1+e^{2 i a} x^{2 i b}\right )^{-p} \, dx,x,c x^n\right )}{n}\\ &=\frac{x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \, _2F_1\left (p,\frac{1}{2} \left (-\frac{2 i}{b n}+p\right );\frac{1}{2} \left (2-\frac{2 i}{b n}+p\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^p\left (a+b \log \left (c x^n\right )\right )}{2+i b n p}\\ \end{align*}
Mathematica [A] time = 0.962605, size = 142, normalized size = 1.34 \[ -\frac{i 2^p x^2 \left (\frac{e^{i a} \left (c x^n\right )^{i b}}{1+e^{2 i a} \left (c x^n\right )^{2 i b}}\right )^p \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \text{Hypergeometric2F1}\left (\frac{p}{2}-\frac{i}{b n},p,-\frac{i}{b n}+\frac{p}{2}+1,-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{b n p-2 i} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.212, size = 0, normalized size = 0. \begin{align*} \int x \left ( \sec \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{p}\, 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 \sec \left (b \log \left (c x^{n}\right ) + a\right )^{p}\,{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 \sec \left (b \log \left (c x^{n}\right ) + a\right )^{p}, 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 \sec ^{p}{\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 \sec \left (b \log \left (c x^{n}\right ) + a\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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