Optimal. Leaf size=70 \[ \frac{(2-p) x \left (1+e^{2 i a} \left (c x^n\right )^{-\frac{2}{n (2-p)}}\right ) \sec ^p\left (a+\frac{i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
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Rubi [A] time = 0.0751668, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4503, 4507, 264} \[ \frac{(2-p) x \left (1+e^{2 i a} \left (c x^n\right )^{-\frac{2}{n (2-p)}}\right ) \sec ^p\left (a+\frac{i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
Antiderivative was successfully verified.
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Rule 4503
Rule 4507
Rule 264
Rubi steps
\begin{align*} \int \sec ^p\left (a-\frac{i \log \left (c x^n\right )}{n (-2+p)}\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \sec ^p\left (a-\frac{i \log (x)}{n (-2+p)}\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-\frac{1}{n}-\frac{p}{n (-2+p)}} \left (1+e^{2 i a} \left (c x^n\right )^{\frac{2}{n (-2+p)}}\right )^p \sec ^p\left (a-\frac{i \log \left (c x^n\right )}{n (-2+p)}\right )\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}+\frac{p}{n (-2+p)}} \left (1+e^{2 i a} x^{\frac{2}{n (-2+p)}}\right )^{-p} \, dx,x,c x^n\right )}{n}\\ &=\frac{(2-p) x \left (1+e^{2 i a} \left (c x^n\right )^{-\frac{2}{n (2-p)}}\right ) \sec ^p\left (a+\frac{i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)}\\ \end{align*}
Mathematica [A] time = 0.791994, size = 62, normalized size = 0.89 \[ \frac{(p-2) x \left (1+e^{2 i a} \left (c x^n\right )^{\frac{2}{n (p-2)}}\right ) \sec ^p\left (a-\frac{i \log \left (c x^n\right )}{n (p-2)}\right )}{2 (p-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.322, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( a-{\frac{i\ln \left ( c{x}^{n} \right ) }{n \left ( p-2 \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 \sec \left (-a + \frac{i \, \log \left (c x^{n}\right )}{n{\left (p - 2\right )}}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.50292, size = 346, normalized size = 4.94 \begin{align*} \frac{{\left ({\left (p - 2\right )} x e^{\left (\frac{2 \,{\left (-i \, a n p + 2 i \, a n - \log \left (c x^{n}\right )\right )}}{n p - 2 \, n}\right )} +{\left (p - 2\right )} x\right )} \left (\frac{2 \, e^{\left (\frac{-i \, a n p + 2 i \, a n - \log \left (c x^{n}\right )}{n p - 2 \, n}\right )}}{e^{\left (\frac{2 \,{\left (-i \, a n p + 2 i \, a n - \log \left (c x^{n}\right )\right )}}{n p - 2 \, n}\right )} + 1}\right )^{p} e^{\left (-\frac{2 \,{\left (-i \, a n p + 2 i \, a n - \log \left (c x^{n}\right )\right )}}{n p - 2 \, n}\right )}}{2 \,{\left (p - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \sec \left (a - \frac{i \, \log \left (c x^{n}\right )}{n{\left (p - 2\right )}}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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