Optimal. Leaf size=130 \[ \frac{2 x^{m+1} \sqrt{1+e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{-b n+2 i m+2 i}{4 b n},-\frac{-5 b n+2 i m+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i b n+2 m+2} \]
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Rubi [A] time = 0.0914406, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4509, 4507, 364} \[ \frac{2 x^{m+1} \sqrt{1+e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac{1}{2},-\frac{2 i m-b n+2 i}{4 b n};-\frac{2 i m-5 b n+2 i}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )}}{i b n+2 m+2} \]
Antiderivative was successfully verified.
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Rule 4509
Rule 4507
Rule 364
Rubi steps
\begin{align*} \int x^m \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\left (x^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1+m}{n}} \sqrt{\sec (a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x^{1+m} \left (c x^n\right )^{-\frac{i b}{2}-\frac{1+m}{n}} \sqrt{1+e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+\frac{i b}{2}+\frac{1+m}{n}}}{\sqrt{1+e^{2 i a} x^{2 i b}}} \, dx,x,c x^n\right )}{n}\\ &=\frac{2 x^{1+m} \sqrt{1+e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac{1}{2},-\frac{2 i+2 i m-b n}{4 b n};-\frac{2 i+2 i m-5 b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )}}{2+2 m+i b n}\\ \end{align*}
Mathematica [A] time = 0.784376, size = 119, normalized size = 0.92 \[ \frac{2 x^{m+1} \left (1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,-\frac{-3 b n+2 i m+2 i}{4 b n},-\frac{-5 b n+2 i m+2 i}{4 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{i b n+2 m+2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.306, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{\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^{m} \sqrt{\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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^{m} \sqrt{\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(-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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