Optimal. Leaf size=130 \[ \frac{2 x^{m+1} \sqrt{1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{-b n+2 i m+2 i}{4 b n},-\frac{-5 b n+2 i m+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i b n+2 m+2} \]
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Rubi [A] time = 0.0924365, 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 = {4510, 4508, 364} \[ \frac{2 x^{m+1} \sqrt{1-e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac{1}{2},-\frac{2 i m-b n+2 i}{4 b n};-\frac{2 i m-5 b n+2 i}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )}}{i b n+2 m+2} \]
Antiderivative was successfully verified.
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Rule 4510
Rule 4508
Rule 364
Rubi steps
\begin{align*} \int x^m \sqrt{\csc \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{\csc (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{\csc \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}} \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} \, _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 )}{2+2 m+i b n}\\ \end{align*}
Mathematica [A] time = 0.969139, size = 138, normalized size = 1.06 \[ \frac{2 e^{-2 i a} x^{m+1} \left (c x^n\right )^{-2 i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,\frac{3 b n+2 i m+2 i}{4 b n},\frac{5 b n+2 i m+2 i}{4 b n},e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{-i b n+2 m+2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.289, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{\csc \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{\csc \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{\csc{\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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