Optimal. Leaf size=129 \[ \frac{2 x^{m+1} \text{Hypergeometric2F1}\left (-\frac{1}{2},-\frac{b n+2 i m+2 i}{4 b n},-\frac{-3 b n+2 i m+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-i b n+2 m+2) \sqrt{1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )}} \]
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Rubi [A] time = 0.0883437, antiderivative size = 126, normalized size of antiderivative = 0.98, 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} \[ \frac{2 x^{m+1} \, _2F_1\left (-\frac{1}{2},\frac{1}{4} \left (-\frac{2 i (m+1)}{b n}-1\right );-\frac{2 i m-3 b n+2 i}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-i b n+2 m+2) \sqrt{1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )}} \]
Antiderivative was successfully verified.
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Rule 4510
Rule 4508
Rule 364
Rubi steps
\begin{align*} \int \frac{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 \frac{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}}\right ) \operatorname{Subst}\left (\int 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 \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 )}}\\ &=\frac{2 x^{1+m} \, _2F_1\left (-\frac{1}{2},\frac{1}{4} \left (-1-\frac{2 i (1+m)}{b n}\right );-\frac{2 i+2 i m-3 b n}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-i b 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 )}}\\ \end{align*}
Mathematica [B] time = 7.34828, size = 441, normalized size = 3.42 \[ \frac{2 x^{m+1} \sin \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{\sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} \left (2 (m+1) \sin \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+b n \cos \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}-\frac{2 e^{i a} b n x^{m+1} \left (c x^n\right )^{i b} \sqrt{2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt{\frac{i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \left ((b n+2 i m+2 i) x^{2 i b n} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{i \left (\frac{3 i b n}{2}+m+1\right )}{2 b n},-\frac{-7 b n+2 i m+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(3 b n-2 i m-2 i) \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{b n+2 i m+2 i}{4 b n},-\frac{-3 b n+2 i m+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )}{(-i b n+2 m+2) (3 i b n+2 m+2) \left (e^{2 i a} (b n-2 i m-2 i) \left (c x^n\right )^{2 i b}+(b n+2 i m+2 i) x^{2 i b n}\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.279, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}{\frac{1}{\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 \frac{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 \frac{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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