Optimal. Leaf size=109 \[ \frac{2 x \sqrt{1-e^{2 i a} \left (c x^n\right )^{2 i b}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{4} \left (1-\frac{2 i}{b n}\right ),\frac{1}{4} \left (5-\frac{2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )}}{2+i b n} \]
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Rubi [A] time = 0.0705614, antiderivative size = 109, 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 = {4504, 4508, 364} \[ \frac{2 x \sqrt{1-e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac{1}{2},\frac{1}{4} \left (1-\frac{2 i}{b n}\right );\frac{1}{4} \left (5-\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )}}{2+i b n} \]
Antiderivative was successfully verified.
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Rule 4504
Rule 4508
Rule 364
Rubi steps
\begin{align*} \int \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \sqrt{\csc (a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-\frac{i b}{2}-\frac{1}{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}{n}}}{\sqrt{1-e^{2 i a} x^{2 i b}}} \, dx,x,c x^n\right )}{n}\\ &=\frac{2 x \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{1}{4} \left (1-\frac{2 i}{b n}\right );\frac{1}{4} \left (5-\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+i b n}\\ \end{align*}
Mathematica [A] time = 0.645026, size = 115, normalized size = 1.06 \[ \frac{2 i e^{-2 i a} x \left (c x^n\right )^{-2 i b} \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} \text{Hypergeometric2F1}\left (1,\frac{3}{4}+\frac{i}{2 b n},\frac{5}{4}+\frac{i}{2 b n},e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{b n+2 i} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.364, size = 0, normalized size = 0. \begin{align*} \int \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 \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 \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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