Optimal. Leaf size=130 \[ \frac{2 x^{m+1} \text{Hypergeometric2F1}\left (-\frac{3}{2},-\frac{3 b n+2 i m+2 i}{4 b n},-\frac{-b n+2 i m+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-3 i b n+2 m+2) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.0943862, antiderivative size = 126, normalized size of antiderivative = 0.97, 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} \[ \frac{2 x^{m+1} \, _2F_1\left (-\frac{3}{2},\frac{1}{4} \left (-\frac{2 i (m+1)}{b n}-3\right );-\frac{2 i m-b n+2 i}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-3 i b n+2 m+2) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 4510
Rule 4508
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m}{\csc ^{\frac{3}{2}}\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}}}{\csc ^{\frac{3}{2}}(a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x^{1+m} \left (c x^n\right )^{\frac{3 i b}{2}-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int x^{-1-\frac{3 i b}{2}+\frac{1+m}{n}} \left (1-e^{2 i a} x^{2 i b}\right )^{3/2} \, dx,x,c x^n\right )}{n \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ &=\frac{2 x^{1+m} \, _2F_1\left (-\frac{3}{2},\frac{1}{4} \left (-3-\frac{2 i (1+m)}{b n}\right );-\frac{2 i+2 i m-b n}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 2.40706, size = 218, normalized size = 1.68 \[ \frac{2 x^{m+1} \left (3 e^{-2 i a} b^2 n^2 \left (c x^n\right )^{-2 i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^2\left (a+b \log \left (c x^n\right )\right ) \text{Hypergeometric2F1}\left (1,\frac{3 b n+2 i m+2 i}{4 b n},\frac{5 b n+2 i m+2 i}{4 b n},e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right )+(-i b n+2 m+2) \left (-3 b n \cot \left (a+b \log \left (c x^n\right )\right )+2 m+2\right )\right )}{(-i b n+2 m+2) (-3 i b n+2 m+2) (3 i b n+2 m+2) \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.307, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( \csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{-{\frac{3}{2}}}}\, 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}}{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}\,{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(-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(-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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