Optimal. Leaf size=85 \[ -\frac{2 e^{i a} \left (c x^n\right )^{i b} \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (1+\frac{2 i}{b n}\right ),\frac{1}{2} \left (3+\frac{2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x^2 (b n+2 i)} \]
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Rubi [A] time = 0.0585942, antiderivative size = 85, 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 = {4510, 4506, 364} \[ -\frac{2 e^{i a} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac{1}{2} \left (1+\frac{2 i}{b n}\right );\frac{1}{2} \left (3+\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x^2 (b n+2 i)} \]
Antiderivative was successfully verified.
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Rule 4510
Rule 4506
Rule 364
Rubi steps
\begin{align*} \int \frac{\csc \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int x^{-1-\frac{2}{n}} \csc (a+b \log (x)) \, dx,x,c x^n\right )}{n x^2}\\ &=-\frac{\left (2 i e^{i a} \left (c x^n\right )^{2/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+i b-\frac{2}{n}}}{1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n x^2}\\ &=-\frac{2 e^{i a} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac{1}{2} \left (1+\frac{2 i}{b n}\right );\frac{1}{2} \left (3+\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2 i+b n) x^2}\\ \end{align*}
Mathematica [A] time = 1.09152, size = 78, normalized size = 0.92 \[ -\frac{2 e^{i a} \left (c x^n\right )^{i b} \text{Hypergeometric2F1}\left (1,\frac{1}{2}+\frac{i}{b n},\frac{3}{2}+\frac{i}{b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{x^2 (b n+2 i)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.454, size = 0, normalized size = 0. \begin{align*} \int{\frac{\csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{{x}^{3}}}\, 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{\csc \left (b \log \left (c x^{n}\right ) + a\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\csc \left (b \log \left (c x^{n}\right ) + a\right )}{x^{3}}, x\right ) \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{\csc{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b \log \left (c x^{n}\right ) + a\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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