Optimal. Leaf size=139 \[ \frac{i e^{i a} \left (-i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x}-\frac{i e^{-i a} \left (i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x} \]
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Rubi [A] time = 0.204583, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6720, 3423, 2218} \[ \frac{i e^{i a} \left (-i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x}-\frac{i e^{-i a} \left (i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3423
Rule 2218
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x^2} \, dx &=\left (\csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}\right ) \int \frac{\sin \left (a+b x^n\right )}{x^2} \, dx\\ &=\frac{1}{2} \left (i \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}\right ) \int \frac{e^{-i a-i b x^n}}{x^2} \, dx-\frac{1}{2} \left (i \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}\right ) \int \frac{e^{i a+i b x^n}}{x^2} \, dx\\ &=\frac{i e^{i a} \left (-i b x^n\right )^{\frac{1}{n}} \csc \left (a+b x^n\right ) \Gamma \left (-\frac{1}{n},-i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x}-\frac{i e^{-i a} \left (i b x^n\right )^{\frac{1}{n}} \csc \left (a+b x^n\right ) \Gamma \left (-\frac{1}{n},i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x}\\ \end{align*}
Mathematica [A] time = 0.159705, size = 110, normalized size = 0.79 \[ \frac{i \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left ((\cos (a)+i \sin (a)) \left (-i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},-i b x^n\right )-(\cos (a)-i \sin (a)) \left (i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (-\frac{1}{n},i b x^n\right )\right )}{2 n x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.185, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt [3]{c \left ( \sin \left ( a+b{x}^{n} \right ) \right ) ^{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{\left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac{1}{3}}}{x^{2}}\,{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{\left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac{1}{3}}}{x^{2}}, x\right ) \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac{1}{3}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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