Optimal. Leaf size=139 \[ \frac {i e^{i a} \left (-i b x^n\right )^{\frac {1}{n}} \csc \left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},-i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x}-\frac {i e^{-i a} \left (i b x^n\right )^{\frac {1}{n}} \csc \left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x} \]
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Rubi [A]
time = 0.14, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6852, 3504,
2250} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 3504
Rule 6852
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*}
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Mathematica [A]
time = 0.11, size = 110, normalized size = 0.79 \begin {gather*} \frac {i \csc \left (a+b x^n\right ) \left (-\left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},i b x^n\right ) (\cos (a)-i \sin (a))+\left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-i b x^n\right ) (\cos (a)+i \sin (a))\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {1}{3}}}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{c \sin ^{3}{\left (a + b x^{n} \right )}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{1/3}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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