Optimal. Leaf size=157 \[ \frac {i e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \csc \left (a+b x^n\right ) \Gamma \left (\frac {m+1}{n},-i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n}-\frac {i e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \csc \left (a+b x^n\right ) \Gamma \left (\frac {m+1}{n},i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n} \]
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Rubi [A] time = 0.38, antiderivative size = 157, 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 = {6720, 3423, 2218} \[ \frac {i e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \csc \left (a+b x^n\right ) \text {Gamma}\left (\frac {m+1}{n},-i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n}-\frac {i e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \csc \left (a+b x^n\right ) \text {Gamma}\left (\frac {m+1}{n},i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 3423
Rule 6720
Rubi steps
\begin {align*} \int x^m \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \, dx &=\left (\csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}\right ) \int x^m \sin \left (a+b x^n\right ) \, 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 e^{-i a-i b x^n} x^m \, 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 e^{i a+i b x^n} x^m \, dx\\ &=\frac {i e^{i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \csc \left (a+b x^n\right ) \Gamma \left (\frac {1+m}{n},-i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n}-\frac {i e^{-i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \csc \left (a+b x^n\right ) \Gamma \left (\frac {1+m}{n},i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 142, normalized size = 0.90 \[ \frac {i x^{m+1} \left (b^2 x^{2 n}\right )^{-\frac {m+1}{n}} \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left ((\cos (a)+i \sin (a)) \left (i b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-i b x^n\right )-(\cos (a)-i \sin (a)) \left (-i b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},i b x^n\right )\right )}{2 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac {1}{3}} x^{m}, x\right ) \]
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 \[ \int \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {1}{3}} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.49, size = 0, normalized size = 0.00 \[ \int x^{m} \left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {1}{3}}\, dx \]
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 \[ \int \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {1}{3}} x^{m}\,{d x} \]
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 \[ \int x^m\,{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{1/3} \,d x \]
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 \[ \int x^{m} \sqrt [3]{c \sin ^{3}{\left (a + b x^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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