Optimal. Leaf size=188 \[ \frac {1}{8} x^4 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {e^{2 i a} 4^{-\frac {2}{n}-1} x^4 \left (-i b x^n\right )^{-4/n} \Gamma \left (\frac {4}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {e^{-2 i a} 4^{-\frac {2}{n}-1} x^4 \left (i b x^n\right )^{-4/n} \Gamma \left (\frac {4}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]
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Rubi [A] time = 0.34, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6720, 3425, 3424, 2218} \[ \frac {e^{2 i a} 4^{-\frac {2}{n}-1} x^4 \left (-i b x^n\right )^{-4/n} \text {Gamma}\left (\frac {4}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {e^{-2 i a} 4^{-\frac {2}{n}-1} x^4 \left (i b x^n\right )^{-4/n} \text {Gamma}\left (\frac {4}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {1}{8} x^4 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 3424
Rule 3425
Rule 6720
Rubi steps
\begin {align*} \int x^3 \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx &=\left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int x^3 \sin ^2\left (a+b x^n\right ) \, dx\\ &=\left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int \left (\frac {x^3}{2}-\frac {1}{2} x^3 \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac {1}{8} x^4 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}-\frac {1}{2} \left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int x^3 \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac {1}{8} x^4 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}-\frac {1}{4} \left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int e^{-2 i a-2 i b x^n} x^3 \, dx-\frac {1}{4} \left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int e^{2 i a+2 i b x^n} x^3 \, dx\\ &=\frac {1}{8} x^4 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {4^{-1-\frac {2}{n}} e^{2 i a} x^4 \left (-i b x^n\right )^{-4/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {4}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {4^{-1-\frac {2}{n}} e^{-2 i a} x^4 \left (i b x^n\right )^{-4/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {4}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 161, normalized size = 0.86 \[ \frac {e^{-2 i a} 2^{-\frac {4}{n}-3} x^4 \left (b^2 x^{2 n}\right )^{-4/n} \csc ^2\left (a+b x^n\right ) \left (e^{2 i a} 16^{\frac {1}{n}} n \left (b^2 x^{2 n}\right )^{4/n}+2 e^{4 i a} \left (i b x^n\right )^{4/n} \Gamma \left (\frac {4}{n},-2 i b x^n\right )+2 \left (-i b x^n\right )^{4/n} \Gamma \left (\frac {4}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac {2}{3}} x^{3}, x\right ) \]
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 \[ \int \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int x^{3} \left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {2}{3}}\, dx \]
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 \[ -\frac {1}{16} \, {\left (x^{4} - 4 \, \int x^{3} \cos \left (2 \, b x^{n} + 2 \, a\right )\,{d x}\right )} c^{\frac {2}{3}} \]
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 \[ \int x^3\,{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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