Optimal. Leaf size=217 \[ \frac {x^{1+m} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (1+m)}+\frac {2^{-\frac {1+m+2 n}{n}} e^{2 i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {1+m}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {2^{-\frac {1+m+2 n}{n}} e^{-2 i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {1+m}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]
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Rubi [A]
time = 0.26, antiderivative size = 217, 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 = {6852, 3506,
3505, 2250} \begin {gather*} \frac {e^{2 i a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \csc ^2\left (a+b x^n\right ) \text {Gamma}\left (\frac {m+1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {e^{-2 i a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \csc ^2\left (a+b x^n\right ) \text {Gamma}\left (\frac {m+1}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {x^{m+1} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 3505
Rule 3506
Rule 6852
Rubi steps
\begin {align*} \int x^m \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^m \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^m}{2}-\frac {1}{2} x^m \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac {x^{1+m} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (1+m)}-\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^m \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac {x^{1+m} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (1+m)}-\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^m \, 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^m \, dx\\ &=\frac {x^{1+m} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (1+m)}+\frac {2^{-\frac {1+m+2 n}{n}} e^{2 i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {1+m}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {2^{-\frac {1+m+2 n}{n}} e^{-2 i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {1+m}{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 = 0.53, size = 194, normalized size = 0.89 \begin {gather*} \frac {2^{-\frac {1+m+2 n}{n}} e^{-2 i a} x^{1+m} \left (b^2 x^{2 n}\right )^{-\frac {1+m}{n}} \csc ^2\left (a+b x^n\right ) \left (2^{\frac {1+m+n}{n}} e^{2 i a} n \left (b^2 x^{2 n}\right )^{\frac {1+m}{n}}+e^{4 i a} (1+m) \left (i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-2 i b x^n\right )+(1+m) \left (-i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{(1+m) n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int x^{m} \left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {2}{3}}\, dx\]
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 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.00 \begin {gather*} \int x^m\,{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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