Optimal. Leaf size=184 \[ -\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 x^2}+\frac {4^{-1+\frac {1}{n}} e^{2 i a} \left (-i b x^n\right )^{2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {2}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}+\frac {4^{-1+\frac {1}{n}} e^{-2 i a} \left (i b x^n\right )^{2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {2}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2} \]
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Rubi [A]
time = 0.19, antiderivative size = 184, 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} 4^{\frac {1}{n}-1} \left (-i b x^n\right )^{2/n} \text {Gamma}\left (-\frac {2}{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 x^2}+\frac {e^{-2 i a} 4^{\frac {1}{n}-1} \left (i b x^n\right )^{2/n} \text {Gamma}\left (-\frac {2}{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 x^2}-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 3505
Rule 3506
Rule 6852
Rubi steps
\begin {align*} \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x^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 \frac {\sin ^2\left (a+b x^n\right )}{x^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 \left (\frac {1}{2 x^3}-\frac {\cos \left (2 a+2 b x^n\right )}{2 x^3}\right ) \, dx\\ &=-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 x^2}-\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 \frac {\cos \left (2 a+2 b x^n\right )}{x^3} \, dx\\ &=-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 x^2}-\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 \frac {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 \frac {e^{2 i a+2 i b x^n}}{x^3} \, dx\\ &=-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 x^2}+\frac {4^{-1+\frac {1}{n}} e^{2 i a} \left (-i b x^n\right )^{2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {2}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}+\frac {4^{-1+\frac {1}{n}} e^{-2 i a} \left (i b x^n\right )^{2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {2}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 129, normalized size = 0.70 \begin {gather*} \frac {e^{-2 i a} \csc ^2\left (a+b x^n\right ) \left (-e^{2 i a} n+4^{\frac {1}{n}} e^{4 i a} \left (-i b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-2 i b x^n\right )+4^{\frac {1}{n}} \left (i b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 n x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {2}{3}}}{x^{3}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \sin ^{3}{\left (a + b x^{n} \right )}\right )^{\frac {2}{3}}}{x^{3}}\, 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 )}^{2/3}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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