Optimal. Leaf size=180 \[ -\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x}+\frac {e^{2 i a} 2^{\frac {1}{n}-2} \left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x}+\frac {e^{-2 i a} 2^{\frac {1}{n}-2} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x} \]
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Rubi [A] time = 0.28, antiderivative size = 180, 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} \[ \frac {e^{2 i a} 2^{\frac {1}{n}-2} \left (-i b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x}+\frac {e^{-2 i a} 2^{\frac {1}{n}-2} \left (i b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x}-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 3424
Rule 3425
Rule 6720
Rubi steps
\begin {align*} \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x^2} \, 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^2} \, 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^2}-\frac {\cos \left (2 a+2 b x^n\right )}{2 x^2}\right ) \, dx\\ &=-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x}-\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^2} \, dx\\ &=-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x}-\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^2} \, 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^2} \, dx\\ &=-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x}+\frac {2^{-2+\frac {1}{n}} e^{2 i a} \left (-i b x^n\right )^{\frac {1}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x}+\frac {2^{-2+\frac {1}{n}} e^{-2 i a} \left (i b x^n\right )^{\frac {1}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 125, normalized size = 0.69 \[ \frac {e^{-2 i a} \csc ^2\left (a+b x^n\right ) \left (e^{4 i a} 2^{\frac {1}{n}} \left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-2 i b x^n\right )-2 e^{2 i a} n+2^{\frac {1}{n}} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 n x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac {2}{3}}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {2}{3}}}{x^{2}}\, dx \]
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 \[ \frac {{\left (x \int \frac {\cos \left (2 \, b x^{n} + 2 \, a\right )}{x^{2}}\,{d x} + 1\right )} c^{\frac {2}{3}}}{4 \, x} \]
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 \[ \int \frac {{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3}}{x^2} \,d x \]
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 \[ \int \frac {\left (c \sin ^{3}{\left (a + b x^{n} \right )}\right )^{\frac {2}{3}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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