Optimal. Leaf size=184 \[ \frac{e^{2 i a} 4^{\frac{1}{n}-1} \left (-i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}+\frac{e^{-2 i a} 4^{\frac{1}{n}-1} \left (i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}-\frac{\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 x^2} \]
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Rubi [A] time = 0.268642, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6720, 3425, 3424, 2218} \[ \frac{e^{2 i a} 4^{\frac{1}{n}-1} \left (-i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}+\frac{e^{-2 i a} 4^{\frac{1}{n}-1} \left (i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}-\frac{\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 3425
Rule 3424
Rule 2218
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*}
Mathematica [A] time = 0.352419, size = 129, normalized size = 0.7 \[ \frac{e^{-2 i a} \csc ^2\left (a+b x^n\right ) \left (e^{4 i a} 4^{\frac{1}{n}} \left (-i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-2 i b x^n\right )+4^{\frac{1}{n}} \left (i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},2 i b x^n\right )-e^{2 i a} n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 n x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( c \left ( \sin \left ( a+b{x}^{n} \right ) \right ) ^{3} \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (2 \, x^{2} \int \frac{\cos \left (2 \, b x^{n} + 2 \, a\right )}{x^{3}}\,{d x} + 1\right )} c^{\frac{2}{3}}}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac{2}{3}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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