Optimal. Leaf size=121 \[ -\frac {\cos (2 a) \text {Ci}\left (2 b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 n}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 n}+\frac {1}{2} \log (x) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \]
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Rubi [A] time = 0.18, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6720, 3425, 3378, 3376, 3375} \[ -\frac {\cos (2 a) \text {CosIntegral}\left (2 b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 n}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 n}+\frac {1}{2} \log (x) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 3375
Rule 3376
Rule 3378
Rule 3425
Rule 6720
Rubi steps
\begin {align*} \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x} \, 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} \, 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}-\frac {\cos \left (2 a+2 b x^n\right )}{2 x}\right ) \, dx\\ &=\frac {1}{2} \csc ^2\left (a+b x^n\right ) \log (x) \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 \frac {\cos \left (2 a+2 b x^n\right )}{x} \, dx\\ &=\frac {1}{2} \csc ^2\left (a+b x^n\right ) \log (x) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}-\frac {1}{2} \left (\cos (2 a) \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 b x^n\right )}{x} \, dx+\frac {1}{2} \left (\csc ^2\left (a+b x^n\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int \frac {\sin \left (2 b x^n\right )}{x} \, dx\\ &=-\frac {\cos (2 a) \text {Ci}\left (2 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}}{2 n}+\frac {1}{2} \csc ^2\left (a+b x^n\right ) \log (x) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {\csc ^2\left (a+b x^n\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \text {Si}\left (2 b x^n\right )}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 63, normalized size = 0.52 \[ \frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \left (-\cos (2 a) \text {Ci}\left (2 b x^n\right )+\sin (2 a) \text {Si}\left (2 b x^n\right )+n \log (x)\right )}{2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 106, normalized size = 0.88 \[ \frac {4^{\frac {2}{3}} {\left (4^{\frac {1}{3}} \cos \left (2 \, a\right ) \operatorname {Ci}\left (2 \, b x^{n}\right ) + 4^{\frac {1}{3}} \cos \left (2 \, a\right ) \operatorname {Ci}\left (-2 \, b x^{n}\right ) - 2 \cdot 4^{\frac {1}{3}} n \log \relax (x) - 2 \cdot 4^{\frac {1}{3}} \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{n}\right )\right )} \left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac {2}{3}}}{16 \, {\left (n \cos \left (b x^{n} + a\right )^{2} - n\right )}} \]
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 \[ \int \frac {\left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.36, size = 343, normalized size = 2.83 \[ \frac {i \left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i b \,x^{n}} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{4 \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{2} n}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i b \,x^{n}} \Si \left (2 b \,x^{n}\right )}{2 \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{2} n}-\frac {\left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i b \,x^{n}} \Ei \left (1, -2 i b \,x^{n}\right )}{4 \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{2} n}-\frac {\Ei \left (1, -2 i b \,x^{n}\right ) \left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b \,x^{n}+2 a \right )}}{4 n \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{2}}-\frac {\ln \relax (x ) \left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (a +b \,x^{n}\right )}}{2 \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.03, size = 153, normalized size = 1.26 \[ \frac {{\left ({\left ({\left (i \, \sqrt {3} + 1\right )} {\rm Ei}\left (2 i \, b x^{n}\right ) + {\left (i \, \sqrt {3} + 1\right )} {\rm Ei}\left (-2 i \, b x^{n}\right ) + {\left (-i \, \sqrt {3} + 1\right )} {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + {\left (-i \, \sqrt {3} + 1\right )} {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \cos \left (2 \, a\right ) - 4 \, n \log \relax (x) - {\left ({\left (\sqrt {3} - i\right )} {\rm Ei}\left (2 i \, b x^{n}\right ) - {\left (\sqrt {3} - i\right )} {\rm Ei}\left (-2 i \, b x^{n}\right ) - {\left (\sqrt {3} + i\right )} {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + {\left (\sqrt {3} + i\right )} {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \sin \left (2 \, a\right )\right )} c^{\frac {2}{3}}}{16 \, n} \]
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 \[ \int \frac {{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3}}{x} \,d x \]
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 \[ \int \frac {\left (c \sin ^{3}{\left (a + b x^{n} \right )}\right )^{\frac {2}{3}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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