Optimal. Leaf size=184 \[ -\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 x^2}+\frac {e^{2 i a} 4^{\frac {1}{n}-1} \left (-i b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}+\frac {e^{-2 i a} 4^{\frac {1}{n}-1} \left (i b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, 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 = {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.
[In]
[Out]
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^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 = 0.36, size = 129, normalized size = 0.70 \[ \frac {e^{-2 i a} \csc ^2\left (a+b x^n\right ) \left (e^{4 i a} 4^{\frac {1}{n}} \left (-i b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},-2 i b x^n\right )-e^{2 i a} n+4^{\frac {1}{n}} \left (i b x^n\right )^{2/n} \Gamma \left (-\frac {2}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 n x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac {2}{3}}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.27, 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.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (2 \, x^{2} \int \frac {\cos \left (2 \, b x^{n} + 2 \, a\right )}{x^{3}}\,{d x} + 1\right )} c^{\frac {2}{3}}}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________