Optimal. Leaf size=188 \[ \frac {1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {e^{2 i a} 4^{-\frac {1}{n}-1} x^2 \left (-i b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {e^{-2 i a} 4^{-\frac {1}{n}-1} x^2 \left (i b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6720, 3425, 3424, 2218} \[ \frac {e^{2 i a} 4^{-\frac {1}{n}-1} x^2 \left (-i b x^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {e^{-2 i a} 4^{-\frac {1}{n}-1} x^2 \left (i b x^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2218
Rule 3424
Rule 3425
Rule 6720
Rubi steps
\begin {align*} \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, 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 x \sin ^2\left (a+b x^n\right ) \, 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 {x}{2}-\frac {1}{2} x \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac {1}{4} x^2 \csc ^2\left (a+b x^n\right ) \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 x \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac {1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}-\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 e^{-2 i a-2 i b x^n} x \, 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 e^{2 i a+2 i b x^n} x \, dx\\ &=\frac {1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {4^{-1-\frac {1}{n}} e^{2 i a} x^2 \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}+\frac {4^{-1-\frac {1}{n}} e^{-2 i a} x^2 \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}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.56, size = 160, normalized size = 0.85 \[ \frac {e^{-2 i a} 4^{-\frac {n+1}{n}} x^2 \left (b^2 x^{2 n}\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \left (e^{2 i a} 4^{\frac {1}{n}} n \left (b^2 x^{2 n}\right )^{2/n}+e^{4 i a} \left (i b x^n\right )^{2/n} \Gamma \left (\frac {2}{n},-2 i b x^n\right )+\left (-i b x^n\right )^{2/n} \Gamma \left (\frac {2}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac {2}{3}} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int x \left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, {\left (x^{2} - 2 \, \int x \cos \left (2 \, b x^{n} + 2 \, a\right )\,{d x}\right )} c^{\frac {2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (c \sin ^{3}{\left (a + b x^{n} \right )}\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________