Optimal. Leaf size=180 \[ -\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x}+\frac {2^{-2+\frac {1}{n}} e^{2 i a} \left (-i b x^n\right )^{\frac {1}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x}+\frac {2^{-2+\frac {1}{n}} e^{-2 i a} \left (i b x^n\right )^{\frac {1}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.19, 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 = {6852, 3506,
3505, 2250} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2250
Rule 3505
Rule 3506
Rule 6852
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.25, size = 125, normalized size = 0.69 \begin {gather*} \frac {e^{-2 i a} \csc ^2\left (a+b x^n\right ) \left (-2 e^{2 i a} n+2^{\frac {1}{n}} e^{4 i a} \left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-2 i b x^n\right )+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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.14, 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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \sin ^{3}{\left (a + b x^{n} \right )}\right )^{\frac {2}{3}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________