Integrand size = 16, antiderivative size = 178 \[ \int \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\frac {1}{2} x \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {2^{-2-\frac {1}{n}} e^{2 i a} x \left (-i b x^n\right )^{-1/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {2^{-2-\frac {1}{n}} e^{-2 i a} x \left (i b x^n\right )^{-1/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {1}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]
1/2*x*csc(a+b*x^n)^2*(c*sin(a+b*x^n)^3)^(2/3)+2^(-2-1/n)*exp(2*I*a)*x*csc( a+b*x^n)^2*GAMMA(1/n,-2*I*b*x^n)*(c*sin(a+b*x^n)^3)^(2/3)/n/((-I*b*x^n)^(1 /n))+2^(-2-1/n)*x*csc(a+b*x^n)^2*GAMMA(1/n,2*I*b*x^n)*(c*sin(a+b*x^n)^3)^( 2/3)/exp(2*I*a)/n/((I*b*x^n)^(1/n))
Time = 0.22 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.84 \[ \int \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\frac {2^{-2-\frac {1}{n}} e^{-2 i a} x \left (b^2 x^{2 n}\right )^{-1/n} \csc ^2\left (a+b x^n\right ) \left (2^{1+\frac {1}{n}} e^{2 i a} n \left (b^2 x^{2 n}\right )^{\frac {1}{n}}+e^{4 i a} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-2 i b x^n\right )+\left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]
(2^(-2 - n^(-1))*x*Csc[a + b*x^n]^2*(2^(1 + n^(-1))*E^((2*I)*a)*n*(b^2*x^( 2*n))^n^(-1) + E^((4*I)*a)*(I*b*x^n)^n^(-1)*Gamma[n^(-1), (-2*I)*b*x^n] + ((-I)*b*x^n)^n^(-1)*Gamma[n^(-1), (2*I)*b*x^n])*(c*Sin[a + b*x^n]^3)^(2/3) )/(E^((2*I)*a)*n*(b^2*x^(2*n))^n^(-1))
Time = 0.34 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.71, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {7271, 3848, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \int \sin ^2\left (b x^n+a\right )dx\) |
\(\Big \downarrow \) 3848 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \int \left (\frac {1}{2}-\frac {1}{2} \cos \left (2 b x^n+2 a\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (\frac {e^{2 i a} 2^{-\frac {1}{n}-2} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i b x^n\right )}{n}+\frac {e^{-2 i a} 2^{-\frac {1}{n}-2} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i b x^n\right )}{n}+\frac {x}{2}\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\) |
Csc[a + b*x^n]^2*(x/2 + (2^(-2 - n^(-1))*E^((2*I)*a)*x*Gamma[n^(-1), (-2*I )*b*x^n])/(n*((-I)*b*x^n)^n^(-1)) + (2^(-2 - n^(-1))*x*Gamma[n^(-1), (2*I) *b*x^n])/(E^((2*I)*a)*n*(I*b*x^n)^n^(-1)))*(c*Sin[a + b*x^n]^3)^(2/3)
3.4.54.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_), x_Sy mbol] :> Int[ExpandTrigReduce[(a + b*Sin[c + d*(e + f*x)^n])^p, x], x] /; F reeQ[{a, b, c, d, e, f, n}, x] && IGtQ[p, 1]
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
\[\int {\left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )}^{\frac {2}{3}}d x\]
\[ \int \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} \,d x } \]
\[ \int \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int \left (c \sin ^{3}{\left (a + b x^{n} \right )}\right )^{\frac {2}{3}}\, dx \]
\[ \int \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} \,d x } \]
\[ \int \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} \,d x } \]
Timed out. \[ \int \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int {\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3} \,d x \]