Integrand size = 18, antiderivative size = 188 \[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\frac {1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {4^{-1-\frac {1}{n}} e^{2 i a} x^2 \left (-i b x^n\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {2}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {4^{-1-\frac {1}{n}} e^{-2 i a} x^2 \left (i b x^n\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {2}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \] Output:
1/4*x^2*csc(a+b*x^n)^2*(c*sin(a+b*x^n)^3)^(2/3)+4^(-1-1/n)*exp(2*I*a)*x^2* csc(a+b*x^n)^2*GAMMA(2/n,-2*I*b*x^n)*(c*sin(a+b*x^n)^3)^(2/3)/n/((-I*b*x^n )^(2/n))+4^(-1-1/n)*x^2*csc(a+b*x^n)^2*GAMMA(2/n,2*I*b*x^n)*(c*sin(a+b*x^n )^3)^(2/3)/exp(2*I*a)/n/((I*b*x^n)^(2/n))
Time = 0.77 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.85 \[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\frac {4^{-\frac {1+n}{n}} e^{-2 i a} x^2 \left (b^2 x^{2 n}\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \left (4^{\frac {1}{n}} e^{2 i a} n \left (b^2 x^{2 n}\right )^{2/n}+e^{4 i a} \left (i b x^n\right )^{2/n} \Gamma \left (\frac {2}{n},-2 i b x^n\right )+\left (-i b x^n\right )^{2/n} \Gamma \left (\frac {2}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \] Input:
Integrate[x*(c*Sin[a + b*x^n]^3)^(2/3),x]
Output:
(x^2*Csc[a + b*x^n]^2*(4^n^(-1)*E^((2*I)*a)*n*(b^2*x^(2*n))^(2/n) + E^((4* I)*a)*(I*b*x^n)^(2/n)*Gamma[2/n, (-2*I)*b*x^n] + ((-I)*b*x^n)^(2/n)*Gamma[ 2/n, (2*I)*b*x^n])*(c*Sin[a + b*x^n]^3)^(2/3))/(4^((1 + n)/n)*E^((2*I)*a)* n*(b^2*x^(2*n))^(2/n))
Time = 0.47 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.73, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {7271, 3906, 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.
\(\displaystyle \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \int x \sin ^2\left (b x^n+a\right )dx\) |
\(\Big \downarrow \) 3906 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \int \left (\frac {x}{2}-\frac {1}{2} x \cos \left (2 b x^n+2 a\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (\frac {e^{2 i a} 4^{-\frac {1}{n}-1} x^2 \left (-i b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-2 i b x^n\right )}{n}+\frac {e^{-2 i a} 4^{-\frac {1}{n}-1} x^2 \left (i b x^n\right )^{-2/n} \Gamma \left (\frac {2}{n},2 i b x^n\right )}{n}+\frac {x^2}{4}\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\) |
Input:
Int[x*(c*Sin[a + b*x^n]^3)^(2/3),x]
Output:
Csc[a + b*x^n]^2*(x^2/4 + (4^(-1 - n^(-1))*E^((2*I)*a)*x^2*Gamma[2/n, (-2* I)*b*x^n])/(n*((-I)*b*x^n)^(2/n)) + (4^(-1 - n^(-1))*x^2*Gamma[2/n, (2*I)* b*x^n])/(E^((2*I)*a)*n*(I*b*x^n)^(2/n)))*(c*Sin[a + b*x^n]^3)^(2/3)
Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x _Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
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])
\[\int x {\left (c \sin \left (a +b \,x^{n}\right )^{3}\right )}^{\frac {2}{3}}d x\]
Input:
int(x*(c*sin(a+b*x^n)^3)^(2/3),x)
Output:
int(x*(c*sin(a+b*x^n)^3)^(2/3),x)
\[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} x \,d x } \] Input:
integrate(x*(c*sin(a+b*x^n)^3)^(2/3),x, algorithm="fricas")
Output:
integral((-(c*cos(b*x^n + a)^2 - c)*sin(b*x^n + a))^(2/3)*x, x)
\[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int x \left (c \sin ^{3}{\left (a + b x^{n} \right )}\right )^{\frac {2}{3}}\, dx \] Input:
integrate(x*(c*sin(a+b*x**n)**3)**(2/3),x)
Output:
Integral(x*(c*sin(a + b*x**n)**3)**(2/3), x)
\[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} x \,d x } \] Input:
integrate(x*(c*sin(a+b*x^n)^3)^(2/3),x, algorithm="maxima")
Output:
-1/8*(x^2 - 2*integrate(x*cos(2*b*x^n + 2*a), x))*c^(2/3)
\[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} x \,d x } \] Input:
integrate(x*(c*sin(a+b*x^n)^3)^(2/3),x, algorithm="giac")
Output:
integrate((c*sin(b*x^n + a)^3)^(2/3)*x, x)
Timed out. \[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int x\,{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3} \,d x \] Input:
int(x*(c*sin(a + b*x^n)^3)^(2/3),x)
Output:
int(x*(c*sin(a + b*x^n)^3)^(2/3), x)
\[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=c^{\frac {2}{3}} \left (\int \sin \left (x^{n} b +a \right )^{2} x d x \right ) \] Input:
int(x*(c*sin(a+b*x^n)^3)^(2/3),x)
Output:
c**(2/3)*int(sin(x**n*b + a)**2*x,x)