\(\int x (c \sin ^3(a+b x^n))^{2/3} \, dx\) [353]

Optimal result

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))
 

Mathematica [A] (verified)

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))
 

Rubi [A] (verified)

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)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3906

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]
 

rule 7271

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])
 

Maple [F]

\[\int x {\left (c \sin \left (a +b \,x^{n}\right )^{3}\right )}^{\frac {2}{3}}d x\]

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)
 

Fricas [F]

\[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} x \,d x } \] 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)
 

Sympy [F]

\[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int x \left (c \sin ^{3}{\left (a + b x^{n} \right )}\right )^{\frac {2}{3}}\, dx \] 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)
 

Maxima [F]

\[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} x \,d x } \] 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)
 

Giac [F]

\[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} x \,d x } \] 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)
 

Mupad [F(-1)]

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)
 

Reduce [F]

\[ \int x \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=c^{\frac {2}{3}} \left (\int \sin \left (x^{n} b +a \right )^{2} x d x \right ) \] 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)