\(\int x^m (c \sin ^3(a+b x))^{2/3} \, dx\) [334]

Optimal result

Integrand size = 18, antiderivative size = 169 \[ \int x^m \left (c \sin ^3(a+b x)\right )^{2/3} \, dx=\frac {x^{1+m} \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 (1+m)}+\frac {i 2^{-3-m} e^{2 i a} x^m (-i b x)^{-m} \csc ^2(a+b x) \Gamma (1+m,-2 i b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{b}-\frac {i 2^{-3-m} e^{-2 i a} x^m (i b x)^{-m} \csc ^2(a+b x) \Gamma (1+m,2 i b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{b} \] Output:

x^(1+m)*csc(b*x+a)^2*(c*sin(b*x+a)^3)^(2/3)/(2+2*m)+I*2^(-3-m)*exp(2*I*a)* 
x^m*csc(b*x+a)^2*GAMMA(1+m,-2*I*b*x)*(c*sin(b*x+a)^3)^(2/3)/b/((-I*b*x)^m) 
-I*2^(-3-m)*x^m*csc(b*x+a)^2*GAMMA(1+m,2*I*b*x)*(c*sin(b*x+a)^3)^(2/3)/b/e 
xp(2*I*a)/((I*b*x)^m)
 

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.84 \[ \int x^m \left (c \sin ^3(a+b x)\right )^{2/3} \, dx=\frac {2^{-3-m} x^m \left (b^2 x^2\right )^{-m} \csc ^2(a+b x) \left (2^{2+m} b x \left (b^2 x^2\right )^m-i (1+m) (-i b x)^m \Gamma (1+m,2 i b x) (\cos (a)-i \sin (a))^2+i (1+m) (i b x)^m \Gamma (1+m,-2 i b x) (\cos (a)+i \sin (a))^2\right ) \left (c \sin ^3(a+b x)\right )^{2/3}}{b (1+m)} \] Input:

Integrate[x^m*(c*Sin[a + b*x]^3)^(2/3),x]
 

Output:

(2^(-3 - m)*x^m*Csc[a + b*x]^2*(2^(2 + m)*b*x*(b^2*x^2)^m - I*(1 + m)*((-I 
)*b*x)^m*Gamma[1 + m, (2*I)*b*x]*(Cos[a] - I*Sin[a])^2 + I*(1 + m)*(I*b*x) 
^m*Gamma[1 + m, (-2*I)*b*x]*(Cos[a] + I*Sin[a])^2)*(c*Sin[a + b*x]^3)^(2/3 
))/(b*(1 + m)*(b^2*x^2)^m)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {7271, 3042, 3793, 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.

\(\displaystyle \int x^m \left (c \sin ^3(a+b x)\right )^{2/3} \, dx\)

\(\Big \downarrow \) 7271

\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \int x^m \sin ^2(a+b x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \int x^m \sin (a+b x)^2dx\)

\(\Big \downarrow \) 3793

\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \int \left (\frac {x^m}{2}-\frac {1}{2} x^m \cos (2 a+2 b x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (\frac {i e^{2 i a} 2^{-m-3} x^m (-i b x)^{-m} \Gamma (m+1,-2 i b x)}{b}-\frac {i e^{-2 i a} 2^{-m-3} x^m (i b x)^{-m} \Gamma (m+1,2 i b x)}{b}+\frac {x^{m+1}}{2 (m+1)}\right )\)

Input:

Int[x^m*(c*Sin[a + b*x]^3)^(2/3),x]
 

Output:

Csc[a + b*x]^2*(x^(1 + m)/(2*(1 + m)) + (I*2^(-3 - m)*E^((2*I)*a)*x^m*Gamm 
a[1 + m, (-2*I)*b*x])/(b*((-I)*b*x)^m) - (I*2^(-3 - m)*x^m*Gamma[1 + m, (2 
*I)*b*x])/(b*E^((2*I)*a)*(I*b*x)^m))*(c*Sin[a + b*x]^3)^(2/3)
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

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]

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

Input:

int(x^m*(c*sin(b*x+a)^3)^(2/3),x)
 

Output:

int(x^m*(c*sin(b*x+a)^3)^(2/3),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.66 \[ \int x^m \left (c \sin ^3(a+b x)\right )^{2/3} \, dx=-\frac {{\left (4 \, b x x^{m} - {\left (i \, m + i\right )} e^{\left (-m \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m + 1, 2 i \, b x\right ) - {\left (-i \, m - i\right )} e^{\left (-m \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m + 1, -2 i \, b x\right )\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {2}{3}}}{8 \, {\left ({\left (b m + b\right )} \cos \left (b x + a\right )^{2} - b m - b\right )}} \] Input:

integrate(x^m*(c*sin(b*x+a)^3)^(2/3),x, algorithm="fricas")
 

Output:

-1/8*(4*b*x*x^m - (I*m + I)*e^(-m*log(2*I*b) - 2*I*a)*gamma(m + 1, 2*I*b*x 
) - (-I*m - I)*e^(-m*log(-2*I*b) + 2*I*a)*gamma(m + 1, -2*I*b*x))*(-(c*cos 
(b*x + a)^2 - c)*sin(b*x + a))^(2/3)/((b*m + b)*cos(b*x + a)^2 - b*m - b)
 

Sympy [F]

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

integrate(x**m*(c*sin(b*x+a)**3)**(2/3),x)
 

Output:

Integral(x**m*(c*sin(a + b*x)**3)**(2/3), x)
 

Maxima [F]

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

integrate(x^m*(c*sin(b*x+a)^3)^(2/3),x, algorithm="maxima")
 

Output:

1/4*((m + 1)*integrate(x^m*cos(2*b*x + 2*a), x) - e^(m*log(x) + log(x)))*c 
^(2/3)/(m + 1)
 

Giac [F]

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

integrate(x^m*(c*sin(b*x+a)^3)^(2/3),x, algorithm="giac")
 

Output:

integrate((c*sin(b*x + a)^3)^(2/3)*x^m, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int x^m \left (c \sin ^3(a+b x)\right )^{2/3} \, dx=\int x^m\,{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{2/3} \,d x \] Input:

int(x^m*(c*sin(a + b*x)^3)^(2/3),x)
 

Output:

int(x^m*(c*sin(a + b*x)^3)^(2/3), x)
 

Reduce [F]

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

int(x^m*(c*sin(b*x+a)^3)^(2/3),x)
 

Output:

c**(2/3)*int(x**m*sin(a + b*x)**2,x)