\(\int x^m \sqrt [3]{c \sin ^3(a+b x^n)} \, dx\) [326]

Optimal result

Integrand size = 20, antiderivative size = 157 \[ \int x^m \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \, dx=\frac {i e^{i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \csc \left (a+b x^n\right ) \Gamma \left (\frac {1+m}{n},-i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n}-\frac {i e^{-i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \csc \left (a+b x^n\right ) \Gamma \left (\frac {1+m}{n},i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n} \] Output:

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

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.90 \[ \int x^m \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \, dx=\frac {i x^{1+m} \left (b^2 x^{2 n}\right )^{-\frac {1+m}{n}} \csc \left (a+b x^n\right ) \left (-\left (-i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},i b x^n\right ) (\cos (a)-i \sin (a))+\left (i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-i b x^n\right ) (\cos (a)+i \sin (a))\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n} \] Input:

Integrate[x^m*(c*Sin[a + b*x^n]^3)^(1/3),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.85, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {7271, 3904, 2648}

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.

\(\displaystyle \int x^m \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \, dx\)

\(\Big \downarrow \) 7271

\(\displaystyle \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \int x^m \sin \left (b x^n+a\right )dx\)

\(\Big \downarrow \) 3904

\(\displaystyle \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left (\frac {1}{2} i \int e^{-i b x^n-i a} x^mdx-\frac {1}{2} i \int e^{i b x^n+i a} x^mdx\right )\)

\(\Big \downarrow \) 2648

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

Input:

Int[x^m*(c*Sin[a + b*x^n]^3)^(1/3),x]
 

Output:

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

Defintions of rubi rules used

rule 2648

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ 
.), x_Symbol] :> Simp[(-F^a)*((e + f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[ 
F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; FreeQ[{F 
, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]
 

rule 3904

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[I/2 
  Int[(e*x)^m*E^((-c)*I - d*I*x^n), x], x] - Simp[I/2   Int[(e*x)^m*E^(c*I 
+ d*I*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]
 

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]

\[\int x^{m} {\left (c \sin \left (a +b \,x^{n}\right )^{3}\right )}^{\frac {1}{3}}d x\]

Input:

int(x^m*(c*sin(a+b*x^n)^3)^(1/3),x)
 

Output:

int(x^m*(c*sin(a+b*x^n)^3)^(1/3),x)
 

Fricas [F]

\[ \int x^m \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {1}{3}} x^{m} \,d x } \] Input:

integrate(x^m*(c*sin(a+b*x^n)^3)^(1/3),x, algorithm="fricas")
 

Output:

integral((-(c*cos(b*x^n + a)^2 - c)*sin(b*x^n + a))^(1/3)*x^m, x)
 

Sympy [F]

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

integrate(x**m*(c*sin(a+b*x**n)**3)**(1/3),x)
 

Output:

Integral(x**m*(c*sin(a + b*x**n)**3)**(1/3), x)
 

Maxima [F]

\[ \int x^m \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {1}{3}} x^{m} \,d x } \] Input:

integrate(x^m*(c*sin(a+b*x^n)^3)^(1/3),x, algorithm="maxima")
 

Output:

integrate((c*sin(b*x^n + a)^3)^(1/3)*x^m, x)
 

Giac [F]

\[ \int x^m \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {1}{3}} x^{m} \,d x } \] Input:

integrate(x^m*(c*sin(a+b*x^n)^3)^(1/3),x, algorithm="giac")
 

Output:

integrate((c*sin(b*x^n + a)^3)^(1/3)*x^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^m \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \, dx=\int x^m\,{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{1/3} \,d x \] Input:

int(x^m*(c*sin(a + b*x^n)^3)^(1/3),x)
 

Output:

int(x^m*(c*sin(a + b*x^n)^3)^(1/3), x)
 

Reduce [F]

\[ \int x^m \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \, dx=c^{\frac {1}{3}} \left (\int x^{m} \sin \left (x^{n} b +a \right )d x \right ) \] Input:

int(x^m*(c*sin(a+b*x^n)^3)^(1/3),x)
 

Output:

c**(1/3)*int(x**m*sin(x**n*b + a),x)