Integrand size = 18, antiderivative size = 115 \[ \int x^m \sqrt [3]{c \sin ^3(a+b x)} \, dx=-\frac {e^{i a} x^m (-i b x)^{-m} \csc (a+b x) \Gamma (1+m,-i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b}-\frac {e^{-i a} x^m (i b x)^{-m} \csc (a+b x) \Gamma (1+m,i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b} \] Output:
-1/2*exp(I*a)*x^m*csc(b*x+a)*GAMMA(1+m,-I*b*x)*(c*sin(b*x+a)^3)^(1/3)/b/(( -I*b*x)^m)-1/2*x^m*csc(b*x+a)*GAMMA(1+m,I*b*x)*(c*sin(b*x+a)^3)^(1/3)/b/ex p(I*a)/((I*b*x)^m)
Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.82 \[ \int x^m \sqrt [3]{c \sin ^3(a+b x)} \, dx=-\frac {e^{-i a} x^m \left (b^2 x^2\right )^{-m} \csc (a+b x) \left (e^{2 i a} (i b x)^m \Gamma (1+m,-i b x)+(-i b x)^m \Gamma (1+m,i b x)\right ) \sqrt [3]{c \sin ^3(a+b x)}}{2 b} \] Input:
Integrate[x^m*(c*Sin[a + b*x]^3)^(1/3),x]
Output:
-1/2*(x^m*Csc[a + b*x]*(E^((2*I)*a)*(I*b*x)^m*Gamma[1 + m, (-I)*b*x] + ((- I)*b*x)^m*Gamma[1 + m, I*b*x])*(c*Sin[a + b*x]^3)^(1/3))/(b*E^(I*a)*(b^2*x ^2)^m)
Time = 0.50 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {7271, 3042, 3789, 2612}
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.
\(\displaystyle \int x^m \sqrt [3]{c \sin ^3(a+b x)} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \int x^m \sin (a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \int x^m \sin (a+b x)dx\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \left (\frac {1}{2} i \int e^{-i (a+b x)} x^mdx-\frac {1}{2} i \int e^{i (a+b x)} x^mdx\right )\) |
\(\Big \downarrow \) 2612 |
\(\displaystyle \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \left (-\frac {e^{i a} x^m (-i b x)^{-m} \Gamma (m+1,-i b x)}{2 b}-\frac {e^{-i a} x^m (i b x)^{-m} \Gamma (m+1,i b x)}{2 b}\right )\) |
Input:
Int[x^m*(c*Sin[a + b*x]^3)^(1/3),x]
Output:
Csc[a + b*x]*(-1/2*(E^(I*a)*x^m*Gamma[1 + m, (-I)*b*x])/(b*((-I)*b*x)^m) - (x^m*Gamma[1 + m, I*b*x])/(2*b*E^(I*a)*(I*b*x)^m))*(c*Sin[a + b*x]^3)^(1/ 3)
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) )^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && !IntegerQ[m]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
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])
\[\int x^{m} \left (c \sin \left (b x +a \right )^{3}\right )^{\frac {1}{3}}d x\]
Input:
int(x^m*(c*sin(b*x+a)^3)^(1/3),x)
Output:
int(x^m*(c*sin(b*x+a)^3)^(1/3),x)
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.70 \[ \int x^m \sqrt [3]{c \sin ^3(a+b x)} \, dx=-\frac {{\left (e^{\left (-m \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m + 1, i \, b x\right ) + e^{\left (-m \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m + 1, -i \, b x\right )\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {1}{3}}}{2 \, b \sin \left (b x + a\right )} \] Input:
integrate(x^m*(c*sin(b*x+a)^3)^(1/3),x, algorithm="fricas")
Output:
-1/2*(e^(-m*log(I*b) - I*a)*gamma(m + 1, I*b*x) + e^(-m*log(-I*b) + I*a)*g amma(m + 1, -I*b*x))*(-(c*cos(b*x + a)^2 - c)*sin(b*x + a))^(1/3)/(b*sin(b *x + a))
\[ \int x^m \sqrt [3]{c \sin ^3(a+b x)} \, dx=\int x^{m} \sqrt [3]{c \sin ^{3}{\left (a + b x \right )}}\, dx \] Input:
integrate(x**m*(c*sin(b*x+a)**3)**(1/3),x)
Output:
Integral(x**m*(c*sin(a + b*x)**3)**(1/3), x)
\[ \int x^m \sqrt [3]{c \sin ^3(a+b x)} \, dx=\int { \left (c \sin \left (b x + a\right )^{3}\right )^{\frac {1}{3}} x^{m} \,d x } \] Input:
integrate(x^m*(c*sin(b*x+a)^3)^(1/3),x, algorithm="maxima")
Output:
integrate((c*sin(b*x + a)^3)^(1/3)*x^m, x)
\[ \int x^m \sqrt [3]{c \sin ^3(a+b x)} \, dx=\int { \left (c \sin \left (b x + a\right )^{3}\right )^{\frac {1}{3}} x^{m} \,d x } \] Input:
integrate(x^m*(c*sin(b*x+a)^3)^(1/3),x, algorithm="giac")
Output:
integrate((c*sin(b*x + a)^3)^(1/3)*x^m, x)
Timed out. \[ \int x^m \sqrt [3]{c \sin ^3(a+b x)} \, dx=\int x^m\,{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{1/3} \,d x \] Input:
int(x^m*(c*sin(a + b*x)^3)^(1/3),x)
Output:
int(x^m*(c*sin(a + b*x)^3)^(1/3), x)
\[ \int x^m \sqrt [3]{c \sin ^3(a+b x)} \, dx=c^{\frac {1}{3}} \left (\int x^{m} \sin \left (b x +a \right )d x \right ) \] Input:
int(x^m*(c*sin(b*x+a)^3)^(1/3),x)
Output:
c**(1/3)*int(x**m*sin(a + b*x),x)