Integrand size = 27, antiderivative size = 289 \[ \int (c e+d e x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {2160 e \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d \sqrt [3]{c+d x}}-\frac {1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d} \]
2160*e*(e*(d*x+c))^(1/3)*cos(a+b*(d*x+c)^(1/3))/b^7/d/(d*x+c)^(1/3)-1080*e *(d*x+c)^(1/3)*(e*(d*x+c))^(1/3)*cos(a+b*(d*x+c)^(1/3))/b^5/d+90*e*(d*x+c) *(e*(d*x+c))^(1/3)*cos(a+b*(d*x+c)^(1/3))/b^3/d-3*e*(d*x+c)^(5/3)*(e*(d*x+ c))^(1/3)*cos(a+b*(d*x+c)^(1/3))/b/d+2160*e*(e*(d*x+c))^(1/3)*sin(a+b*(d*x +c)^(1/3))/b^6/d-360*e*(d*x+c)^(2/3)*(e*(d*x+c))^(1/3)*sin(a+b*(d*x+c)^(1/ 3))/b^4/d+18*e*(d*x+c)^(4/3)*(e*(d*x+c))^(1/3)*sin(a+b*(d*x+c)^(1/3))/b^2/ d
Time = 0.51 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.78 \[ \int (c e+d e x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 (e (c+d x))^{4/3} \left (-\cos \left (b \sqrt [3]{c+d x}\right ) \left (\left (-720+360 b^2 (c+d x)^{2/3}-30 b^4 (c+d x)^{4/3}+b^6 (c+d x)^2\right ) \cos (a)-6 b \left (120 \sqrt [3]{c+d x}-20 b^2 (c+d x)+b^4 (c+d x)^{5/3}\right ) \sin (a)\right )+\left (6 b \left (120 \sqrt [3]{c+d x}-20 b^2 (c+d x)+b^4 (c+d x)^{5/3}\right ) \cos (a)+\left (-720+360 b^2 (c+d x)^{2/3}-30 b^4 (c+d x)^{4/3}+b^6 (c+d x)^2\right ) \sin (a)\right ) \sin \left (b \sqrt [3]{c+d x}\right )\right )}{b^7 d (c+d x)^{4/3}} \]
(3*(e*(c + d*x))^(4/3)*(-(Cos[b*(c + d*x)^(1/3)]*((-720 + 360*b^2*(c + d*x )^(2/3) - 30*b^4*(c + d*x)^(4/3) + b^6*(c + d*x)^2)*Cos[a] - 6*b*(120*(c + d*x)^(1/3) - 20*b^2*(c + d*x) + b^4*(c + d*x)^(5/3))*Sin[a])) + (6*b*(120 *(c + d*x)^(1/3) - 20*b^2*(c + d*x) + b^4*(c + d*x)^(5/3))*Cos[a] + (-720 + 360*b^2*(c + d*x)^(2/3) - 30*b^4*(c + d*x)^(4/3) + b^6*(c + d*x)^2)*Sin[ a])*Sin[b*(c + d*x)^(1/3)]))/(b^7*d*(c + d*x)^(4/3))
Time = 0.93 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.81, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {3912, 30, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3118}
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.
| \(\displaystyle \int (c e+d e x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 3912 |
| \(\displaystyle \frac {3 \int (c+d x)^{2/3} (e (c+d x))^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{d}\) |
| \(\Big \downarrow \) 30 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \int (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \int (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \int (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \int (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}+\frac {\pi }{2}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {5 \int -(c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}+\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \int (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \int (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \left (\frac {4 \int (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \left (\frac {4 \int (c+d x) \sin \left (a+b \sqrt [3]{c+d x}+\frac {\pi }{2}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \left (\frac {4 \left (\frac {3 \int -(c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}+\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \int (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \int (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \int \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \int \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}+\frac {\pi }{2}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \left (\frac {\int -\sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}+\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \left (\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {\int \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \left (\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {\int \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {3 e \sqrt [3]{e (c+d x)} \left (\frac {6 \left (\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {5 \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \left (\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2}+\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^2 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
(3*e*(e*(c + d*x))^(1/3)*(-(((c + d*x)^2*Cos[a + b*(c + d*x)^(1/3)])/b) + (6*(((c + d*x)^(5/3)*Sin[a + b*(c + d*x)^(1/3)])/b - (5*(-(((c + d*x)^(4/3 )*Cos[a + b*(c + d*x)^(1/3)])/b) + (4*(((c + d*x)*Sin[a + b*(c + d*x)^(1/3 )])/b - (3*(-(((c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/b) + (2*(Cos[a + b*(c + d*x)^(1/3)]/b^2 + ((c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/b) )/b))/b))/b))/b))/b))/(d*(c + d*x)^(1/3))
3.3.27.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & & !IntegerQ[p]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/(n*f) Subst[Int[ExpandIntegra nd[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - e*(h/f) + h*(x^(1/n)/f))^m, x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p , 0] && IntegerQ[1/n]
\[\int \left (d e x +c e \right )^{\frac {4}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )d x\]
Time = 0.72 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.81 \[ \int (c e+d e x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \, {\left ({\left (30 \, b^{4} d^{2} e x^{2} + 60 \, b^{4} c d e x + 30 \, b^{4} c^{2} e - {\left (b^{6} d^{2} e x^{2} + 2 \, b^{6} c d e x + {\left (b^{6} c^{2} - 720\right )} e\right )} {\left (d x + c\right )}^{\frac {2}{3}} - 360 \, {\left (b^{2} d e x + b^{2} c e\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} {\left (d e x + c e\right )}^{\frac {1}{3}} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + 6 \, {\left (120 \, b d e x + 120 \, b c e - 20 \, {\left (b^{3} d e x + b^{3} c e\right )} {\left (d x + c\right )}^{\frac {2}{3}} + {\left (b^{5} d^{2} e x^{2} + 2 \, b^{5} c d e x + b^{5} c^{2} e\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} {\left (d e x + c e\right )}^{\frac {1}{3}} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{7} d^{2} x + b^{7} c d} \]
3*((30*b^4*d^2*e*x^2 + 60*b^4*c*d*e*x + 30*b^4*c^2*e - (b^6*d^2*e*x^2 + 2* b^6*c*d*e*x + (b^6*c^2 - 720)*e)*(d*x + c)^(2/3) - 360*(b^2*d*e*x + b^2*c* e)*(d*x + c)^(1/3))*(d*e*x + c*e)^(1/3)*cos((d*x + c)^(1/3)*b + a) + 6*(12 0*b*d*e*x + 120*b*c*e - 20*(b^3*d*e*x + b^3*c*e)*(d*x + c)^(2/3) + (b^5*d^ 2*e*x^2 + 2*b^5*c*d*e*x + b^5*c^2*e)*(d*x + c)^(1/3))*(d*e*x + c*e)^(1/3)* sin((d*x + c)^(1/3)*b + a))/(b^7*d^2*x + b^7*c*d)
Timed out. \[ \int (c e+d e x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\text {Timed out} \]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.38 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.61 \[ \int (c e+d e x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \, {\left (3 \, {\left ({\left (\Gamma \left (6, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (6, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (6, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + \Gamma \left (6, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \cos \left (a\right ) + {\left (-i \, \Gamma \left (6, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (6, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, \Gamma \left (6, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + i \, \Gamma \left (6, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \sin \left (a\right )\right )} e - 2 \, {\left (b^{6} d^{2} e x^{2} + 2 \, b^{6} c d e x + b^{6} c^{2} e\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} e^{\frac {1}{3}}}{2 \, b^{7} d} \]
3/2*(3*((gamma(6, I*b*conjugate((d*x + c)^(1/3))) + gamma(6, -I*b*conjugat e((d*x + c)^(1/3))) + gamma(6, I*(d*x + c)^(1/3)*b) + gamma(6, -I*(d*x + c )^(1/3)*b))*cos(a) + (-I*gamma(6, I*b*conjugate((d*x + c)^(1/3))) + I*gamm a(6, -I*b*conjugate((d*x + c)^(1/3))) - I*gamma(6, I*(d*x + c)^(1/3)*b) + I*gamma(6, -I*(d*x + c)^(1/3)*b))*sin(a))*e - 2*(b^6*d^2*e*x^2 + 2*b^6*c*d *e*x + b^6*c^2*e)*cos((d*x + c)^(1/3)*b + a))*e^(1/3)/(b^7*d)
Time = 0.39 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.68 \[ \int (c e+d e x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left (d^{2} e {\left (\frac {{\left (b^{6} c^{2} e^{7} - 2 \, {\left (d e x + c e\right )} b^{6} c e^{6} + {\left (d e x + c e\right )}^{2} b^{6} e^{5} + 12 \, {\left (d e x + c e\right )}^{\frac {1}{3}} b^{4} c e^{6} {\left | e \right |}^{\frac {2}{3}} - 30 \, {\left (d e x + c e\right )}^{\frac {4}{3}} b^{4} e^{5} {\left | e \right |}^{\frac {2}{3}} + 360 \, {\left (d e x + c e\right )}^{\frac {2}{3}} b^{2} e^{5} {\left | e \right |}^{\frac {4}{3}} - 720 \, e^{7}\right )} \cos \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b^{7} d^{2} e^{6} {\left | e \right |}^{\frac {2}{3}}} + \frac {6 \, {\left ({\left (d e x + c e\right )}^{\frac {2}{3}} b^{5} c e^{5} {\left | e \right |}^{\frac {4}{3}} - {\left (d e x + c e\right )}^{\frac {5}{3}} b^{5} e^{4} {\left | e \right |}^{\frac {4}{3}} - 2 \, b^{3} c e^{7} + 20 \, {\left (d e x + c e\right )} b^{3} e^{6} - 120 \, {\left (d e x + c e\right )}^{\frac {1}{3}} b e^{6} {\left | e \right |}^{\frac {2}{3}}\right )} \sin \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b^{7} d^{2} e^{6} {\left | e \right |}^{\frac {2}{3}}}\right )} + \frac {c^{2} e^{2} \cos \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b {\left | e \right |}^{\frac {2}{3}}} - 2 \, c {\left (\frac {{\left (b^{3} c e^{4} - {\left (d e x + c e\right )} b^{3} e^{3} + 6 \, {\left (d e x + c e\right )}^{\frac {1}{3}} b e^{3} {\left | e \right |}^{\frac {2}{3}}\right )} \cos \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b^{4} e^{2} {\left | e \right |}^{\frac {2}{3}}} + \frac {3 \, {\left ({\left (d e x + c e\right )}^{\frac {2}{3}} b^{2} e^{2} {\left | e \right |}^{\frac {4}{3}} - 2 \, e^{4}\right )} \sin \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b^{4} e^{2} {\left | e \right |}^{\frac {2}{3}}}\right )}\right )}}{d} \]
-3*(d^2*e*((b^6*c^2*e^7 - 2*(d*e*x + c*e)*b^6*c*e^6 + (d*e*x + c*e)^2*b^6* e^5 + 12*(d*e*x + c*e)^(1/3)*b^4*c*e^6*abs(e)^(2/3) - 30*(d*e*x + c*e)^(4/ 3)*b^4*e^5*abs(e)^(2/3) + 360*(d*e*x + c*e)^(2/3)*b^2*e^5*abs(e)^(4/3) - 7 20*e^7)*cos((a*e + (d*e*x + c*e)^(1/3)*b*abs(e)^(2/3))/e)/(b^7*d^2*e^6*abs (e)^(2/3)) + 6*((d*e*x + c*e)^(2/3)*b^5*c*e^5*abs(e)^(4/3) - (d*e*x + c*e) ^(5/3)*b^5*e^4*abs(e)^(4/3) - 2*b^3*c*e^7 + 20*(d*e*x + c*e)*b^3*e^6 - 120 *(d*e*x + c*e)^(1/3)*b*e^6*abs(e)^(2/3))*sin((a*e + (d*e*x + c*e)^(1/3)*b* abs(e)^(2/3))/e)/(b^7*d^2*e^6*abs(e)^(2/3))) + c^2*e^2*cos((a*e + (d*e*x + c*e)^(1/3)*b*abs(e)^(2/3))/e)/(b*abs(e)^(2/3)) - 2*c*((b^3*c*e^4 - (d*e*x + c*e)*b^3*e^3 + 6*(d*e*x + c*e)^(1/3)*b*e^3*abs(e)^(2/3))*cos((a*e + (d* e*x + c*e)^(1/3)*b*abs(e)^(2/3))/e)/(b^4*e^2*abs(e)^(2/3)) + 3*((d*e*x + c *e)^(2/3)*b^2*e^2*abs(e)^(4/3) - 2*e^4)*sin((a*e + (d*e*x + c*e)^(1/3)*b*a bs(e)^(2/3))/e)/(b^4*e^2*abs(e)^(2/3))))/d
Timed out. \[ \int (c e+d e x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int \sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c\,e+d\,e\,x\right )}^{4/3} \,d x \]