Integrand size = 20, antiderivative size = 243 \[ \int (e+f x) \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\frac {3 f \cos \left (a+b (c+d x)^{2/3}\right )}{b^3 d^2}-\frac {3 (d e-c f) \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}-\frac {3 f (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d^2}+\frac {3 (d e-c f) \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d^2}-\frac {3 (d e-c f) \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d^2}+\frac {3 f (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2 d^2} \]
3*f*cos(a+b*(d*x+c)^(2/3))/b^3/d^2-3/2*(-c*f+d*e)*(d*x+c)^(1/3)*cos(a+b*(d *x+c)^(2/3))/b/d^2-3/2*f*(d*x+c)^(4/3)*cos(a+b*(d*x+c)^(2/3))/b/d^2+3*f*(d *x+c)^(2/3)*sin(a+b*(d*x+c)^(2/3))/b^2/d^2+3/4*(-c*f+d*e)*cos(a)*FresnelC( (d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/d^2-3/4*( -c*f+d*e)*FresnelS((d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*2^(1/2)* Pi^(1/2)/b^(3/2)/d^2
Time = 0.70 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.88 \[ \int (e+f x) \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\frac {3 \left (4 f \cos \left (a+b (c+d x)^{2/3}\right )-2 b^2 d e \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )-2 b^2 d f x \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )+b^{3/2} (d e-c f) \sqrt {2 \pi } \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )-b^{3/2} (d e-c f) \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)+4 b f (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )\right )}{4 b^3 d^2} \]
(3*(4*f*Cos[a + b*(c + d*x)^(2/3)] - 2*b^2*d*e*(c + d*x)^(1/3)*Cos[a + b*( c + d*x)^(2/3)] - 2*b^2*d*f*x*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(2/3)] + b^(3/2)*(d*e - c*f)*Sqrt[2*Pi]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d* x)^(1/3)] - b^(3/2)*(d*e - c*f)*Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a] + 4*b*f*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(2/3)]))/ (4*b^3*d^2)
Time = 0.38 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3914, 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.
| \(\displaystyle \int (e+f x) \sin \left (a+b (c+d x)^{2/3}\right ) \, dx\) |
| \(\Big \downarrow \) 3914 |
| \(\displaystyle \frac {3 \int \left (f \sin \left (a+b (c+d x)^{2/3}\right ) (c+d x)^{5/3}+(d e-c f) \sin \left (a+b (c+d x)^{2/3}\right ) (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {\frac {\pi }{2}} \cos (a) (d e-c f) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \sin (a) (d e-c f) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2}}+\frac {f \cos \left (a+b (c+d x)^{2/3}\right )}{b^3}+\frac {f (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{b^2}-\frac {\sqrt [3]{c+d x} (d e-c f) \cos \left (a+b (c+d x)^{2/3}\right )}{2 b}-\frac {f (c+d x)^{4/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b}\right )}{d^2}\) |
(3*((f*Cos[a + b*(c + d*x)^(2/3)])/b^3 - ((d*e - c*f)*(c + d*x)^(1/3)*Cos[ a + b*(c + d*x)^(2/3)])/(2*b) - (f*(c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(2/ 3)])/(2*b) + ((d*e - c*f)*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(2*b^(3/2)) - ((d*e - c*f)*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqr t[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(2*b^(3/2)) + (f*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(2/3)])/b^2))/d^2
3.3.13.3.1 Defintions of rubi rules used
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Module[{k = If[FractionQ[n], Denominat or[n], 1]}, Simp[k/f^(m + 1) Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x ^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)], x ]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]
Time = 0.41 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.72
| method | result | size |
| derivativedivides | \(\frac {-\frac {3 f \left (d x +c \right )^{\frac {4}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {6 f \left (\frac {\left (d x +c \right )^{\frac {2}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {\cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b^{2}}\right )}{b}+\frac {3 \left (c f -d e \right ) \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}-\frac {3 \left (c f -d e \right ) \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}}{d^{2}}\) | \(175\) |
| default | \(\frac {-\frac {3 f \left (d x +c \right )^{\frac {4}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {6 f \left (\frac {\left (d x +c \right )^{\frac {2}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {\cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b^{2}}\right )}{b}+\frac {3 \left (c f -d e \right ) \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}-\frac {3 \left (c f -d e \right ) \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}}{d^{2}}\) | \(175\) |
| parts | \(-\frac {3 \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right ) f x}{2 d b}-\frac {3 \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right ) e}{2 d b}+\frac {3 \sqrt {2}\, \sqrt {\pi }\, \cos \left (a \right ) \operatorname {C}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right ) f x}{4 d \,b^{\frac {3}{2}}}+\frac {3 \sqrt {2}\, \sqrt {\pi }\, \cos \left (a \right ) \operatorname {C}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right ) e}{4 d \,b^{\frac {3}{2}}}-\frac {3 \sqrt {2}\, \sqrt {\pi }\, \sin \left (a \right ) \operatorname {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right ) f x}{4 d \,b^{\frac {3}{2}}}-\frac {3 \sqrt {2}\, \sqrt {\pi }\, \sin \left (a \right ) \operatorname {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right ) e}{4 d \,b^{\frac {3}{2}}}-\frac {3 f \left (\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\frac {3 \cos \left (a \right ) \pi ^{\frac {3}{2}} \sqrt {2}\, \left (\frac {2 \,\operatorname {C}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right ) \sqrt {2}\, b^{\frac {3}{2}} \left (d x +c \right )}{3 \pi ^{\frac {3}{2}}}-\frac {2 b \left (d x +c \right )^{\frac {2}{3}} \sin \left (b \left (d x +c \right )^{\frac {2}{3}}\right )}{3 \pi ^{2}}-\frac {2 \cos \left (b \left (d x +c \right )^{\frac {2}{3}}\right )}{3 \pi ^{2}}\right )}{4 d \,b^{\frac {3}{2}}}-\frac {3 \sin \left (a \right ) \pi ^{\frac {3}{2}} \sqrt {2}\, \left (\frac {2 \,\operatorname {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right ) \sqrt {2}\, b^{\frac {3}{2}} \left (d x +c \right )}{3 \pi ^{\frac {3}{2}}}+\frac {2 \cos \left (b \left (d x +c \right )^{\frac {2}{3}}\right ) b \left (d x +c \right )^{\frac {2}{3}}}{3 \pi ^{2}}-\frac {2 \sin \left (b \left (d x +c \right )^{\frac {2}{3}}\right )}{3 \pi ^{2}}\right )}{4 d \,b^{\frac {3}{2}}}\right )}{b^{\frac {3}{2}}}-\frac {3 \left (\cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )+\left (a +b \left (d x +c \right )^{\frac {2}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )-a \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )\right )}{b^{3} d}\right )}{4 d}\) | \(457\) |
3/d^2*(-1/2*f/b*(d*x+c)^(4/3)*cos(a+b*(d*x+c)^(2/3))+2*f/b*(1/2/b*(d*x+c)^ (2/3)*sin(a+b*(d*x+c)^(2/3))+1/2/b^2*cos(a+b*(d*x+c)^(2/3)))+1/2*(c*f-d*e) /b*(d*x+c)^(1/3)*cos(a+b*(d*x+c)^(2/3))-1/4*(c*f-d*e)/b^(3/2)*2^(1/2)*Pi^( 1/2)*(cos(a)*FresnelC((d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))-sin(a)*Fresn elS((d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))))
Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.65 \[ \int (e+f x) \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\frac {3 \, {\left (\sqrt {2} \pi {\left (b d e - b c f\right )} \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {C}\left (\sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} \sqrt {\frac {b}{\pi }}\right ) - \sqrt {2} \pi {\left (b d e - b c f\right )} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) + 4 \, {\left (d x + c\right )}^{\frac {2}{3}} b f \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) - 2 \, {\left ({\left (b^{2} d f x + b^{2} d e\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 2 \, f\right )} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )}}{4 \, b^{3} d^{2}} \]
3/4*(sqrt(2)*pi*(b*d*e - b*c*f)*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*(d*x + c)^(1/3)*sqrt(b/pi)) - sqrt(2)*pi*(b*d*e - b*c*f)*sqrt(b/pi)*fresnel_si n(sqrt(2)*(d*x + c)^(1/3)*sqrt(b/pi))*sin(a) + 4*(d*x + c)^(2/3)*b*f*sin(( d*x + c)^(2/3)*b + a) - 2*((b^2*d*f*x + b^2*d*e)*(d*x + c)^(1/3) - 2*f)*co s((d*x + c)^(2/3)*b + a))/(b^3*d^2)
\[ \int (e+f x) \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\int \left (e + f x\right ) \sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.02 \[ \int (e+f x) \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=-\frac {3 \, {\left (\frac {{\left (\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {i \, b}\right ) + {\left (-\left (i + 1\right ) \, \cos \left (a\right ) - \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {-i \, b}\right )\right )} b^{\frac {3}{2}} + 8 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )} e}{b^{3}} - \frac {{\left (\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {i \, b}\right ) + {\left (-\left (i + 1\right ) \, \cos \left (a\right ) - \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {-i \, b}\right )\right )} b^{\frac {3}{2}} + 8 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )} c f}{b^{3} d} - \frac {8 \, {\left (2 \, {\left (d x + c\right )}^{\frac {2}{3}} b \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) - {\left ({\left (d x + c\right )}^{\frac {4}{3}} b^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )} f}{b^{3} d}\right )}}{16 \, d} \]
-3/16*((sqrt(2)*sqrt(pi)*(((I - 1)*cos(a) + (I + 1)*sin(a))*erf((d*x + c)^ (1/3)*sqrt(I*b)) + (-(I + 1)*cos(a) - (I - 1)*sin(a))*erf((d*x + c)^(1/3)* sqrt(-I*b)))*b^(3/2) + 8*(d*x + c)^(1/3)*b^2*cos((d*x + c)^(2/3)*b + a))*e /b^3 - (sqrt(2)*sqrt(pi)*(((I - 1)*cos(a) + (I + 1)*sin(a))*erf((d*x + c)^ (1/3)*sqrt(I*b)) + (-(I + 1)*cos(a) - (I - 1)*sin(a))*erf((d*x + c)^(1/3)* sqrt(-I*b)))*b^(3/2) + 8*(d*x + c)^(1/3)*b^2*cos((d*x + c)^(2/3)*b + a))*c *f/(b^3*d) - 8*(2*(d*x + c)^(2/3)*b*sin((d*x + c)^(2/3)*b + a) - ((d*x + c )^(4/3)*b^2 - 2)*cos((d*x + c)^(2/3)*b + a))*f/(b^3*d))/d
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.68 \[ \int (e+f x) \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=-\frac {3 \, {\left (e {\left (-\frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{b {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{b {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} e^{\left (i \, {\left (d x + c\right )}^{\frac {2}{3}} b + i \, a\right )}}{b} + \frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} e^{\left (-i \, {\left (d x + c\right )}^{\frac {2}{3}} b - i \, a\right )}}{b}\right )} + \frac {{\left (\frac {i \, \sqrt {2} \sqrt {\pi } c \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{b {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {i \, \sqrt {2} \sqrt {\pi } c \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{b {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {2 i \, {\left (i \, {\left (d x + c\right )}^{\frac {4}{3}} b^{2} - i \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} c - 2 \, {\left (d x + c\right )}^{\frac {2}{3}} b - 2 i\right )} e^{\left (i \, {\left (d x + c\right )}^{\frac {2}{3}} b + i \, a\right )}}{b^{3}} - \frac {2 i \, {\left (i \, {\left (d x + c\right )}^{\frac {4}{3}} b^{2} - i \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} c + 2 \, {\left (d x + c\right )}^{\frac {2}{3}} b - 2 i\right )} e^{\left (-i \, {\left (d x + c\right )}^{\frac {2}{3}} b - i \, a\right )}}{b^{3}}\right )} f}{d}\right )}}{8 \, d} \]
-3/8*(e*(-I*sqrt(2)*sqrt(pi)*erf(-1/2*I*sqrt(2)*(d*x + c)^(1/3)*(I*b/abs(b ) + 1)*sqrt(abs(b)))*e^(I*a)/(b*(I*b/abs(b) + 1)*sqrt(abs(b))) + I*sqrt(2) *sqrt(pi)*erf(1/2*I*sqrt(2)*(d*x + c)^(1/3)*(-I*b/abs(b) + 1)*sqrt(abs(b)) )*e^(-I*a)/(b*(-I*b/abs(b) + 1)*sqrt(abs(b))) + 2*(d*x + c)^(1/3)*e^(I*(d* x + c)^(2/3)*b + I*a)/b + 2*(d*x + c)^(1/3)*e^(-I*(d*x + c)^(2/3)*b - I*a) /b) + (I*sqrt(2)*sqrt(pi)*c*erf(-1/2*I*sqrt(2)*(d*x + c)^(1/3)*(I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I*a)/(b*(I*b/abs(b) + 1)*sqrt(abs(b))) - I*sqrt(2)* sqrt(pi)*c*erf(1/2*I*sqrt(2)*(d*x + c)^(1/3)*(-I*b/abs(b) + 1)*sqrt(abs(b) ))*e^(-I*a)/(b*(-I*b/abs(b) + 1)*sqrt(abs(b))) - 2*I*(I*(d*x + c)^(4/3)*b^ 2 - I*(d*x + c)^(1/3)*b^2*c - 2*(d*x + c)^(2/3)*b - 2*I)*e^(I*(d*x + c)^(2 /3)*b + I*a)/b^3 - 2*I*(I*(d*x + c)^(4/3)*b^2 - I*(d*x + c)^(1/3)*b^2*c + 2*(d*x + c)^(2/3)*b - 2*I)*e^(-I*(d*x + c)^(2/3)*b - I*a)/b^3)*f/d)/d
Timed out. \[ \int (e+f x) \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\int \sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )\,\left (e+f\,x\right ) \,d x \]