Integrand size = 20, antiderivative size = 341 \[ \int (e+f x)^3 \sin \left (a+b (c+d x)^2\right ) \, dx=-\frac {3 f (d e-c f)^2 \cos \left (a+b (c+d x)^2\right )}{2 b d^4}-\frac {3 f^2 (d e-c f) (c+d x) \cos \left (a+b (c+d x)^2\right )}{2 b d^4}-\frac {f^3 (c+d x)^2 \cos \left (a+b (c+d x)^2\right )}{2 b d^4}+\frac {3 f^2 (d e-c f) \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2} d^4}+\frac {(d e-c f)^3 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^4}+\frac {(d e-c f)^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \sin (a)}{\sqrt {b} d^4}-\frac {3 f^2 (d e-c f) \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \sin (a)}{2 b^{3/2} d^4}+\frac {f^3 \sin \left (a+b (c+d x)^2\right )}{2 b^2 d^4} \]
-3/2*f*(-c*f+d*e)^2*cos(a+b*(d*x+c)^2)/b/d^4-3/2*f^2*(-c*f+d*e)*(d*x+c)*co s(a+b*(d*x+c)^2)/b/d^4-1/2*f^3*(d*x+c)^2*cos(a+b*(d*x+c)^2)/b/d^4+1/2*f^3* sin(a+b*(d*x+c)^2)/b^2/d^4+3/4*f^2*(-c*f+d*e)*cos(a)*FresnelC((d*x+c)*b^(1 /2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/d^4-3/4*f^2*(-c*f+d*e)*Fres nelS((d*x+c)*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*2^(1/2)*Pi^(1/2)/b^(3/2)/d^4 +1/2*(-c*f+d*e)^3*cos(a)*FresnelS((d*x+c)*b^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2 )*Pi^(1/2)/d^4/b^(1/2)+1/2*(-c*f+d*e)^3*FresnelC((d*x+c)*b^(1/2)*2^(1/2)/P i^(1/2))*sin(a)*2^(1/2)*Pi^(1/2)/d^4/b^(1/2)
Time = 1.84 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.64 \[ \int (e+f x)^3 \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {-4 b f \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right ) \cos \left (a+b (c+d x)^2\right )+2 \sqrt {b} (d e-c f) \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \left (2 b (d e-c f)^2 \cos (a)-3 f^2 \sin (a)\right )+2 \sqrt {b} (d e-c f) \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \left (3 f^2 \cos (a)+2 b (d e-c f)^2 \sin (a)\right )+4 f^3 \sin \left (a+b (c+d x)^2\right )}{8 b^2 d^4} \]
(-4*b*f*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2))*Co s[a + b*(c + d*x)^2] + 2*Sqrt[b]*(d*e - c*f)*Sqrt[2*Pi]*FresnelS[Sqrt[b]*S qrt[2/Pi]*(c + d*x)]*(2*b*(d*e - c*f)^2*Cos[a] - 3*f^2*Sin[a]) + 2*Sqrt[b] *(d*e - c*f)*Sqrt[2*Pi]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*(3*f^2*Cos[ a] + 2*b*(d*e - c*f)^2*Sin[a]) + 4*f^3*Sin[a + b*(c + d*x)^2])/(8*b^2*d^4)
Time = 0.55 (sec) , antiderivative size = 321, 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.
| \(\displaystyle \int (e+f x)^3 \sin \left (a+b (c+d x)^2\right ) \, dx\) |
| \(\Big \downarrow \) 3914 |
| \(\displaystyle \frac {\int \left (\sin \left (b (c+d x)^2+a\right ) (d e-c f)^3+3 f (c+d x) \sin \left (b (c+d x)^2+a\right ) (d e-c f)^2+3 f^2 (c+d x)^2 \sin \left (b (c+d x)^2+a\right ) (d e-c f)+f^3 (c+d x)^3 \sin \left (b (c+d x)^2+a\right )\right )d(c+d x)}{d^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 \sqrt {\frac {\pi }{2}} f^2 \cos (a) (d e-c f) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} f^2 \sin (a) (d e-c f) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2}}+\frac {f^3 \sin \left (a+b (c+d x)^2\right )}{2 b^2}-\frac {3 f^2 (c+d x) (d e-c f) \cos \left (a+b (c+d x)^2\right )}{2 b}+\frac {\sqrt {\frac {\pi }{2}} \sin (a) (d e-c f)^3 \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) (d e-c f)^3 \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b}}-\frac {3 f (d e-c f)^2 \cos \left (a+b (c+d x)^2\right )}{2 b}-\frac {f^3 (c+d x)^2 \cos \left (a+b (c+d x)^2\right )}{2 b}}{d^4}\) |
((-3*f*(d*e - c*f)^2*Cos[a + b*(c + d*x)^2])/(2*b) - (3*f^2*(d*e - c*f)*(c + d*x)*Cos[a + b*(c + d*x)^2])/(2*b) - (f^3*(c + d*x)^2*Cos[a + b*(c + d* x)^2])/(2*b) + (3*f^2*(d*e - c*f)*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[ 2/Pi]*(c + d*x)])/(2*b^(3/2)) + ((d*e - c*f)^3*Sqrt[Pi/2]*Cos[a]*FresnelS[ Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/Sqrt[b] + ((d*e - c*f)^3*Sqrt[Pi/2]*Fresnel C[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/Sqrt[b] - (3*f^2*(d*e - c*f)*Sqrt[ Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(2*b^(3/2)) + (f^3*Si n[a + b*(c + d*x)^2])/(2*b^2))/d^4
3.2.65.3.1 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]
Result contains complex when optimal does not.
Time = 1.05 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.02
| method | result | size |
| risch | \(\frac {i \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right ) \sqrt {\pi }\, e^{3} {\mathrm e}^{i a}}{4 \sqrt {-i b}\, d}-\frac {i f^{3} {\mathrm e}^{i a} c^{3} \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d^{4} \sqrt {-i b}}+\frac {3 f^{3} {\mathrm e}^{i a} c \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{8 d^{4} b \sqrt {-i b}}-\frac {3 i f \,e^{2} {\mathrm e}^{i a} c \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d^{2} \sqrt {-i b}}+\frac {3 i f^{2} e \,{\mathrm e}^{i a} c^{2} \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d^{3} \sqrt {-i b}}-\frac {3 f^{2} e \,{\mathrm e}^{i a} \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{8 b \,d^{3} \sqrt {-i b}}+\frac {i {\mathrm e}^{-i a} e^{3} \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d \sqrt {i b}}-\frac {i f^{3} {\mathrm e}^{-i a} c^{3} \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d^{4} \sqrt {i b}}-\frac {3 f^{3} {\mathrm e}^{-i a} c \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{8 d^{4} b \sqrt {i b}}-\frac {3 i f \,e^{2} {\mathrm e}^{-i a} c \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d^{2} \sqrt {i b}}+\frac {3 i f^{2} e \,{\mathrm e}^{-i a} c^{2} \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d^{3} \sqrt {i b}}+\frac {3 f^{2} e \,{\mathrm e}^{-i a} \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{8 b \,d^{3} \sqrt {i b}}-\frac {f \left (d^{2} x^{2} f^{2}-c d \,f^{2} x +3 f e x \,d^{2}+c^{2} f^{2}-3 c d e f +3 d^{2} e^{2}\right ) \cos \left (d^{2} x^{2} b +2 c d x b +c^{2} b +a \right )}{2 b \,d^{4}}+\frac {f^{3} \sin \left (d^{2} x^{2} b +2 c d x b +c^{2} b +a \right )}{2 b^{2} d^{4}}\) | \(690\) |
| default | \(\text {Expression too large to display}\) | \(1248\) |
| parts | \(\text {Expression too large to display}\) | \(2249\) |
1/4*I*erf(-d*(-I*b)^(1/2)*x+I*b*c/(-I*b)^(1/2))/(-I*b)^(1/2)/d*Pi^(1/2)*e^ 3*exp(I*a)-1/4*I*f^3*exp(I*a)*c^3/d^4*Pi^(1/2)/(-I*b)^(1/2)*erf(-d*(-I*b)^ (1/2)*x+I*b*c/(-I*b)^(1/2))+3/8*f^3*exp(I*a)*c/d^4/b*Pi^(1/2)/(-I*b)^(1/2) *erf(-d*(-I*b)^(1/2)*x+I*b*c/(-I*b)^(1/2))-3/4*I*f*e^2*exp(I*a)*c/d^2*Pi^( 1/2)/(-I*b)^(1/2)*erf(-d*(-I*b)^(1/2)*x+I*b*c/(-I*b)^(1/2))+3/4*I*f^2*e*ex p(I*a)*c^2/d^3*Pi^(1/2)/(-I*b)^(1/2)*erf(-d*(-I*b)^(1/2)*x+I*b*c/(-I*b)^(1 /2))-3/8*f^2*e*exp(I*a)/b/d^3*Pi^(1/2)/(-I*b)^(1/2)*erf(-d*(-I*b)^(1/2)*x+ I*b*c/(-I*b)^(1/2))+1/4*I*exp(-I*a)*e^3*Pi^(1/2)/d/(I*b)^(1/2)*erf(d*(I*b) ^(1/2)*x+I*b*c/(I*b)^(1/2))-1/4*I*f^3*exp(-I*a)*c^3/d^4*Pi^(1/2)/(I*b)^(1/ 2)*erf(d*(I*b)^(1/2)*x+I*b*c/(I*b)^(1/2))-3/8*f^3*exp(-I*a)*c/d^4/b*Pi^(1/ 2)/(I*b)^(1/2)*erf(d*(I*b)^(1/2)*x+I*b*c/(I*b)^(1/2))-3/4*I*f*e^2*exp(-I*a )*c/d^2*Pi^(1/2)/(I*b)^(1/2)*erf(d*(I*b)^(1/2)*x+I*b*c/(I*b)^(1/2))+3/4*I* f^2*e*exp(-I*a)*c^2/d^3*Pi^(1/2)/(I*b)^(1/2)*erf(d*(I*b)^(1/2)*x+I*b*c/(I* b)^(1/2))+3/8*f^2*e*exp(-I*a)/b/d^3*Pi^(1/2)/(I*b)^(1/2)*erf(d*(I*b)^(1/2) *x+I*b*c/(I*b)^(1/2))-1/2*f/b*(d^2*f^2*x^2-c*d*f^2*x+3*d^2*e*f*x+c^2*f^2-3 *c*d*e*f+3*d^2*e^2)/d^4*cos(b*d^2*x^2+2*b*c*d*x+b*c^2+a)+1/2*f^3/b^2/d^4*s in(b*d^2*x^2+2*b*c*d*x+b*c^2+a)
Time = 0.34 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.96 \[ \int (e+f x)^3 \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {2 \, d f^{3} \sin \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right ) + \sqrt {2} {\left (3 \, \pi {\left (d e f^{2} - c f^{3}\right )} \cos \left (a\right ) + 2 \, \pi {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \sin \left (a\right )\right )} \sqrt {\frac {b d^{2}}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) + \sqrt {2} {\left (2 \, \pi {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \cos \left (a\right ) - 3 \, \pi {\left (d e f^{2} - c f^{3}\right )} \sin \left (a\right )\right )} \sqrt {\frac {b d^{2}}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (b d^{3} f^{3} x^{2} + 3 \, b d^{3} e^{2} f - 3 \, b c d^{2} e f^{2} + b c^{2} d f^{3} + {\left (3 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x\right )} \cos \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}{4 \, b^{2} d^{5}} \]
1/4*(2*d*f^3*sin(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a) + sqrt(2)*(3*pi*(d*e*f ^2 - c*f^3)*cos(a) + 2*pi*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*sin(a))*sqrt(b*d^2/pi)*fresnel_cos(sqrt(2)*sqrt(b*d^2/pi)*(d*x + c)/d) + sqrt(2)*(2*pi*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*cos(a) - 3*pi*(d*e*f^2 - c*f^3)*sin(a))*sqrt(b*d^2/pi)*fresnel_ sin(sqrt(2)*sqrt(b*d^2/pi)*(d*x + c)/d) - 2*(b*d^3*f^3*x^2 + 3*b*d^3*e^2*f - 3*b*c*d^2*e*f^2 + b*c^2*d*f^3 + (3*b*d^3*e*f^2 - b*c*d^2*f^3)*x)*cos(b* d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/(b^2*d^5)
\[ \int (e+f x)^3 \sin \left (a+b (c+d x)^2\right ) \, dx=\int \left (e + f x\right )^{3} \sin {\left (a + b c^{2} + 2 b c d x + b d^{2} x^{2} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 1.84 (sec) , antiderivative size = 1824, normalized size of antiderivative = 5.35 \[ \int (e+f x)^3 \sin \left (a+b (c+d x)^2\right ) \, dx=\text {Too large to display} \]
-1/8*sqrt(2)*sqrt(pi)*((-(I + 1)*cos(a) + (I - 1)*sin(a))*erf((I*b*d*x + I *b*c)/sqrt(I*b)) + (-(I - 1)*cos(a) + (I + 1)*sin(a))*erf((I*b*d*x + I*b*c )/sqrt(-I*b)))*e^3/(sqrt(b)*d) - 3/8*(2*((e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I *b*c^2) + e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - (-I*e^(I*b*d^ 2*x^2 + 2*I*b*c*d*x + I*b*c^2) + I*e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2 ))*sin(a))*d*x - sqrt(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*((-(I + 1)*sqrt(2)*sq rt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)) - 1) + (I - 1)*sqrt (2)*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*cos(a) + ((I - 1)*sqrt(2)*sqrt(pi)*(erf(sqrt(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2 )) - 1) - (I + 1)*sqrt(2)*sqrt(pi)*(erf(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*sin(a))*c + 2*((e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - (-I*e^(I*b*d^2*x^2 + 2* I*b*c*d*x + I*b*c^2) + I*e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a)) *c)*e^2*f/(b*d^3*x + b*c*d^2) + 3/8*(4*((e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I* b*c^2) + e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - (-I*e^(I*b*d^2 *x^2 + 2*I*b*c*d*x + I*b*c^2) + I*e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2) )*sin(a))*b*c*d*x + 4*((e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + e^(-I*b* d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*cos(a) - (-I*e^(I*b*d^2*x^2 + 2*I*b*c*d* x + I*b*c^2) + I*e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2))*sin(a))*b*c^2 - sqrt(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*(((-(I + 1)*sqrt(2)*sqrt(pi)*(erf(...
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.53 \[ \int (e+f x)^3 \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 \, b d^{3} e^{3} - 6 \, b c d^{2} e^{2} f + 6 \, b c^{2} d e f^{2} - 2 \, b c^{3} f^{3} + 3 i \, d e f^{2} - 3 i \, c f^{3}\right )} \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{\left (i \, a\right )}}{\sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} b} - \frac {2 i \, {\left (-i \, b d^{2} f^{3} {\left (x + \frac {c}{d}\right )}^{2} - 3 \, b d^{2} e f^{2} {\left (i \, x + \frac {i \, c}{d}\right )} - 3 \, b c d f^{3} {\left (-i \, x - \frac {i \, c}{d}\right )} - 3 i \, b d^{2} e^{2} f + 6 i \, b c d e f^{2} - 3 i \, b c^{2} f^{3} + f^{3}\right )} e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2} + i \, a\right )}}{b^{2} d}}{8 \, d^{3}} + \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 \, b d^{3} e^{3} - 6 \, b c d^{2} e^{2} f + 6 \, b c^{2} d e f^{2} - 2 \, b c^{3} f^{3} - 3 i \, d e f^{2} + 3 i \, c f^{3}\right )} \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {2} \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{\left (-i \, a\right )}}{\sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} b} - \frac {2 i \, {\left (-i \, b d^{2} f^{3} {\left (x + \frac {c}{d}\right )}^{2} - 3 \, b d^{2} e f^{2} {\left (i \, x + \frac {i \, c}{d}\right )} - 3 \, b c d f^{3} {\left (-i \, x - \frac {i \, c}{d}\right )} - 3 i \, b d^{2} e^{2} f + 6 i \, b c d e f^{2} - 3 i \, b c^{2} f^{3} - f^{3}\right )} e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2} - i \, a\right )}}{b^{2} d}}{8 \, d^{3}} \]
1/8*(sqrt(2)*sqrt(pi)*(2*b*d^3*e^3 - 6*b*c*d^2*e^2*f + 6*b*c^2*d*e*f^2 - 2 *b*c^3*f^3 + 3*I*d*e*f^2 - 3*I*c*f^3)*erf(-1/2*I*sqrt(2)*sqrt(b*d^2)*(I*b* d^2/sqrt(b^2*d^4) + 1)*(x + c/d))*e^(I*a)/(sqrt(b*d^2)*(I*b*d^2/sqrt(b^2*d ^4) + 1)*b) - 2*I*(-I*b*d^2*f^3*(x + c/d)^2 - 3*b*d^2*e*f^2*(I*x + I*c/d) - 3*b*c*d*f^3*(-I*x - I*c/d) - 3*I*b*d^2*e^2*f + 6*I*b*c*d*e*f^2 - 3*I*b*c ^2*f^3 + f^3)*e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2 + I*a)/(b^2*d))/d^3 + 1/8*(sqrt(2)*sqrt(pi)*(2*b*d^3*e^3 - 6*b*c*d^2*e^2*f + 6*b*c^2*d*e*f^2 - 2*b*c^3*f^3 - 3*I*d*e*f^2 + 3*I*c*f^3)*erf(1/2*I*sqrt(2)*sqrt(b*d^2)*(-I*b *d^2/sqrt(b^2*d^4) + 1)*(x + c/d))*e^(-I*a)/(sqrt(b*d^2)*(-I*b*d^2/sqrt(b^ 2*d^4) + 1)*b) - 2*I*(-I*b*d^2*f^3*(x + c/d)^2 - 3*b*d^2*e*f^2*(I*x + I*c/ d) - 3*b*c*d*f^3*(-I*x - I*c/d) - 3*I*b*d^2*e^2*f + 6*I*b*c*d*e*f^2 - 3*I* b*c^2*f^3 - f^3)*e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2 - I*a)/(b^2*d))/d ^3
Timed out. \[ \int (e+f x)^3 \sin \left (a+b (c+d x)^2\right ) \, dx=\int \sin \left (a+b\,{\left (c+d\,x\right )}^2\right )\,{\left (e+f\,x\right )}^3 \,d x \]