Integrand size = 18, antiderivative size = 223 \[ \int (e+f x)^3 \sin \left (b (c+d x)^2\right ) \, dx=-\frac {3 f (d e-c f)^2 \cos \left (b (c+d x)^2\right )}{2 b d^4}-\frac {3 f^2 (d e-c f) (c+d x) \cos \left (b (c+d x)^2\right )}{2 b d^4}-\frac {f^3 (c+d x)^2 \cos \left (b (c+d x)^2\right )}{2 b d^4}+\frac {3 f^2 (d e-c f) \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2} d^4}+\frac {(d e-c f)^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^4}+\frac {f^3 \sin \left (b (c+d x)^2\right )}{2 b^2 d^4} \]
-3/2*f*(-c*f+d*e)^2*cos(b*(d*x+c)^2)/b/d^4-3/2*f^2*(-c*f+d*e)*(d*x+c)*cos( b*(d*x+c)^2)/b/d^4-1/2*f^3*(d*x+c)^2*cos(b*(d*x+c)^2)/b/d^4+1/2*f^3*sin(b* (d*x+c)^2)/b^2/d^4+3/4*f^2*(-c*f+d*e)*FresnelC((d*x+c)*b^(1/2)*2^(1/2)/Pi^ (1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/d^4+1/2*(-c*f+d*e)^3*FresnelS((d*x+c)*b^(1 /2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)/d^4/b^(1/2)
Time = 0.67 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.78 \[ \int (e+f x)^3 \sin \left (b (c+d x)^2\right ) \, dx=\frac {-4 b f \left (c^2 f^2-c d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right ) \cos \left (b (c+d x)^2\right )-6 \sqrt {b} f^2 (-d e+c f) \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )+4 b^{3/2} (d e-c f)^3 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )+4 f^3 \sin \left (b (c+d x)^2\right )}{8 b^2 d^4} \]
(-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[b*(c + d*x)^2] - 6*Sqrt[b]*f^2*(-(d*e) + c*f)*Sqrt[2*Pi]*FresnelC[Sqrt[b ]*Sqrt[2/Pi]*(c + d*x)] + 4*b^(3/2)*(d*e - c*f)^3*Sqrt[2*Pi]*FresnelS[Sqrt [b]*Sqrt[2/Pi]*(c + d*x)] + 4*f^3*Sin[b*(c + d*x)^2])/(8*b^2*d^4)
Time = 0.41 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int (e+f x)^3 \sin \left (b (c+d x)^2\right ) \, dx\) |
| \(\Big \downarrow \) 3914 |
| \(\displaystyle \frac {\int \left (\sin \left (b (c+d x)^2\right ) (d e-c f)^3+3 f (c+d x) \sin \left (b (c+d x)^2\right ) (d e-c f)^2+3 f^2 (c+d x)^2 \sin \left (b (c+d x)^2\right ) (d e-c f)+f^3 (c+d x)^3 \sin \left (b (c+d x)^2\right )\right )d(c+d x)}{d^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 \sqrt {\frac {\pi }{2}} f^2 (d e-c f) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2}}+\frac {f^3 \sin \left (b (c+d x)^2\right )}{2 b^2}-\frac {3 f^2 (c+d x) (d e-c f) \cos \left (b (c+d x)^2\right )}{2 b}+\frac {\sqrt {\frac {\pi }{2}} (d e-c f)^3 \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b}}-\frac {3 f (d e-c f)^2 \cos \left (b (c+d x)^2\right )}{2 b}-\frac {f^3 (c+d x)^2 \cos \left (b (c+d x)^2\right )}{2 b}}{d^4}\) |
((-3*f*(d*e - c*f)^2*Cos[b*(c + d*x)^2])/(2*b) - (3*f^2*(d*e - c*f)*(c + d *x)*Cos[b*(c + d*x)^2])/(2*b) - (f^3*(c + d*x)^2*Cos[b*(c + d*x)^2])/(2*b) + (3*f^2*(d*e - c*f)*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/( 2*b^(3/2)) + ((d*e - c*f)^3*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d* x)])/Sqrt[b] + (f^3*Sin[b*(c + d*x)^2])/(2*b^2))/d^4
3.2.53.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(585\) vs. \(2(194)=388\).
Time = 1.14 (sec) , antiderivative size = 586, normalized size of antiderivative = 2.63
| method | result | size |
| default | \(-\frac {f^{3} x^{2} \cos \left (d^{2} x^{2} b +2 c d x b +c^{2} b \right )}{2 b \,d^{2}}-\frac {f^{3} c \left (-\frac {x \cos \left (d^{2} x^{2} b +2 c d x b +c^{2} b \right )}{2 b \,d^{2}}-\frac {c \left (-\frac {\cos \left (d^{2} x^{2} b +2 c d x b +c^{2} b \right )}{2 b \,d^{2}}-\frac {c \sqrt {2}\, \sqrt {\pi }\, \operatorname {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{2 d \sqrt {b \,d^{2}}}\right )}{d}+\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {C}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{4 b \,d^{2} \sqrt {b \,d^{2}}}\right )}{d}+\frac {f^{3} \left (\frac {\sin \left (d^{2} x^{2} b +2 c d x b +c^{2} b \right )}{2 b \,d^{2}}-\frac {c \sqrt {2}\, \sqrt {\pi }\, \operatorname {C}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{2 d \sqrt {b \,d^{2}}}\right )}{b \,d^{2}}-\frac {3 e \,f^{2} x \cos \left (d^{2} x^{2} b +2 c d x b +c^{2} b \right )}{2 b \,d^{2}}-\frac {3 e \,f^{2} c \left (-\frac {\cos \left (d^{2} x^{2} b +2 c d x b +c^{2} b \right )}{2 b \,d^{2}}-\frac {c \sqrt {2}\, \sqrt {\pi }\, \operatorname {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{2 d \sqrt {b \,d^{2}}}\right )}{d}+\frac {3 e \,f^{2} \sqrt {2}\, \sqrt {\pi }\, \operatorname {C}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{4 b \,d^{2} \sqrt {b \,d^{2}}}-\frac {3 e^{2} f \cos \left (d^{2} x^{2} b +2 c d x b +c^{2} b \right )}{2 b \,d^{2}}-\frac {3 e^{2} f c \sqrt {2}\, \sqrt {\pi }\, \operatorname {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{2 d \sqrt {b \,d^{2}}}+\frac {\sqrt {2}\, \sqrt {\pi }\, e^{3} \operatorname {S}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {b \,d^{2}}}\right )}{2 \sqrt {b \,d^{2}}}\) | \(586\) |
| risch | \(\frac {i \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right ) \sqrt {\pi }\, e^{3}}{4 \sqrt {-i b}\, d}-\frac {i f^{3} c^{3} \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d^{4} \sqrt {-i b}}+\frac {3 f^{3} c \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{8 d^{4} b \sqrt {-i b}}-\frac {3 i e^{2} f c \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d^{2} \sqrt {-i b}}+\frac {3 i e \,f^{2} c^{2} \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d^{3} \sqrt {-i b}}-\frac {3 e \,f^{2} \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right ) \sqrt {\pi }}{8 \sqrt {-i b}\, d^{3} b}+\frac {i e^{3} \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d \sqrt {i b}}-\frac {i f^{3} c^{3} \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d^{4} \sqrt {i b}}-\frac {3 f^{3} c \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{8 d^{4} b \sqrt {i b}}-\frac {3 i e^{2} f c \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d^{2} \sqrt {i b}}+\frac {3 i e \,f^{2} c^{2} \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d^{3} \sqrt {i b}}+\frac {3 e \,f^{2} \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right ) \sqrt {\pi }}{8 \sqrt {i b}\, d^{3} b}-\frac {f \left (d^{2} x^{2} f^{2}-c d \,f^{2} x +3 f e x \,d^{2}+c^{2} f^{2}-3 c d e f +3 d^{2} e^{2}\right ) \cos \left (b \left (d x +c \right )^{2}\right )}{2 b \,d^{4}}+\frac {f^{3} \sin \left (b \left (d x +c \right )^{2}\right )}{2 b^{2} d^{4}}\) | \(606\) |
| parts | \(\text {Expression too large to display}\) | \(1002\) |
-1/2*f^3/b/d^2*x^2*cos(b*d^2*x^2+2*b*c*d*x+b*c^2)-f^3*c/d*(-1/2/b/d^2*x*co s(b*d^2*x^2+2*b*c*d*x+b*c^2)-c/d*(-1/2/b/d^2*cos(b*d^2*x^2+2*b*c*d*x+b*c^2 )-1/2*c/d*2^(1/2)*Pi^(1/2)/(b*d^2)^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)/(b*d^2) ^(1/2)*(b*d^2*x+b*c*d)))+1/4/b/d^2*2^(1/2)*Pi^(1/2)/(b*d^2)^(1/2)*FresnelC (2^(1/2)/Pi^(1/2)/(b*d^2)^(1/2)*(b*d^2*x+b*c*d)))+f^3/b/d^2*(1/2/b/d^2*sin (b*d^2*x^2+2*b*c*d*x+b*c^2)-1/2*c/d*2^(1/2)*Pi^(1/2)/(b*d^2)^(1/2)*Fresnel C(2^(1/2)/Pi^(1/2)/(b*d^2)^(1/2)*(b*d^2*x+b*c*d)))-3/2*e*f^2/b/d^2*x*cos(b *d^2*x^2+2*b*c*d*x+b*c^2)-3*e*f^2*c/d*(-1/2/b/d^2*cos(b*d^2*x^2+2*b*c*d*x+ b*c^2)-1/2*c/d*2^(1/2)*Pi^(1/2)/(b*d^2)^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)/(b *d^2)^(1/2)*(b*d^2*x+b*c*d)))+3/4*e*f^2/b/d^2*2^(1/2)*Pi^(1/2)/(b*d^2)^(1/ 2)*FresnelC(2^(1/2)/Pi^(1/2)/(b*d^2)^(1/2)*(b*d^2*x+b*c*d))-3/2*e^2*f/b/d^ 2*cos(b*d^2*x^2+2*b*c*d*x+b*c^2)-3/2*e^2*f*c/d*2^(1/2)*Pi^(1/2)/(b*d^2)^(1 /2)*FresnelS(2^(1/2)/Pi^(1/2)/(b*d^2)^(1/2)*(b*d^2*x+b*c*d))+1/2*2^(1/2)*P i^(1/2)/(b*d^2)^(1/2)*e^3*FresnelS(2^(1/2)/Pi^(1/2)/(b*d^2)^(1/2)*(b*d^2*x +b*c*d))
Time = 0.30 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.14 \[ \int (e+f x)^3 \sin \left (b (c+d x)^2\right ) \, dx=\frac {2 \, d f^{3} \sin \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right ) + 3 \, \sqrt {2} \pi {\left (d e f^{2} - c f^{3}\right )} \sqrt {\frac {b d^{2}}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) + 2 \, \sqrt {2} \pi {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + 3 \, b c^{2} d e f^{2} - b c^{3} f^{3}\right )} \sqrt {\frac {b d^{2}}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (b d^{3} f^{3} x^{2} + 3 \, b d^{3} e^{2} f - 3 \, b c d^{2} e f^{2} + b c^{2} d f^{3} + {\left (3 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x\right )} \cos \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )}{4 \, b^{2} d^{5}} \]
1/4*(2*d*f^3*sin(b*d^2*x^2 + 2*b*c*d*x + b*c^2) + 3*sqrt(2)*pi*(d*e*f^2 - c*f^3)*sqrt(b*d^2/pi)*fresnel_cos(sqrt(2)*sqrt(b*d^2/pi)*(d*x + c)/d) + 2* sqrt(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)*sqr t(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))/(b^2*d^5)
\[ \int (e+f x)^3 \sin \left (b (c+d x)^2\right ) \, dx=\int \left (e + f x\right )^{3} \sin {\left (b c^{2} + 2 b c d x + b d^{2} x^{2} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 1.20 (sec) , antiderivative size = 974, normalized size of antiderivative = 4.37 \[ \int (e+f x)^3 \sin \left (b (c+d x)^2\right ) \, dx=\text {Too large to display} \]
1/8*sqrt(2)*sqrt(pi)*e^3*((I + 1)*erf((I*b*d*x + I*b*c)/sqrt(I*b)) + (I - 1)*erf((I*b*d*x + I*b*c)/sqrt(-I*b)))/(sqrt(b)*d) - 3/8*(2*d*x*(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)) - sqrt(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*(-(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)*(er f(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*c + 2*c*(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)))*e^ 2*f/(b*d^3*x + b*c*d^2) + 3/8*(4*b*c*d*x*(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)) + 4*b*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)) - sqrt(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*((-(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)*(er f(sqrt(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)) - 1))*b*c^2 - (I - 1)*sqrt(2 )*gamma(3/2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + (I + 1)*sqrt(2)*gamma( 3/2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)))*e*f^2/(b^2*d^4*x + b^2*c*d^3) - 1/8*(6*b*c^3*(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)) + 2*(3*b*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)) - I*gamma(2, I*b*d^2*x^ 2 + 2*I*b*c*d*x + I*b*c^2) + I*gamma(2, -I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c ^2))*d*x + 2*c*(-I*gamma(2, I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2) + I*ga...
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.27 \[ \int (e+f x)^3 \sin \left (b (c+d x)^2\right ) \, dx=\frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 \, b d^{3} e^{3} - 6 \, b c d^{2} e^{2} f + 6 \, b c^{2} d e f^{2} - 2 \, b c^{3} f^{3} + 3 i \, d e f^{2} - 3 i \, c f^{3}\right )} \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right )}{\sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} b} - \frac {2 i \, {\left (-i \, b d^{2} f^{3} {\left (x + \frac {c}{d}\right )}^{2} - 3 \, b d^{2} e f^{2} {\left (i \, x + \frac {i \, c}{d}\right )} - 3 \, b c d f^{3} {\left (-i \, x - \frac {i \, c}{d}\right )} - 3 i \, b d^{2} e^{2} f + 6 i \, b c d e f^{2} - 3 i \, b c^{2} f^{3} + f^{3}\right )} e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right )}}{b^{2} d}}{8 \, d^{3}} + \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 \, b d^{3} e^{3} - 6 \, b c d^{2} e^{2} f + 6 \, b c^{2} d e f^{2} - 2 \, b c^{3} f^{3} - 3 i \, d e f^{2} + 3 i \, c f^{3}\right )} \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {2} \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right )}{\sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} b} - \frac {2 i \, {\left (-i \, b d^{2} f^{3} {\left (x + \frac {c}{d}\right )}^{2} - 3 \, b d^{2} e f^{2} {\left (i \, x + \frac {i \, c}{d}\right )} - 3 \, b c d f^{3} {\left (-i \, x - \frac {i \, c}{d}\right )} - 3 i \, b d^{2} e^{2} f + 6 i \, b c d e f^{2} - 3 i \, b c^{2} f^{3} - f^{3}\right )} e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )}}{b^{2} d}}{8 \, d^{3}} \]
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))/(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)/(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))/(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)/(b^2*d))/d^3
Time = 6.31 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.04 \[ \int (e+f x)^3 \sin \left (b (c+d x)^2\right ) \, dx=\frac {f^3\,\sin \left (b\,{\left (c+d\,x\right )}^2\right )}{2\,b^2\,d^4}-\frac {\cos \left (b\,{\left (c+d\,x\right )}^2\right )\,\left (c^2\,f^3-3\,c\,d\,e\,f^2+3\,d^2\,e^2\,f\right )}{2\,b\,d^4}-\frac {f^3\,x^2\,\cos \left (b\,{\left (c+d\,x\right )}^2\right )}{2\,b\,d^2}+\frac {x\,\cos \left (b\,{\left (c+d\,x\right )}^2\right )\,\left (c\,f^3-3\,d\,e\,f^2\right )}{2\,b\,d^3}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {S}\left (\frac {\sqrt {2}\,\sqrt {b}\,\left (c+d\,x\right )}{\sqrt {\pi }}\right )\,\left (c^3\,f^3-3\,c^2\,d\,e\,f^2+3\,c\,d^2\,e^2\,f-d^3\,e^3\right )}{2\,\sqrt {b}\,d^4}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {b}\,\left (c+d\,x\right )}{\sqrt {\pi }}\right )\,\left (3\,c\,f^3-3\,d\,e\,f^2\right )}{4\,b^{3/2}\,d^4} \]
(f^3*sin(b*(c + d*x)^2))/(2*b^2*d^4) - (cos(b*(c + d*x)^2)*(c^2*f^3 + 3*d^ 2*e^2*f - 3*c*d*e*f^2))/(2*b*d^4) - (f^3*x^2*cos(b*(c + d*x)^2))/(2*b*d^2) + (x*cos(b*(c + d*x)^2)*(c*f^3 - 3*d*e*f^2))/(2*b*d^3) - (2^(1/2)*pi^(1/2 )*fresnels((2^(1/2)*b^(1/2)*(c + d*x))/pi^(1/2))*(c^3*f^3 - d^3*e^3 + 3*c* d^2*e^2*f - 3*c^2*d*e*f^2))/(2*b^(1/2)*d^4) - (2^(1/2)*pi^(1/2)*fresnelc(( 2^(1/2)*b^(1/2)*(c + d*x))/pi^(1/2))*(3*c*f^3 - 3*d*e*f^2))/(4*b^(3/2)*d^4 )