Integrand size = 20, antiderivative size = 256 \[ \int (e+f x)^2 \sin \left (a+b (c+d x)^2\right ) \, dx=-\frac {f (d e-c f) \cos \left (a+b (c+d x)^2\right )}{b d^3}-\frac {f^2 (c+d x) \cos \left (a+b (c+d x)^2\right )}{2 b d^3}+\frac {f^2 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2} d^3}+\frac {(d e-c f)^2 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^3}+\frac {(d e-c f)^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \sin (a)}{\sqrt {b} d^3}-\frac {f^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \sin (a)}{2 b^{3/2} d^3} \] Output:
-f*(-c*f+d*e)*cos(a+b*(d*x+c)^2)/b/d^3-1/2*f^2*(d*x+c)*cos(a+b*(d*x+c)^2)/ b/d^3+1/4*f^2*2^(1/2)*Pi^(1/2)*cos(a)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d *x+c))/b^(3/2)/d^3+1/2*(-c*f+d*e)^2*2^(1/2)*Pi^(1/2)*cos(a)*FresnelS(b^(1/ 2)*2^(1/2)/Pi^(1/2)*(d*x+c))/b^(1/2)/d^3+1/2*(-c*f+d*e)^2*2^(1/2)*Pi^(1/2) *FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c))*sin(a)/b^(1/2)/d^3-1/4*f^2*2^( 1/2)*Pi^(1/2)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c))*sin(a)/b^(3/2)/d^ 3
Time = 1.97 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.59 \[ \int (e+f x)^2 \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {-4 \sqrt {b} f (2 d e-c f+d f x) \cos \left (a+b (c+d x)^2\right )+2 \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \left (2 b (d e-c f)^2 \cos (a)-f^2 \sin (a)\right )+2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \left (f^2 \cos (a)+2 b (d e-c f)^2 \sin (a)\right )}{8 b^{3/2} d^3} \] Input:
Integrate[(e + f*x)^2*Sin[a + b*(c + d*x)^2],x]
Output:
(-4*Sqrt[b]*f*(2*d*e - c*f + d*f*x)*Cos[a + b*(c + d*x)^2] + 2*Sqrt[2*Pi]* FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*(2*b*(d*e - c*f)^2*Cos[a] - f^2*Sin [a]) + 2*Sqrt[2*Pi]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*(f^2*Cos[a] + 2 *b*(d*e - c*f)^2*Sin[a]))/(8*b^(3/2)*d^3)
Time = 0.41 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.95, 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.
| \(\displaystyle \int (e+f x)^2 \sin \left (a+b (c+d x)^2\right ) \, dx\) |
| \(\Big \downarrow \) 3914 |
| \(\displaystyle \frac {\int \left (\sin \left (b (c+d x)^2+a\right ) (d e-c f)^2+2 f (c+d x) \sin \left (b (c+d x)^2+a\right ) (d e-c f)+f^2 (c+d x)^2 \sin \left (b (c+d x)^2+a\right )\right )d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sqrt {\frac {\pi }{2}} f^2 \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} f^2 \sin (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{2 b^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} \sin (a) (d e-c f)^2 \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) (d e-c f)^2 \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b}}-\frac {f (d e-c f) \cos \left (a+b (c+d x)^2\right )}{b}-\frac {f^2 (c+d x) \cos \left (a+b (c+d x)^2\right )}{2 b}}{d^3}\) |
Input:
Int[(e + f*x)^2*Sin[a + b*(c + d*x)^2],x]
Output:
(-((f*(d*e - c*f)*Cos[a + b*(c + d*x)^2])/b) - (f^2*(c + d*x)*Cos[a + b*(c + d*x)^2])/(2*b) + (f^2*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/(2*b^(3/2)) + ((d*e - c*f)^2*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*S qrt[2/Pi]*(c + d*x)])/Sqrt[b] + ((d*e - c*f)^2*Sqrt[Pi/2]*FresnelC[Sqrt[b] *Sqrt[2/Pi]*(c + d*x)]*Sin[a])/Sqrt[b] - (f^2*Sqrt[Pi/2]*FresnelS[Sqrt[b]* Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(2*b^(3/2)))/d^3
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 = 1.96 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.71
| method | result | size |
| risch | \(\frac {i \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right ) \sqrt {\pi }\, e^{2} {\mathrm e}^{i a}}{4 \sqrt {-i b}\, d}+\frac {i f^{2} {\mathrm e}^{i a} c^{2} \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d^{3} \sqrt {-i b}}-\frac {f^{2} {\mathrm e}^{i a} \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{8 b \,d^{3} \sqrt {-i b}}-\frac {i f e \,{\mathrm e}^{i a} c \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{2 d^{2} \sqrt {-i b}}+\frac {i {\mathrm e}^{-i a} e^{2} \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d \sqrt {i b}}+\frac {i f^{2} {\mathrm e}^{-i a} c^{2} \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d^{3} \sqrt {i b}}+\frac {f^{2} {\mathrm e}^{-i a} \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{8 b \,d^{3} \sqrt {i b}}-\frac {i f e \,{\mathrm e}^{-i a} c \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{2 d^{2} \sqrt {i b}}+2 \left (\frac {i f^{2} \left (\frac {i x}{2 d^{2} b}-\frac {i c}{2 b \,d^{3}}\right )}{2}-\frac {e f}{2 d^{2} b}\right ) \cos \left (x^{2} d^{2} b +2 c d x b +b \,c^{2}+a \right )\) | \(438\) |
| default | \(-\frac {f^{2} x \cos \left (x^{2} d^{2} b +2 c d x b +b \,c^{2}+a \right )}{2 d^{2} b}-\frac {f^{2} c \left (-\frac {\cos \left (x^{2} d^{2} b +2 c d x b +b \,c^{2}+a \right )}{2 d^{2} b}-\frac {c \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )-\sin \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )\right )}{2 d \sqrt {d^{2} b}}\right )}{d}+\frac {f^{2} \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )+\sin \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )\right )}{4 d^{2} b \sqrt {d^{2} b}}-\frac {e f \cos \left (x^{2} d^{2} b +2 c d x b +b \,c^{2}+a \right )}{d^{2} b}-\frac {e f c \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )-\sin \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )\right )}{d \sqrt {d^{2} b}}+\frac {\sqrt {2}\, \sqrt {\pi }\, e^{2} \left (\cos \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )-\sin \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )\right )}{2 \sqrt {d^{2} b}}\) | \(669\) |
| parts | \(\text {Expression too large to display}\) | \(1282\) |
Input:
int((f*x+e)^2*sin(a+b*(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
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^ 2*exp(I*a)+1/4*I*f^2*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))-1/8*f^2*exp(I*a)/b/d^3*Pi^(1/2)/(-I*b)^(1/2)*e rf(-d*(-I*b)^(1/2)*x+I*b*c/(-I*b)^(1/2))-1/2*I*f*e*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))+1/4*I*exp(-I*a)*e^ 2*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^2* 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))+1/8*f^2*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*I*f*e*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))+2*(1/2*I*f^2*(1/2*I*x/d^2/b-1/2*I/b/d^3*c)-1/2*e *f/d^2/b)*cos(b*d^2*x^2+2*b*c*d*x+b*c^2+a)
Time = 0.09 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.81 \[ \int (e+f x)^2 \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {\sqrt {2} {\left (\pi f^{2} \cos \left (a\right ) + 2 \, \pi {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \sin \left (a\right )\right )} \sqrt {\frac {b d^{2}}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) - \sqrt {2} {\left (\pi f^{2} \sin \left (a\right ) - 2 \, \pi {\left (b d^{2} e^{2} - 2 \, b c d e f + b c^{2} f^{2}\right )} \cos \left (a\right )\right )} \sqrt {\frac {b d^{2}}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (b d^{2} f^{2} x + 2 \, b d^{2} e f - b c d f^{2}\right )} \cos \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}{4 \, b^{2} d^{4}} \] Input:
integrate((f*x+e)^2*sin(a+b*(d*x+c)^2),x, algorithm="fricas")
Output:
1/4*(sqrt(2)*(pi*f^2*cos(a) + 2*pi*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*s in(a))*sqrt(b*d^2/pi)*fresnel_cos(sqrt(2)*sqrt(b*d^2/pi)*(d*x + c)/d) - sq rt(2)*(pi*f^2*sin(a) - 2*pi*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*cos(a))* sqrt(b*d^2/pi)*fresnel_sin(sqrt(2)*sqrt(b*d^2/pi)*(d*x + c)/d) - 2*(b*d^2* f^2*x + 2*b*d^2*e*f - b*c*d*f^2)*cos(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a))/( b^2*d^4)
\[ \int (e+f x)^2 \sin \left (a+b (c+d x)^2\right ) \, dx=\int \left (e + f x\right )^{2} \sin {\left (a + b c^{2} + 2 b c d x + b d^{2} x^{2} \right )}\, dx \] Input:
integrate((f*x+e)**2*sin(a+b*(d*x+c)**2),x)
Output:
Integral((e + f*x)**2*sin(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2), x)
Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 1038, normalized size of antiderivative = 4.05 \[ \int (e+f x)^2 \sin \left (a+b (c+d x)^2\right ) \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*sin(a+b*(d*x+c)^2),x, algorithm="maxima")
Output:
-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^2/(sqrt(b)*d) - 1/4*(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*f/(b*d^3*x + b*c*d^2) + 1/8*(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 - s qrt(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*(((-(I + 1)*sqrt(2)*sqrt(pi)*(erf(sq...
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.35 \[ \int (e+f x)^2 \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 i \, b d^{2} e^{2} - 4 i \, b c d e f + 2 i \, b c^{2} f^{2} - f^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{\left (i \, a\right )}}{\sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} b} - \frac {2 \, {\left (d f^{2} {\left (x + \frac {c}{d}\right )} + 2 \, d e f - 2 \, c f^{2}\right )} e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2} + i \, a\right )}}{b d}}{8 \, d^{2}} + \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (-2 i \, b d^{2} e^{2} + 4 i \, b c d e f - 2 i \, b c^{2} f^{2} - f^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{\left (-i \, a\right )}}{\sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} b} - \frac {2 \, {\left (d f^{2} {\left (x + \frac {c}{d}\right )} + 2 \, d e f - 2 \, c f^{2}\right )} e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2} - i \, a\right )}}{b d}}{8 \, d^{2}} \] Input:
integrate((f*x+e)^2*sin(a+b*(d*x+c)^2),x, algorithm="giac")
Output:
1/8*(sqrt(2)*sqrt(pi)*(2*I*b*d^2*e^2 - 4*I*b*c*d*e*f + 2*I*b*c^2*f^2 - f^2 )*erf(-1/2*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*(d*f^2*(x + c/d) + 2 *d*e*f - 2*c*f^2)*e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2 + I*a)/(b*d))/d^2 + 1/8*(sqrt(2)*sqrt(pi)*(-2*I*b*d^2*e^2 + 4*I*b*c*d*e*f - 2*I*b*c^2*f^2 - f^2)*erf(-1/2*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*(d*f^2*(x + c/d) + 2*d*e*f - 2*c*f^2)*e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2 - I*a)/(b*d)) /d^2
Timed out. \[ \int (e+f x)^2 \sin \left (a+b (c+d x)^2\right ) \, dx=\int \sin \left (a+b\,{\left (c+d\,x\right )}^2\right )\,{\left (e+f\,x\right )}^2 \,d x \] Input:
int(sin(a + b*(c + d*x)^2)*(e + f*x)^2,x)
Output:
int(sin(a + b*(c + d*x)^2)*(e + f*x)^2, x)
\[ \int (e+f x)^2 \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {24 b^{2} c^{3} d^{2} e f x -8 b^{2} c \,d^{4} e f \,x^{3}-12 \sin \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right ) b c \,d^{2} e f x -3 d \,f^{2} x +6 d e f +6 \left (\int \frac {1}{\tan \left (\frac {1}{2} b \,d^{2} x^{2}+b c d x +\frac {1}{2} b \,c^{2}+\frac {1}{2} a \right )^{2}+1}d x \right ) d \,f^{2}-12 b^{2} c^{4} d \,f^{2} x -24 \left (\int \frac {x^{2}}{\tan \left (\frac {1}{2} b \,d^{2} x^{2}+b c d x +\frac {1}{2} b \,c^{2}+\frac {1}{2} a \right )^{2}+1}d x \right ) b^{2} d^{5} e^{2}-6 \sin \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right ) b c \,d^{2} e^{2}+6 \sin \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right ) b \,d^{3} e^{2} x -3 c \,f^{2}+48 \left (\int \frac {x^{2}}{\tan \left (\frac {1}{2} b \,d^{2} x^{2}+b c d x +\frac {1}{2} b \,c^{2}+\frac {1}{2} a \right )^{2}+1}d x \right ) b^{2} c \,d^{4} e f -48 \left (\int \frac {1}{\tan \left (\frac {1}{2} b \,d^{2} x^{2}+b c d x +\frac {1}{2} b \,c^{2}+\frac {1}{2} a \right )^{2}+1}d x \right ) b^{2} c^{3} d^{2} e f +4 b^{2} d^{5} e^{2} x^{3}-12 b^{2} c^{2} d^{3} e^{2} x +4 b^{2} c^{2} d^{3} f^{2} x^{3}-6 \cos \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right ) d e f -3 \cos \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right ) d \,f^{2} x -6 \sin \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right ) b \,c^{3} f^{2}+24 \left (\int \frac {1}{\tan \left (\frac {1}{2} b \,d^{2} x^{2}+b c d x +\frac {1}{2} b \,c^{2}+\frac {1}{2} a \right )^{2}+1}d x \right ) b^{2} c^{4} d \,f^{2}+24 \left (\int \frac {1}{\tan \left (\frac {1}{2} b \,d^{2} x^{2}+b c d x +\frac {1}{2} b \,c^{2}+\frac {1}{2} a \right )^{2}+1}d x \right ) b^{2} c^{2} d^{3} e^{2}+3 \cos \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right ) c \,f^{2}-24 \left (\int \frac {x^{2}}{\tan \left (\frac {1}{2} b \,d^{2} x^{2}+b c d x +\frac {1}{2} b \,c^{2}+\frac {1}{2} a \right )^{2}+1}d x \right ) b^{2} c^{2} d^{3} f^{2}+12 \sin \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right ) b \,c^{2} d e f +6 \sin \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right ) b \,c^{2} d \,f^{2} x}{6 b \,d^{3}} \] Input:
int((f*x+e)^2*sin(a+b*(d*x+c)^2),x)
Output:
(3*cos(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)*c*f**2 - 6*cos(a + b*c**2 + 2 *b*c*d*x + b*d**2*x**2)*d*e*f - 3*cos(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2 )*d*f**2*x - 24*int(x**2/(tan((a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)/2)**2 + 1),x)*b**2*c**2*d**3*f**2 + 48*int(x**2/(tan((a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)/2)**2 + 1),x)*b**2*c*d**4*e*f - 24*int(x**2/(tan((a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)/2)**2 + 1),x)*b**2*d**5*e**2 + 24*int(1/(tan((a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)/2)**2 + 1),x)*b**2*c**4*d*f**2 - 48*i nt(1/(tan((a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)/2)**2 + 1),x)*b**2*c**3*d **2*e*f + 24*int(1/(tan((a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)/2)**2 + 1), x)*b**2*c**2*d**3*e**2 + 6*int(1/(tan((a + b*c**2 + 2*b*c*d*x + b*d**2*x** 2)/2)**2 + 1),x)*d*f**2 - 6*sin(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)*b*c* *3*f**2 + 12*sin(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)*b*c**2*d*e*f + 6*si n(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)*b*c**2*d*f**2*x - 6*sin(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)*b*c*d**2*e**2 - 12*sin(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)*b*c*d**2*e*f*x + 6*sin(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2 )*b*d**3*e**2*x - 12*b**2*c**4*d*f**2*x + 24*b**2*c**3*d**2*e*f*x - 12*b** 2*c**2*d**3*e**2*x + 4*b**2*c**2*d**3*f**2*x**3 - 8*b**2*c*d**4*e*f*x**3 + 4*b**2*d**5*e**2*x**3 - 3*c*f**2 + 6*d*e*f - 3*d*f**2*x)/(6*b*d**3)