\(\int (e+f x) \sin (a+b (c+d x)^2) \, dx\) [167]

Optimal result

Integrand size = 18, antiderivative size = 122 \[ \int (e+f x) \sin \left (a+b (c+d x)^2\right ) \, dx=-\frac {f \cos \left (a+b (c+d x)^2\right )}{2 b d^2}+\frac {(d e-c f) \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b} d^2}+\frac {(d e-c f) \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \sin (a)}{\sqrt {b} d^2} \] Output:

-1/2*f*cos(a+b*(d*x+c)^2)/b/d^2+1/2*(-c*f+d*e)*2^(1/2)*Pi^(1/2)*cos(a)*Fre 
snelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c))/b^(1/2)/d^2+1/2*(-c*f+d*e)*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^2
 

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.93 \[ \int (e+f x) \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {-f \cos \left (a+b (c+d x)^2\right )+\sqrt {b} (d e-c f) \sqrt {2 \pi } \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )+\sqrt {b} (d e-c f) \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right ) \sin (a)}{2 b d^2} \] Input:

Integrate[(e + f*x)*Sin[a + b*(c + d*x)^2],x]
 

Output:

(-(f*Cos[a + b*(c + d*x)^2]) + Sqrt[b]*(d*e - c*f)*Sqrt[2*Pi]*Cos[a]*Fresn 
elS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)] + Sqrt[b]*(d*e - c*f)*Sqrt[2*Pi]*Fresnel 
C[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/(2*b*d^2)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.96, 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.

\(\displaystyle \int (e+f x) \sin \left (a+b (c+d x)^2\right ) \, dx\)

\(\Big \downarrow \) 3914

\(\displaystyle \frac {\int \left ((d e-c f) \sin \left (b (c+d x)^2+a\right )+f (c+d x) \sin \left (b (c+d x)^2+a\right )\right )d(c+d x)}{d^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\sqrt {\frac {\pi }{2}} \sin (a) (d e-c f) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) (d e-c f) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} (c+d x)\right )}{\sqrt {b}}-\frac {f \cos \left (a+b (c+d x)^2\right )}{2 b}}{d^2}\)

Input:

Int[(e + f*x)*Sin[a + b*(c + d*x)^2],x]
 

Output:

(-1/2*(f*Cos[a + b*(c + d*x)^2])/b + ((d*e - c*f)*Sqrt[Pi/2]*Cos[a]*Fresne 
lS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)])/Sqrt[b] + ((d*e - c*f)*Sqrt[Pi/2]*Fresne 
lC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)]*Sin[a])/Sqrt[b])/d^2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3914

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.29 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.71

method	result	size
risch	\(\frac {i \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right ) \sqrt {\pi }\, e \,{\mathrm e}^{i a}}{4 \sqrt {-i b}\, d}-\frac {i f \,{\mathrm e}^{i a} c \sqrt {\pi }\, \operatorname {erf}\left (-d \sqrt {-i b}\, x +\frac {i b c}{\sqrt {-i b}}\right )}{4 d^{2} \sqrt {-i b}}+\frac {i {\mathrm e}^{-i a} e \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d \sqrt {i b}}-\frac {i f \,{\mathrm e}^{-i a} c \sqrt {\pi }\, \operatorname {erf}\left (d \sqrt {i b}\, x +\frac {i b c}{\sqrt {i b}}\right )}{4 d^{2} \sqrt {i b}}-\frac {f \cos \left (x^{2} d^{2} b +2 c d x b +b \,c^{2}+a \right )}{2 d^{2} b}\)	\(209\)
default	\(-\frac {f \cos \left (x^{2} d^{2} b +2 c d x b +b \,c^{2}+a \right )}{2 d^{2} b}-\frac {f c \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )-\sin \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )\right )}{2 d \sqrt {d^{2} b}}+\frac {\sqrt {2}\, \sqrt {\pi }\, e \left (\cos \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )-\sin \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (b \,d^{2} x +c d b \right )}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )\right )}{2 \sqrt {d^{2} b}}\)	\(309\)
parts	\(\frac {\sqrt {2}\, \sqrt {\pi }\, \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 ) f x}{2 \sqrt {d^{2} b}}-\frac {\sqrt {2}\, \sqrt {\pi }\, \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 ) f x}{2 \sqrt {d^{2} b}}+\frac {\sqrt {2}\, \sqrt {\pi }\, \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 ) e}{2 \sqrt {d^{2} b}}-\frac {\sqrt {2}\, \sqrt {\pi }\, \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 ) e}{2 \sqrt {d^{2} b}}-\frac {\sqrt {2}\, \sqrt {\pi }\, f \left (\frac {\cos \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {d^{2} b}\, \left (\operatorname {FresnelS}\left (\frac {\sqrt {2}\, d^{2} b x}{\sqrt {\pi }\, \sqrt {d^{2} b}}+\frac {\sqrt {2}\, c d b}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right ) \left (\frac {\sqrt {2}\, d^{2} b x}{\sqrt {\pi }\, \sqrt {d^{2} b}}+\frac {\sqrt {2}\, c d b}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )+\frac {\cos \left (\frac {\pi \left (\frac {\sqrt {2}\, d^{2} b x}{\sqrt {\pi }\, \sqrt {d^{2} b}}+\frac {\sqrt {2}\, c d b}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )^{2}}{2}\right )}{\pi }\right )}{2 d^{2} b}-\frac {\sin \left (\frac {b^{2} c^{2} d^{2}-d^{2} b \left (b \,c^{2}+a \right )}{d^{2} b}\right ) \sqrt {2}\, \sqrt {\pi }\, \sqrt {d^{2} b}\, \left (\operatorname {FresnelC}\left (\frac {\sqrt {2}\, d^{2} b x}{\sqrt {\pi }\, \sqrt {d^{2} b}}+\frac {\sqrt {2}\, c d b}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right ) \left (\frac {\sqrt {2}\, d^{2} b x}{\sqrt {\pi }\, \sqrt {d^{2} b}}+\frac {\sqrt {2}\, c d b}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )-\frac {\sin \left (\frac {\pi \left (\frac {\sqrt {2}\, d^{2} b x}{\sqrt {\pi }\, \sqrt {d^{2} b}}+\frac {\sqrt {2}\, c d b}{\sqrt {\pi }\, \sqrt {d^{2} b}}\right )^{2}}{2}\right )}{\pi }\right )}{2 d^{2} b}\right )}{2 \sqrt {d^{2} b}}\)	\(672\)

Input:

int((f*x+e)*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* 
exp(I*a)-1/4*I*f*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*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*exp(-I*a)*c/d^2*Pi^(1/2)/(I*b)^(1/2)*e 
rf(d*(I*b)^(1/2)*x+I*b*c/(I*b)^(1/2))-1/2*f/d^2/b*cos(b*d^2*x^2+2*b*c*d*x+ 
b*c^2+a)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07 \[ \int (e+f x) \sin \left (a+b (c+d x)^2\right ) \, dx=\frac {\sqrt {2} \pi \sqrt {\frac {b d^{2}}{\pi }} {\left (d e - c f\right )} \cos \left (a\right ) \operatorname {S}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) + \sqrt {2} \pi \sqrt {\frac {b d^{2}}{\pi }} {\left (d e - c f\right )} \operatorname {C}\left (\frac {\sqrt {2} \sqrt {\frac {b d^{2}}{\pi }} {\left (d x + c\right )}}{d}\right ) \sin \left (a\right ) - d f \cos \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}{2 \, b d^{3}} \] Input:

integrate((f*x+e)*sin(a+b*(d*x+c)^2),x, algorithm="fricas")
 

Output:

1/2*(sqrt(2)*pi*sqrt(b*d^2/pi)*(d*e - c*f)*cos(a)*fresnel_sin(sqrt(2)*sqrt 
(b*d^2/pi)*(d*x + c)/d) + sqrt(2)*pi*sqrt(b*d^2/pi)*(d*e - c*f)*fresnel_co 
s(sqrt(2)*sqrt(b*d^2/pi)*(d*x + c)/d)*sin(a) - d*f*cos(b*d^2*x^2 + 2*b*c*d 
*x + b*c^2 + a))/(b*d^3)
 

Sympy [F]

\[ \int (e+f x) \sin \left (a+b (c+d x)^2\right ) \, dx=\int \left (e + f x\right ) \sin {\left (a + b c^{2} + 2 b c d x + b d^{2} x^{2} \right )}\, dx \] Input:

integrate((f*x+e)*sin(a+b*(d*x+c)**2),x)
 

Output:

Integral((e + f*x)*sin(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2), x)
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.44 (sec) , antiderivative size = 483, normalized size of antiderivative = 3.96 \[ \int (e+f x) \sin \left (a+b (c+d x)^2\right ) \, dx=-\frac {\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 (\frac {i \, b d x + i \, b c}{\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 (\frac {i \, b d x + i \, b c}{\sqrt {-i \, b}}\right )\right )} e}{8 \, \sqrt {b} d} - \frac {{\left (2 \, {\left ({\left (e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right )} + e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )}\right )} \cos \left (a\right ) - {\left (-i \, e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right )} + i \, e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )}\right )} \sin \left (a\right )\right )} d x - \sqrt {b d^{2} x^{2} + 2 \, b c d x + b c^{2}} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}}\right ) - 1\right )} + \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}}\right ) - 1\right )}\right )} \cos \left (a\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}}\right ) - 1\right )} - \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}}\right ) - 1\right )}\right )} \sin \left (a\right )\right )} c + 2 \, {\left ({\left (e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right )} + e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )}\right )} \cos \left (a\right ) - {\left (-i \, e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right )} + i \, e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )}\right )} \sin \left (a\right )\right )} c\right )} f}{8 \, {\left (b d^{3} x + b c d^{2}\right )}} \] Input:

integrate((f*x+e)*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/(sqrt(b)*d) - 1/8*(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)*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))*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 
)*f/(b*d^3*x + b*c*d^2)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.02 \[ \int (e+f x) \sin \left (a+b (c+d x)^2\right ) \, dx=-\frac {\frac {\sqrt {2} \sqrt {\pi } {\left (-i \, d e + i \, c f\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{\left (i \, a\right )}}{\sqrt {b d^{2}} {\left (-\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )}} + \frac {f e^{\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2} + i \, a\right )}}{b d}}{4 \, d} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (i \, d e - i \, c f\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )} {\left (x + \frac {c}{d}\right )}\right ) e^{\left (-i \, a\right )}}{\sqrt {b d^{2}} {\left (\frac {i \, b d^{2}}{\sqrt {b^{2} d^{4}}} + 1\right )}} + \frac {f e^{\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2} - i \, a\right )}}{b d}}{4 \, d} \] Input:

integrate((f*x+e)*sin(a+b*(d*x+c)^2),x, algorithm="giac")
 

Output:

-1/4*(sqrt(2)*sqrt(pi)*(-I*d*e + I*c*f)*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)) + f*e^(I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2 + I*a)/(b*d))/d - 1 
/4*(sqrt(2)*sqrt(pi)*(I*d*e - I*c*f)*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)) + f*e^(-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2 - I*a)/(b*d))/d
 

Mupad [F(-1)]

Timed out. \[ \int (e+f x) \sin \left (a+b (c+d x)^2\right ) \, dx=\int \sin \left (a+b\,{\left (c+d\,x\right )}^2\right )\,\left (e+f\,x\right ) \,d x \] Input:

int(sin(a + b*(c + d*x)^2)*(e + f*x),x)
 

Output:

int(sin(a + b*(c + d*x)^2)*(e + f*x), x)
 

Reduce [F]

\[ \int (e+f x) \sin \left (a+b (c+d x)^2\right ) \, dx=\left (\int \sin \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )d x \right ) e +\left (\int \sin \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right ) x d x \right ) f \] Input:

int((f*x+e)*sin(a+b*(d*x+c)^2),x)
 

Output:

int(sin(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2),x)*e + int(sin(a + b*c**2 + 
2*b*c*d*x + b*d**2*x**2)*x,x)*f