\(\int x \cos (a+b x+c x^2) \, dx\) [2]

Optimal result

Integrand size = 13, antiderivative size = 123 \[ \int x \cos \left (a+b x+c x^2\right ) \, dx=-\frac {b \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{2 c^{3/2}}+\frac {b \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{2 c^{3/2}}+\frac {\sin \left (a+b x+c x^2\right )}{2 c} \] Output:

-1/4*b*2^(1/2)*Pi^(1/2)*cos(a-1/4*b^2/c)*FresnelC(1/2*(2*c*x+b)/c^(1/2)*2^ 
(1/2)/Pi^(1/2))/c^(3/2)+1/4*b*2^(1/2)*Pi^(1/2)*FresnelS(1/2*(2*c*x+b)/c^(1 
/2)*2^(1/2)/Pi^(1/2))*sin(a-1/4*b^2/c)/c^(3/2)+1/2*sin(c*x^2+b*x+a)/c
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.91 \[ \int x \cos \left (a+b x+c x^2\right ) \, dx=\frac {-b \sqrt {2 \pi } \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )+b \sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )+2 \sqrt {c} \sin (a+x (b+c x))}{4 c^{3/2}} \] Input:

Integrate[x*Cos[a + b*x + c*x^2],x]
 

Output:

(-(b*Sqrt[2*Pi]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi 
])]) + b*Sqrt[2*Pi]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2 
/(4*c)] + 2*Sqrt[c]*Sin[a + x*(b + c*x)])/(4*c^(3/2))
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3943, 3929, 3832, 3833}

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.

\(\displaystyle \int x \cos \left (a+b x+c x^2\right ) \, dx\)

\(\Big \downarrow \) 3943

\(\displaystyle \frac {\sin \left (a+b x+c x^2\right )}{2 c}-\frac {b \int \cos \left (c x^2+b x+a\right )dx}{2 c}\)

\(\Big \downarrow \) 3929

\(\displaystyle \frac {\sin \left (a+b x+c x^2\right )}{2 c}-\frac {b \left (\cos \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx-\sin \left (a-\frac {b^2}{4 c}\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right )dx\right )}{2 c}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {\sin \left (a+b x+c x^2\right )}{2 c}-\frac {b \left (\cos \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {\sin \left (a+b x+c x^2\right )}{2 c}-\frac {b \left (\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}\)

Input:

Int[x*Cos[a + b*x + c*x^2],x]
 

Output:

-1/2*(b*((Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt 
[2*Pi])])/Sqrt[c] - (Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])] 
*Sin[a - b^2/(4*c)])/Sqrt[c]))/c + Sin[a + b*x + c*x^2]/(2*c)
 

Defintions of rubi rules used

rule 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3929

Int[Cos[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Cos[(b^2 - 4* 
a*c)/(4*c)]   Int[Cos[(b + 2*c*x)^2/(4*c)], x], x] + Simp[Sin[(b^2 - 4*a*c) 
/(4*c)]   Int[Sin[(b + 2*c*x)^2/(4*c)], x], x] /; FreeQ[{a, b, c}, x] && Ne 
Q[b^2 - 4*a*c, 0]
 

rule 3943

Int[Cos[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]*((d_.) + (e_.)*(x_)), x_Symbol] 
:> Simp[e*(Sin[a + b*x + c*x^2]/(2*c)), x] + Simp[(2*c*d - b*e)/(2*c)   Int 
[Cos[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b 
*e, 0]
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.80

method	result	size
default	\(\frac {\sin \left (c \,x^{2}+b x +a \right )}{2 c}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )+\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{4 c^{\frac {3}{2}}}\)	\(99\)
risch	\(-\frac {b \sqrt {\pi }\, {\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}} \operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right )}{8 c \sqrt {i c}}+\frac {b \sqrt {\pi }\, {\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}} \operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right )}{8 c \sqrt {-i c}}+\frac {\sin \left (c \,x^{2}+b x +a \right )}{2 c}\)	\(123\)
parts	\(\frac {\sqrt {2}\, \sqrt {\pi }\, x \cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )}{2 \sqrt {c}}+\frac {\sqrt {2}\, \sqrt {\pi }\, x \sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\frac {\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (\operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )-\frac {\sin \left (\frac {\pi \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )^{2}}{2}\right )}{\pi }\right )}{2 \sqrt {c}}+\frac {\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )+\frac {\cos \left (\frac {\pi \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )^{2}}{2}\right )}{\pi }\right )}{2 \sqrt {c}}\right )}{2 \sqrt {c}}\)	\(327\)

Input:

int(x*cos(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/2*sin(c*x^2+b*x+a)/c-1/4*b/c^(3/2)*2^(1/2)*Pi^(1/2)*(cos((1/4*b^2-a*c)/c 
)*FresnelC(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b))+sin((1/4*b^2-a*c)/c)*Fres 
nelS(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.97 \[ \int x \cos \left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {2} \pi b \sqrt {\frac {c}{\pi }} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - \sqrt {2} \pi b \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - 2 \, c \sin \left (c x^{2} + b x + a\right )}{4 \, c^{2}} \] Input:

integrate(x*cos(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

-1/4*(sqrt(2)*pi*b*sqrt(c/pi)*cos(-1/4*(b^2 - 4*a*c)/c)*fresnel_cos(1/2*sq 
rt(2)*(2*c*x + b)*sqrt(c/pi)/c) - sqrt(2)*pi*b*sqrt(c/pi)*fresnel_sin(1/2* 
sqrt(2)*(2*c*x + b)*sqrt(c/pi)/c)*sin(-1/4*(b^2 - 4*a*c)/c) - 2*c*sin(c*x^ 
2 + b*x + a))/c^2
 

Sympy [F]

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

integrate(x*cos(c*x**2+b*x+a),x)
 

Output:

Integral(x*cos(a + b*x + c*x**2), x)
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 580, normalized size of antiderivative = 4.72 \[ \int x \cos \left (a+b x+c x^2\right ) \, dx =\text {Too large to display} \] Input:

integrate(x*cos(c*x^2+b*x+a),x, algorithm="maxima")
 

Output:

1/16*(((I - 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I 
*b^2)/c)) - 1) - (I + 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4* 
I*b*c*x + I*b^2)/c)) - 1))*b^2*cos(-1/4*(b^2 - 4*a*c)/c) + ((I + 1)*sqrt(2 
)*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) - (I - 
 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) 
- 1))*b^2*sin(-1/4*(b^2 - 4*a*c)/c) - 2*((-(I - 1)*sqrt(2)*sqrt(pi)*(erf(1 
/2*sqrt((4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1) + (I + 1)*sqrt(2)*sqrt( 
pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b*c*cos(-1/ 
4*(b^2 - 4*a*c)/c) + (-(I + 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt((4*I*c^2*x^2 
 + 4*I*b*c*x + I*b^2)/c)) - 1) + (I - 1)*sqrt(2)*sqrt(pi)*(erf(1/2*sqrt(-( 
4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c)) - 1))*b*c*sin(-1/4*(b^2 - 4*a*c)/c))* 
x - 4*(c*(I*e^(1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) - I*e^(-1/4*(4*I*c 
^2*x^2 + 4*I*b*c*x + I*b^2)/c))*cos(-1/4*(b^2 - 4*a*c)/c) - c*(e^(1/4*(4*I 
*c^2*x^2 + 4*I*b*c*x + I*b^2)/c) + e^(-1/4*(4*I*c^2*x^2 + 4*I*b*c*x + I*b^ 
2)/c))*sin(-1/4*(b^2 - 4*a*c)/c))*sqrt((4*c^2*x^2 + 4*b*c*x + b^2)/c))/(c^ 
2*sqrt((4*c^2*x^2 + 4*b*c*x + b^2)/c))
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.47 \[ \int x \cos \left (a+b x+c x^2\right ) \, dx=\frac {\frac {\sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{{\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} - 2 i \, e^{\left (i \, c x^{2} + i \, b x + i \, a\right )}}{8 \, c} + \frac {\frac {\sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{{\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 i \, e^{\left (-i \, c x^{2} - i \, b x - i \, a\right )}}{8 \, c} \] Input:

integrate(x*cos(c*x^2+b*x+a),x, algorithm="giac")
 

Output:

1/8*(sqrt(2)*sqrt(pi)*b*erf(-1/4*sqrt(2)*(2*x + b/c)*(-I*c/abs(c) + 1)*sqr 
t(abs(c)))*e^(-1/4*(I*b^2 - 4*I*a*c)/c)/((-I*c/abs(c) + 1)*sqrt(abs(c))) - 
 2*I*e^(I*c*x^2 + I*b*x + I*a))/c + 1/8*(sqrt(2)*sqrt(pi)*b*erf(-1/4*sqrt( 
2)*(2*x + b/c)*(I*c/abs(c) + 1)*sqrt(abs(c)))*e^(-1/4*(-I*b^2 + 4*I*a*c)/c 
)/((I*c/abs(c) + 1)*sqrt(abs(c))) + 2*I*e^(-I*c*x^2 - I*b*x - I*a))/c
 

Mupad [F(-1)]

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

int(x*cos(a + b*x + c*x^2),x)
 

Output:

int(x*cos(a + b*x + c*x^2), x)
 

Reduce [F]

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

int(x*cos(c*x^2+b*x+a),x)
 

Output:

( - int(cos(a + b*x + c*x**2),x)*b + sin(a + b*x + c*x**2))/(2*c)