\(\int x^2 \cos (a+b x-c x^2) \, dx\) [6]

Optimal result

Integrand size = 16, antiderivative size = 251 \[ \int x^2 \cos \left (a+b x-c x^2\right ) \, dx=-\frac {b^2 \sqrt {\frac {\pi }{2}} \cos \left (a+\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{4 c^{5/2}}+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a+\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{2 c^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a+\frac {b^2}{4 c}\right )}{2 c^{3/2}}-\frac {b^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a+\frac {b^2}{4 c}\right )}{4 c^{5/2}}-\frac {b \sin \left (a+b x-c x^2\right )}{4 c^2}-\frac {x \sin \left (a+b x-c x^2\right )}{2 c} \] Output:

-1/8*b^2*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^(5/2)+1/4*2^(1/2)*Pi^(1/2)*cos(a+1/4*b^2/c)*FresnelS( 
1/2*(-2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))/c^(3/2)-1/4*2^(1/2)*Pi^(1/2)*Fres 
nelC(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)-1/8 
*b^2*2^(1/2)*Pi^(1/2)*FresnelS(1/2*(-2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))*si 
n(a+1/4*b^2/c)/c^(5/2)-1/4*b*sin(-c*x^2+b*x+a)/c^2-1/2*x*sin(-c*x^2+b*x+a) 
/c
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.65 \[ \int x^2 \cos \left (a+b x-c x^2\right ) \, dx=\frac {-\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {-b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \left (2 c \cos \left (a+\frac {b^2}{4 c}\right )-b^2 \sin \left (a+\frac {b^2}{4 c}\right )\right )+\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {-b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \left (b^2 \cos \left (a+\frac {b^2}{4 c}\right )+2 c \sin \left (a+\frac {b^2}{4 c}\right )\right )-2 \sqrt {c} (b+2 c x) \sin (a+x (b-c x))}{8 c^{5/2}} \] Input:

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

Output:

(-(Sqrt[2*Pi]*FresnelS[(-b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*(2*c*Cos[a + b^2 
/(4*c)] - b^2*Sin[a + b^2/(4*c)])) + Sqrt[2*Pi]*FresnelC[(-b + 2*c*x)/(Sqr 
t[c]*Sqrt[2*Pi])]*(b^2*Cos[a + b^2/(4*c)] + 2*c*Sin[a + b^2/(4*c)]) - 2*Sq 
rt[c]*(b + 2*c*x)*Sin[a + x*(b - c*x)])/(8*c^(5/2))
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3945, 3928, 25, 3832, 3833, 3943, 3929, 25, 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.

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

\(\Big \downarrow \) 3945

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

\(\Big \downarrow \) 3928

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

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

\(\Big \downarrow \) 3943

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

\(\Big \downarrow \) 3929

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3832

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

\(\Big \downarrow \) 3833

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3928

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

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]
 

rule 3945

Int[Cos[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]*((d_.) + (e_.)*(x_))^(m_), x_Sym 
bol] :> Simp[e*(d + e*x)^(m - 1)*(Sin[a + b*x + c*x^2]/(2*c)), x] + (-Simp[ 
(b*e - 2*c*d)/(2*c)   Int[(d + e*x)^(m - 1)*Cos[a + b*x + c*x^2], x], x] - 
Simp[e^2*((m - 1)/(2*c))   Int[(d + e*x)^(m - 2)*Sin[a + b*x + c*x^2], x], 
x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
 

Maple [A] (verified)

Time = 1.73 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {x \sin \left (-c \,x^{2}+b x +a \right )}{2 c}+\frac {b \left (-\frac {\sin \left (-c \,x^{2}+b x +a \right )}{2 c}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}+a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (-c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {-c}}\right )-\sin \left (\frac {\frac {b^{2}}{4}+a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (-c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {-c}}\right )\right )}{4 c \sqrt {-c}}\right )}{2 c}+\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}+a c}{c}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \left (-c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {-c}}\right )+\sin \left (\frac {\frac {b^{2}}{4}+a c}{c}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \left (-c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {-c}}\right )\right )}{4 c \sqrt {-c}}\)	\(224\)
risch	\(\frac {b^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {i \left (4 a c +b^{2}\right )}{4 c}} \operatorname {erf}\left (\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right )}{16 c^{2} \sqrt {-i c}}+\frac {i \sqrt {\pi }\, {\mathrm e}^{-\frac {i \left (4 a c +b^{2}\right )}{4 c}} \operatorname {erf}\left (\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right )}{8 c \sqrt {-i c}}-\frac {b^{2} \sqrt {\pi }\, {\mathrm e}^{\frac {i \left (4 a c +b^{2}\right )}{4 c}} \operatorname {erf}\left (-\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right )}{16 c^{2} \sqrt {i c}}+\frac {i \sqrt {\pi }\, {\mathrm e}^{\frac {i \left (4 a c +b^{2}\right )}{4 c}} \operatorname {erf}\left (-\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right )}{8 c \sqrt {i c}}-2 i \left (-\frac {i x}{4 c}-\frac {i b}{8 c^{2}}\right ) \sin \left (-c \,x^{2}+b x +a \right )\)	\(238\)
parts	\(\frac {\sqrt {2}\, \sqrt {\pi }\, x^{2} \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^{2} \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}\, \pi ^{\frac {3}{2}} \left (\frac {\cos \left (\frac {4 a c +b^{2}}{4 c}\right ) \left (\operatorname {FresnelC}\left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right ) \left (\left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right )^{2} \sqrt {\pi }\, \sqrt {-c}-\sqrt {2}\, b \left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right )\right )-\frac {\sqrt {-c}\, \left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right ) \sin \left (\frac {\pi \left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right )^{2}}{2}\right )}{\sqrt {\pi }}+\frac {\sqrt {-c}\, \operatorname {FresnelS}\left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right )}{\sqrt {\pi }}+\frac {\sqrt {2}\, b \sin \left (\frac {\pi \left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right )^{2}}{2}\right )}{\pi }\right )}{\sqrt {\pi }\, \sqrt {-c}}-\frac {\sin \left (\frac {4 a c +b^{2}}{4 c}\right ) \left (\operatorname {FresnelS}\left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right ) \left (\left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right )^{2} \sqrt {\pi }\, \sqrt {-c}-\sqrt {2}\, b \left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right )\right )+\frac {\sqrt {-c}\, \left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right ) \cos \left (\frac {\pi \left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right )^{2}}{2}\right )}{\sqrt {\pi }}-\frac {\sqrt {-c}\, \operatorname {FresnelC}\left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right )}{\sqrt {\pi }}-\frac {\sqrt {2}\, b \cos \left (\frac {\pi \left (-\frac {\sqrt {2}\, c x}{\sqrt {\pi }\, \sqrt {-c}}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {-c}}\right )^{2}}{2}\right )}{\pi }\right )}{\sqrt {\pi }\, \sqrt {-c}}\right )}{4 \sqrt {-c}\, c}\)	\(711\)

Input:

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

Output:

-1/2*x*sin(-c*x^2+b*x+a)/c+1/2*b/c*(-1/2*sin(-c*x^2+b*x+a)/c+1/4*b/c*2^(1/ 
2)*Pi^(1/2)/(-c)^(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)*FresnelS(2^(1/2)/Pi^(1/2)/(-c)^ 
(1/2)*(-c*x+1/2*b))))+1/4/c*2^(1/2)*Pi^(1/2)/(-c)^(1/2)*(cos((1/4*b^2+a*c) 
/c)*FresnelS(2^(1/2)/Pi^(1/2)/(-c)^(1/2)*(-c*x+1/2*b))+sin((1/4*b^2+a*c)/c 
)*FresnelC(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 = 178, normalized size of antiderivative = 0.71 \[ \int x^2 \cos \left (a+b x-c x^2\right ) \, dx=\frac {\sqrt {2} {\left (\pi b^{2} \cos \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) + 2 \, \pi c \sin \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) + \sqrt {2} {\left (\pi b^{2} \sin \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) - 2 \, \pi c \cos \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) + 2 \, {\left (2 \, c^{2} x + b c\right )} \sin \left (c x^{2} - b x - a\right )}{8 \, c^{3}} \] Input:

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

Output:

1/8*(sqrt(2)*(pi*b^2*cos(1/4*(b^2 + 4*a*c)/c) + 2*pi*c*sin(1/4*(b^2 + 4*a* 
c)/c))*sqrt(c/pi)*fresnel_cos(1/2*sqrt(2)*(2*c*x - b)*sqrt(c/pi)/c) + sqrt 
(2)*(pi*b^2*sin(1/4*(b^2 + 4*a*c)/c) - 2*pi*c*cos(1/4*(b^2 + 4*a*c)/c))*sq 
rt(c/pi)*fresnel_sin(1/2*sqrt(2)*(2*c*x - b)*sqrt(c/pi)/c) + 2*(2*c^2*x + 
b*c)*sin(c*x^2 - b*x - a))/c^3
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.91 (sec) , antiderivative size = 1570, normalized size of antiderivative = 6.25 \[ \int x^2 \cos \left (a+b x-c x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/32*(8*(((-(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*c^3 + 4*((I + 1)*sqrt(2)*gamma(3/2, 1/ 
4*(4*I*c^2*x^2 - 4*I*b*c*x + I*b^2)/c) - (I - 1)*sqrt(2)*gamma(3/2, -1/4*( 
4*I*c^2*x^2 - 4*I*b*c*x + I*b^2)/c))*c^4)*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*c^3 + 4*((I - 1)*sqrt(2)*gamma(3/2, 1/4*(4*I*c^2*x^2 
- 4*I*b*c*x + I*b^2)/c) - (I + 1)*sqrt(2)*gamma(3/2, -1/4*(4*I*c^2*x^2 - 4 
*I*b*c*x + I*b^2)/c))*c^4)*sin(1/4*(b^2 + 4*a*c)/c))*x^3 + 12*((((I - 1)*s 
qrt(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^3*c^2 + 4*(-(I + 1)*sqrt(2)*gamma(3/2, 1/4*(4*I*c^2*x^2 - 4*I 
*b*c*x + I*b^2)/c) + (I - 1)*sqrt(2)*gamma(3/2, -1/4*(4*I*c^2*x^2 - 4*I*b* 
c*x + I*b^2)/c))*b*c^3)*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)*sq 
rt(2)*sqrt(pi)*(erf(1/2*sqrt(-(4*I*c^2*x^2 - 4*I*b*c*x + I*b^2)/c)) - 1))* 
b^3*c^2 + 4*(-(I - 1)*sqrt(2)*gamma(3/2, 1/4*(4*I*c^2*x^2 - 4*I*b*c*x + I* 
b^2)/c) + (I + 1)*sqrt(2)*gamma(3/2, -1/4*(4*I*c^2*x^2 - 4*I*b*c*x + I*b^2 
)/c))*b*c^3)*sin(1/4*(b^2 + 4*a*c)/c))*x^2 + 8*(b*c^2*(-I*e^(1/4*(4*I*c...
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.90 \[ \int x^2 \cos \left (a+b x-c x^2\right ) \, dx=-\frac {\frac {\sqrt {2} \sqrt {\pi } {\left (b^{2} + 2 i \, c\right )} \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x - \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{{\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} - 2 \, {\left (c {\left (-2 i \, x + \frac {i \, b}{c}\right )} - 2 i \, b\right )} e^{\left (i \, c x^{2} - i \, b x - i \, a\right )}}{16 \, c^{2}} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (b^{2} - 2 i \, c\right )} \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x - \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{{\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} - 2 \, {\left (c {\left (2 i \, x - \frac {i \, b}{c}\right )} + 2 i \, b\right )} e^{\left (-i \, c x^{2} + i \, b x + i \, a\right )}}{16 \, c^{2}} \] Input:

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

Output:

-1/16*(sqrt(2)*sqrt(pi)*(b^2 + 2*I*c)*erf(-1/4*sqrt(2)*(2*x - b/c)*(-I*c/a 
bs(c) + 1)*sqrt(abs(c)))*e^(-1/4*(I*b^2 + 4*I*a*c)/c)/((-I*c/abs(c) + 1)*s 
qrt(abs(c))) - 2*(c*(-2*I*x + I*b/c) - 2*I*b)*e^(I*c*x^2 - I*b*x - I*a))/c 
^2 - 1/16*(sqrt(2)*sqrt(pi)*(b^2 - 2*I*c)*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*(c*(2*I*x - I*b/c) + 2*I*b)*e^(-I*c*x^2 + I*b*x + I*a) 
)/c^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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