Integrand size = 15, antiderivative size = 249 \[ \int x^2 \sin \left (a+b x+c x^2\right ) \, dx=\frac {b \cos \left (a+b x+c x^2\right )}{4 c^2}-\frac {x \cos \left (a+b x+c x^2\right )}{2 c}+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{2 c^{3/2}}+\frac {b^2 \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{4 c^{5/2}}+\frac {b^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{4 c^{5/2}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{2 c^{3/2}} \]
1/4*b*cos(c*x^2+b*x+a)/c^2-1/2*x*cos(c*x^2+b*x+a)/c+1/4*cos(a-1/4*b^2/c)*F resnelC(1/2*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)/c^(3/2)+1 /8*b^2*cos(a-1/4*b^2/c)*FresnelS(1/2*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))*2 ^(1/2)*Pi^(1/2)/c^(5/2)+1/8*b^2*FresnelC(1/2*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^ (1/2))*sin(a-1/4*b^2/c)*2^(1/2)*Pi^(1/2)/c^(5/2)-1/4*FresnelS(1/2*(2*c*x+b )/c^(1/2)*2^(1/2)/Pi^(1/2))*sin(a-1/4*b^2/c)*2^(1/2)*Pi^(1/2)/c^(3/2)
Time = 0.48 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.63 \[ \int x^2 \sin \left (a+b x+c x^2\right ) \, dx=\frac {2 \sqrt {c} (b-2 c x) \cos (a+x (b+c x))+\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \left (2 c \cos \left (a-\frac {b^2}{4 c}\right )+b^2 \sin \left (a-\frac {b^2}{4 c}\right )\right )+\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \left (b^2 \cos \left (a-\frac {b^2}{4 c}\right )-2 c \sin \left (a-\frac {b^2}{4 c}\right )\right )}{8 c^{5/2}} \]
(2*Sqrt[c]*(b - 2*c*x)*Cos[a + x*(b + c*x)] + Sqrt[2*Pi]*FresnelC[(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]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*(b^2*Cos[a - b^2 /(4*c)] - 2*c*Sin[a - b^2/(4*c)]))/(8*c^(5/2))
Time = 0.68 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3944, 3929, 3832, 3833, 3942, 3928, 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.
| \(\displaystyle \int x^2 \sin \left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 3944 |
| \(\displaystyle -\frac {b \int x \sin \left (c x^2+b x+a\right )dx}{2 c}+\frac {\int \cos \left (c x^2+b x+a\right )dx}{2 c}-\frac {x \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3929 |
| \(\displaystyle \frac {\cos \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx-\sin \left (a-\frac {b^2}{4 c}\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right )dx}{2 c}-\frac {b \int x \sin \left (c x^2+b x+a\right )dx}{2 c}-\frac {x \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {\cos \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}}{2 c}-\frac {b \int x \sin \left (c x^2+b x+a\right )dx}{2 c}-\frac {x \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle -\frac {b \int x \sin \left (c x^2+b x+a\right )dx}{2 c}+\frac {\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}}{2 c}-\frac {x \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3942 |
| \(\displaystyle -\frac {b \left (-\frac {b \int \sin \left (c x^2+b x+a\right )dx}{2 c}-\frac {\cos \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}+\frac {\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}}{2 c}-\frac {x \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3928 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\sin \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx+\cos \left (a-\frac {b^2}{4 c}\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right )dx\right )}{2 c}-\frac {\cos \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}+\frac {\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}}{2 c}-\frac {x \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\sin \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {\cos \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}+\frac {\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}}{2 c}-\frac {x \cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle -\frac {b \left (-\frac {b \left (\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {\cos \left (a+b x+c x^2\right )}{2 c}\right )}{2 c}+\frac {\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}}{2 c}-\frac {x \cos \left (a+b x+c x^2\right )}{2 c}\) |
-1/2*(x*Cos[a + b*x + c*x^2])/c + ((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) - (b*(-1/2*C os[a + b*x + c*x^2]/c - (b*((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)/(S qrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/Sqrt[c]))/(2*c)))/(2*c)
3.1.1.3.1 Defintions of rubi rules used
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]
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]
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]
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]
Int[((d_.) + (e_.)*(x_))*Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[(-e)*(Cos[a + b*x + c*x^2]/(2*c)), x] + Simp[(2*c*d - b*e)/(2*c) Int[Sin[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*Sin[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sym bol] :> Simp[(-e)*(d + e*x)^(m - 1)*(Cos[a + b*x + c*x^2]/(2*c)), x] + (-Si mp[(b*e - 2*c*d)/(2*c) Int[(d + e*x)^(m - 1)*Sin[a + b*x + c*x^2], x], x] + Simp[e^2*((m - 1)/(2*c)) Int[(d + e*x)^(m - 2)*Cos[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]
Time = 0.98 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.82
| method | result | size |
| default | \(-\frac {x \cos \left (c \,x^{2}+b x +a \right )}{2 c}-\frac {b \left (-\frac {\cos \left (c \,x^{2}+b x +a \right )}{2 c}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )-\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{4 c^{\frac {3}{2}}}\right )}{2 c}+\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )+\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{4 c^{\frac {3}{2}}}\) | \(204\) |
| risch | \(\frac {i b^{2} \sqrt {\pi }\, {\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}} \operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right )}{16 c^{2} \sqrt {-i c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}} \operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right )}{8 c \sqrt {-i c}}+\frac {i b^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}} \operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right )}{16 c^{2} \sqrt {i c}}+\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}} \operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right )}{8 c \sqrt {i c}}+2 \left (-\frac {x}{4 c}+\frac {b}{8 c^{2}}\right ) \cos \left (c \,x^{2}+b x +a \right )\) | \(242\) |
| parts | \(\frac {\sqrt {2}\, \sqrt {\pi }\, x^{2} \cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {S}\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 {C}\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 {S}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \left (-\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )^{2} \sqrt {\pi }\, \sqrt {c}+\sqrt {2}\, b \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )\right )-\frac {\sqrt {c}\, \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \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 )}{\sqrt {\pi }}+\frac {\sqrt {c}\, \operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )}{\sqrt {\pi }}+\frac {\sqrt {2}\, b \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 )}{\sqrt {\pi }\, \sqrt {c}}-\frac {\sin \left (\frac {4 a c -b^{2}}{4 c}\right ) \left (\operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \left (-\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )^{2} \sqrt {\pi }\, \sqrt {c}+\sqrt {2}\, b \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )\right )+\frac {\sqrt {c}\, \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \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 )}{\sqrt {\pi }}-\frac {\sqrt {c}\, \operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )}{\sqrt {\pi }}-\frac {\sqrt {2}\, b \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 )}{\sqrt {\pi }\, \sqrt {c}}\right )}{4 c^{\frac {3}{2}}}\) | \(603\) |
-1/2*x*cos(c*x^2+b*x+a)/c-1/2*b/c*(-1/2*cos(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)*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))))+1/4/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)*FresnelS(2^(1/2)/Pi^(1 /2)/c^(1/2)*(c*x+1/2*b)))
Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.69 \[ \int x^2 \sin \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {2} {\left (\pi b^{2} \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + 2 \, \pi c \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) + \sqrt {2} {\left (\pi b^{2} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - 2 \, \pi c \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - 2 \, {\left (2 \, c^{2} x - b c\right )} \cos \left (c x^{2} + b x + a\right )}{8 \, c^{3}} \]
1/8*(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))*sqrt(c/pi)*fresnel_cos(1/2*sqrt(2)*(2*c*x + b)*sqrt(c/pi)/c) + sq rt(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_sin(1/2*sqrt(2)*(2*c*x + b)*sqrt(c/pi)/c) - 2*(2*c^2* x - b*c)*cos(c*x^2 + b*x + a))/c^3
\[ \int x^2 \sin \left (a+b x+c x^2\right ) \, dx=\int x^{2} \sin {\left (a + b x + c x^{2} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 1569, normalized size of antiderivative = 6.30 \[ \int x^2 \sin \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \]
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) *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^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)*sq rt(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)*sin(-1/4*(b^2 - 4*a*c)/c))*x^2 + 8*(b*c^2*(e^(1/4*(4*I*c^...
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.90 \[ \int x^2 \sin \left (a+b x+c x^2\right ) \, dx=\frac {\frac {\sqrt {2} \sqrt {\pi } {\left (b^{2} + 2 i \, c\right )} \operatorname {erf}\left (-\frac {1}{4} i \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{{\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 i \, {\left (c {\left (2 i \, x + \frac {i \, b}{c}\right )} - 2 i \, b\right )} e^{\left (i \, c x^{2} + i \, b x + i \, a\right )}}{16 \, c^{2}} + \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (b^{2} - 2 i \, c\right )} \operatorname {erf}\left (\frac {1}{4} i \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{{\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 i \, {\left (c {\left (2 i \, x + \frac {i \, b}{c}\right )} - 2 i \, b\right )} e^{\left (-i \, c x^{2} - i \, b x - i \, a\right )}}{16 \, c^{2}} \]
1/16*(sqrt(2)*sqrt(pi)*(b^2 + 2*I*c)*erf(-1/4*I*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)*sq rt(abs(c))) + 2*I*(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*I*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*(c*(2*I*x + I*b/c) - 2*I*b)*e^(-I*c*x^2 - I*b*x - I*a))/c^2
Timed out. \[ \int x^2 \sin \left (a+b x+c x^2\right ) \, dx=\int x^2\,\sin \left (c\,x^2+b\,x+a\right ) \,d x \]