Integrand size = 13, antiderivative size = 123 \[ \int x \sin \left (a+b x+c x^2\right ) \, dx=-\frac {\cos \left (a+b x+c x^2\right )}{2 c}-\frac {b \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{2 c^{3/2}}-\frac {b \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{2 c^{3/2}} \]
-1/2*cos(c*x^2+b*x+a)/c-1/4*b*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^(3/2)-1/4*b*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^(3/2)
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int x \sin \left (a+b x+c x^2\right ) \, dx=-\frac {2 \sqrt {c} \cos (a+x (b+c x))+b \sqrt {2 \pi } \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )+b \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{4 c^{3/2}} \]
-1/4*(2*Sqrt[c]*Cos[a + x*(b + c*x)] + b*Sqrt[2*Pi]*Cos[a - b^2/(4*c)]*Fre snelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])] + b*Sqrt[2*Pi]*FresnelC[(b + 2*c*x )/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/c^(3/2)
Time = 0.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {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.
| \(\displaystyle \int x \sin \left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 3942 |
| \(\displaystyle -\frac {b \int \sin \left (c x^2+b x+a\right )dx}{2 c}-\frac {\cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3928 |
| \(\displaystyle -\frac {b \left (\sin \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx+\cos \left (a-\frac {b^2}{4 c}\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right )dx\right )}{2 c}-\frac {\cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle -\frac {b \left (\sin \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {\cos \left (a+b x+c x^2\right )}{2 c}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle -\frac {b \left (\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}+\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\right )}{2 c}-\frac {\cos \left (a+b x+c x^2\right )}{2 c}\) |
-1/2*Cos[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)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/Sqrt[c]))/(2*c)
3.1.2.3.1 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[((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]
Time = 0.46 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.81
| method | result | size |
| default | \(-\frac {\cos \left (c \,x^{2}+b x +a \right )}{2 c}-\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )-\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{4 c^{\frac {3}{2}}}\) | \(100\) |
| risch | \(-\frac {i b \sqrt {\pi }\, {\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}} \operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right )}{8 c \sqrt {-i c}}-\frac {i b \sqrt {\pi }\, {\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}} \operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right )}{8 c \sqrt {i c}}-\frac {\cos \left (c \,x^{2}+b x +a \right )}{2 c}\) | \(125\) |
| parts | \(\frac {\sqrt {2}\, \sqrt {\pi }\, x \cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2}\, \sqrt {\pi }\, x \sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )}{2 \sqrt {c}}-\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\frac {\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (\operatorname {S}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )+\frac {\cos \left (\frac {\pi \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )^{2}}{2}\right )}{\pi }\right )}{2 \sqrt {c}}-\frac {\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \sqrt {2}\, \sqrt {\pi }\, \left (\operatorname {C}\left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right ) \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )-\frac {\sin \left (\frac {\pi \left (\frac {\sqrt {2}\, \sqrt {c}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}\, b}{2 \sqrt {\pi }\, \sqrt {c}}\right )^{2}}{2}\right )}{\pi }\right )}{2 \sqrt {c}}\right )}{2 \sqrt {c}}\) | \(327\) |
-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)*Fre snelC(2^(1/2)/Pi^(1/2)/c^(1/2)*(c*x+1/2*b)))
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.96 \[ \int x \sin \left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {2} \pi b \sqrt {\frac {c}{\pi }} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) + \sqrt {2} \pi b \sqrt {\frac {c}{\pi }} \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + 2 \, c \cos \left (c x^{2} + b x + a\right )}{4 \, c^{2}} \]
-1/4*(sqrt(2)*pi*b*sqrt(c/pi)*cos(-1/4*(b^2 - 4*a*c)/c)*fresnel_sin(1/2*sq rt(2)*(2*c*x + b)*sqrt(c/pi)/c) + sqrt(2)*pi*b*sqrt(c/pi)*fresnel_cos(1/2* sqrt(2)*(2*c*x + b)*sqrt(c/pi)/c)*sin(-1/4*(b^2 - 4*a*c)/c) + 2*c*cos(c*x^ 2 + b*x + a))/c^2
\[ \int x \sin \left (a+b x+c x^2\right ) \, dx=\int x \sin {\left (a + b x + c x^{2} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 579, normalized size of antiderivative = 4.71 \[ \int x \sin \left (a+b x+c x^2\right ) \, dx=\frac {{\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} + \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b^{2} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} - \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b^{2} \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - 2 \, {\left ({\left (\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} - \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b c \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (-\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )} + \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{c}}\right ) - 1\right )}\right )} b c \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} x - 4 \, {\left (c {\left (e^{\left (\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{4 \, c}\right )} + e^{\left (-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{4 \, c}\right )}\right )} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + c {\left (i \, e^{\left (\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{4 \, c}\right )} - i \, e^{\left (-\frac {4 i \, c^{2} x^{2} + 4 i \, b c x + i \, b^{2}}{4 \, c}\right )}\right )} \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{16 \, c^{2} \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}} \]
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*(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))*cos(-1/4*(b^2 - 4*a*c)/c) + 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))*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))
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.47 \[ \int x \sin \left (a+b x+c x^2\right ) \, dx=-\frac {\frac {\sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{4} i \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{{\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 \, e^{\left (i \, c x^{2} + i \, b x + i \, a\right )}}{8 \, c} - \frac {\frac {\sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (\frac {1}{4} i \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{{\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} + 2 \, e^{\left (-i \, c x^{2} - i \, b x - i \, a\right )}}{8 \, c} \]
-1/8*(sqrt(2)*sqrt(pi)*b*erf(-1/4*I*sqrt(2)*(2*x + b/c)*(I*c/abs(c) + 1)*s qrt(abs(c)))*e^(-1/4*(I*b^2 - 4*I*a*c)/c)/((I*c/abs(c) + 1)*sqrt(abs(c))) + 2*e^(I*c*x^2 + I*b*x + I*a))/c - 1/8*(sqrt(2)*sqrt(pi)*b*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*e^(-I*c*x^2 - I*b*x - I*a))/c
Timed out. \[ \int x \sin \left (a+b x+c x^2\right ) \, dx=\int x\,\sin \left (c\,x^2+b\,x+a\right ) \,d x \]