3.8 \(\int \cos (a+b x-c x^2) \, dx\)

Optimal. Leaf size=99 \[ -\frac {\sqrt {\frac {\pi }{2}} \cos \left (a+\frac {b^2}{4 c}\right ) C\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a+\frac {b^2}{4 c}\right ) S\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}} \]

[Out]

-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))*2^(1/2)*Pi^(1/2)/c^(1/2)-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)*2^(1/2)*Pi^(1/2)/c^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3448, 3352, 3351} \[ -\frac {\sqrt {\frac {\pi }{2}} \cos \left (a+\frac {b^2}{4 c}\right ) \text {FresnelC}\left (\frac {b-2 c x}{\sqrt {2 \pi } \sqrt {c}}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a+\frac {b^2}{4 c}\right ) S\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x - c*x^2],x]

[Out]

-((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]

Rule 3351

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

Rule 3352

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

Rule 3448

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

Rubi steps

\begin {align*} \int \cos \left (a+b x-c x^2\right ) \, dx &=\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\\ &=-\frac {\sqrt {\frac {\pi }{2}} \cos \left (a+\frac {b^2}{4 c}\right ) C\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} S\left (\frac {b-2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a+\frac {b^2}{4 c}\right )}{\sqrt {c}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 88, normalized size = 0.89 \[ \frac {\sqrt {\frac {\pi }{2}} \left (\cos \left (a+\frac {b^2}{4 c}\right ) C\left (\frac {2 c x-b}{\sqrt {c} \sqrt {2 \pi }}\right )+\sin \left (a+\frac {b^2}{4 c}\right ) S\left (\frac {2 c x-b}{\sqrt {c} \sqrt {2 \pi }}\right )\right )}{\sqrt {c}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x - c*x^2],x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.20, size = 106, normalized size = 1.07 \[ \frac {\sqrt {2} \pi \sqrt {\frac {c}{\pi }} \cos \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) + \sqrt {2} \pi \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) \sin \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [C]  time = 0.45, size = 137, normalized size = 1.38 \[ -\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x - \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{4 \, {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} {\left (2 \, x - \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{4 \, {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.02, size = 79, normalized size = 0.80 \[ \frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}+c a}{c}\right ) \FresnelC \left (\frac {\sqrt {2}\, \left (c x -\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )+\sin \left (\frac {\frac {b^{2}}{4}+c a}{c}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \left (c x -\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{2 \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [C]  time = 0.77, size = 112, normalized size = 1.13 \[ \frac {\sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i - 1\right ) \, \cos \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) + \left (i + 1\right ) \, \sin \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x - i \, b}{2 \, \sqrt {i \, c}}\right ) + {\left (-\left (i + 1\right ) \, \cos \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right ) + \left (i - 1\right ) \, \sin \left (\frac {b^{2} + 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x - i \, b}{2 \, \sqrt {-i \, c}}\right )\right )}}{8 \, \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2)*sqrt(pi)*((-(I - 1)*cos(1/4*(b^2 + 4*a*c)/c) + (I + 1)*sin(1/4*(b^2 + 4*a*c)/c))*erf(1/2*(2*I*c*x
- I*b)/sqrt(I*c)) + (-(I + 1)*cos(1/4*(b^2 + 4*a*c)/c) + (I - 1)*sin(1/4*(b^2 + 4*a*c)/c))*erf(1/2*(2*I*c*x -
I*b)/sqrt(-I*c)))/sqrt(c)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 105, normalized size = 1.06 \[ \frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\left (\frac {b}{2}-c\,x\right )\,\sqrt {-\frac {1}{c}}}{\sqrt {\pi }}\right )\,\cos \left (\frac {b^2+4\,a\,c}{4\,c}\right )\,\sqrt {-\frac {1}{c}}}{2}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {S}\left (\frac {\sqrt {2}\,\left (\frac {b}{2}-c\,x\right )\,\sqrt {-\frac {1}{c}}}{\sqrt {\pi }}\right )\,\sin \left (\frac {b^2+4\,a\,c}{4\,c}\right )\,\sqrt {-\frac {1}{c}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x - c*x^2),x)

[Out]

(2^(1/2)*pi^(1/2)*fresnelc((2^(1/2)*(b/2 - c*x)*(-1/c)^(1/2))/pi^(1/2))*cos((4*a*c + b^2)/(4*c))*(-1/c)^(1/2))
/2 - (2^(1/2)*pi^(1/2)*fresnels((2^(1/2)*(b/2 - c*x)*(-1/c)^(1/2))/pi^(1/2))*sin((4*a*c + b^2)/(4*c))*(-1/c)^(
1/2))/2

________________________________________________________________________________________

sympy [A]  time = 0.49, size = 94, normalized size = 0.95 \[ \frac {\sqrt {2} \sqrt {\pi } \sqrt {- \frac {1}{c}} \left (- \sin {\left (a + \frac {b^{2}}{4 c} \right )} S\left (\frac {\sqrt {2} \left (b - 2 c x\right )}{2 \sqrt {\pi } \sqrt {- c}}\right ) + \cos {\left (a + \frac {b^{2}}{4 c} \right )} C\left (\frac {\sqrt {2} \left (b - 2 c x\right )}{2 \sqrt {\pi } \sqrt {- c}}\right )\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*sqrt(pi)*sqrt(-1/c)*(-sin(a + b**2/(4*c))*fresnels(sqrt(2)*(b - 2*c*x)/(2*sqrt(pi)*sqrt(-c))) + cos(a
+ b**2/(4*c))*fresnelc(sqrt(2)*(b - 2*c*x)/(2*sqrt(pi)*sqrt(-c))))/2

________________________________________________________________________________________