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

Optimal. Leaf size=98 \[ \frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) C\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) S\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}} \]

[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)

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Rubi [A]  time = 0.03, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, 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*}

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Mathematica [A]  time = 0.17, size = 85, normalized size = 0.87 \[ \frac {\sqrt {\frac {\pi }{2}} \left (\cos \left (a-\frac {b^2}{4 c}\right ) C\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )-\sin \left (a-\frac {b^2}{4 c}\right ) S\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )\right )}{\sqrt {c}} \]

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]*Sqr
t[2*Pi])]*Sin[a - b^2/(4*c)]))/Sqrt[c]

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fricas [A]  time = 1.16, size = 103, normalized size = 1.05 \[ \frac {\sqrt {2} \pi \sqrt {\frac {c}{\pi }} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - \sqrt {2} \pi \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )}{2 \, c} \]

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

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giac [C]  time = 0.52, size = 135, 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)))

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maple [A]  time = 0.03, size = 81, normalized size = 0.83 \[ \frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-c a}{c}\right ) \FresnelC \left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )+\sin \left (\frac {\frac {b^{2}}{4}-c a}{c}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{2 \sqrt {c}} \]

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)))

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maxima [C]  time = 0.70, size = 112, normalized size = 1.14 \[ -\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + \left (i + 1\right ) \, \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + i \, b}{2 \, \sqrt {i \, c}}\right ) + {\left (\left (i + 1\right ) \, \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + \left (i - 1\right ) \, \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + i \, b}{2 \, \sqrt {-i \, c}}\right )\right )}}{8 \, \sqrt {c}} \]

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)

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mupad [B]  time = 0.06, size = 99, normalized size = 1.01 \[ \frac {\sqrt {2}\,\sqrt {\pi }\,\cos \left (\frac {4\,a\,c-b^2}{4\,c}\right )\,\mathrm {C}\left (\frac {\sqrt {2}\,\left (\frac {b}{2}+c\,x\right )\,\sqrt {\frac {1}{c}}}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{c}}}{2}-\frac {\sqrt {2}\,\sqrt {\pi }\,\sin \left (\frac {4\,a\,c-b^2}{4\,c}\right )\,\mathrm {S}\left (\frac {\sqrt {2}\,\left (\frac {b}{2}+c\,x\right )\,\sqrt {\frac {1}{c}}}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{c}}}{2} \]

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

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sympy [A]  time = 0.50, size = 88, normalized size = 0.90 \[ \frac {\sqrt {2} \sqrt {\pi } \left (- \sin {\left (a - \frac {b^{2}}{4 c} \right )} S\left (\frac {\sqrt {2} \left (b + 2 c x\right )}{2 \sqrt {\pi } \sqrt {c}}\right ) + \cos {\left (a - \frac {b^{2}}{4 c} \right )} C\left (\frac {\sqrt {2} \left (b + 2 c x\right )}{2 \sqrt {\pi } \sqrt {c}}\right )\right ) \sqrt {\frac {1}{c}}}{2} \]

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)*(-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))))*sqrt(1/c)/2

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