3.13 \(\int \cos (\frac{1}{4}+x+x^2) \, dx\)

Optimal. Leaf size=24 \[ \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\frac{2 x+1}{\sqrt{2 \pi }}\right ) \]

[Out]

Sqrt[Pi/2]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]]

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Rubi [A]  time = 0.0056943, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3446, 3352} \[ \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\frac{2 x+1}{\sqrt{2 \pi }}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[1/4 + x + x^2],x]

[Out]

Sqrt[Pi/2]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]]

Rule 3446

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

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]

Rubi steps

\begin{align*} \int \cos \left (\frac{1}{4}+x+x^2\right ) \, dx &=\int \cos \left (\frac{1}{4} (1+2 x)^2\right ) \, dx\\ &=\sqrt{\frac{\pi }{2}} C\left (\frac{1+2 x}{\sqrt{2 \pi }}\right )\\ \end{align*}

Mathematica [A]  time = 0.0261537, size = 24, normalized size = 1. \[ \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\frac{2 x+1}{\sqrt{2 \pi }}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[1/4 + x + x^2],x]

[Out]

Sqrt[Pi/2]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]]

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Maple [A]  time = 0.026, size = 20, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{2}{\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }} \left ( x+{\frac{1}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/4+x+x^2),x)

[Out]

1/2*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*(x+1/2))

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Maxima [C]  time = 2.82259, size = 95, normalized size = 3.96 \begin{align*} -\frac{1}{16} \, \sqrt{\pi }{\left (\left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\frac{1}{2} \, \left (-1\right )^{\frac{3}{4}}{\left (2 i \, x + i\right )}\right ) + \left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2}{\left (2 i \, x + i\right )}\right ) - \left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2}{\left (2 i \, x + i\right )}\right ) + \left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (\frac{2 i \, x + i}{2 \, \sqrt{-i}}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/4+x+x^2),x, algorithm="maxima")

[Out]

-1/16*sqrt(pi)*((I - 1)*sqrt(2)*erf(-1/2*(-1)^(3/4)*(2*I*x + I)) + (I - 1)*sqrt(2)*erf(-(1/4*I - 1/4)*sqrt(2)*
(2*I*x + I)) - (I + 1)*sqrt(2)*erf(-(1/4*I + 1/4)*sqrt(2)*(2*I*x + I)) + (I + 1)*sqrt(2)*erf(1/2*(2*I*x + I)/s
qrt(-I)))

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Fricas [A]  time = 1.273, size = 89, normalized size = 3.71 \begin{align*} \frac{1}{2} \, \sqrt{2} \sqrt{\pi } \operatorname{C}\left (\frac{\sqrt{2}{\left (2 \, x + 1\right )}}{2 \, \sqrt{\pi }}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/4+x+x^2),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*sqrt(pi)*fresnel_cos(1/2*sqrt(2)*(2*x + 1)/sqrt(pi))

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Sympy [A]  time = 0.567033, size = 29, normalized size = 1.21 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } C\left (\frac{\sqrt{2} \left (2 x + 1\right )}{2 \sqrt{\pi }}\right )}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/4+x+x**2),x)

[Out]

sqrt(2)*sqrt(pi)*fresnelc(sqrt(2)*(2*x + 1)/(2*sqrt(pi)))/2

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Giac [C]  time = 1.11824, size = 53, normalized size = 2.21 \begin{align*} -\left (\frac{1}{8} i + \frac{1}{8}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) + \left (\frac{1}{8} i - \frac{1}{8}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2}{\left (2 \, x + 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/4+x+x^2),x, algorithm="giac")

[Out]

-(1/8*I + 1/8)*sqrt(2)*sqrt(pi)*erf((1/4*I - 1/4)*sqrt(2)*(2*x + 1)) + (1/8*I - 1/8)*sqrt(2)*sqrt(pi)*erf(-(1/
4*I + 1/4)*sqrt(2)*(2*x + 1))