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

Optimal. Leaf size=41 \[ \frac{1}{2} \sin \left (x^2+x+\frac{1}{4}\right )-\frac{1}{2} \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\frac{2 x+1}{\sqrt{2 \pi }}\right ) \]

[Out]

-(Sqrt[Pi/2]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]])/2 + Sin[1/4 + x + x^2]/2

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Rubi [A]  time = 0.0137269, antiderivative size = 41, normalized size of antiderivative = 1., 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 = {3462, 3446, 3352} \[ \frac{1}{2} \sin \left (x^2+x+\frac{1}{4}\right )-\frac{1}{2} \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\frac{2 x+1}{\sqrt{2 \pi }}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[1/4 + x + x^2],x]

[Out]

-(Sqrt[Pi/2]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]])/2 + Sin[1/4 + x + x^2]/2

Rule 3462

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

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

Mathematica [A]  time = 0.0546599, size = 39, normalized size = 0.95 \[ \frac{1}{4} \left (2 \sin \left (x^2+x+\frac{1}{4}\right )-\sqrt{2 \pi } \text{FresnelC}\left (\frac{2 x+1}{\sqrt{2 \pi }}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[1/4 + x + x^2],x]

[Out]

(-(Sqrt[2*Pi]*FresnelC[(1 + 2*x)/Sqrt[2*Pi]]) + 2*Sin[1/4 + x + x^2])/4

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Maple [A]  time = 0.024, size = 30, normalized size = 0.7 \begin{align*}{\frac{1}{2}\sin \left ({\frac{1}{4}}+x+{x}^{2} \right ) }-{\frac{\sqrt{2}\sqrt{\pi }}{4}{\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(x*cos(1/4+x+x^2),x)

[Out]

1/2*sin(1/4+x+x^2)-1/4*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*(x+1/2))

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Maxima [C]  time = 2.67135, size = 169, normalized size = 4.12 \begin{align*} -\frac{x{\left (2048 i \, e^{\left (i \, x^{2} + i \, x + \frac{1}{4} i\right )} - 2048 i \, e^{\left (-i \, x^{2} - i \, x - \frac{1}{4} i\right )}\right )} + \sqrt{4 \, x^{2} + 4 \, x + 1}{\left (-\left (256 i - 256\right ) \, \sqrt{2} \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{i \, x^{2} + i \, x + \frac{1}{4} i}\right ) - 1\right )} + \left (256 i + 256\right ) \, \sqrt{2} \sqrt{\pi }{\left (\operatorname{erf}\left (\sqrt{-i \, x^{2} - i \, x - \frac{1}{4} i}\right ) - 1\right )}\right )} + 1024 i \, e^{\left (i \, x^{2} + i \, x + \frac{1}{4} i\right )} - 1024 i \, e^{\left (-i \, x^{2} - i \, x - \frac{1}{4} i\right )}}{4096 \,{\left (2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4096*(x*(2048*I*e^(I*x^2 + I*x + 1/4*I) - 2048*I*e^(-I*x^2 - I*x - 1/4*I)) + sqrt(4*x^2 + 4*x + 1)*(-(256*I
- 256)*sqrt(2)*sqrt(pi)*(erf(sqrt(I*x^2 + I*x + 1/4*I)) - 1) + (256*I + 256)*sqrt(2)*sqrt(pi)*(erf(sqrt(-I*x^
2 - I*x - 1/4*I)) - 1)) + 1024*I*e^(I*x^2 + I*x + 1/4*I) - 1024*I*e^(-I*x^2 - I*x - 1/4*I))/(2*x + 1)

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Fricas [A]  time = 1.39762, size = 124, normalized size = 3.02 \begin{align*} -\frac{1}{4} \, \sqrt{2} \sqrt{\pi } \operatorname{C}\left (\frac{\sqrt{2}{\left (2 \, x + 1\right )}}{2 \, \sqrt{\pi }}\right ) + \frac{1}{2} \, \sin \left (x^{2} + x + \frac{1}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*sqrt(pi)*fresnel_cos(1/2*sqrt(2)*(2*x + 1)/sqrt(pi)) + 1/2*sin(x^2 + x + 1/4)

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Sympy [B]  time = 1.35827, size = 155, normalized size = 3.78 \begin{align*} - \frac{\sqrt{2} \sqrt{\pi } x C\left (\frac{\sqrt{2} x}{\sqrt{\pi }} + \frac{\sqrt{2}}{2 \sqrt{\pi }}\right ) \Gamma \left (\frac{1}{4}\right )}{8 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{2} \sqrt{\pi } x C\left (\frac{\sqrt{2} x}{\sqrt{\pi }} + \frac{\sqrt{2}}{2 \sqrt{\pi }}\right )}{2} + \frac{\sin{\left (\left (x + \frac{1}{2}\right )^{2} \right )} \Gamma \left (\frac{1}{4}\right )}{8 \Gamma \left (\frac{5}{4}\right )} - \frac{\sqrt{2} \sqrt{\pi } C\left (\frac{\sqrt{2} x}{\sqrt{\pi }} + \frac{\sqrt{2}}{2 \sqrt{\pi }}\right ) \Gamma \left (\frac{1}{4}\right )}{16 \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*sqrt(pi)*x*fresnelc(sqrt(2)*x/sqrt(pi) + sqrt(2)/(2*sqrt(pi)))*gamma(1/4)/(8*gamma(5/4)) + sqrt(2)*sq
rt(pi)*x*fresnelc(sqrt(2)*x/sqrt(pi) + sqrt(2)/(2*sqrt(pi)))/2 + sin((x + 1/2)**2)*gamma(1/4)/(8*gamma(5/4)) -
sqrt(2)*sqrt(pi)*fresnelc(sqrt(2)*x/sqrt(pi) + sqrt(2)/(2*sqrt(pi)))*gamma(1/4)/(16*gamma(5/4))

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Giac [C]  time = 1.17573, size = 88, normalized size = 2.15 \begin{align*} \left (\frac{1}{16} i + \frac{1}{16}\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}{16} i - \frac{1}{16}\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 ) - \frac{1}{4} i \, e^{\left (i \, x^{2} + i \, x + \frac{1}{4} i\right )} + \frac{1}{4} i \, e^{\left (-i \, x^{2} - i \, x - \frac{1}{4} i\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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