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

Optimal. Leaf size=27 \[ \frac {1}{4} \sqrt {\pi } C\left (\frac {2 x+1}{\sqrt {\pi }}\right )+\frac {x}{2} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3450, 3446, 3352} \[ \frac {1}{4} \sqrt {\pi } \text {FresnelC}\left (\frac {2 x+1}{\sqrt {\pi }}\right )+\frac {x}{2} \]

Antiderivative was successfully verified.

[In]

Int[Cos[1/4 + x + x^2]^2,x]

[Out]

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

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 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 3450

Int[Cos[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]^(n_), x_Symbol] :> Int[ExpandTrigReduce[Cos[a + b*x + c*x^2]^n, x],
x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 1]

Rubi steps

\begin {align*} \int \cos ^2\left (\frac {1}{4}+x+x^2\right ) \, dx &=\int \left (\frac {1}{2}+\frac {1}{2} \cos \left (\frac {1}{2}+2 x+2 x^2\right )\right ) \, dx\\ &=\frac {x}{2}+\frac {1}{2} \int \cos \left (\frac {1}{2}+2 x+2 x^2\right ) \, dx\\ &=\frac {x}{2}+\frac {1}{2} \int \cos \left (\frac {1}{8} (2+4 x)^2\right ) \, dx\\ &=\frac {x}{2}+\frac {1}{4} \sqrt {\pi } C\left (\frac {1+2 x}{\sqrt {\pi }}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.96 \[ \frac {1}{4} \left (\sqrt {\pi } C\left (\frac {2 x+1}{\sqrt {\pi }}\right )+2 x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[1/4 + x + x^2]^2,x]

[Out]

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

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fricas [A]  time = 0.80, size = 19, normalized size = 0.70 \[ \frac {1}{4} \, \sqrt {\pi } \operatorname {C}\left (\frac {2 \, x + 1}{\sqrt {\pi }}\right ) + \frac {1}{2} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [C]  time = 0.97, size = 26, normalized size = 0.96 \[ -\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, x + \frac {1}{2} i - \frac {1}{2}\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, x - \frac {1}{2} i - \frac {1}{2}\right ) + \frac {1}{2} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 20, normalized size = 0.74 \[ \frac {x}{2}+\frac {\FresnelC \left (\frac {1+2 x}{\sqrt {\pi }}\right ) \sqrt {\pi }}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 1.28, size = 34, normalized size = 1.26 \[ -\frac {1}{16} \, \sqrt {\pi } {\left (\left (i - 1\right ) \, \operatorname {erf}\left (\frac {2 i \, x + i}{\sqrt {2 i}}\right ) + \left (i + 1\right ) \, \operatorname {erf}\left (\frac {2 i \, x + i}{\sqrt {-2 i}}\right )\right )} + \frac {1}{2} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\cos \left (x^2+x+\frac {1}{4}\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x + x^2 + 1/4)^2,x)

[Out]

int(cos(x + x^2 + 1/4)^2, x)

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sympy [A]  time = 0.79, size = 22, normalized size = 0.81 \[ \frac {x}{2} + \frac {\sqrt {\pi } C\left (\frac {4 x + 2}{2 \sqrt {\pi }}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/2 + sqrt(pi)*fresnelc((4*x + 2)/(2*sqrt(pi)))/4

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