\(\int x \cos (\frac {1}{4}+x+x^2) \, dx\) [12]

Optimal result

Integrand size = 11, antiderivative size = 41 \[ \int x \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=-\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {1+2 x}{\sqrt {2 \pi }}\right )+\frac {1}{2} \sin \left (\frac {1}{4}+x+x^2\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3543, 3527, 3433} \[ \int x \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{2} \sin \left (x^2+x+\frac {1}{4}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {2 x+1}{\sqrt {2 \pi }}\right ) \]

[In]

Int[x*Cos[1/4 + x + x^2],x]

[Out]

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

Rule 3433

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

Rule 3527

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 3543

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]

Rubi steps \begin{align*} \text {integral}& = \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}} \operatorname {FresnelC}\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] (verified)

Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95 \[ \int x \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{4} \left (-\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {1+2 x}{\sqrt {2 \pi }}\right )+2 \sin \left (\frac {1}{4}+x+x^2\right )\right ) \]

[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

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.73

method	result	size
default	\(\frac {\sin \left (\frac {1}{4}+x +x^{2}\right )}{2}-\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {C}\left (\frac {\sqrt {2}\, \left (x +\frac {1}{2}\right )}{\sqrt {\pi }}\right )}{4}\)	\(30\)
risch	\(\frac {\sqrt {\pi }\, \left (-1\right )^{\frac {3}{4}} \operatorname {erf}\left (\left (-1\right )^{\frac {1}{4}} x +\frac {\left (-1\right )^{\frac {1}{4}}}{2}\right )}{8}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-i}\, x -\frac {i}{2 \sqrt {-i}}\right )}{8 \sqrt {-i}}+\frac {\sin \left (\frac {\left (1+2 x \right )^{2}}{4}\right )}{2}\)	\(58\)
parts	\(\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {C}\left (\frac {\sqrt {2}\, \left (x +\frac {1}{2}\right )}{\sqrt {\pi }}\right ) x}{2}-\frac {\pi \left (\operatorname {C}\left (\frac {\sqrt {2}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}}{2 \sqrt {\pi }}\right ) \left (\frac {\sqrt {2}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}}{2 \sqrt {\pi }}\right )-\frac {\sin \left (\frac {\pi \left (\frac {\sqrt {2}\, x}{\sqrt {\pi }}+\frac {\sqrt {2}}{2 \sqrt {\pi }}\right )^{2}}{2}\right )}{\pi }\right )}{2}\)	\(90\)

[In]

int(x*cos(1/4+x+x^2),x,method=_RETURNVERBOSE)

[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))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.78 \[ \int x \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=-\frac {1}{4} \, \sqrt {2} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, x + 1\right )}}{2 \, \sqrt {\pi }}\right ) + \frac {1}{2} \, \sin \left (x^{2} + x + \frac {1}{4}\right ) \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (39) = 78\).

Time = 0.66 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.78 \[ \int x \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=- \frac {\sqrt {2} \sqrt {\pi } x C\left (\frac {\sqrt {2} x}{\sqrt {\pi }} + \frac {\sqrt {2}}{2 \sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{8 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {2} \sqrt {\pi } x C\left (\frac {\sqrt {2} x}{\sqrt {\pi }} + \frac {\sqrt {2}}{2 \sqrt {\pi }}\right )}{2} + \frac {\sin {\left (\left (x + \frac {1}{2}\right )^{2} \right )} \Gamma \left (\frac {1}{4}\right )}{8 \Gamma \left (\frac {5}{4}\right )} - \frac {\sqrt {2} \sqrt {\pi } C\left (\frac {\sqrt {2} x}{\sqrt {\pi }} + \frac {\sqrt {2}}{2 \sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{16 \Gamma \left (\frac {5}{4}\right )} \]

[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))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.07 \[ \int x \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {8 \, x {\left (-i \, e^{\left (i \, x^{2} + i \, x + \frac {1}{4} i\right )} + i \, e^{\left (-i \, x^{2} - i \, x - \frac {1}{4} i\right )}\right )} + \sqrt {4 \, x^{2} + 4 \, x + 1} {\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, x^{2} + i \, x + \frac {1}{4} i}\right ) - 1\right )} - \left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, x^{2} - i \, x - \frac {1}{4} i}\right ) - 1\right )}\right )} - 4 i \, e^{\left (i \, x^{2} + i \, x + \frac {1}{4} i\right )} + 4 i \, e^{\left (-i \, x^{2} - i \, x - \frac {1}{4} i\right )}}{16 \, {\left (2 \, x + 1\right )}} \]

[In]

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

[Out]

1/16*(8*x*(-I*e^(I*x^2 + I*x + 1/4*I) + I*e^(-I*x^2 - I*x - 1/4*I)) + sqrt(4*x^2 + 4*x + 1)*((I - 1)*sqrt(2)*s
qrt(pi)*(erf(sqrt(I*x^2 + I*x + 1/4*I)) - 1) - (I + 1)*sqrt(2)*sqrt(pi)*(erf(sqrt(-I*x^2 - I*x - 1/4*I)) - 1))
- 4*I*e^(I*x^2 + I*x + 1/4*I) + 4*I*e^(-I*x^2 - I*x - 1/4*I))/(2*x + 1)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.59 \[ \int x \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) - \frac {1}{4} i \, e^{\left (i \, x^{2} + i \, x + \frac {1}{4} i\right )} + \frac {1}{4} i \, e^{\left (-i \, x^{2} - i \, x - \frac {1}{4} i\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 14.53 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.78 \[ \int x \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {\sin \left (x^2+x+\frac {1}{4}\right )}{2}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\left (2\,x+1\right )}{2\,\sqrt {\pi }}\right )}{4} \]

[In]

int(x*cos(x + x^2 + 1/4),x)

[Out]

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