\(\int \frac {x}{1+x^2+x^4} \, dx\) [915]

Optimal result

Integrand size = 12, antiderivative size = 20 \[ \int \frac {x}{1+x^2+x^4} \, dx=\frac {\arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{\sqrt {3}} \]

[Out]

1/3*arctan(1/3*(2*x^2+1)*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1121, 632, 210} \[ \int \frac {x}{1+x^2+x^4} \, dx=\frac {\arctan \left (\frac {2 x^2+1}{\sqrt {3}}\right )}{\sqrt {3}} \]

[In]

Int[x/(1 + x^2 + x^4),x]

[Out]

ArcTan[(1 + 2*x^2)/Sqrt[3]]/Sqrt[3]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1121

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,x^2\right ) \\ & = -\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x^2\right ) \\ & = \frac {\tan ^{-1}\left (\frac {1+2 x^2}{\sqrt {3}}\right )}{\sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {x}{1+x^2+x^4} \, dx=\frac {\arctan \left (\frac {1+2 x^2}{\sqrt {3}}\right )}{\sqrt {3}} \]

[In]

Integrate[x/(1 + x^2 + x^4),x]

[Out]

ArcTan[(1 + 2*x^2)/Sqrt[3]]/Sqrt[3]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {\arctan \left (\frac {\left (2 x^{2}+1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}\)	\(19\)
risch	\(\frac {\arctan \left (\frac {\left (2 x^{2}+1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}\)	\(19\)

[In]

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

[Out]

1/3*arctan(1/3*(2*x^2+1)*3^(1/2))*3^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {x}{1+x^2+x^4} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) \]

[In]

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {x}{1+x^2+x^4} \, dx=\frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{2}}{3} + \frac {\sqrt {3}}{3} \right )}}{3} \]

[In]

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

[Out]

sqrt(3)*atan(2*sqrt(3)*x**2/3 + sqrt(3)/3)/3

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {x}{1+x^2+x^4} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) \]

[In]

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {x}{1+x^2+x^4} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) \]

[In]

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {x}{1+x^2+x^4} \, dx=\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^2}{3}+\frac {\sqrt {3}}{3}\right )}{3} \]

[In]

int(x/(x^2 + x^4 + 1),x)

[Out]

(3^(1/2)*atan(3^(1/2)/3 + (2*3^(1/2)*x^2)/3))/3