\(\int \frac {2+3 x^2}{4+9 x^4} \, dx\) [5]

Optimal result

Integrand size = 17, antiderivative size = 40 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=-\frac {\arctan \left (1-\sqrt {3} x\right )}{2 \sqrt {3}}+\frac {\arctan \left (1+\sqrt {3} x\right )}{2 \sqrt {3}} \]

[Out]

1/6*arctan(-1+x*3^(1/2))*3^(1/2)+1/6*arctan(1+x*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1176, 631, 210} \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {\arctan \left (\sqrt {3} x+1\right )}{2 \sqrt {3}}-\frac {\arctan \left (1-\sqrt {3} x\right )}{2 \sqrt {3}} \]

[In]

Int[(2 + 3*x^2)/(4 + 9*x^4),x]

[Out]

-1/2*ArcTan[1 - Sqrt[3]*x]/Sqrt[3] + ArcTan[1 + Sqrt[3]*x]/(2*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 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \int \frac {1}{\frac {2}{3}-\frac {2 x}{\sqrt {3}}+x^2} \, dx+\frac {1}{6} \int \frac {1}{\frac {2}{3}+\frac {2 x}{\sqrt {3}}+x^2} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {3} x\right )}{2 \sqrt {3}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {3} x\right )}{2 \sqrt {3}} \\ & = -\frac {\tan ^{-1}\left (1-\sqrt {3} x\right )}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (1+\sqrt {3} x\right )}{2 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {-\arctan \left (1-\sqrt {3} x\right )+\arctan \left (1+\sqrt {3} x\right )}{2 \sqrt {3}} \]

[In]

Integrate[(2 + 3*x^2)/(4 + 9*x^4),x]

[Out]

(-ArcTan[1 - Sqrt[3]*x] + ArcTan[1 + Sqrt[3]*x])/(2*Sqrt[3])

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88

method	result	size
risch	\(\frac {\sqrt {3}\, \arctan \left (\frac {x \sqrt {3}}{2}\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {3 x^{3} \sqrt {3}}{4}+\frac {x \sqrt {3}}{2}\right )}{6}\)	\(35\)
default	\(\frac {\sqrt {6}\, \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {6}\, x \sqrt {2}}{3}+\frac {2}{3}}{x^{2}-\frac {\sqrt {6}\, x \sqrt {2}}{3}+\frac {2}{3}}\right )+2 \arctan \left (\frac {\sqrt {6}\, x \sqrt {2}}{2}+1\right )+2 \arctan \left (\frac {\sqrt {6}\, x \sqrt {2}}{2}-1\right )\right )}{48}+\frac {\sqrt {6}\, \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\frac {\sqrt {6}\, x \sqrt {2}}{3}+\frac {2}{3}}{x^{2}+\frac {\sqrt {6}\, x \sqrt {2}}{3}+\frac {2}{3}}\right )+2 \arctan \left (\frac {\sqrt {6}\, x \sqrt {2}}{2}+1\right )+2 \arctan \left (\frac {\sqrt {6}\, x \sqrt {2}}{2}-1\right )\right )}{48}\)	\(140\)
meijerg	\(\frac {\sqrt {6}\, \left (-\frac {x \sqrt {2}\, \ln \left (1-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \ln \left (1+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{24}+\frac {\sqrt {6}\, \left (\frac {x^{3} \sqrt {2}\, \ln \left (1-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}\right )}{24}\)	\(284\)

[In]

int((3*x^2+2)/(9*x^4+4),x,method=_RETURNVERBOSE)

[Out]

1/6*3^(1/2)*arctan(1/2*x*3^(1/2))+1/6*3^(1/2)*arctan(3/4*x^3*3^(1/2)+1/2*x*3^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{4} \, \sqrt {3} {\left (3 \, x^{3} + 2 \, x\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{2} \, \sqrt {3} x\right ) \]

[In]

integrate((3*x^2+2)/(9*x^4+4),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*arctan(1/4*sqrt(3)*(3*x^3 + 2*x)) + 1/6*sqrt(3)*arctan(1/2*sqrt(3)*x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {\sqrt {3} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {3} x}{2} \right )} + 2 \operatorname {atan}{\left (\frac {3 \sqrt {3} x^{3}}{4} + \frac {\sqrt {3} x}{2} \right )}\right )}{12} \]

[In]

integrate((3*x**2+2)/(9*x**4+4),x)

[Out]

sqrt(3)*(2*atan(sqrt(3)*x/2) + 2*atan(3*sqrt(3)*x**3/4 + sqrt(3)*x/2))/12

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (3 \, x + \sqrt {3}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (3 \, x - \sqrt {3}\right )}\right ) \]

[In]

integrate((3*x^2+2)/(9*x^4+4),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(3*x + sqrt(3))) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(3*x - sqrt(3)))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.30 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {9}{8} \, \sqrt {2} \left (\frac {4}{9}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {4}{9}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {9}{8} \, \sqrt {2} \left (\frac {4}{9}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {4}{9}\right )^{\frac {1}{4}}\right )}\right ) \]

[In]

integrate((3*x^2+2)/(9*x^4+4),x, algorithm="giac")

[Out]

1/6*sqrt(3)*arctan(9/8*sqrt(2)*(4/9)^(3/4)*(2*x + sqrt(2)*(4/9)^(1/4))) + 1/6*sqrt(3)*arctan(9/8*sqrt(2)*(4/9)
^(3/4)*(2*x - sqrt(2)*(4/9)^(1/4)))

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {\sqrt {3}\,\left (\mathrm {atan}\left (\frac {3\,\sqrt {3}\,x^3}{4}+\frac {\sqrt {3}\,x}{2}\right )+\mathrm {atan}\left (\frac {\sqrt {3}\,x}{2}\right )\right )}{6} \]

[In]

int((3*x^2 + 2)/(9*x^4 + 4),x)

[Out]

(3^(1/2)*(atan((3^(1/2)*x)/2 + (3*3^(1/2)*x^3)/4) + atan((3^(1/2)*x)/2)))/6