Integrand size = 17, antiderivative size = 40 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=-\frac {\arctan \left (1-\sqrt {3} x\right )}{2 \sqrt {3}}+\frac {\arctan \left (1+\sqrt {3} x\right )}{2 \sqrt {3}} \] Output:
1/6*arctan(-1+x*3^(1/2))*3^(1/2)+1/6*arctan(1+x*3^(1/2))*3^(1/2)
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {-\arctan \left (1-\sqrt {3} x\right )+\arctan \left (1+\sqrt {3} x\right )}{2 \sqrt {3}} \] Input:
Integrate[(2 + 3*x^2)/(4 + 9*x^4),x]
Output:
(-ArcTan[1 - Sqrt[3]*x] + ArcTan[1 + Sqrt[3]*x])/(2*Sqrt[3])
Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1476, 1082, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {3 x^2+2}{9 x^4+4} \, dx\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {1}{6} \int \frac {1}{x^2-\frac {2 x}{\sqrt {3}}+\frac {2}{3}}dx+\frac {1}{6} \int \frac {1}{x^2+\frac {2 x}{\sqrt {3}}+\frac {2}{3}}dx\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\int \frac {1}{-\left (1-\sqrt {3} x\right )^2-1}d\left (1-\sqrt {3} x\right )}{2 \sqrt {3}}-\frac {\int \frac {1}{-\left (\sqrt {3} x+1\right )^2-1}d\left (\sqrt {3} x+1\right )}{2 \sqrt {3}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\arctan \left (\sqrt {3} x+1\right )}{2 \sqrt {3}}-\frac {\arctan \left (1-\sqrt {3} x\right )}{2 \sqrt {3}}\) |
Input:
Int[(2 + 3*x^2)/(4 + 9*x^4),x]
Output:
-1/2*ArcTan[1 - Sqrt[3]*x]/Sqrt[3] + ArcTan[1 + Sqrt[3]*x]/(2*Sqrt[3])
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 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]
Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{2}\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {3 x^{3} \sqrt {3}}{4}+\frac {\sqrt {3}\, x}{2}\right )}{6}\) | \(35\) |
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 | \(\frac {\sqrt {6}\, \left (-\frac {x \sqrt {2}\, \ln \left (1-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \ln \left (1+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{24}+\frac {\sqrt {6}\, \left (\frac {x^{3} \sqrt {2}\, \ln \left (1-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2-\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}+\frac {3 \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{2+\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}\right )}{24}\) | \(284\) |
Input:
int((3*x^2+2)/(9*x^4+4),x,method=_RETURNVERBOSE)
Output:
1/6*3^(1/2)*arctan(1/2*3^(1/2)*x)+1/6*3^(1/2)*arctan(3/4*x^3*3^(1/2)+1/2*3 ^(1/2)*x)
Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{4} \, \sqrt {3} {\left (3 \, x^{3} + 2 \, x\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{2} \, \sqrt {3} x\right ) \] Input:
integrate((3*x^2+2)/(9*x^4+4),x, algorithm="fricas")
Output:
1/6*sqrt(3)*arctan(1/4*sqrt(3)*(3*x^3 + 2*x)) + 1/6*sqrt(3)*arctan(1/2*sqr t(3)*x)
Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {\sqrt {3} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {3} x}{2} \right )} + 2 \operatorname {atan}{\left (\frac {3 \sqrt {3} x^{3}}{4} + \frac {\sqrt {3} x}{2} \right )}\right )}{12} \] Input:
integrate((3*x**2+2)/(9*x**4+4),x)
Output:
sqrt(3)*(2*atan(sqrt(3)*x/2) + 2*atan(3*sqrt(3)*x**3/4 + sqrt(3)*x/2))/12
Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (3 \, x + \sqrt {3}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (3 \, x - \sqrt {3}\right )}\right ) \] Input:
integrate((3*x^2+2)/(9*x^4+4),x, algorithm="maxima")
Output:
1/6*sqrt(3)*arctan(1/3*sqrt(3)*(3*x + sqrt(3))) + 1/6*sqrt(3)*arctan(1/3*s qrt(3)*(3*x - sqrt(3)))
Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.30 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {9}{8} \, \sqrt {2} \left (\frac {4}{9}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {4}{9}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {9}{8} \, \sqrt {2} \left (\frac {4}{9}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {4}{9}\right )^{\frac {1}{4}}\right )}\right ) \] Input:
integrate((3*x^2+2)/(9*x^4+4),x, algorithm="giac")
Output:
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)))
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {\sqrt {3}\,\left (\mathrm {atan}\left (\frac {3\,\sqrt {3}\,x^3}{4}+\frac {\sqrt {3}\,x}{2}\right )+\mathrm {atan}\left (\frac {\sqrt {3}\,x}{2}\right )\right )}{6} \] Input:
int((3*x^2 + 2)/(9*x^4 + 4),x)
Output:
(3^(1/2)*(atan((3^(1/2)*x)/2 + (3*3^(1/2)*x^3)/4) + atan((3^(1/2)*x)/2)))/ 6
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78 \[ \int \frac {2+3 x^2}{4+9 x^4} \, dx=\frac {\sqrt {3}\, \left (-\mathit {atan} \left (\frac {\sqrt {3}-3 x}{\sqrt {3}}\right )+\mathit {atan} \left (\frac {\sqrt {3}+3 x}{\sqrt {3}}\right )\right )}{6} \] Input:
int((3*x^2+2)/(9*x^4+4),x)
Output:
(sqrt(3)*( - atan((sqrt(3) - 3*x)/sqrt(3)) + atan((sqrt(3) + 3*x)/sqrt(3)) ))/6