\(\int \frac {a+b x^2}{1+x^2+x^4} \, dx\) [98]

Optimal result

Integrand size = 18, antiderivative size = 83 \[ \int \frac {a+b x^2}{1+x^2+x^4} \, dx=-\frac {(a+b) \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(a+b) \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} (a-b) \log \left (1-x+x^2\right )+\frac {1}{4} (a-b) \log \left (1+x+x^2\right ) \]

[Out]

-1/4*(a-b)*ln(x^2-x+1)+1/4*(a-b)*ln(x^2+x+1)-1/6*(a+b)*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)+1/6*(a+b)*arctan(1/
3*(1+2*x)*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1183, 648, 632, 210, 642} \[ \int \frac {a+b x^2}{1+x^2+x^4} \, dx=-\frac {(a+b) \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(a+b) \arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} (a-b) \log \left (x^2-x+1\right )+\frac {1}{4} (a-b) \log \left (x^2+x+1\right ) \]

[In]

Int[(a + b*x^2)/(1 + x^2 + x^4),x]

[Out]

-1/2*((a + b)*ArcTan[(1 - 2*x)/Sqrt[3]])/Sqrt[3] + ((a + b)*ArcTan[(1 + 2*x)/Sqrt[3]])/(2*Sqrt[3]) - ((a - b)*
Log[1 - x + x^2])/4 + ((a - b)*Log[1 + x + x^2])/4

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 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

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

Rule 1183

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {a-(a-b) x}{1-x+x^2} \, dx+\frac {1}{2} \int \frac {a+(a-b) x}{1+x+x^2} \, dx \\ & = \frac {1}{4} (a-b) \int \frac {1+2 x}{1+x+x^2} \, dx+\frac {1}{4} (-a+b) \int \frac {-1+2 x}{1-x+x^2} \, dx+\frac {1}{4} (a+b) \int \frac {1}{1-x+x^2} \, dx+\frac {1}{4} (a+b) \int \frac {1}{1+x+x^2} \, dx \\ & = -\frac {1}{4} (a-b) \log \left (1-x+x^2\right )+\frac {1}{4} (a-b) \log \left (1+x+x^2\right )+\frac {1}{2} (-a-b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{2} (-a-b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right ) \\ & = -\frac {(a+b) \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {(a+b) \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} (a-b) \log \left (1-x+x^2\right )+\frac {1}{4} (a-b) \log \left (1+x+x^2\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2}{1+x^2+x^4} \, dx=\frac {\left (2 i a+\left (-i+\sqrt {3}\right ) b\right ) \arctan \left (\frac {1}{2} \left (-i+\sqrt {3}\right ) x\right )}{\sqrt {6+6 i \sqrt {3}}}+\frac {\left (-2 i a+\left (i+\sqrt {3}\right ) b\right ) \arctan \left (\frac {1}{2} \left (i+\sqrt {3}\right ) x\right )}{\sqrt {6-6 i \sqrt {3}}} \]

[In]

Integrate[(a + b*x^2)/(1 + x^2 + x^4),x]

[Out]

(((2*I)*a + (-I + Sqrt[3])*b)*ArcTan[((-I + Sqrt[3])*x)/2])/Sqrt[6 + (6*I)*Sqrt[3]] + (((-2*I)*a + (I + Sqrt[3
])*b)*ArcTan[((I + Sqrt[3])*x)/2])/Sqrt[6 - (6*I)*Sqrt[3]]

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\left (-a +b \right ) \ln \left (x^{2}-x +1\right )}{4}+\frac {\left (\frac {a}{2}+\frac {b}{2}\right ) \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {\left (a -b \right ) \ln \left (x^{2}+x +1\right )}{4}+\frac {\left (\frac {a}{2}+\frac {b}{2}\right ) \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}\)	\(78\)
risch	\(\frac {\sqrt {3}\, a \arctan \left (\frac {2 b^{2} \sqrt {3}\, x}{3 \left (a^{2}-a b +b^{2}\right )}+\frac {b \sqrt {3}\, a}{3 a^{2}-3 a b +3 b^{2}}+\frac {2 \sqrt {3}\, a^{2} x}{3 \left (a^{2}-a b +b^{2}\right )}-\frac {2 \sqrt {3}\, a b x}{3 \left (a^{2}-a b +b^{2}\right )}-\frac {\sqrt {3}\, a^{2}}{3 \left (a^{2}-a b +b^{2}\right )}-\frac {\sqrt {3}\, b^{2}}{3 \left (a^{2}-a b +b^{2}\right )}\right )}{6}+\frac {\sqrt {3}\, b \arctan \left (\frac {2 b^{2} \sqrt {3}\, x}{3 \left (a^{2}-a b +b^{2}\right )}+\frac {b \sqrt {3}\, a}{3 a^{2}-3 a b +3 b^{2}}+\frac {2 \sqrt {3}\, a^{2} x}{3 \left (a^{2}-a b +b^{2}\right )}-\frac {2 \sqrt {3}\, a b x}{3 \left (a^{2}-a b +b^{2}\right )}-\frac {\sqrt {3}\, a^{2}}{3 \left (a^{2}-a b +b^{2}\right )}-\frac {\sqrt {3}\, b^{2}}{3 \left (a^{2}-a b +b^{2}\right )}\right )}{6}+\frac {a \ln \left (4 a^{2} x^{2}-4 a b \,x^{2}+4 b^{2} x^{2}+4 a^{2} x -4 a b x +4 b^{2} x +4 a^{2}-4 a b +4 b^{2}\right )}{4}-\frac {b \ln \left (4 a^{2} x^{2}-4 a b \,x^{2}+4 b^{2} x^{2}+4 a^{2} x -4 a b x +4 b^{2} x +4 a^{2}-4 a b +4 b^{2}\right )}{4}+\frac {\sqrt {3}\, b \arctan \left (\frac {2 b^{2} \sqrt {3}\, x}{3 \left (a^{2}-a b +b^{2}\right )}-\frac {b \sqrt {3}\, a}{3 \left (a^{2}-a b +b^{2}\right )}+\frac {2 \sqrt {3}\, a^{2} x}{3 \left (a^{2}-a b +b^{2}\right )}-\frac {2 \sqrt {3}\, a b x}{3 \left (a^{2}-a b +b^{2}\right )}+\frac {\sqrt {3}\, a^{2}}{3 a^{2}-3 a b +3 b^{2}}+\frac {\sqrt {3}\, b^{2}}{3 a^{2}-3 a b +3 b^{2}}\right )}{6}-\frac {a \ln \left (4 a^{2} x^{2}-4 a b \,x^{2}+4 b^{2} x^{2}-4 a^{2} x +4 a b x -4 b^{2} x +4 a^{2}-4 a b +4 b^{2}\right )}{4}+\frac {b \ln \left (4 a^{2} x^{2}-4 a b \,x^{2}+4 b^{2} x^{2}-4 a^{2} x +4 a b x -4 b^{2} x +4 a^{2}-4 a b +4 b^{2}\right )}{4}+\frac {\sqrt {3}\, a \arctan \left (\frac {2 b^{2} \sqrt {3}\, x}{3 \left (a^{2}-a b +b^{2}\right )}-\frac {b \sqrt {3}\, a}{3 \left (a^{2}-a b +b^{2}\right )}+\frac {2 \sqrt {3}\, a^{2} x}{3 \left (a^{2}-a b +b^{2}\right )}-\frac {2 \sqrt {3}\, a b x}{3 \left (a^{2}-a b +b^{2}\right )}+\frac {\sqrt {3}\, a^{2}}{3 a^{2}-3 a b +3 b^{2}}+\frac {\sqrt {3}\, b^{2}}{3 a^{2}-3 a b +3 b^{2}}\right )}{6}\)	\(778\)

[In]

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

[Out]

1/4*(-a+b)*ln(x^2-x+1)+1/3*(1/2*a+1/2*b)*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))+1/4*(a-b)*ln(x^2+x+1)+1/3*(1/2*a+
1/2*b)*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int \frac {a+b x^2}{1+x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} - x + 1\right ) \]

[In]

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

[Out]

1/6*sqrt(3)*(a + b)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sqrt(3)*(a + b)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/4*(a
- b)*log(x^2 + x + 1) - 1/4*(a - b)*log(x^2 - x + 1)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 740, normalized size of antiderivative = 8.92 \[ \int \frac {a+b x^2}{1+x^2+x^4} \, dx=\left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) \log {\left (x + \frac {2 a^{3} \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 6 a^{2} b \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 12 a b^{2} \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 24 a \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3} + 2 b^{3} \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 48 b \left (- \frac {a}{4} + \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3}}{a^{4} - a^{3} b + a b^{3} - b^{4}} \right )} + \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) \log {\left (x + \frac {2 a^{3} \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 6 a^{2} b \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 12 a b^{2} \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 24 a \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3} + 2 b^{3} \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 48 b \left (- \frac {a}{4} + \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3}}{a^{4} - a^{3} b + a b^{3} - b^{4}} \right )} + \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) \log {\left (x + \frac {2 a^{3} \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 6 a^{2} b \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 12 a b^{2} \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 24 a \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3} + 2 b^{3} \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 48 b \left (\frac {a}{4} - \frac {b}{4} - \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3}}{a^{4} - a^{3} b + a b^{3} - b^{4}} \right )} + \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) \log {\left (x + \frac {2 a^{3} \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 6 a^{2} b \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 12 a b^{2} \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) + 24 a \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3} + 2 b^{3} \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right ) - 48 b \left (\frac {a}{4} - \frac {b}{4} + \frac {\sqrt {3} i \left (a + b\right )}{12}\right )^{3}}{a^{4} - a^{3} b + a b^{3} - b^{4}} \right )} \]

[In]

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

[Out]

(-a/4 + b/4 - sqrt(3)*I*(a + b)/12)*log(x + (2*a**3*(-a/4 + b/4 - sqrt(3)*I*(a + b)/12) + 6*a**2*b*(-a/4 + b/4
- sqrt(3)*I*(a + b)/12) - 12*a*b**2*(-a/4 + b/4 - sqrt(3)*I*(a + b)/12) + 24*a*(-a/4 + b/4 - sqrt(3)*I*(a + b
)/12)**3 + 2*b**3*(-a/4 + b/4 - sqrt(3)*I*(a + b)/12) - 48*b*(-a/4 + b/4 - sqrt(3)*I*(a + b)/12)**3)/(a**4 - a
**3*b + a*b**3 - b**4)) + (-a/4 + b/4 + sqrt(3)*I*(a + b)/12)*log(x + (2*a**3*(-a/4 + b/4 + sqrt(3)*I*(a + b)/
12) + 6*a**2*b*(-a/4 + b/4 + sqrt(3)*I*(a + b)/12) - 12*a*b**2*(-a/4 + b/4 + sqrt(3)*I*(a + b)/12) + 24*a*(-a/
4 + b/4 + sqrt(3)*I*(a + b)/12)**3 + 2*b**3*(-a/4 + b/4 + sqrt(3)*I*(a + b)/12) - 48*b*(-a/4 + b/4 + sqrt(3)*I
*(a + b)/12)**3)/(a**4 - a**3*b + a*b**3 - b**4)) + (a/4 - b/4 - sqrt(3)*I*(a + b)/12)*log(x + (2*a**3*(a/4 -
b/4 - sqrt(3)*I*(a + b)/12) + 6*a**2*b*(a/4 - b/4 - sqrt(3)*I*(a + b)/12) - 12*a*b**2*(a/4 - b/4 - sqrt(3)*I*(
a + b)/12) + 24*a*(a/4 - b/4 - sqrt(3)*I*(a + b)/12)**3 + 2*b**3*(a/4 - b/4 - sqrt(3)*I*(a + b)/12) - 48*b*(a/
4 - b/4 - sqrt(3)*I*(a + b)/12)**3)/(a**4 - a**3*b + a*b**3 - b**4)) + (a/4 - b/4 + sqrt(3)*I*(a + b)/12)*log(
x + (2*a**3*(a/4 - b/4 + sqrt(3)*I*(a + b)/12) + 6*a**2*b*(a/4 - b/4 + sqrt(3)*I*(a + b)/12) - 12*a*b**2*(a/4
- b/4 + sqrt(3)*I*(a + b)/12) + 24*a*(a/4 - b/4 + sqrt(3)*I*(a + b)/12)**3 + 2*b**3*(a/4 - b/4 + sqrt(3)*I*(a
+ b)/12) - 48*b*(a/4 - b/4 + sqrt(3)*I*(a + b)/12)**3)/(a**4 - a**3*b + a*b**3 - b**4))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int \frac {a+b x^2}{1+x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} - x + 1\right ) \]

[In]

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

[Out]

1/6*sqrt(3)*(a + b)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sqrt(3)*(a + b)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/4*(a
- b)*log(x^2 + x + 1) - 1/4*(a - b)*log(x^2 - x + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.83 \[ \int \frac {a+b x^2}{1+x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {1}{6} \, \sqrt {3} {\left (a + b\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} + x + 1\right ) - \frac {1}{4} \, {\left (a - b\right )} \log \left (x^{2} - x + 1\right ) \]

[In]

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

[Out]

1/6*sqrt(3)*(a + b)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sqrt(3)*(a + b)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/4*(a
- b)*log(x^2 + x + 1) - 1/4*(a - b)*log(x^2 - x + 1)

Mupad [B] (verification not implemented)

Time = 13.44 (sec) , antiderivative size = 827, normalized size of antiderivative = 9.96 \[ \int \frac {a+b x^2}{1+x^2+x^4} \, dx=\text {Too large to display} \]

[In]

int((a + b*x^2)/(x^2 + x^4 + 1),x)

[Out]

- atan(((x*(4*a*b - 4*a^2 + 2*b^2) + (12*a + 24*x*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 -
a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12)*1i + (x*(4*a
*b - 4*a^2 + 2*b^2) - (12*a - 24*x*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*
a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12)*1i)/((x*(4*a*b - 4*a^2 + 2*
b^2) + (12*a + 24*x*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(
1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12) - (x*(4*a*b - 4*a^2 + 2*b^2) - (12*a - 24*
x*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(b
/4 - a/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12) - 2*a*b^2 + 2*a^2*b + 2*b^3))*((a*1i)/2 - (b*1i)/2 + (3^(1/2
)*a)/6 + (3^(1/2)*b)/6) - atan(((x*(4*a*b - 4*a^2 + 2*b^2) + (12*a + 24*x*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^
(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2
)*b*1i)/12)*1i + (x*(4*a*b - 4*a^2 + 2*b^2) - (12*a - 24*x*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12)
)*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12)*1i)
/((x*(4*a*b - 4*a^2 + 2*b^2) + (12*a + 24*x*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(a/4 - b/4 +
(3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12) - (x*(4*a*b - 4*a^
2 + 2*b^2) - (12*a - 24*x*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12
+ (3^(1/2)*b*1i)/12))*(a/4 - b/4 + (3^(1/2)*a*1i)/12 + (3^(1/2)*b*1i)/12) - 2*a*b^2 + 2*a^2*b + 2*b^3))*((b*1i
)/2 - (a*1i)/2 + (3^(1/2)*a)/6 + (3^(1/2)*b)/6)