\(\int \frac {1}{x^2 (1+x^4+x^8)} \, dx\) [343]

Optimal result

Integrand size = 14, antiderivative size = 145 \[ \int \frac {1}{x^2 \left (1+x^4+x^8\right )} \, dx=-\frac {1}{x}+\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \arctan \left (\sqrt {3}-2 x\right )-\frac {\arctan \left (\frac {1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \arctan \left (\sqrt {3}+2 x\right )-\frac {1}{8} \log \left (1-x+x^2\right )+\frac {1}{8} \log \left (1+x+x^2\right )-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{8 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{8 \sqrt {3}} \]

[Out]

-1/x-1/4*arctan(2*x-3^(1/2))-1/4*arctan(2*x+3^(1/2))-1/8*ln(x^2-x+1)+1/8*ln(x^2+x+1)+1/12*arctan(1/3*(1-2*x)*3
^(1/2))*3^(1/2)-1/12*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/24*ln(1+x^2-x*3^(1/2))*3^(1/2)+1/24*ln(1+x^2+x*3^(1
/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1382, 1520, 1141, 1175, 632, 210, 1178, 642} \[ \int \frac {1}{x^2 \left (1+x^4+x^8\right )} \, dx=\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \arctan \left (\sqrt {3}-2 x\right )-\frac {\arctan \left (\frac {2 x+1}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \arctan \left (2 x+\sqrt {3}\right )-\frac {1}{8} \log \left (x^2-x+1\right )+\frac {1}{8} \log \left (x^2+x+1\right )-\frac {\log \left (x^2-\sqrt {3} x+1\right )}{8 \sqrt {3}}+\frac {\log \left (x^2+\sqrt {3} x+1\right )}{8 \sqrt {3}}-\frac {1}{x} \]

[In]

Int[1/(x^2*(1 + x^4 + x^8)),x]

[Out]

-x^(-1) + ArcTan[(1 - 2*x)/Sqrt[3]]/(4*Sqrt[3]) + ArcTan[Sqrt[3] - 2*x]/4 - ArcTan[(1 + 2*x)/Sqrt[3]]/(4*Sqrt[
3]) - ArcTan[Sqrt[3] + 2*x]/4 - Log[1 - x + x^2]/8 + Log[1 + x + x^2]/8 - Log[1 - Sqrt[3]*x + x^2]/(8*Sqrt[3])
+ Log[1 + Sqrt[3]*x + x^2]/(8*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 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 1141

Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, Dist[1/2, Int[(q + x^2)/(
a + b*x^2 + c*x^4), x], x] - Dist[1/2, Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && Lt
Q[b^2 - 4*a*c, 0] && PosQ[a*c]

Rule 1175

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e) - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[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, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !Lt
Q[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rule 1178

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

Rule 1382

Int[((d_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(d*x)^(m + 1)*((a +
b*x^n + c*x^(2*n))^(p + 1)/(a*d*(m + 1))), x] - Dist[1/(a*d^n*(m + 1)), Int[(d*x)^(m + n)*(b*(m + n*(p + 1) +
1) + c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2
*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]

Rule 1520

Int[(((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wit
h[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q - b*c, 2]}, Dist[c/(2*q*r), Int[(f*x)^m*(Simp[d*r - (c*d - e*q)*x^(n/2)
, x]/(q - r*x^(n/2) + c*x^n)), x], x] + Dist[c/(2*q*r), Int[(f*x)^m*(Simp[d*r + (c*d - e*q)*x^(n/2), x]/(q + r
*x^(n/2) + c*x^n)), x], x]] /;  !LtQ[2*c*q - b*c, 0]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && LtQ[b
^2 - 4*a*c, 0] && IntegersQ[m, n/2] && LtQ[0, m, n] && PosQ[a*c]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{x}+\int \frac {x^2 \left (-1-x^4\right )}{1+x^4+x^8} \, dx \\ & = -\frac {1}{x}-\frac {1}{2} \int \frac {x^2}{1-x^2+x^4} \, dx-\frac {1}{2} \int \frac {x^2}{1+x^2+x^4} \, dx \\ & = -\frac {1}{x}+\frac {1}{4} \int \frac {1-x^2}{1-x^2+x^4} \, dx-\frac {1}{4} \int \frac {1+x^2}{1-x^2+x^4} \, dx+\frac {1}{4} \int \frac {1-x^2}{1+x^2+x^4} \, dx-\frac {1}{4} \int \frac {1+x^2}{1+x^2+x^4} \, dx \\ & = -\frac {1}{x}-\frac {1}{8} \int \frac {1+2 x}{-1-x-x^2} \, dx-\frac {1}{8} \int \frac {1-2 x}{-1+x-x^2} \, dx-\frac {1}{8} \int \frac {1}{1-x+x^2} \, dx-\frac {1}{8} \int \frac {1}{1+x+x^2} \, dx-\frac {1}{8} \int \frac {1}{1-\sqrt {3} x+x^2} \, dx-\frac {1}{8} \int \frac {1}{1+\sqrt {3} x+x^2} \, dx-\frac {\int \frac {\sqrt {3}+2 x}{-1-\sqrt {3} x-x^2} \, dx}{8 \sqrt {3}}-\frac {\int \frac {\sqrt {3}-2 x}{-1+\sqrt {3} x-x^2} \, dx}{8 \sqrt {3}} \\ & = -\frac {1}{x}-\frac {1}{8} \log \left (1-x+x^2\right )+\frac {1}{8} \log \left (1+x+x^2\right )-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{8 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{8 \sqrt {3}}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 x\right ) \\ & = -\frac {1}{x}+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{4} \tan ^{-1}\left (\sqrt {3}-2 x\right )-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{4} \tan ^{-1}\left (\sqrt {3}+2 x\right )-\frac {1}{8} \log \left (1-x+x^2\right )+\frac {1}{8} \log \left (1+x+x^2\right )-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{8 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{8 \sqrt {3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^2 \left (1+x^4+x^8\right )} \, dx=\frac {1}{24} \left (-\frac {24}{x}+2 i \sqrt {-6+6 i \sqrt {3}} \arctan \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) x\right )-2 i \sqrt {-6-6 i \sqrt {3}} \arctan \left (\frac {1}{2} \left (1+i \sqrt {3}\right ) x\right )-2 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )-3 \log \left (1-x+x^2\right )+3 \log \left (1+x+x^2\right )\right ) \]

[In]

Integrate[1/(x^2*(1 + x^4 + x^8)),x]

[Out]

(-24/x + (2*I)*Sqrt[-6 + (6*I)*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x)/2] - (2*I)*Sqrt[-6 - (6*I)*Sqrt[3]]*ArcTan[
((1 + I*Sqrt[3])*x)/2] - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 3*Log[1
- x + x^2] + 3*Log[1 + x + x^2])/24

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.13 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.66

method	result	size
risch	\(-\frac {1}{x}+\frac {\ln \left (4 x^{2}+4 x +4\right )}{8}-\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}-\frac {\ln \left (4 x^{2}-4 x +4\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-6 \textit {\_R}^{3}-\textit {\_R} +x \right )\right )}{4}\)	\(96\)
default	\(-\frac {\ln \left (x^{2}-x +1\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}+\frac {\ln \left (x^{2}+x +1\right )}{8}-\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}-\frac {1}{x}+\frac {\sqrt {3}\, \left (-\frac {\ln \left (1+x^{2}-x \sqrt {3}\right )}{2}-\sqrt {3}\, \arctan \left (2 x -\sqrt {3}\right )\right )}{12}+\frac {\sqrt {3}\, \left (\frac {\ln \left (1+x^{2}+x \sqrt {3}\right )}{2}-\sqrt {3}\, \arctan \left (2 x +\sqrt {3}\right )\right )}{12}\)	\(126\)

[In]

int(1/x^2/(x^8+x^4+1),x,method=_RETURNVERBOSE)

[Out]

-1/x+1/8*ln(4*x^2+4*x+4)-1/12*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/8*ln(4*x^2-4*x+4)-1/12*3^(1/2)*arctan(1/3*
(2*x-1)*3^(1/2))+1/4*sum(_R*ln(-6*_R^3-_R+x),_R=RootOf(9*_Z^4+3*_Z^2+1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.44 \[ \int \frac {1}{x^2 \left (1+x^4+x^8\right )} \, dx=-\frac {\sqrt {6} x \sqrt {i \, \sqrt {3} - 1} \log \left (i \, \sqrt {6} \sqrt {3} \sqrt {i \, \sqrt {3} - 1} + 6 \, x\right ) - \sqrt {6} x \sqrt {i \, \sqrt {3} - 1} \log \left (-i \, \sqrt {6} \sqrt {3} \sqrt {i \, \sqrt {3} - 1} + 6 \, x\right ) - \sqrt {6} x \sqrt {-i \, \sqrt {3} - 1} \log \left (i \, \sqrt {6} \sqrt {3} \sqrt {-i \, \sqrt {3} - 1} + 6 \, x\right ) + \sqrt {6} x \sqrt {-i \, \sqrt {3} - 1} \log \left (-i \, \sqrt {6} \sqrt {3} \sqrt {-i \, \sqrt {3} - 1} + 6 \, x\right ) + 2 \, \sqrt {3} x \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + 2 \, \sqrt {3} x \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 3 \, x \log \left (x^{2} + x + 1\right ) + 3 \, x \log \left (x^{2} - x + 1\right ) + 24}{24 \, x} \]

[In]

integrate(1/x^2/(x^8+x^4+1),x, algorithm="fricas")

[Out]

-1/24*(sqrt(6)*x*sqrt(I*sqrt(3) - 1)*log(I*sqrt(6)*sqrt(3)*sqrt(I*sqrt(3) - 1) + 6*x) - sqrt(6)*x*sqrt(I*sqrt(
3) - 1)*log(-I*sqrt(6)*sqrt(3)*sqrt(I*sqrt(3) - 1) + 6*x) - sqrt(6)*x*sqrt(-I*sqrt(3) - 1)*log(I*sqrt(6)*sqrt(
3)*sqrt(-I*sqrt(3) - 1) + 6*x) + sqrt(6)*x*sqrt(-I*sqrt(3) - 1)*log(-I*sqrt(6)*sqrt(3)*sqrt(-I*sqrt(3) - 1) +
6*x) + 2*sqrt(3)*x*arctan(1/3*sqrt(3)*(2*x + 1)) + 2*sqrt(3)*x*arctan(1/3*sqrt(3)*(2*x - 1)) - 3*x*log(x^2 + x
+ 1) + 3*x*log(x^2 - x + 1) + 24)/x

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.50 \[ \int \frac {1}{x^2 \left (1+x^4+x^8\right )} \, dx=\left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right ) \log {\left (x - 442368 \left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{7} - 384 \left (- \frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{3} \right )} + \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right ) \log {\left (x - 384 \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{3} - 442368 \left (- \frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{7} \right )} + \left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right ) \log {\left (x - 442368 \left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{7} - 384 \left (\frac {1}{8} - \frac {\sqrt {3} i}{24}\right )^{3} \right )} + \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right ) \log {\left (x - 384 \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{3} - 442368 \left (\frac {1}{8} + \frac {\sqrt {3} i}{24}\right )^{7} \right )} + \operatorname {RootSum} {\left (2304 t^{4} + 48 t^{2} + 1, \left ( t \mapsto t \log {\left (- 442368 t^{7} - 384 t^{3} + x \right )} \right )\right )} - \frac {1}{x} \]

[In]

integrate(1/x**2/(x**8+x**4+1),x)

[Out]

(-1/8 - sqrt(3)*I/24)*log(x - 442368*(-1/8 - sqrt(3)*I/24)**7 - 384*(-1/8 - sqrt(3)*I/24)**3) + (-1/8 + sqrt(3
)*I/24)*log(x - 384*(-1/8 + sqrt(3)*I/24)**3 - 442368*(-1/8 + sqrt(3)*I/24)**7) + (1/8 - sqrt(3)*I/24)*log(x -
442368*(1/8 - sqrt(3)*I/24)**7 - 384*(1/8 - sqrt(3)*I/24)**3) + (1/8 + sqrt(3)*I/24)*log(x - 384*(1/8 + sqrt(
3)*I/24)**3 - 442368*(1/8 + sqrt(3)*I/24)**7) + RootSum(2304*_t**4 + 48*_t**2 + 1, Lambda(_t, _t*log(-442368*_
t**7 - 384*_t**3 + x))) - 1/x

Maxima [F]

\[ \int \frac {1}{x^2 \left (1+x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} + x^{4} + 1\right )} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(x^8+x^4+1),x, algorithm="maxima")

[Out]

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/x - 1/2*integrate
(x^2/(x^4 - x^2 + 1), x) + 1/8*log(x^2 + x + 1) - 1/8*log(x^2 - x + 1)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \left (1+x^4+x^8\right )} \, dx=-\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{24} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) - \frac {1}{24} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) - \frac {1}{x} - \frac {1}{4} \, \arctan \left (2 \, x + \sqrt {3}\right ) - \frac {1}{4} \, \arctan \left (2 \, x - \sqrt {3}\right ) + \frac {1}{8} \, \log \left (x^{2} + x + 1\right ) - \frac {1}{8} \, \log \left (x^{2} - x + 1\right ) \]

[In]

integrate(1/x^2/(x^8+x^4+1),x, algorithm="giac")

[Out]

-1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/24*sqrt(3)*log(x^
2 + sqrt(3)*x + 1) - 1/24*sqrt(3)*log(x^2 - sqrt(3)*x + 1) - 1/x - 1/4*arctan(2*x + sqrt(3)) - 1/4*arctan(2*x
- sqrt(3)) + 1/8*log(x^2 + x + 1) - 1/8*log(x^2 - x + 1)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^2 \left (1+x^4+x^8\right )} \, dx=\mathrm {atan}\left (\frac {2\,x}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\mathrm {atan}\left (\frac {2\,x}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{12}-\frac {1}{4}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,2{}\mathrm {i}}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{12}+\frac {1}{4}{}\mathrm {i}\right )-\frac {1}{x} \]

[In]

int(1/(x^2*(x^4 + x^8 + 1)),x)

[Out]

atan((2*x)/(3^(1/2)*1i - 1))*((3^(1/2)*1i)/12 + 1/4) + atan((2*x)/(3^(1/2)*1i + 1))*((3^(1/2)*1i)/12 - 1/4) -
atan((x*2i)/(3^(1/2)*1i - 1))*(3^(1/2)/12 - 1i/4) - atan((x*2i)/(3^(1/2)*1i + 1))*(3^(1/2)/12 + 1i/4) - 1/x