Integrand size = 28, antiderivative size = 106 \[ \int \frac {x \left (1+2 x^2\right )}{\sqrt {1+x^2} \left (1+x^2+x^4\right )} \, dx=-\frac {1}{2} \arctan \left (\sqrt {3}-2 \sqrt {1+x^2}\right )+\frac {1}{2} \arctan \left (\sqrt {3}+2 \sqrt {1+x^2}\right )+\frac {1}{4} \sqrt {3} \log \left (2+x^2-\sqrt {3} \sqrt {1+x^2}\right )-\frac {1}{4} \sqrt {3} \log \left (2+x^2+\sqrt {3} \sqrt {1+x^2}\right ) \]
1/2*arctan(-3^(1/2)+2*(x^2+1)^(1/2))+1/2*arctan(3^(1/2)+2*(x^2+1)^(1/2))+1 /4*ln(2+x^2-3^(1/2)*(x^2+1)^(1/2))*3^(1/2)-1/4*ln(2+x^2+3^(1/2)*(x^2+1)^(1 /2))*3^(1/2)
Result contains complex when optimal does not.
Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int \frac {x \left (1+2 x^2\right )}{\sqrt {1+x^2} \left (1+x^2+x^4\right )} \, dx=\frac {1}{2} \left (1-i \sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {1+x^2}\right )+\frac {1}{2} \left (1+i \sqrt {3}\right ) \arctan \left (\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt {1+x^2}\right ) \]
((1 - I*Sqrt[3])*ArcTan[((1 - I*Sqrt[3])*Sqrt[1 + x^2])/2])/2 + ((1 + I*Sq rt[3])*ArcTan[((1 + I*Sqrt[3])*Sqrt[1 + x^2])/2])/2
Time = 0.38 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2238, 1197, 25, 1483, 27, 1142, 25, 1083, 217, 1103}
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.
\(\displaystyle \int \frac {x \left (2 x^2+1\right )}{\sqrt {x^2+1} \left (x^4+x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 2238 |
\(\displaystyle \frac {1}{2} \int \frac {2 x^2+1}{\sqrt {x^2+1} \left (x^4+x^2+1\right )}dx^2\) |
\(\Big \downarrow \) 1197 |
\(\displaystyle \int -\frac {1-2 x^4}{x^8-x^4+1}d\sqrt {x^2+1}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {1-2 x^4}{x^8-x^4+1}d\sqrt {x^2+1}\) |
\(\Big \downarrow \) 1483 |
\(\displaystyle -\frac {\int \frac {\sqrt {3}-3 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}-\frac {\int \frac {\sqrt {3} \left (\sqrt {3} \sqrt {x^2+1}+1\right )}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {\sqrt {3}-3 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}-\frac {1}{2} \int \frac {\sqrt {3} \sqrt {x^2+1}+1}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt {x^2+1}+\sqrt {3}}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}\right )-\frac {-\frac {1}{2} \sqrt {3} \int \frac {1}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}-\frac {3}{2} \int -\frac {\sqrt {3}-2 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt {x^2+1}+\sqrt {3}}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}\right )-\frac {\frac {3}{2} \int \frac {\sqrt {3}-2 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}-\frac {1}{2} \sqrt {3} \int \frac {1}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {1}{-x^4-1}d\left (2 \sqrt {x^2+1}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt {x^2+1}+\sqrt {3}}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}\right )-\frac {\sqrt {3} \int \frac {1}{-x^4-1}d\left (2 \sqrt {x^2+1}-\sqrt {3}\right )+\frac {3}{2} \int \frac {\sqrt {3}-2 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}}{2 \sqrt {3}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (\arctan \left (2 \sqrt {x^2+1}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt {x^2+1}+\sqrt {3}}{x^4+\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}\right )-\frac {\frac {3}{2} \int \frac {\sqrt {3}-2 \sqrt {x^2+1}}{x^4-\sqrt {3} \sqrt {x^2+1}+1}d\sqrt {x^2+1}+\sqrt {3} \arctan \left (\sqrt {3}-2 \sqrt {x^2+1}\right )}{2 \sqrt {3}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (\arctan \left (2 \sqrt {x^2+1}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (x^4+\sqrt {3} \sqrt {x^2+1}+1\right )\right )-\frac {\sqrt {3} \arctan \left (\sqrt {3}-2 \sqrt {x^2+1}\right )-\frac {3}{2} \log \left (x^4-\sqrt {3} \sqrt {x^2+1}+1\right )}{2 \sqrt {3}}\) |
-1/2*(Sqrt[3]*ArcTan[Sqrt[3] - 2*Sqrt[1 + x^2]] - (3*Log[1 + x^4 - Sqrt[3] *Sqrt[1 + x^2]])/2)/Sqrt[3] + (ArcTan[Sqrt[3] + 2*Sqrt[1 + x^2]] - (Sqrt[3 ]*Log[1 + x^4 + Sqrt[3]*Sqrt[1 + x^2]])/2)/2
3.1.34.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
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]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[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] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[(Px_)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(Px /. x -> Sqrt[x])*(d + e*x )^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && PolyQ[Px, x^2]
Time = 1.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {\arctan \left (-\sqrt {3}+2 \sqrt {x^{2}+1}\right )}{2}+\frac {\arctan \left (\sqrt {3}+2 \sqrt {x^{2}+1}\right )}{2}+\frac {\ln \left (2+x^{2}-\sqrt {3}\, \sqrt {x^{2}+1}\right ) \sqrt {3}}{4}-\frac {\ln \left (2+x^{2}+\sqrt {3}\, \sqrt {x^{2}+1}\right ) \sqrt {3}}{4}\) | \(81\) |
default | \(-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x -1\right )^{2}}{\left (-x -1\right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x -1\right )^{2}}{\left (-x -1\right )^{2}}+2}\, \sqrt {3}}{2}\right )+\arctan \left (\frac {\sqrt {\frac {2 \left (x -1\right )^{2}}{\left (-x -1\right )^{2}}+2}\, \left (x -1\right )}{\left (\frac {\left (x -1\right )^{2}}{\left (-x -1\right )^{2}}+1\right ) \left (-x -1\right )}\right )\right )}{4 \sqrt {\frac {\frac {\left (x -1\right )^{2}}{\left (-x -1\right )^{2}}+1}{\left (\frac {x -1}{-x -1}+1\right )^{2}}}\, \left (\frac {x -1}{-x -1}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \sqrt {3}}{2}\right )+\arctan \left (\frac {\sqrt {\frac {2 \left (x +1\right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (x +1\right )}{\left (\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )\right )}{4 \sqrt {\frac {\frac {\left (x +1\right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {x +1}{1-x}+1\right )^{2}}}\, \left (\frac {x +1}{1-x}+1\right )}\) | \(296\) |
trager | \(-4 \ln \left (\frac {-16 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{5} x^{2}+16 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{3} x^{2}+12 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{3}-4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right ) x^{2}+3 \sqrt {x^{2}+1}-6 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )}{\left (4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{2} x -x +1\right ) \left (4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{2} x -x -1\right )}\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{3}+\ln \left (\frac {-16 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{5} x^{2}+16 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{3} x^{2}+12 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{3}-4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right ) x^{2}+3 \sqrt {x^{2}+1}-6 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )}{\left (4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{2} x -x +1\right ) \left (4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{2} x -x -1\right )}\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )-\operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-16 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{5} x^{2}+4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{3} x^{2}-12 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{3}+2 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right ) x^{2}+3 \sqrt {x^{2}+1}+6 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )}{\left (4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{2} x -1\right ) \left (4 \operatorname {RootOf}\left (16 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )^{2} x +1\right )}\right )\) | \(499\) |
1/2*arctan(-3^(1/2)+2*(x^2+1)^(1/2))+1/2*arctan(3^(1/2)+2*(x^2+1)^(1/2))+1 /4*ln(2+x^2-3^(1/2)*(x^2+1)^(1/2))*3^(1/2)-1/4*ln(2+x^2+3^(1/2)*(x^2+1)^(1 /2))*3^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (80) = 160\).
Time = 0.33 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.70 \[ \int \frac {x \left (1+2 x^2\right )}{\sqrt {1+x^2} \left (1+x^2+x^4\right )} \, dx=\frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {-3} + 1} \log \left (4 \, x^{2} + \sqrt {2} {\left (\sqrt {-3} x + x\right )} \sqrt {-\sqrt {-3} + 1} - \sqrt {x^{2} + 1} {\left (\sqrt {2} {\left (\sqrt {-3} + 1\right )} \sqrt {-\sqrt {-3} + 1} + 4 \, x\right )} + 4\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {-3} + 1} \log \left (4 \, x^{2} - \sqrt {2} {\left (\sqrt {-3} x + x\right )} \sqrt {-\sqrt {-3} + 1} + \sqrt {x^{2} + 1} {\left (\sqrt {2} {\left (\sqrt {-3} + 1\right )} \sqrt {-\sqrt {-3} + 1} - 4 \, x\right )} + 4\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {-3} + 1} \log \left (4 \, x^{2} - 4 \, \sqrt {x^{2} + 1} x + {\left (\sqrt {2} \sqrt {x^{2} + 1} {\left (\sqrt {-3} - 1\right )} - \sqrt {2} {\left (\sqrt {-3} x - x\right )}\right )} \sqrt {\sqrt {-3} + 1} + 4\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {-3} + 1} \log \left (4 \, x^{2} - 4 \, \sqrt {x^{2} + 1} x - {\left (\sqrt {2} \sqrt {x^{2} + 1} {\left (\sqrt {-3} - 1\right )} - \sqrt {2} {\left (\sqrt {-3} x - x\right )}\right )} \sqrt {\sqrt {-3} + 1} + 4\right ) \]
1/4*sqrt(2)*sqrt(-sqrt(-3) + 1)*log(4*x^2 + sqrt(2)*(sqrt(-3)*x + x)*sqrt( -sqrt(-3) + 1) - sqrt(x^2 + 1)*(sqrt(2)*(sqrt(-3) + 1)*sqrt(-sqrt(-3) + 1) + 4*x) + 4) - 1/4*sqrt(2)*sqrt(-sqrt(-3) + 1)*log(4*x^2 - sqrt(2)*(sqrt(- 3)*x + x)*sqrt(-sqrt(-3) + 1) + sqrt(x^2 + 1)*(sqrt(2)*(sqrt(-3) + 1)*sqrt (-sqrt(-3) + 1) - 4*x) + 4) + 1/4*sqrt(2)*sqrt(sqrt(-3) + 1)*log(4*x^2 - 4 *sqrt(x^2 + 1)*x + (sqrt(2)*sqrt(x^2 + 1)*(sqrt(-3) - 1) - sqrt(2)*(sqrt(- 3)*x - x))*sqrt(sqrt(-3) + 1) + 4) - 1/4*sqrt(2)*sqrt(sqrt(-3) + 1)*log(4* x^2 - 4*sqrt(x^2 + 1)*x - (sqrt(2)*sqrt(x^2 + 1)*(sqrt(-3) - 1) - sqrt(2)* (sqrt(-3)*x - x))*sqrt(sqrt(-3) + 1) + 4)
\[ \int \frac {x \left (1+2 x^2\right )}{\sqrt {1+x^2} \left (1+x^2+x^4\right )} \, dx=\int \frac {x \left (2 x^{2} + 1\right )}{\sqrt {x^{2} + 1} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
\[ \int \frac {x \left (1+2 x^2\right )}{\sqrt {1+x^2} \left (1+x^2+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{2} + 1\right )} x}{{\left (x^{4} + x^{2} + 1\right )} \sqrt {x^{2} + 1}} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.75 \[ \int \frac {x \left (1+2 x^2\right )}{\sqrt {1+x^2} \left (1+x^2+x^4\right )} \, dx=-\frac {1}{4} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} \sqrt {x^{2} + 1} + 2\right ) + \frac {1}{4} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} \sqrt {x^{2} + 1} + 2\right ) + \frac {1}{2} \, \arctan \left (\sqrt {3} + 2 \, \sqrt {x^{2} + 1}\right ) + \frac {1}{2} \, \arctan \left (-\sqrt {3} + 2 \, \sqrt {x^{2} + 1}\right ) \]
-1/4*sqrt(3)*log(x^2 + sqrt(3)*sqrt(x^2 + 1) + 2) + 1/4*sqrt(3)*log(x^2 - sqrt(3)*sqrt(x^2 + 1) + 2) + 1/2*arctan(sqrt(3) + 2*sqrt(x^2 + 1)) + 1/2*a rctan(-sqrt(3) + 2*sqrt(x^2 + 1))
Time = 8.63 (sec) , antiderivative size = 397, normalized size of antiderivative = 3.75 \[ \int \frac {x \left (1+2 x^2\right )}{\sqrt {1+x^2} \left (1+x^2+x^4\right )} \, dx=\frac {\left (\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (\frac {x}{2}+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}+1+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left (2\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}\,\left (4\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3+1+\sqrt {3}\,1{}\mathrm {i}\right )}+\frac {\left (\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left (2\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}\,\left (4\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3-1+\sqrt {3}\,1{}\mathrm {i}\right )}+\frac {\left (\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (\frac {x}{2}+\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}+1-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left (2\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{\sqrt {{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}\,\left (4\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3-1+\sqrt {3}\,1{}\mathrm {i}\right )}+\frac {\left (\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-\ln \left (1+\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\sqrt {x^2+1}-\frac {x}{2}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{2}\right )\right )\,\left (2\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{\sqrt {{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2+1}\,\left (4\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^3+1+\sqrt {3}\,1{}\mathrm {i}\right )} \]
((log(x - (3^(1/2)*1i)/2 - 1/2) - log(x/2 + (3^(1/2)/2 + 1i/2)*(x^2 + 1)^( 1/2) + (3^(1/2)*x*1i)/2 + 1))*((3^(1/2)*1i)/2 + 2*((3^(1/2)*1i)/2 + 1/2)^3 + 1/2))/((((3^(1/2)*1i)/2 + 1/2)^2 + 1)^(1/2)*(3^(1/2)*1i + 4*((3^(1/2)*1 i)/2 + 1/2)^3 + 1)) + ((log(x - (3^(1/2)*1i)/2 + 1/2) - log((3^(1/2)/2 - 1 i/2)*(x^2 + 1)^(1/2) - x/2 + (3^(1/2)*x*1i)/2 + 1))*((3^(1/2)*1i)/2 + 2*(( 3^(1/2)*1i)/2 - 1/2)^3 - 1/2))/((((3^(1/2)*1i)/2 - 1/2)^2 + 1)^(1/2)*(3^(1 /2)*1i + 4*((3^(1/2)*1i)/2 - 1/2)^3 - 1)) + ((log(x + (3^(1/2)*1i)/2 - 1/2 ) - log(x/2 + (3^(1/2)/2 - 1i/2)*(x^2 + 1)^(1/2) - (3^(1/2)*x*1i)/2 + 1))* ((3^(1/2)*1i)/2 + 2*((3^(1/2)*1i)/2 - 1/2)^3 - 1/2))/((((3^(1/2)*1i)/2 - 1 /2)^2 + 1)^(1/2)*(3^(1/2)*1i + 4*((3^(1/2)*1i)/2 - 1/2)^3 - 1)) + ((log(x + (3^(1/2)*1i)/2 + 1/2) - log((3^(1/2)/2 + 1i/2)*(x^2 + 1)^(1/2) - x/2 - ( 3^(1/2)*x*1i)/2 + 1))*((3^(1/2)*1i)/2 + 2*((3^(1/2)*1i)/2 + 1/2)^3 + 1/2)) /((((3^(1/2)*1i)/2 + 1/2)^2 + 1)^(1/2)*(3^(1/2)*1i + 4*((3^(1/2)*1i)/2 + 1 /2)^3 + 1))