Integrand size = 16, antiderivative size = 119 \[ \int \frac {x^7}{3+4 x^3+x^6} \, dx=\frac {x^2}{2}-\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {3}{2} \sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-\frac {1}{6} \log (1+x)+\frac {1}{2} 3^{2/3} \log \left (\sqrt [3]{3}+x\right )+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {1}{4} 3^{2/3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right ) \]
1/2*x^2+3/2*3^(1/6)*arctan(1/3*(3^(1/3)-2*x)*3^(1/6))-1/6*ln(1+x)+1/2*3^(2 /3)*ln(3^(1/3)+x)+1/12*ln(x^2-x+1)-1/4*3^(2/3)*ln(3^(2/3)-3^(1/3)*x+x^2)-1 /6*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)
Time = 0.03 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int \frac {x^7}{3+4 x^3+x^6} \, dx=\frac {1}{12} \left (6 x^2+18 \sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+2 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )-2 \log (1+x)+6\ 3^{2/3} \log \left (3+3^{2/3} x\right )+\log \left (1-x+x^2\right )-3\ 3^{2/3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )\right ) \]
(6*x^2 + 18*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Log[1 + x] + 6*3^(2/3)*Log[3 + 3^(2/3)*x] + Log[1 - x + x^2] - 3*3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12
Time = 0.35 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {1703, 27, 1834, 821, 16, 1142, 25, 1082, 217, 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.
| \(\displaystyle \int \frac {x^7}{x^6+4 x^3+3} \, dx\) |
| \(\Big \downarrow \) 1703 |
| \(\displaystyle \frac {x^2}{2}-\frac {1}{2} \int \frac {2 x \left (4 x^3+3\right )}{x^6+4 x^3+3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^2}{2}-\int \frac {x \left (4 x^3+3\right )}{x^6+4 x^3+3}dx\) |
| \(\Big \downarrow \) 1834 |
| \(\displaystyle \frac {1}{2} \int \frac {x}{x^3+1}dx-\frac {9}{2} \int \frac {x}{x^3+3}dx+\frac {x^2}{2}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {x+1}{x^2-x+1}dx-\frac {1}{3} \int \frac {1}{x+1}dx\right )-\frac {9}{2} \left (\frac {\int \frac {x+\sqrt [3]{3}}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\int \frac {1}{x+\sqrt [3]{3}}dx}{3 \sqrt [3]{3}}\right )+\frac {x^2}{2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {x+1}{x^2-x+1}dx-\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {\int \frac {x+\sqrt [3]{3}}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^2}{2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx+\frac {1}{2} \int -\frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx+\frac {1}{2} \int -\frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^2}{2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^2}{2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {3 \int \frac {1}{-\left (1-\frac {2 x}{\sqrt [3]{3}}\right )^2-3}d\left (1-\frac {2 x}{\sqrt [3]{3}}\right )-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^2}{2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {9}{2} \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {1}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )+\frac {x^2}{2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {9}{2} \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {1}{2} \left (\frac {1}{3} \left (-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-3 \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)\right )-\frac {1}{3} \log (x+1)\right )+\frac {x^2}{2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^2}{2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (x^2-x+1\right )\right )-\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {\frac {1}{2} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^2}{2}\) |
x^2/2 + (-1/3*Log[1 + x] + (Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + Log[1 - x + x^2]/2)/3)/2 - (9*(-1/3*Log[3^(1/3) + x]/3^(1/3) + (-(Sqrt[3]*ArcTan[(1 - (2*x)/3^(1/3))/Sqrt[3]]) + Log[3^(2/3) - 3^(1/3)*x + x^2]/2)/(3*3^(1/3) )))/2
3.2.60.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
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[((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[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x _Symbol] :> Simp[d^(2*n - 1)*(d*x)^(m - 2*n + 1)*((a + b*x^n + c*x^(2*n))^( p + 1)/(c*(m + 2*n*p + 1))), x] - Simp[d^(2*n)/(c*(m + 2*n*p + 1)) Int[(d *x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x ^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && N eQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n*p + 1, 0 ] && IntegerQ[p]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ [{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n , 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.53
| method | result | size |
| risch | \(\frac {x^{2}}{2}+\frac {\ln \left (4 x^{2}-4 x +4\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-9\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{2}+3 x \right )\right )}{2}-\frac {\ln \left (x +1\right )}{6}\) | \(63\) |
| default | \(\frac {x^{2}}{2}-\frac {\ln \left (x +1\right )}{6}+\frac {3^{\frac {2}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{2}-\frac {3^{\frac {2}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{4}-\frac {3 \,3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{2}+\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\) | \(89\) |
1/2*x^2+1/12*ln(4*x^2-4*x+4)+1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))+1/2*s um(_R*ln(_R^2+3*x),_R=RootOf(_Z^3-9))-1/6*ln(x+1)
Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.83 \[ \int \frac {x^7}{3+4 x^3+x^6} \, dx=\frac {1}{2} \, x^{2} - \frac {1}{2} \cdot 9^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {2}{9} \cdot 9^{\frac {1}{3}} \sqrt {3} x - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{4} \cdot 9^{\frac {1}{3}} \log \left (3 \, x^{2} - 9^{\frac {2}{3}} x + 3 \cdot 9^{\frac {1}{3}}\right ) + \frac {1}{2} \cdot 9^{\frac {1}{3}} \log \left (3 \, x + 9^{\frac {2}{3}}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]
1/2*x^2 - 1/2*9^(1/3)*sqrt(3)*arctan(2/9*9^(1/3)*sqrt(3)*x - 1/3*sqrt(3)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/4*9^(1/3)*log(3*x^2 - 9^(2 /3)*x + 3*9^(1/3)) + 1/2*9^(1/3)*log(3*x + 9^(2/3)) + 1/12*log(x^2 - x + 1 ) - 1/6*log(x + 1)
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.13 \[ \int \frac {x^7}{3+4 x^3+x^6} \, dx=\frac {x^{2}}{2} - \frac {\log {\left (x + 1 \right )}}{6} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {6562 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{2}}{183} - \frac {1872 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{5}}{61} \right )} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x - \frac {1872 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{5}}{61} + \frac {6562 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{2}}{183} \right )} + \operatorname {RootSum} {\left (8 t^{3} - 9, \left ( t \mapsto t \log {\left (- \frac {1872 t^{5}}{61} + \frac {6562 t^{2}}{183} + x \right )} \right )\right )} \]
x**2/2 - log(x + 1)/6 + (1/12 - sqrt(3)*I/12)*log(x + 6562*(1/12 - sqrt(3) *I/12)**2/183 - 1872*(1/12 - sqrt(3)*I/12)**5/61) + (1/12 + sqrt(3)*I/12)* log(x - 1872*(1/12 + sqrt(3)*I/12)**5/61 + 6562*(1/12 + sqrt(3)*I/12)**2/1 83) + RootSum(8*_t**3 - 9, Lambda(_t, _t*log(-1872*_t**5/61 + 6562*_t**2/1 83 + x)))
Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.75 \[ \int \frac {x^7}{3+4 x^3+x^6} \, dx=\frac {1}{2} \, x^{2} - \frac {1}{4} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{2} \cdot 3^{\frac {2}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {3}{2} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]
1/2*x^2 - 1/4*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/2*3^(2/3)*log(x + 3^(1/3)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 3/2*3^(1/6)*arctan (1/3*3^(1/6)*(2*x - 3^(1/3))) + 1/12*log(x^2 - x + 1) - 1/6*log(x + 1)
Time = 0.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.76 \[ \int \frac {x^7}{3+4 x^3+x^6} \, dx=\frac {1}{2} \, x^{2} - \frac {1}{4} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{2} \cdot 3^{\frac {2}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {3}{2} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]
1/2*x^2 - 1/4*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/2*3^(2/3)*log(abs (x + 3^(1/3))) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 3/2*3^(1/6)*a rctan(1/3*3^(1/6)*(2*x - 3^(1/3))) + 1/12*log(x^2 - x + 1) - 1/6*log(abs(x + 1))
Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.99 \[ \int \frac {x^7}{3+4 x^3+x^6} \, dx=\frac {3^{2/3}\,\ln \left (x+3^{1/3}\right )}{2}-\frac {\ln \left (x+1\right )}{6}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {x^2}{2}-\ln \left (x-\frac {3^{1/3}}{2}-\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{2/3}}{4}-\frac {3^{1/6}\,3{}\mathrm {i}}{4}\right )-\ln \left (x-\frac {3^{1/3}}{2}+\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{2/3}}{4}+\frac {3^{1/6}\,3{}\mathrm {i}}{4}\right ) \]
(3^(2/3)*log(x + 3^(1/3)))/2 - log(x + 1)/6 - log(x - (3^(1/2)*1i)/2 - 1/2 )*((3^(1/2)*1i)/12 - 1/12) + log(x + (3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/1 2 + 1/12) + x^2/2 - log(x - 3^(1/3)/2 - (3^(5/6)*1i)/2)*(3^(2/3)/4 - (3^(1 /6)*3i)/4) - log(x - 3^(1/3)/2 + (3^(5/6)*1i)/2)*(3^(2/3)/4 + (3^(1/6)*3i) /4)