Integrand size = 16, antiderivative size = 112 \[ \int \frac {x^4}{3+4 x^3+x^6} \, dx=\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{2} \sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{2 \sqrt [3]{3}}-\frac {1}{12} \log \left (1-x+x^2\right )+\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{4 \sqrt [3]{3}} \] Output:
1/6*arctan(1/3*(1-2*x)*3^(1/2))*3^(1/2)-1/2*3^(1/6)*arctan(1/3*(3^(1/3)-2* x)*3^(1/6))+1/6*ln(1+x)-1/6*3^(2/3)*ln(3^(1/3)+x)-1/12*ln(x^2-x+1)+1/12*3^ (2/3)*ln(3^(2/3)-3^(1/3)*x+x^2)
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {x^4}{3+4 x^3+x^6} \, dx=\frac {1}{12} \left (-6 \sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-2 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )+2 \log (1+x)-2\ 3^{2/3} \log \left (3+3^{2/3} x\right )-\log \left (1-x+x^2\right )+3^{2/3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )\right ) \] Input:
Integrate[x^4/(3 + 4*x^3 + x^6),x]
Output:
(-6*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] - 2*3^(2/3)*Log[3 + 3^(2/3)*x] - Log[1 - x + x^2] + 3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12
Time = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {1710, 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.
| \(\displaystyle \int \frac {x^4}{x^6+4 x^3+3} \, dx\) |
| \(\Big \downarrow \) 1710 |
| \(\displaystyle \frac {3}{2} \int \frac {x}{x^3+3}dx-\frac {1}{2} \int \frac {x}{x^3+1}dx\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {1}{x+1}dx-\frac {1}{3} \int \frac {x+1}{x^2-x+1}dx\right )+\frac {3}{2} \left (\frac {\int \frac {x+\sqrt [3]{3}}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\int \frac {1}{x+\sqrt [3]{3}}dx}{3 \sqrt [3]{3}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \log (x+1)-\frac {1}{3} \int \frac {x+1}{x^2-x+1}dx\right )+\frac {3}{2} \left (\frac {\int \frac {x+\sqrt [3]{3}}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int -\frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx+\frac {1}{2} \int -\frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-\frac {3}{2} \int \frac {1}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-\frac {3}{2} \int \frac {1}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {3 \int \frac {1}{-\left (1-\frac {2 x}{\sqrt [3]{3}}\right )^2-3}d\left (1-\frac {2 x}{\sqrt [3]{3}}\right )-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{2} \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-\frac {3}{2} \int \frac {1}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {3}{2} \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx+3 \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)\right )+\frac {1}{3} \log (x+1)\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )\right )+\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (x^2-x+1\right )\right )+\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {\frac {1}{2} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )\) |
Input:
Int[x^4/(3 + 4*x^3 + x^6),x]
Output:
(Log[1 + x]/3 + (-(Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]]) - Log[1 - x + x^2]/ 2)/3)/2 + (3*(-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
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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_)), x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^n/2)*(b/q + 1) Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Simp[(d^n/2)*(b/q - 1) Int[(d*x)^(m - n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] & & NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[m, n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.54
| method | result | size |
| risch | \(-\frac {\ln \left (4 x^{2}-4 x +4\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\ln \left (x +1\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left (3 \textit {\_R}^{2}+x \right )\right )}{2}\) | \(60\) |
| default | \(-\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}-\frac {3^{\frac {2}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{6}+\frac {3^{\frac {2}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{12}+\frac {3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{2}+\frac {\ln \left (x +1\right )}{6}\) | \(84\) |
Input:
int(x^4/(x^6+4*x^3+3),x,method=_RETURNVERBOSE)
Output:
-1/12*ln(4*x^2-4*x+4)-1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))+1/6*ln(x+1)+ 1/2*sum(_R*ln(3*_R^2+x),_R=RootOf(3*_Z^3+1))
Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.75 \[ \int \frac {x^4}{3+4 x^3+x^6} \, dx=\frac {1}{12} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 3^{\frac {2}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{2} \cdot 3^{\frac {1}{6}} \arctan \left (-\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \] Input:
integrate(x^4/(x^6+4*x^3+3),x, algorithm="fricas")
Output:
1/12*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/6*3^(2/3)*log(x + 3^(1/3)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/2*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)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.20 \[ \int \frac {x^4}{3+4 x^3+x^6} \, dx=\frac {\log {\left (x + 1 \right )}}{6} + \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {2592 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{5}}{5} + \frac {168 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{2}}{5} \right )} + \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {168 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{2}}{5} + \frac {2592 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{5}}{5} \right )} + \operatorname {RootSum} {\left (24 t^{3} + 1, \left ( t \mapsto t \log {\left (\frac {2592 t^{5}}{5} + \frac {168 t^{2}}{5} + x \right )} \right )\right )} \] Input:
integrate(x**4/(x**6+4*x**3+3),x)
Output:
log(x + 1)/6 + (-1/12 - sqrt(3)*I/12)*log(x + 2592*(-1/12 - sqrt(3)*I/12)* *5/5 + 168*(-1/12 - sqrt(3)*I/12)**2/5) + (-1/12 + sqrt(3)*I/12)*log(x + 1 68*(-1/12 + sqrt(3)*I/12)**2/5 + 2592*(-1/12 + sqrt(3)*I/12)**5/5) + RootS um(24*_t**3 + 1, Lambda(_t, _t*log(2592*_t**5/5 + 168*_t**2/5 + x)))
Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.75 \[ \int \frac {x^4}{3+4 x^3+x^6} \, dx=\frac {1}{12} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 3^{\frac {2}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{2} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \] Input:
integrate(x^4/(x^6+4*x^3+3),x, algorithm="maxima")
Output:
1/12*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/6*3^(2/3)*log(x + 3^(1/3)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/2*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 = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77 \[ \int \frac {x^4}{3+4 x^3+x^6} \, dx=\frac {1}{12} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{6} \cdot 3^{\frac {2}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{2} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \] Input:
integrate(x^4/(x^6+4*x^3+3),x, algorithm="giac")
Output:
1/12*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/6*3^(2/3)*log(abs(x + 3^(1 /3))) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/2*3^(1/6)*arctan(1/3 *3^(1/6)*(2*x - 3^(1/3))) - 1/12*log(x^2 - x + 1) + 1/6*log(abs(x + 1))
Time = 0.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02 \[ \int \frac {x^4}{3+4 x^3+x^6} \, dx=\frac {\ln \left (x+1\right )}{6}-\frac {3^{2/3}\,\ln \left (x+3^{1/3}\right )}{6}+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\frac {{\left (-1\right )}^{1/3}\,\ln \left (x-\frac {{\left (-1\right )}^{1/3}\,3^{1/3}}{2}-\frac {{\left (-1\right )}^{1/6}\,3^{5/6}}{2}+\frac {3^{1/3}}{2}\right )\,\left (3^{2/3}+3^{1/6}\,3{}\mathrm {i}\right )}{12}+\frac {{\left (-1\right )}^{1/3}\,3^{2/3}\,\ln \left (x+{\left (-1\right )}^{2/3}\,3^{1/3}\right )}{6} \] Input:
int(x^4/(4*x^3 + x^6 + 3),x)
Output:
log(x + 1)/6 - (3^(2/3)*log(x + 3^(1/3)))/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) - ((-1)^(1/3)*log(x - ((-1)^(1/3)*3^(1/3))/2 - ((-1)^(1/6)*3^(5/ 6))/2 + 3^(1/3)/2)*(3^(2/3) + 3^(1/6)*3i))/12 + ((-1)^(1/3)*3^(2/3)*log(x + (-1)^(2/3)*3^(1/3)))/6
Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.72 \[ \int \frac {x^4}{3+4 x^3+x^6} \, dx=-\frac {3^{\frac {1}{6}} \mathit {atan} \left (\frac {\left (3^{\frac {1}{3}}-2 x \right ) 3^{\frac {1}{6}}}{3}\right )}{2}-\frac {\sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right )}{6}+\frac {3^{\frac {2}{3}} \mathrm {log}\left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{12}-\frac {3^{\frac {2}{3}} \mathrm {log}\left (3^{\frac {1}{3}}+x \right )}{6}-\frac {\mathrm {log}\left (x^{2}-x +1\right )}{12}+\frac {\mathrm {log}\left (x +1\right )}{6} \] Input:
int(x^4/(x^6+4*x^3+3),x)
Output:
( - 6*3**(1/6)*atan((3**(1/3) - 2*x)/(3**(2/3)*3**(1/6))) - 2*sqrt(3)*atan ((2*x - 1)/sqrt(3)) + 3**(2/3)*log(3**(2/3) - 3**(1/3)*x + x**2) - 2*3**(2 /3)*log(3**(1/3) + x) - log(x**2 - x + 1) + 2*log(x + 1))/12