Integrand size = 16, antiderivative size = 124 \[ \int \frac {x^{10}}{3+4 x^3+x^6} \, dx=-2 x^2+\frac {x^5}{5}+\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {9}{2} \sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+\frac {1}{6} \log (1+x)-\frac {3}{2} 3^{2/3} \log \left (\sqrt [3]{3}+x\right )-\frac {1}{12} \log \left (1-x+x^2\right )+\frac {3}{4} 3^{2/3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right ) \]
-2*x^2+1/5*x^5-9/2*3^(1/6)*arctan(1/3*(3^(1/3)-2*x)*3^(1/6))+1/6*ln(1+x)-3 /2*3^(2/3)*ln(3^(1/3)+x)-1/12*ln(x^2-x+1)+3/4*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 = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.95 \[ \int \frac {x^{10}}{3+4 x^3+x^6} \, dx=\frac {1}{60} \left (-120 x^2+12 x^5-270 \sqrt [6]{3} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-10 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )+10 \log (1+x)-90\ 3^{2/3} \log \left (3+3^{2/3} x\right )-5 \log \left (1-x+x^2\right )+45\ 3^{2/3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )\right ) \]
(-120*x^2 + 12*x^5 - 270*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] - 10*Sqrt [3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 10*Log[1 + x] - 90*3^(2/3)*Log[3 + 3^(2/3 )*x] - 5*Log[1 - x + x^2] + 45*3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/6 0
Time = 0.40 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {1703, 27, 1826, 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.
| \(\displaystyle \int \frac {x^{10}}{x^6+4 x^3+3} \, dx\) |
| \(\Big \downarrow \) 1703 |
| \(\displaystyle \frac {x^5}{5}-\frac {1}{5} \int \frac {5 x^4 \left (4 x^3+3\right )}{x^6+4 x^3+3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^5}{5}-\int \frac {x^4 \left (4 x^3+3\right )}{x^6+4 x^3+3}dx\) |
| \(\Big \downarrow \) 1826 |
| \(\displaystyle \frac {1}{2} \int \frac {2 x \left (13 x^3+12\right )}{x^6+4 x^3+3}dx+\frac {x^5}{5}-2 x^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {x \left (13 x^3+12\right )}{x^6+4 x^3+3}dx+\frac {x^5}{5}-2 x^2\) |
| \(\Big \downarrow \) 1834 |
| \(\displaystyle -\frac {1}{2} \int \frac {x}{x^3+1}dx+\frac {27}{2} \int \frac {x}{x^3+3}dx+\frac {x^5}{5}-2 x^2\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {1}{x+1}dx-\frac {1}{3} \int \frac {x+1}{x^2-x+1}dx\right )+\frac {27}{2} \left (\frac {\int \frac {x+\sqrt [3]{3}}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\int \frac {1}{x+\sqrt [3]{3}}dx}{3 \sqrt [3]{3}}\right )+\frac {x^5}{5}-2 x^2\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \log (x+1)-\frac {1}{3} \int \frac {x+1}{x^2-x+1}dx\right )+\frac {27}{2} \left (\frac {\int \frac {x+\sqrt [3]{3}}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^5}{5}-2 x^2\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int -\frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )+\frac {27}{2} \left (\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx+\frac {1}{2} \int -\frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^5}{5}-2 x^2\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-\frac {3}{2} \int \frac {1}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )+\frac {27}{2} \left (\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^5}{5}-2 x^2\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-\frac {3}{2} \int \frac {1}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )+\frac {27}{2} \left (\frac {3 \int \frac {1}{-\left (1-\frac {2 x}{\sqrt [3]{3}}\right )^2-3}d\left (1-\frac {2 x}{\sqrt [3]{3}}\right )-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^5}{5}-2 x^2\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {27}{2} \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-\frac {3}{2} \int \frac {1}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )+\frac {x^5}{5}-2 x^2\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {27}{2} \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx+3 \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)\right )+\frac {1}{3} \log (x+1)\right )+\frac {x^5}{5}-2 x^2\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )\right )+\frac {1}{3} \log (x+1)\right )+\frac {27}{2} \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^5}{5}-2 x^2\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (x^2-x+1\right )\right )+\frac {1}{3} \log (x+1)\right )+\frac {27}{2} \left (\frac {\frac {1}{2} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3 \sqrt [3]{3}}\right )+\frac {x^5}{5}-2 x^2\) |
-2*x^2 + x^5/5 + (Log[1 + x]/3 + (-(Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]]) - Log[1 - x + x^2]/2)/3)/2 + (27*(-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
3.2.58.3.1 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_))^(p_), x_Symbol] :> Simp[e*f^(n - 1)*(f*x)^(m - n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(m + n*(2*p + 1) + 1))), x] - Simp[f^n/(c*( m + n*(2*p + 1) + 1)) Int[(f*x)^(m - n)*(a + b*x^n + c*x^(2*n))^p*Simp[a* e*(m - n + 1) + (b*e*(m + n*p + 1) - c*d*(m + n*(2*p + 1) + 1))*x^n, x], x] , x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*(2*p + 1) + 1, 0] && Intege rQ[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 = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.55
| method | result | size |
| risch | \(\frac {x^{5}}{5}-2 x^{2}-\frac {\ln \left (4 x^{2}-4 x +4\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}+9\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{2}+3 x \right )\right )}{2}+\frac {\ln \left (x +1\right )}{6}\) | \(68\) |
| default | \(\frac {x^{5}}{5}-2 x^{2}+\frac {\ln \left (x +1\right )}{6}-\frac {3 \,3^{\frac {2}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{2}+\frac {3 \,3^{\frac {2}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{4}+\frac {9 \,3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{2}-\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\) | \(94\) |
1/5*x^5-2*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)) +3/2*sum(_R*ln(_R^2+3*x),_R=RootOf(_Z^3+9))+1/6*ln(x+1)
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.82 \[ \int \frac {x^{10}}{3+4 x^3+x^6} \, dx=\frac {1}{5} \, x^{5} - 2 \, x^{2} + \frac {3}{2} \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (2 \, \left (-9\right )^{\frac {1}{3}} x + 3\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {3}{4} \, \left (-9\right )^{\frac {1}{3}} \log \left (3 \, x^{2} - \left (-9\right )^{\frac {2}{3}} x - 3 \, \left (-9\right )^{\frac {1}{3}}\right ) + \frac {3}{2} \, \left (-9\right )^{\frac {1}{3}} \log \left (3 \, x + \left (-9\right )^{\frac {2}{3}}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]
1/5*x^5 - 2*x^2 + 3/2*sqrt(3)*(-9)^(1/3)*arctan(1/9*sqrt(3)*(2*(-9)^(1/3)* x + 3)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 3/4*(-9)^(1/3)*log(3 *x^2 - (-9)^(2/3)*x - 3*(-9)^(1/3)) + 3/2*(-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 = 0.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.16 \[ \int \frac {x^{10}}{3+4 x^3+x^6} \, dx=\frac {x^{5}}{5} - 2 x^{2} + \frac {\log {\left (x + 1 \right )}}{6} + \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {3872 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{5}}{3281} + \frac {3188648 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{2}}{88587} \right )} + \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {3188648 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{2}}{88587} + \frac {3872 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{5}}{3281} \right )} + \operatorname {RootSum} {\left (8 t^{3} + 243, \left ( t \mapsto t \log {\left (\frac {3872 t^{5}}{3281} + \frac {3188648 t^{2}}{88587} + x \right )} \right )\right )} \]
x**5/5 - 2*x**2 + log(x + 1)/6 + (-1/12 - sqrt(3)*I/12)*log(x + 3872*(-1/1 2 - sqrt(3)*I/12)**5/3281 + 3188648*(-1/12 - sqrt(3)*I/12)**2/88587) + (-1 /12 + sqrt(3)*I/12)*log(x + 3188648*(-1/12 + sqrt(3)*I/12)**2/88587 + 3872 *(-1/12 + sqrt(3)*I/12)**5/3281) + RootSum(8*_t**3 + 243, Lambda(_t, _t*lo g(3872*_t**5/3281 + 3188648*_t**2/88587 + x)))
Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76 \[ \int \frac {x^{10}}{3+4 x^3+x^6} \, dx=\frac {1}{5} \, x^{5} - 2 \, x^{2} + \frac {3}{4} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {3}{2} \cdot 3^{\frac {2}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {9}{2} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]
1/5*x^5 - 2*x^2 + 3/4*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 3/2*3^(2/3) *log(x + 3^(1/3)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 9/2*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 = 0.33 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.77 \[ \int \frac {x^{10}}{3+4 x^3+x^6} \, dx=\frac {1}{5} \, x^{5} - 2 \, x^{2} + \frac {3}{4} \cdot 3^{\frac {2}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {3}{2} \cdot 3^{\frac {2}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {9}{2} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]
1/5*x^5 - 2*x^2 + 3/4*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 3/2*3^(2/3) *log(abs(x + 3^(1/3))) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 9/2*3 ^(1/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) - 1/12*log(x^2 - x + 1) + 1/6*l og(abs(x + 1))
Time = 0.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00 \[ \int \frac {x^{10}}{3+4 x^3+x^6} \, dx=\frac {\ln \left (x+1\right )}{6}-\frac {3\,3^{2/3}\,\ln \left (x+3^{1/3}\right )}{2}+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-2\,x^2+\frac {x^5}{5}-\frac {3\,{\left (-1\right )}^{1/3}\,\ln \left (x-\frac {{\left (-1\right )}^{1/3}\,3^{1/3}}{2}-\frac {{\left (-1\right )}^{1/6}\,3^{5/6}}{2}+\frac {3^{1/3}}{2}\right )\,\left (3^{2/3}+3^{1/6}\,3{}\mathrm {i}\right )}{4}+\frac {3\,{\left (-1\right )}^{1/3}\,3^{2/3}\,\ln \left (x+{\left (-1\right )}^{2/3}\,3^{1/3}\right )}{2} \]
log(x + 1)/6 - (3*3^(2/3)*log(x + 3^(1/3)))/2 + 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) /12 + 1/12) - 2*x^2 + x^5/5 - (3*(-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))/4 + (3*(-1) ^(1/3)*3^(2/3)*log(x + (-1)^(2/3)*3^(1/3)))/2