Integrand size = 16, antiderivative size = 113 \[ \int \frac {x^6}{3+4 x^3+x^6} \, dx=x-\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{2} 3^{5/6} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+\frac {1}{6} \log (1+x)-\frac {1}{2} \sqrt [3]{3} \log \left (\sqrt [3]{3}+x\right )-\frac {1}{12} \log \left (1-x+x^2\right )+\frac {1}{4} \sqrt [3]{3} \log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right ) \]
x+1/2*3^(5/6)*arctan(1/3*(3^(1/3)-2*x)*3^(1/6))+1/6*ln(1+x)-1/2*3^(1/3)*ln (3^(1/3)+x)-1/12*ln(x^2-x+1)+1/4*3^(1/3)*ln(3^(2/3)-3^(1/3)*x+x^2)-1/6*arc tan(1/3*(1-2*x)*3^(1/2))*3^(1/2)
Time = 0.02 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.98 \[ \int \frac {x^6}{3+4 x^3+x^6} \, dx=\frac {1}{12} \left (12 x+6\ 3^{5/6} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+2 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )+2 \log (1+x)-6 \sqrt [3]{3} \log \left (3+3^{2/3} x\right )-\log \left (1-x+x^2\right )+3 \sqrt [3]{3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )\right ) \]
(12*x + 6*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 2*Log[1 + x] - 6*3^(1/3)*Log[3 + 3^(2/3)*x] - Log[1 - x + x^2] + 3*3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12
Time = 0.34 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {1703, 1752, 750, 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.
| \(\displaystyle \int \frac {x^6}{x^6+4 x^3+3} \, dx\) |
| \(\Big \downarrow \) 1703 |
| \(\displaystyle x-\int \frac {4 x^3+3}{x^6+4 x^3+3}dx\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^3+1}dx-\frac {9}{2} \int \frac {1}{x^3+3}dx+x\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {2-x}{x^2-x+1}dx+\frac {1}{3} \int \frac {1}{x+1}dx\right )-\frac {9}{2} \left (\frac {\int \frac {2 \sqrt [3]{3}-x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}+\frac {\int \frac {1}{x+\sqrt [3]{3}}dx}{3\ 3^{2/3}}\right )+x\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {2-x}{x^2-x+1}dx+\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {\int \frac {2 \sqrt [3]{3}-x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+x\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int -\frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\frac {1}{2} \int -\frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+x\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx+\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+x\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx+\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx+3 \int \frac {1}{-\left (1-\frac {2 x}{\sqrt [3]{3}}\right )^2-3}d\left (1-\frac {2 x}{\sqrt [3]{3}}\right )}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+x\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {9}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+\frac {1}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx+\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )+x\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {9}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-3 \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)\right )+\frac {1}{3} \log (x+1)\right )+x\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx+\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )\right )+\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+x\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (x^2-x+1\right )\right )+\frac {1}{3} \log (x+1)\right )-\frac {9}{2} \left (\frac {-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+x\) |
x + (Log[1 + x]/3 + (Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - Log[1 - x + x^2] /2)/3)/2 - (9*(Log[3^(1/3) + x]/(3*3^(2/3)) + (-(Sqrt[3]*ArcTan[(1 - (2*x) /3^(1/3))/Sqrt[3]]) - Log[3^(2/3) - 3^(1/3)*x + x^2]/2)/(3*3^(2/3))))/2
3.2.61.3.1 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_) + (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_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*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[((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[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) I nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 , 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.47
| method | result | size |
| risch | \(x +\frac {\ln \left (x +1\right )}{6}-\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x -\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}+3\right )}{\sum }\textit {\_R} \ln \left (x -\textit {\_R} \right )\right )}{2}\) | \(53\) |
| default | \(x +\frac {\ln \left (x +1\right )}{6}-\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{2}+\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{4}-\frac {3^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{2}-\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\) | \(85\) |
x+1/6*ln(x+1)-1/12*ln(x^2-x+1)+1/6*3^(1/2)*arctan(2/3*(x-1/2)*3^(1/2))+1/2 *sum(_R*ln(x-_R),_R=RootOf(_Z^3+3))
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78 \[ \int \frac {x^6}{3+4 x^3+x^6} \, dx=\frac {1}{2} \, \sqrt {3} \left (-3\right )^{\frac {1}{3}} \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (2 \, \left (-3\right )^{\frac {2}{3}} x - 3\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{4} \, \left (-3\right )^{\frac {1}{3}} \log \left (x^{2} + \left (-3\right )^{\frac {1}{3}} x + \left (-3\right )^{\frac {2}{3}}\right ) + \frac {1}{2} \, \left (-3\right )^{\frac {1}{3}} \log \left (x - \left (-3\right )^{\frac {1}{3}}\right ) + x - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]
1/2*sqrt(3)*(-3)^(1/3)*arctan(1/9*sqrt(3)*(2*(-3)^(2/3)*x - 3)) + 1/6*sqrt (3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/4*(-3)^(1/3)*log(x^2 + (-3)^(1/3)*x + (-3)^(2/3)) + 1/2*(-3)^(1/3)*log(x - (-3)^(1/3)) + x - 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 = 126, normalized size of antiderivative = 1.12 \[ \int \frac {x^6}{3+4 x^3+x^6} \, dx=x + \frac {\log {\left (x + 1 \right )}}{6} + \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x - \frac {121}{246} - \frac {121 \sqrt {3} i}{246} + \frac {864 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{4}}{41} \right )} + \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x - \frac {121}{246} + \frac {864 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{4}}{41} + \frac {121 \sqrt {3} i}{246} \right )} + \operatorname {RootSum} {\left (8 t^{3} + 3, \left ( t \mapsto t \log {\left (\frac {864 t^{4}}{41} + \frac {242 t}{41} + x \right )} \right )\right )} \]
x + log(x + 1)/6 + (-1/12 - sqrt(3)*I/12)*log(x - 121/246 - 121*sqrt(3)*I/ 246 + 864*(-1/12 - sqrt(3)*I/12)**4/41) + (-1/12 + sqrt(3)*I/12)*log(x - 1 21/246 + 864*(-1/12 + sqrt(3)*I/12)**4/41 + 121*sqrt(3)*I/246) + RootSum(8 *_t**3 + 3, Lambda(_t, _t*log(864*_t**4/41 + 242*_t/41 + x)))
Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.75 \[ \int \frac {x^6}{3+4 x^3+x^6} \, dx=-\frac {1}{2} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{2} \cdot 3^{\frac {1}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) + x - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]
-1/2*3^(5/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) + 1/6*sqrt(3)*arctan(1/3* sqrt(3)*(2*x - 1)) + 1/4*3^(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/2*3^(1 /3)*log(x + 3^(1/3)) + x - 1/12*log(x^2 - x + 1) + 1/6*log(x + 1)
Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.77 \[ \int \frac {x^6}{3+4 x^3+x^6} \, dx=-\frac {1}{2} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{4} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{2} \cdot 3^{\frac {1}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) + x - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]
-1/2*3^(5/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) + 1/6*sqrt(3)*arctan(1/3* sqrt(3)*(2*x - 1)) + 1/4*3^(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/2*3^(1 /3)*log(abs(x + 3^(1/3))) + x - 1/12*log(x^2 - x + 1) + 1/6*log(abs(x + 1) )
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.92 \[ \int \frac {x^6}{3+4 x^3+x^6} \, dx=x+\frac {\ln \left (x+1\right )}{6}-\frac {3^{1/3}\,\ln \left (x+3^{1/3}\right )}{2}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {3^{1/3}}{2}+\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{1/3}}{4}-\frac {3^{5/6}\,1{}\mathrm {i}}{4}\right )+\frac {{\left (-1\right )}^{1/3}\,3^{1/3}\,\ln \left (x-{\left (-1\right )}^{1/3}\,3^{1/3}\right )}{2} \]