Integrand size = 16, antiderivative size = 112 \[ \int \frac {x^3}{3+4 x^3+x^6} \, dx=\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{2 \sqrt [6]{3}}-\frac {1}{6} \log (1+x)+\frac {\log \left (\sqrt [3]{3}+x\right )}{2\ 3^{2/3}}+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{4\ 3^{2/3}} \]
-1/6*3^(5/6)*arctan(1/3*(3^(1/3)-2*x)*3^(1/6))-1/6*ln(1+x)+1/6*3^(1/3)*ln( 3^(1/3)+x)+1/12*ln(x^2-x+1)-1/12*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 = 106, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{3+4 x^3+x^6} \, dx=\frac {1}{12} \left (-2 3^{5/6} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )-2 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )-2 \log (1+x)+2 \sqrt [3]{3} \log \left (3+3^{2/3} x\right )+\log \left (1-x+x^2\right )-\sqrt [3]{3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )\right ) \]
(-2*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] + 2*3^(1/3)*Log[3 + 3^(2/3)*x] + Log[1 - x + x^2] - 3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12
Time = 0.32 (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, 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.
| \(\displaystyle \int \frac {x^3}{x^6+4 x^3+3} \, dx\) |
| \(\Big \downarrow \) 1710 |
| \(\displaystyle \frac {3}{2} \int \frac {1}{x^3+3}dx-\frac {1}{2} \int \frac {1}{x^3+1}dx\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{3} \int \frac {2-x}{x^2-x+1}dx-\frac {1}{3} \int \frac {1}{x+1}dx\right )+\frac {3}{2} \left (\frac {\int \frac {2 \sqrt [3]{3}-x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}+\frac {\int \frac {1}{x+\sqrt [3]{3}}dx}{3\ 3^{2/3}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{3} \int \frac {2-x}{x^2-x+1}dx-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {\int \frac {2 \sqrt [3]{3}-x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int -\frac {1-2 x}{x^2-x+1}dx-\frac {3}{2} \int \frac {1}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\frac {1}{2} \int -\frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx+3 \int \frac {1}{-\left (1-\frac {2 x}{\sqrt [3]{3}}\right )^2-3}d\left (1-\frac {2 x}{\sqrt [3]{3}}\right )}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+\frac {1}{2} \left (\frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {3}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+\frac {1}{2} \left (\frac {1}{3} \left (3 \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )-\frac {1}{3} \log (x+1)\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )\right )-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{2} \log \left (x^2-x+1\right )-\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )\right )-\frac {1}{3} \log (x+1)\right )+\frac {3}{2} \left (\frac {-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{3\ 3^{2/3}}+\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\) |
(-1/3*Log[1 + x] + (-(Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]]) + Log[1 - x + x^ 2]/2)/3)/2 + (3*(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
3.2.63.3.1 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_)), 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 = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (x +3 \textit {\_R} \right )\right )}{2}-\frac {\ln \left (x +1\right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x -\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}\) | \(54\) |
| default | \(-\frac {\ln \left (x +1\right )}{6}+\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{6}-\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{12}+\frac {3^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\) | \(84\) |
1/2*sum(_R*ln(x+3*_R),_R=RootOf(9*_Z^3-1))-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))
Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{3+4 x^3+x^6} \, dx=\frac {1}{6} \cdot 9^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {1}{27} \cdot 9^{\frac {1}{6}} {\left (2 \cdot 9^{\frac {2}{3}} \sqrt {3} x - 3 \cdot 9^{\frac {1}{3}} \sqrt {3}\right )}\right ) - \frac {1}{36} \cdot 9^{\frac {2}{3}} \log \left (3 \, x^{2} - 9^{\frac {2}{3}} x + 3 \cdot 9^{\frac {1}{3}}\right ) + \frac {1}{18} \cdot 9^{\frac {2}{3}} \log \left (3 \, x + 9^{\frac {2}{3}}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]
1/6*9^(1/6)*sqrt(3)*arctan(1/27*9^(1/6)*(2*9^(2/3)*sqrt(3)*x - 3*9^(1/3)*s qrt(3))) - 1/36*9^(2/3)*log(3*x^2 - 9^(2/3)*x + 3*9^(1/3)) + 1/18*9^(2/3)* log(3*x + 9^(2/3)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/12*log( x^2 - x + 1) - 1/6*log(x + 1)
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.98 \[ \int \frac {x^3}{3+4 x^3+x^6} \, dx=- \frac {\log {\left (x + 1 \right )}}{6} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x - \frac {1}{4} + 648 \left (\frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{4} + \frac {\sqrt {3} i}{4} \right )} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x - \frac {1}{4} + 648 \left (\frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{4} - \frac {\sqrt {3} i}{4} \right )} + \operatorname {RootSum} {\left (72 t^{3} - 1, \left ( t \mapsto t \log {\left (648 t^{4} - 3 t + x \right )} \right )\right )} \]
-log(x + 1)/6 + (1/12 - sqrt(3)*I/12)*log(x - 1/4 + 648*(1/12 - sqrt(3)*I/ 12)**4 + sqrt(3)*I/4) + (1/12 + sqrt(3)*I/12)*log(x - 1/4 + 648*(1/12 + sq rt(3)*I/12)**4 - sqrt(3)*I/4) + RootSum(72*_t**3 - 1, Lambda(_t, _t*log(64 8*_t**4 - 3*_t + x)))
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.75 \[ \int \frac {x^3}{3+4 x^3+x^6} \, dx=\frac {1}{6} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{12} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 3^{\frac {1}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) \]
1/6*3^(5/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) - 1/6*sqrt(3)*arctan(1/3*s qrt(3)*(2*x - 1)) - 1/12*3^(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/6*3^(1 /3)*log(x + 3^(1/3)) + 1/12*log(x^2 - x + 1) - 1/6*log(x + 1)
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77 \[ \int \frac {x^3}{3+4 x^3+x^6} \, dx=\frac {1}{6} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{12} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) + \frac {1}{6} \cdot 3^{\frac {1}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]
1/6*3^(5/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) - 1/6*sqrt(3)*arctan(1/3*s qrt(3)*(2*x - 1)) - 1/12*3^(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/6*3^(1 /3)*log(abs(x + 3^(1/3))) + 1/12*log(x^2 - x + 1) - 1/6*log(abs(x + 1))
Time = 8.37 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{3+4 x^3+x^6} \, dx=\frac {3^{1/3}\,\ln \left (x+3^{1/3}\right )}{6}-\frac {\ln \left (x+1\right )}{6}+\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {3^{1/3}}{2}-\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{1/3}}{12}+\frac {3^{5/6}\,1{}\mathrm {i}}{12}\right )-\ln \left (x-\frac {3^{1/3}}{2}+\frac {3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3^{1/3}}{12}-\frac {3^{5/6}\,1{}\mathrm {i}}{12}\right ) \]
(3^(1/3)*log(x + 3^(1/3)))/6 - 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) - log(x - 3^(1/3)/2 - (3^(5/6)*1i)/2)*(3^(1/3)/12 + (3^(5/6)*1i) /12) - log(x - 3^(1/3)/2 + (3^(5/6)*1i)/2)*(3^(1/3)/12 - (3^(5/6)*1i)/12)