Integrand size = 16, antiderivative size = 126 \[ \int \frac {1}{x^6 \left (3+4 x^3+x^6\right )} \, dx=-\frac {1}{15 x^5}+\frac {2}{9 x^2}-\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{54 \sqrt [6]{3}}+\frac {1}{6} \log (1+x)-\frac {\log \left (\sqrt [3]{3}+x\right )}{54\ 3^{2/3}}-\frac {1}{12} \log \left (1-x+x^2\right )+\frac {\log \left (3^{2/3}-\sqrt [3]{3} x+x^2\right )}{108\ 3^{2/3}} \]
-1/15/x^5+2/9/x^2+1/162*3^(5/6)*arctan(1/3*(3^(1/3)-2*x)*3^(1/6))+1/6*ln(1 +x)-1/162*3^(1/3)*ln(3^(1/3)+x)-1/12*ln(x^2-x+1)+1/324*3^(1/3)*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 = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^6 \left (3+4 x^3+x^6\right )} \, dx=\frac {-\frac {108}{x^5}+\frac {360}{x^2}+10\ 3^{5/6} \arctan \left (\frac {\sqrt [3]{3}-2 x}{3^{5/6}}\right )+270 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )+270 \log (1+x)-10 \sqrt [3]{3} \log \left (3+3^{2/3} x\right )-135 \log \left (1-x+x^2\right )+5 \sqrt [3]{3} \log \left (3-3^{2/3} x+\sqrt [3]{3} x^2\right )}{1620} \]
(-108/x^5 + 360/x^2 + 10*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 270*Sqr t[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 270*Log[1 + x] - 10*3^(1/3)*Log[3 + 3^(2 /3)*x] - 135*Log[1 - x + x^2] + 5*3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2] )/1620
Time = 0.39 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.21, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {1704, 27, 1828, 27, 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.
| \(\displaystyle \int \frac {1}{x^6 \left (x^6+4 x^3+3\right )} \, dx\) |
| \(\Big \downarrow \) 1704 |
| \(\displaystyle \frac {1}{15} \int -\frac {5 \left (x^3+4\right )}{x^3 \left (x^6+4 x^3+3\right )}dx-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{3} \int \frac {x^3+4}{x^3 \left (x^6+4 x^3+3\right )}dx-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 1828 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{6} \int \frac {2 \left (4 x^3+13\right )}{x^6+4 x^3+3}dx+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \int \frac {4 x^3+13}{x^6+4 x^3+3}dx+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 1752 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \int \frac {1}{x^3+1}dx-\frac {1}{2} \int \frac {1}{x^3+3}dx\right )+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \int \frac {2-x}{x^2-x+1}dx+\frac {1}{3} \int \frac {1}{x+1}dx\right )+\frac {1}{2} \left (-\frac {\int \frac {2 \sqrt [3]{3}-x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}-\frac {\int \frac {1}{x+\sqrt [3]{3}}dx}{3\ 3^{2/3}}\right )\right )+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \int \frac {2-x}{x^2-x+1}dx+\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (-\frac {\int \frac {2 \sqrt [3]{3}-x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\right )+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx-\frac {1}{2} \int -\frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (-\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\frac {1}{2} \int -\frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\right )+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx+\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (-\frac {\frac {3}{2} \sqrt [3]{3} \int \frac {1}{x^2-\sqrt [3]{3} x+3^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx}{3\ 3^{2/3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\right )+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx+\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (-\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx+3 \int \frac {1}{-\left (1-\frac {2 x}{\sqrt [3]{3}}\right )^2-3}d\left (1-\frac {2 x}{\sqrt [3]{3}}\right )}{3\ 3^{2/3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\right )+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3\ 3^{2/3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+\frac {9}{2} \left (\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{x^2-x+1}dx+\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx\right )+\frac {1}{3} \log (x+1)\right )\right )+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3\ 3^{2/3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )+\frac {9}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx-3 \int \frac {1}{-(2 x-1)^2-3}d(2 x-1)\right )+\frac {1}{3} \log (x+1)\right )\right )+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {1-2 x}{x^2-x+1}dx+\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )\right )+\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (-\frac {\frac {1}{2} \int \frac {\sqrt [3]{3}-2 x}{x^2-\sqrt [3]{3} x+3^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )}{3\ 3^{2/3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\right )+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {9}{2} \left (\frac {1}{3} \left (\sqrt {3} \arctan \left (\frac {2 x-1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (x^2-x+1\right )\right )+\frac {1}{3} \log (x+1)\right )+\frac {1}{2} \left (-\frac {-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{3}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{3\ 3^{2/3}}-\frac {\log \left (x+\sqrt [3]{3}\right )}{3\ 3^{2/3}}\right )\right )+\frac {2}{3 x^2}\right )-\frac {1}{15 x^5}\) |
-1/15*1/x^5 + (2/(3*x^2) + ((9*(Log[1 + x]/3 + (Sqrt[3]*ArcTan[(-1 + 2*x)/ Sqrt[3]] - Log[1 - x + x^2]/2)/3))/2 + (-1/3*Log[3^(1/3) + x]/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)/3
3.2.69.3.1 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[((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*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*d*(m + 1) )), x] - Simp[1/(a*d^n*(m + 1)) Int[(d*x)^(m + n)*(b*(m + n*(p + 1) + 1) + c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{ a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && 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])
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^n + c*x^ (2*n))^(p + 1)/(a*f*(m + 1))), x] + Simp[1/(a*f^n*(m + 1)) Int[(f*x)^(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(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] && LtQ[m, -1] && Int egerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(\frac {\frac {2 x^{3}}{9}-\frac {1}{15}}{x^{5}}+\frac {\ln \left (x +1\right )}{6}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left (x -3 \textit {\_R} \right )\right )}{54}-\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x -\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}\) | \(65\) |
| default | \(-\frac {1}{15 x^{5}}+\frac {2}{9 x^{2}}+\frac {\ln \left (x +1\right )}{6}-\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {1}{3}}+x \right )}{162}+\frac {3^{\frac {1}{3}} \ln \left (3^{\frac {2}{3}}-3^{\frac {1}{3}} x +x^{2}\right )}{324}-\frac {3^{\frac {5}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3}-1\right )}{3}\right )}{162}-\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\) | \(94\) |
(2/9*x^3-1/15)/x^5+1/6*ln(x+1)+1/54*sum(_R*ln(x-3*_R),_R=RootOf(9*_Z^3+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 = 153, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^6 \left (3+4 x^3+x^6\right )} \, dx=\frac {30 \cdot 9^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {1}{27} \cdot 9^{\frac {1}{6}} {\left (2 \cdot 9^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} x - 3 \cdot 9^{\frac {1}{3}} \sqrt {3}\right )}\right ) - 5 \cdot 9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x + 3 \, x^{2} + 3 \cdot 9^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}}\right ) + 10 \cdot 9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-9^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 3 \, x\right ) + 810 \, \sqrt {3} x^{5} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 405 \, x^{5} \log \left (x^{2} - x + 1\right ) + 810 \, x^{5} \log \left (x + 1\right ) + 1080 \, x^{3} - 324}{4860 \, x^{5}} \]
1/4860*(30*9^(1/6)*sqrt(3)*(-1)^(1/3)*x^5*arctan(1/27*9^(1/6)*(2*9^(2/3)*s qrt(3)*(-1)^(2/3)*x - 3*9^(1/3)*sqrt(3))) - 5*9^(2/3)*(-1)^(1/3)*x^5*log(9 ^(2/3)*(-1)^(1/3)*x + 3*x^2 + 3*9^(1/3)*(-1)^(2/3)) + 10*9^(2/3)*(-1)^(1/3 )*x^5*log(-9^(2/3)*(-1)^(1/3) + 3*x) + 810*sqrt(3)*x^5*arctan(1/3*sqrt(3)* (2*x - 1)) - 405*x^5*log(x^2 - x + 1) + 810*x^5*log(x + 1) + 1080*x^3 - 32 4)/x^5
Result contains complex when optimal does not.
Time = 1.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^6 \left (3+4 x^3+x^6\right )} \, dx=\frac {\log {\left (x + 1 \right )}}{6} + \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {88573}{6562} - \frac {88573 \sqrt {3} i}{6562} + \frac {119042784 \left (- \frac {1}{12} + \frac {\sqrt {3} i}{12}\right )^{4}}{3281} \right )} + \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right ) \log {\left (x + \frac {88573}{6562} + \frac {119042784 \left (- \frac {1}{12} - \frac {\sqrt {3} i}{12}\right )^{4}}{3281} + \frac {88573 \sqrt {3} i}{6562} \right )} + \operatorname {RootSum} {\left (1417176 t^{3} + 1, \left ( t \mapsto t \log {\left (\frac {119042784 t^{4}}{3281} - \frac {531438 t}{3281} + x \right )} \right )\right )} + \frac {10 x^{3} - 3}{45 x^{5}} \]
log(x + 1)/6 + (-1/12 + sqrt(3)*I/12)*log(x + 88573/6562 - 88573*sqrt(3)*I /6562 + 119042784*(-1/12 + sqrt(3)*I/12)**4/3281) + (-1/12 - sqrt(3)*I/12) *log(x + 88573/6562 + 119042784*(-1/12 - sqrt(3)*I/12)**4/3281 + 88573*sqr t(3)*I/6562) + RootSum(1417176*_t**3 + 1, Lambda(_t, _t*log(119042784*_t** 4/3281 - 531438*_t/3281 + x))) + (10*x**3 - 3)/(45*x**5)
Time = 0.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^6 \left (3+4 x^3+x^6\right )} \, dx=-\frac {1}{162} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{324} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{162} \cdot 3^{\frac {1}{3}} \log \left (x + 3^{\frac {1}{3}}\right ) + \frac {10 \, x^{3} - 3}{45 \, x^{5}} - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) \]
-1/162*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)) + 1/324*3^(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/16 2*3^(1/3)*log(x + 3^(1/3)) + 1/45*(10*x^3 - 3)/x^5 - 1/12*log(x^2 - x + 1) + 1/6*log(x + 1)
Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^6 \left (3+4 x^3+x^6\right )} \, dx=-\frac {1}{162} \cdot 3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (2 \, x - 3^{\frac {1}{3}}\right )}\right ) + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{324} \cdot 3^{\frac {1}{3}} \log \left (x^{2} - 3^{\frac {1}{3}} x + 3^{\frac {2}{3}}\right ) - \frac {1}{162} \cdot 3^{\frac {1}{3}} \log \left ({\left | x + 3^{\frac {1}{3}} \right |}\right ) + \frac {10 \, x^{3} - 3}{45 \, x^{5}} - \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) \]
-1/162*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)) + 1/324*3^(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/16 2*3^(1/3)*log(abs(x + 3^(1/3))) + 1/45*(10*x^3 - 3)/x^5 - 1/12*log(x^2 - x + 1) + 1/6*log(abs(x + 1))
Time = 8.50 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^6 \left (3+4 x^3+x^6\right )} \, dx=\frac {\ln \left (x+1\right )}{6}-\frac {3^{1/3}\,\ln \left (x+3^{1/3}\right )}{162}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\frac {\frac {2\,x^3}{9}-\frac {1}{15}}{x^5}+\frac {{\left (-1\right )}^{1/3}\,3^{1/3}\,\ln \left (x-{\left (-1\right )}^{1/3}\,3^{1/3}\right )}{162}-\frac {{\left (-1\right )}^{1/3}\,\ln \left (x+\frac {{\left (-1\right )}^{1/3}\,3^{1/3}}{2}+\frac {{\left (-1\right )}^{1/3}\,3^{5/6}\,1{}\mathrm {i}}{2}\right )\,\left (3^{1/3}+3^{5/6}\,1{}\mathrm {i}\right )}{324} \]
log(x + 1)/6 - (3^(1/3)*log(x + 3^(1/3)))/162 - 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^3)/9 - 1/15)/x^5 + ((-1)^(1/3)*3^(1/3)*log(x - (-1)^(1 /3)*3^(1/3)))/162 - ((-1)^(1/3)*log(x + ((-1)^(1/3)*3^(1/3))/2 + ((-1)^(1/ 3)*3^(5/6)*1i)/2)*(3^(1/3) + 3^(5/6)*1i))/324