Integrand size = 14, antiderivative size = 399 \[ \int \frac {x^3}{2+x^3+x^6} \, dx=-\frac {i \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {i \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21}}+\frac {\left (7+i \sqrt {7}\right ) \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}+\frac {\left (7-i \sqrt {7}\right ) \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{21 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}}-\frac {\left (7+i \sqrt {7}\right ) \log \left (\left (1-i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {\left (7-i \sqrt {7}\right ) \log \left (\left (1+i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{42 \sqrt [3]{2} \left (1+i \sqrt {7}\right )^{2/3}} \]
1/42*ln(2^(1/3)*x+(1+I*7^(1/2))^(1/3))*(7-I*7^(1/2))*2^(2/3)/(1+I*7^(1/2)) ^(2/3)-1/84*ln(2^(2/3)*x^2-2^(1/3)*x*(1+I*7^(1/2))^(1/3)+(1+I*7^(1/2))^(2/ 3))*(7-I*7^(1/2))*2^(2/3)/(1+I*7^(1/2))^(2/3)+1/42*ln(2^(1/3)*x+(1-I*7^(1/ 2))^(1/3))*(7+I*7^(1/2))*2^(2/3)/(1-I*7^(1/2))^(2/3)-1/84*ln(2^(2/3)*x^2-2 ^(1/3)*x*(1-I*7^(1/2))^(1/3)+(1-I*7^(1/2))^(2/3))*(7+I*7^(1/2))*2^(2/3)/(1 -I*7^(1/2))^(2/3)-1/42*I*arctan(1/3*(1-2*2^(1/3)*x/(1-I*7^(1/2))^(1/3))*3^ (1/2))*(1-I*7^(1/2))^(1/3)*2^(2/3)*21^(1/2)+1/42*I*arctan(1/3*(1-2*2^(1/3) *x/(1+I*7^(1/2))^(1/3))*3^(1/2))*(1+I*7^(1/2))^(1/3)*2^(2/3)*21^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.09 \[ \int \frac {x^3}{2+x^3+x^6} \, dx=\frac {1}{3} \text {RootSum}\left [2+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{1+2 \text {$\#$1}^3}\&\right ] \]
Time = 0.52 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.88, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1710, 750, 16, 1142, 25, 1082, 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.
| \(\displaystyle \int \frac {x^3}{x^6+x^3+2} \, dx\) |
| \(\Big \downarrow \) 1710 |
| \(\displaystyle \frac {1}{14} \left (7+i \sqrt {7}\right ) \int \frac {1}{x^3+\frac {1}{2} \left (1-i \sqrt {7}\right )}dx+\frac {1}{14} \left (7-i \sqrt {7}\right ) \int \frac {1}{x^3+\frac {1}{2} \left (1+i \sqrt {7}\right )}dx\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{14} \left (7+i \sqrt {7}\right ) \left (\frac {\int \frac {2^{2/3} \sqrt [3]{1-i \sqrt {7}}-x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\int \frac {1}{x+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )+\frac {1}{14} \left (7-i \sqrt {7}\right ) \left (\frac {\int \frac {2^{2/3} \sqrt [3]{1+i \sqrt {7}}-x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\int \frac {1}{x+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{14} \left (7+i \sqrt {7}\right ) \left (\frac {\int \frac {2^{2/3} \sqrt [3]{1-i \sqrt {7}}-x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )+\frac {1}{14} \left (7-i \sqrt {7}\right ) \left (\frac {\int \frac {2^{2/3} \sqrt [3]{1+i \sqrt {7}}-x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{14} \left (7+i \sqrt {7}\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \int \frac {1}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx-\frac {1}{2} \int -\frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )+\frac {1}{14} \left (7-i \sqrt {7}\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \int \frac {1}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx-\frac {1}{2} \int -\frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{14} \left (7+i \sqrt {7}\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \int \frac {1}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )+\frac {1}{14} \left (7-i \sqrt {7}\right ) \left (\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \int \frac {1}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{14} \left (7+i \sqrt {7}\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx+3 \int \frac {1}{-\left (1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )^2-3}d\left (1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )+\frac {1}{14} \left (7-i \sqrt {7}\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx+3 \int \frac {1}{-\left (1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )^2-3}d\left (1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{14} \left (7+i \sqrt {7}\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )+\frac {1}{14} \left (7-i \sqrt {7}\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{14} \left (7+i \sqrt {7}\right ) \left (\frac {-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+\left (1-i \sqrt {7}\right )^{2/3}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )+\frac {1}{14} \left (7-i \sqrt {7}\right ) \left (\frac {-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+\left (1+i \sqrt {7}\right )^{2/3}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )\) |
((7 + I*Sqrt[7])*(Log[(1 - I*Sqrt[7])^(1/3) + 2^(1/3)*x]/(3*((1 - I*Sqrt[7 ])/2)^(2/3)) + (-(Sqrt[3]*ArcTan[(1 - (2*x)/((1 - I*Sqrt[7])/2)^(1/3))/Sqr t[3]]) - Log[(1 - I*Sqrt[7])^(2/3) - (2*(1 - I*Sqrt[7]))^(1/3)*x + 2^(2/3) *x^2]/2)/(3*((1 - I*Sqrt[7])/2)^(2/3))))/14 + ((7 - I*Sqrt[7])*(Log[(1 + I *Sqrt[7])^(1/3) + 2^(1/3)*x]/(3*((1 + I*Sqrt[7])/2)^(2/3)) + (-(Sqrt[3]*Ar cTan[(1 - (2*x)/((1 + I*Sqrt[7])/2)^(1/3))/Sqrt[3]]) - Log[(1 + I*Sqrt[7]) ^(2/3) - (2*(1 + I*Sqrt[7]))^(1/3)*x + 2^(2/3)*x^2]/2)/(3*((1 + I*Sqrt[7]) /2)^(2/3))))/14
3.2.84.3.1 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[((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 = 36, normalized size of antiderivative = 0.09
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{3}\) | \(36\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{3}\) | \(36\) |
Time = 0.26 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.65 \[ \int \frac {x^3}{2+x^3+x^6} \, dx=-\frac {1}{588} \cdot 98^{\frac {2}{3}} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (98^{\frac {2}{3}} \sqrt {7} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (i \, \sqrt {-3} + i\right )} + 196 \, x\right ) + \frac {1}{588} \cdot 98^{\frac {2}{3}} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (98^{\frac {2}{3}} \sqrt {7} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (i \, \sqrt {-3} - i\right )} + 196 \, x\right ) + \frac {1}{588} \cdot 98^{\frac {2}{3}} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (98^{\frac {2}{3}} \sqrt {7} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (-i \, \sqrt {-3} + i\right )} + 196 \, x\right ) - \frac {1}{588} \cdot 98^{\frac {2}{3}} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (98^{\frac {2}{3}} \sqrt {7} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (-i \, \sqrt {-3} - i\right )} + 196 \, x\right ) + \frac {1}{294} \cdot 98^{\frac {2}{3}} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} \log \left (i \cdot 98^{\frac {2}{3}} \sqrt {7} {\left (i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} + 98 \, x\right ) + \frac {1}{294} \cdot 98^{\frac {2}{3}} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} \log \left (-i \cdot 98^{\frac {2}{3}} \sqrt {7} {\left (-i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} + 98 \, x\right ) \]
-1/588*98^(2/3)*(-I*sqrt(7) - 7)^(1/3)*(sqrt(-3) + 1)*log(98^(2/3)*sqrt(7) *(-I*sqrt(7) - 7)^(1/3)*(I*sqrt(-3) + I) + 196*x) + 1/588*98^(2/3)*(I*sqrt (7) - 7)^(1/3)*(sqrt(-3) - 1)*log(98^(2/3)*sqrt(7)*(I*sqrt(7) - 7)^(1/3)*( I*sqrt(-3) - I) + 196*x) + 1/588*98^(2/3)*(-I*sqrt(7) - 7)^(1/3)*(sqrt(-3) - 1)*log(98^(2/3)*sqrt(7)*(-I*sqrt(7) - 7)^(1/3)*(-I*sqrt(-3) + I) + 196* x) - 1/588*98^(2/3)*(I*sqrt(7) - 7)^(1/3)*(sqrt(-3) + 1)*log(98^(2/3)*sqrt (7)*(I*sqrt(7) - 7)^(1/3)*(-I*sqrt(-3) - I) + 196*x) + 1/294*98^(2/3)*(I*s qrt(7) - 7)^(1/3)*log(I*98^(2/3)*sqrt(7)*(I*sqrt(7) - 7)^(1/3) + 98*x) + 1 /294*98^(2/3)*(-I*sqrt(7) - 7)^(1/3)*log(-I*98^(2/3)*sqrt(7)*(-I*sqrt(7) - 7)^(1/3) + 98*x)
Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.06 \[ \int \frac {x^3}{2+x^3+x^6} \, dx=\operatorname {RootSum} {\left (250047 t^{6} + 1323 t^{3} + 2, \left ( t \mapsto t \log {\left (7938 t^{4} + 21 t + x \right )} \right )\right )} \]
\[ \int \frac {x^3}{2+x^3+x^6} \, dx=\int { \frac {x^{3}}{x^{6} + x^{3} + 2} \,d x } \]
Exception generated. \[ \int \frac {x^3}{2+x^3+x^6} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument ValueDone
Time = 9.29 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{2+x^3+x^6} \, dx=\frac {\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{14}\right )\,{\left (-196-\sqrt {7}\,28{}\mathrm {i}\right )}^{1/3}}{42}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{14}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{42}-\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}-\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x+\frac {2^{2/3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}+\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}+\frac {2^{2/3}\,7^{1/3}\,\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}-\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84}-\frac {2^{2/3}\,7^{1/3}\,\ln \left (x-\frac {2^{2/3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}+\frac {2^{2/3}\,\sqrt {3}\,7^{5/6}\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{28}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,1{}\mathrm {i}\right )}^{1/3}}{84} \]
(log(x - (2^(2/3)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3)*1i)/14)*(- 7^(1/2)*28i - 196)^(1/3))/42 + (2^(2/3)*7^(1/3)*log(x + (2^(2/3)*7^(5/6)*(7^(1/2)*1i - 7)^(1/3)*1i)/14)*(7^(1/2)*1i - 7)^(1/3))/42 - (2^(2/3)*7^(1/3)*log(x + (2 ^(2/3)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3)*1i)/28 - (2^(2/3)*3^(1/2)*7^(5/6)* (- 7^(1/2)*1i - 7)^(1/3))/28)*(3^(1/2)*1i + 1)*(- 7^(1/2)*1i - 7)^(1/3))/8 4 + (2^(2/3)*7^(1/3)*log(x + (2^(2/3)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3)*1i) /28 + (2^(2/3)*3^(1/2)*7^(5/6)*(- 7^(1/2)*1i - 7)^(1/3))/28)*(3^(1/2)*1i - 1)*(- 7^(1/2)*1i - 7)^(1/3))/84 + (2^(2/3)*7^(1/3)*log(x - (2^(2/3)*7^(5/ 6)*(7^(1/2)*1i - 7)^(1/3)*1i)/28 - (2^(2/3)*3^(1/2)*7^(5/6)*(7^(1/2)*1i - 7)^(1/3))/28)*(3^(1/2)*1i - 1)*(7^(1/2)*1i - 7)^(1/3))/84 - (2^(2/3)*7^(1/ 3)*log(x - (2^(2/3)*7^(5/6)*(7^(1/2)*1i - 7)^(1/3)*1i)/28 + (2^(2/3)*3^(1/ 2)*7^(5/6)*(7^(1/2)*1i - 7)^(1/3))/28)*(3^(1/2)*1i + 1)*(7^(1/2)*1i - 7)^( 1/3))/84