Integrand size = 10, antiderivative size = 381 \[ \int \frac {1}{2+x^3+x^6} \, dx=\frac {i \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21} \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}-\frac {i \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{\sqrt {21} \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}-\frac {i \log \left (\sqrt [3]{1-i \sqrt {7}}+\sqrt [3]{2} x\right )}{3 \sqrt {7} \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {i \log \left (\sqrt [3]{1+i \sqrt {7}}+\sqrt [3]{2} x\right )}{3 \sqrt {7} \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {i \log \left (\left (1-i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {7} \left (1-i \sqrt {7}\right )^{2/3}}-\frac {i \log \left (\left (1+i \sqrt {7}\right )^{2/3}-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+2^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt {7} \left (1+i \sqrt {7}\right )^{2/3}} \]
-1/21*I*2^(2/3)*ln(2^(1/3)*x+(1-I*7^(1/2))^(1/3))/(1-I*7^(1/2))^(2/3)*7^(1 /2)+1/42*I*ln(2^(2/3)*x^2-2^(1/3)*x*(1-I*7^(1/2))^(1/3)+(1-I*7^(1/2))^(2/3 ))*2^(2/3)/(1-I*7^(1/2))^(2/3)*7^(1/2)+1/21*I*2^(2/3)*ln(2^(1/3)*x+(1+I*7^ (1/2))^(1/3))/(1+I*7^(1/2))^(2/3)*7^(1/2)-1/42*I*ln(2^(2/3)*x^2-2^(1/3)*x* (1+I*7^(1/2))^(1/3)+(1+I*7^(1/2))^(2/3))*2^(2/3)/(1+I*7^(1/2))^(2/3)*7^(1/ 2)+1/21*I*2^(2/3)*arctan(1/3*(1-2*2^(1/3)*x/(1-I*7^(1/2))^(1/3))*3^(1/2))/ (1-I*7^(1/2))^(2/3)*21^(1/2)-1/21*I*2^(2/3)*arctan(1/3*(1-2*2^(1/3)*x/(1+I *7^(1/2))^(1/3))*3^(1/2))/(1+I*7^(1/2))^(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 = 38, normalized size of antiderivative = 0.10 \[ \int \frac {1}{2+x^3+x^6} \, dx=\frac {1}{3} \text {RootSum}\left [2+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{\text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \]
Time = 0.60 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.89, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {1685, 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.
| \(\displaystyle \int \frac {1}{x^6+x^3+2} \, dx\) |
| \(\Big \downarrow \) 1685 |
| \(\displaystyle \frac {i \int \frac {1}{x^3+\frac {1}{2} \left (1+i \sqrt {7}\right )}dx}{\sqrt {7}}-\frac {i \int \frac {1}{x^3+\frac {1}{2} \left (1-i \sqrt {7}\right )}dx}{\sqrt {7}}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {i \left (\frac {\int \frac {2^{2/3} \sqrt [3]{1+i \sqrt {7}}-x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\int \frac {1}{x+\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}-\frac {i \left (\frac {\int \frac {2^{2/3} \sqrt [3]{1-i \sqrt {7}}-x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\int \frac {1}{x+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {i \left (\frac {\int \frac {2^{2/3} \sqrt [3]{1+i \sqrt {7}}-x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}-\frac {i \left (\frac {\int \frac {2^{2/3} \sqrt [3]{1-i \sqrt {7}}-x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {i \left (\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \int \frac {1}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx-\frac {1}{2} \int -\frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}-\frac {i \left (\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \int \frac {1}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx-\frac {1}{2} \int -\frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i \left (\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} \int \frac {1}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}-\frac {i \left (\frac {\frac {3}{2} \sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} \int \frac {1}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {i \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx+3 \int \frac {1}{-\left (1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )^2-3}d\left (1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}-\frac {i \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx+3 \int \frac {1}{-\left (1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )^2-3}d\left (1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {i \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}-\frac {i \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}-2 x}{x^2-\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )} x+\left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {i \left (\frac {-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1+i \sqrt {7}\right )}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1+i \sqrt {7}\right )} x+\left (1+i \sqrt {7}\right )^{2/3}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1+i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}-\frac {i \left (\frac {-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {7}\right )}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (2^{2/3} x^2-\sqrt [3]{2 \left (1-i \sqrt {7}\right )} x+\left (1-i \sqrt {7}\right )^{2/3}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}+\frac {\log \left (\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt {7}}\right )}{3 \left (\frac {1}{2} \left (1-i \sqrt {7}\right )\right )^{2/3}}\right )}{\sqrt {7}}\) |
((-I)*(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))/Sqrt[3]]) - Lo g[(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))))/Sqrt[7] + (I*(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))/Sqrt[3]]) - Log[(1 + I*Sqrt[7])^(2/3) - (2*(1 + I*S qrt[7]))^(1/3)*x + 2^(2/3)*x^2]/2)/(3*((1 + I*Sqrt[7])/2)^(2/3))))/Sqrt[7]
3.2.82.3.1 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[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^n), x], x] - Simp[ c/q Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.09
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{3}\) | \(33\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}+2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{3}\) | \(33\) |
Time = 0.28 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.78 \[ \int \frac {1}{2+x^3+x^6} \, dx=-\frac {1}{588} \cdot 49^{\frac {2}{3}} {\left (3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (49^{\frac {2}{3}} {\left (\sqrt {7} {\left (i \, \sqrt {-3} + i\right )} - 7 \, \sqrt {-3} - 7\right )} {\left (3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} + 392 \, x\right ) + \frac {1}{588} \cdot 49^{\frac {2}{3}} {\left (3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (49^{\frac {2}{3}} {\left (\sqrt {7} {\left (-i \, \sqrt {-3} + i\right )} + 7 \, \sqrt {-3} - 7\right )} {\left (3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} + 392 \, x\right ) + \frac {1}{588} \cdot 49^{\frac {2}{3}} {\left (-3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (49^{\frac {2}{3}} {\left (\sqrt {7} {\left (i \, \sqrt {-3} - i\right )} + 7 \, \sqrt {-3} - 7\right )} {\left (-3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} + 392 \, x\right ) - \frac {1}{588} \cdot 49^{\frac {2}{3}} {\left (-3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (49^{\frac {2}{3}} {\left (\sqrt {7} {\left (-i \, \sqrt {-3} - i\right )} - 7 \, \sqrt {-3} - 7\right )} {\left (-3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} + 392 \, x\right ) + \frac {1}{294} \cdot 49^{\frac {2}{3}} {\left (3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} \log \left (49^{\frac {2}{3}} {\left (3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} {\left (-i \, \sqrt {7} + 7\right )} + 196 \, x\right ) + \frac {1}{294} \cdot 49^{\frac {2}{3}} {\left (-3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} \log \left (49^{\frac {2}{3}} {\left (i \, \sqrt {7} + 7\right )} {\left (-3 i \, \sqrt {7} - 7\right )}^{\frac {1}{3}} + 196 \, x\right ) \]
-1/588*49^(2/3)*(3*I*sqrt(7) - 7)^(1/3)*(sqrt(-3) + 1)*log(49^(2/3)*(sqrt( 7)*(I*sqrt(-3) + I) - 7*sqrt(-3) - 7)*(3*I*sqrt(7) - 7)^(1/3) + 392*x) + 1 /588*49^(2/3)*(3*I*sqrt(7) - 7)^(1/3)*(sqrt(-3) - 1)*log(49^(2/3)*(sqrt(7) *(-I*sqrt(-3) + I) + 7*sqrt(-3) - 7)*(3*I*sqrt(7) - 7)^(1/3) + 392*x) + 1/ 588*49^(2/3)*(-3*I*sqrt(7) - 7)^(1/3)*(sqrt(-3) - 1)*log(49^(2/3)*(sqrt(7) *(I*sqrt(-3) - I) + 7*sqrt(-3) - 7)*(-3*I*sqrt(7) - 7)^(1/3) + 392*x) - 1/ 588*49^(2/3)*(-3*I*sqrt(7) - 7)^(1/3)*(sqrt(-3) + 1)*log(49^(2/3)*(sqrt(7) *(-I*sqrt(-3) - I) - 7*sqrt(-3) - 7)*(-3*I*sqrt(7) - 7)^(1/3) + 392*x) + 1 /294*49^(2/3)*(3*I*sqrt(7) - 7)^(1/3)*log(49^(2/3)*(3*I*sqrt(7) - 7)^(1/3) *(-I*sqrt(7) + 7) + 196*x) + 1/294*49^(2/3)*(-3*I*sqrt(7) - 7)^(1/3)*log(4 9^(2/3)*(I*sqrt(7) + 7)*(-3*I*sqrt(7) - 7)^(1/3) + 196*x)
Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.06 \[ \int \frac {1}{2+x^3+x^6} \, dx=\operatorname {RootSum} {\left (1000188 t^{6} + 1323 t^{3} + 1, \left ( t \mapsto t \log {\left (- 5292 t^{4} + 7 t + x \right )} \right )\right )} \]
\[ \int \frac {1}{2+x^3+x^6} \, dx=\int { \frac {1}{x^{6} + x^{3} + 2} \,d x } \]
Exception generated. \[ \int \frac {1}{2+x^3+x^6} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument Valuein tegrate(1/(sageVARx^6+sageVARx^3+2),sageVARx)
Time = 9.36 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.35 \[ \int \frac {1}{2+x^3+x^6} \, dx=\frac {\ln \left (x+\frac {7^{1/3}\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{4}+\frac {7^{5/6}\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}\right )\,{\left (-49-\sqrt {7}\,21{}\mathrm {i}\right )}^{1/3}}{42}+\frac {\ln \left (x+\frac {7^{1/3}\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{4}-\frac {7^{5/6}\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,1{}\mathrm {i}}{28}\right )\,{\left (-49+\sqrt {7}\,21{}\mathrm {i}\right )}^{1/3}}{42}+\frac {7^{1/3}\,\ln \left (6\,x+\frac {7^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,\left (\frac {7^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{2/3}\,\left (3969\,x+\frac {567\,7^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{2}\right )}{7056}+63\right )}{84}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{84}+\frac {7^{1/3}\,\ln \left (6\,x+\frac {7^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,\left (\frac {7^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{2/3}\,\left (3969\,x+\frac {567\,7^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{2}\right )}{7056}+63\right )}{84}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{84}-\frac {7^{1/3}\,\ln \left (6\,x-\frac {7^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,\left (\frac {7^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{2/3}\,\left (3969\,x-\frac {567\,7^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{2}\right )}{7056}+63\right )}{84}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7-\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{84}-\frac {7^{1/3}\,\ln \left (6\,x-\frac {7^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}\,\left (\frac {7^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{2/3}\,\left (3969\,x-\frac {567\,7^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{2}\right )}{7056}+63\right )}{84}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-7+\sqrt {7}\,3{}\mathrm {i}\right )}^{1/3}}{84} \]
(log(x + (7^(1/3)*(- 7^(1/2)*3i - 7)^(1/3))/4 + (7^(5/6)*(- 7^(1/2)*3i - 7 )^(1/3)*1i)/28)*(- 7^(1/2)*21i - 49)^(1/3))/42 + (log(x + (7^(1/3)*(7^(1/2 )*3i - 7)^(1/3))/4 - (7^(5/6)*(7^(1/2)*3i - 7)^(1/3)*1i)/28)*(7^(1/2)*21i - 49)^(1/3))/42 + (7^(1/3)*log(6*x + (7^(1/3)*(3^(1/2)*1i - 1)*(- 7^(1/2)* 3i - 7)^(1/3)*((7^(2/3)*(3^(1/2)*1i - 1)^2*(- 7^(1/2)*3i - 7)^(2/3)*(3969* x + (567*7^(1/3)*(3^(1/2)*1i - 1)*(- 7^(1/2)*3i - 7)^(1/3))/2))/7056 + 63) )/84)*(3^(1/2)*1i - 1)*(- 7^(1/2)*3i - 7)^(1/3))/84 + (7^(1/3)*log(6*x + ( 7^(1/3)*(3^(1/2)*1i - 1)*(7^(1/2)*3i - 7)^(1/3)*((7^(2/3)*(3^(1/2)*1i - 1) ^2*(7^(1/2)*3i - 7)^(2/3)*(3969*x + (567*7^(1/3)*(3^(1/2)*1i - 1)*(7^(1/2) *3i - 7)^(1/3))/2))/7056 + 63))/84)*(3^(1/2)*1i - 1)*(7^(1/2)*3i - 7)^(1/3 ))/84 - (7^(1/3)*log(6*x - (7^(1/3)*(3^(1/2)*1i + 1)*(- 7^(1/2)*3i - 7)^(1 /3)*((7^(2/3)*(3^(1/2)*1i + 1)^2*(- 7^(1/2)*3i - 7)^(2/3)*(3969*x - (567*7 ^(1/3)*(3^(1/2)*1i + 1)*(- 7^(1/2)*3i - 7)^(1/3))/2))/7056 + 63))/84)*(3^( 1/2)*1i + 1)*(- 7^(1/2)*3i - 7)^(1/3))/84 - (7^(1/3)*log(6*x - (7^(1/3)*(3 ^(1/2)*1i + 1)*(7^(1/2)*3i - 7)^(1/3)*((7^(2/3)*(3^(1/2)*1i + 1)^2*(7^(1/2 )*3i - 7)^(2/3)*(3969*x - (567*7^(1/3)*(3^(1/2)*1i + 1)*(7^(1/2)*3i - 7)^( 1/3))/2))/7056 + 63))/84)*(3^(1/2)*1i + 1)*(7^(1/2)*3i - 7)^(1/3))/84