Integrand size = 16, antiderivative size = 427 \[ \int \frac {x^2}{1+3 x^4+x^8} \, dx=-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3-\sqrt {5}\right )}}+\frac {\arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3-\sqrt {5}\right )}}+\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}-\frac {\arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}+\frac {\log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {5} \sqrt [4]{2 \left (3-\sqrt {5}\right )}}-\frac {\log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {5} \sqrt [4]{2 \left (3-\sqrt {5}\right )}}-\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}+\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}} \]
1/20*arctan(-1+2^(3/4)*x/(3-5^(1/2))^(1/4))*2^(3/4)/(3-5^(1/2))^(1/4)*5^(1 /2)+1/20*arctan(1+2^(3/4)*x/(3-5^(1/2))^(1/4))*2^(3/4)/(3-5^(1/2))^(1/4)*5 ^(1/2)+1/40*ln(2*x^2-2*2^(1/4)*x*(3-5^(1/2))^(1/4)+5^(1/2)-1)*2^(3/4)/(3-5 ^(1/2))^(1/4)*5^(1/2)-1/40*ln(2*x^2+2*2^(1/4)*x*(3-5^(1/2))^(1/4)+5^(1/2)- 1)*2^(3/4)/(3-5^(1/2))^(1/4)*5^(1/2)-1/20*arctan(-1+2^(3/4)*x/(3+5^(1/2))^ (1/4))*2^(3/4)*5^(1/2)/(3+5^(1/2))^(1/4)-1/20*arctan(1+2^(3/4)*x/(3+5^(1/2 ))^(1/4))*2^(3/4)*5^(1/2)/(3+5^(1/2))^(1/4)-1/40*ln(2*x^2-2*2^(1/4)*x*(3+5 ^(1/2))^(1/4)+5^(1/2)+1)*2^(3/4)*5^(1/2)/(3+5^(1/2))^(1/4)+1/40*ln(2*x^2+2 *2^(1/4)*x*(3+5^(1/2))^(1/4)+5^(1/2)+1)*2^(3/4)*5^(1/2)/(3+5^(1/2))^(1/4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.09 \[ \int \frac {x^2}{1+3 x^4+x^8} \, dx=\frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{3 \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \]
Time = 0.64 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1711, 27, 826, 1476, 1082, 217, 1479, 25, 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.
| \(\displaystyle \int \frac {x^2}{x^8+3 x^4+1} \, dx\) |
| \(\Big \downarrow \) 1711 |
| \(\displaystyle \frac {\int \frac {2 x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {5}}-\frac {\int \frac {2 x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {5}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {5}}-\frac {2 \int \frac {x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {5}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {\sqrt {2} x^2+\sqrt {3-\sqrt {5}}}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {2}}\right )}{\sqrt {5}}-\frac {2 \left (\frac {\int \frac {\sqrt {2} x^2+\sqrt {3+\sqrt {5}}}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {2}}\right )}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {2}}\right )}{\sqrt {5}}-\frac {2 \left (\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {2}}\right )}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {2}}\right )}{\sqrt {5}}-\frac {2 \left (\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {2}}\right )}{\sqrt {5}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {2}}\right )}{\sqrt {5}}-\frac {2 \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {2}}\right )}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{2 \sqrt {2}}-\frac {-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}\right )}{\sqrt {5}}-\frac {2 \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}-\frac {-\frac {\int -\frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int -\frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}\right )}{\sqrt {5}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{2 \sqrt {2}}-\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx+\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}\right )}{\sqrt {5}}-\frac {2 \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}-\frac {\frac {\int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}+\frac {\int \frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}\right )}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{2 \sqrt {2}}-\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{2 \sqrt {2}}\right )}{\sqrt {5}}-\frac {2 \left (\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}-\frac {\frac {\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{2 \sqrt {2}}\right )}{\sqrt {5}}\) |
(2*((-(ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)]/(2^(3/4)*(3 - Sqrt[5])^ (1/4))) + ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)]/(2^(3/4)*(3 - Sqrt[5 ])^(1/4)))/(2*Sqrt[2]) - (-1/4*(((3 + Sqrt[5])/2)^(1/4)*Log[Sqrt[2*(3 - Sq rt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2]) + (((3 + Sqrt[5])/2)^(1/4) *Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/4)/(2*S qrt[2])))/Sqrt[5] - (2*((-(ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(2^ (3/4)*(3 + Sqrt[5])^(1/4))) + ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/ (2^(3/4)*(3 + Sqrt[5])^(1/4)))/(2*Sqrt[2]) - (-1/2*Log[Sqrt[2*(3 + Sqrt[5] )] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2]/(2^(3/4)*(3 + Sqrt[5])^(1/4)) + Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2]/(2*2^(3/4 )*(3 + Sqrt[5])^(1/4)))/(2*Sqrt[2])))/Sqrt[5]
3.4.81.3.1 Defintions of rubi rules used
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_.)*(x_))^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symb ol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c/q Int[(d*x)^m/(b/2 - q/2 + c *x^n), x], x] - Simp[c/q Int[(d*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; Free Q[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.09
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(40\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(40\) |
Time = 0.25 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.23 \[ \int \frac {x^2}{1+3 x^4+x^8} \, dx=-\frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {\sqrt {5} - 3}} \log \left (\sqrt {10} {\left (3 \, \sqrt {5} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {5} - 3}} \sqrt {\sqrt {5} - 3} + 40 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {\sqrt {5} - 3}} \log \left (-\sqrt {10} {\left (3 \, \sqrt {5} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {5} - 3}} \sqrt {\sqrt {5} - 3} + 40 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {\sqrt {5} - 3}} \log \left (\sqrt {10} {\left (3 \, \sqrt {5} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {-\sqrt {2} \sqrt {\sqrt {5} - 3}} \sqrt {\sqrt {5} - 3} + 40 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {\sqrt {5} - 3}} \log \left (-\sqrt {10} {\left (3 \, \sqrt {5} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {-\sqrt {2} \sqrt {\sqrt {5} - 3}} \sqrt {\sqrt {5} - 3} + 40 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {-\sqrt {5} - 3}} \log \left (\sqrt {10} {\left (3 \, \sqrt {5} \sqrt {2} - 5 \, \sqrt {2}\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {5} - 3}} \sqrt {-\sqrt {5} - 3} + 40 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {2} \sqrt {-\sqrt {5} - 3}} \log \left (-\sqrt {10} {\left (3 \, \sqrt {5} \sqrt {2} - 5 \, \sqrt {2}\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {5} - 3}} \sqrt {-\sqrt {5} - 3} + 40 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {-\sqrt {5} - 3}} \log \left (\sqrt {10} {\left (3 \, \sqrt {5} \sqrt {2} - 5 \, \sqrt {2}\right )} \sqrt {-\sqrt {2} \sqrt {-\sqrt {5} - 3}} \sqrt {-\sqrt {5} - 3} + 40 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {2} \sqrt {-\sqrt {5} - 3}} \log \left (-\sqrt {10} {\left (3 \, \sqrt {5} \sqrt {2} - 5 \, \sqrt {2}\right )} \sqrt {-\sqrt {2} \sqrt {-\sqrt {5} - 3}} \sqrt {-\sqrt {5} - 3} + 40 \, x\right ) \]
-1/40*sqrt(10)*sqrt(sqrt(2)*sqrt(sqrt(5) - 3))*log(sqrt(10)*(3*sqrt(5)*sqr t(2) + 5*sqrt(2))*sqrt(sqrt(2)*sqrt(sqrt(5) - 3))*sqrt(sqrt(5) - 3) + 40*x ) + 1/40*sqrt(10)*sqrt(sqrt(2)*sqrt(sqrt(5) - 3))*log(-sqrt(10)*(3*sqrt(5) *sqrt(2) + 5*sqrt(2))*sqrt(sqrt(2)*sqrt(sqrt(5) - 3))*sqrt(sqrt(5) - 3) + 40*x) + 1/40*sqrt(10)*sqrt(-sqrt(2)*sqrt(sqrt(5) - 3))*log(sqrt(10)*(3*sqr t(5)*sqrt(2) + 5*sqrt(2))*sqrt(-sqrt(2)*sqrt(sqrt(5) - 3))*sqrt(sqrt(5) - 3) + 40*x) - 1/40*sqrt(10)*sqrt(-sqrt(2)*sqrt(sqrt(5) - 3))*log(-sqrt(10)* (3*sqrt(5)*sqrt(2) + 5*sqrt(2))*sqrt(-sqrt(2)*sqrt(sqrt(5) - 3))*sqrt(sqrt (5) - 3) + 40*x) + 1/40*sqrt(10)*sqrt(sqrt(2)*sqrt(-sqrt(5) - 3))*log(sqrt (10)*(3*sqrt(5)*sqrt(2) - 5*sqrt(2))*sqrt(sqrt(2)*sqrt(-sqrt(5) - 3))*sqrt (-sqrt(5) - 3) + 40*x) - 1/40*sqrt(10)*sqrt(sqrt(2)*sqrt(-sqrt(5) - 3))*lo g(-sqrt(10)*(3*sqrt(5)*sqrt(2) - 5*sqrt(2))*sqrt(sqrt(2)*sqrt(-sqrt(5) - 3 ))*sqrt(-sqrt(5) - 3) + 40*x) - 1/40*sqrt(10)*sqrt(-sqrt(2)*sqrt(-sqrt(5) - 3))*log(sqrt(10)*(3*sqrt(5)*sqrt(2) - 5*sqrt(2))*sqrt(-sqrt(2)*sqrt(-sqr t(5) - 3))*sqrt(-sqrt(5) - 3) + 40*x) + 1/40*sqrt(10)*sqrt(-sqrt(2)*sqrt(- sqrt(5) - 3))*log(-sqrt(10)*(3*sqrt(5)*sqrt(2) - 5*sqrt(2))*sqrt(-sqrt(2)* sqrt(-sqrt(5) - 3))*sqrt(-sqrt(5) - 3) + 40*x)
Time = 0.88 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.06 \[ \int \frac {x^2}{1+3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (40960000 t^{8} + 19200 t^{4} + 1, \left ( t \mapsto t \log {\left (- 6144000 t^{7} - 2240 t^{3} + x \right )} \right )\right )} \]
RootSum(40960000*_t**8 + 19200*_t**4 + 1, Lambda(_t, _t*log(-6144000*_t**7 - 2240*_t**3 + x)))
\[ \int \frac {x^2}{1+3 x^4+x^8} \, dx=\int { \frac {x^{2}}{x^{8} + 3 \, x^{4} + 1} \,d x } \]
Time = 0.43 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.56 \[ \int \frac {x^2}{1+3 x^4+x^8} \, dx=\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {5 \, \sqrt {5} + 5} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} - 1\right )\right )} \sqrt {5 \, \sqrt {5} + 5} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {5 \, \sqrt {5} - 5} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} + 1\right )\right )} \sqrt {5 \, \sqrt {5} - 5} + \frac {1}{40} \, \sqrt {5 \, \sqrt {5} - 5} \log \left (16900 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 16900 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {5 \, \sqrt {5} - 5} \log \left (16900 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 16900 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {5 \, \sqrt {5} + 5} \log \left (2500 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 2500 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {5 \, \sqrt {5} + 5} \log \left (2500 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 2500 \, x^{2}\right ) \]
1/80*(pi + 4*arctan(x*sqrt(sqrt(5) + 1) - 1))*sqrt(5*sqrt(5) + 5) - 1/80*( pi + 4*arctan(-x*sqrt(sqrt(5) + 1) - 1))*sqrt(5*sqrt(5) + 5) - 1/80*(pi + 4*arctan(x*sqrt(sqrt(5) - 1) + 1))*sqrt(5*sqrt(5) - 5) + 1/80*(pi + 4*arct an(-x*sqrt(sqrt(5) - 1) + 1))*sqrt(5*sqrt(5) - 5) + 1/40*sqrt(5*sqrt(5) - 5)*log(16900*(x + sqrt(sqrt(5) + 1))^2 + 16900*x^2) - 1/40*sqrt(5*sqrt(5) - 5)*log(16900*(x - sqrt(sqrt(5) + 1))^2 + 16900*x^2) - 1/40*sqrt(5*sqrt(5 ) + 5)*log(2500*(x + sqrt(sqrt(5) - 1))^2 + 2500*x^2) + 1/40*sqrt(5*sqrt(5 ) + 5)*log(2500*(x - sqrt(sqrt(5) - 1))^2 + 2500*x^2)
Time = 8.29 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.64 \[ \int \frac {x^2}{1+3 x^4+x^8} \, dx=\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {7\,2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (3\,\sqrt {5}-7\right )}-\frac {3\,2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (3\,\sqrt {5}-7\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,7{}\mathrm {i}}{2\,\left (3\,\sqrt {5}-7\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,3{}\mathrm {i}}{2\,\left (3\,\sqrt {5}-7\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {7\,2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (3\,\sqrt {5}+7\right )}+\frac {3\,2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (3\,\sqrt {5}+7\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,7{}\mathrm {i}}{2\,\left (3\,\sqrt {5}+7\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,3{}\mathrm {i}}{2\,\left (3\,\sqrt {5}+7\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{20} \]
(2^(3/4)*5^(1/2)*atan((7*2^(3/4)*x*(5^(1/2) - 3)^(1/4))/(2*(3*5^(1/2) - 7) ) - (3*2^(3/4)*5^(1/2)*x*(5^(1/2) - 3)^(1/4))/(2*(3*5^(1/2) - 7)))*(5^(1/2 ) - 3)^(1/4))/20 + (2^(3/4)*5^(1/2)*atan((2^(3/4)*x*(5^(1/2) - 3)^(1/4)*7i )/(2*(3*5^(1/2) - 7)) - (2^(3/4)*5^(1/2)*x*(5^(1/2) - 3)^(1/4)*3i)/(2*(3*5 ^(1/2) - 7)))*(5^(1/2) - 3)^(1/4)*1i)/20 + (2^(3/4)*5^(1/2)*atan((7*2^(3/4 )*x*(- 5^(1/2) - 3)^(1/4))/(2*(3*5^(1/2) + 7)) + (3*2^(3/4)*5^(1/2)*x*(- 5 ^(1/2) - 3)^(1/4))/(2*(3*5^(1/2) + 7)))*(- 5^(1/2) - 3)^(1/4))/20 + (2^(3/ 4)*5^(1/2)*atan((2^(3/4)*x*(- 5^(1/2) - 3)^(1/4)*7i)/(2*(3*5^(1/2) + 7)) + (2^(3/4)*5^(1/2)*x*(- 5^(1/2) - 3)^(1/4)*3i)/(2*(3*5^(1/2) + 7)))*(- 5^(1 /2) - 3)^(1/4)*1i)/20