Integrand size = 12, antiderivative size = 414 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=-\frac {\sqrt [4]{9+4 \sqrt {5}} \arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {10}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {10}}+\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\arctan \left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\sqrt [4]{9+4 \sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {10}}+\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}} \]
-1/20*arctan(-1+x*(5^(1/2)-1)^(1/2))*(-20+10*5^(1/2))^(1/2)-1/20*arctan(1+ x*(5^(1/2)-1)^(1/2))*(-20+10*5^(1/2))^(1/2)+1/40*ln(1+2*x^2+5^(1/2)-2*x*(5 ^(1/2)+1)^(1/2))*(-20+10*5^(1/2))^(1/2)-1/40*ln(1+2*x^2+5^(1/2)+2*x*(5^(1/ 2)+1)^(1/2))*(-20+10*5^(1/2))^(1/2)+1/20*arctan(-1+x*(5^(1/2)+1)^(1/2))*(2 0+10*5^(1/2))^(1/2)+1/20*arctan(1+x*(5^(1/2)+1)^(1/2))*(20+10*5^(1/2))^(1/ 2)-1/40*ln(-1+2*x^2+5^(1/2)-2*x*(5^(1/2)-1)^(1/2))*(20+10*5^(1/2))^(1/2)+1 /40*ln(-1+2*x^2+5^(1/2)+2*x*(5^(1/2)-1)^(1/2))*(20+10*5^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.10 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=\frac {1}{4} \text {RootSum}\left [1+3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
Time = 0.66 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {1685, 755, 27, 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.
| \(\displaystyle \int \frac {1}{x^8+3 x^4+1} \, dx\) |
| \(\Big \downarrow \) 1685 |
| \(\displaystyle \frac {\int \frac {1}{x^4+\frac {1}{2} \left (3-\sqrt {5}\right )}dx}{\sqrt {5}}-\frac {\int \frac {1}{x^4+\frac {1}{2} \left (3+\sqrt {5}\right )}dx}{\sqrt {5}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (\sqrt {3-\sqrt {5}}-\sqrt {2} x^2\right )}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {3-\sqrt {5}}}+\frac {\int \frac {2 \left (\sqrt {2} x^2+\sqrt {3-\sqrt {5}}\right )}{2 x^4-\sqrt {5}+3}dx}{2 \sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\int \frac {2 \left (\sqrt {3+\sqrt {5}}-\sqrt {2} x^2\right )}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {3+\sqrt {5}}}+\frac {\int \frac {2 \left (\sqrt {2} x^2+\sqrt {3+\sqrt {5}}\right )}{2 x^4+\sqrt {5}+3}dx}{2 \sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\int \frac {\sqrt {2} x^2+\sqrt {3-\sqrt {5}}}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\int \frac {\sqrt {2} x^2+\sqrt {3+\sqrt {5}}}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{\sqrt {3-\sqrt {5}}}+\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\frac {\int \frac {1}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}+\frac {\int \frac {1}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2 \sqrt {2}}}{\sqrt {3+\sqrt {5}}}+\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )^2-1}d\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int \frac {1}{-\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )^2-1}d\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{2 x^4-\sqrt {5}+3}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{2 x^4+\sqrt {5}+3}dx}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int -\frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {-\frac {\int -\frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\int -\frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx+\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \int \frac {2 x+\sqrt [4]{2 \left (3-\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}}dx}{\sqrt {3-\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\frac {\int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{x^2-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}+\frac {\int \frac {2 x+\sqrt [4]{2 \left (3+\sqrt {5}\right )}}{x^2+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}}dx}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3-\sqrt {5}}}}{\sqrt {3-\sqrt {5}}}+\frac {\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )-\frac {1}{4} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}-\frac {\frac {\frac {\arctan \left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\arctan \left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}+\frac {\frac {\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}-\frac {\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2\ 2^{3/4} \sqrt [4]{3+\sqrt {5}}}}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}\) |
((-(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)))/Sqrt[3 - Sqrt[5]] + (-1/4*(((3 + Sqrt[5])/2)^(1/4)*Log[Sqrt[2*(3 - Sqrt[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)/S qrt[3 - Sqrt[5]])/Sqrt[5] - ((-(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)))/Sqrt[3 + Sqrt[5]] + (-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)))/Sqrt[3 + Sqrt[5]])/Sqrt[5]
3.4.82.3.1 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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[((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 = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.09
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(37\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}+3 \textit {\_R}^{3}}\right )}{4}\) | \(37\) |
Time = 0.26 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.75 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=-\frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {4 \, \sqrt {5} - 9}} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {-\sqrt {4 \, \sqrt {5} - 9}} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {4 \, \sqrt {5} - 9}} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {-\sqrt {4 \, \sqrt {5} - 9}} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {-4 \, \sqrt {5} - 9}} \log \left ({\left (\sqrt {5} - 3\right )} \sqrt {-\sqrt {-4 \, \sqrt {5} - 9}} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {-4 \, \sqrt {5} - 9}} \log \left (-{\left (\sqrt {5} - 3\right )} \sqrt {-\sqrt {-4 \, \sqrt {5} - 9}} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} {\left (4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} \log \left ({\left (4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} {\left (\sqrt {5} + 3\right )} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} {\left (4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} \log \left (-{\left (4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} {\left (\sqrt {5} + 3\right )} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} {\left (-4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} \log \left ({\left (\sqrt {5} - 3\right )} {\left (-4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} {\left (-4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} \log \left (-{\left (\sqrt {5} - 3\right )} {\left (-4 \, \sqrt {5} - 9\right )}^{\frac {1}{4}} + 2 \, x\right ) \]
-1/20*sqrt(5)*sqrt(-sqrt(4*sqrt(5) - 9))*log((sqrt(5) + 3)*sqrt(-sqrt(4*sq rt(5) - 9)) + 2*x) + 1/20*sqrt(5)*sqrt(-sqrt(4*sqrt(5) - 9))*log(-(sqrt(5) + 3)*sqrt(-sqrt(4*sqrt(5) - 9)) + 2*x) - 1/20*sqrt(5)*sqrt(-sqrt(-4*sqrt( 5) - 9))*log((sqrt(5) - 3)*sqrt(-sqrt(-4*sqrt(5) - 9)) + 2*x) + 1/20*sqrt( 5)*sqrt(-sqrt(-4*sqrt(5) - 9))*log(-(sqrt(5) - 3)*sqrt(-sqrt(-4*sqrt(5) - 9)) + 2*x) - 1/20*sqrt(5)*(4*sqrt(5) - 9)^(1/4)*log((4*sqrt(5) - 9)^(1/4)* (sqrt(5) + 3) + 2*x) + 1/20*sqrt(5)*(4*sqrt(5) - 9)^(1/4)*log(-(4*sqrt(5) - 9)^(1/4)*(sqrt(5) + 3) + 2*x) - 1/20*sqrt(5)*(-4*sqrt(5) - 9)^(1/4)*log( (sqrt(5) - 3)*(-4*sqrt(5) - 9)^(1/4) + 2*x) + 1/20*sqrt(5)*(-4*sqrt(5) - 9 )^(1/4)*log(-(sqrt(5) - 3)*(-4*sqrt(5) - 9)^(1/4) + 2*x)
Time = 0.92 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.06 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (40960000 t^{8} + 115200 t^{4} + 1, \left ( t \mapsto t \log {\left (- 9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} \]
\[ \int \frac {1}{1+3 x^4+x^8} \, dx=\int { \frac {1}{x^{8} + 3 \, x^{4} + 1} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.58 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=\frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {10 \, \sqrt {5} + 20} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {10 \, \sqrt {5} + 20} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {10 \, \sqrt {5} - 20} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {10 \, \sqrt {5} - 20} - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 20} \log \left (10000 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 10000 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 20} \log \left (10000 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 10000 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 20} \log \left (400 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 400 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 20} \log \left (400 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 400 \, x^{2}\right ) \]
1/80*(pi + 4*arctan(x*sqrt(sqrt(5) + 1) + 1))*sqrt(10*sqrt(5) + 20) - 1/80 *(pi + 4*arctan(-x*sqrt(sqrt(5) + 1) + 1))*sqrt(10*sqrt(5) + 20) - 1/80*(p i + 4*arctan(x*sqrt(sqrt(5) - 1) - 1))*sqrt(10*sqrt(5) - 20) + 1/80*(pi + 4*arctan(-x*sqrt(sqrt(5) - 1) - 1))*sqrt(10*sqrt(5) - 20) - 1/40*sqrt(10*s qrt(5) - 20)*log(10000*(x + sqrt(sqrt(5) + 1))^2 + 10000*x^2) + 1/40*sqrt( 10*sqrt(5) - 20)*log(10000*(x - sqrt(sqrt(5) + 1))^2 + 10000*x^2) + 1/40*s qrt(10*sqrt(5) + 20)*log(400*(x + sqrt(sqrt(5) - 1))^2 + 400*x^2) - 1/40*s qrt(10*sqrt(5) + 20)*log(400*(x - sqrt(sqrt(5) - 1))^2 + 400*x^2)
Time = 0.05 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.97 \[ \int \frac {1}{1+3 x^4+x^8} \, dx=\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {144\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}+\frac {64\,\sqrt {5}\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}\right )\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {144\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}-\frac {64\,\sqrt {5}\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}\right )\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,144{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}+\frac {\sqrt {5}\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,64{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}\right )\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,1{}\mathrm {i}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,144{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}-\frac {\sqrt {5}\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,64{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}\right )\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,1{}\mathrm {i}}{10} \]
(5^(1/2)*atan((144*x*(- 4*5^(1/2) - 9)^(1/4))/(24*5^(1/2)*(- 4*5^(1/2) - 9 )^(1/2) + 56*(- 4*5^(1/2) - 9)^(1/2)) + (64*5^(1/2)*x*(- 4*5^(1/2) - 9)^(1 /4))/(24*5^(1/2)*(- 4*5^(1/2) - 9)^(1/2) + 56*(- 4*5^(1/2) - 9)^(1/2)))*(- 4*5^(1/2) - 9)^(1/4))/10 + (5^(1/2)*atan((144*x*(4*5^(1/2) - 9)^(1/4))/(2 4*5^(1/2)*(4*5^(1/2) - 9)^(1/2) - 56*(4*5^(1/2) - 9)^(1/2)) - (64*5^(1/2)* x*(4*5^(1/2) - 9)^(1/4))/(24*5^(1/2)*(4*5^(1/2) - 9)^(1/2) - 56*(4*5^(1/2) - 9)^(1/2)))*(4*5^(1/2) - 9)^(1/4))/10 - (5^(1/2)*atan((x*(- 4*5^(1/2) - 9)^(1/4)*144i)/(24*5^(1/2)*(- 4*5^(1/2) - 9)^(1/2) + 56*(- 4*5^(1/2) - 9)^ (1/2)) + (5^(1/2)*x*(- 4*5^(1/2) - 9)^(1/4)*64i)/(24*5^(1/2)*(- 4*5^(1/2) - 9)^(1/2) + 56*(- 4*5^(1/2) - 9)^(1/2)))*(- 4*5^(1/2) - 9)^(1/4)*1i)/10 - (5^(1/2)*atan((x*(4*5^(1/2) - 9)^(1/4)*144i)/(24*5^(1/2)*(4*5^(1/2) - 9)^ (1/2) - 56*(4*5^(1/2) - 9)^(1/2)) - (5^(1/2)*x*(4*5^(1/2) - 9)^(1/4)*64i)/ (24*5^(1/2)*(4*5^(1/2) - 9)^(1/2) - 56*(4*5^(1/2) - 9)^(1/2)))*(4*5^(1/2) - 9)^(1/4)*1i)/10