Integrand size = 18, antiderivative size = 331 \[ \int \frac {1+x^4}{1-x^4+x^8} \, dx=-\frac {1}{4} \sqrt {2-\sqrt {3}} \arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {2+\sqrt {3}} \arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {2-\sqrt {3}} \arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {2+\sqrt {3}} \arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {2-\sqrt {3}}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{8 \sqrt {2-\sqrt {3}}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {2+\sqrt {3}}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{8 \sqrt {2+\sqrt {3}}} \]
-1/8*ln(1+x^2-x*(1/2*6^(1/2)-1/2*2^(1/2)))/(1/2*6^(1/2)-1/2*2^(1/2))+1/8*l n(1+x^2+x*(1/2*6^(1/2)-1/2*2^(1/2)))/(1/2*6^(1/2)-1/2*2^(1/2))-1/4*arctan( (-2*x+1/2*6^(1/2)-1/2*2^(1/2))/(1/2*6^(1/2)+1/2*2^(1/2)))*(1/2*6^(1/2)-1/2 *2^(1/2))+1/4*arctan((2*x+1/2*6^(1/2)-1/2*2^(1/2))/(1/2*6^(1/2)+1/2*2^(1/2 )))*(1/2*6^(1/2)-1/2*2^(1/2))-1/8*ln(1+x^2-x*(1/2*6^(1/2)+1/2*2^(1/2)))/(1 /2*6^(1/2)+1/2*2^(1/2))+1/8*ln(1+x^2+x*(1/2*6^(1/2)+1/2*2^(1/2)))/(1/2*6^( 1/2)+1/2*2^(1/2))-1/4*arctan((-2*x+1/2*6^(1/2)+1/2*2^(1/2))/(1/2*6^(1/2)-1 /2*2^(1/2)))*(1/2*6^(1/2)+1/2*2^(1/2))+1/4*arctan((2*x+1/2*6^(1/2)+1/2*2^( 1/2))/(1/2*6^(1/2)-1/2*2^(1/2)))*(1/2*6^(1/2)+1/2*2^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.17 \[ \int \frac {1+x^4}{1-x^4+x^8} \, dx=\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
Time = 0.54 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1749, 1407, 1142, 25, 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.
| \(\displaystyle \int \frac {x^4+1}{x^8-x^4+1} \, dx\) |
| \(\Big \downarrow \) 1749 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4-\sqrt {3} x^2+1}dx+\frac {1}{2} \int \frac {1}{x^4+\sqrt {3} x^2+1}dx\) |
| \(\Big \downarrow \) 1407 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\sqrt {2-\sqrt {3}}-x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\int \frac {x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\int \frac {\sqrt {2+\sqrt {3}}-x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\int \frac {x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \int -\frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \int -\frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2-\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x-\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2-\sqrt {3}} \int \frac {1}{-\left (2 x+\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x+\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x-\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x+\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx+\sqrt {\frac {2-\sqrt {3}}{2+\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx+\sqrt {\frac {2-\sqrt {3}}{2+\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\frac {1}{2} \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {\frac {2+\sqrt {3}}{2-\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {\frac {2+\sqrt {3}}{2-\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {\sqrt {\frac {2-\sqrt {3}}{2+\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\sqrt {\frac {2-\sqrt {3}}{2+\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{2} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\sqrt {\frac {2+\sqrt {3}}{2-\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\sqrt {\frac {2+\sqrt {3}}{2-\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}\right )\) |
((Sqrt[(2 - Sqrt[3])/(2 + Sqrt[3])]*ArcTan[(-Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt [2 + Sqrt[3]]] - Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2]/2)/(2*Sqrt[2 - Sqrt[3] ]) + (Sqrt[(2 - Sqrt[3])/(2 + Sqrt[3])]*ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/S qrt[2 + Sqrt[3]]] + Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2]/2)/(2*Sqrt[2 - Sqrt [3]]))/2 + ((Sqrt[(2 + Sqrt[3])/(2 - Sqrt[3])]*ArcTan[(-Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]] - Log[1 - Sqrt[2 + Sqrt[3]]*x + x^2]/2)/(2*Sqrt[ 2 + Sqrt[3]]) + (Sqrt[(2 + Sqrt[3])/(2 - Sqrt[3])]*ArcTan[(Sqrt[2 + Sqrt[3 ]] + 2*x)/Sqrt[2 - Sqrt[3]]] + Log[1 + Sqrt[2 + Sqrt[3]]*x + x^2]/2)/(2*Sq rt[2 + Sqrt[3]]))/2
3.1.14.3.1 Defintions of rubi rules used
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_) + (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[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[2*(d/e) - b/c, 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x^(n/2) + x^n, x], x], x] + Simp[e/(2*c) Int[1/Simp[d/e - q*x^(n/2) + x^n, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d, e*Rt[a/c, 2]]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.13
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(42\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(42\) |
Time = 0.27 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.18 \[ \int \frac {1+x^4}{1-x^4+x^8} \, dx=\frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {\sqrt {-3} + 1}} \log \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {\sqrt {-3} + 1}} {\left (\sqrt {-3} + 1\right )} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {\sqrt {-3} + 1}} \log \left (-\sqrt {2} \sqrt {\sqrt {2} \sqrt {\sqrt {-3} + 1}} {\left (\sqrt {-3} + 1\right )} + 4 \, x\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {2} \sqrt {\sqrt {-3} + 1}} \log \left (\sqrt {2} \sqrt {-\sqrt {2} \sqrt {\sqrt {-3} + 1}} {\left (\sqrt {-3} + 1\right )} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {2} \sqrt {\sqrt {-3} + 1}} \log \left (-\sqrt {2} \sqrt {-\sqrt {2} \sqrt {\sqrt {-3} + 1}} {\left (\sqrt {-3} + 1\right )} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {-\sqrt {-3} + 1}} \log \left (\sqrt {2} \sqrt {\sqrt {2} \sqrt {-\sqrt {-3} + 1}} {\left (\sqrt {-3} - 1\right )} + 4 \, x\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {2} \sqrt {-\sqrt {-3} + 1}} \log \left (-\sqrt {2} \sqrt {\sqrt {2} \sqrt {-\sqrt {-3} + 1}} {\left (\sqrt {-3} - 1\right )} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {2} \sqrt {-\sqrt {-3} + 1}} \log \left (\sqrt {2} \sqrt {-\sqrt {2} \sqrt {-\sqrt {-3} + 1}} {\left (\sqrt {-3} - 1\right )} + 4 \, x\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {2} \sqrt {-\sqrt {-3} + 1}} \log \left (-\sqrt {2} \sqrt {-\sqrt {2} \sqrt {-\sqrt {-3} + 1}} {\left (\sqrt {-3} - 1\right )} + 4 \, x\right ) \]
1/8*sqrt(2)*sqrt(sqrt(2)*sqrt(sqrt(-3) + 1))*log(sqrt(2)*sqrt(sqrt(2)*sqrt (sqrt(-3) + 1))*(sqrt(-3) + 1) + 4*x) - 1/8*sqrt(2)*sqrt(sqrt(2)*sqrt(sqrt (-3) + 1))*log(-sqrt(2)*sqrt(sqrt(2)*sqrt(sqrt(-3) + 1))*(sqrt(-3) + 1) + 4*x) + 1/8*sqrt(2)*sqrt(-sqrt(2)*sqrt(sqrt(-3) + 1))*log(sqrt(2)*sqrt(-sqr t(2)*sqrt(sqrt(-3) + 1))*(sqrt(-3) + 1) + 4*x) - 1/8*sqrt(2)*sqrt(-sqrt(2) *sqrt(sqrt(-3) + 1))*log(-sqrt(2)*sqrt(-sqrt(2)*sqrt(sqrt(-3) + 1))*(sqrt( -3) + 1) + 4*x) - 1/8*sqrt(2)*sqrt(sqrt(2)*sqrt(-sqrt(-3) + 1))*log(sqrt(2 )*sqrt(sqrt(2)*sqrt(-sqrt(-3) + 1))*(sqrt(-3) - 1) + 4*x) + 1/8*sqrt(2)*sq rt(sqrt(2)*sqrt(-sqrt(-3) + 1))*log(-sqrt(2)*sqrt(sqrt(2)*sqrt(-sqrt(-3) + 1))*(sqrt(-3) - 1) + 4*x) - 1/8*sqrt(2)*sqrt(-sqrt(2)*sqrt(-sqrt(-3) + 1) )*log(sqrt(2)*sqrt(-sqrt(2)*sqrt(-sqrt(-3) + 1))*(sqrt(-3) - 1) + 4*x) + 1 /8*sqrt(2)*sqrt(-sqrt(2)*sqrt(-sqrt(-3) + 1))*log(-sqrt(2)*sqrt(-sqrt(2)*s qrt(-sqrt(-3) + 1))*(sqrt(-3) - 1) + 4*x)
Time = 1.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.06 \[ \int \frac {1+x^4}{1-x^4+x^8} \, dx=\operatorname {RootSum} {\left (65536 t^{8} - 256 t^{4} + 1, \left ( t \mapsto t \log {\left (1024 t^{5} + x \right )} \right )\right )} \]
\[ \int \frac {1+x^4}{1-x^4+x^8} \, dx=\int { \frac {x^{4} + 1}{x^{8} - x^{4} + 1} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.74 \[ \int \frac {1+x^4}{1-x^4+x^8} \, dx=\frac {1}{8} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{8} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{16} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{16} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{16} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{16} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \]
1/8*(sqrt(6) - sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2 ))) + 1/8*(sqrt(6) - sqrt(2))*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2))) + 1/8*(sqrt(6) + sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt (6) - sqrt(2))) + 1/8*(sqrt(6) + sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2)) /(sqrt(6) - sqrt(2))) + 1/16*(sqrt(6) - sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/16*(sqrt(6) - sqrt(2))*log(x^2 - 1/2*x*(sqrt(6) + sqrt (2)) + 1) + 1/16*(sqrt(6) + sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/16*(sqrt(6) + sqrt(2))*log(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)
Time = 0.14 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.44 \[ \int \frac {1+x^4}{1-x^4+x^8} \, dx=-\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (27-27{}\mathrm {i}\right )}{27\,\sqrt {3}-81{}\mathrm {i}}\right )\,\left (\sqrt {2}\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )+\sqrt {6}\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )\right )-\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (27+27{}\mathrm {i}\right )}{27\,\sqrt {3}-81{}\mathrm {i}}\right )\,\left (\sqrt {2}\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )+\sqrt {6}\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )\right )-\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (27-27{}\mathrm {i}\right )}{27\,\sqrt {3}+81{}\mathrm {i}}\right )\,\left (\sqrt {2}\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )+\sqrt {6}\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )\right )-\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (27+27{}\mathrm {i}\right )}{27\,\sqrt {3}+81{}\mathrm {i}}\right )\,\left (\sqrt {2}\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )+\sqrt {6}\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )\right ) \]
- atan((6^(1/2)*x*(27 - 27i))/(27*3^(1/2) - 81i))*(2^(1/2)*(1/8 + 1i/8) - 6^(1/2)*(1/8 - 1i/8)) - atan((6^(1/2)*x*(27 + 27i))/(27*3^(1/2) - 81i))*(2 ^(1/2)*(1/8 - 1i/8) + 6^(1/2)*(1/8 + 1i/8)) - atan((6^(1/2)*x*(27 - 27i))/ (27*3^(1/2) + 81i))*(2^(1/2)*(1/8 + 1i/8) + 6^(1/2)*(1/8 - 1i/8)) - atan(( 6^(1/2)*x*(27 + 27i))/(27*3^(1/2) + 81i))*(2^(1/2)*(1/8 - 1i/8) - 6^(1/2)* (1/8 + 1i/8))