Integrand size = 20, antiderivative size = 355 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right ) \]
1/8*ln(1+x^2-x*(1/2*6^(1/2)-1/2*2^(1/2)))*(1/2*2^(1/2)-1/6*6^(1/2))-1/8*ln (1+x^2+x*(1/2*6^(1/2)-1/2*2^(1/2)))*(1/2*2^(1/2)-1/6*6^(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)))/(3/2*2^(1/2)-1/2* 6^(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) ))/(3/2*2^(1/2)-1/2*6^(1/2))-1/8*ln(1+x^2-x*(1/2*6^(1/2)+1/2*2^(1/2)))*(1/ 2*2^(1/2)+1/6*6^(1/2))+1/8*ln(1+x^2+x*(1/2*6^(1/2)+1/2*2^(1/2)))*(1/2*2^(1 /2)+1/6*6^(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)))/(3/2*2^(1/2)+1/2*6^(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)))/(3/2*2^(1/2)+1/2*6^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.16 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
Time = 0.61 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1751, 25, 1483, 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.
| \(\displaystyle \int \frac {1-x^4}{x^8-x^4+1} \, dx\) |
| \(\Big \downarrow \) 1751 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {3}-2 x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {\int -\frac {2 x^2+\sqrt {3}}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sqrt {3}-2 x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+\frac {\int \frac {2 x^2+\sqrt {3}}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\frac {\int \frac {\left (2-\sqrt {3}\right ) x+\sqrt {3 \left (2-\sqrt {3}\right )}}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}-\left (2-\sqrt {3}\right ) x}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\int \frac {\sqrt {3 \left (2+\sqrt {3}\right )}-\left (2+\sqrt {3}\right ) x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\int \frac {\left (2+\sqrt {3}\right ) x+\sqrt {3 \left (2+\sqrt {3}\right )}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx+\frac {1}{2} \left (2-\sqrt {3}\right ) \int -\frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \left (2+\sqrt {3}\right ) \int -\frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x-\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \int \frac {1}{-\left (2 x+\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x+\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {2-\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x-\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {2-\sqrt {3}} \int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x+\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{2} \left (2-\sqrt {3}\right ) \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (2-\sqrt {3}\right ) \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {-\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \left (2+\sqrt {3}\right ) \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )-\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
((ArcTan[(-Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]] + ((2 - Sqrt[3])*Lo g[1 - Sqrt[2 - Sqrt[3]]*x + x^2])/2)/(2*Sqrt[2 - Sqrt[3]]) + (ArcTan[(Sqrt [2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]] - ((2 - Sqrt[3])*Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2])/2)/(2*Sqrt[2 - Sqrt[3]]))/(2*Sqrt[3]) + ((-ArcTan[(-Sqr t[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]] - ((2 + Sqrt[3])*Log[1 - Sqrt[2 + Sqrt[3]]*x + x^2])/2)/(2*Sqrt[2 + Sqrt[3]]) + (-ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]] + ((2 + Sqrt[3])*Log[1 + Sqrt[2 + Sqrt[3]]*x + x^2])/2)/(2*Sqrt[2 + Sqrt[3]]))/(2*Sqrt[3])
3.1.25.3.1 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[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 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*q) Int[(q - 2*x ^(n/2))/Simp[d/e + q*x^(n/2) - x^n, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x^(n/2))/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] && IGt Q[n/2, 0] && !GtQ[b^2 - 4*a*c, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.12
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(44\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(44\) |
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.17 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=\frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} {\left (i \, \sqrt {3} + 3\right )} + 12 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} {\left (i \, \sqrt {3} + 3\right )} + 12 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} {\left (i \, \sqrt {3} - 3\right )} + 12 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} {\left (i \, \sqrt {3} - 3\right )} + 12 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} {\left (-i \, \sqrt {3} + 3\right )} + 12 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} {\left (-i \, \sqrt {3} + 3\right )} + 12 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} {\left (-i \, \sqrt {3} - 3\right )} + 12 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} {\left (-i \, \sqrt {3} - 3\right )} + 12 \, x\right ) \]
1/24*sqrt(6)*sqrt(sqrt(2)*sqrt(-I*sqrt(3) + 1))*log(sqrt(6)*sqrt(sqrt(2)*s qrt(-I*sqrt(3) + 1))*(I*sqrt(3) + 3) + 12*x) + 1/24*sqrt(6)*sqrt(-sqrt(2)* sqrt(-I*sqrt(3) + 1))*log(sqrt(6)*sqrt(-sqrt(2)*sqrt(-I*sqrt(3) + 1))*(I*s qrt(3) + 3) + 12*x) - 1/24*sqrt(6)*sqrt(sqrt(2)*sqrt(I*sqrt(3) + 1))*log(s qrt(6)*sqrt(sqrt(2)*sqrt(I*sqrt(3) + 1))*(I*sqrt(3) - 3) + 12*x) - 1/24*sq rt(6)*sqrt(-sqrt(2)*sqrt(I*sqrt(3) + 1))*log(sqrt(6)*sqrt(-sqrt(2)*sqrt(I* sqrt(3) + 1))*(I*sqrt(3) - 3) + 12*x) + 1/24*sqrt(6)*sqrt(sqrt(2)*sqrt(I*s qrt(3) + 1))*log(sqrt(6)*sqrt(sqrt(2)*sqrt(I*sqrt(3) + 1))*(-I*sqrt(3) + 3 ) + 12*x) + 1/24*sqrt(6)*sqrt(-sqrt(2)*sqrt(I*sqrt(3) + 1))*log(sqrt(6)*sq rt(-sqrt(2)*sqrt(I*sqrt(3) + 1))*(-I*sqrt(3) + 3) + 12*x) - 1/24*sqrt(6)*s qrt(sqrt(2)*sqrt(-I*sqrt(3) + 1))*log(sqrt(6)*sqrt(sqrt(2)*sqrt(-I*sqrt(3) + 1))*(-I*sqrt(3) - 3) + 12*x) - 1/24*sqrt(6)*sqrt(-sqrt(2)*sqrt(-I*sqrt( 3) + 1))*log(sqrt(6)*sqrt(-sqrt(2)*sqrt(-I*sqrt(3) + 1))*(-I*sqrt(3) - 3) + 12*x)
Time = 1.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.07 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=- \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (9216 t^{5} - 8 t + x \right )} \right )\right )} \]
\[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=\int { -\frac {x^{4} - 1}{x^{8} - x^{4} + 1} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.71 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=\frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \]
1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqr t(2))) + 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt (6) + sqrt(2))) + 1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) + sqrt( 2))/(sqrt(6) - sqrt(2))) + 1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x - sqrt(6 ) - sqrt(2))/(sqrt(6) - sqrt(2))) + 1/48*(sqrt(6) + 3*sqrt(2))*log(x^2 + 1 /2*x*(sqrt(6) + sqrt(2)) + 1) - 1/48*(sqrt(6) + 3*sqrt(2))*log(x^2 - 1/2*x *(sqrt(6) + sqrt(2)) + 1) + 1/48*(sqrt(6) - 3*sqrt(2))*log(x^2 + 1/2*x*(sq rt(6) - sqrt(2)) + 1) - 1/48*(sqrt(6) - 3*sqrt(2))*log(x^2 - 1/2*x*(sqrt(6 ) - sqrt(2)) + 1)
Time = 8.44 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.59 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}-\frac {\sqrt {3}\,x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {3}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \]
(2^(3/4)*3^(1/2)*atan((2^(1/4)*x)/(2*(3^(1/2)*1i + 1)^(1/4)) - (2^(1/4)*3^ (1/2)*x*1i)/(2*(3^(1/2)*1i + 1)^(1/4)))*(3^(1/2)*1i + 1)^(1/4)*1i)/12 - (3 ^(1/2)*atan((x*1i)/(8 - 3^(1/2)*8i)^(1/4) - (3^(1/2)*x)/(8 - 3^(1/2)*8i)^( 1/4))*(8 - 3^(1/2)*8i)^(1/4))/12 - (3^(1/2)*atan(x/(8 - 3^(1/2)*8i)^(1/4) + (3^(1/2)*x*1i)/(8 - 3^(1/2)*8i)^(1/4))*(8 - 3^(1/2)*8i)^(1/4)*1i)/12 + ( 2^(3/4)*3^(1/2)*atan((2^(1/4)*x*1i)/(2*(3^(1/2)*1i + 1)^(1/4)) + (2^(1/4)* 3^(1/2)*x)/(2*(3^(1/2)*1i + 1)^(1/4)))*(3^(1/2)*1i + 1)^(1/4))/12