Integrand size = 12, antiderivative size = 275 \[ \int \frac {1}{1-x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}} \] Output:
-1/12*arctan((-2*x+1/2*6^(1/2)-1/2*2^(1/2))/(1/2*6^(1/2)+1/2*2^(1/2)))*6^( 1/2)+1/12*arctan((2*x+1/2*6^(1/2)-1/2*2^(1/2))/(1/2*6^(1/2)+1/2*2^(1/2)))* 6^(1/2)-1/12*arctan((-2*x+1/2*6^(1/2)+1/2*2^(1/2))/(1/2*6^(1/2)-1/2*2^(1/2 )))*6^(1/2)+1/12*arctan((2*x+1/2*6^(1/2)+1/2*2^(1/2))/(1/2*6^(1/2)-1/2*2^( 1/2)))*6^(1/2)-1/24*ln(1+x^2-x*(1/2*6^(1/2)-1/2*2^(1/2)))*6^(1/2)+1/24*ln( 1+x^2+x*(1/2*6^(1/2)-1/2*2^(1/2)))*6^(1/2)-1/24*ln(1+x^2-x*(1/2*6^(1/2)+1/ 2*2^(1/2)))*6^(1/2)+1/24*ln(1+x^2+x*(1/2*6^(1/2)+1/2*2^(1/2)))*6^(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 = 0.15 \[ \int \frac {1}{1-x^4+x^8} \, dx=\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \] Input:
Integrate[(1 - x^4 + x^8)^(-1),x]
Output:
RootSum[1 - #1^4 + #1^8 & , Log[x - #1]/(-#1^3 + 2*#1^7) & ]/4
Time = 0.55 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.49, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {1684, 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.
| \(\displaystyle \int \frac {1}{x^8-x^4+1} \, dx\) |
| \(\Big \downarrow \) 1684 |
| \(\displaystyle \frac {\int \frac {\sqrt {3}-x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+\frac {\int \frac {x^2+\sqrt {3}}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\frac {\int \frac {\left (1-\sqrt {3}\right ) x+\sqrt {3 \left (2-\sqrt {3}\right )}}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}-\left (1-\sqrt {3}\right ) x}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\int \frac {\sqrt {3 \left (2+\sqrt {3}\right )}-\left (1+\sqrt {3}\right ) x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\int \frac {\left (1+\sqrt {3}\right ) x+\sqrt {3 \left (2+\sqrt {3}\right )}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{\sqrt {2}}+\frac {1}{2} \left (1-\sqrt {3}\right ) \int -\frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {\int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{\sqrt {2}}-\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {\int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{\sqrt {2}}-\frac {1}{2} \left (1+\sqrt {3}\right ) \int -\frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {\int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{\sqrt {2}}+\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{\sqrt {2}}-\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {\int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{\sqrt {2}}-\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {\int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{\sqrt {2}}+\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {\int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{\sqrt {2}}+\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2} \int \frac {1}{-\left (2 x-\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x-\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {-\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2} \int \frac {1}{-\left (2 x+\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x+\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\sqrt {2} \int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x-\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\sqrt {2} \int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x+\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\sqrt {\frac {2}{2+\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\sqrt {\frac {2}{2+\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {\frac {2}{2-\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {\frac {2}{2-\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\sqrt {\frac {2}{2+\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{2} \left (1-\sqrt {3}\right ) \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\sqrt {\frac {2}{2+\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (1-\sqrt {3}\right ) \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\sqrt {\frac {2}{2-\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \left (1+\sqrt {3}\right ) \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\sqrt {\frac {2}{2-\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (1+\sqrt {3}\right ) \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}\) |
Input:
Int[(1 - x^4 + x^8)^(-1),x]
Output:
((Sqrt[2/(2 + Sqrt[3])]*ArcTan[(-Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3] ]] + ((1 - Sqrt[3])*Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2])/2)/(2*Sqrt[2 - Sqr t[3]]) + (Sqrt[2/(2 + Sqrt[3])]*ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]] - ((1 - Sqrt[3])*Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2])/2)/(2*Sqrt[ 2 - Sqrt[3]]))/(2*Sqrt[3]) + ((Sqrt[2/(2 - Sqrt[3])]*ArcTan[(-Sqrt[2 + Sqr t[3]] + 2*x)/Sqrt[2 - Sqrt[3]]] - ((1 + Sqrt[3])*Log[1 - Sqrt[2 + Sqrt[3]] *x + x^2])/2)/(2*Sqrt[2 + Sqrt[3]]) + (Sqrt[2/(2 - Sqrt[3])]*ArcTan[(Sqrt[ 2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]] + ((1 + Sqrt[3])*Log[1 + Sqrt[2 + S qrt[3]]*x + x^2])/2)/(2*Sqrt[2 + Sqrt[3]]))/(2*Sqrt[3])
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[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-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^( n/2))/(q - r*x^(n/2) + x^n), x], x] + Simp[1/(2*c*q*r) Int[(r + x^(n/2))/ (q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && N eQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && NegQ[b^2 - 4*a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.11
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (3 \textit {\_R}^{2}+3 \textit {\_R} x +x^{2}\right )\right )}{4}\) | \(30\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (3 \textit {\_R}^{2}+3 \textit {\_R} x +x^{2}\right )\right )}{4}\) | \(30\) |
Input:
int(1/(x^8-x^4+1),x,method=_RETURNVERBOSE)
Output:
1/4*sum(_R*ln(3*_R^2+3*_R*x+x^2),_R=RootOf(9*_Z^4+1))
Time = 0.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.47 \[ \int \frac {1}{1-x^4+x^8} \, dx=\frac {1}{4} \, \sqrt {\frac {2}{3}} \arctan \left (4 \, x^{2} + 3 \, \sqrt {\frac {2}{3}} {\left (x^{3} + 2 \, x\right )} + 3\right ) + \frac {1}{4} \, \sqrt {\frac {2}{3}} \arctan \left (-4 \, x^{2} + 3 \, \sqrt {\frac {2}{3}} {\left (x^{3} + 2 \, x\right )} - 3\right ) + \frac {1}{4} \, \sqrt {\frac {2}{3}} \arctan \left (\sqrt {\frac {2}{3}} x + \frac {1}{3}\right ) + \frac {1}{4} \, \sqrt {\frac {2}{3}} \arctan \left (\sqrt {\frac {2}{3}} x - \frac {1}{3}\right ) + \frac {1}{8} \, \sqrt {\frac {2}{3}} \log \left (x^{4} + 3 \, x^{2} + 3 \, \sqrt {\frac {2}{3}} {\left (x^{3} + x\right )} + 1\right ) - \frac {1}{8} \, \sqrt {\frac {2}{3}} \log \left (x^{4} + 3 \, x^{2} - 3 \, \sqrt {\frac {2}{3}} {\left (x^{3} + x\right )} + 1\right ) \] Input:
integrate(1/(x^8-x^4+1),x, algorithm="fricas")
Output:
1/4*sqrt(2/3)*arctan(4*x^2 + 3*sqrt(2/3)*(x^3 + 2*x) + 3) + 1/4*sqrt(2/3)* arctan(-4*x^2 + 3*sqrt(2/3)*(x^3 + 2*x) - 3) + 1/4*sqrt(2/3)*arctan(sqrt(2 /3)*x + 1/3) + 1/4*sqrt(2/3)*arctan(sqrt(2/3)*x - 1/3) + 1/8*sqrt(2/3)*log (x^4 + 3*x^2 + 3*sqrt(2/3)*(x^3 + x) + 1) - 1/8*sqrt(2/3)*log(x^4 + 3*x^2 - 3*sqrt(2/3)*(x^3 + x) + 1)
Time = 0.09 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.60 \[ \int \frac {1}{1-x^4+x^8} \, dx=\frac {\sqrt {6} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} - \frac {1}{3} \right )} + 2 \operatorname {atan}{\left (\sqrt {6} x^{3} - 4 x^{2} + 2 \sqrt {6} x - 3 \right )}\right )}{24} + \frac {\sqrt {6} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} + \frac {1}{3} \right )} + 2 \operatorname {atan}{\left (\sqrt {6} x^{3} + 4 x^{2} + 2 \sqrt {6} x + 3 \right )}\right )}{24} - \frac {\sqrt {6} \log {\left (x^{4} - \sqrt {6} x^{3} + 3 x^{2} - \sqrt {6} x + 1 \right )}}{24} + \frac {\sqrt {6} \log {\left (x^{4} + \sqrt {6} x^{3} + 3 x^{2} + \sqrt {6} x + 1 \right )}}{24} \] Input:
integrate(1/(x**8-x**4+1),x)
Output:
sqrt(6)*(2*atan(sqrt(6)*x/3 - 1/3) + 2*atan(sqrt(6)*x**3 - 4*x**2 + 2*sqrt (6)*x - 3))/24 + sqrt(6)*(2*atan(sqrt(6)*x/3 + 1/3) + 2*atan(sqrt(6)*x**3 + 4*x**2 + 2*sqrt(6)*x + 3))/24 - sqrt(6)*log(x**4 - sqrt(6)*x**3 + 3*x**2 - sqrt(6)*x + 1)/24 + sqrt(6)*log(x**4 + sqrt(6)*x**3 + 3*x**2 + sqrt(6)* x + 1)/24
\[ \int \frac {1}{1-x^4+x^8} \, dx=\int { \frac {1}{x^{8} - x^{4} + 1} \,d x } \] Input:
integrate(1/(x^8-x^4+1),x, algorithm="maxima")
Output:
integrate(1/(x^8 - x^4 + 1), x)
Time = 0.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.75 \[ \int \frac {1}{1-x^4+x^8} \, dx=\frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, \sqrt {6} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{24} \, \sqrt {6} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{24} \, \sqrt {6} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{24} \, \sqrt {6} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \] Input:
integrate(1/(x^8-x^4+1),x, algorithm="giac")
Output:
1/12*sqrt(6)*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))) + 1/12* sqrt(6)*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2))) + 1/12*sqrt( 6)*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqrt(2))) + 1/12*sqrt(6)*ar ctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2))) + 1/24*sqrt(6)*log(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/24*sqrt(6)*log(x^2 - 1/2*x*(sqrt(6) + sqrt(2)) + 1) + 1/24*sqrt(6)*log(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/24*sqrt(6)*log(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.19 \[ \int \frac {1}{1-x^4+x^8} \, dx=\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (\frac {1}{3}+\frac {1}{3}{}\mathrm {i}\right )}{\frac {2\,x^2}{3}-\frac {2}{3}{}\mathrm {i}}\right )\,\left (-\frac {1}{12}-\frac {1}{12}{}\mathrm {i}\right )+\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (\frac {1}{3}-\frac {1}{3}{}\mathrm {i}\right )}{\frac {2\,x^2}{3}+\frac {2}{3}{}\mathrm {i}}\right )\,\left (-\frac {1}{12}+\frac {1}{12}{}\mathrm {i}\right ) \] Input:
int(1/(x^8 - x^4 + 1),x)
Output:
- 6^(1/2)*atan((6^(1/2)*x*(1/3 + 1i/3))/((2*x^2)/3 - 2i/3))*(1/12 + 1i/12) - 6^(1/2)*atan((6^(1/2)*x*(1/3 - 1i/3))/((2*x^2)/3 + 2i/3))*(1/12 - 1i/12 )
Time = 0.16 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.15 \[ \int \frac {1}{1-x^4+x^8} \, dx=-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{12}-\frac {\sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{4}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{12}+\frac {\sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{4}-\frac {\sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right )}{12}+\frac {\sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right )}{12}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{24}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{24}-\frac {\sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{8}+\frac {\sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{8}-\frac {\sqrt {6}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{24}+\frac {\sqrt {6}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{24} \] Input:
int(1/(x^8-x^4+1),x)
Output:
( - 2*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((sqrt(6) + sqrt(2) - 4*x)/(2*sqrt( - sqrt(3) + 2))) - 6*sqrt( - sqrt(3) + 2)*atan((sqrt(6) + sqrt(2) - 4*x)/ (2*sqrt( - sqrt(3) + 2))) + 2*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((sqrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2))) + 6*sqrt( - sqrt(3) + 2)*atan((s qrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2))) - 2*sqrt(6)*atan((2*sqrt ( - sqrt(3) + 2) - 4*x)/(sqrt(6) + sqrt(2))) + 2*sqrt(6)*atan((2*sqrt( - s qrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2))) - sqrt( - sqrt(3) + 2)*sqrt(3)*log ( - sqrt( - sqrt(3) + 2)*x + x**2 + 1) + sqrt( - sqrt(3) + 2)*sqrt(3)*log( sqrt( - sqrt(3) + 2)*x + x**2 + 1) - 3*sqrt( - sqrt(3) + 2)*log( - sqrt( - sqrt(3) + 2)*x + x**2 + 1) + 3*sqrt( - sqrt(3) + 2)*log(sqrt( - sqrt(3) + 2)*x + x**2 + 1) - sqrt(6)*log(( - sqrt(6)*x - sqrt(2)*x + 2*x**2 + 2)/2) + sqrt(6)*log((sqrt(6)*x + sqrt(2)*x + 2*x**2 + 2)/2))/24