Integrand size = 16, antiderivative size = 286 \[ \int \frac {x^8}{1-x^4+x^8} \, dx=x+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {3}} x}{1+x^2}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {3}} x}{1+x^2}\right ) \] Output:
x+1/4*(1/2*2^(1/2)+1/6*6^(1/2))*arctan((1/2*6^(1/2)-1/2*2^(1/2)-2*x)/(1/2* 6^(1/2)+1/2*2^(1/2)))-1/4*(1/2*2^(1/2)-1/6*6^(1/2))*arctan((1/2*6^(1/2)+1/ 2*2^(1/2)-2*x)/(1/2*6^(1/2)-1/2*2^(1/2)))-1/4*(1/2*2^(1/2)+1/6*6^(1/2))*ar ctan((1/2*6^(1/2)-1/2*2^(1/2)+2*x)/(1/2*6^(1/2)+1/2*2^(1/2)))+1/4*(1/2*2^( 1/2)-1/6*6^(1/2))*arctan((1/2*6^(1/2)+1/2*2^(1/2)+2*x)/(1/2*6^(1/2)-1/2*2^ (1/2)))+1/4*(1/2*2^(1/2)-1/6*6^(1/2))*arctanh((1/2*6^(1/2)-1/2*2^(1/2))*x/ (x^2+1))-1/4*(1/2*2^(1/2)+1/6*6^(1/2))*arctanh((1/2*6^(1/2)+1/2*2^(1/2))*x /(x^2+1))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.21 \[ \int \frac {x^8}{1-x^4+x^8} \, dx=x+\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \] Input:
Integrate[x^8/(1 - x^4 + x^8),x]
Output:
x + RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4
Time = 0.68 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1703, 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.
| \(\displaystyle \int \frac {x^8}{x^8-x^4+1} \, dx\) |
| \(\Big \downarrow \) 1703 |
| \(\displaystyle x-\int \frac {1-x^4}{x^8-x^4+1}dx\) |
| \(\Big \downarrow \) 1751 |
| \(\displaystyle \frac {\int -\frac {\sqrt {3}-2 x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+\frac {\int -\frac {2 x^2+\sqrt {3}}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+x\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\sqrt {3}-2 x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {\int \frac {2 x^2+\sqrt {3}}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+x\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle -\frac {\frac {\int \frac {\left (2-\sqrt {3}\right ) x+\sqrt {3 \left (2-\sqrt {3}\right )}}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}-\left (2-\sqrt {3}\right ) x}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}-\frac {\frac {\int \frac {\sqrt {3 \left (2+\sqrt {3}\right )}-\left (2+\sqrt {3}\right ) x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\int \frac {\left (2+\sqrt {3}\right ) x+\sqrt {3 \left (2+\sqrt {3}\right )}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}+x\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx+\frac {1}{2} \left (2-\sqrt {3}\right ) \int -\frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}-\frac {\frac {-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \left (2+\sqrt {3}\right ) \int -\frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}+x\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}-\frac {\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}+x\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x-\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \int \frac {1}{-\left (2 x+\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x+\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}-\frac {\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {2-\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x-\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {2-\sqrt {3}} \int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x+\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}+x\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}-\frac {\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}+x\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{2} \left (2-\sqrt {3}\right ) \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (2-\sqrt {3}\right ) \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}-\frac {\frac {-\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \left (2+\sqrt {3}\right ) \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )-\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}+x\) |
Input:
Int[x^8/(1 - x^4 + x^8),x]
Output:
x - ((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]]) + (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[( -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]]) + (-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])
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_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x _Symbol] :> Simp[d^(2*n - 1)*(d*x)^(m - 2*n + 1)*((a + b*x^n + c*x^(2*n))^( p + 1)/(c*(m + 2*n*p + 1))), x] - Simp[d^(2*n)/(c*(m + 2*n*p + 1)) Int[(d *x)^(m - 2*n)*Simp[a*(m - 2*n + 1) + b*(m + n*(p - 1) + 1)*x^n, x]*(a + b*x ^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2*n] && N eQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1] && NeQ[m + 2*n*p + 1, 0 ] && IntegerQ[p]
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 = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.15
| method | result | size |
| default | \(x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(44\) |
| risch | \(x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(44\) |
Input:
int(x^8/(x^8-x^4+1),x,method=_RETURNVERBOSE)
Output:
x+1/4*sum((_R^4-1)/(2*_R^7-_R^3)*ln(x-_R),_R=RootOf(_Z^8-_Z^4+1))
Time = 0.07 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.18 \[ \int \frac {x^8}{1-x^4+x^8} \, dx =\text {Too large to display} \] Input:
integrate(x^8/(x^8-x^4+1),x, algorithm="fricas")
Output:
1/4*sqrt(1/3)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2))*log(3*sqrt(1/3)*(sqrt(-1/3 ) - 1)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2)) + 2*x) - 1/4*sqrt(1/3)*sqrt(-sqrt (3/2*sqrt(-1/3) + 1/2))*log(-3*sqrt(1/3)*(sqrt(-1/3) - 1)*sqrt(-sqrt(3/2*s qrt(-1/3) + 1/2)) + 2*x) - 1/4*sqrt(1/3)*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2) )*log(3*sqrt(1/3)*(sqrt(-1/3) + 1)*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2)) + 2* x) + 1/4*sqrt(1/3)*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2))*log(-3*sqrt(1/3)*(sq rt(-1/3) + 1)*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2)) + 2*x) + 1/4*sqrt(1/3)*(3 /2*sqrt(-1/3) + 1/2)^(1/4)*log(3*sqrt(1/3)*(3/2*sqrt(-1/3) + 1/2)^(1/4)*(s qrt(-1/3) - 1) + 2*x) - 1/4*sqrt(1/3)*(3/2*sqrt(-1/3) + 1/2)^(1/4)*log(-3* sqrt(1/3)*(3/2*sqrt(-1/3) + 1/2)^(1/4)*(sqrt(-1/3) - 1) + 2*x) - 1/4*sqrt( 1/3)*(-3/2*sqrt(-1/3) + 1/2)^(1/4)*log(3*sqrt(1/3)*(sqrt(-1/3) + 1)*(-3/2* sqrt(-1/3) + 1/2)^(1/4) + 2*x) + 1/4*sqrt(1/3)*(-3/2*sqrt(-1/3) + 1/2)^(1/ 4)*log(-3*sqrt(1/3)*(sqrt(-1/3) + 1)*(-3/2*sqrt(-1/3) + 1/2)^(1/4) + 2*x) + x
Time = 1.51 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.09 \[ \int \frac {x^8}{1-x^4+x^8} \, dx=x + \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (9216 t^{5} - 8 t + x \right )} \right )\right )} \] Input:
integrate(x**8/(x**8-x**4+1),x)
Output:
x + RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(9216*_t**5 - 8*_t + x)))
\[ \int \frac {x^8}{1-x^4+x^8} \, dx=\int { \frac {x^{8}}{x^{8} - x^{4} + 1} \,d x } \] Input:
integrate(x^8/(x^8-x^4+1),x, algorithm="maxima")
Output:
x + integrate((x^4 - 1)/(x^8 - x^4 + 1), x)
Time = 0.12 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.89 \[ \int \frac {x^8}{1-x^4+x^8} \, dx=-\frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) + x \] Input:
integrate(x^8/(x^8-x^4+1),x, algorithm="giac")
Output:
-1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sq rt(2))) - 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x - sqrt(6) + sqrt(2))/(sqr t(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*(s qrt(6) - sqrt(2)) + 1) + 1/48*(sqrt(6) - 3*sqrt(2))*log(x^2 - 1/2*x*(sqrt( 6) - sqrt(2)) + 1) + x
Time = 19.01 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.73 \[ \int \frac {x^8}{1-x^4+x^8} \, dx=x+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}-\frac {\sqrt {3}\,x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {3}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \] Input:
int(x^8/(x^8 - x^4 + 1),x)
Output:
x + (3^(1/2)*atan(x/(8 - 3^(1/2)*8i)^(1/4) + (3^(1/2)*x*1i)/(8 - 3^(1/2)*8 i)^(1/4))*(8 - 3^(1/2)*8i)^(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 - (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 - (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
Time = 0.18 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.05 \[ \int \frac {x^8}{1-x^4+x^8} \, dx=-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{12}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{12}+\frac {\sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right )}{24}+\frac {\sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}-\frac {\sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right )}{24}-\frac {\sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{24}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{24}+\frac {\sqrt {6}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{48}-\frac {\sqrt {6}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{48}+\frac {\sqrt {2}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{16}-\frac {\sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{16}+x \] Input:
int(x^8/(x^8-x^4+1),x)
Output:
( - 4*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((sqrt(6) + sqrt(2) - 4*x)/(2*sqrt( - sqrt(3) + 2))) + 4*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((sqrt(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))) + 6*sqrt(2)*atan((2*sqrt( - sqrt(3) + 2) - 4* x)/(sqrt(6) + sqrt(2))) - 2*sqrt(6)*atan((2*sqrt( - sqrt(3) + 2) + 4*x)/(s qrt(6) + sqrt(2))) - 6*sqrt(2)*atan((2*sqrt( - sqrt(3) + 2) + 4*x)/(sqrt(6 ) + sqrt(2))) - 2*sqrt( - sqrt(3) + 2)*sqrt(3)*log( - sqrt( - sqrt(3) + 2) *x + x**2 + 1) + 2*sqrt( - sqrt(3) + 2)*sqrt(3)*log(sqrt( - sqrt(3) + 2)*x + x**2 + 1) + sqrt(6)*log(( - sqrt(6)*x - sqrt(2)*x + 2*x**2 + 2)/2) - sq rt(6)*log((sqrt(6)*x + sqrt(2)*x + 2*x**2 + 2)/2) + 3*sqrt(2)*log(( - sqrt (6)*x - sqrt(2)*x + 2*x**2 + 2)/2) - 3*sqrt(2)*log((sqrt(6)*x + sqrt(2)*x + 2*x**2 + 2)/2) + 48*x)/48