Integrand size = 23, antiderivative size = 280 \[ \int \frac {1-x^4}{x^4 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{3 x^3}-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\text {arctanh}\left (\frac {\sqrt {2-\sqrt {3}} x}{1+x^2}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\text {arctanh}\left (\frac {\sqrt {2+\sqrt {3}} x}{1+x^2}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}} \] Output:
-1/3/x^3-1/4*arctan((1/2*6^(1/2)-1/2*2^(1/2)-2*x)/(1/2*6^(1/2)+1/2*2^(1/2) ))/(3/2*2^(1/2)+1/2*6^(1/2))+1/4*arctan((1/2*6^(1/2)+1/2*2^(1/2)-2*x)/(1/2 *6^(1/2)-1/2*2^(1/2)))/(3/2*2^(1/2)-1/2*6^(1/2))+1/4*arctan((1/2*6^(1/2)-1 /2*2^(1/2)+2*x)/(1/2*6^(1/2)+1/2*2^(1/2)))/(3/2*2^(1/2)+1/2*6^(1/2))-1/4*a rctan((1/2*6^(1/2)+1/2*2^(1/2)+2*x)/(1/2*6^(1/2)-1/2*2^(1/2)))/(3/2*2^(1/2 )-1/2*6^(1/2))-1/4*arctanh((1/2*6^(1/2)-1/2*2^(1/2))*x/(x^2+1))/(3/2*2^(1/ 2)-1/2*6^(1/2))+1/4*arctanh((1/2*6^(1/2)+1/2*2^(1/2))*x/(x^2+1))/(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 = 47, normalized size of antiderivative = 0.17 \[ \int \frac {1-x^4}{x^4 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{3 x^3}-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\&\right ] \] Input:
Integrate[(1 - x^4)/(x^4*(1 - x^4 + x^8)),x]
Output:
-1/3*1/x^3 - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1)/(-1 + 2*#1^4) & ]/4
Time = 0.58 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.32, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1828, 27, 1709, 1447, 1475, 1083, 217, 1478, 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.
| \(\displaystyle \int \frac {1-x^4}{x^4 \left (x^8-x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 1828 |
| \(\displaystyle -\frac {1}{3} \int \frac {3 x^4}{x^8-x^4+1}dx-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \frac {x^4}{x^8-x^4+1}dx-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1709 |
| \(\displaystyle -\frac {\int \frac {x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+\frac {\int \frac {x^2}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1447 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {x^2+1}{x^4-\sqrt {3} x^2+1}dx-\frac {1}{2} \int \frac {1-x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+\frac {\frac {1}{2} \int \frac {x^2+1}{x^4+\sqrt {3} x^2+1}dx-\frac {1}{2} \int \frac {1-x^2}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1475 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx\right )-\frac {1}{2} \int \frac {1-x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+\frac {\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx+\frac {1}{2} \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx\right )-\frac {1}{2} \int \frac {1-x^2}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\int \frac {1}{-\left (2 x-\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x-\sqrt {2-\sqrt {3}}\right )-\int \frac {1}{-\left (2 x+\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x+\sqrt {2-\sqrt {3}}\right )\right )-\frac {1}{2} \int \frac {1-x^2}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {\frac {1}{2} \left (-\int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x-\sqrt {2+\sqrt {3}}\right )-\int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x+\sqrt {2+\sqrt {3}}\right )\right )-\frac {1}{2} \int \frac {1-x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \int \frac {1-x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+\frac {\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {2+\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \int \frac {1-x^2}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1478 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\int -\frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {2+\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3}}-\frac {\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\int -\frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}-\frac {\int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {2+\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3}}-\frac {\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}-\frac {\int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {2+\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}-\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3}}-\frac {\frac {1}{2} \left (\frac {\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \left (\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}-\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
Input:
Int[(1 - x^4)/(x^4*(1 - x^4 + x^8)),x]
Output:
-1/3*1/x^3 + ((ArcTan[(-Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]]/Sqrt[2 + Sqrt[3]] + ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]]/Sqrt[2 + Sqrt[3]])/2 + (Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2]/(2*Sqrt[2 - Sqrt[3]]) - Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2]/(2*Sqrt[2 - Sqrt[3]]))/2)/(2*Sqrt[3]) - ((ArcTan[(-Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/Sqrt[2 - Sqrt[3]] + ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/Sqrt[2 - Sqrt[3]])/ 2 + (Log[1 - Sqrt[2 + Sqrt[3]]*x + x^2]/(2*Sqrt[2 + Sqrt[3]]) - Log[1 + Sq rt[2 + Sqrt[3]]*x + x^2]/(2*Sqrt[2 + Sqrt[3]]))/2)/(2*Sqrt[3])
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_) + (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[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a/c, 2]}, Simp[1/2 Int[(q + x^2)/(a + b*x^2 + c*x^4), x], x] - Simp[1/2 Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && LtQ[b ^2 - 4*a*c, 0] && PosQ[a*c]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[2*(d/e) - b/c, 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]] /; F reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] , 0]))
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[-2*(d/e) - b/c, 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]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ [c*d^2 - a*e^2, 0] && !GtQ[b^2 - 4*a*c, 0]
Int[(x_)^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> W ith[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*r) Int[x^( m - n/2)/(q - r*x^(n/2) + x^n), x], x] - Simp[1/(2*c*r) Int[x^(m - n/2)/( q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && IGtQ[m, 0] && GeQ[m, n/2] && LtQ[m, 3* (n/2)] && NegQ[b^2 - 4*a*c]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^n + c*x^ (2*n))^(p + 1)/(a*f*(m + 1))), x] + Simp[1/(a*f^n*(m + 1)) Int[(f*x)^(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x ] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && Int egerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.14
| method | result | size |
| risch | \(-\frac {1}{3 x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (81 \textit {\_Z}^{8}-9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (18 \textit {\_R}^{5}-\textit {\_R} +x \right )\right )}{4}\) | \(38\) |
| default | \(-\frac {1}{3 x^{3}}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(46\) |
Input:
int((-x^4+1)/x^4/(x^8-x^4+1),x,method=_RETURNVERBOSE)
Output:
-1/3/x^3+1/4*sum(_R*ln(18*_R^5-_R+x),_R=RootOf(81*_Z^8-9*_Z^4+1))
Time = 0.07 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.20 \[ \int \frac {1-x^4}{x^4 \left (1-x^4+x^8\right )} \, dx=\frac {3 \, \sqrt {\frac {1}{3}} x^{3} \sqrt {-\sqrt {\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}}} \log \left (3 \, \sqrt {\frac {1}{3}} \sqrt {-\frac {1}{3}} \sqrt {-\sqrt {\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}}} + x\right ) - 3 \, \sqrt {\frac {1}{3}} x^{3} \sqrt {-\sqrt {\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}}} \log \left (-3 \, \sqrt {\frac {1}{3}} \sqrt {-\frac {1}{3}} \sqrt {-\sqrt {\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}}} + x\right ) - 3 \, \sqrt {\frac {1}{3}} x^{3} \sqrt {-\sqrt {-\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}}} \log \left (3 \, \sqrt {\frac {1}{3}} \sqrt {-\frac {1}{3}} \sqrt {-\sqrt {-\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}}} + x\right ) + 3 \, \sqrt {\frac {1}{3}} x^{3} \sqrt {-\sqrt {-\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}}} \log \left (-3 \, \sqrt {\frac {1}{3}} \sqrt {-\frac {1}{3}} \sqrt {-\sqrt {-\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}}} + x\right ) + 3 \, \sqrt {\frac {1}{3}} x^{3} {\left (\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}\right )}^{\frac {1}{4}} \log \left (3 \, \sqrt {\frac {1}{3}} \sqrt {-\frac {1}{3}} {\left (\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}\right )}^{\frac {1}{4}} + x\right ) - 3 \, \sqrt {\frac {1}{3}} x^{3} {\left (\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}\right )}^{\frac {1}{4}} \log \left (-3 \, \sqrt {\frac {1}{3}} \sqrt {-\frac {1}{3}} {\left (\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}\right )}^{\frac {1}{4}} + x\right ) - 3 \, \sqrt {\frac {1}{3}} x^{3} {\left (-\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}\right )}^{\frac {1}{4}} \log \left (3 \, \sqrt {\frac {1}{3}} \sqrt {-\frac {1}{3}} {\left (-\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}\right )}^{\frac {1}{4}} + x\right ) + 3 \, \sqrt {\frac {1}{3}} x^{3} {\left (-\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}\right )}^{\frac {1}{4}} \log \left (-3 \, \sqrt {\frac {1}{3}} \sqrt {-\frac {1}{3}} {\left (-\frac {3}{2} \, \sqrt {-\frac {1}{3}} + \frac {1}{2}\right )}^{\frac {1}{4}} + x\right ) - 4}{12 \, x^{3}} \] Input:
integrate((-x^4+1)/x^4/(x^8-x^4+1),x, algorithm="fricas")
Output:
1/12*(3*sqrt(1/3)*x^3*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2))*log(3*sqrt(1/3)*sq rt(-1/3)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2)) + x) - 3*sqrt(1/3)*x^3*sqrt(-sq rt(3/2*sqrt(-1/3) + 1/2))*log(-3*sqrt(1/3)*sqrt(-1/3)*sqrt(-sqrt(3/2*sqrt( -1/3) + 1/2)) + x) - 3*sqrt(1/3)*x^3*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2))*lo g(3*sqrt(1/3)*sqrt(-1/3)*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2)) + x) + 3*sqrt( 1/3)*x^3*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2))*log(-3*sqrt(1/3)*sqrt(-1/3)*sq rt(-sqrt(-3/2*sqrt(-1/3) + 1/2)) + x) + 3*sqrt(1/3)*x^3*(3/2*sqrt(-1/3) + 1/2)^(1/4)*log(3*sqrt(1/3)*sqrt(-1/3)*(3/2*sqrt(-1/3) + 1/2)^(1/4) + x) - 3*sqrt(1/3)*x^3*(3/2*sqrt(-1/3) + 1/2)^(1/4)*log(-3*sqrt(1/3)*sqrt(-1/3)*( 3/2*sqrt(-1/3) + 1/2)^(1/4) + x) - 3*sqrt(1/3)*x^3*(-3/2*sqrt(-1/3) + 1/2) ^(1/4)*log(3*sqrt(1/3)*sqrt(-1/3)*(-3/2*sqrt(-1/3) + 1/2)^(1/4) + x) + 3*s qrt(1/3)*x^3*(-3/2*sqrt(-1/3) + 1/2)^(1/4)*log(-3*sqrt(1/3)*sqrt(-1/3)*(-3 /2*sqrt(-1/3) + 1/2)^(1/4) + x) - 4)/x^3
Time = 1.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.11 \[ \int \frac {1-x^4}{x^4 \left (1-x^4+x^8\right )} \, dx=- \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (- 18432 t^{5} + 4 t + x \right )} \right )\right )} - \frac {1}{3 x^{3}} \] Input:
integrate((-x**4+1)/x**4/(x**8-x**4+1),x)
Output:
-RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-18432*_t**5 + 4*_t + x))) - 1/(3*x**3)
\[ \int \frac {1-x^4}{x^4 \left (1-x^4+x^8\right )} \, dx=\int { -\frac {x^{4} - 1}{{\left (x^{8} - x^{4} + 1\right )} x^{4}} \,d x } \] Input:
integrate((-x^4+1)/x^4/(x^8-x^4+1),x, algorithm="maxima")
Output:
-1/3/x^3 - integrate(x^4/(x^8 - x^4 + 1), x)
Time = 0.14 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.92 \[ \int \frac {1-x^4}{x^4 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{3 \, x^{3}} \] Input:
integrate((-x^4+1)/x^4/(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) - 1/3/x^3
Time = 20.51 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.71 \[ \int \frac {1-x^4}{x^4 \left (1-x^4+x^8\right )} \, dx =\text {Too large to display} \] Input:
int(-(x^4 - 1)/(x^4*(x^8 - x^4 + 1)),x)
Output:
(3^(1/2)*atan((x*(8 - 3^(1/2)*8i)^(1/4))/(2*((3^(1/2)*(8 - 3^(1/2)*8i)^(1/ 2)*1i)/4 + (8 - 3^(1/2)*8i)^(1/2)/4)) + (3^(1/2)*x*(8 - 3^(1/2)*8i)^(1/4)* 1i)/(2*((3^(1/2)*(8 - 3^(1/2)*8i)^(1/2)*1i)/4 + (8 - 3^(1/2)*8i)^(1/2)/4)) )*(8 - 3^(1/2)*8i)^(1/4)*1i)/12 - 1/(3*x^3) + (3^(1/2)*atan((x*(8 - 3^(1/2 )*8i)^(1/4)*1i)/(2*((3^(1/2)*(8 - 3^(1/2)*8i)^(1/2)*1i)/4 + (8 - 3^(1/2)*8 i)^(1/2)/4)) - (3^(1/2)*x*(8 - 3^(1/2)*8i)^(1/4))/(2*((3^(1/2)*(8 - 3^(1/2 )*8i)^(1/2)*1i)/4 + (8 - 3^(1/2)*8i)^(1/2)/4)))*(8 - 3^(1/2)*8i)^(1/4))/12 - (2^(3/4)*3^(1/2)*atan((2^(3/4)*x*(3^(1/2)*1i + 1)^(1/4))/(2*((2^(1/2)*( 3^(1/2)*1i + 1)^(1/2))/2 - (2^(1/2)*3^(1/2)*(3^(1/2)*1i + 1)^(1/2)*1i)/2)) - (2^(3/4)*3^(1/2)*x*(3^(1/2)*1i + 1)^(1/4)*1i)/(2*((2^(1/2)*(3^(1/2)*1i + 1)^(1/2))/2 - (2^(1/2)*3^(1/2)*(3^(1/2)*1i + 1)^(1/2)*1i)/2)))*(3^(1/2)* 1i + 1)^(1/4)*1i)/12 - (2^(3/4)*3^(1/2)*atan((2^(3/4)*x*(3^(1/2)*1i + 1)^( 1/4)*1i)/(2*((2^(1/2)*(3^(1/2)*1i + 1)^(1/2))/2 - (2^(1/2)*3^(1/2)*(3^(1/2 )*1i + 1)^(1/2)*1i)/2)) + (2^(3/4)*3^(1/2)*x*(3^(1/2)*1i + 1)^(1/4))/(2*(( 2^(1/2)*(3^(1/2)*1i + 1)^(1/2))/2 - (2^(1/2)*3^(1/2)*(3^(1/2)*1i + 1)^(1/2 )*1i)/2)))*(3^(1/2)*1i + 1)^(1/4))/12
Time = 0.18 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.64 \[ \int \frac {1-x^4}{x^4 \left (1-x^4+x^8\right )} \, dx=\frac {8 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x^{3}+12 \sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x^{3}-8 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x^{3}-12 \sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x^{3}+2 \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{3}-6 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{3}-2 \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{3}+6 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{3}+4 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x^{3}-4 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x^{3}+6 \sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x^{3}-6 \sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x^{3}+\sqrt {6}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x^{3}-\sqrt {6}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x^{3}-3 \sqrt {2}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x^{3}+3 \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x^{3}-16}{48 x^{3}} \] Input:
int((-x^4+1)/x^4/(x^8-x^4+1),x)
Output:
(8*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((sqrt(6) + sqrt(2) - 4*x)/(2*sqrt( - sqrt(3) + 2)))*x**3 + 12*sqrt( - sqrt(3) + 2)*atan((sqrt(6) + sqrt(2) - 4* x)/(2*sqrt( - sqrt(3) + 2)))*x**3 - 8*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((s qrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2)))*x**3 - 12*sqrt( - sqrt(3 ) + 2)*atan((sqrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2)))*x**3 + 2*s qrt(6)*atan((2*sqrt( - sqrt(3) + 2) - 4*x)/(sqrt(6) + sqrt(2)))*x**3 - 6*s qrt(2)*atan((2*sqrt( - sqrt(3) + 2) - 4*x)/(sqrt(6) + sqrt(2)))*x**3 - 2*s qrt(6)*atan((2*sqrt( - sqrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2)))*x**3 + 6*s qrt(2)*atan((2*sqrt( - sqrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2)))*x**3 + 4*s qrt( - sqrt(3) + 2)*sqrt(3)*log( - sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x**3 - 4*sqrt( - sqrt(3) + 2)*sqrt(3)*log(sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x **3 + 6*sqrt( - sqrt(3) + 2)*log( - sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x** 3 - 6*sqrt( - sqrt(3) + 2)*log(sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x**3 + s qrt(6)*log(( - sqrt(6)*x - sqrt(2)*x + 2*x**2 + 2)/2)*x**3 - sqrt(6)*log(( sqrt(6)*x + sqrt(2)*x + 2*x**2 + 2)/2)*x**3 - 3*sqrt(2)*log(( - sqrt(6)*x - sqrt(2)*x + 2*x**2 + 2)/2)*x**3 + 3*sqrt(2)*log((sqrt(6)*x + sqrt(2)*x + 2*x**2 + 2)/2)*x**3 - 16)/(48*x**3)