Integrand size = 16, antiderivative size = 290 \[ \int \frac {1}{x^2 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{x}+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {3}} x}{1+x^2}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {3}} x}{1+x^2}\right ) \] Output:
-1/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)) *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))*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 = 61, normalized size of antiderivative = 0.21 \[ \int \frac {1}{x^2 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{x}-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \] Input:
Integrate[1/(x^2*(1 - x^4 + x^8)),x]
Output:
-x^(-1) - RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(- #1 + 2*#1^5) & ]/4
Time = 0.67 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.26, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {1704, 1830, 1602, 25, 1483, 27, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (x^8-x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 1704 |
| \(\displaystyle \int \frac {x^2 \left (1-x^4\right )}{x^8-x^4+1}dx-\frac {1}{x}\) |
| \(\Big \downarrow \) 1830 |
| \(\displaystyle \frac {\int \frac {x^2 \left (\sqrt {3}-2 x^2\right )}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+\frac {\int \frac {x^2 \left (2 x^2+\sqrt {3}\right )}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {-\int -\frac {2-\sqrt {3} x^2}{x^4-\sqrt {3} x^2+1}dx-2 x}{2 \sqrt {3}}+\frac {2 x-\int \frac {\sqrt {3} x^2+2}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2-\sqrt {3} x^2}{x^4-\sqrt {3} x^2+1}dx-2 x}{2 \sqrt {3}}+\frac {2 x-\int \frac {\sqrt {3} x^2+2}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {-\frac {\int \frac {2 \sqrt {2-\sqrt {3}}-\left (2-\sqrt {3}\right ) x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}-\frac {\int \frac {\sqrt {2-\sqrt {3}} \left (\sqrt {2-\sqrt {3}} x+2\right )}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+2 x}{2 \sqrt {3}}+\frac {\frac {\int \frac {2 \sqrt {2+\sqrt {3}}-\left (2+\sqrt {3}\right ) x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\int \frac {\sqrt {2+\sqrt {3}} \left (\sqrt {2+\sqrt {3}} x+2\right )}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}-2 x}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {2 \sqrt {2-\sqrt {3}}-\left (2-\sqrt {3}\right ) x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}-\frac {1}{2} \int \frac {\sqrt {2-\sqrt {3}} x+2}{x^2+\sqrt {2-\sqrt {3}} x+1}dx+2 x}{2 \sqrt {3}}+\frac {\frac {\int \frac {2 \sqrt {2+\sqrt {3}}-\left (2+\sqrt {3}\right ) x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {1}{2} \int \frac {\sqrt {2+\sqrt {3}} x+2}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-2 x}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \left (2-\sqrt {3}\right ) \int -\frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {1}{2} \left (-\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx\right )+2 x}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \left (2+\sqrt {3}\right ) \int -\frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {1}{2} \left (\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx\right )-2 x}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx+\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {1}{2} \left (-\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx\right )+2 x}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {1}{2} \left (\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx\right )-2 x}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x-\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {1}{2} \left (\left (2+\sqrt {3}\right ) \int \frac {1}{-\left (2 x+\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x+\sqrt {2-\sqrt {3}}\right )-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx\right )+2 x}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\sqrt {2-\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x-\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {1}{2} \left (\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\left (2-\sqrt {3}\right ) \int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x+\sqrt {2+\sqrt {3}}\right )\right )-2 x}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx+\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {1}{2} \left (-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )\right )+2 x}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {1}{2} \left (\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {2-\sqrt {3}} \arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )\right )-2 x}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (2-\sqrt {3}\right ) \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {1}{2} \left (-\sqrt {2+\sqrt {3}} \arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \sqrt {2-\sqrt {3}} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )\right )+2 x}{2 \sqrt {3}}+\frac {\frac {\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \left (2+\sqrt {3}\right ) \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {1}{2} \left (\sqrt {2-\sqrt {3}} \arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{2} \sqrt {2+\sqrt {3}} \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )\right )-2 x}{2 \sqrt {3}}-\frac {1}{x}\) |
Input:
Int[1/(x^2*(1 - x^4 + x^8)),x]
Output:
-x^(-1) + (2*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]]) + (-(Sqrt[2 + Sqrt[3]]*ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]] ]) - (Sqrt[2 - Sqrt[3]]*Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2])/2)/2)/(2*Sqrt[ 3]) + (-2*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]]) + (Sqrt[2 - Sqrt[3]]*ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]] + ( Sqrt[2 + Sqrt[3]]*Log[1 + Sqrt[2 + Sqrt[3]]*x + x^2])/2)/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[((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[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3)) Int[(f*x)^(m - 2)* (a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c , 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | | IntegerQ[m])
Int[((d_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_ Symbol] :> Simp[(d*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*d*(m + 1) )), x] - Simp[1/(a*d^n*(m + 1)) Int[(d*x)^(m + n)*(b*(m + n*(p + 1) + 1) + c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{ a, b, c, d, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]
Int[(((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q - b*c, 2]}, Simp[c/(2*q*r) Int[(f*x)^m*(Simp[d*r - (c*d - e*q)*x^(n/2), x] /(q - r*x^(n/2) + c*x^n)), x], x] + Simp[c/(2*q*r) Int[(f*x)^m*(Simp[d*r + (c*d - e*q)*x^(n/2), x]/(q + r*x^(n/2) + c*x^n)), x], x]] /; !LtQ[2*c*q - b*c, 0]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && LtQ[b^2 - 4*a *c, 0] && IntegersQ[m, n/2] && LtQ[0, m, n] && PosQ[a*c]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.14
| method | result | size |
| risch | \(-\frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (81 \textit {\_Z}^{8}-9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (-27 \textit {\_R}^{7}+6 \textit {\_R}^{3}+x \right )\right )}{4}\) | \(40\) |
| default | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}-\textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}-\frac {1}{x}\) | \(52\) |
Input:
int(1/x^2/(x^8-x^4+1),x,method=_RETURNVERBOSE)
Output:
-1/x+1/4*sum(_R*ln(-27*_R^7+6*_R^3+x),_R=RootOf(81*_Z^8-9*_Z^4+1))
Time = 0.07 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^2 \left (1-x^4+x^8\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/x^2/(x^8-x^4+1),x, algorithm="fricas")
Output:
1/4*(sqrt(1/3)*x*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2))*log(3*sqrt(1/3)*sqrt(3/ 2*sqrt(-1/3) + 1/2)*(sqrt(-1/3) - 1)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2)) + 2 *x) - sqrt(1/3)*x*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2))*log(-3*sqrt(1/3)*sqrt( 3/2*sqrt(-1/3) + 1/2)*(sqrt(-1/3) - 1)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2)) + 2*x) - sqrt(1/3)*x*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2))*log(3*sqrt(1/3)*(sq rt(-1/3) + 1)*sqrt(-3/2*sqrt(-1/3) + 1/2)*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2 )) + 2*x) + sqrt(1/3)*x*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2))*log(-3*sqrt(1/3 )*(sqrt(-1/3) + 1)*sqrt(-3/2*sqrt(-1/3) + 1/2)*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2)) + 2*x) - sqrt(1/3)*x*(3/2*sqrt(-1/3) + 1/2)^(1/4)*log(3*sqrt(1/3)* (3/2*sqrt(-1/3) + 1/2)^(3/4)*(sqrt(-1/3) - 1) + 2*x) + sqrt(1/3)*x*(3/2*sq rt(-1/3) + 1/2)^(1/4)*log(-3*sqrt(1/3)*(3/2*sqrt(-1/3) + 1/2)^(3/4)*(sqrt( -1/3) - 1) + 2*x) + sqrt(1/3)*x*(-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)^(3/4) + 2*x) - sqrt(1/3)*x*(- 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)^(3/4) + 2*x) - 4)/x
Time = 1.66 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.10 \[ \int \frac {1}{x^2 \left (1-x^4+x^8\right )} \, dx=\operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (- 442368 t^{7} + 384 t^{3} + x \right )} \right )\right )} - \frac {1}{x} \] Input:
integrate(1/x**2/(x**8-x**4+1),x)
Output:
RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-442368*_t**7 + 384*_t**3 + x))) - 1/x
\[ \int \frac {1}{x^2 \left (1-x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - x^{4} + 1\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(x^8-x^4+1),x, algorithm="maxima")
Output:
-1/x - integrate((x^6 - x^2)/(x^8 - x^4 + 1), x)
Time = 0.14 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{x} \] Input:
integrate(1/x^2/(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/x
Time = 19.36 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^2 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{x}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}+\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}-\frac {\sqrt {3}\,x\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{2\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{3/4}}-\frac {2^{3/4}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{3/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{3/4}}+\frac {2^{3/4}\,\sqrt {3}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{3/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \] Input:
int(1/(x^2*(x^8 - x^4 + 1)),x)
Output:
(3^(1/2)*atan((x*(8 - 3^(1/2)*8i)^(1/4))/(2*(3^(1/2)*1i - 1)) + (3^(1/2)*x *(8 - 3^(1/2)*8i)^(1/4)*1i)/(2*(3^(1/2)*1i - 1)))*(8 - 3^(1/2)*8i)^(1/4)*1 i)/12 - 1/x - (3^(1/2)*atan((x*(8 - 3^(1/2)*8i)^(1/4)*1i)/(2*(3^(1/2)*1i - 1)) - (3^(1/2)*x*(8 - 3^(1/2)*8i)^(1/4))/(2*(3^(1/2)*1i - 1)))*(8 - 3^(1/ 2)*8i)^(1/4))/12 + (2^(3/4)*3^(1/2)*atan((2^(3/4)*x)/(2*(3^(1/2)*1i + 1)^( 3/4)) - (2^(3/4)*3^(1/2)*x*1i)/(2*(3^(1/2)*1i + 1)^(3/4)))*(3^(1/2)*1i + 1 )^(1/4)*1i)/12 - (2^(3/4)*3^(1/2)*atan((2^(3/4)*x*1i)/(2*(3^(1/2)*1i + 1)^ (3/4)) + (2^(3/4)*3^(1/2)*x)/(2*(3^(1/2)*1i + 1)^(3/4)))*(3^(1/2)*1i + 1)^ (1/4))/12
Time = 0.18 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \left (1-x^4+x^8\right )} \, dx=\frac {-4 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x +4 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x +2 \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right ) x +6 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right ) x -2 \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right ) x -6 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right ) x +2 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x -2 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x -\sqrt {6}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x +\sqrt {6}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x -3 \sqrt {2}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x +3 \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x -48}{48 x} \] Input:
int(1/x^2/(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)))*x + 4*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((sqrt(6) + sqrt( 2) + 4*x)/(2*sqrt( - sqrt(3) + 2)))*x + 2*sqrt(6)*atan((2*sqrt( - sqrt(3) + 2) - 4*x)/(sqrt(6) + sqrt(2)))*x + 6*sqrt(2)*atan((2*sqrt( - sqrt(3) + 2 ) - 4*x)/(sqrt(6) + sqrt(2)))*x - 2*sqrt(6)*atan((2*sqrt( - sqrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2)))*x - 6*sqrt(2)*atan((2*sqrt( - sqrt(3) + 2) + 4* x)/(sqrt(6) + sqrt(2)))*x + 2*sqrt( - sqrt(3) + 2)*sqrt(3)*log( - sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x - 2*sqrt( - sqrt(3) + 2)*sqrt(3)*log(sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x - sqrt(6)*log(( - sqrt(6)*x - sqrt(2)*x + 2*x **2 + 2)/2)*x + sqrt(6)*log((sqrt(6)*x + sqrt(2)*x + 2*x**2 + 2)/2)*x - 3* sqrt(2)*log(( - sqrt(6)*x - sqrt(2)*x + 2*x**2 + 2)/2)*x + 3*sqrt(2)*log(( sqrt(6)*x + sqrt(2)*x + 2*x**2 + 2)/2)*x - 48)/(48*x)