Integrand size = 23, antiderivative size = 224 \[ \int \frac {1-x^4}{x^2 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{x}+\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\text {arctanh}\left (\frac {\sqrt {2-\sqrt {3}} x}{1+x^2}\right )}{2 \sqrt {6}}+\frac {\text {arctanh}\left (\frac {\sqrt {2+\sqrt {3}} x}{1+x^2}\right )}{2 \sqrt {6}} \] Output:
-1/x+1/12*arctan((1/2*6^(1/2)-1/2*2^(1/2)-2*x)/(1/2*6^(1/2)+1/2*2^(1/2)))* 6^(1/2)+1/12*arctan((1/2*6^(1/2)+1/2*2^(1/2)-2*x)/(1/2*6^(1/2)-1/2*2^(1/2) ))*6^(1/2)-1/12*arctan((1/2*6^(1/2)-1/2*2^(1/2)+2*x)/(1/2*6^(1/2)+1/2*2^(1 /2)))*6^(1/2)-1/12*arctan((1/2*6^(1/2)+1/2*2^(1/2)+2*x)/(1/2*6^(1/2)-1/2*2 ^(1/2)))*6^(1/2)+1/12*arctanh((1/2*6^(1/2)-1/2*2^(1/2))*x/(x^2+1))*6^(1/2) +1/12*arctanh((1/2*6^(1/2)+1/2*2^(1/2))*x/(x^2+1))*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 = 0.21 \[ \int \frac {1-x^4}{x^2 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{x}-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^3}{-1+2 \text {$\#$1}^4}\&\right ] \] Input:
Integrate[(1 - x^4)/(x^2*(1 - x^4 + x^8)),x]
Output:
-x^(-1) - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^3)/(-1 + 2*#1^4) & ] /4
Time = 0.62 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.87, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {1828, 1708, 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.
| \(\displaystyle \int \frac {1-x^4}{x^2 \left (x^8-x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 1828 |
| \(\displaystyle -\int \frac {x^6}{x^8-x^4+1}dx-\frac {1}{x}\) |
| \(\Big \downarrow \) 1708 |
| \(\displaystyle \frac {\int \frac {1-\sqrt {3} x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {\int \frac {\sqrt {3} x^2+1}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle -\frac {\frac {\int \frac {\sqrt {2-\sqrt {3}}-\left (1-\sqrt {3}\right ) x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\int \frac {\left (1-\sqrt {3}\right ) x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\int \frac {\sqrt {2+\sqrt {3}}-\left (1+\sqrt {3}\right ) x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\int \frac {\left (1+\sqrt {3}\right ) x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{\sqrt {2}}-\frac {1}{2} \left (1-\sqrt {3}\right ) \int -\frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {\int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{\sqrt {2}}+\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {-\frac {\int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{\sqrt {2}}-\frac {1}{2} \left (1+\sqrt {3}\right ) \int -\frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\frac {\int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{\sqrt {2}}}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{\sqrt {2}}+\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {\int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{\sqrt {2}}+\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\frac {\int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{\sqrt {2}}}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\frac {\int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{\sqrt {2}}}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2} \int \frac {1}{-\left (2 x-\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x-\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2} \int \frac {1}{-\left (2 x+\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x+\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {2} \int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x-\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {2} \int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x+\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx+\sqrt {\frac {2}{2+\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx+\sqrt {\frac {2}{2+\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\sqrt {\frac {2}{2-\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\sqrt {\frac {2}{2-\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{x}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\frac {\sqrt {\frac {2}{2+\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (1-\sqrt {3}\right ) \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\sqrt {\frac {2}{2+\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{2} \left (1-\sqrt {3}\right ) \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {-\sqrt {\frac {2}{2-\sqrt {3}}} \arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \left (1+\sqrt {3}\right ) \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (1+\sqrt {3}\right ) \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )-\sqrt {\frac {2}{2-\sqrt {3}}} \arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{x}\) |
Input:
Int[(1 - x^4)/(x^2*(1 - x^4 + x^8)),x]
Output:
-x^(-1) - ((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*Sq rt[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 - Sqrt[3]]*x + x^2])/2 )/(2*Sqrt[2 - Sqrt[3]]))/(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 + 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])*Lo g[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[(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 - 3*(n/2))*((q - r*x^(n/2))/(q - r*x^(n/2) + x^n)), x], x] + Simp[1/(2*c *r) Int[x^(m - 3*(n/2))*((q + r*x^(n/2))/(q + r*x^(n/2) + x^n)), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && IGtQ[m, 0] && GeQ[m, 3*(n/2)] && LtQ[m, 2*n] && 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 = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.17
| method | result | size |
| default | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (9 x \,\textit {\_R}^{3}-3 \textit {\_R}^{2}+x^{2}\right )\right )}{4}-\frac {1}{x}\) | \(38\) |
| risch | \(-\frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (-9 x \,\textit {\_R}^{3}-3 \textit {\_R}^{2}+x^{2}\right )\right )}{4}\) | \(38\) |
Input:
int((-x^4+1)/x^2/(x^8-x^4+1),x,method=_RETURNVERBOSE)
Output:
-1/4*sum(_R*ln(9*_R^3*x-3*_R^2+x^2),_R=RootOf(9*_Z^4+1))-1/x
Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.62 \[ \int \frac {1-x^4}{x^2 \left (1-x^4+x^8\right )} \, dx=-\frac {2 \, \sqrt {\frac {2}{3}} x \arctan \left (4 \, x^{2} + 3 \, \sqrt {\frac {2}{3}} {\left (x^{3} + 2 \, x\right )} + 3\right ) + 2 \, \sqrt {\frac {2}{3}} x \arctan \left (-4 \, x^{2} + 3 \, \sqrt {\frac {2}{3}} {\left (x^{3} + 2 \, x\right )} - 3\right ) + 2 \, \sqrt {\frac {2}{3}} x \arctan \left (\sqrt {\frac {2}{3}} x + \frac {1}{3}\right ) + 2 \, \sqrt {\frac {2}{3}} x \arctan \left (\sqrt {\frac {2}{3}} x - \frac {1}{3}\right ) - \sqrt {\frac {2}{3}} x \log \left (x^{4} + 3 \, x^{2} + 3 \, \sqrt {\frac {2}{3}} {\left (x^{3} + x\right )} + 1\right ) + \sqrt {\frac {2}{3}} x \log \left (x^{4} + 3 \, x^{2} - 3 \, \sqrt {\frac {2}{3}} {\left (x^{3} + x\right )} + 1\right ) + 8}{8 \, x} \] Input:
integrate((-x^4+1)/x^2/(x^8-x^4+1),x, algorithm="fricas")
Output:
-1/8*(2*sqrt(2/3)*x*arctan(4*x^2 + 3*sqrt(2/3)*(x^3 + 2*x) + 3) + 2*sqrt(2 /3)*x*arctan(-4*x^2 + 3*sqrt(2/3)*(x^3 + 2*x) - 3) + 2*sqrt(2/3)*x*arctan( sqrt(2/3)*x + 1/3) + 2*sqrt(2/3)*x*arctan(sqrt(2/3)*x - 1/3) - sqrt(2/3)*x *log(x^4 + 3*x^2 + 3*sqrt(2/3)*(x^3 + x) + 1) + sqrt(2/3)*x*log(x^4 + 3*x^ 2 - 3*sqrt(2/3)*(x^3 + x) + 1) + 8)/x
Time = 0.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.75 \[ \int \frac {1-x^4}{x^2 \left (1-x^4+x^8\right )} \, dx=- \frac {\sqrt {6} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} - \frac {1}{3} \right )} + 2 \operatorname {atan}{\left (\sqrt {6} x^{3} - 4 x^{2} + 2 \sqrt {6} x - 3 \right )}\right )}{24} - \frac {\sqrt {6} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} + \frac {1}{3} \right )} + 2 \operatorname {atan}{\left (\sqrt {6} x^{3} + 4 x^{2} + 2 \sqrt {6} x + 3 \right )}\right )}{24} - \frac {\sqrt {6} \log {\left (x^{4} - \sqrt {6} x^{3} + 3 x^{2} - \sqrt {6} x + 1 \right )}}{24} + \frac {\sqrt {6} \log {\left (x^{4} + \sqrt {6} x^{3} + 3 x^{2} + \sqrt {6} x + 1 \right )}}{24} - \frac {1}{x} \] Input:
integrate((-x**4+1)/x**2/(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*sqr t(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 - 1/x
\[ \int \frac {1-x^4}{x^2 \left (1-x^4+x^8\right )} \, dx=\int { -\frac {x^{4} - 1}{{\left (x^{8} - x^{4} + 1\right )} x^{2}} \,d x } \] Input:
integrate((-x^4+1)/x^2/(x^8-x^4+1),x, algorithm="maxima")
Output:
-1/x - integrate(x^6/(x^8 - x^4 + 1), x)
Time = 0.14 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.94 \[ \int \frac {1-x^4}{x^2 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, \sqrt {6} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{24} \, \sqrt {6} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{24} \, \sqrt {6} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{24} \, \sqrt {6} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{x} \] Input:
integrate((-x^4+1)/x^2/(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)*a rctan((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) - 1/x
Time = 20.85 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.26 \[ \int \frac {1-x^4}{x^2 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{x}+\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (\frac {1}{3}+\frac {1}{3}{}\mathrm {i}\right )}{\frac {2\,x^2}{3}-\frac {2}{3}{}\mathrm {i}}\right )\,\left (\frac {1}{12}-\frac {1}{12}{}\mathrm {i}\right )+\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (\frac {1}{3}-\frac {1}{3}{}\mathrm {i}\right )}{\frac {2\,x^2}{3}+\frac {2}{3}{}\mathrm {i}}\right )\,\left (\frac {1}{12}+\frac {1}{12}{}\mathrm {i}\right ) \] Input:
int(-(x^4 - 1)/(x^2*(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) - 1/x
Time = 0.18 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.48 \[ \int \frac {1-x^4}{x^2 \left (1-x^4+x^8\right )} \, dx=\frac {2 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x +6 \sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x -2 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x -6 \sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x +2 \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right ) x -2 \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right ) x -\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x +\sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x -3 \sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x +3 \sqrt {-\sqrt {3}+2}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x -\sqrt {6}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x +\sqrt {6}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x -24}{24 x} \] Input:
int((-x^4+1)/x^2/(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)))*x + 6*sqrt( - sqrt(3) + 2)*atan((sqrt(6) + sqrt(2) - 4*x)/( 2*sqrt( - sqrt(3) + 2)))*x - 2*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((sqrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2)))*x - 6*sqrt( - sqrt(3) + 2)*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 - 2*sqrt(6)*atan((2*sq rt( - sqrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2)))*x - sqrt( - sqrt(3) + 2)*sq rt(3)*log( - sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x + sqrt( - sqrt(3) + 2)*s qrt(3)*log(sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x - 3*sqrt( - sqrt(3) + 2)*l og( - sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x + 3*sqrt( - sqrt(3) + 2)*log(sq rt( - 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 - 24)/(24*x)