Integrand size = 16, antiderivative size = 292 \[ \int \frac {1}{x^4 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{3 x^3}-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {3}} x}{1+x^2}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {3}} x}{1+x^2}\right ) \] Output:
-1/3/x^3-1/4*(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 = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^4 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{3 x^3}-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \] Input:
Integrate[1/(x^4*(1 - x^4 + x^8)),x]
Output:
-1/3*1/x^3 - RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4) /(-#1^3 + 2*#1^7) & ]/4
Time = 0.67 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.21, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {1704, 27, 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.
| \(\displaystyle \int \frac {1}{x^4 \left (x^8-x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 1704 |
| \(\displaystyle \frac {1}{3} \int \frac {3 \left (1-x^4\right )}{x^8-x^4+1}dx-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1-x^4}{x^8-x^4+1}dx-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1751 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {3}-2 x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {\int -\frac {2 x^2+\sqrt {3}}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sqrt {3}-2 x^2}{x^4-\sqrt {3} x^2+1}dx}{2 \sqrt {3}}+\frac {\int \frac {2 x^2+\sqrt {3}}{x^4+\sqrt {3} x^2+1}dx}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\frac {\int \frac {\left (2-\sqrt {3}\right ) x+\sqrt {3 \left (2-\sqrt {3}\right )}}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}-\left (2-\sqrt {3}\right ) x}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\int \frac {\sqrt {3 \left (2+\sqrt {3}\right )}-\left (2+\sqrt {3}\right ) x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\int \frac {\left (2+\sqrt {3}\right ) x+\sqrt {3 \left (2+\sqrt {3}\right )}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx+\frac {1}{2} \left (2-\sqrt {3}\right ) \int -\frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \left (2+\sqrt {3}\right ) \int -\frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\frac {1}{2} \sqrt {2+\sqrt {3}} \int \frac {1}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2-\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\frac {1}{2} \sqrt {2-\sqrt {3}} \int \frac {1}{x^2+\sqrt {2+\sqrt {3}} x+1}dx}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x-\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx-\sqrt {2+\sqrt {3}} \int \frac {1}{-\left (2 x+\sqrt {2-\sqrt {3}}\right )^2-\sqrt {3}-2}d\left (2 x+\sqrt {2-\sqrt {3}}\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {2-\sqrt {3}} \int \frac {1}{-\left (2 x-\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x-\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx+\sqrt {2-\sqrt {3}} \int \frac {1}{-\left (2 x+\sqrt {2+\sqrt {3}}\right )^2+\sqrt {3}-2}d\left (2 x+\sqrt {2+\sqrt {3}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}-2 x}{x^2-\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {2 x+\sqrt {2-\sqrt {3}}}{x^2+\sqrt {2-\sqrt {3}} x+1}dx}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {\sqrt {2+\sqrt {3}}-2 x}{x^2-\sqrt {2+\sqrt {3}} x+1}dx-\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \int \frac {2 x+\sqrt {2+\sqrt {3}}}{x^2+\sqrt {2+\sqrt {3}} x+1}dx-\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {2 x-\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{2} \left (2-\sqrt {3}\right ) \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{2} \left (2-\sqrt {3}\right ) \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{2 \sqrt {2-\sqrt {3}}}}{2 \sqrt {3}}+\frac {\frac {-\arctan \left (\frac {2 x-\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{2} \left (2+\sqrt {3}\right ) \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{2 \sqrt {2+\sqrt {3}}}+\frac {\frac {1}{2} \left (2+\sqrt {3}\right ) \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )-\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {2+\sqrt {3}}}}{2 \sqrt {3}}-\frac {1}{3 x^3}\) |
Input:
Int[1/(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]]] + ((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_)*(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[((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[((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 = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.13
| method | result | size |
| risch | \(-\frac {1}{3 x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (81 \textit {\_Z}^{8}-9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (-9 \textit {\_R}^{5}+2 \textit {\_R} +x \right )\right )}{4}\) | \(38\) |
| default | \(-\frac {1}{3 x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(50\) |
Input:
int(1/x^4/(x^8-x^4+1),x,method=_RETURNVERBOSE)
Output:
-1/3/x^3+1/4*sum(_R*ln(-9*_R^5+2*_R+x),_R=RootOf(81*_Z^8-9*_Z^4+1))
Time = 0.08 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.26 \[ \int \frac {1}{x^4 \left (1-x^4+x^8\right )} \, dx =\text {Too large to display} \] Input:
integrate(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)*( sqrt(-1/3) - 1)*sqrt(-sqrt(3/2*sqrt(-1/3) + 1/2)) + 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) - 1)*sqrt( -sqrt(3/2*sqrt(-1/3) + 1/2)) + 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) + 1)*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2)) + 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) + 1)*sqrt(-sqrt(-3/2*sqrt(-1/3) + 1/2)) + 2*x) + 3 *sqrt(1/3)*x^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) - 3*sqrt(1/3)*x^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) - 3*sqrt(1/3)*x^3*(-3/2*sqrt(-1/3) + 1/2)^(1/4)*log(3*sqrt(1/3)*(s qrt(-1/3) + 1)*(-3/2*sqrt(-1/3) + 1/2)^(1/4) + 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) + 1)*(-3/2*sqrt(-1/ 3) + 1/2)^(1/4) + 2*x) + 4)/x^3
Time = 1.57 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^4 \left (1-x^4+x^8\right )} \, dx=\operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (- 9216 t^{5} + 8 t + x \right )} \right )\right )} - \frac {1}{3 x^{3}} \] Input:
integrate(1/x**4/(x**8-x**4+1),x)
Output:
RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-9216*_t**5 + 8* _t + x))) - 1/(3*x**3)
\[ \int \frac {1}{x^4 \left (1-x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - x^{4} + 1\right )} x^{4}} \,d x } \] Input:
integrate(1/x^4/(x^8-x^4+1),x, algorithm="maxima")
Output:
-1/3/x^3 - integrate((x^4 - 1)/(x^8 - x^4 + 1), x)
Time = 0.13 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^4 \left (1-x^4+x^8\right )} \, dx=\frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{3 \, x^{3}} \] Input:
integrate(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) + sqr t(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))) + 1/24*(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*(sq rt(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 = 0.06 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^4 \left (1-x^4+x^8\right )} \, dx=-\frac {1}{3\,x^3}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}-\frac {\sqrt {3}\,x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {3}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \] Input:
int(1/(x^4*(x^8 - x^4 + 1)),x)
Output:
(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 - (3 ^(1/2)*atan(x/(8 - 3^(1/2)*8i)^(1/4) + (3^(1/2)*x*1i)/(8 - 3^(1/2)*8i)^(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 - 1 /(3*x^3) + (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 = 341, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^4 \left (1-x^4+x^8\right )} \, dx=\frac {4 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x^{3}-4 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right ) x^{3}-2 \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{3}-6 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{3}+2 \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{3}+6 \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right ) x^{3}+2 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x^{3}-2 \sqrt {-\sqrt {3}+2}\, \sqrt {3}\, \mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right ) x^{3}-\sqrt {6}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x^{3}+\sqrt {6}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x^{3}-3 \sqrt {2}\, \mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x^{3}+3 \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right ) x^{3}-16}{48 x^{3}} \] Input:
int(1/x^4/(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**3 - 4*sqrt( - sqrt(3) + 2)*sqrt(3)*atan((sqrt(6) + sqrt( 2) + 4*x)/(2*sqrt( - sqrt(3) + 2)))*x**3 - 2*sqrt(6)*atan((2*sqrt( - sqrt( 3) + 2) - 4*x)/(sqrt(6) + sqrt(2)))*x**3 - 6*sqrt(2)*atan((2*sqrt( - sqrt( 3) + 2) - 4*x)/(sqrt(6) + sqrt(2)))*x**3 + 2*sqrt(6)*atan((2*sqrt( - sqrt( 3) + 2) + 4*x)/(sqrt(6) + sqrt(2)))*x**3 + 6*sqrt(2)*atan((2*sqrt( - sqrt( 3) + 2) + 4*x)/(sqrt(6) + sqrt(2)))*x**3 + 2*sqrt( - sqrt(3) + 2)*sqrt(3)* log( - sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x**3 - 2*sqrt( - sqrt(3) + 2)*sq rt(3)*log(sqrt( - sqrt(3) + 2)*x + x**2 + 1)*x**3 - sqrt(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)