Integrand size = 16, antiderivative size = 41 \[ \int \frac {1}{x \left (1-x^4+x^8\right )} \, dx=-\frac {\arctan \left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\log (x)-\frac {1}{8} \log \left (1-x^4+x^8\right ) \] Output:
-1/12*arctan(1/3*(-2*x^4+1)*3^(1/2))*3^(1/2)+ln(x)-1/8*ln(x^8-x^4+1)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x \left (1-x^4+x^8\right )} \, dx=\log (x)-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-1+2 \text {$\#$1}^4}\&\right ] \] Input:
Integrate[1/(x*(1 - x^4 + x^8)),x]
Output:
Log[x] - RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-1 + 2*#1^4) & ]/4
Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1693, 1144, 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.
\(\displaystyle \int \frac {1}{x \left (x^8-x^4+1\right )} \, dx\) |
\(\Big \downarrow \) 1693 |
\(\displaystyle \frac {1}{4} \int \frac {1}{x^4 \left (x^8-x^4+1\right )}dx^4\) |
\(\Big \downarrow \) 1144 |
\(\displaystyle \frac {1}{4} \left (\int \frac {1-x^4}{x^8-x^4+1}dx^4+\log \left (x^4\right )\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {1}{x^8-x^4+1}dx^4-\frac {1}{2} \int -\frac {1-2 x^4}{x^8-x^4+1}dx^4+\log \left (x^4\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {1}{x^8-x^4+1}dx^4+\frac {1}{2} \int \frac {1-2 x^4}{x^8-x^4+1}dx^4+\log \left (x^4\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{4} \left (-\int \frac {1}{-x^8-3}d\left (2 x^4-1\right )+\frac {1}{2} \int \frac {1-2 x^4}{x^8-x^4+1}dx^4+\log \left (x^4\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {1-2 x^4}{x^8-x^4+1}dx^4+\frac {\arctan \left (\frac {2 x^4-1}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (x^4\right )\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{4} \left (\frac {\arctan \left (\frac {2 x^4-1}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (x^4\right )-\frac {1}{2} \log \left (x^8-x^4+1\right )\right )\) |
Input:
Int[1/(x*(1 - x^4 + x^8)),x]
Output:
(ArcTan[(-1 + 2*x^4)/Sqrt[3]]/Sqrt[3] + Log[x^4] - Log[1 - x^4 + x^8]/2)/4
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[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[e*(Log[RemoveContent[d + e*x, x]]/(c*d^2 - b*d*e + a*e^2)), x] + S imp[1/(c*d^2 - b*d*e + a*e^2) Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ [Simplify[(m + 1)/n]]
Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\ln \left (x \right )+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{4}-\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{12}-\frac {\ln \left (x^{8}-x^{4}+1\right )}{8}\) | \(33\) |
default | \(-\frac {\ln \left (x^{8}-x^{4}+1\right )}{8}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{4}-1\right ) \sqrt {3}}{3}\right )}{12}+\ln \left (x \right )\) | \(35\) |
Input:
int(1/x/(x^8-x^4+1),x,method=_RETURNVERBOSE)
Output:
ln(x)+1/12*3^(1/2)*arctan(2/3*(x^4-1/2)*3^(1/2))-1/8*ln(x^8-x^4+1)
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x \left (1-x^4+x^8\right )} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) + \log \left (x\right ) \] Input:
integrate(1/x/(x^8-x^4+1),x, algorithm="fricas")
Output:
1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) - 1/8*log(x^8 - x^4 + 1) + lo g(x)
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1-x^4+x^8\right )} \, dx=\log {\left (x \right )} - \frac {\log {\left (x^{8} - x^{4} + 1 \right )}}{8} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} - \frac {\sqrt {3}}{3} \right )}}{12} \] Input:
integrate(1/x/(x**8-x**4+1),x)
Output:
log(x) - log(x**8 - x**4 + 1)/8 + sqrt(3)*atan(2*sqrt(3)*x**4/3 - sqrt(3)/ 3)/12
Time = 0.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x \left (1-x^4+x^8\right )} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \] Input:
integrate(1/x/(x^8-x^4+1),x, algorithm="maxima")
Output:
1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) - 1/8*log(x^8 - x^4 + 1) + 1/ 4*log(x^4)
Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x \left (1-x^4+x^8\right )} \, dx=\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \] Input:
integrate(1/x/(x^8-x^4+1),x, algorithm="giac")
Output:
1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) - 1/8*log(x^8 - x^4 + 1) + 1/ 4*log(x^4)
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x \left (1-x^4+x^8\right )} \, dx=\ln \left (x\right )-\frac {\ln \left (x^8-x^4+1\right )}{8}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,x^4}{3}\right )}{12} \] Input:
int(1/(x*(x^8 - x^4 + 1)),x)
Output:
log(x) - log(x^8 - x^4 + 1)/8 - (3^(1/2)*atan(3^(1/2)/3 - (2*3^(1/2)*x^4)/ 3))/12
Time = 0.16 (sec) , antiderivative size = 330, normalized size of antiderivative = 8.05 \[ \int \frac {1}{x \left (1-x^4+x^8\right )} \, dx=-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{24}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}-4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{8}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{24}-\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {6}+\sqrt {2}+4 x}{2 \sqrt {-\sqrt {3}+2}}\right )}{8}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right )}{24}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right )}{24}+\frac {\sqrt {-\sqrt {3}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 x}{\sqrt {6}+\sqrt {2}}\right )}{8}-\frac {\mathrm {log}\left (-\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{8}-\frac {\mathrm {log}\left (\sqrt {-\sqrt {3}+2}\, x +x^{2}+1\right )}{8}-\frac {\mathrm {log}\left (-\frac {\sqrt {6}\, x}{2}-\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{8}-\frac {\mathrm {log}\left (\frac {\sqrt {6}\, x}{2}+\frac {\sqrt {2}\, x}{2}+x^{2}+1\right )}{8}+\mathrm {log}\left (x \right ) \] Input:
int(1/x/(x^8-x^4+1),x)
Output:
( - sqrt( - sqrt(3) + 2)*sqrt(6)*atan((sqrt(6) + sqrt(2) - 4*x)/(2*sqrt( - sqrt(3) + 2))) - 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((sqrt(6) + sqrt(2) - 4*x)/(2*sqrt( - sqrt(3) + 2))) - sqrt( - sqrt(3) + 2)*sqrt(6)*atan((sqrt( 6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2))) - 3*sqrt( - sqrt(3) + 2)*sqr t(2)*atan((sqrt(6) + sqrt(2) + 4*x)/(2*sqrt( - sqrt(3) + 2))) + sqrt( - sq rt(3) + 2)*sqrt(6)*atan((2*sqrt( - sqrt(3) + 2) - 4*x)/(sqrt(6) + sqrt(2)) ) + 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((2*sqrt( - sqrt(3) + 2) - 4*x)/(sq rt(6) + sqrt(2))) + sqrt( - sqrt(3) + 2)*sqrt(6)*atan((2*sqrt( - sqrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2))) + 3*sqrt( - sqrt(3) + 2)*sqrt(2)*atan((2*s qrt( - sqrt(3) + 2) + 4*x)/(sqrt(6) + sqrt(2))) - 3*log( - sqrt( - sqrt(3) + 2)*x + x**2 + 1) - 3*log(sqrt( - sqrt(3) + 2)*x + x**2 + 1) - 3*log(( - sqrt(6)*x - sqrt(2)*x + 2*x**2 + 2)/2) - 3*log((sqrt(6)*x + sqrt(2)*x + 2 *x**2 + 2)/2) + 24*log(x))/24