Integrand size = 14, antiderivative size = 39 \[ \int \frac {1}{x \left (1+x^4+x^8\right )} \, dx=-\frac {\arctan \left (\frac {1+2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\log (x)-\frac {1}{8} \log \left (1+x^4+x^8\right ) \] 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 complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.41 \[ \int \frac {1}{x \left (1+x^4+x^8\right )} \, dx=\frac {1}{24} \left (2 \sqrt {3} \arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+2 x}{\sqrt {3}}\right )+24 \log (x)-\sqrt {3} \left (i+\sqrt {3}\right ) \log \left (i+\sqrt {3}-2 i x^2\right )-\sqrt {3} \left (-i+\sqrt {3}\right ) \log \left (-i+\sqrt {3}+2 i x^2\right )-3 \log \left (1-x+x^2\right )-3 \log \left (1+x+x^2\right )\right ) \] Input:
Integrate[1/(x*(1 + x^4 + x^8)),x]
Output:
(2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3] ] + 24*Log[x] - Sqrt[3]*(I + Sqrt[3])*Log[I + Sqrt[3] - (2*I)*x^2] - Sqrt[ 3]*(-I + Sqrt[3])*Log[-I + Sqrt[3] + (2*I)*x^2] - 3*Log[1 - x + x^2] - 3*L og[1 + x + x^2])/24
Time = 0.21 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1693, 1144, 25, 1142, 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 |
\(\displaystyle \frac {1}{4} \left (\int -\frac {x^4+1}{x^8+x^4+1}dx^4+\log \left (x^4\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{4} \left (\log \left (x^4\right )-\int \frac {x^4+1}{x^8+x^4+1}dx^4\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int \frac {1}{x^8+x^4+1}dx^4-\frac {1}{2} \int \frac {2 x^4+1}{x^8+x^4+1}dx^4+\log \left (x^4\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{4} \left (\int \frac {1}{-x^8-3}d\left (2 x^4+1\right )-\frac {1}{2} \int \frac {2 x^4+1}{x^8+x^4+1}dx^4+\log \left (x^4\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{4} \left (-\frac {1}{2} \int \frac {2 x^4+1}{x^8+x^4+1}dx^4-\frac {\arctan \left (\frac {2 x^4+1}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (x^4\right )\right )\) |
\(\Big \downarrow \) 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 = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\ln \left (x \right )-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{4}+\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{12}-\frac {\ln \left (x^{8}+x^{4}+1\right )}{8}\) | \(31\) |
default | \(-\frac {\ln \left (x^{2}-x +1\right )}{8}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{12}-\frac {\ln \left (x^{2}+x +1\right )}{8}-\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}-\frac {\ln \left (x^{4}-x^{2}+1\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{12}+\ln \left (x \right )\) | \(87\) |
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 = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x \left (1+x^4+x^8\right )} \, dx=-\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) + \log \left (x\right ) \] 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) + l og(x)
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x \left (1+x^4+x^8\right )} \, dx=\log {\left (x \right )} - \frac {\log {\left (x^{8} + x^{4} + 1 \right )}}{8} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} + \frac {\sqrt {3}}{3} \right )}}{12} \] 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 = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \left (1+x^4+x^8\right )} \, dx=-\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \] 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 = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \left (1+x^4+x^8\right )} \, dx=-\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} + 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} + x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \] 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 = 18.94 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \left (1+x^4+x^8\right )} \, dx=\ln \left (x\right )-\frac {\ln \left (x^8+x^4+1\right )}{8}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^4}{3}+\frac {\sqrt {3}}{3}\right )}{12} \] Input:
int(1/(x*(x^4 + x^8 + 1)),x)
Output:
log(x) - log(x^4 + x^8 + 1)/8 - (3^(1/2)*atan(3^(1/2)/3 + (2*3^(1/2)*x^4)/ 3))/12
Time = 0.15 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.56 \[ \int \frac {1}{x \left (1+x^4+x^8\right )} \, dx=\frac {\sqrt {3}\, \mathit {atan} \left (\sqrt {3}-2 x \right )}{12}+\frac {\sqrt {3}\, \mathit {atan} \left (\sqrt {3}+2 x \right )}{12}+\frac {\sqrt {3}\, \mathit {atan} \left (\frac {2 x -1}{\sqrt {3}}\right )}{12}-\frac {\sqrt {3}\, \mathit {atan} \left (\frac {2 x +1}{\sqrt {3}}\right )}{12}-\frac {\mathrm {log}\left (x^{2}-x +1\right )}{8}-\frac {\mathrm {log}\left (x^{2}+x +1\right )}{8}-\frac {\mathrm {log}\left (-\sqrt {3}\, x +x^{2}+1\right )}{8}-\frac {\mathrm {log}\left (\sqrt {3}\, x +x^{2}+1\right )}{8}+\mathrm {log}\left (x \right ) \] Input:
int(1/x/(x^8+x^4+1),x)
Output:
(2*sqrt(3)*atan(sqrt(3) - 2*x) + 2*sqrt(3)*atan(sqrt(3) + 2*x) + 2*sqrt(3) *atan((2*x - 1)/sqrt(3)) - 2*sqrt(3)*atan((2*x + 1)/sqrt(3)) - 3*log(x**2 - x + 1) - 3*log(x**2 + x + 1) - 3*log( - sqrt(3)*x + x**2 + 1) - 3*log(sq rt(3)*x + x**2 + 1) + 24*log(x))/24