Integrand size = 12, antiderivative size = 169 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}} \] Output:
-1/10*arctan(2^(1/4)*(1/(3+5^(1/2)))^(1/4)*x)*2^(3/4)*5^(1/2)/(3+5^(1/2))^ (3/4)+1/20*(3+5^(1/2))^(3/4)*arctan((3/2+1/2*5^(1/2))^(1/4)*x)*2^(1/4)*5^( 1/2)-1/10*arctanh(2^(1/4)*(1/(3+5^(1/2)))^(1/4)*x)*2^(3/4)*5^(1/2)/(3+5^(1 /2))^(3/4)+1/20*(3+5^(1/2))^(3/4)*arctanh((3/2+1/2*5^(1/2))^(1/4)*x)*2^(1/ 4)*5^(1/2)
Time = 0.10 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.95 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=\frac {\frac {\left (1+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (-1+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}+\frac {\left (1+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (-1+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}}{2 \sqrt {10}} \] Input:
Integrate[(1 - 3*x^4 + x^8)^(-1),x]
Output:
(((1 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((- 1 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]] + ((1 + Sq rt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[-1 + Sqrt[5]] - ((-1 + Sqrt [5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[1 + Sqrt[5]])/(2*Sqrt[10])
Time = 0.25 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1685, 756, 216, 219}
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.
| \(\displaystyle \int \frac {1}{x^8-3 x^4+1} \, dx\) |
| \(\Big \downarrow \) 1685 |
| \(\displaystyle \frac {\int \frac {1}{x^4+\frac {1}{2} \left (-3-\sqrt {5}\right )}dx}{\sqrt {5}}-\frac {\int \frac {1}{x^4+\frac {1}{2} \left (-3+\sqrt {5}\right )}dx}{\sqrt {5}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {-\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3+\sqrt {5}}}-\frac {\int \frac {1}{\sqrt {2} x^2+\sqrt {3+\sqrt {5}}}dx}{\sqrt {3+\sqrt {5}}}}{\sqrt {5}}-\frac {-\frac {\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3-\sqrt {5}}}-\frac {\int \frac {1}{\sqrt {2} x^2+\sqrt {3-\sqrt {5}}}dx}{\sqrt {3-\sqrt {5}}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {-\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3+\sqrt {5}}}-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \left (3+\sqrt {5}\right )^{3/4}}}{\sqrt {5}}-\frac {-\frac {\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}dx}{\sqrt {3-\sqrt {5}}}-\frac {\arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt [4]{2} \left (3-\sqrt {5}\right )^{3/4}}}{\sqrt {5}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \left (3+\sqrt {5}\right )^{3/4}}-\frac {\text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \left (3+\sqrt {5}\right )^{3/4}}}{\sqrt {5}}-\frac {-\frac {\arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt [4]{2} \left (3-\sqrt {5}\right )^{3/4}}-\frac {\text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{\sqrt [4]{2} \left (3-\sqrt {5}\right )^{3/4}}}{\sqrt {5}}\) |
Input:
Int[(1 - 3*x^4 + x^8)^(-1),x]
Output:
(-(ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*(3 + Sqrt[5])^(3/4))) - ArcT anh[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*(3 + Sqrt[5])^(3/4)))/Sqrt[5] - (- (ArcTan[((3 + Sqrt[5])/2)^(1/4)*x]/(2^(1/4)*(3 - Sqrt[5])^(3/4))) - ArcTan h[((3 + Sqrt[5])/2)^(1/4)*x]/(2^(1/4)*(3 - Sqrt[5])^(3/4)))/Sqrt[5]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^n), x], x] - Simp[ c/q Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}-20 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (15 \textit {\_R}^{3}-11 \textit {\_R} +2 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-15 \textit {\_R}^{3}-11 \textit {\_R} +2 x \right )\right )}{4}\) | \(68\) |
| default | \(-\frac {\sqrt {5}\, \left (\sqrt {5}-1\right ) \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}+\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}+\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{10 \sqrt {-2+2 \sqrt {5}}}-\frac {\sqrt {5}\, \left (\sqrt {5}-1\right ) \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{10 \sqrt {2+2 \sqrt {5}}}\) | \(130\) |
Input:
int(1/(x^8-3*x^4+1),x,method=_RETURNVERBOSE)
Output:
1/4*sum(_R*ln(15*_R^3-11*_R+2*x),_R=RootOf(25*_Z^4-20*_Z^2-1))+1/4*sum(_R* ln(-15*_R^3-11*_R+2*x),_R=RootOf(25*_Z^4+20*_Z^2-1))
Time = 0.08 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.18 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {1}{2} \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}} \arctan \left (\frac {1}{2} \, {\left (\sqrt {5} x - 5 \, x\right )} \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}} \arctan \left (\frac {1}{2} \, {\left (\sqrt {5} x + 5 \, x\right )} \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}} \log \left ({\left (3 \, \sqrt {5} - 5\right )} \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}} + 2 \, x\right ) - \frac {1}{4} \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}} \log \left (-{\left (3 \, \sqrt {5} - 5\right )} \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {2}{5}} + 2 \, x\right ) - \frac {1}{4} \, \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}} \log \left ({\left (3 \, \sqrt {5} + 5\right )} \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}} + 2 \, x\right ) + \frac {1}{4} \, \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}} \log \left (-{\left (3 \, \sqrt {5} + 5\right )} \sqrt {\frac {1}{5} \, \sqrt {5} - \frac {2}{5}} + 2 \, x\right ) \] Input:
integrate(1/(x^8-3*x^4+1),x, algorithm="fricas")
Output:
-1/2*sqrt(1/5*sqrt(5) + 2/5)*arctan(1/2*(sqrt(5)*x - 5*x)*sqrt(1/5*sqrt(5) + 2/5)) - 1/2*sqrt(1/5*sqrt(5) - 2/5)*arctan(1/2*(sqrt(5)*x + 5*x)*sqrt(1 /5*sqrt(5) - 2/5)) + 1/4*sqrt(1/5*sqrt(5) + 2/5)*log((3*sqrt(5) - 5)*sqrt( 1/5*sqrt(5) + 2/5) + 2*x) - 1/4*sqrt(1/5*sqrt(5) + 2/5)*log(-(3*sqrt(5) - 5)*sqrt(1/5*sqrt(5) + 2/5) + 2*x) - 1/4*sqrt(1/5*sqrt(5) - 2/5)*log((3*sqr t(5) + 5)*sqrt(1/5*sqrt(5) - 2/5) + 2*x) + 1/4*sqrt(1/5*sqrt(5) - 2/5)*log (-(3*sqrt(5) + 5)*sqrt(1/5*sqrt(5) - 2/5) + 2*x)
Time = 0.82 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.31 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (6400 t^{4} - 320 t^{2} - 1, \left ( t \mapsto t \log {\left (9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 320 t^{2} - 1, \left ( t \mapsto t \log {\left (9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} \] Input:
integrate(1/(x**8-3*x**4+1),x)
Output:
RootSum(6400*_t**4 - 320*_t**2 - 1, Lambda(_t, _t*log(9600*_t**5 - 47*_t/2 + x))) + RootSum(6400*_t**4 + 320*_t**2 - 1, Lambda(_t, _t*log(9600*_t**5 - 47*_t/2 + x)))
\[ \int \frac {1}{1-3 x^4+x^8} \, dx=\int { \frac {1}{x^{8} - 3 \, x^{4} + 1} \,d x } \] Input:
integrate(1/(x^8-3*x^4+1),x, algorithm="maxima")
Output:
integrate(1/(x^8 - 3*x^4 + 1), x)
Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.87 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {1}{10} \, \sqrt {5 \, \sqrt {5} - 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{10} \, \sqrt {5 \, \sqrt {5} + 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \] Input:
integrate(1/(x^8-3*x^4+1),x, algorithm="giac")
Output:
-1/10*sqrt(5*sqrt(5) - 10)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/10*sqrt(5 *sqrt(5) + 10)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/20*sqrt(5*sqrt(5) - 1 0)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(5*sqrt(5) - 10)*log(a bs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(5*sqrt(5) + 10)*log(abs(x + s qrt(1/2*sqrt(5) - 1/2))) - 1/20*sqrt(5*sqrt(5) + 10)*log(abs(x - sqrt(1/2* sqrt(5) - 1/2)))
Time = 0.09 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.45 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {2-\sqrt {5}}\,144{}\mathrm {i}}{104\,\sqrt {5}-232}-\frac {\sqrt {5}\,x\,\sqrt {2-\sqrt {5}}\,64{}\mathrm {i}}{104\,\sqrt {5}-232}\right )\,\sqrt {2-\sqrt {5}}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {5}-2}\,144{}\mathrm {i}}{104\,\sqrt {5}+232}+\frac {\sqrt {5}\,x\,\sqrt {-\sqrt {5}-2}\,64{}\mathrm {i}}{104\,\sqrt {5}+232}\right )\,\sqrt {-\sqrt {5}-2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {5}-2}\,144{}\mathrm {i}}{104\,\sqrt {5}-232}-\frac {\sqrt {5}\,x\,\sqrt {\sqrt {5}-2}\,64{}\mathrm {i}}{104\,\sqrt {5}-232}\right )\,\sqrt {\sqrt {5}-2}\,1{}\mathrm {i}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {5}+2}\,144{}\mathrm {i}}{104\,\sqrt {5}+232}+\frac {\sqrt {5}\,x\,\sqrt {\sqrt {5}+2}\,64{}\mathrm {i}}{104\,\sqrt {5}+232}\right )\,\sqrt {\sqrt {5}+2}\,1{}\mathrm {i}}{10} \] Input:
int(1/(x^8 - 3*x^4 + 1),x)
Output:
(5^(1/2)*atan((x*(- 5^(1/2) - 2)^(1/2)*144i)/(104*5^(1/2) + 232) + (5^(1/2 )*x*(- 5^(1/2) - 2)^(1/2)*64i)/(104*5^(1/2) + 232))*(- 5^(1/2) - 2)^(1/2)* 1i)/10 - (5^(1/2)*atan((x*(2 - 5^(1/2))^(1/2)*144i)/(104*5^(1/2) - 232) - (5^(1/2)*x*(2 - 5^(1/2))^(1/2)*64i)/(104*5^(1/2) - 232))*(2 - 5^(1/2))^(1/ 2)*1i)/10 + (5^(1/2)*atan((x*(5^(1/2) - 2)^(1/2)*144i)/(104*5^(1/2) - 232) - (5^(1/2)*x*(5^(1/2) - 2)^(1/2)*64i)/(104*5^(1/2) - 232))*(5^(1/2) - 2)^ (1/2)*1i)/10 - (5^(1/2)*atan((x*(5^(1/2) + 2)^(1/2)*144i)/(104*5^(1/2) + 2 32) + (5^(1/2)*x*(5^(1/2) + 2)^(1/2)*64i)/(104*5^(1/2) + 232))*(5^(1/2) + 2)^(1/2)*1i)/10
\[ \int \frac {1}{1-3 x^4+x^8} \, dx=\int \frac {1}{x^{8}-3 x^{4}+1}d x \] Input:
int(1/(x^8-3*x^4+1),x)
Output:
int(1/(x**8 - 3*x**4 + 1),x)