\(\int \frac {1}{1-3 x^4+x^8} \, dx\) [400]

Optimal result

Integrand size = 12, antiderivative size = 169 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}} \]

[Out]

-1/10*arctan(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*2^(3/4)*5^(1/2)/(3+5^(1/2))^(3/4)-1/10*arctanh(2^(1/4)*x*(1/(3+5
^(1/2)))^(1/4))*2^(3/4)*5^(1/2)/(3+5^(1/2))^(3/4)+1/10*arctan(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(9+4*5^(1/2))^(
1/4)*5^(1/2)+1/10*arctanh(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(9+4*5^(1/2))^(1/4)*5^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1361, 218, 212, 209} \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}} \]

[In]

Int[(1 - 3*x^4 + x^8)^(-1),x]

[Out]

-(ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*Sqrt[5]*(3 + Sqrt[5])^(3/4))) + ((3 + Sqrt[5])^(3/4)*ArcTan[((3 +
Sqrt[5])/2)^(1/4)*x])/(2*2^(3/4)*Sqrt[5]) - ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x]/(2^(1/4)*Sqrt[5]*(3 + Sqrt[5])
^(3/4)) + ((3 + Sqrt[5])^(3/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(2*2^(3/4)*Sqrt[5])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 1361

Int[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, In
t[1/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n
2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}}-\frac {\int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}} \\ & = \frac {\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx}{\sqrt {5 \left (3-\sqrt {5}\right )}}+\frac {\int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx}{\sqrt {5 \left (3-\sqrt {5}\right )}}-\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{\sqrt {5 \left (3+\sqrt {5}\right )}}-\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{\sqrt {5 \left (3+\sqrt {5}\right )}} \\ & = -\frac {\tan ^{-1}\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} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\tanh ^{-1}\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} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.95 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=\frac {\frac {\left (1+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (-1+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}+\frac {\left (1+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (-1+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}}{2 \sqrt {10}} \]

[In]

Integrate[(1 - 3*x^4 + x^8)^(-1),x]

[Out]

(((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 + Sqrt[5])*ArcTanh[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]])/(2*Sqrt[10])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.08 (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	\(-\frac {\sqrt {5}\, \left (\sqrt {5}-1\right ) \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}-\frac {\sqrt {5}\, \left (\sqrt {5}-1\right ) \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}\)	\(130\)

[In]

int(1/(x^8-3*x^4+1),x,method=_RETURNVERBOSE)

[Out]

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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (113) = 226\).

Time = 0.27 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.54 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 3\right )} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (-\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 3\right )} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} + 2} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {-\sqrt {5} + 2} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} + 2} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {-\sqrt {5} + 2} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} - 2} \log \left ({\left (\sqrt {5} - 3\right )} \sqrt {-\sqrt {5} - 2} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} - 2} \log \left (-{\left (\sqrt {5} - 3\right )} \sqrt {-\sqrt {5} - 2} + 2 \, x\right ) \]

[In]

integrate(1/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

-1/20*sqrt(5)*sqrt(sqrt(5) - 2)*log((sqrt(5) + 3)*sqrt(sqrt(5) - 2) + 2*x) + 1/20*sqrt(5)*sqrt(sqrt(5) - 2)*lo
g(-(sqrt(5) + 3)*sqrt(sqrt(5) - 2) + 2*x) - 1/20*sqrt(5)*sqrt(sqrt(5) + 2)*log(sqrt(sqrt(5) + 2)*(sqrt(5) - 3)
+ 2*x) + 1/20*sqrt(5)*sqrt(sqrt(5) + 2)*log(-sqrt(sqrt(5) + 2)*(sqrt(5) - 3) + 2*x) - 1/20*sqrt(5)*sqrt(-sqrt
(5) + 2)*log((sqrt(5) + 3)*sqrt(-sqrt(5) + 2) + 2*x) + 1/20*sqrt(5)*sqrt(-sqrt(5) + 2)*log(-(sqrt(5) + 3)*sqrt
(-sqrt(5) + 2) + 2*x) - 1/20*sqrt(5)*sqrt(-sqrt(5) - 2)*log((sqrt(5) - 3)*sqrt(-sqrt(5) - 2) + 2*x) + 1/20*sqr
t(5)*sqrt(-sqrt(5) - 2)*log(-(sqrt(5) - 3)*sqrt(-sqrt(5) - 2) + 2*x)

Sympy [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.31 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (6400 t^{4} - 320 t^{2} - 1, \left ( t \mapsto t \log {\left (9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 320 t^{2} - 1, \left ( t \mapsto t \log {\left (9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} \]

[In]

integrate(1/(x**8-3*x**4+1),x)

[Out]

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)))

Maxima [F]

\[ \int \frac {1}{1-3 x^4+x^8} \, dx=\int { \frac {1}{x^{8} - 3 \, x^{4} + 1} \,d x } \]

[In]

integrate(1/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

integrate(1/(x^8 - 3*x^4 + 1), x)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.87 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {1}{10} \, \sqrt {5 \, \sqrt {5} - 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{10} \, \sqrt {5 \, \sqrt {5} + 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \]

[In]

integrate(1/(x^8-3*x^4+1),x, algorithm="giac")

[Out]

-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*sqr
t(5) - 1/2)) - 1/20*sqrt(5*sqrt(5) - 10)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(5*sqrt(5) - 10)*log
(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/20*sqrt(5*sqrt(5) + 10)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/20*sq
rt(5*sqrt(5) + 10)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2)))

Mupad [B] (verification not implemented)

Time = 8.38 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.45 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {2-\sqrt {5}}\,144{}\mathrm {i}}{104\,\sqrt {5}-232}-\frac {\sqrt {5}\,x\,\sqrt {2-\sqrt {5}}\,64{}\mathrm {i}}{104\,\sqrt {5}-232}\right )\,\sqrt {2-\sqrt {5}}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {5}-2}\,144{}\mathrm {i}}{104\,\sqrt {5}+232}+\frac {\sqrt {5}\,x\,\sqrt {-\sqrt {5}-2}\,64{}\mathrm {i}}{104\,\sqrt {5}+232}\right )\,\sqrt {-\sqrt {5}-2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {5}-2}\,144{}\mathrm {i}}{104\,\sqrt {5}-232}-\frac {\sqrt {5}\,x\,\sqrt {\sqrt {5}-2}\,64{}\mathrm {i}}{104\,\sqrt {5}-232}\right )\,\sqrt {\sqrt {5}-2}\,1{}\mathrm {i}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {5}+2}\,144{}\mathrm {i}}{104\,\sqrt {5}+232}+\frac {\sqrt {5}\,x\,\sqrt {\sqrt {5}+2}\,64{}\mathrm {i}}{104\,\sqrt {5}+232}\right )\,\sqrt {\sqrt {5}+2}\,1{}\mathrm {i}}{10} \]

[In]

int(1/(x^8 - 3*x^4 + 1),x)

[Out]

(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) + 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