\(\int \frac {x^4}{1-3 x^4+x^8} \, dx\) [398]

Optimal result

Integrand size = 16, antiderivative size = 173 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \]

[Out]

1/20*arctan(1/2*x*(3+5^(1/2))^(1/4)*2^(3/4))*(3-5^(1/2))^(1/4)*2^(3/4)*5^(1/2)+1/20*arctanh(1/2*x*(3+5^(1/2))^
(1/4)*2^(3/4))*(3-5^(1/2))^(1/4)*2^(3/4)*5^(1/2)-1/20*arctan(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*(3+5^(1/2))^(1/4
)*2^(3/4)*5^(1/2)-1/20*arctanh(2^(1/4)*x*(1/(3+5^(1/2)))^(1/4))*(3+5^(1/2))^(1/4)*2^(3/4)*5^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1388, 218, 212, 209} \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \]

[In]

Int[x^4/(1 - 3*x^4 + x^8),x]

[Out]

-1/2*(((3 + Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/Sqrt[5] + (((3 - Sqrt[5])/2)^(1/4)*ArcTan[((3
+ Sqrt[5])/2)^(1/4)*x])/(2*Sqrt[5]) - (((3 + Sqrt[5])/2)^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]
) + (((3 - Sqrt[5])/2)^(1/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(2*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 1388

Int[((d_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[(d^n/2)*(b/q + 1), Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n/2)*(b/q - 1), Int[(d*x)^(m
- n)/(b/2 - q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n,
0] && GeQ[m, n]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{10} \left (5-3 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx+\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx \\ & = \frac {1}{2} \sqrt {\frac {1}{5} \left (3-\sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx+\frac {1}{2} \sqrt {\frac {1}{5} \left (3-\sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx-\frac {1}{2} \sqrt {\frac {1}{5} \left (3+\sqrt {5}\right )} \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx-\frac {1}{2} \sqrt {\frac {1}{5} \left (3+\sqrt {5}\right )} \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx \\ & = -\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.76 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=\frac {\sqrt {-1+\sqrt {5}} \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-\sqrt {1+\sqrt {5}} \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+\sqrt {-1+\sqrt {5}} \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-\sqrt {1+\sqrt {5}} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{2 \sqrt {10}} \]

[In]

Integrate[x^4/(1 - 3*x^4 + x^8),x]

[Out]

(Sqrt[-1 + Sqrt[5]]*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x] - Sqrt[1 + Sqrt[5]]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + Sqr
t[-1 + Sqrt[5]]*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x] - Sqrt[1 + Sqrt[5]]*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/(2*Sqr
t[10])

Maple [C] (verified)

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

Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.35

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (10 \textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-10 \textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{4}\)	\(60\)
default	\(-\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\sqrt {5}\, \left (\sqrt {5}-1\right ) \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}-\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\sqrt {5}\, \left (\sqrt {5}-1\right ) \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}\)	\(130\)

[In]

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

[Out]

1/4*sum(_R*ln(10*_R^3+_R+x),_R=RootOf(25*_Z^4+5*_Z^2-1))+1/4*sum(_R*ln(-10*_R^3+_R+x),_R=RootOf(25*_Z^4-5*_Z^2
-1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (119) = 238\).

Time = 0.28 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.55 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=-\frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (\sqrt {10} \sqrt {5} \sqrt {\sqrt {5} + 1} + 10 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (-\sqrt {10} \sqrt {5} \sqrt {\sqrt {5} + 1} + 10 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (\sqrt {10} \sqrt {5} \sqrt {\sqrt {5} - 1} + 10 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (-\sqrt {10} \sqrt {5} \sqrt {\sqrt {5} - 1} + 10 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} + 1} \log \left (\sqrt {10} \sqrt {5} \sqrt {-\sqrt {5} + 1} + 10 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} + 1} \log \left (-\sqrt {10} \sqrt {5} \sqrt {-\sqrt {5} + 1} + 10 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} - 1} \log \left (\sqrt {10} \sqrt {5} \sqrt {-\sqrt {5} - 1} + 10 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} - 1} \log \left (-\sqrt {10} \sqrt {5} \sqrt {-\sqrt {5} - 1} + 10 \, x\right ) \]

[In]

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

[Out]

-1/40*sqrt(10)*sqrt(sqrt(5) + 1)*log(sqrt(10)*sqrt(5)*sqrt(sqrt(5) + 1) + 10*x) + 1/40*sqrt(10)*sqrt(sqrt(5) +
1)*log(-sqrt(10)*sqrt(5)*sqrt(sqrt(5) + 1) + 10*x) + 1/40*sqrt(10)*sqrt(sqrt(5) - 1)*log(sqrt(10)*sqrt(5)*sqr
t(sqrt(5) - 1) + 10*x) - 1/40*sqrt(10)*sqrt(sqrt(5) - 1)*log(-sqrt(10)*sqrt(5)*sqrt(sqrt(5) - 1) + 10*x) + 1/4
0*sqrt(10)*sqrt(-sqrt(5) + 1)*log(sqrt(10)*sqrt(5)*sqrt(-sqrt(5) + 1) + 10*x) - 1/40*sqrt(10)*sqrt(-sqrt(5) +
1)*log(-sqrt(10)*sqrt(5)*sqrt(-sqrt(5) + 1) + 10*x) - 1/40*sqrt(10)*sqrt(-sqrt(5) - 1)*log(sqrt(10)*sqrt(5)*sq
rt(-sqrt(5) - 1) + 10*x) + 1/40*sqrt(10)*sqrt(-sqrt(5) - 1)*log(-sqrt(10)*sqrt(5)*sqrt(-sqrt(5) - 1) + 10*x)

Sympy [A] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.28 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (6400 t^{4} - 80 t^{2} - 1, \left ( t \mapsto t \log {\left (- 51200 t^{5} + 12 t + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 80 t^{2} - 1, \left ( t \mapsto t \log {\left (- 51200 t^{5} + 12 t + x \right )} \right )\right )} \]

[In]

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

[Out]

RootSum(6400*_t**4 - 80*_t**2 - 1, Lambda(_t, _t*log(-51200*_t**5 + 12*_t + x))) + RootSum(6400*_t**4 + 80*_t*
*2 - 1, Lambda(_t, _t*log(-51200*_t**5 + 12*_t + x)))

Maxima [F]

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

[In]

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

[Out]

integrate(x^4/(x^8 - 3*x^4 + 1), x)

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=-\frac {1}{20} \, \sqrt {10 \, \sqrt {5} + 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {10 \, \sqrt {5} - 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \]

[In]

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

[Out]

-1/20*sqrt(10*sqrt(5) + 10)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/20*sqrt(10*sqrt(5) - 10)*arctan(x/sqrt(1/2*s
qrt(5) - 1/2)) - 1/40*sqrt(10*sqrt(5) + 10)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(10*sqrt(5) + 10)
*log(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(10*sqrt(5) - 10)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/
40*sqrt(10*sqrt(5) - 10)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2)))

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.55 \[ \int \frac {x^4}{1-3 x^4+x^8} \, dx=\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {-\sqrt {5}-1}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}-1\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {-\sqrt {5}-1}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}-1\right )}\right )\,\sqrt {-\sqrt {5}-1}\,1{}\mathrm {i}}{20}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {1-\sqrt {5}}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}+1\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {1-\sqrt {5}}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}+1\right )}\right )\,\sqrt {1-\sqrt {5}}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {\sqrt {5}+1}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}-1\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {\sqrt {5}+1}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}-1\right )}\right )\,\sqrt {\sqrt {5}+1}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {\sqrt {5}-1}\,1{}\mathrm {i}}{2\,\left (\sqrt {5}+1\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {\sqrt {5}-1}\,3{}\mathrm {i}}{10\,\left (\sqrt {5}+1\right )}\right )\,\sqrt {\sqrt {5}-1}\,1{}\mathrm {i}}{20} \]

[In]

int(x^4/(x^8 - 3*x^4 + 1),x)

[Out]

(10^(1/2)*atan((10^(1/2)*x*(- 5^(1/2) - 1)^(1/2)*1i)/(2*(5^(1/2) - 1)) - (5^(1/2)*10^(1/2)*x*(- 5^(1/2) - 1)^(
1/2)*3i)/(10*(5^(1/2) - 1)))*(- 5^(1/2) - 1)^(1/2)*1i)/20 + (10^(1/2)*atan((10^(1/2)*x*(1 - 5^(1/2))^(1/2)*1i)
/(2*(5^(1/2) + 1)) + (5^(1/2)*10^(1/2)*x*(1 - 5^(1/2))^(1/2)*3i)/(10*(5^(1/2) + 1)))*(1 - 5^(1/2))^(1/2)*1i)/2
0 - (10^(1/2)*atan((10^(1/2)*x*(5^(1/2) + 1)^(1/2)*1i)/(2*(5^(1/2) - 1)) - (5^(1/2)*10^(1/2)*x*(5^(1/2) + 1)^(
1/2)*3i)/(10*(5^(1/2) - 1)))*(5^(1/2) + 1)^(1/2)*1i)/20 - (10^(1/2)*atan((10^(1/2)*x*(5^(1/2) - 1)^(1/2)*1i)/(
2*(5^(1/2) + 1)) + (5^(1/2)*10^(1/2)*x*(5^(1/2) - 1)^(1/2)*3i)/(10*(5^(1/2) + 1)))*(5^(1/2) - 1)^(1/2)*1i)/20