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

Optimal. Leaf size=173 \[ -\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (3-\sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (3-\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

[Out]

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

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Rubi [A]  time = 0.075094, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1374, 212, 206, 203} \[ -\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (3-\sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (3-\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(1 - 3*x^4 + x^8),x]

[Out]

-(((3 + Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*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 1374

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*(b/q + 1))/2, Int[(d*x)^(m - n)/(b/2 + q/2 + c*x^n), x], x] - Dist[(d^n*(b/q - 1))/2, 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]

Rule 212

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] &&
!GtQ[a/b, 0]

Rule 206

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{x^4}{1-3 x^4+x^8} \, 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}{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]  time = 0.196874, size = 132, normalized size = 0.76 \[ \frac{\sqrt{\sqrt{5}-1} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )-\sqrt{1+\sqrt{5}} \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )+\sqrt{\sqrt{5}-1} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )-\sqrt{1+\sqrt{5}} \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

[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])

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Maple [A]  time = 0.034, size = 206, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{10\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{2\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{10\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{2\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{10\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(2+2*5^(1/2))^(1/2)*arctanh(2*x/(2+2*5^(1/2))^(1/2))-1/10*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctanh(2*x/(2+2*5^
(1/2))^(1/2))+1/2/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))-1/10*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arct
an(2*x/(-2+2*5^(1/2))^(1/2))+1/2/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))-1/10*5^(1/2)/(-2+2*5^(
1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))-1/2/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2+2*5^(1/2))^(1/2))-1/10*5^(
1/2)/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2+2*5^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/(x^8 - 3*x^4 + 1), x)

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Fricas [B]  time = 1.9188, size = 932, normalized size = 5.39 \begin{align*} -\frac{1}{10} \, \sqrt{10} \sqrt{\sqrt{5} + 1} \arctan \left (\frac{1}{40} \, \sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} + 1}{\left (\sqrt{5} \sqrt{2} - 5 \, \sqrt{2}\right )} \sqrt{\sqrt{5} + 1} - \frac{1}{20} \, \sqrt{10}{\left (\sqrt{5} x - 5 \, x\right )} \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{10} \, \sqrt{10} \sqrt{\sqrt{5} - 1} \arctan \left (\frac{1}{40} \, \sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} - 1}{\left (\sqrt{5} \sqrt{2} + 5 \, \sqrt{2}\right )} \sqrt{\sqrt{5} - 1} - \frac{1}{20} \, \sqrt{10}{\left (\sqrt{5} x + 5 \, x\right )} \sqrt{\sqrt{5} - 1}\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*sqrt(10)*sqrt(sqrt(5) + 1)*arctan(1/40*sqrt(10)*sqrt(2*x^2 + sqrt(5) + 1)*(sqrt(5)*sqrt(2) - 5*sqrt(2))*
sqrt(sqrt(5) + 1) - 1/20*sqrt(10)*(sqrt(5)*x - 5*x)*sqrt(sqrt(5) + 1)) - 1/10*sqrt(10)*sqrt(sqrt(5) - 1)*arcta
n(1/40*sqrt(10)*sqrt(2*x^2 + sqrt(5) - 1)*(sqrt(5)*sqrt(2) + 5*sqrt(2))*sqrt(sqrt(5) - 1) - 1/20*sqrt(10)*(sqr
t(5)*x + 5*x)*sqrt(sqrt(5) - 1)) - 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(sqr
t(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)

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Sympy [A]  time = 0.883801, size = 49, normalized size = 0.28 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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Giac [A]  time = 1.24809, size = 198, normalized size = 1.14 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))