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

Optimal. Leaf size=169 \[ -\frac{\tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt [4]{2} \sqrt{5} \left (3+\sqrt{5}\right )^{3/4}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt [4]{2} \sqrt{5} \left (3+\sqrt{5}\right )^{3/4}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2\ 2^{3/4} \sqrt{5}} \]

[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])

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Rubi [A]  time = 0.0596203, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1347, 212, 206, 203} \[ -\frac{\tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt [4]{2} \sqrt{5} \left (3+\sqrt{5}\right )^{3/4}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2\ 2^{3/4} \sqrt{5}}-\frac{\tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt [4]{2} \sqrt{5} \left (3+\sqrt{5}\right )^{3/4}}+\frac{\left (3+\sqrt{5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2\ 2^{3/4} \sqrt{5}} \]

Antiderivative was successfully verified.

[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 1347

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]

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{1}{1-3 x^4+x^8} \, dx &=\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]  time = 0.17215, size = 160, normalized size = 0.95 \[ \frac{\frac{\left (1+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{\sqrt{5}-1}}-\frac{\left (\sqrt{5}-1\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{1+\sqrt{5}}}+\frac{\left (1+\sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{\sqrt{5}-1}}-\frac{\left (\sqrt{5}-1\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{1+\sqrt{5}}}}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

[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])

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Maple [A]  time = 0.021, 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(1/(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{1}{x^{8} - 3 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x^8 - 3*x^4 + 1), x)

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Fricas [B]  time = 1.87717, size = 834, normalized size = 4.93 \begin{align*} -\frac{1}{5} \, \sqrt{5} \sqrt{\sqrt{5} + 2} \arctan \left (\frac{1}{4} \, \sqrt{2 \, x^{2} + \sqrt{5} - 1}{\left (\sqrt{5} \sqrt{2} - \sqrt{2}\right )} \sqrt{\sqrt{5} + 2} - \frac{1}{2} \,{\left (\sqrt{5} x - x\right )} \sqrt{\sqrt{5} + 2}\right ) + \frac{1}{5} \, \sqrt{5} \sqrt{\sqrt{5} - 2} \arctan \left (\frac{1}{4} \, \sqrt{2 \, x^{2} + \sqrt{5} + 1}{\left (\sqrt{5} \sqrt{2} + \sqrt{2}\right )} \sqrt{\sqrt{5} - 2} - \frac{1}{2} \,{\left (\sqrt{5} x + x\right )} \sqrt{\sqrt{5} - 2}\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 (\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*sqrt(5)*sqrt(sqrt(5) + 2)*arctan(1/4*sqrt(2*x^2 + sqrt(5) - 1)*(sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) +
2) - 1/2*(sqrt(5)*x - x)*sqrt(sqrt(5) + 2)) + 1/5*sqrt(5)*sqrt(sqrt(5) - 2)*arctan(1/4*sqrt(2*x^2 + sqrt(5) +
1)*(sqrt(5)*sqrt(2) + sqrt(2))*sqrt(sqrt(5) - 2) - 1/2*(sqrt(5)*x + x)*sqrt(sqrt(5) - 2)) - 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)*s
qrt(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*sqr
t(5)*sqrt(sqrt(5) + 2)*log(-sqrt(sqrt(5) + 2)*(sqrt(5) - 3) + 2*x)

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Sympy [A]  time = 0.90778, size = 53, normalized size = 0.31 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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Giac [A]  time = 1.20588, size = 198, normalized size = 1.17 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))