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

Optimal. Leaf size=131 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}} \]

[Out]

ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTan[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt[5])
] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sq
rt[5])]

________________________________________________________________________________________

Rubi [A]  time = 0.0862422, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1419, 1093, 203, 207} \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(1 - 3*x^4 + x^8),x]

[Out]

ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTan[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt[5])
] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sq
rt[5])]

Rule 1419

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

Rule 1093

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

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])

Rule 207

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

Rubi steps

\begin{align*} \int \frac{1+x^4}{1-3 x^4+x^8} \, dx &=\frac{1}{2} \int \frac{1}{1-\sqrt{5} x^2+x^4} \, dx+\frac{1}{2} \int \frac{1}{1+\sqrt{5} x^2+x^4} \, dx\\ &=\frac{1}{2} \int \frac{1}{-\frac{1}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx-\frac{1}{2} \int \frac{1}{\frac{1}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx+\frac{1}{2} \int \frac{1}{-\frac{1}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx-\frac{1}{2} \int \frac{1}{\frac{1}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx\\ &=\frac{\tan ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{5}}} x\right )}{\sqrt{2 \left (-1+\sqrt{5}\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{5}}} x\right )}{\sqrt{2 \left (-1+\sqrt{5}\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}}\\ \end{align*}

Mathematica [A]  time = 0.0798201, size = 131, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)/(1 - 3*x^4 + x^8),x]

[Out]

ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTan[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt[5])
] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sq
rt[5])]

________________________________________________________________________________________

Maple [A]  time = 0.029, size = 96, normalized size = 0.7 \begin{align*} -{\frac{1}{\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{\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+1)/(x^8-3*x^4+1),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.37286, size = 815, normalized size = 6.22 \begin{align*} -\frac{1}{2} \, \sqrt{2} \sqrt{\sqrt{5} + 1} \arctan \left (-\frac{1}{2} \, \sqrt{2} x \sqrt{\sqrt{5} + 1} + \frac{1}{2} \, \sqrt{2 \, x^{2} + \sqrt{5} - 1} \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{2} \, \sqrt{2} \sqrt{\sqrt{5} - 1} \arctan \left (-\frac{1}{2} \, \sqrt{2} x \sqrt{\sqrt{5} - 1} + \frac{1}{2} \, \sqrt{2 \, x^{2} + \sqrt{5} + 1} \sqrt{\sqrt{5} - 1}\right ) + \frac{1}{8} \, \sqrt{2} \sqrt{\sqrt{5} + 1} \log \left ({\left (\sqrt{5} \sqrt{2} - \sqrt{2}\right )} \sqrt{\sqrt{5} + 1} + 4 \, x\right ) - \frac{1}{8} \, \sqrt{2} \sqrt{\sqrt{5} + 1} \log \left (-{\left (\sqrt{5} \sqrt{2} - \sqrt{2}\right )} \sqrt{\sqrt{5} + 1} + 4 \, x\right ) - \frac{1}{8} \, \sqrt{2} \sqrt{\sqrt{5} - 1} \log \left ({\left (\sqrt{5} \sqrt{2} + \sqrt{2}\right )} \sqrt{\sqrt{5} - 1} + 4 \, x\right ) + \frac{1}{8} \, \sqrt{2} \sqrt{\sqrt{5} - 1} \log \left (-{\left (\sqrt{5} \sqrt{2} + \sqrt{2}\right )} \sqrt{\sqrt{5} - 1} + 4 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*sqrt(sqrt(5) + 1)*arctan(-1/2*sqrt(2)*x*sqrt(sqrt(5) + 1) + 1/2*sqrt(2*x^2 + sqrt(5) - 1)*sqrt(sq
rt(5) + 1)) + 1/2*sqrt(2)*sqrt(sqrt(5) - 1)*arctan(-1/2*sqrt(2)*x*sqrt(sqrt(5) - 1) + 1/2*sqrt(2*x^2 + sqrt(5)
+ 1)*sqrt(sqrt(5) - 1)) + 1/8*sqrt(2)*sqrt(sqrt(5) + 1)*log((sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) + 1) + 4
*x) - 1/8*sqrt(2)*sqrt(sqrt(5) + 1)*log(-(sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) + 1) + 4*x) - 1/8*sqrt(2)*sq
rt(sqrt(5) - 1)*log((sqrt(5)*sqrt(2) + sqrt(2))*sqrt(sqrt(5) - 1) + 4*x) + 1/8*sqrt(2)*sqrt(sqrt(5) - 1)*log(-
(sqrt(5)*sqrt(2) + sqrt(2))*sqrt(sqrt(5) - 1) + 4*x)

________________________________________________________________________________________

Sympy [A]  time = 0.882531, size = 49, normalized size = 0.37 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} - 16 t^{2} - 1, \left ( t \mapsto t \log{\left (1024 t^{5} - 8 t + x \right )} \right )\right )} + \operatorname{RootSum}{\left (256 t^{4} + 16 t^{2} - 1, \left ( t \mapsto t \log{\left (1024 t^{5} - 8 t + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(256*_t**4 - 16*_t**2 - 1, Lambda(_t, _t*log(1024*_t**5 - 8*_t + x))) + RootSum(256*_t**4 + 16*_t**2 -
1, Lambda(_t, _t*log(1024*_t**5 - 8*_t + x)))

________________________________________________________________________________________

Giac [A]  time = 1.23239, size = 198, normalized size = 1.51 \begin{align*} -\frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) + \frac{1}{4} \, \sqrt{2 \, \sqrt{5} + 2} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{8} \, \sqrt{2 \, \sqrt{5} - 2} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{8} \, \sqrt{2 \, \sqrt{5} - 2} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{8} \, \sqrt{2 \, \sqrt{5} + 2} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{8} \, \sqrt{2 \, \sqrt{5} + 2} \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+1)/(x^8-3*x^4+1),x, algorithm="giac")

[Out]

-1/4*sqrt(2*sqrt(5) - 2)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/4*sqrt(2*sqrt(5) + 2)*arctan(x/sqrt(1/2*sqrt(5)
- 1/2)) - 1/8*sqrt(2*sqrt(5) - 2)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/8*sqrt(2*sqrt(5) - 2)*log(abs(x -
sqrt(1/2*sqrt(5) + 1/2))) + 1/8*sqrt(2*sqrt(5) + 2)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/8*sqrt(2*sqrt(5
) + 2)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2)))