3.342 \(\int \frac{1}{1+x^4+x^8} \, dx\)

Optimal. Leaf size=88 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0516773, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {1346, 1164, 628, 1161, 618, 204} \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4 + x^8)^(-1),x]

[Out]

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

Rule 1346

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

Rule 1164

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1161

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

Rule 618

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

Rule 204

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

Rubi steps

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

Mathematica [A]  time = 0.0167882, size = 68, normalized size = 0.77 \[ \frac{-\log \left (-x^2+\sqrt{3} x-1\right )+\log \left (x^2+\sqrt{3} x+1\right )+2 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4 + x^8)^(-1),x]

[Out]

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

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Maple [A]  time = 0.009, size = 67, normalized size = 0.8 \begin{align*}{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}}-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8+x^4+1),x)

[Out]

1/6*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)+1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))+1/12*ln(1+x^2+x*3^(1/2))*3^(1/
2)-1/12*ln(1+x^2-x*3^(1/2))*3^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{2} \, \int \frac{x^{2} - 1}{x^{4} - x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/2*integrate((x^2 - 1
)/(x^4 - x^2 + 1), x)

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Fricas [A]  time = 1.48196, size = 212, normalized size = 2.41 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x^{3} + 2 \, x\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) + \frac{1}{12} \, \sqrt{3} \log \left (\frac{x^{4} + 5 \, x^{2} + 2 \, \sqrt{3}{\left (x^{3} + x\right )} + 1}{x^{4} - x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(x^3 + 2*x)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*x) + 1/12*sqrt(3)*log((x^4 + 5*x^
2 + 2*sqrt(3)*(x^3 + x) + 1)/(x^4 - x^2 + 1))

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Sympy [A]  time = 0.171143, size = 82, normalized size = 0.93 \begin{align*} \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )} + 2 \operatorname{atan}{\left (\frac{\sqrt{3} x^{3}}{3} + \frac{2 \sqrt{3} x}{3} \right )}\right )}{12} - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*(2*atan(sqrt(3)*x/3) + 2*atan(sqrt(3)*x**3/3 + 2*sqrt(3)*x/3))/12 - sqrt(3)*log(x**2 - sqrt(3)*x + 1)/
12 + sqrt(3)*log(x**2 + sqrt(3)*x + 1)/12

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Giac [A]  time = 1.11414, size = 97, normalized size = 1.1 \begin{align*} \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{12} \, \sqrt{3} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{3} + \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{3} + \frac{2}{x} \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/12*sqrt(3)*log(abs(2
*x - 2*sqrt(3) + 2/x)/abs(2*x + 2*sqrt(3) + 2/x))