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

Optimal. Leaf size=140 \[ -\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )-\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

-ArcTan[(1 - 2*x)/Sqrt[3]]/(4*Sqrt[3]) - ArcTan[Sqrt[3] - 2*x]/4 + ArcTan[(1 + 2*x)/Sqrt[3]]/(4*Sqrt[3]) + Arc
Tan[Sqrt[3] + 2*x]/4 - Log[1 - x + x^2]/8 + Log[1 + x + x^2]/8 - Log[1 - Sqrt[3]*x + x^2]/(8*Sqrt[3]) + Log[1
+ Sqrt[3]*x + x^2]/(8*Sqrt[3])

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Rubi [A]  time = 0.0954812, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {1419, 1094, 634, 618, 204, 628} \[ -\frac{1}{8} \log \left (x^2-x+1\right )+\frac{1}{8} \log \left (x^2+x+1\right )-\frac{\log \left (x^2-\sqrt{3} x+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{8 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{4 \sqrt{3}}+\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(1 + x^4 + x^8),x]

[Out]

-ArcTan[(1 - 2*x)/Sqrt[3]]/(4*Sqrt[3]) - ArcTan[Sqrt[3] - 2*x]/4 + ArcTan[(1 + 2*x)/Sqrt[3]]/(4*Sqrt[3]) + Arc
Tan[Sqrt[3] + 2*x]/4 - Log[1 - x + x^2]/8 + Log[1 + x + x^2]/8 - Log[1 - Sqrt[3]*x + x^2]/(8*Sqrt[3]) + Log[1
+ Sqrt[3]*x + x^2]/(8*Sqrt[3])

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 1094

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-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)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 634

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

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])

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]

Rubi steps

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

Mathematica [C]  time = 0.184966, size = 135, normalized size = 0.96 \[ \frac{1}{48} \left (-6 \log \left (x^2-x+1\right )+6 \log \left (x^2+x+1\right )+4 i \sqrt{-6-6 i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x\right )-4 i \sqrt{-6+6 i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((4*I)*Sqrt[-6 - (6*I)*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x)/2] - (4*I)*Sqrt[-6 + (6*I)*Sqrt[3]]*ArcTan[((1 + I*
Sqrt[3])*x)/2] + 4*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]] + 4*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 6*Log[1 - x + x^
2] + 6*Log[1 + x + x^2])/48

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Maple [A]  time = 0.016, size = 109, normalized size = 0.8 \begin{align*}{\frac{\ln \left ({x}^{2}+x+1 \right ) }{8}}+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{8}}+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{4}}-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*ln(x^2+x+1)+1/12*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)-1/8*ln(x^2-x+1)+1/12*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/
2))+1/24*ln(1+x^2+x*3^(1/2))*3^(1/2)+1/4*arctan(2*x+3^(1/2))-1/24*ln(1+x^2-x*3^(1/2))*3^(1/2)+1/4*arctan(2*x-3
^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{2} \, \int \frac{1}{x^{4} - x^{2} + 1}\,{d x} + \frac{1}{8} \, \log \left (x^{2} + x + 1\right ) - \frac{1}{8} \, \log \left (x^{2} - x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/2*integrate(1/(x^4
- x^2 + 1), x) + 1/8*log(x^2 + x + 1) - 1/8*log(x^2 - x + 1)

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Fricas [A]  time = 1.43924, size = 716, normalized size = 5.11 \begin{align*} -\frac{1}{12} \, \sqrt{6} \sqrt{3} \sqrt{2} \arctan \left (-\frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2} x + \frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{\sqrt{6} \sqrt{2} x + 2 \, x^{2} + 2} - \sqrt{3}\right ) - \frac{1}{12} \, \sqrt{6} \sqrt{3} \sqrt{2} \arctan \left (-\frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2} x + \frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{-\sqrt{6} \sqrt{2} x + 2 \, x^{2} + 2} + \sqrt{3}\right ) + \frac{1}{48} \, \sqrt{6} \sqrt{2} \log \left (\sqrt{6} \sqrt{2} x + 2 \, x^{2} + 2\right ) - \frac{1}{48} \, \sqrt{6} \sqrt{2} \log \left (-\sqrt{6} \sqrt{2} x + 2 \, x^{2} + 2\right ) + \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{8} \, \log \left (x^{2} + x + 1\right ) - \frac{1}{8} \, \log \left (x^{2} - x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(6)*sqrt(3)*sqrt(2)*arctan(-1/3*sqrt(6)*sqrt(3)*sqrt(2)*x + 1/3*sqrt(6)*sqrt(3)*sqrt(sqrt(6)*sqrt(2)
*x + 2*x^2 + 2) - sqrt(3)) - 1/12*sqrt(6)*sqrt(3)*sqrt(2)*arctan(-1/3*sqrt(6)*sqrt(3)*sqrt(2)*x + 1/3*sqrt(6)*
sqrt(3)*sqrt(-sqrt(6)*sqrt(2)*x + 2*x^2 + 2) + sqrt(3)) + 1/48*sqrt(6)*sqrt(2)*log(sqrt(6)*sqrt(2)*x + 2*x^2 +
2) - 1/48*sqrt(6)*sqrt(2)*log(-sqrt(6)*sqrt(2)*x + 2*x^2 + 2) + 1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) +
1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/8*log(x^2 + x + 1) - 1/8*log(x^2 - x + 1)

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Sympy [C]  time = 0.692687, size = 190, normalized size = 1.36 \begin{align*} \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x - 1 - \frac{\sqrt{3} i}{3} + 9216 \left (- \frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{5} \right )} + \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x - 1 + 9216 \left (- \frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{5} + \frac{\sqrt{3} i}{3} \right )} + \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right ) \log{\left (x + 1 - \frac{\sqrt{3} i}{3} + 9216 \left (\frac{1}{8} - \frac{\sqrt{3} i}{24}\right )^{5} \right )} + \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right ) \log{\left (x + 1 + 9216 \left (\frac{1}{8} + \frac{\sqrt{3} i}{24}\right )^{5} + \frac{\sqrt{3} i}{3} \right )} + \operatorname{RootSum}{\left (2304 t^{4} + 48 t^{2} + 1, \left ( t \mapsto t \log{\left (9216 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+x**4+1),x)

[Out]

(-1/8 - sqrt(3)*I/24)*log(x - 1 - sqrt(3)*I/3 + 9216*(-1/8 - sqrt(3)*I/24)**5) + (-1/8 + sqrt(3)*I/24)*log(x -
1 + 9216*(-1/8 + sqrt(3)*I/24)**5 + sqrt(3)*I/3) + (1/8 - sqrt(3)*I/24)*log(x + 1 - sqrt(3)*I/3 + 9216*(1/8 -
sqrt(3)*I/24)**5) + (1/8 + sqrt(3)*I/24)*log(x + 1 + 9216*(1/8 + sqrt(3)*I/24)**5 + sqrt(3)*I/3) + RootSum(23
04*_t**4 + 48*_t**2 + 1, Lambda(_t, _t*log(9216*_t**5 + 8*_t + x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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