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

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

[Out]

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

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Rubi [A]  time = 0.0987737, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1421, 1169, 634, 618, 204, 628} \[ \frac{1}{8} \log \left (x^2-x+1\right )-\frac{1}{8} \log \left (x^2+x+1\right )-\frac{1}{8} \sqrt{3} \log \left (x^2-\sqrt{3} x+1\right )+\frac{1}{8} \sqrt{3} \log \left (x^2+\sqrt{3} x+1\right )-\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )-\frac{1}{4} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)/(1 + x^4 + x^8),x]

[Out]

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

Rule 1421

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*q), Int[(q - 2*x^(n/2))/Simp[d/e + q*x^(n/2) - x^n, x], x], x] + Dist[e/(2*c*q), Int[(q
+ 2*x^(n/2))/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[b^2 - 4*a*c, 0]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 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 &=-\left (\frac{1}{2} \int \frac{1+2 x^2}{-1-x^2-x^4} \, dx\right )-\frac{1}{2} \int \frac{1-2 x^2}{-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}-3 x}{1-\sqrt{3} x+x^2} \, dx}{4 \sqrt{3}}+\frac{\int \frac{\sqrt{3}+3 x}{1+\sqrt{3} x+x^2} \, dx}{4 \sqrt{3}}\\ &=\frac{1}{8} \int \frac{-1+2 x}{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{3}{8} \int \frac{1}{1-x+x^2} \, dx+\frac{3}{8} \int \frac{1}{1+x+x^2} \, dx-\frac{1}{8} \sqrt{3} \int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx+\frac{1}{8} \sqrt{3} \int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx\\ &=\frac{1}{8} \log \left (1-x+x^2\right )-\frac{1}{8} \log \left (1+x+x^2\right )-\frac{1}{8} \sqrt{3} \log \left (1-\sqrt{3} x+x^2\right )+\frac{1}{8} \sqrt{3} \log \left (1+\sqrt{3} x+x^2\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{3}{4} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )+\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )-\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{1}{8} \sqrt{3} \log \left (1-\sqrt{3} x+x^2\right )+\frac{1}{8} \sqrt{3} \log \left (1+\sqrt{3} x+x^2\right )\\ \end{align*}

Mathematica [C]  time = 0.177305, size = 129, normalized size = 0.92 \[ \frac{1}{8} \left (\log \left (x^2-x+1\right )-\log \left (x^2+x+1\right )-2 \sqrt{-2-2 i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x\right )-2 \sqrt{-2+2 i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 109, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ({x}^{2}+x+1 \right ) }{8}}+{\frac{\sqrt{3}}{4}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{8}}+{\frac{\sqrt{3}}{4}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{8}}-{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{4}}-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{8}}-{\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)+1/4*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)+1/8*ln(x^2-x+1)+1/4*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2
))+1/8*ln(1+x^2+x*3^(1/2))*3^(1/2)-1/4*arctan(2*x+3^(1/2))-1/8*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}{4} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{4} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{2} \, \int \frac{2 \, x^{2} - 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]

1/4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) + 1/4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/2*integrate((2*x^2 -
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 = 1.36462, size = 444, normalized size = 3.17 \begin{align*} \frac{1}{4} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{4} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{8} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) - \frac{1}{8} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) + \frac{1}{2} \, \arctan \left (-2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}\right ) + \frac{1}{2} \, \arctan \left (-2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}\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]

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

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Sympy [C]  time = 0.612263, size = 148, normalized size = 1.06 \begin{align*} - \left (- \frac{1}{8} - \frac{\sqrt{3} i}{8}\right ) \log{\left (x + 1024 \left (- \frac{1}{8} - \frac{\sqrt{3} i}{8}\right )^{5} \right )} - \left (- \frac{1}{8} + \frac{\sqrt{3} i}{8}\right ) \log{\left (x + 1024 \left (- \frac{1}{8} + \frac{\sqrt{3} i}{8}\right )^{5} \right )} - \left (\frac{1}{8} - \frac{\sqrt{3} i}{8}\right ) \log{\left (x + 1024 \left (\frac{1}{8} - \frac{\sqrt{3} i}{8}\right )^{5} \right )} - \left (\frac{1}{8} + \frac{\sqrt{3} i}{8}\right ) \log{\left (x + 1024 \left (\frac{1}{8} + \frac{\sqrt{3} i}{8}\right )^{5} \right )} - \operatorname{RootSum}{\left (256 t^{4} - 16 t^{2} + 1, \left ( t \mapsto t \log{\left (1024 t^{5} + 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/8)*log(x + 1024*(-1/8 - sqrt(3)*I/8)**5) - (-1/8 + sqrt(3)*I/8)*log(x + 1024*(-1/8 + sqrt(3
)*I/8)**5) - (1/8 - sqrt(3)*I/8)*log(x + 1024*(1/8 - sqrt(3)*I/8)**5) - (1/8 + sqrt(3)*I/8)*log(x + 1024*(1/8
+ sqrt(3)*I/8)**5) - RootSum(256*_t**4 - 16*_t**2 + 1, Lambda(_t, _t*log(1024*_t**5 + 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)