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

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

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Rubi [A]  time = 0.10, antiderivative size = 140, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 ) \end {gather*}

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 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 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 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 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 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 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]

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*}

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Mathematica [C]  time = 0.17, size = 129, normalized size = 0.92 \begin {gather*} \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 ) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^4}{1+x^4+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.16, size = 137, normalized size = 0.98 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.37, size = 108, normalized size = 0.77 \begin {gather*} \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}{4} \, \arctan \left (2 \, x + \sqrt {3}\right ) - \frac {1}{4} \, \arctan \left (2 \, x - \sqrt {3}\right ) - \frac {1}{8} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{8} \, \log \left (x^{2} - x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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maple [A]  time = 0.01, size = 109, normalized size = 0.78 \begin {gather*} \frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{4}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{4}-\frac {\arctan \left (2 x -\sqrt {3}\right )}{4}-\frac {\arctan \left (2 x +\sqrt {3}\right )}{4}-\frac {\sqrt {3}\, \ln \left (x^{2}-\sqrt {3}\, x +1\right )}{8}+\frac {\sqrt {3}\, \ln \left (x^{2}+\sqrt {3}\, x +1\right )}{8}+\frac {\ln \left (x^{2}-x +1\right )}{8}-\frac {\ln \left (x^{2}+x +1\right )}{8} \end {gather*}

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.19, size = 109, normalized size = 0.78 \begin {gather*} -\mathrm {atan}\left (\frac {54\,\sqrt {3}\,x}{-81+\sqrt {3}\,27{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{4}+\frac {1}{4}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {54\,\sqrt {3}\,x}{81+\sqrt {3}\,27{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{4}-\frac {1}{4}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\sqrt {3}\,x\,54{}\mathrm {i}}{-81+\sqrt {3}\,27{}\mathrm {i}}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )-\mathrm {atan}\left (\frac {\sqrt {3}\,x\,54{}\mathrm {i}}{81+\sqrt {3}\,27{}\mathrm {i}}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4 - 1)/(x^4 + x^8 + 1),x)

[Out]

atan((54*3^(1/2)*x)/(3^(1/2)*27i + 81))*(3^(1/2)/4 - 1i/4) - atan((54*3^(1/2)*x)/(3^(1/2)*27i - 81))*(3^(1/2)/
4 + 1i/4) + atan((3^(1/2)*x*54i)/(3^(1/2)*27i - 81))*((3^(1/2)*1i)/4 - 1/4) - atan((3^(1/2)*x*54i)/(3^(1/2)*27
i + 81))*((3^(1/2)*1i)/4 + 1/4)

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sympy [C]  time = 0.62, size = 148, normalized size = 1.06 \begin {gather*} - \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 {gather*}

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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