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

Optimal. Leaf size=41 \[ \frac {\tan ^{-1}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{8} \log \left (x^8-x^4+1\right )+\log (x) \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1474, 800, 634, 618, 204, 628} \[ -\frac {1}{8} \log \left (x^8-x^4+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)/(x*(1 - x^4 + x^8)),x]

[Out]

ArcTan[(1 - 2*x^4)/Sqrt[3]]/(4*Sqrt[3]) + Log[x] - Log[1 - x^4 + x^8]/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 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1474

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a,
b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [C]  time = 0.01, size = 44, normalized size = 1.07 \[ \log (x)-\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})}{2 \text {$\#$1}^4-1}\& \right ] \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1^4)/(-1 + 2*#1^4) & ]/4

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fricas [A]  time = 0.79, size = 34, normalized size = 0.83 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) + \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.45, size = 38, normalized size = 0.93 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 35, normalized size = 0.85 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{4}-1\right ) \sqrt {3}}{3}\right )}{12}+\ln \relax (x )-\frac {\ln \left (x^{8}-x^{4}+1\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.97, size = 38, normalized size = 0.93 \[ -\frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) - \frac {1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) + \frac {1}{4} \, \log \left (x^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.89, size = 36, normalized size = 0.88 \[ \ln \relax (x)-\frac {\ln \left (x^8-x^4+1\right )}{8}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,x^4}{3}\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4 - 1)/(x*(x^8 - x^4 + 1)),x)

[Out]

log(x) - log(x^8 - x^4 + 1)/8 + (3^(1/2)*atan(3^(1/2)/3 - (2*3^(1/2)*x^4)/3))/12

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sympy [A]  time = 0.16, size = 41, normalized size = 1.00 \[ \log {\relax (x )} - \frac {\log {\left (x^{8} - x^{4} + 1 \right )}}{8} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} - \frac {\sqrt {3}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(x**8 - x**4 + 1)/8 - sqrt(3)*atan(2*sqrt(3)*x**4/3 - sqrt(3)/3)/12

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