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

Optimal. Leaf size=370 \[ -\frac{1}{3 x^3}+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.269125, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {1504, 12, 1373, 1127, 1161, 618, 204, 1164, 628} \[ -\frac{1}{3 x^3}+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1504

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[(d*(f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*f*(m + 1)), x] + Dist[1/(a*f^n*(m + 1)), Int[(f*x)^
(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n,
x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -
1] && IntegerQ[p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1373

Int[(x_)^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[
2*q - b/c, 2]}, Dist[1/(2*c*r), Int[x^(m - n/2)/(q - r*x^(n/2) + x^n), x], x] - Dist[1/(2*c*r), Int[x^(m - 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
] && IGtQ[m, 0] && GeQ[m, n/2] && LtQ[m, (3*n)/2] && NegQ[b^2 - 4*a*c]

Rule 1127

Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, Dist[1/2, Int[(q + x^2)/(
a + b*x^2 + c*x^4), x], x] - Dist[1/2, Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && Lt
Q[b^2 - 4*a*c, 0] && PosQ[a*c]

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])

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]

Rubi steps

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

Mathematica [C]  time = 0.0146211, size = 47, normalized size = 0.13 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^4-1}\& \right ]-\frac{1}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*x^3) - RootSum[1 - #1^4 + #1^8 & , (Log[x - #1]*#1)/(-1 + 2*#1^4) & ]/4

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Maple [C]  time = 0.01, size = 46, normalized size = 0.1 \begin{align*} -{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{{{\it \_R}}^{4}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}}-{\frac{1}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sum(_R^4/(2*_R^7-_R^3)*ln(x-_R),_R=RootOf(_Z^8-_Z^4+1))-1/3/x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{3 \, x^{3}} - \int \frac{x^{4}}{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^4/(x^8-x^4+1),x, algorithm="maxima")

[Out]

-1/3/x^3 - integrate(x^4/(x^8 - x^4 + 1), x)

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Fricas [B]  time = 1.79793, size = 1962, normalized size = 5.3 \begin{align*} \frac{8 \, \sqrt{6} \sqrt{2} x^{3} \sqrt{\sqrt{3} + 2} \arctan \left (-\frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + \frac{1}{6} \, \sqrt{6} \sqrt{2} \sqrt{2 \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + 12 \, x^{2} + 12} \sqrt{\sqrt{3} + 2} - \sqrt{3} - 2\right ) + 8 \, \sqrt{6} \sqrt{2} x^{3} \sqrt{\sqrt{3} + 2} \arctan \left (-\frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + \frac{1}{6} \, \sqrt{6} \sqrt{2} \sqrt{-2 \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + 12 \, x^{2} + 12} \sqrt{\sqrt{3} + 2} + \sqrt{3} + 2\right ) - 4 \, \sqrt{6} \sqrt{2} x^{3} \sqrt{-4 \, \sqrt{3} + 8} \arctan \left (-\frac{1}{6} \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + \frac{1}{12} \, \sqrt{6} \sqrt{2} \sqrt{\sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + 12 \, x^{2} + 12} \sqrt{-4 \, \sqrt{3} + 8} + \sqrt{3} - 2\right ) - 4 \, \sqrt{6} \sqrt{2} x^{3} \sqrt{-4 \, \sqrt{3} + 8} \arctan \left (-\frac{1}{6} \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + \frac{1}{12} \, \sqrt{6} \sqrt{2} \sqrt{-\sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + 12 \, x^{2} + 12} \sqrt{-4 \, \sqrt{3} + 8} - \sqrt{3} + 2\right ) - 2 \, \sqrt{6}{\left (\sqrt{3} \sqrt{2} x^{3} - 2 \, \sqrt{2} x^{3}\right )} \sqrt{\sqrt{3} + 2} \log \left (2 \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + 12 \, x^{2} + 12\right ) + 2 \, \sqrt{6}{\left (\sqrt{3} \sqrt{2} x^{3} - 2 \, \sqrt{2} x^{3}\right )} \sqrt{\sqrt{3} + 2} \log \left (-2 \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + 12 \, x^{2} + 12\right ) - \sqrt{6}{\left (\sqrt{3} \sqrt{2} x^{3} + 2 \, \sqrt{2} x^{3}\right )} \sqrt{-4 \, \sqrt{3} + 8} \log \left (\sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + 12 \, x^{2} + 12\right ) + \sqrt{6}{\left (\sqrt{3} \sqrt{2} x^{3} + 2 \, \sqrt{2} x^{3}\right )} \sqrt{-4 \, \sqrt{3} + 8} \log \left (-\sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + 12 \, x^{2} + 12\right ) - 32}{96 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(8*sqrt(6)*sqrt(2)*x^3*sqrt(sqrt(3) + 2)*arctan(-1/3*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(sqrt(3) + 2) + 1/6*sq
rt(6)*sqrt(2)*sqrt(2*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(sqrt(3) + 2) + 12*x^2 + 12)*sqrt(sqrt(3) + 2) - sqrt(3) -
2) + 8*sqrt(6)*sqrt(2)*x^3*sqrt(sqrt(3) + 2)*arctan(-1/3*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(sqrt(3) + 2) + 1/6*sqr
t(6)*sqrt(2)*sqrt(-2*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(sqrt(3) + 2) + 12*x^2 + 12)*sqrt(sqrt(3) + 2) + sqrt(3) +
2) - 4*sqrt(6)*sqrt(2)*x^3*sqrt(-4*sqrt(3) + 8)*arctan(-1/6*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(-4*sqrt(3) + 8) + 1
/12*sqrt(6)*sqrt(2)*sqrt(sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(-4*sqrt(3) + 8) + 12*x^2 + 12)*sqrt(-4*sqrt(3) + 8) +
sqrt(3) - 2) - 4*sqrt(6)*sqrt(2)*x^3*sqrt(-4*sqrt(3) + 8)*arctan(-1/6*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(-4*sqrt(3
) + 8) + 1/12*sqrt(6)*sqrt(2)*sqrt(-sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(-4*sqrt(3) + 8) + 12*x^2 + 12)*sqrt(-4*sqrt
(3) + 8) - sqrt(3) + 2) - 2*sqrt(6)*(sqrt(3)*sqrt(2)*x^3 - 2*sqrt(2)*x^3)*sqrt(sqrt(3) + 2)*log(2*sqrt(6)*sqrt
(3)*sqrt(2)*x*sqrt(sqrt(3) + 2) + 12*x^2 + 12) + 2*sqrt(6)*(sqrt(3)*sqrt(2)*x^3 - 2*sqrt(2)*x^3)*sqrt(sqrt(3)
+ 2)*log(-2*sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(sqrt(3) + 2) + 12*x^2 + 12) - sqrt(6)*(sqrt(3)*sqrt(2)*x^3 + 2*sqrt
(2)*x^3)*sqrt(-4*sqrt(3) + 8)*log(sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(-4*sqrt(3) + 8) + 12*x^2 + 12) + sqrt(6)*(sqr
t(3)*sqrt(2)*x^3 + 2*sqrt(2)*x^3)*sqrt(-4*sqrt(3) + 8)*log(-sqrt(6)*sqrt(3)*sqrt(2)*x*sqrt(-4*sqrt(3) + 8) + 1
2*x^2 + 12) - 32)/x^3

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Sympy [A]  time = 1.59861, size = 32, normalized size = 0.09 \begin{align*} - \operatorname{RootSum}{\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log{\left (- 18432 t^{5} + 4 t + x \right )} \right )\right )} - \frac{1}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-18432*_t**5 + 4*_t + x))) - 1/(3*x**3)

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Giac [A]  time = 1.16982, size = 348, normalized size = 0.94 \begin{align*} -\frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) - \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) + \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))) - 1/24*(sqrt(6) - 3*sqrt(2))
*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2))) - 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt(6) + sq
rt(2))/(sqrt(6) - sqrt(2))) - 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2)))
- 1/48*(sqrt(6) - 3*sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) + 1/48*(sqrt(6) - 3*sqrt(2))*log(x^2 -
1/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/48*(sqrt(6) + 3*sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) + 1/48*(s
qrt(6) + 3*sqrt(2))*log(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/3/x^3