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

Optimal. Leaf size=331 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{2-\sqrt{3}}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{2-\sqrt{3}}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{2+\sqrt{3}}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{2+\sqrt{3}}}-\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.234564, antiderivative size = 331, 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 = {1419, 1094, 634, 618, 204, 628} \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{2-\sqrt{3}}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{2-\sqrt{3}}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{2+\sqrt{3}}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{2+\sqrt{3}}}-\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )+\frac{1}{4} \sqrt{2-\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{2+\sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(1 - x^4 + x^8),x]

[Out]

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

Mathematica [C]  time = 0.0155387, size = 55, normalized size = 0.17 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\& \right ] \]

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[1 - #1^4 + #1^8 & , (Log[x - #1] + Log[x - #1]*#1^4)/(-#1^3 + 2*#1^7) & ]/4

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Maple [C]  time = 0.006, size = 42, normalized size = 0.1 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ({{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [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="maxima")

[Out]

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

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Fricas [A]  time = 1.46206, size = 1196, normalized size = 3.61 \begin{align*} -\frac{1}{8} \, \sqrt{\sqrt{3} + 2}{\left (\sqrt{3} - 2\right )} \log \left (2 \, x^{2} + 2 \, x \sqrt{\sqrt{3} + 2} + 2\right ) + \frac{1}{8} \, \sqrt{\sqrt{3} + 2}{\left (\sqrt{3} - 2\right )} \log \left (2 \, x^{2} - 2 \, x \sqrt{\sqrt{3} + 2} + 2\right ) + \frac{1}{16} \,{\left (\sqrt{3} + 2\right )} \sqrt{-4 \, \sqrt{3} + 8} \log \left (2 \, x^{2} + x \sqrt{-4 \, \sqrt{3} + 8} + 2\right ) - \frac{1}{16} \,{\left (\sqrt{3} + 2\right )} \sqrt{-4 \, \sqrt{3} + 8} \log \left (2 \, x^{2} - x \sqrt{-4 \, \sqrt{3} + 8} + 2\right ) - \frac{1}{2} \, \sqrt{\sqrt{3} + 2} \arctan \left (\sqrt{2} \sqrt{2 \, x^{2} + 2 \, x \sqrt{\sqrt{3} + 2} + 2} \sqrt{\sqrt{3} + 2} - 2 \, x \sqrt{\sqrt{3} + 2} - \sqrt{3} - 2\right ) - \frac{1}{2} \, \sqrt{\sqrt{3} + 2} \arctan \left (\sqrt{2} \sqrt{2 \, x^{2} - 2 \, x \sqrt{\sqrt{3} + 2} + 2} \sqrt{\sqrt{3} + 2} - 2 \, x \sqrt{\sqrt{3} + 2} + \sqrt{3} + 2\right ) - \frac{1}{4} \, \sqrt{-4 \, \sqrt{3} + 8} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{2 \, x^{2} + x \sqrt{-4 \, \sqrt{3} + 8} + 2} \sqrt{-4 \, \sqrt{3} + 8} - x \sqrt{-4 \, \sqrt{3} + 8} + \sqrt{3} - 2\right ) - \frac{1}{4} \, \sqrt{-4 \, \sqrt{3} + 8} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{2 \, x^{2} - x \sqrt{-4 \, \sqrt{3} + 8} + 2} \sqrt{-4 \, \sqrt{3} + 8} - x \sqrt{-4 \, \sqrt{3} + 8} - \sqrt{3} + 2\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]

-1/8*sqrt(sqrt(3) + 2)*(sqrt(3) - 2)*log(2*x^2 + 2*x*sqrt(sqrt(3) + 2) + 2) + 1/8*sqrt(sqrt(3) + 2)*(sqrt(3) -
2)*log(2*x^2 - 2*x*sqrt(sqrt(3) + 2) + 2) + 1/16*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8)*log(2*x^2 + x*sqrt(-4*sqr
t(3) + 8) + 2) - 1/16*(sqrt(3) + 2)*sqrt(-4*sqrt(3) + 8)*log(2*x^2 - x*sqrt(-4*sqrt(3) + 8) + 2) - 1/2*sqrt(sq
rt(3) + 2)*arctan(sqrt(2)*sqrt(2*x^2 + 2*x*sqrt(sqrt(3) + 2) + 2)*sqrt(sqrt(3) + 2) - 2*x*sqrt(sqrt(3) + 2) -
sqrt(3) - 2) - 1/2*sqrt(sqrt(3) + 2)*arctan(sqrt(2)*sqrt(2*x^2 - 2*x*sqrt(sqrt(3) + 2) + 2)*sqrt(sqrt(3) + 2)
- 2*x*sqrt(sqrt(3) + 2) + sqrt(3) + 2) - 1/4*sqrt(-4*sqrt(3) + 8)*arctan(1/2*sqrt(2)*sqrt(2*x^2 + x*sqrt(-4*sq
rt(3) + 8) + 2)*sqrt(-4*sqrt(3) + 8) - x*sqrt(-4*sqrt(3) + 8) + sqrt(3) - 2) - 1/4*sqrt(-4*sqrt(3) + 8)*arctan
(1/2*sqrt(2)*sqrt(2*x^2 - x*sqrt(-4*sqrt(3) + 8) + 2)*sqrt(-4*sqrt(3) + 8) - x*sqrt(-4*sqrt(3) + 8) - sqrt(3)
+ 2)

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Sympy [A]  time = 1.2298, size = 20, normalized size = 0.06 \begin{align*} \operatorname{RootSum}{\left (65536 t^{8} - 256 t^{4} + 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]

RootSum(65536*_t**8 - 256*_t**4 + 1, Lambda(_t, _t*log(1024*_t**5 + x)))

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Giac [A]  time = 1.1562, size = 331, normalized size = 1. \begin{align*} \frac{1}{8} \,{\left (\sqrt{6} - \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{8} \,{\left (\sqrt{6} - \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{8} \,{\left (\sqrt{6} + \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{8} \,{\left (\sqrt{6} + \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{16} \,{\left (\sqrt{6} - \sqrt{2}\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{16} \,{\left (\sqrt{6} - \sqrt{2}\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{16} \,{\left (\sqrt{6} + \sqrt{2}\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{16} \,{\left (\sqrt{6} + \sqrt{2}\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 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="giac")

[Out]

1/8*(sqrt(6) - sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))) + 1/8*(sqrt(6) - sqrt(2))*arctan
((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2))) + 1/8*(sqrt(6) + sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sq
rt(6) - sqrt(2))) + 1/8*(sqrt(6) + sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2))) + 1/16*(sqrt
(6) - sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/16*(sqrt(6) - sqrt(2))*log(x^2 - 1/2*x*(sqrt(6) +
sqrt(2)) + 1) + 1/16*(sqrt(6) + sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/16*(sqrt(6) + sqrt(2))*l
og(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)