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

Optimal. Leaf size=347 \[ \frac{1}{8} \sqrt{\frac{1}{2} \left (2-\sqrt{2}\right )} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{2} \left (2-\sqrt{2}\right )} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2-\sqrt{2}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2+\sqrt{2}}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2-\sqrt{2}}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2+\sqrt{2}}} \]

[Out]

-ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2 - Sqrt[2]]) + ArcTan[(Sqrt[2 + Sqrt[2]] - 2*x)/
Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2 + Sqrt[2]]) + ArcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2 - Sqr
t[2]]) - ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2 + Sqrt[2]]) + (Sqrt[(2 - Sqrt[2])/2]*Lo
g[1 - Sqrt[2 - Sqrt[2]]*x + x^2])/8 - (Sqrt[(2 - Sqrt[2])/2]*Log[1 + Sqrt[2 - Sqrt[2]]*x + x^2])/8 - (Sqrt[(2
+ Sqrt[2])/2]*Log[1 - Sqrt[2 + Sqrt[2]]*x + x^2])/8 + (Sqrt[(2 + Sqrt[2])/2]*Log[1 + Sqrt[2 + Sqrt[2]]*x + x^2
])/8

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Rubi [A]  time = 0.270287, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {1414, 1169, 634, 618, 204, 628} \[ \frac{1}{8} \sqrt{\frac{1}{2} \left (2-\sqrt{2}\right )} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{2} \left (2-\sqrt{2}\right )} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \log \left (x^2-\sqrt{2+\sqrt{2}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \log \left (x^2+\sqrt{2+\sqrt{2}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2-\sqrt{2}}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2+\sqrt{2}}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{4 \sqrt{2-\sqrt{2}}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right )}{4 \sqrt{2+\sqrt{2}}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)/(1 + x^8),x]

[Out]

-ArcTan[(Sqrt[2 - Sqrt[2]] - 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2 - Sqrt[2]]) + ArcTan[(Sqrt[2 + Sqrt[2]] - 2*x)/
Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2 + Sqrt[2]]) + ArcTan[(Sqrt[2 - Sqrt[2]] + 2*x)/Sqrt[2 + Sqrt[2]]]/(4*Sqrt[2 - Sqr
t[2]]) - ArcTan[(Sqrt[2 + Sqrt[2]] + 2*x)/Sqrt[2 - Sqrt[2]]]/(4*Sqrt[2 + Sqrt[2]]) + (Sqrt[(2 - Sqrt[2])/2]*Lo
g[1 - Sqrt[2 - Sqrt[2]]*x + x^2])/8 - (Sqrt[(2 - Sqrt[2])/2]*Log[1 + Sqrt[2 - Sqrt[2]]*x + x^2])/8 - (Sqrt[(2
+ Sqrt[2])/2]*Log[1 - Sqrt[2 + Sqrt[2]]*x + x^2])/8 + (Sqrt[(2 + Sqrt[2])/2]*Log[1 + Sqrt[2 + Sqrt[2]]*x + x^2
])/8

Rule 1414

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[-2*d*e, 2]}, Dist[d/(2*a), I
nt[(d - q*x^(n/2))/(d - q*x^(n/2) - e*x^n), x], x] + Dist[d/(2*a), Int[(d + q*x^(n/2))/(d + q*x^(n/2) - e*x^n)
, x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - a*e^2, 0] && IGtQ[n/2, 0] && NegQ[d*e]

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

Mathematica [A]  time = 0.164174, size = 257, normalized size = 0.74 \[ \frac{1}{8} \left (-\left (\sin \left (\frac{\pi }{8}\right )+\cos \left (\frac{\pi }{8}\right )\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{8}\right )+1\right )+\left (\sin \left (\frac{\pi }{8}\right )+\cos \left (\frac{\pi }{8}\right )\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{8}\right )+1\right )+\left (\sin \left (\frac{\pi }{8}\right )-\cos \left (\frac{\pi }{8}\right )\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{8}\right )+1\right )+\left (\cos \left (\frac{\pi }{8}\right )-\sin \left (\frac{\pi }{8}\right )\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{8}\right )+1\right )+2 \left (\sin \left (\frac{\pi }{8}\right )-\cos \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{8}\right ) \left (x+\cos \left (\frac{\pi }{8}\right )\right )\right )+2 \left (\sin \left (\frac{\pi }{8}\right )+\cos \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{8}\right ) \left (x+\sin \left (\frac{\pi }{8}\right )\right )\right )+2 \left (\sin \left (\frac{\pi }{8}\right )+\cos \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (x \sec \left (\frac{\pi }{8}\right )-\tan \left (\frac{\pi }{8}\right )\right )+2 \left (\cos \left (\frac{\pi }{8}\right )-\sin \left (\frac{\pi }{8}\right )\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-x \csc \left (\frac{\pi }{8}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^4)/(1 + x^8),x]

[Out]

(2*ArcTan[Cot[Pi/8] - x*Csc[Pi/8]]*(Cos[Pi/8] - Sin[Pi/8]) + Log[1 + x^2 - 2*x*Sin[Pi/8]]*(Cos[Pi/8] - Sin[Pi/
8]) + 2*ArcTan[(x + Cos[Pi/8])*Csc[Pi/8]]*(-Cos[Pi/8] + Sin[Pi/8]) + Log[1 + x^2 + 2*x*Sin[Pi/8]]*(-Cos[Pi/8]
+ Sin[Pi/8]) + 2*ArcTan[Sec[Pi/8]*(x + Sin[Pi/8])]*(Cos[Pi/8] + Sin[Pi/8]) + 2*ArcTan[x*Sec[Pi/8] - Tan[Pi/8]]
*(Cos[Pi/8] + Sin[Pi/8]) - Log[1 + x^2 - 2*x*Cos[Pi/8]]*(Cos[Pi/8] + Sin[Pi/8]) + Log[1 + x^2 + 2*x*Cos[Pi/8]]
*(Cos[Pi/8] + Sin[Pi/8]))/8

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Maple [C]  time = 0.006, size = 29, normalized size = 0.1 \begin{align*}{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{7}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{4} - 1}{x^{8} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((x^4 - 1)/(x^8 + 1), x)

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Fricas [B]  time = 1.46893, size = 3092, normalized size = 8.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))*arctan(-(2*x - 2*sqrt(x^2 + x*sqrt(-sqrt(2) + 2) + 1) + sqrt(-sq
rt(2) + 2))/sqrt(sqrt(2) + 2)) - 1/8*(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))*arctan(-(2*x - 2*sqrt(x^2 - x*sq
rt(-sqrt(2) + 2) + 1) - sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 1/8*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))*
arctan(-(2*x - 2*sqrt(x^2 + x*sqrt(sqrt(2) + 2) + 1) + sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) + 1/8*(sqrt(sqrt
(2) + 2) - sqrt(-sqrt(2) + 2))*arctan(-(2*x - 2*sqrt(x^2 - x*sqrt(sqrt(2) + 2) + 1) - sqrt(sqrt(2) + 2))/sqrt(
-sqrt(2) + 2)) - 1/8*sqrt(2)*sqrt(sqrt(2) + 2)*arctan(-(2*sqrt(2)*x - 2*sqrt(2)*sqrt(x^2 + 1/2*sqrt(2)*x*sqrt(
sqrt(2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) + sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) +
2) + sqrt(-sqrt(2) + 2))) - 1/8*sqrt(2)*sqrt(sqrt(2) + 2)*arctan(-(2*sqrt(2)*x - 2*sqrt(2)*sqrt(x^2 - 1/2*sqr
t(2)*x*sqrt(sqrt(2) + 2) + 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) - sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))/(sq
rt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))) - 1/8*sqrt(2)*sqrt(-sqrt(2) + 2)*arctan((2*sqrt(2)*x - 2*sqrt(2)*sqrt(x
^2 + 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) + sqrt(sqrt(2) + 2) + sqrt(-sqrt(
2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) - 1/8*sqrt(2)*sqrt(-sqrt(2) + 2)*arctan((2*sqrt(2)*x - 2*sq
rt(2)*sqrt(x^2 - 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) - sqrt(sqrt(2) + 2) -
sqrt(-sqrt(2) + 2))/(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))) + 1/32*sqrt(2)*sqrt(sqrt(2) + 2)*log(x^2 + 1/2*
sqrt(2)*x*sqrt(sqrt(2) + 2) + 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) - 1/32*sqrt(2)*sqrt(-sqrt(2) + 2)*log(x^2
+ 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) + 1/32*sqrt(2)*sqrt(-sqrt(2) + 2)*lo
g(x^2 - 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) + 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) - 1/32*sqrt(2)*sqrt(sqrt(2) +
2)*log(x^2 - 1/2*sqrt(2)*x*sqrt(sqrt(2) + 2) - 1/2*sqrt(2)*x*sqrt(-sqrt(2) + 2) + 1) + 1/32*(sqrt(sqrt(2) + 2)
+ sqrt(-sqrt(2) + 2))*log(x^2 + x*sqrt(sqrt(2) + 2) + 1) - 1/32*(sqrt(sqrt(2) + 2) + sqrt(-sqrt(2) + 2))*log(
x^2 - x*sqrt(sqrt(2) + 2) + 1) - 1/32*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))*log(x^2 + x*sqrt(-sqrt(2) + 2)
+ 1) + 1/32*(sqrt(sqrt(2) + 2) - sqrt(-sqrt(2) + 2))*log(x^2 - x*sqrt(-sqrt(2) + 2) + 1)

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Sympy [A]  time = 1.12129, size = 20, normalized size = 0.06 \begin{align*} - \operatorname{RootSum}{\left (1048576 t^{8} + 1, \left ( t \mapsto t \log{\left (4096 t^{5} - 4 t + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(1048576*_t**8 + 1, Lambda(_t, _t*log(4096*_t**5 - 4*_t + x)))

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Giac [A]  time = 1.19706, size = 333, normalized size = 0.96 \begin{align*} \frac{1}{8} \, \sqrt{2 \, \sqrt{2} + 4} \arctan \left (\frac{2 \, x + \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) + \frac{1}{8} \, \sqrt{2 \, \sqrt{2} + 4} \arctan \left (\frac{2 \, x - \sqrt{-\sqrt{2} + 2}}{\sqrt{\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{-2 \, \sqrt{2} + 4} \arctan \left (\frac{2 \, x + \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) - \frac{1}{8} \, \sqrt{-2 \, \sqrt{2} + 4} \arctan \left (\frac{2 \, x - \sqrt{\sqrt{2} + 2}}{\sqrt{-\sqrt{2} + 2}}\right ) + \frac{1}{16} \, \sqrt{2 \, \sqrt{2} + 4} \log \left (x^{2} + x \sqrt{\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{2 \, \sqrt{2} + 4} \log \left (x^{2} - x \sqrt{\sqrt{2} + 2} + 1\right ) - \frac{1}{16} \, \sqrt{-2 \, \sqrt{2} + 4} \log \left (x^{2} + x \sqrt{-\sqrt{2} + 2} + 1\right ) + \frac{1}{16} \, \sqrt{-2 \, \sqrt{2} + 4} \log \left (x^{2} - x \sqrt{-\sqrt{2} + 2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2*sqrt(2) + 4)*arctan((2*x + sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 1/8*sqrt(2*sqrt(2) + 4)*arctan(
(2*x - sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) - 1/8*sqrt(-2*sqrt(2) + 4)*arctan((2*x + sqrt(sqrt(2) + 2))/sqrt
(-sqrt(2) + 2)) - 1/8*sqrt(-2*sqrt(2) + 4)*arctan((2*x - sqrt(sqrt(2) + 2))/sqrt(-sqrt(2) + 2)) + 1/16*sqrt(2*
sqrt(2) + 4)*log(x^2 + x*sqrt(sqrt(2) + 2) + 1) - 1/16*sqrt(2*sqrt(2) + 4)*log(x^2 - x*sqrt(sqrt(2) + 2) + 1)
- 1/16*sqrt(-2*sqrt(2) + 4)*log(x^2 + x*sqrt(-sqrt(2) + 2) + 1) + 1/16*sqrt(-2*sqrt(2) + 4)*log(x^2 - x*sqrt(-
sqrt(2) + 2) + 1)