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

Optimal. Leaf size=511 \[ \frac {\sqrt {2-\sqrt {2-b}} \log \left (-\sqrt {2-\sqrt {2-b}} x+x^2+1\right )}{8 \sqrt {2-b}}-\frac {\sqrt {2-\sqrt {2-b}} \log \left (\sqrt {2-\sqrt {2-b}} x+x^2+1\right )}{8 \sqrt {2-b}}-\frac {\sqrt {\sqrt {2-b}+2} \log \left (-\sqrt {\sqrt {2-b}+2} x+x^2+1\right )}{8 \sqrt {2-b}}+\frac {\sqrt {\sqrt {2-b}+2} \log \left (\sqrt {\sqrt {2-b}+2} x+x^2+1\right )}{8 \sqrt {2-b}}-\frac {\sqrt {b+2} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2-b}}-2 x}{\sqrt {\sqrt {2-b}+2}}\right )}{4 \sqrt {2-\sqrt {2-b}} \sqrt {2-b}}+\frac {\sqrt {b+2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {2-b}+2}-2 x}{\sqrt {2-\sqrt {2-b}}}\right )}{4 \sqrt {\sqrt {2-b}+2} \sqrt {2-b}}+\frac {\sqrt {b+2} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2-b}}+2 x}{\sqrt {\sqrt {2-b}+2}}\right )}{4 \sqrt {2-\sqrt {2-b}} \sqrt {2-b}}-\frac {\sqrt {b+2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {2-b}+2}+2 x}{\sqrt {2-\sqrt {2-b}}}\right )}{4 \sqrt {\sqrt {2-b}+2} \sqrt {2-b}} \]

[Out]

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

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Rubi [A]  time = 0.36, antiderivative size = 511, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1421, 1169, 634, 618, 204, 628} \[ \frac {\sqrt {2-\sqrt {2-b}} \log \left (-\sqrt {2-\sqrt {2-b}} x+x^2+1\right )}{8 \sqrt {2-b}}-\frac {\sqrt {2-\sqrt {2-b}} \log \left (\sqrt {2-\sqrt {2-b}} x+x^2+1\right )}{8 \sqrt {2-b}}-\frac {\sqrt {\sqrt {2-b}+2} \log \left (-\sqrt {\sqrt {2-b}+2} x+x^2+1\right )}{8 \sqrt {2-b}}+\frac {\sqrt {\sqrt {2-b}+2} \log \left (\sqrt {\sqrt {2-b}+2} x+x^2+1\right )}{8 \sqrt {2-b}}-\frac {\sqrt {b+2} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2-b}}-2 x}{\sqrt {\sqrt {2-b}+2}}\right )}{4 \sqrt {2-\sqrt {2-b}} \sqrt {2-b}}+\frac {\sqrt {b+2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {2-b}+2}-2 x}{\sqrt {2-\sqrt {2-b}}}\right )}{4 \sqrt {\sqrt {2-b}+2} \sqrt {2-b}}+\frac {\sqrt {b+2} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2-b}}+2 x}{\sqrt {\sqrt {2-b}+2}}\right )}{4 \sqrt {2-\sqrt {2-b}} \sqrt {2-b}}-\frac {\sqrt {b+2} \tan ^{-1}\left (\frac {\sqrt {\sqrt {2-b}+2}+2 x}{\sqrt {2-\sqrt {2-b}}}\right )}{4 \sqrt {\sqrt {2-b}+2} \sqrt {2-b}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)/(1 + b*x^4 + x^8),x]

[Out]

-(Sqrt[2 + b]*ArcTan[(Sqrt[2 - Sqrt[2 - b]] - 2*x)/Sqrt[2 + Sqrt[2 - b]]])/(4*Sqrt[2 - Sqrt[2 - b]]*Sqrt[2 - b
]) + (Sqrt[2 + b]*ArcTan[(Sqrt[2 + Sqrt[2 - b]] - 2*x)/Sqrt[2 - Sqrt[2 - b]]])/(4*Sqrt[2 + Sqrt[2 - b]]*Sqrt[2
- b]) + (Sqrt[2 + b]*ArcTan[(Sqrt[2 - Sqrt[2 - b]] + 2*x)/Sqrt[2 + Sqrt[2 - b]]])/(4*Sqrt[2 - Sqrt[2 - b]]*Sq
rt[2 - b]) - (Sqrt[2 + b]*ArcTan[(Sqrt[2 + Sqrt[2 - b]] + 2*x)/Sqrt[2 - Sqrt[2 - b]]])/(4*Sqrt[2 + Sqrt[2 - b]
]*Sqrt[2 - b]) + (Sqrt[2 - Sqrt[2 - b]]*Log[1 - Sqrt[2 - Sqrt[2 - b]]*x + x^2])/(8*Sqrt[2 - b]) - (Sqrt[2 - Sq
rt[2 - b]]*Log[1 + Sqrt[2 - Sqrt[2 - b]]*x + x^2])/(8*Sqrt[2 - b]) - (Sqrt[2 + Sqrt[2 - b]]*Log[1 - Sqrt[2 + S
qrt[2 - b]]*x + x^2])/(8*Sqrt[2 - b]) + (Sqrt[2 + Sqrt[2 - b]]*Log[1 + Sqrt[2 + Sqrt[2 - b]]*x + x^2])/(8*Sqrt
[2 - b])

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

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Mathematica [C]  time = 0.03, size = 57, normalized size = 0.11 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+\text {$\#$1}^4 b+1\& ,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})-\log (x-\text {$\#$1})}{2 \text {$\#$1}^7+\text {$\#$1}^3 b}\& \right ] \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.85, size = 1443, normalized size = 2.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(sqrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b + 4)))*arctan(1/2
*sqrt(1/2)*(b^2 + (b^3 - 6*b^2 + 12*b - 8)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - 4*b + 4)*sqrt(x^2 + 1/2*sq
rt(1/2)*(b^2 + (b^3 - 6*b^2 + 12*b - 8)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - 2*b)*sqrt(((b^2 - 4*b + 4)*sq
rt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b + 4)))*sqrt(sqrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)/
(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b + 4)))*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8))
- b)/(b^2 - 4*b + 4)) - 1/2*sqrt(1/2)*((b^3 - 6*b^2 + 12*b - 8)*x*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + (b^
2 - 4*b + 4)*x)*sqrt(sqrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b +
4)))*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b + 4))) + sqrt(sqrt(1/2)*sqrt
(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + b)/(b^2 - 4*b + 4)))*arctan(-1/2*(sqrt(1/2)*(b^2 -
(b^3 - 6*b^2 + 12*b - 8)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - 4*b + 4)*sqrt(x^2 + 1/2*sqrt(1/2)*(b^2 - (b
^3 - 6*b^2 + 12*b - 8)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - 2*b)*sqrt(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3
- 6*b^2 + 12*b - 8)) + b)/(b^2 - 4*b + 4)))*sqrt(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + b)
/(b^2 - 4*b + 4)) + sqrt(1/2)*((b^3 - 6*b^2 + 12*b - 8)*x*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - (b^2 - 4*b
+ 4)*x)*sqrt(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + b)/(b^2 - 4*b + 4)))*sqrt(sqrt(1/2)*sq
rt(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + b)/(b^2 - 4*b + 4)))) + 1/4*sqrt(sqrt(1/2)*sqrt(
-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + b)/(b^2 - 4*b + 4)))*log(1/2*((b^2 - 4*b + 4)*sqrt(
(b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b + 2)*sqrt(sqrt(1/2)*sqrt(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 1
2*b - 8)) + b)/(b^2 - 4*b + 4))) + x) - 1/4*sqrt(sqrt(1/2)*sqrt(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 +
12*b - 8)) + b)/(b^2 - 4*b + 4)))*log(-1/2*((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b + 2)*sq
rt(sqrt(1/2)*sqrt(-((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + b)/(b^2 - 4*b + 4))) + x) - 1/4*s
qrt(sqrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b + 4)))*log(1/2*((b^
2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) + b - 2)*sqrt(sqrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)
/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b + 4))) + x) + 1/4*sqrt(sqrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)
/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b + 4)))*log(-1/2*((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b -
8)) + b - 2)*sqrt(sqrt(1/2)*sqrt(((b^2 - 4*b + 4)*sqrt((b + 2)/(b^3 - 6*b^2 + 12*b - 8)) - b)/(b^2 - 4*b + 4)
)) + x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.75Unable to convert to real 1/4 Error: Bad Argument Value

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maple [C]  time = 0.00, size = 44, normalized size = 0.09 \[ \frac {\left (-\RootOf \left (\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+1\right )^{4}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+1\right )^{7}+4 \RootOf \left (\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+1\right )^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x^{4} - 1}{x^{8} + b x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.74, size = 5341, normalized size = 10.45 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4 - 1)/(b*x^4 + x^8 + 1),x)

[Out]

- atan((((-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(25
6*b + ((-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(2621
44*b - 196608*b^2 - 196608*b^3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144) + x*(32768*b - 65536*b^
2 - 32768*b^3 + 20480*b^4 + 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b
^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(3/4) - 64*b^3 - 16*b^4 + 256) - x*(32*b + 48*b^2 + 24*b^3
+ 4*b^4))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*1
i - ((-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(256*b
+ ((-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b
- 196608*b^2 - 196608*b^3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144) - x*(32768*b - 65536*b^2 -
32768*b^3 + 20480*b^4 + 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 +
b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(3/4) - 64*b^3 - 16*b^4 + 256) + x*(32*b + 48*b^2 + 24*b^3 + 4
*b^4))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*1i)/(
((-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((
-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 1
96608*b^2 - 196608*b^3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144) + x*(32768*b - 65536*b^2 - 3276
8*b^3 + 20480*b^4 + 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3
)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(3/4) - 64*b^3 - 16*b^4 + 256) - x*(32*b + 48*b^2 + 24*b^3 + 4*b^4
))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4) + ((-(4*b
+ ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*b +
((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b
^2 - 196608*b^3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144) - x*(32768*b - 65536*b^2 - 32768*b^3 +
20480*b^4 + 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*
(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(3/4) - 64*b^3 - 16*b^4 + 256) + x*(32*b + 48*b^2 + 24*b^3 + 4*b^4))*(-(4
*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)))*(-(4*b + ((b -
2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*2i - 2*atan((((-(4*b + ((b
- 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b + ((b - 2)^5*(b
+ 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b^2 - 196608*b^
3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144)*1i + x*(32768*b - 65536*b^2 - 32768*b^3 + 20480*b^4
+ 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 3
2*b - 8*b^3 + b^4 + 16)))^(3/4)*1i - 256*b + 64*b^3 + 16*b^4 - 256)*1i + x*(32*b + 48*b^2 + 24*b^3 + 4*b^4))*(
-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4) - ((-(4*b + (
(b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b + ((b - 2)^5*
(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b^2 - 196608*
b^3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144)*1i - x*(32768*b - 65536*b^2 - 32768*b^3 + 20480*b^
4 + 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 -
32*b - 8*b^3 + b^4 + 16)))^(3/4)*1i - 256*b + 64*b^3 + 16*b^4 - 256)*1i - x*(32*b + 48*b^2 + 24*b^3 + 4*b^4))
*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4))/(((-(4*b +
((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b + ((b - 2)^
5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b^2 - 19660
8*b^3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144)*1i + x*(32768*b - 65536*b^2 - 32768*b^3 + 20480*
b^4 + 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2
- 32*b - 8*b^3 + b^4 + 16)))^(3/4)*1i - 256*b + 64*b^3 + 16*b^4 - 256)*1i + x*(32*b + 48*b^2 + 24*b^3 + 4*b^4
))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*1i + ((-(
4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b + ((b
- 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b^2 -
196608*b^3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144)*1i - x*(32768*b - 65536*b^2 - 32768*b^3 + 2
0480*b^4 + 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(2
4*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(3/4)*1i - 256*b + 64*b^3 + 16*b^4 - 256)*1i - x*(32*b + 48*b^2 + 24*b^3 +
4*b^4))*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*1i))
*(-(4*b + ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4) - atan((((-
(4*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4
*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 1966
08*b^2 - 196608*b^3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144) + x*(32768*b - 65536*b^2 - 32768*b
^3 + 20480*b^4 + 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(
512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(3/4) - 64*b^3 - 16*b^4 + 256) - x*(32*b + 48*b^2 + 24*b^3 + 4*b^4))*
(-(4*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*1i - ((-(4*b
- ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*b -
((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b
^2 - 196608*b^3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144) - x*(32768*b - 65536*b^2 - 32768*b^3 +
20480*b^4 + 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*
(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(3/4) - 64*b^3 - 16*b^4 + 256) + x*(32*b + 48*b^2 + 24*b^3 + 4*b^4))*(-(4
*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*1i)/(((-(4*b - (
(b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*b - ((b
- 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b^2 -
196608*b^3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144) + x*(32768*b - 65536*b^2 - 32768*b^3 + 204
80*b^4 + 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*
b^2 - 32*b - 8*b^3 + b^4 + 16)))^(3/4) - 64*b^3 - 16*b^4 + 256) - x*(32*b + 48*b^2 + 24*b^3 + 4*b^4))*(-(4*b -
((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4) + ((-(4*b - ((b - 2)
^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*b - ((b - 2)^5
*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b^2 - 196608
*b^3 + 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144) - x*(32768*b - 65536*b^2 - 32768*b^3 + 20480*b^4
+ 10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 3
2*b - 8*b^3 + b^4 + 16)))^(3/4) - 64*b^3 - 16*b^4 + 256) + x*(32*b + 48*b^2 + 24*b^3 + 4*b^4))*(-(4*b - ((b -
2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)))*(-(4*b - ((b - 2)^5*(b +
2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*2i - 2*atan((((-(4*b - ((b - 2)^5*(b
+ 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b - ((b - 2)^5*(b + 2))^(1/2
) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b^2 - 196608*b^3 + 49152*b
^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144)*1i + x*(32768*b - 65536*b^2 - 32768*b^3 + 20480*b^4 + 10240*b^5
- 2048*b^6 - 1024*b^7 + 65536))*(-(4*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3
+ b^4 + 16)))^(3/4)*1i - 256*b + 64*b^3 + 16*b^4 - 256)*1i + x*(32*b + 48*b^2 + 24*b^3 + 4*b^4))*(-(4*b - ((b
- 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4) - ((-(4*b - ((b - 2)^5*(
b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b - ((b - 2)^5*(b + 2))^(1
/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b^2 - 196608*b^3 + 49152
*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144)*1i - x*(32768*b - 65536*b^2 - 32768*b^3 + 20480*b^4 + 10240*b
^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b
^3 + b^4 + 16)))^(3/4)*1i - 256*b + 64*b^3 + 16*b^4 - 256)*1i - x*(32*b + 48*b^2 + 24*b^3 + 4*b^4))*(-(4*b - (
(b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4))/(((-(4*b - ((b - 2)^5
*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b - ((b - 2)^5*(b + 2))^
(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b^2 - 196608*b^3 + 491
52*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144)*1i + x*(32768*b - 65536*b^2 - 32768*b^3 + 20480*b^4 + 10240
*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8
*b^3 + b^4 + 16)))^(3/4)*1i - 256*b + 64*b^3 + 16*b^4 - 256)*1i + x*(32*b + 48*b^2 + 24*b^3 + 4*b^4))*(-(4*b -
((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*1i + ((-(4*b - ((b -
2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b - ((b - 2)^5*(b +
2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*(262144*b - 196608*b^2 - 196608*b^3
+ 49152*b^4 + 49152*b^5 - 4096*b^6 - 4096*b^7 + 262144)*1i - x*(32768*b - 65536*b^2 - 32768*b^3 + 20480*b^4 +
10240*b^5 - 2048*b^6 - 1024*b^7 + 65536))*(-(4*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*
b - 8*b^3 + b^4 + 16)))^(3/4)*1i - 256*b + 64*b^3 + 16*b^4 - 256)*1i - x*(32*b + 48*b^2 + 24*b^3 + 4*b^4))*(-(
4*b - ((b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)*1i))*(-(4*b - (
(b - 2)^5*(b + 2))^(1/2) - 4*b^2 + b^3)/(512*(24*b^2 - 32*b - 8*b^3 + b^4 + 16)))^(1/4)

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sympy [A]  time = 3.63, size = 76, normalized size = 0.15 \[ - \operatorname {RootSum} {\left (t^{8} \left (65536 b^{4} - 524288 b^{3} + 1572864 b^{2} - 2097152 b + 1048576\right ) + t^{4} \left (256 b^{3} - 1024 b^{2} + 1024 b\right ) + 1, \left (t \mapsto t \log {\left (1024 t^{5} b^{2} - 4096 t^{5} b + 4096 t^{5} + 4 t b - 4 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(_t**8*(65536*b**4 - 524288*b**3 + 1572864*b**2 - 2097152*b + 1048576) + _t**4*(256*b**3 - 1024*b**2 +
1024*b) + 1, Lambda(_t, _t*log(1024*_t**5*b**2 - 4096*_t**5*b + 4096*_t**5 + 4*_t*b - 4*_t + x)))

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