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

Optimal. Leaf size=411 \[ -\frac {\log \left (-\sqrt {2-\sqrt {2-b}} x+x^2+1\right )}{8 \sqrt {2-\sqrt {2-b}}}+\frac {\log \left (\sqrt {2-\sqrt {2-b}} x+x^2+1\right )}{8 \sqrt {2-\sqrt {2-b}}}-\frac {\log \left (-\sqrt {\sqrt {2-b}+2} x+x^2+1\right )}{8 \sqrt {\sqrt {2-b}+2}}+\frac {\log \left (\sqrt {\sqrt {2-b}+2} x+x^2+1\right )}{8 \sqrt {\sqrt {2-b}+2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2-b}}-2 x}{\sqrt {\sqrt {2-b}+2}}\right )}{4 \sqrt {\sqrt {2-b}+2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2-b}+2}-2 x}{\sqrt {2-\sqrt {2-b}}}\right )}{4 \sqrt {2-\sqrt {2-b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2-b}}+2 x}{\sqrt {\sqrt {2-b}+2}}\right )}{4 \sqrt {\sqrt {2-b}+2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2-b}+2}+2 x}{\sqrt {2-\sqrt {2-b}}}\right )}{4 \sqrt {2-\sqrt {2-b}}} \]

[Out]

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

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Rubi [A]  time = 0.29, antiderivative size = 411, normalized size of antiderivative = 1.00, 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} \[ -\frac {\log \left (-\sqrt {2-\sqrt {2-b}} x+x^2+1\right )}{8 \sqrt {2-\sqrt {2-b}}}+\frac {\log \left (\sqrt {2-\sqrt {2-b}} x+x^2+1\right )}{8 \sqrt {2-\sqrt {2-b}}}-\frac {\log \left (-\sqrt {\sqrt {2-b}+2} x+x^2+1\right )}{8 \sqrt {\sqrt {2-b}+2}}+\frac {\log \left (\sqrt {\sqrt {2-b}+2} x+x^2+1\right )}{8 \sqrt {\sqrt {2-b}+2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2-b}}-2 x}{\sqrt {\sqrt {2-b}+2}}\right )}{4 \sqrt {\sqrt {2-b}+2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2-b}+2}-2 x}{\sqrt {2-\sqrt {2-b}}}\right )}{4 \sqrt {2-\sqrt {2-b}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2-b}}+2 x}{\sqrt {\sqrt {2-b}+2}}\right )}{4 \sqrt {\sqrt {2-b}+2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {2-b}+2}+2 x}{\sqrt {2-\sqrt {2-b}}}\right )}{4 \sqrt {2-\sqrt {2-b}}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(1 + b*x^4 + x^8),x]

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.03, size = 55, normalized size = 0.13 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+\text {$\#$1}^4 b+1\& ,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})+\log (x-\text {$\#$1})}{2 \text {$\#$1}^7+\text {$\#$1}^3 b}\& \right ] \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.18, size = 1443, normalized size = 3.51 \[ \text {result too large to display} \]

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*sqr
t(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)*sqr
t((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)*sqr
t(-((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 + 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 + 1
2*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)*sqr
t(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*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)))*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 = 0.06, size = 42, normalized size = 0.10 \[ \frac {\left (\RootOf \left (\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+1\right )^{4}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+1\right )^{7}+4 \RootOf \left (\textit {\_Z}^{8}+b \,\textit {\_Z}^{4}+1\right )^{3} b} \]

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 = 3.68, size = 5341, normalized size = 13.00 \[ \text {result too large to display} \]

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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(((
-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3
)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(3/4) - 256*b + 64*b^3 - 16*b^4 + 256) + x*(32*b - 48*b^2 + 24*b^3
- 4*b^4))*(-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*1
i - ((-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*
b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(262144*b + 19660
8*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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(5
12*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(3/4) - 256*b + 64*b^3 - 16*b^4 + 256) - x*(32*b - 48*b^2 + 24*b^3 - 4
*b^4))*(-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*1i)/(
((-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b +
((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(
32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(3/4) - 256*b + 64*b^3 - 16*b^4 + 256) + x*(32*b - 48*b^2 + 24*b^3 - 4*b^4
))*(-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4) + ((-(4*b
+ ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b + ((b - 2
)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(262144*b + 196608*b^2 - 196
608*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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b +
24*b^2 + 8*b^3 + b^4 + 16)))^(3/4) - 256*b + 64*b^3 - 16*b^4 + 256) - x*(32*b - 48*b^2 + 24*b^3 - 4*b^4))*(-(4
*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)))*(-(4*b + ((b -
2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*2i - 2*atan((((-(4*b + ((b
- 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*b + ((b -
2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(262144*b + 196608*b^2 - 1
96608*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 - 20
480*b^4 + 10240*b^5 + 2048*b^6 - 1024*b^7 - 65536))*(-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32
*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(3/4)*1i - 64*b^3 + 16*b^4 - 256)*1i - x*(32*b - 48*b^2 + 24*b^3 - 4*b^4))*(
-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4) - ((-(4*b + (
(b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*b + ((b
- 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(
32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(3/4)*1i - 64*b^3 + 16*b^4 - 256)*1i + x*(32*b - 48*b^2 + 24*b^3 - 4*b^4))
*(-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4))/(((-(4*b +
((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*b + (
(b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512
*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(3/4)*1i - 64*b^3 + 16*b^4 - 256)*1i - x*(32*b - 48*b^2 + 24*b^3 - 4*b^4
))*(-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*1i + ((-(
4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*
b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(262144*b + 19660
8*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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)
/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(3/4)*1i - 64*b^3 + 16*b^4 - 256)*1i + x*(32*b - 48*b^2 + 24*b^3 -
4*b^4))*(-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*1i))
*(-(4*b + ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4) - atan((((-
(4*b - ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b - ((b
- 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*
b + 24*b^2 + 8*b^3 + b^4 + 16)))^(3/4) - 256*b + 64*b^3 - 16*b^4 + 256) + x*(32*b - 48*b^2 + 24*b^3 - 4*b^4))*
(-(4*b - ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*1i - ((-(4*b
- ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b - ((b - 2
)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(262144*b + 196608*b^2 - 196
608*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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b +
24*b^2 + 8*b^3 + b^4 + 16)))^(3/4) - 256*b + 64*b^3 - 16*b^4 + 256) - x*(32*b - 48*b^2 + 24*b^3 - 4*b^4))*(-(4
*b - ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*1i)/(((-(4*b - (
(b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b - ((b - 2)*(b
+ 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b
^2 + 8*b^3 + b^4 + 16)))^(3/4) - 256*b + 64*b^3 - 16*b^4 + 256) + x*(32*b - 48*b^2 + 24*b^3 - 4*b^4))*(-(4*b -
((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4) + ((-(4*b - ((b - 2)
*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(((-(4*b - ((b - 2)*(b + 2)^5
)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(262144*b + 196608*b^2 - 196608*b^3 - 4
9152*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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*
b^3 + b^4 + 16)))^(3/4) - 256*b + 64*b^3 - 16*b^4 + 256) - x*(32*b - 48*b^2 + 24*b^3 - 4*b^4))*(-(4*b - ((b -
2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)))*(-(4*b - ((b - 2)*(b + 2)
^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*2i - 2*atan((((-(4*b - ((b - 2)*(b +
2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*b - ((b - 2)*(b + 2)
^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 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 + 1
0240*b^5 + 2048*b^6 - 1024*b^7 - 65536))*(-(4*b - ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2
+ 8*b^3 + b^4 + 16)))^(3/4)*1i - 64*b^3 + 16*b^4 - 256)*1i - x*(32*b - 48*b^2 + 24*b^3 - 4*b^4))*(-(4*b - ((b
- 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4) - ((-(4*b - ((b - 2)*(b
+ 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*b - ((b - 2)*(b +
2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b
^2 + 8*b^3 + b^4 + 16)))^(3/4)*1i - 64*b^3 + 16*b^4 - 256)*1i + x*(32*b - 48*b^2 + 24*b^3 - 4*b^4))*(-(4*b - (
(b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4))/(((-(4*b - ((b - 2)*(
b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*b - ((b - 2)*(b
+ 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 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)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24
*b^2 + 8*b^3 + b^4 + 16)))^(3/4)*1i - 64*b^3 + 16*b^4 - 256)*1i - x*(32*b - 48*b^2 + 24*b^3 - 4*b^4))*(-(4*b -
((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*1i + ((-(4*b - ((b -
2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(256*b + ((-(4*b - ((b - 2
)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*(262144*b + 196608*b^2 - 196
608*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 - 2048
0*b^4 + 10240*b^5 + 2048*b^6 - 1024*b^7 - 65536))*(-(4*b - ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b
+ 24*b^2 + 8*b^3 + b^4 + 16)))^(3/4)*1i - 64*b^3 + 16*b^4 - 256)*1i + x*(32*b - 48*b^2 + 24*b^3 - 4*b^4))*(-(
4*b - ((b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)*1i))*(-(4*b - (
(b - 2)*(b + 2)^5)^(1/2) + 4*b^2 + b^3)/(512*(32*b + 24*b^2 + 8*b^3 + b^4 + 16)))^(1/4)

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sympy [A]  time = 3.67, size = 75, normalized size = 0.18 \[ \operatorname {RootSum} {\left (t^{8} \left (65536 b^{4} + 524288 b^{3} + 1572864 b^{2} + 2097152 b + 1048576\right ) + t^{4} \left (256 b^{3} + 1024 b^{2} + 1024 b\right ) + 1, \left (t \mapsto t \log {\left (1024 t^{5} b^{2} + 4096 t^{5} b + 4096 t^{5} + 4 t b + 4 t + x \right )} \right )\right )} \]

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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