3.1.10 \(\int \frac {1+x^4}{1+3 x^4+x^8} \, dx\)

Optimal. Leaf size=451 \[ -\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}} \]

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Rubi [A]  time = 0.41, antiderivative size = 451, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1420, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{2\ 2^{3/4} \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(1 + 3*x^4 + x^8),x]

[Out]

-((3 + Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((3 + Sqrt[5])^(1/4)*
ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) - ((3 - Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)
/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)*Sqrt[5]) + ((3 - Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])
/(2*2^(3/4)*Sqrt[5]) - ((3 + Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/
(4*2^(3/4)*Sqrt[5]) + ((3 + Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(
4*2^(3/4)*Sqrt[5]) - ((3 - Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4
*2^(3/4)*Sqrt[5]) + ((3 - Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*
2^(3/4)*Sqrt[5])

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 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[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, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 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, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1420

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, 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+3 x^4+x^8} \, dx &=\frac {1}{10} \left (5-\sqrt {5}\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx+\frac {1}{10} \left (5+\sqrt {5}\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx\\ &=\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {10}}+\frac {\int \frac {\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {10}}+\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {10}}+\frac {\int \frac {\sqrt {3+\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {10}}\\ &=\frac {\int \frac {1}{\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+x^2} \, dx}{4 \sqrt {5}}+\frac {\int \frac {1}{\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x+x^2} \, dx}{4 \sqrt {5}}+\frac {\int \frac {1}{\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+x^2} \, dx}{4 \sqrt {5}}+\frac {\int \frac {1}{\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x+x^2} \, dx}{4 \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}-\sqrt [4]{2 \left (3-\sqrt {5}\right )} x-x^2} \, dx}{4\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{3+\sqrt {5}} \int \frac {\sqrt [4]{2 \left (3-\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (3-\sqrt {5}\right )}+\sqrt [4]{2 \left (3-\sqrt {5}\right )} x-x^2} \, dx}{4\ 2^{3/4} \sqrt {5}}-\frac {\int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}+2 x}{-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}-\sqrt [4]{2 \left (3+\sqrt {5}\right )} x-x^2} \, dx}{4 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}-\frac {\int \frac {\sqrt [4]{2 \left (3+\sqrt {5}\right )}-2 x}{-\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )}+\sqrt [4]{2 \left (3+\sqrt {5}\right )} x-x^2} \, dx}{4 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}\\ &=-\frac {\sqrt [4]{3+\sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}+\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3-\sqrt {5}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3-\sqrt {5}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}\\ &=-\frac {\tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3-\sqrt {5}\right )}}+\frac {\tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3-\sqrt {5}\right )}}-\frac {\tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}+\frac {\tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{2 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}-\frac {\sqrt [4]{3+\sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{3+\sqrt {5}} \log \left (\sqrt {2 \left (3-\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+2 x^2\right )}{4\ 2^{3/4} \sqrt {5}}-\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}+\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{4 \sqrt {5} \sqrt [4]{2 \left (3+\sqrt {5}\right )}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 55, normalized size = 0.12 \begin {gather*} \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+3 \text {$\#$1}^4+1\&,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})+\log (x-\text {$\#$1})}{2 \text {$\#$1}^7+3 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)/(1 + 3*x^4 + x^8),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^4}{1+3 x^4+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 + x^4)/(1 + 3*x^4 + x^8),x]

[Out]

IntegrateAlgebraic[(1 + x^4)/(1 + 3*x^4 + x^8), x]

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fricas [B]  time = 1.24, size = 951, normalized size = 2.11

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^8+3*x^4+1),x, algorithm="fricas")

[Out]

1/80*sqrt(10)*(2*sqrt(5) + 6)^(3/4)*sqrt(sqrt(5) + 3)*(sqrt(5) - 3)*arctan(1/80*sqrt(10)*sqrt(20*x^2 + sqrt(10
)*(sqrt(5)*sqrt(2)*x - 5*sqrt(2)*x)*(2*sqrt(5) + 6)^(1/4) - 5*sqrt(2*sqrt(5) + 6)*(sqrt(5) - 3))*(2*sqrt(5) +
6)^(5/4)*sqrt(sqrt(5) + 3)*(sqrt(5) - 2) + 1/40*sqrt(10)*(2*sqrt(5)*x - 5*x)*(2*sqrt(5) + 6)^(5/4)*sqrt(sqrt(5
) + 3) - 1/8*(sqrt(5)*sqrt(2) - 3*sqrt(2))*sqrt(2*sqrt(5) + 6)*sqrt(sqrt(5) + 3)) + 1/80*sqrt(10)*(2*sqrt(5) +
6)^(3/4)*sqrt(sqrt(5) + 3)*(sqrt(5) - 3)*arctan(1/80*sqrt(10)*sqrt(20*x^2 - sqrt(10)*(sqrt(5)*sqrt(2)*x - 5*s
qrt(2)*x)*(2*sqrt(5) + 6)^(1/4) - 5*sqrt(2*sqrt(5) + 6)*(sqrt(5) - 3))*(2*sqrt(5) + 6)^(5/4)*sqrt(sqrt(5) + 3)
*(sqrt(5) - 2) + 1/40*sqrt(10)*(2*sqrt(5)*x - 5*x)*(2*sqrt(5) + 6)^(5/4)*sqrt(sqrt(5) + 3) + 1/8*(sqrt(5)*sqrt
(2) - 3*sqrt(2))*sqrt(2*sqrt(5) + 6)*sqrt(sqrt(5) + 3)) - 1/80*sqrt(10)*(sqrt(5) + 3)*sqrt(-sqrt(5) + 3)*(-2*s
qrt(5) + 6)^(3/4)*arctan(1/80*sqrt(10)*sqrt(20*x^2 + sqrt(10)*(sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(-2*sqrt(5) +
6)^(1/4) + 5*(sqrt(5) + 3)*sqrt(-2*sqrt(5) + 6))*(sqrt(5) + 2)*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(5/4) - 1/4
0*(sqrt(10)*(2*sqrt(5)*x + 5*x)*(-2*sqrt(5) + 6)^(5/4) + 5*(sqrt(5)*sqrt(2) + 3*sqrt(2))*sqrt(-2*sqrt(5) + 6))
*sqrt(-sqrt(5) + 3)) - 1/80*sqrt(10)*(sqrt(5) + 3)*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(3/4)*arctan(1/80*sqrt(
10)*sqrt(20*x^2 - sqrt(10)*(sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(-2*sqrt(5) + 6)^(1/4) + 5*(sqrt(5) + 3)*sqrt(-2*
sqrt(5) + 6))*(sqrt(5) + 2)*sqrt(-sqrt(5) + 3)*(-2*sqrt(5) + 6)^(5/4) - 1/40*(sqrt(10)*(2*sqrt(5)*x + 5*x)*(-2
*sqrt(5) + 6)^(5/4) - 5*(sqrt(5)*sqrt(2) + 3*sqrt(2))*sqrt(-2*sqrt(5) + 6))*sqrt(-sqrt(5) + 3)) - 1/80*sqrt(10
)*sqrt(2)*(2*sqrt(5) + 6)^(1/4)*log(20*x^2 + sqrt(10)*(sqrt(5)*sqrt(2)*x - 5*sqrt(2)*x)*(2*sqrt(5) + 6)^(1/4)
- 5*sqrt(2*sqrt(5) + 6)*(sqrt(5) - 3)) + 1/80*sqrt(10)*sqrt(2)*(2*sqrt(5) + 6)^(1/4)*log(20*x^2 - sqrt(10)*(sq
rt(5)*sqrt(2)*x - 5*sqrt(2)*x)*(2*sqrt(5) + 6)^(1/4) - 5*sqrt(2*sqrt(5) + 6)*(sqrt(5) - 3)) + 1/80*sqrt(10)*sq
rt(2)*(-2*sqrt(5) + 6)^(1/4)*log(20*x^2 + sqrt(10)*(sqrt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(-2*sqrt(5) + 6)^(1/4) +
5*(sqrt(5) + 3)*sqrt(-2*sqrt(5) + 6)) - 1/80*sqrt(10)*sqrt(2)*(-2*sqrt(5) + 6)^(1/4)*log(20*x^2 - sqrt(10)*(sq
rt(5)*sqrt(2)*x + 5*sqrt(2)*x)*(-2*sqrt(5) + 6)^(1/4) + 5*(sqrt(5) + 3)*sqrt(-2*sqrt(5) + 6))

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giac [A]  time = 0.93, size = 239, normalized size = 0.53 \begin {gather*} \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {5 \, \sqrt {5} + 5} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {5 \, \sqrt {5} + 5} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {5 \, \sqrt {5} - 5} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {5 \, \sqrt {5} - 5} + \frac {1}{40} \, \sqrt {5 \, \sqrt {5} - 5} \log \left (16900 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 16900 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {5 \, \sqrt {5} - 5} \log \left (16900 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 16900 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {5 \, \sqrt {5} + 5} \log \left (2500 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 2500 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {5 \, \sqrt {5} + 5} \log \left (2500 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 2500 \, x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^8+3*x^4+1),x, algorithm="giac")

[Out]

1/80*(pi + 4*arctan(x*sqrt(sqrt(5) + 1) + 1))*sqrt(5*sqrt(5) + 5) - 1/80*(pi + 4*arctan(-x*sqrt(sqrt(5) + 1) +
1))*sqrt(5*sqrt(5) + 5) + 1/80*(pi + 4*arctan(x*sqrt(sqrt(5) - 1) - 1))*sqrt(5*sqrt(5) - 5) - 1/80*(pi + 4*ar
ctan(-x*sqrt(sqrt(5) - 1) - 1))*sqrt(5*sqrt(5) - 5) + 1/40*sqrt(5*sqrt(5) - 5)*log(16900*(x + sqrt(sqrt(5) + 1
))^2 + 16900*x^2) - 1/40*sqrt(5*sqrt(5) - 5)*log(16900*(x - sqrt(sqrt(5) + 1))^2 + 16900*x^2) + 1/40*sqrt(5*sq
rt(5) + 5)*log(2500*(x + sqrt(sqrt(5) - 1))^2 + 2500*x^2) - 1/40*sqrt(5*sqrt(5) + 5)*log(2500*(x - sqrt(sqrt(5
) - 1))^2 + 2500*x^2)

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maple [C]  time = 0.01, size = 42, normalized size = 0.09 \begin {gather*} \frac {\left (\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{4}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{7}+12 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)/(x^8+3*x^4+1),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/(x^8+3*x^4+1),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.18, size = 459, normalized size = 1.02 \begin {gather*} \frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {7\,2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (2\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}+\frac {3\,2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}}{2\,\left (2\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,7{}\mathrm {i}}{2\,\left (2\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}+\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (-\sqrt {5}-3\right )}^{1/4}\,3{}\mathrm {i}}{2\,\left (2\,\sqrt {2}\,\sqrt {-\sqrt {5}-3}+\sqrt {2}\,\sqrt {5}\,\sqrt {-\sqrt {5}-3}\right )}\right )\,{\left (-\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{20}-\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {7\,2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (2\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}-\frac {3\,2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}}{2\,\left (2\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}}{20}+\frac {2^{3/4}\,\sqrt {5}\,\mathrm {atan}\left (\frac {2^{3/4}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,7{}\mathrm {i}}{2\,\left (2\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}-\frac {2^{3/4}\,\sqrt {5}\,x\,{\left (\sqrt {5}-3\right )}^{1/4}\,3{}\mathrm {i}}{2\,\left (2\,\sqrt {2}\,\sqrt {\sqrt {5}-3}-\sqrt {2}\,\sqrt {5}\,\sqrt {\sqrt {5}-3}\right )}\right )\,{\left (\sqrt {5}-3\right )}^{1/4}\,1{}\mathrm {i}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 + 1)/(3*x^4 + x^8 + 1),x)

[Out]

(2^(3/4)*5^(1/2)*atan((7*2^(3/4)*x*(- 5^(1/2) - 3)^(1/4))/(2*(2*2^(1/2)*(- 5^(1/2) - 3)^(1/2) + 2^(1/2)*5^(1/2
)*(- 5^(1/2) - 3)^(1/2))) + (3*2^(3/4)*5^(1/2)*x*(- 5^(1/2) - 3)^(1/4))/(2*(2*2^(1/2)*(- 5^(1/2) - 3)^(1/2) +
2^(1/2)*5^(1/2)*(- 5^(1/2) - 3)^(1/2))))*(- 5^(1/2) - 3)^(1/4))/20 - (2^(3/4)*5^(1/2)*atan((2^(3/4)*x*(- 5^(1/
2) - 3)^(1/4)*7i)/(2*(2*2^(1/2)*(- 5^(1/2) - 3)^(1/2) + 2^(1/2)*5^(1/2)*(- 5^(1/2) - 3)^(1/2))) + (2^(3/4)*5^(
1/2)*x*(- 5^(1/2) - 3)^(1/4)*3i)/(2*(2*2^(1/2)*(- 5^(1/2) - 3)^(1/2) + 2^(1/2)*5^(1/2)*(- 5^(1/2) - 3)^(1/2)))
)*(- 5^(1/2) - 3)^(1/4)*1i)/20 - (2^(3/4)*5^(1/2)*atan((7*2^(3/4)*x*(5^(1/2) - 3)^(1/4))/(2*(2*2^(1/2)*(5^(1/2
) - 3)^(1/2) - 2^(1/2)*5^(1/2)*(5^(1/2) - 3)^(1/2))) - (3*2^(3/4)*5^(1/2)*x*(5^(1/2) - 3)^(1/4))/(2*(2*2^(1/2)
*(5^(1/2) - 3)^(1/2) - 2^(1/2)*5^(1/2)*(5^(1/2) - 3)^(1/2))))*(5^(1/2) - 3)^(1/4))/20 + (2^(3/4)*5^(1/2)*atan(
(2^(3/4)*x*(5^(1/2) - 3)^(1/4)*7i)/(2*(2*2^(1/2)*(5^(1/2) - 3)^(1/2) - 2^(1/2)*5^(1/2)*(5^(1/2) - 3)^(1/2))) -
(2^(3/4)*5^(1/2)*x*(5^(1/2) - 3)^(1/4)*3i)/(2*(2*2^(1/2)*(5^(1/2) - 3)^(1/2) - 2^(1/2)*5^(1/2)*(5^(1/2) - 3)^
(1/2))))*(5^(1/2) - 3)^(1/4)*1i)/20

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sympy [A]  time = 1.48, size = 24, normalized size = 0.05 \begin {gather*} \operatorname {RootSum} {\left (40960000 t^{8} + 19200 t^{4} + 1, \left (t \mapsto t \log {\left (25600 t^{5} + 16 t + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)/(x**8+3*x**4+1),x)

[Out]

RootSum(40960000*_t**8 + 19200*_t**4 + 1, Lambda(_t, _t*log(25600*_t**5 + 16*_t + x)))

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