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

Optimal. Leaf size=414 \[ -\frac {\sqrt [4]{9+4 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\sqrt [4]{9+4 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {10}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2 \sqrt {10}}+\frac {\tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}} \]

[Out]

-1/20*arctan(-1+x*(5^(1/2)-1)^(1/2))*(-20+10*5^(1/2))^(1/2)-1/20*arctan(1+x*(5^(1/2)-1)^(1/2))*(-20+10*5^(1/2)
)^(1/2)+1/40*ln(1+2*x^2+5^(1/2)-2*x*(5^(1/2)+1)^(1/2))*(-20+10*5^(1/2))^(1/2)-1/40*ln(1+2*x^2+5^(1/2)+2*x*(5^(
1/2)+1)^(1/2))*(-20+10*5^(1/2))^(1/2)+1/20*arctan(-1+x*(5^(1/2)+1)^(1/2))*(20+10*5^(1/2))^(1/2)+1/20*arctan(1+
x*(5^(1/2)+1)^(1/2))*(20+10*5^(1/2))^(1/2)-1/40*ln(-1+2*x^2+5^(1/2)-2*x*(5^(1/2)-1)^(1/2))*(20+10*5^(1/2))^(1/
2)+1/40*ln(-1+2*x^2+5^(1/2)+2*x*(5^(1/2)-1)^(1/2))*(20+10*5^(1/2))^(1/2)

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Rubi [A]  time = 0.26, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {1347, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\sqrt [4]{9+4 \sqrt {5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt {5}\right )} x+\sqrt {2 \left (3-\sqrt {5}\right )}\right )}{4 \sqrt {10}}+\frac {\log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )}\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\sqrt [4]{9+4 \sqrt {5}} \tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{2 \sqrt {10}}+\frac {\sqrt [4]{9+4 \sqrt {5}} \tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}+1\right )}{2 \sqrt {10}}+\frac {\tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}+1\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 3*x^4 + x^8)^(-1),x]

[Out]

-((9 + 4*Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*Sqrt[10]) + ((9 + 4*Sqrt[5])^(1/4)*Arc
Tan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*Sqrt[10]) + ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(Sqrt[5]*
(2*(3 + Sqrt[5]))^(3/4)) - ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)]/(Sqrt[5]*(2*(3 + Sqrt[5]))^(3/4)) - ((9
+ 4*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*Sqrt[10]) + ((9 + 4*S
qrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(4*Sqrt[10]) + Log[Sqrt[2*(3 +
Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2]/(2*Sqrt[5]*(2*(3 + Sqrt[5]))^(3/4)) - Log[Sqrt[2*(3 + Sqrt[5
])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2]/(2*Sqrt[5]*(2*(3 + Sqrt[5]))^(3/4))

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 1347

Int[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, In
t[1/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n
2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {1}{1+3 x^4+x^8} \, dx &=\frac {\int \frac {1}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}}-\frac {\int \frac {1}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}}\\ &=\frac {\int \frac {\sqrt {3-\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {5 \left (3-\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {3-\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {5 \left (3-\sqrt {5}\right )}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}-\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {5 \left (3+\sqrt {5}\right )}}-\frac {\int \frac {\sqrt {3+\sqrt {5}}+\sqrt {2} x^2}{\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{2 \sqrt {5 \left (3+\sqrt {5}\right )}}\\ &=\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}{2 \sqrt {10 \left (3-\sqrt {5}\right )}}+\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}{2 \sqrt {10 \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}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}+\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}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\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}{2 \sqrt {10 \left (3+\sqrt {5}\right )}}-\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}{2 \sqrt {10 \left (3+\sqrt {5}\right )}}-\frac {\sqrt [4]{9+4 \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 \sqrt {10}}-\frac {\sqrt [4]{9+4 \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 \sqrt {10}}\\ &=-\frac {\sqrt [4]{9+4 \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 \sqrt {10}}+\frac {\sqrt [4]{9+4 \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 \sqrt {10}}+\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3-\sqrt {5}\right )\right )^{3/4}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3-\sqrt {5}\right )\right )^{3/4}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}\\ &=-\frac {\tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3-\sqrt {5}\right )\right )^{3/4}}+\frac {\tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3-\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3-\sqrt {5}\right )\right )^{3/4}}+\frac {\tan ^{-1}\left (1-\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\tan ^{-1}\left (1+\frac {2^{3/4} x}{\sqrt [4]{3+\sqrt {5}}}\right )}{\sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\sqrt [4]{9+4 \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 \sqrt {10}}+\frac {\sqrt [4]{9+4 \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 \sqrt {10}}+\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}-2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}-\frac {\log \left (\sqrt {2 \left (3+\sqrt {5}\right )}+2 \sqrt [4]{2 \left (3+\sqrt {5}\right )} x+2 x^2\right )}{2 \sqrt {5} \left (2 \left (3+\sqrt {5}\right )\right )^{3/4}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 42, normalized size = 0.10 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8+3 \text {$\#$1}^4+1\& ,\frac {\log (x-\text {$\#$1})}{2 \text {$\#$1}^7+3 \text {$\#$1}^3}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 3*x^4 + x^8)^(-1),x]

[Out]

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

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fricas [B]  time = 0.99, size = 733, normalized size = 1.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*sqrt(5)*sqrt(2)*(4*sqrt(5) + 9)^(1/4)*arctan(1/2*sqrt(2*x^2 - sqrt(4*sqrt(5) + 9)*(3*sqrt(5) - 7) + (sqrt
(5)*sqrt(2)*x - 3*sqrt(2)*x)*(4*sqrt(5) + 9)^(1/4))*(4*sqrt(5) + 9)^(3/4)*(3*sqrt(5) - 7) - 1/2*(3*sqrt(5)*sqr
t(2)*x - 7*sqrt(2)*x)*(4*sqrt(5) + 9)^(3/4) - 1) + 1/10*sqrt(5)*sqrt(2)*(4*sqrt(5) + 9)^(1/4)*arctan(1/2*sqrt(
2*x^2 - sqrt(4*sqrt(5) + 9)*(3*sqrt(5) - 7) - (sqrt(5)*sqrt(2)*x - 3*sqrt(2)*x)*(4*sqrt(5) + 9)^(1/4))*(4*sqrt
(5) + 9)^(3/4)*(3*sqrt(5) - 7) - 1/2*(3*sqrt(5)*sqrt(2)*x - 7*sqrt(2)*x)*(4*sqrt(5) + 9)^(3/4) + 1) + 1/10*sqr
t(5)*sqrt(2)*(-4*sqrt(5) + 9)^(1/4)*arctan(1/2*sqrt(2*x^2 + (3*sqrt(5) + 7)*sqrt(-4*sqrt(5) + 9) + (sqrt(5)*sq
rt(2)*x + 3*sqrt(2)*x)*(-4*sqrt(5) + 9)^(1/4))*(3*sqrt(5) + 7)*(-4*sqrt(5) + 9)^(3/4) - 1/2*(3*sqrt(5)*sqrt(2)
*x + 7*sqrt(2)*x)*(-4*sqrt(5) + 9)^(3/4) - 1) + 1/10*sqrt(5)*sqrt(2)*(-4*sqrt(5) + 9)^(1/4)*arctan(1/2*sqrt(2*
x^2 + (3*sqrt(5) + 7)*sqrt(-4*sqrt(5) + 9) - (sqrt(5)*sqrt(2)*x + 3*sqrt(2)*x)*(-4*sqrt(5) + 9)^(1/4))*(3*sqrt
(5) + 7)*(-4*sqrt(5) + 9)^(3/4) - 1/2*(3*sqrt(5)*sqrt(2)*x + 7*sqrt(2)*x)*(-4*sqrt(5) + 9)^(3/4) + 1) - 1/40*s
qrt(5)*sqrt(2)*(4*sqrt(5) + 9)^(1/4)*log(2*x^2 - sqrt(4*sqrt(5) + 9)*(3*sqrt(5) - 7) + (sqrt(5)*sqrt(2)*x - 3*
sqrt(2)*x)*(4*sqrt(5) + 9)^(1/4)) + 1/40*sqrt(5)*sqrt(2)*(4*sqrt(5) + 9)^(1/4)*log(2*x^2 - sqrt(4*sqrt(5) + 9)
*(3*sqrt(5) - 7) - (sqrt(5)*sqrt(2)*x - 3*sqrt(2)*x)*(4*sqrt(5) + 9)^(1/4)) - 1/40*sqrt(5)*sqrt(2)*(-4*sqrt(5)
+ 9)^(1/4)*log(2*x^2 + (3*sqrt(5) + 7)*sqrt(-4*sqrt(5) + 9) + (sqrt(5)*sqrt(2)*x + 3*sqrt(2)*x)*(-4*sqrt(5) +
9)^(1/4)) + 1/40*sqrt(5)*sqrt(2)*(-4*sqrt(5) + 9)^(1/4)*log(2*x^2 + (3*sqrt(5) + 7)*sqrt(-4*sqrt(5) + 9) - (s
qrt(5)*sqrt(2)*x + 3*sqrt(2)*x)*(-4*sqrt(5) + 9)^(1/4))

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giac [A]  time = 0.53, size = 239, normalized size = 0.58 \[ \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {10 \, \sqrt {5} + 20} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} + 1} + 1\right )\right )} \sqrt {10 \, \sqrt {5} + 20} - \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {10 \, \sqrt {5} - 20} + \frac {1}{80} \, {\left (\pi + 4 \, \arctan \left (-x \sqrt {\sqrt {5} - 1} - 1\right )\right )} \sqrt {10 \, \sqrt {5} - 20} - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 20} \log \left (10000 \, {\left (x + \sqrt {\sqrt {5} + 1}\right )}^{2} + 10000 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 20} \log \left (10000 \, {\left (x - \sqrt {\sqrt {5} + 1}\right )}^{2} + 10000 \, x^{2}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 20} \log \left (400 \, {\left (x + \sqrt {\sqrt {5} - 1}\right )}^{2} + 400 \, x^{2}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 20} \log \left (400 \, {\left (x - \sqrt {\sqrt {5} - 1}\right )}^{2} + 400 \, x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80*(pi + 4*arctan(x*sqrt(sqrt(5) + 1) + 1))*sqrt(10*sqrt(5) + 20) - 1/80*(pi + 4*arctan(-x*sqrt(sqrt(5) + 1)
+ 1))*sqrt(10*sqrt(5) + 20) - 1/80*(pi + 4*arctan(x*sqrt(sqrt(5) - 1) - 1))*sqrt(10*sqrt(5) - 20) + 1/80*(pi
+ 4*arctan(-x*sqrt(sqrt(5) - 1) - 1))*sqrt(10*sqrt(5) - 20) - 1/40*sqrt(10*sqrt(5) - 20)*log(10000*(x + sqrt(s
qrt(5) + 1))^2 + 10000*x^2) + 1/40*sqrt(10*sqrt(5) - 20)*log(10000*(x - sqrt(sqrt(5) + 1))^2 + 10000*x^2) + 1/
40*sqrt(10*sqrt(5) + 20)*log(400*(x + sqrt(sqrt(5) - 1))^2 + 400*x^2) - 1/40*sqrt(10*sqrt(5) + 20)*log(400*(x
- sqrt(sqrt(5) - 1))^2 + 400*x^2)

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maple [C]  time = 0.01, size = 37, normalized size = 0.09 \[ \frac {\ln \left (-\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{7}+12 \RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{4}+1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sum(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 \[ \int \frac {1}{x^{8} + 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x^8 + 3*x^4 + 1), x)

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mupad [B]  time = 0.08, size = 403, normalized size = 0.97 \[ \frac {\sqrt {5}\,\mathrm {atan}\left (\frac {144\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}+\frac {64\,\sqrt {5}\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}\right )\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {144\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}-\frac {64\,\sqrt {5}\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}\right )\,{\left (4\,\sqrt {5}-9\right )}^{1/4}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,144{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}+\frac {\sqrt {5}\,x\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,64{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {-4\,\sqrt {5}-9}+56\,\sqrt {-4\,\sqrt {5}-9}}\right )\,{\left (-4\,\sqrt {5}-9\right )}^{1/4}\,1{}\mathrm {i}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,144{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}-\frac {\sqrt {5}\,x\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,64{}\mathrm {i}}{24\,\sqrt {5}\,\sqrt {4\,\sqrt {5}-9}-56\,\sqrt {4\,\sqrt {5}-9}}\right )\,{\left (4\,\sqrt {5}-9\right )}^{1/4}\,1{}\mathrm {i}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^4 + x^8 + 1),x)

[Out]

(5^(1/2)*atan((144*x*(- 4*5^(1/2) - 9)^(1/4))/(24*5^(1/2)*(- 4*5^(1/2) - 9)^(1/2) + 56*(- 4*5^(1/2) - 9)^(1/2)
) + (64*5^(1/2)*x*(- 4*5^(1/2) - 9)^(1/4))/(24*5^(1/2)*(- 4*5^(1/2) - 9)^(1/2) + 56*(- 4*5^(1/2) - 9)^(1/2)))*
(- 4*5^(1/2) - 9)^(1/4))/10 + (5^(1/2)*atan((144*x*(4*5^(1/2) - 9)^(1/4))/(24*5^(1/2)*(4*5^(1/2) - 9)^(1/2) -
56*(4*5^(1/2) - 9)^(1/2)) - (64*5^(1/2)*x*(4*5^(1/2) - 9)^(1/4))/(24*5^(1/2)*(4*5^(1/2) - 9)^(1/2) - 56*(4*5^(
1/2) - 9)^(1/2)))*(4*5^(1/2) - 9)^(1/4))/10 - (5^(1/2)*atan((x*(- 4*5^(1/2) - 9)^(1/4)*144i)/(24*5^(1/2)*(- 4*
5^(1/2) - 9)^(1/2) + 56*(- 4*5^(1/2) - 9)^(1/2)) + (5^(1/2)*x*(- 4*5^(1/2) - 9)^(1/4)*64i)/(24*5^(1/2)*(- 4*5^
(1/2) - 9)^(1/2) + 56*(- 4*5^(1/2) - 9)^(1/2)))*(- 4*5^(1/2) - 9)^(1/4)*1i)/10 - (5^(1/2)*atan((x*(4*5^(1/2) -
9)^(1/4)*144i)/(24*5^(1/2)*(4*5^(1/2) - 9)^(1/2) - 56*(4*5^(1/2) - 9)^(1/2)) - (5^(1/2)*x*(4*5^(1/2) - 9)^(1/
4)*64i)/(24*5^(1/2)*(4*5^(1/2) - 9)^(1/2) - 56*(4*5^(1/2) - 9)^(1/2)))*(4*5^(1/2) - 9)^(1/4)*1i)/10

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sympy [A]  time = 1.52, size = 26, normalized size = 0.06 \[ \operatorname {RootSum} {\left (40960000 t^{8} + 115200 t^{4} + 1, \left (t \mapsto t \log {\left (- 9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(40960000*_t**8 + 115200*_t**4 + 1, Lambda(_t, _t*log(-9600*_t**5 - 47*_t/2 + x)))

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