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]

-((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))

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Rubi [A]  time = 0.256883, antiderivative size = 414, normalized size of antiderivative = 1., 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 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]

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 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 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 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 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])

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*}

Mathematica [C]  time = 0.0114351, size = 42, normalized size = 0.1 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\& ,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\& \right ] \]

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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Maple [C]  time = 0.006, size = 37, normalized size = 0.1 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+3\,{{\it \_Z}}^{4}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+3\,{{\it \_R}}^{3}}}} \end{align*}

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(x-_R),_R=RootOf(_Z^8+3*_Z^4+1))

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

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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Fricas [B]  time = 1.77359, size = 2304, normalized size = 5.57 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [A]  time = 1.10694, size = 26, normalized size = 0.06 \begin{align*} \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 )} \end{align*}

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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Giac [A]  time = 1.24306, size = 342, normalized size = 0.83 \begin{align*} -\frac{1}{40} \,{\left (i + 1\right )} \sqrt{10 \, \sqrt{5} - 20} \log \left (100 \,{\left (i + 1\right )} x + 100 \, i \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i + 1\right )} \sqrt{10 \, \sqrt{5} - 20} \log \left (100 \,{\left (i + 1\right )} x - 100 \, i \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i - 1\right )} \sqrt{10 \, \sqrt{5} - 20} \log \left (100 \,{\left (i + 1\right )} x + 100 \, \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{40} \,{\left (i - 1\right )} \sqrt{10 \, \sqrt{5} - 20} \log \left (100 \,{\left (i + 1\right )} x - 100 \, \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i + 1\right )} \sqrt{10 \, \sqrt{5} + 20} \log \left (20 \,{\left (i + 1\right )} x + 20 \, i \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \,{\left (i + 1\right )} \sqrt{10 \, \sqrt{5} + 20} \log \left (20 \,{\left (i + 1\right )} x - 20 \, i \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \,{\left (i - 1\right )} \sqrt{10 \, \sqrt{5} + 20} \log \left (20 \,{\left (i + 1\right )} x + 20 \, \sqrt{\sqrt{5} - 1}\right ) + \frac{1}{40} \,{\left (i - 1\right )} \sqrt{10 \, \sqrt{5} + 20} \log \left (20 \,{\left (i + 1\right )} x - 20 \, \sqrt{\sqrt{5} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(i + 1)*sqrt(10*sqrt(5) - 20)*log(100*(i + 1)*x + 100*i*sqrt(sqrt(5) + 1)) + 1/40*(i + 1)*sqrt(10*sqrt(5
) - 20)*log(100*(i + 1)*x - 100*i*sqrt(sqrt(5) + 1)) + 1/40*(i - 1)*sqrt(10*sqrt(5) - 20)*log(100*(i + 1)*x +
100*sqrt(sqrt(5) + 1)) - 1/40*(i - 1)*sqrt(10*sqrt(5) - 20)*log(100*(i + 1)*x - 100*sqrt(sqrt(5) + 1)) + 1/40*
(i + 1)*sqrt(10*sqrt(5) + 20)*log(20*(i + 1)*x + 20*i*sqrt(sqrt(5) - 1)) - 1/40*(i + 1)*sqrt(10*sqrt(5) + 20)*
log(20*(i + 1)*x - 20*i*sqrt(sqrt(5) - 1)) - 1/40*(i - 1)*sqrt(10*sqrt(5) + 20)*log(20*(i + 1)*x + 20*sqrt(sqr
t(5) - 1)) + 1/40*(i - 1)*sqrt(10*sqrt(5) + 20)*log(20*(i + 1)*x - 20*sqrt(sqrt(5) - 1))