3.383 \(\int \frac{1}{x^2 (1+3 x^4+x^8)} \, dx\)

Optimal. Leaf size=416 \[ -\frac{\left (3+\sqrt{5}\right )^{5/4} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{8\ 2^{3/4} \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{5/4} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{8\ 2^{3/4} \sqrt{5}}+\frac{1}{40} \sqrt [4]{6150-2750 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )-\frac{1}{40} \sqrt [4]{6150-2750 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )-\frac{1}{x}+\frac{\left (3+\sqrt{5}\right )^{5/4} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{5/4} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{1}{20} \sqrt [4]{6150-2750 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )+\frac{1}{20} \sqrt [4]{6150-2750 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right ) \]

[Out]

-x^(-1) + ((3 + Sqrt[5])^(5/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(4*2^(3/4)*Sqrt[5]) - ((3 + Sqrt[5
])^(5/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(4*2^(3/4)*Sqrt[5]) - ((6150 - 2750*Sqrt[5])^(1/4)*ArcTa
n[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/20 + ((6150 - 2750*Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])
^(1/4)])/20 - ((3 + Sqrt[5])^(5/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(8*2^(3/4
)*Sqrt[5]) + ((3 + Sqrt[5])^(5/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(8*2^(3/4)
*Sqrt[5]) + ((6150 - 2750*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/40
- ((6150 - 2750*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/40

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Rubi [A]  time = 0.285476, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1368, 1510, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\left (3+\sqrt{5}\right )^{5/4} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{8\ 2^{3/4} \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{5/4} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{8\ 2^{3/4} \sqrt{5}}+\frac{1}{40} \sqrt [4]{6150-2750 \sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )-\frac{1}{40} \sqrt [4]{6150-2750 \sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )-\frac{1}{x}+\frac{\left (3+\sqrt{5}\right )^{5/4} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{5/4} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{4\ 2^{3/4} \sqrt{5}}-\frac{1}{20} \sqrt [4]{6150-2750 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )+\frac{1}{20} \sqrt [4]{6150-2750 \sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(1 + 3*x^4 + x^8)),x]

[Out]

-x^(-1) + ((3 + Sqrt[5])^(5/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(4*2^(3/4)*Sqrt[5]) - ((3 + Sqrt[5
])^(5/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(4*2^(3/4)*Sqrt[5]) - ((6150 - 2750*Sqrt[5])^(1/4)*ArcTa
n[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/20 + ((6150 - 2750*Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])
^(1/4)])/20 - ((3 + Sqrt[5])^(5/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(8*2^(3/4
)*Sqrt[5]) + ((3 + Sqrt[5])^(5/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 - Sqrt[5]))^(1/4)*x + 2*x^2])/(8*2^(3/4)
*Sqrt[5]) + ((6150 - 2750*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/40
- ((6150 - 2750*Sqrt[5])^(1/4)*Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/40

Rule 1368

Int[((d_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((d*x)^(m + 1)*(a +
b*x^n + c*x^(2*n))^(p + 1))/(a*d*(m + 1)), x] - Dist[1/(a*d^n*(m + 1)), Int[(d*x)^(m + n)*(b*(m + n*(p + 1) +
1) + c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2
*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]

Rule 1510

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Dist[e/
2 - (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2
, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

Rule 297

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

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])

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]

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (1+3 x^4+x^8\right )} \, dx &=-\frac{1}{x}+\int \frac{x^2 \left (-3-x^4\right )}{1+3 x^4+x^8} \, dx\\ &=-\frac{1}{x}+\frac{1}{10} \left (-5+3 \sqrt{5}\right ) \int \frac{x^2}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx-\frac{1}{10} \left (5+3 \sqrt{5}\right ) \int \frac{x^2}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx\\ &=-\frac{1}{x}-\frac{\left (3-\sqrt{5}\right ) \int \frac{\sqrt{3+\sqrt{5}}-\sqrt{2} x^2}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx}{4 \sqrt{10}}+\frac{\left (3-\sqrt{5}\right ) \int \frac{\sqrt{3+\sqrt{5}}+\sqrt{2} x^2}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx}{4 \sqrt{10}}+\frac{\left (3+\sqrt{5}\right ) \int \frac{\sqrt{3-\sqrt{5}}-\sqrt{2} x^2}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx}{4 \sqrt{10}}-\frac{\left (3+\sqrt{5}\right ) \int \frac{\sqrt{3-\sqrt{5}}+\sqrt{2} x^2}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx}{4 \sqrt{10}}\\ &=-\frac{1}{x}-\frac{\left (3+\sqrt{5}\right )^{5/4} \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}{8\ 2^{3/4} \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{5/4} \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}{8\ 2^{3/4} \sqrt{5}}+\frac{\left (3-\sqrt{5}\right ) \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}{8 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}+\frac{\left (3-\sqrt{5}\right ) \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}{8 \sqrt{5} \sqrt [4]{2 \left (3+\sqrt{5}\right )}}+\frac{1}{40} \left (-5+3 \sqrt{5}\right ) \int \frac{1}{\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )}-\sqrt [4]{2 \left (3+\sqrt{5}\right )} x+x^2} \, dx+\frac{1}{40} \left (-5+3 \sqrt{5}\right ) \int \frac{1}{\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )}+\sqrt [4]{2 \left (3+\sqrt{5}\right )} x+x^2} \, dx-\frac{1}{40} \left (5+3 \sqrt{5}\right ) \int \frac{1}{\sqrt{\frac{1}{2} \left (3-\sqrt{5}\right )}-\sqrt [4]{2 \left (3-\sqrt{5}\right )} x+x^2} \, dx-\frac{1}{40} \left (5+3 \sqrt{5}\right ) \int \frac{1}{\sqrt{\frac{1}{2} \left (3-\sqrt{5}\right )}+\sqrt [4]{2 \left (3-\sqrt{5}\right )} x+x^2} \, dx\\ &=-\frac{1}{x}-\frac{\left (3+\sqrt{5}\right )^{5/4} \log \left (\sqrt{2 \left (3-\sqrt{5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{5/4} \log \left (\sqrt{2 \left (3-\sqrt{5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt{5}}+\frac{1}{40} \sqrt [4]{6150-2750 \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 )-\frac{1}{40} \sqrt [4]{6150-2750 \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 )+-\frac{\left (-5-3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{20 \sqrt [4]{2 \left (3-\sqrt{5}\right )}}+\frac{\left (5-3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{20 \sqrt [4]{2 \left (3+\sqrt{5}\right )}}+\frac{\left (-5+3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{20 \sqrt [4]{2 \left (3+\sqrt{5}\right )}}-\frac{\left (5+3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{20 \sqrt [4]{2 \left (3-\sqrt{5}\right )}}\\ &=-\frac{1}{x}+\frac{\sqrt [4]{246+110 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{4 \sqrt{5}}-\frac{\sqrt [4]{246+110 \sqrt{5}} \tan ^{-1}\left (1+\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{4 \sqrt{5}}-\frac{1}{20} \sqrt [4]{6150-2750 \sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )+\frac{\sqrt [4]{246-110 \sqrt{5}} \tan ^{-1}\left (1+\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{4 \sqrt{5}}-\frac{\left (3+\sqrt{5}\right )^{5/4} \log \left (\sqrt{2 \left (3-\sqrt{5}\right )}-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt{5}}+\frac{\left (3+\sqrt{5}\right )^{5/4} \log \left (\sqrt{2 \left (3-\sqrt{5}\right )}+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+2 x^2\right )}{8\ 2^{3/4} \sqrt{5}}+\frac{1}{40} \sqrt [4]{6150-2750 \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 )-\frac{1}{40} \sqrt [4]{6150-2750 \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 )\\ \end{align*}

Mathematica [C]  time = 0.0166801, size = 61, normalized size = 0.15 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^5+3 \text{$\#$1}}\& \right ]-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(1 + 3*x^4 + x^8)),x]

[Out]

-x^(-1) - RootSum[1 + 3*#1^4 + #1^8 & , (3*Log[x - #1] + Log[x - #1]*#1^4)/(3*#1 + 2*#1^5) & ]/4

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Maple [C]  time = 0.009, size = 52, 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{ \left ({{\it \_R}}^{6}+3\,{{\it \_R}}^{2} \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+3\,{{\it \_R}}^{3}}}}-{x}^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(x^8+3*x^4+1),x)

[Out]

-1/4*sum((_R^6+3*_R^2)/(2*_R^7+3*_R^3)*ln(x-_R),_R=RootOf(_Z^8+3*_Z^4+1))-1/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{x} - \int \frac{x^{6} + 3 \, x^{2}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8+3*x^4+1),x, algorithm="maxima")

[Out]

-1/x - integrate((x^6 + 3*x^2)/(x^8 + 3*x^4 + 1), x)

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Fricas [B]  time = 1.85182, size = 3545, normalized size = 8.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8+3*x^4+1),x, algorithm="fricas")

[Out]

-1/80*(sqrt(10)*(55*sqrt(5)*x - 123*x)*(110*sqrt(5) + 246)^(3/4)*sqrt(55*sqrt(5) + 123)*arctan(-1/20*sqrt(10)*
(161*sqrt(5)*x - 360*x)*(110*sqrt(5) + 246)^(3/4)*sqrt(55*sqrt(5) + 123) + 1/40*sqrt(sqrt(10)*(47*sqrt(5)*sqrt
(2)*x - 105*sqrt(2)*x)*(110*sqrt(5) + 246)^(3/4) + 40*x^2 - 20*sqrt(110*sqrt(5) + 246)*(4*sqrt(5) - 9))*(161*s
qrt(5) - 360)*(110*sqrt(5) + 246)^(3/4)*sqrt(55*sqrt(5) + 123) + 1/8*(55*sqrt(5)*sqrt(2) - 123*sqrt(2))*sqrt(1
10*sqrt(5) + 246)*sqrt(55*sqrt(5) + 123)) + sqrt(10)*(55*sqrt(5)*x - 123*x)*(110*sqrt(5) + 246)^(3/4)*sqrt(55*
sqrt(5) + 123)*arctan(-1/20*sqrt(10)*(161*sqrt(5)*x - 360*x)*(110*sqrt(5) + 246)^(3/4)*sqrt(55*sqrt(5) + 123)
+ 1/40*sqrt(-sqrt(10)*(47*sqrt(5)*sqrt(2)*x - 105*sqrt(2)*x)*(110*sqrt(5) + 246)^(3/4) + 40*x^2 - 20*sqrt(110*
sqrt(5) + 246)*(4*sqrt(5) - 9))*(161*sqrt(5) - 360)*(110*sqrt(5) + 246)^(3/4)*sqrt(55*sqrt(5) + 123) - 1/8*(55
*sqrt(5)*sqrt(2) - 123*sqrt(2))*sqrt(110*sqrt(5) + 246)*sqrt(55*sqrt(5) + 123)) + sqrt(10)*(55*sqrt(5)*x + 123
*x)*sqrt(-55*sqrt(5) + 123)*(-110*sqrt(5) + 246)^(3/4)*arctan(1/40*sqrt(sqrt(10)*(47*sqrt(5)*sqrt(2)*x + 105*s
qrt(2)*x)*(-110*sqrt(5) + 246)^(3/4) + 40*x^2 + 20*(4*sqrt(5) + 9)*sqrt(-110*sqrt(5) + 246))*(161*sqrt(5) + 36
0)*sqrt(-55*sqrt(5) + 123)*(-110*sqrt(5) + 246)^(3/4) - 1/40*(2*sqrt(10)*(161*sqrt(5)*x + 360*x)*(-110*sqrt(5)
+ 246)^(3/4) + 5*(55*sqrt(5)*sqrt(2) + 123*sqrt(2))*sqrt(-110*sqrt(5) + 246))*sqrt(-55*sqrt(5) + 123)) + sqrt
(10)*(55*sqrt(5)*x + 123*x)*sqrt(-55*sqrt(5) + 123)*(-110*sqrt(5) + 246)^(3/4)*arctan(1/40*sqrt(-sqrt(10)*(47*
sqrt(5)*sqrt(2)*x + 105*sqrt(2)*x)*(-110*sqrt(5) + 246)^(3/4) + 40*x^2 + 20*(4*sqrt(5) + 9)*sqrt(-110*sqrt(5)
+ 246))*(161*sqrt(5) + 360)*sqrt(-55*sqrt(5) + 123)*(-110*sqrt(5) + 246)^(3/4) - 1/40*(2*sqrt(10)*(161*sqrt(5)
*x + 360*x)*(-110*sqrt(5) + 246)^(3/4) - 5*(55*sqrt(5)*sqrt(2) + 123*sqrt(2))*sqrt(-110*sqrt(5) + 246))*sqrt(-
55*sqrt(5) + 123)) - sqrt(10)*sqrt(2)*x*(110*sqrt(5) + 246)^(1/4)*log(sqrt(10)*(47*sqrt(5)*sqrt(2)*x - 105*sqr
t(2)*x)*(110*sqrt(5) + 246)^(3/4) + 40*x^2 - 20*sqrt(110*sqrt(5) + 246)*(4*sqrt(5) - 9)) + sqrt(10)*sqrt(2)*x*
(110*sqrt(5) + 246)^(1/4)*log(-sqrt(10)*(47*sqrt(5)*sqrt(2)*x - 105*sqrt(2)*x)*(110*sqrt(5) + 246)^(3/4) + 40*
x^2 - 20*sqrt(110*sqrt(5) + 246)*(4*sqrt(5) - 9)) + sqrt(10)*sqrt(2)*x*(-110*sqrt(5) + 246)^(1/4)*log(sqrt(10)
*(47*sqrt(5)*sqrt(2)*x + 105*sqrt(2)*x)*(-110*sqrt(5) + 246)^(3/4) + 40*x^2 + 20*(4*sqrt(5) + 9)*sqrt(-110*sqr
t(5) + 246)) - sqrt(10)*sqrt(2)*x*(-110*sqrt(5) + 246)^(1/4)*log(-sqrt(10)*(47*sqrt(5)*sqrt(2)*x + 105*sqrt(2)
*x)*(-110*sqrt(5) + 246)^(3/4) + 40*x^2 + 20*(4*sqrt(5) + 9)*sqrt(-110*sqrt(5) + 246)) + 80)/x

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Sympy [A]  time = 1.24525, size = 32, normalized size = 0.08 \begin{align*} \operatorname{RootSum}{\left (40960000 t^{8} + 787200 t^{4} + 1, \left ( t \mapsto t \log{\left (\frac{19251200 t^{7}}{11} + \frac{369792 t^{3}}{11} + x \right )} \right )\right )} - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(x**8+3*x**4+1),x)

[Out]

RootSum(40960000*_t**8 + 787200*_t**4 + 1, Lambda(_t, _t*log(19251200*_t**7/11 + 369792*_t**3/11 + x))) - 1/x

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Giac [A]  time = 1.32929, size = 348, normalized size = 0.84 \begin{align*} \frac{1}{40} \,{\left (i - 1\right )} \sqrt{25 \, \sqrt{5} - 55} \log \left (865 \,{\left (i + 1\right )} x + 865 \, i \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{40} \,{\left (i - 1\right )} \sqrt{25 \, \sqrt{5} - 55} \log \left (865 \,{\left (i + 1\right )} x - 865 \, i \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{40} \,{\left (i + 1\right )} \sqrt{25 \, \sqrt{5} - 55} \log \left (865 \,{\left (i + 1\right )} x + 865 \, \sqrt{\sqrt{5} + 1}\right ) + \frac{1}{40} \,{\left (i + 1\right )} \sqrt{25 \, \sqrt{5} - 55} \log \left (865 \,{\left (i + 1\right )} x - 865 \, \sqrt{\sqrt{5} + 1}\right ) - \frac{1}{40} \,{\left (i - 1\right )} \sqrt{25 \, \sqrt{5} + 55} \log \left (425 \,{\left (i + 1\right )} x + 425 \, i \sqrt{\sqrt{5} - 1}\right ) + \frac{1}{40} \,{\left (i - 1\right )} \sqrt{25 \, \sqrt{5} + 55} \log \left (425 \,{\left (i + 1\right )} x - 425 \, i \sqrt{\sqrt{5} - 1}\right ) + \frac{1}{40} \,{\left (i + 1\right )} \sqrt{25 \, \sqrt{5} + 55} \log \left (425 \,{\left (i + 1\right )} x + 425 \, \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{40} \,{\left (i + 1\right )} \sqrt{25 \, \sqrt{5} + 55} \log \left (425 \,{\left (i + 1\right )} x - 425 \, \sqrt{\sqrt{5} - 1}\right ) - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(x^8+3*x^4+1),x, algorithm="giac")

[Out]

1/40*(i - 1)*sqrt(25*sqrt(5) - 55)*log(865*(i + 1)*x + 865*i*sqrt(sqrt(5) + 1)) - 1/40*(i - 1)*sqrt(25*sqrt(5)
- 55)*log(865*(i + 1)*x - 865*i*sqrt(sqrt(5) + 1)) - 1/40*(i + 1)*sqrt(25*sqrt(5) - 55)*log(865*(i + 1)*x + 8
65*sqrt(sqrt(5) + 1)) + 1/40*(i + 1)*sqrt(25*sqrt(5) - 55)*log(865*(i + 1)*x - 865*sqrt(sqrt(5) + 1)) - 1/40*(
i - 1)*sqrt(25*sqrt(5) + 55)*log(425*(i + 1)*x + 425*i*sqrt(sqrt(5) - 1)) + 1/40*(i - 1)*sqrt(25*sqrt(5) + 55)
*log(425*(i + 1)*x - 425*i*sqrt(sqrt(5) - 1)) + 1/40*(i + 1)*sqrt(25*sqrt(5) + 55)*log(425*(i + 1)*x + 425*sqr
t(sqrt(5) - 1)) - 1/40*(i + 1)*sqrt(25*sqrt(5) + 55)*log(425*(i + 1)*x - 425*sqrt(sqrt(5) - 1)) - 1/x