3.149 \(\int \frac{1}{x (216+108 x^2+324 x^3+18 x^4+x^6)} \, dx\)

Optimal. Leaf size=415 \[ -\frac{\left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt{3}\right )\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{46656}-\frac{\left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{23328}-\frac{\left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{23328}+\frac{\log (x)}{216}+\frac{(-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{216 \sqrt [3]{2} 3^{5/6} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac{(-1)^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{216 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (1-\sqrt [3]{2} 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{216 \sqrt [6]{2} 3^{5/6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]

[Out]

((-1)^(2/3)*((-2)^(2/3) - 2*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/
(216*2^(1/3)*3^(5/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3)]) - ((-1)^(2/3)*((-3)^(1/3) + 3*2^(1/3
))*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(216*6^(1/6)*(1 + (-1)^(1/
3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((1 - 2^(1/3)*3^(2/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[
3*(-4 + 3*2^(1/3)*3^(2/3))]])/(216*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + Log[x]/216 - ((36 + 2^(2/3)
*3^(1/3)*(1 + I*Sqrt[3]))*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/46656 - ((18 - (-2)^(2/3)*3^(1/3))*Log[6 + 3*
(-2)^(2/3)*3^(1/3)*x + x^2])/23328 - ((18 - 2^(2/3)*3^(1/3))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/23328

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Rubi [A]  time = 0.901607, antiderivative size = 415, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2097, 634, 618, 204, 628, 206} \[ -\frac{\left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt{3}\right )\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{46656}-\frac{\left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{23328}-\frac{\left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{23328}+\frac{\log (x)}{216}+\frac{(-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{216 \sqrt [3]{2} 3^{5/6} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac{(-1)^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{216 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (1-\sqrt [3]{2} 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{216 \sqrt [6]{2} 3^{5/6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)),x]

[Out]

((-1)^(2/3)*((-2)^(2/3) - 2*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/
(216*2^(1/3)*3^(5/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3)]) - ((-1)^(2/3)*((-3)^(1/3) + 3*2^(1/3
))*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(216*6^(1/6)*(1 + (-1)^(1/
3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((1 - 2^(1/3)*3^(2/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[
3*(-4 + 3*2^(1/3)*3^(2/3))]])/(216*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + Log[x]/216 - ((36 + 2^(2/3)
*3^(1/3)*(1 + I*Sqrt[3]))*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/46656 - ((18 - (-2)^(2/3)*3^(1/3))*Log[6 + 3*
(-2)^(2/3)*3^(1/3)*x + x^2])/23328 - ((18 - 2^(2/3)*3^(1/3))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/23328

Rule 2097

Int[(Q6_)^(p_)*(u_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coe
ff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Dist[1/(3^(3*p)*a^(2*p)), Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c,
3]*x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3*(-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x
+ b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] &&
EqQ[Coeff[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]

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 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 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 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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx &=1259712 \int \left (\frac{1}{272097792 x}+\frac{(-1)^{2/3} \left (6 \left (9+\sqrt [3]{-3} 2^{2/3}\right )-\left (1-3 (-3)^{2/3} \sqrt [3]{2}\right ) x\right )}{816293376 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac{(-1)^{2/3} \left (-6 \left (9-(-2)^{2/3} \sqrt [3]{3}\right )+\left (1+3 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{816293376 \sqrt [3]{2} 3^{2/3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac{-6 \sqrt [3]{6} \left (9 \sqrt [3]{2}-2 \sqrt [3]{3}\right )-\left (18-2^{2/3} \sqrt [3]{3}\right ) x}{14693280768 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=\frac{\log (x)}{216}+\frac{\int \frac{-6 \sqrt [3]{6} \left (9 \sqrt [3]{2}-2 \sqrt [3]{3}\right )-\left (18-2^{2/3} \sqrt [3]{3}\right ) x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{11664}+\frac{(-1)^{2/3} \int \frac{-6 \left (9-(-2)^{2/3} \sqrt [3]{3}\right )+\left (1+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1944 \sqrt [3]{2} 3^{2/3}}+\frac{(-1)^{2/3} \int \frac{6 \left (9+\sqrt [3]{-3} 2^{2/3}\right )-\left (1-3 (-3)^{2/3} \sqrt [3]{2}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{648 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=\frac{\log (x)}{216}+\frac{\left (\left (-\frac{1}{6}\right )^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right )\right ) \int \frac{1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{72 \left (1+\sqrt [3]{-1}\right )^2}+\frac{\left ((-1)^{2/3} \left (-1+3 (-3)^{2/3} \sqrt [3]{2}\right )\right ) \int \frac{-3 \sqrt [3]{-3} 2^{2/3}+2 x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\left (-18+(-2)^{2/3} \sqrt [3]{3}\right ) \int \frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{23328}+\frac{\left (-18+2^{2/3} \sqrt [3]{3}\right ) \int \frac{3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{23328}+\frac{\left ((-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right )\right ) \int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{432 \sqrt [3]{6}}+\frac{\left (1-\sqrt [3]{2} 3^{2/3}\right ) \int \frac{1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{216\ 2^{2/3} \sqrt [3]{3}}\\ &=\frac{\log (x)}{216}-\frac{(-1)^{2/3} \left (1-3 (-3)^{2/3} \sqrt [3]{2}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{\left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{23328}-\frac{\left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{23328}-\frac{\left (\left (-\frac{1}{6}\right )^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )-x^2} \, dx,x,-3 \sqrt [3]{-3} 2^{2/3}+2 x\right )}{36 \left (1+\sqrt [3]{-1}\right )^2}-\frac{\left ((-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )-x^2} \, dx,x,3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )}{216 \sqrt [3]{6}}-\frac{\left (1-\sqrt [3]{2} 3^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-6 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )-x^2} \, dx,x,3\ 2^{2/3} \sqrt [3]{3}+2 x\right )}{108\ 2^{2/3} \sqrt [3]{3}}\\ &=\frac{(-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right ) \tan ^{-1}\left (\frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{216\ 6^{5/6} \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac{(-1)^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{216 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (1-\sqrt [3]{2} 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt{3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{216 \sqrt [6]{2} 3^{5/6} \sqrt{-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac{\log (x)}{216}-\frac{(-1)^{2/3} \left (1-3 (-3)^{2/3} \sqrt [3]{2}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{\left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{23328}-\frac{\left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{23328}\\ \end{align*}

Mathematica [C]  time = 0.0180972, size = 103, normalized size = 0.25 \[ \frac{\log (x)}{216}-\frac{\text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+18 \text{$\#$1}^2 \log (x-\text{$\#$1})+324 \text{$\#$1} \log (x-\text{$\#$1})+108 \log (x-\text{$\#$1})}{\text{$\#$1}^4+12 \text{$\#$1}^2+162 \text{$\#$1}+36}\& \right ]}{1296} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)),x]

[Out]

Log[x]/216 - RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (108*Log[x - #1] + 324*Log[x - #1]*#1 + 18
*Log[x - #1]*#1^2 + Log[x - #1]*#1^4)/(36 + 162*#1 + 12*#1^2 + #1^4) & ]/1296

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Maple [C]  time = 0.007, size = 75, normalized size = 0.2 \begin{align*} -{\frac{1}{1296}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{ \left ({{\it \_R}}^{5}+18\,{{\it \_R}}^{3}+324\,{{\it \_R}}^{2}+108\,{\it \_R} \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}}+{\frac{\ln \left ( x \right ) }{216}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^6+18*x^4+324*x^3+108*x^2+216),x)

[Out]

-1/1296*sum((_R^5+18*_R^3+324*_R^2+108*_R)/(_R^5+12*_R^3+162*_R^2+36*_R)*ln(x-_R),_R=RootOf(_Z^6+18*_Z^4+324*_
Z^3+108*_Z^2+216))+1/216*ln(x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{216} \, \int \frac{x^{5} + 18 \, x^{3} + 324 \, x^{2} + 108 \, x}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} + \frac{1}{216} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima")

[Out]

-1/216*integrate((x^5 + 18*x^3 + 324*x^2 + 108*x)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x) + 1/216*log(x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 0.366844, size = 82, normalized size = 0.2 \begin{align*} \frac{\log{\left (x \right )}}{216} + \operatorname{RootSum}{\left (7379637425677839491923968 t^{6} + 34164988081841849499648 t^{5} + 52598809250685370368 t^{4} + 26673506015311872 t^{3} - 309171116160 t^{2} + 944784 t - 1, \left ( t \mapsto t \log{\left (\frac{8145570099668817936783362115119297360560128 t^{6}}{143425799309052440063} + \frac{977068766770806381087358257564745728 t^{5}}{143425799309052440063} - \frac{116529526608851264288400971539061538816 t^{4}}{143425799309052440063} - \frac{239359794985242202542501440710766592 t^{3}}{143425799309052440063} - \frac{136678312638137094439887341418240 t^{2}}{143425799309052440063} + \frac{1563115569067663795735413696 t}{143425799309052440063} + x - \frac{3164446315075236190044}{143425799309052440063} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**6+18*x**4+324*x**3+108*x**2+216),x)

[Out]

log(x)/216 + RootSum(7379637425677839491923968*_t**6 + 34164988081841849499648*_t**5 + 52598809250685370368*_t
**4 + 26673506015311872*_t**3 - 309171116160*_t**2 + 944784*_t - 1, Lambda(_t, _t*log(814557009966881793678336
2115119297360560128*_t**6/143425799309052440063 + 977068766770806381087358257564745728*_t**5/14342579930905244
0063 - 116529526608851264288400971539061538816*_t**4/143425799309052440063 - 239359794985242202542501440710766
592*_t**3/143425799309052440063 - 136678312638137094439887341418240*_t**2/143425799309052440063 + 156311556906
7663795735413696*_t/143425799309052440063 + x - 3164446315075236190044/143425799309052440063)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="giac")

[Out]

integrate(1/((x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)*x), x)