\(\int \frac {1}{x (216+108 x^2+324 x^3+18 x^4+x^6)} \, dx\) [149]

Optimal result

Integrand size = 26, antiderivative size = 415 \[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\frac {(-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{216 \sqrt [3]{2} 3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{216 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{216 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\log (x)}{216}-\frac {\left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt {3}\right )\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{46656}-\frac {\left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{23328}-\frac {\left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{23328} \]

[Out]

1/216*ln(x)-1/23328*(18-(-2)^(2/3)*3^(1/3))*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)-1/23328*(18-2^(2/3)*3^(1/3))*ln(6
+3*2^(2/3)*3^(1/3)*x+x^2)-1/46656*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*(36+2^(2/3)*3^(1/3)*(1+I*3^(1/2)))-1/1296*(
-1)^(2/3)*((-3)^(1/3)+3*2^(1/3))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/(12-9*(-3)^(2/3)*2^(1/3))^(1/2))*6^(5
/6)/(1+(-1)^(1/3))^2/(4-3*(-3)^(2/3)*2^(1/3))^(1/2)-1/1296*(1-2^(1/3)*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1
/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*2^(5/6)*3^(1/6)/(-4+3*2^(1/3)*3^(2/3))^(1/2)+1/1296*(-1)^(2/3)*((-2)^(2/
3)-2*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*2^(2/3)*3^(1/6)/(8+9*I*2^(1/
3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(1/2)

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2122, 648, 632, 210, 642, 212} \[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\frac {(-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{216 \sqrt [3]{2} 3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{216 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{216 \sqrt [6]{2} 3^{5/6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}-\frac {\left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt {3}\right )\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{46656}-\frac {\left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{23328}-\frac {\left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{23328}+\frac {\log (x)}{216} \]

[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 210

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

Rule 212

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

Rule 632

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 642

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 648

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 2122

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]

Rubi steps \begin{align*} \text {integral}& = 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 ) \text {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 ) \text {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 ) \text {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] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.01 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\frac {\log (x)}{216}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {108 \log (x-\text {$\#$1})+324 \log (x-\text {$\#$1}) \text {$\#$1}+18 \log (x-\text {$\#$1}) \text {$\#$1}^2+\log (x-\text {$\#$1}) \text {$\#$1}^4}{36+162 \text {$\#$1}+12 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{1296} \]

[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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.18

method	result	size
risch	\(\frac {\ln \left (x \right )}{216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (136728 \textit {\_Z}^{6}+1230552 \textit {\_Z}^{5}+3682908 \textit {\_Z}^{4}+3630708 \textit {\_Z}^{3}-81810 \textit {\_Z}^{2}+486 \textit {\_Z} -1\right )}{\sum }\textit {\_R} \ln \left (-23672342955240 \textit {\_R}^{5}-213056277916248 \textit {\_R}^{4}-637689647288592 \textit {\_R}^{3}-628763677061560 \textit {\_R}^{2}[phone] \textit {\_R} [phone] x -55133083786\right )\right )}{1944}\)	\(73\)
default	\(\frac {\ln \left (x \right )}{216}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (\textit {\_R}^{5}+18 \textit {\_R}^{3}+324 \textit {\_R}^{2}+108 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{1296}\)	\(75\)

[In]

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

[Out]

1/216*ln(x)+1/1944*sum(_R*ln(-23672342955240*_R^5-213056277916248*_R^4-637689647288592*_R^3-628763677061560*_R
^2+14004611129596*_R+2499731391*x-55133083786),_R=RootOf(136728*_Z^6+1230552*_Z^5+3682908*_Z^4+3630708*_Z^3-81
810*_Z^2+486*_Z-1))

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.20 \[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\frac {\log {\left (x \right )}}{216} + \operatorname {RootSum} {\left (7379637425677839491923968 t^{6} + 34164988081841849499648 t^{5} + 52598809250685370368 t^{4} + 26673506015311872 t^{3} - 309171116160 t^{2} + 944784 t - 1, \left ( t \mapsto t \log {\left (\frac {8145570099668817936783362115119297360560128 t^{6}}{143425799309052440063} + \frac {977068766770806381087358257564745728 t^{5}}{143425799309052440063} - \frac {116529526608851264288400971539061538816 t^{4}}{143425799309052440063} - \frac {239359794985242202542501440710766592 t^{3}}{143425799309052440063} - \frac {136678312638137094439887341418240 t^{2}}{143425799309052440063} + \frac {1563115569067663795735413696 t}{143425799309052440063} + x - \frac {3164446315075236190044}{143425799309052440063} \right )} \right )\right )} \]

[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)))

Maxima [F]

\[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} x} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 9.44 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\frac {\ln \left (x\right )}{216}+\left (\sum _{k=1}^6\ln \left (\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )\,x\,7-{\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )}^2\,x\,5670000+{\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )}^3\,x\,1546875947520-{\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )}^4\,x\,106961147905609728-{\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )}^5\,x\,140511995854134018048-{\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )}^6\,x\,45607290567387619000320+839808\,{\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )}^2+594896472576\,{\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )}^3-8483430130458624\,{\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )}^4-3831425535283494912\,{\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )}^5+1217393817906599165952\,{\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )}^6\right )\,\mathrm {root}\left (z^6+\frac {z^5}{216}+\frac {421\,z^4}{59066496}+\frac {100853\,z^3}{27902540178432}-\frac {505\,z^2}{12053897357082624}+\frac {z}{7810925487389540352}-\frac {1}{7379637425677839491923968},z,k\right )\right ) \]

[In]

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

[Out]

log(x)/216 + symsum(log(7*root(z^6 + z^5/216 + (421*z^4)/59066496 + (100853*z^3)/27902540178432 - (505*z^2)/12
053897357082624 + z/7810925487389540352 - 1/7379637425677839491923968, z, k)*x - 5670000*root(z^6 + z^5/216 +
(421*z^4)/59066496 + (100853*z^3)/27902540178432 - (505*z^2)/12053897357082624 + z/7810925487389540352 - 1/737
9637425677839491923968, z, k)^2*x + 1546875947520*root(z^6 + z^5/216 + (421*z^4)/59066496 + (100853*z^3)/27902
540178432 - (505*z^2)/12053897357082624 + z/7810925487389540352 - 1/7379637425677839491923968, z, k)^3*x - 106
961147905609728*root(z^6 + z^5/216 + (421*z^4)/59066496 + (100853*z^3)/27902540178432 - (505*z^2)/120538973570
82624 + z/7810925487389540352 - 1/7379637425677839491923968, z, k)^4*x - 140511995854134018048*root(z^6 + z^5/
216 + (421*z^4)/59066496 + (100853*z^3)/27902540178432 - (505*z^2)/12053897357082624 + z/7810925487389540352 -
1/7379637425677839491923968, z, k)^5*x - 45607290567387619000320*root(z^6 + z^5/216 + (421*z^4)/59066496 + (1
00853*z^3)/27902540178432 - (505*z^2)/12053897357082624 + z/7810925487389540352 - 1/7379637425677839491923968,
z, k)^6*x + 839808*root(z^6 + z^5/216 + (421*z^4)/59066496 + (100853*z^3)/27902540178432 - (505*z^2)/12053897
357082624 + z/7810925487389540352 - 1/7379637425677839491923968, z, k)^2 + 594896472576*root(z^6 + z^5/216 + (
421*z^4)/59066496 + (100853*z^3)/27902540178432 - (505*z^2)/12053897357082624 + z/7810925487389540352 - 1/7379
637425677839491923968, z, k)^3 - 8483430130458624*root(z^6 + z^5/216 + (421*z^4)/59066496 + (100853*z^3)/27902
540178432 - (505*z^2)/12053897357082624 + z/7810925487389540352 - 1/7379637425677839491923968, z, k)^4 - 38314
25535283494912*root(z^6 + z^5/216 + (421*z^4)/59066496 + (100853*z^3)/27902540178432 - (505*z^2)/1205389735708
2624 + z/7810925487389540352 - 1/7379637425677839491923968, z, k)^5 + 1217393817906599165952*root(z^6 + z^5/21
6 + (421*z^4)/59066496 + (100853*z^3)/27902540178432 - (505*z^2)/12053897357082624 + z/7810925487389540352 - 1
/7379637425677839491923968, z, k)^6)*root(z^6 + z^5/216 + (421*z^4)/59066496 + (100853*z^3)/27902540178432 - (
505*z^2)/12053897357082624 + z/7810925487389540352 - 1/7379637425677839491923968, z, k), k, 1, 6)