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

Optimal. Leaf size=377 \[ -\frac {\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{648\ 2^{2/3}}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{648\ 2^{2/3} \sqrt [3]{3}}+\frac {(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{324 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac {\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{972 \sqrt {3 \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\left (9-2^{2/3} \sqrt [3]{3}\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{972 \sqrt {6 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}} \]

[Out]

-1/1296*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(1/3)*3^(2/3)/(1+(-1)^(1/3))^2-1/3888*(-1)^(1/3)*3^(2/3)*ln(6+3*(-2
)^(2/3)*3^(1/3)*x+x^2)*2^(1/3)+1/3888*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)*2^(1/3)*3^(2/3)+1/972*(-1)^(2/3)*(3*(-3)^(
2/3)-2^(2/3))*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2))*3^(5/6)/(1+(-1)^(1/3))^2/(8-
6*(-3)^(2/3)*2^(1/3))^(1/2)-1/972*(9-2^(2/3)*3^(1/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(
2/3))^(1/2))/(-24+18*2^(1/3)*3^(2/3))^(1/2)+1/972*(9-(-2)^(2/3)*3^(1/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24
+18*(-2)^(1/3)*3^(2/3))^(1/2))/(24+27*I*2^(1/3)*3^(1/6)+9*2^(1/3)*3^(2/3))^(1/2)

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Rubi [A]  time = 0.72, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2070, 634, 618, 204, 628, 206} \[ -\frac {\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{648\ 2^{2/3}}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{648\ 2^{2/3} \sqrt [3]{3}}+\frac {(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{324 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac {\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{972 \sqrt {3 \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\left (9-2^{2/3} \sqrt [3]{3}\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{972 \sqrt {6 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1)^(2/3)*(3*(-3)^(2/3) - 2^(2/3))*ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]])/
(324*3^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[2*(4 - 3*(-3)^(2/3)*2^(1/3))]) + ((9 - (-2)^(2/3)*3^(1/3))*ArcTan[(3*(-2)
^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(972*Sqrt[3*(8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*
3^(2/3))]) - ((9 - 2^(2/3)*3^(1/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]
])/(972*Sqrt[6*(-4 + 3*2^(1/3)*3^(2/3))]) - Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2]/(216*2^(2/3)*3^(1/3)*(1 + (-
1)^(1/3))^2) - ((-1/3)^(1/3)*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(648*2^(2/3)) + Log[6 + 3*2^(2/3)*3^(1/3)*
x + x^2]/(648*2^(2/3)*3^(1/3))

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 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])

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 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 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 2070

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

Rubi steps

\begin {align*} \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx &=1259712 \int \left (\frac {(-1)^{2/3} \left (-2+6 (-3)^{2/3} \sqrt [3]{2}-\sqrt [3]{-3} 2^{2/3} x\right )}{272097792 \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 {2 (-1)^{2/3}-6 \sqrt [3]{2} 3^{2/3}+\sqrt [3]{-3} 2^{2/3} x}{272097792 \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 {18-2^{2/3} \sqrt [3]{3}+\sqrt [3]{2} 3^{2/3} x}{2448880128 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {18-2^{2/3} \sqrt [3]{3}+\sqrt [3]{2} 3^{2/3} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1944}-\frac {\int \frac {2 (-1)^{2/3}-6 \sqrt [3]{2} 3^{2/3}+\sqrt [3]{-3} 2^{2/3} x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{648 \sqrt [3]{2} 3^{2/3}}+\frac {(-1)^{2/3} \int \frac {-2+6 (-3)^{2/3} \sqrt [3]{2}-\sqrt [3]{-3} 2^{2/3} x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=-\frac {\sqrt [3]{-\frac {1}{3}} \int \frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{648\ 2^{2/3}}+\frac {\int \frac {3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{648\ 2^{2/3} \sqrt [3]{3}}-\frac {\int \frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\left (\sqrt [3]{-1} \left (9+\sqrt [3]{-3} 2^{2/3}\right )\right ) \int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{648 \left (1+\sqrt [3]{-1}\right )^2}-\frac {\left (-9+(-2)^{2/3} \sqrt [3]{3}\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1944}+\frac {\left (9-2^{2/3} \sqrt [3]{3}\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1944}\\ &=-\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3} \sqrt [3]{3}}+\frac {\left (\sqrt [3]{-1} \left (9+\sqrt [3]{-3} 2^{2/3}\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 )}{324 \left (1+\sqrt [3]{-1}\right )^2}-\frac {1}{972} \left (9-(-2)^{2/3} \sqrt [3]{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 )+\frac {1}{972} \left (-9+2^{2/3} \sqrt [3]{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 )\\ &=-\frac {\sqrt [3]{-1} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{324 \left (1+\sqrt [3]{-1}\right )^2 \sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac {\left ((-2)^{2/3}-3\ 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 )}{972 \sqrt [6]{3} \sqrt {2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}+\frac {\left (2^{2/3}-3\ 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 )}{972 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3} \sqrt [3]{3}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 62, normalized size = 0.16 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6+18 \text {$\#$1}^4+324 \text {$\#$1}^3+108 \text {$\#$1}^2+216\& ,\frac {\log (x-\text {$\#$1})}{\text {$\#$1}^5+12 \text {$\#$1}^3+162 \text {$\#$1}^2+36 \text {$\#$1}}\& \right ] \]

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , Log[x - #1]/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/6

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

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maple [C]  time = 0.01, size = 53, normalized size = 0.14 \[ \frac {\ln \left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+x \right )}{6 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5}+72 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}+972 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}+216 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.67, size = 306, normalized size = 0.81 \[ \sum _{k=1}^6\ln \left (-\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )\,x\,6+{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^2\,x\,349920-{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^3\,x\,6122200320-{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^4\,x\,258263796059136-{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^5\,x\,6940988288557056+944784\,{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^2-16529940864\,{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^3-33192121254912\,{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^4-168897381688221696\,{\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right )}^5\right )\,\mathrm {root}\left (z^6-\frac {z^4}{9844416}-\frac {217\,z^3}{86118951168}-\frac {5\,z^2}{111610160713728}+\frac {z}{488182842961846272}-\frac {1}{34164988081841849499648},z,k\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(349920*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/111610160713728 + z/488182842961846
272 - 1/34164988081841849499648, z, k)^2*x - 6*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/111610
160713728 + z/488182842961846272 - 1/34164988081841849499648, z, k)*x - 6122200320*root(z^6 - z^4/9844416 - (2
17*z^3)/86118951168 - (5*z^2)/111610160713728 + z/488182842961846272 - 1/34164988081841849499648, z, k)^3*x -
258263796059136*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/111610160713728 + z/48818284296184627
2 - 1/34164988081841849499648, z, k)^4*x - 6940988288557056*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (
5*z^2)/111610160713728 + z/488182842961846272 - 1/34164988081841849499648, z, k)^5*x + 944784*root(z^6 - z^4/9
844416 - (217*z^3)/86118951168 - (5*z^2)/111610160713728 + z/488182842961846272 - 1/34164988081841849499648, z
, k)^2 - 16529940864*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/111610160713728 + z/488182842961
846272 - 1/34164988081841849499648, z, k)^3 - 33192121254912*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 -
(5*z^2)/111610160713728 + z/488182842961846272 - 1/34164988081841849499648, z, k)^4 - 168897381688221696*root(
z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/111610160713728 + z/488182842961846272 - 1/341649880818418
49499648, z, k)^5)*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/111610160713728 + z/48818284296184
6272 - 1/34164988081841849499648, z, k), k, 1, 6)

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sympy [A]  time = 0.27, size = 65, normalized size = 0.17 \[ \operatorname {RootSum} {\left (34164988081841849499648 t^{6} - 3470494144278528 t^{4} - 86087932019712 t^{3} - 1530550080 t^{2} + 69984 t - 1, \left (t \mapsto t \log {\left (\frac {185904446699109611410573787136 t^{5}}{57121295165} + \frac {6377301253267917382766592 t^{4}}{57121295165} - \frac {18904636002388564311552 t^{3}}{57121295165} - \frac {469080552915181723968 t^{2}}{57121295165} - \frac {24358640509989936 t}{57121295165} + x + \frac {152427895956}{57121295165} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(34164988081841849499648*_t**6 - 3470494144278528*_t**4 - 86087932019712*_t**3 - 1530550080*_t**2 + 699
84*_t - 1, Lambda(_t, _t*log(185904446699109611410573787136*_t**5/57121295165 + 6377301253267917382766592*_t**
4/57121295165 - 18904636002388564311552*_t**3/57121295165 - 469080552915181723968*_t**2/57121295165 - 24358640
509989936*_t/57121295165 + x + 152427895956/57121295165)))

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