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

Optimal. Leaf size=448 \[ -\frac {(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\left (3 (-6)^{2/3}+2 \sqrt [3]{-2}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{7776 \sqrt [3]{3}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{3888 \sqrt [3]{6}}-\frac {1}{216 x}-\frac {\left (27 \sqrt [3]{-6}-(-2)^{2/3}+12\ 3^{2/3}\right ) \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{5832 \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \tan ^{-1}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{1944 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{5832 \sqrt [6]{6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}} \]

[Out]

-1/216/x-1/7776*(-1)^(2/3)*(9+(-3)^(1/3)*2^(2/3))*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(2/3)*3^(1/3)/(1+(-1)^(1/
3))^2+1/23328*(3*(-6)^(2/3)+2*(-2)^(1/3))*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)*3^(2/3)-1/23328*(2^(2/3)-3*3^(2/3))
*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)*6^(2/3)-1/11664*(-1)^(2/3)*(6*(-6)^(2/3)+27*(-3)^(1/3)-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)-1/34992*(2^(1/3)+27*3^(1/3)-6*6^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2)
)*6^(5/6)/(-4+3*2^(1/3)*3^(2/3))^(1/2)-1/17496*(27*(-6)^(1/3)-(-2)^(2/3)+12*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1
/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*3^(5/6)/(8+9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(1/2)

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Rubi [A]  time = 1.10, antiderivative size = 448, normalized size of antiderivative = 1.00, 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} \[ -\frac {(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\left (3 (-6)^{2/3}+2 \sqrt [3]{-2}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{7776 \sqrt [3]{3}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{3888 \sqrt [3]{6}}-\frac {1}{216 x}-\frac {\left (27 \sqrt [3]{-6}-(-2)^{2/3}+12\ 3^{2/3}\right ) \tan ^{-1}\left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{5832 \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \tan ^{-1}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{1944 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{2/3}\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{5832 \sqrt [6]{6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(216*x) - ((27*(-6)^(1/3) - (-2)^(2/3) + 12*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)
^(1/3)*3^(2/3))]])/(5832*3^(1/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3)]) - ((-1)^(2/3)*(6*(-6)^(2
/3) + 27*(-3)^(1/3) - 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))]]
)/(1944*6^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((2^(1/3) + 27*3^(1/3) - 6*6^(2/3))*ArcTa
nh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]])/(5832*6^(1/6)*Sqrt[-4 + 3*2^(1/3)*3^(2
/3)]) - ((-1)^(2/3)*(9 + (-3)^(1/3)*2^(2/3))*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(1296*2^(1/3)*3^(2/3)*(1 +
(-1)^(1/3))^2) + ((3*(-6)^(2/3) + 2*(-2)^(1/3))*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(7776*3^(1/3)) - ((2^(
2/3) - 3*3^(2/3))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/(3888*6^(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 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]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx &=1259712 \int \left (\frac {1}{272097792 x^2}+\frac {(-1)^{2/3} \left (-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}-\left (9+\sqrt [3]{-3} 2^{2/3}\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 (1+27 (-2)^{2/3} \sqrt [3]{3}+9 \sqrt [3]{-2} 3^{2/3}+\left (9-(-2)^{2/3} \sqrt [3]{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 {-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [3]{2} 3^{2/3}-6^{2/3} \left (2^{2/3}-3\ 3^{2/3}\right ) x}{14693280768 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=-\frac {1}{216 x}+\frac {\int \frac {-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [3]{2} 3^{2/3}-6^{2/3} \left (2^{2/3}-3\ 3^{2/3}\right ) x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{11664}+\frac {(-1)^{2/3} \int \frac {1+27 (-2)^{2/3} \sqrt [3]{3}+9 \sqrt [3]{-2} 3^{2/3}+\left (9-(-2)^{2/3} \sqrt [3]{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 {-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}-\left (9+\sqrt [3]{-3} 2^{2/3}\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 {1}{216 x}-\frac {\left ((-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\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 ((-1)^{2/3} \left ((-2)^{2/3}-3\ 3^{2/3}\right )\right ) \int \frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{3888 \sqrt [3]{6}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \int \frac {3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{3888 \sqrt [3]{6}}+\frac {\left ((-1)^{2/3} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right )\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{3888 \sqrt [3]{2} 3^{2/3}}+\frac {\left ((-1)^{2/3} \left (3 \sqrt [3]{-3} 2^{2/3} \left (-9-\sqrt [3]{-3} 2^{2/3}\right )+2 \left (-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}\right )\right )\right ) \int \frac {1}{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 \sqrt [3]{2} \left (2^{2/3}-3\ 3^{2/3}\right )+2 \left (-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [3]{2} 3^{2/3}\right )\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{23328}\\ &=-\frac {1}{216 x}-\frac {(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\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 {(-1)^{2/3} \left ((-2)^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}}-\frac {\left ((-1)^{2/3} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [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 )}{1944 \sqrt [3]{2} 3^{2/3}}-\frac {\left ((-1)^{2/3} \left (3 \sqrt [3]{-3} 2^{2/3} \left (-9-\sqrt [3]{-3} 2^{2/3}\right )+2 \left (-1+9 (-3)^{2/3} \sqrt [3]{2}+27 \sqrt [3]{-3} 2^{2/3}\right )\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 )}{648 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\left (18 \sqrt [3]{2} \left (2^{2/3}-3\ 3^{2/3}\right )+2 \left (-54+2^{2/3} \sqrt [3]{3}+54 \sqrt [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 )}{11664}\\ &=-\frac {1}{216 x}+\frac {(-1)^{2/3} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [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 )}{5832\ 2^{5/6} \sqrt [6]{3} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {(-1)^{2/3} \left (2-12 (-3)^{2/3} \sqrt [3]{2}-27 \sqrt [3]{-3} 2^{2/3}\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 )}{1944\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\left (18\ 2^{2/3}-27\ 3^{2/3}-\sqrt [3]{6}\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 )}{5832 \sqrt [6]{2} \sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\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 {(-1)^{2/3} \left ((-2)^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 109, normalized size = 0.24 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^6+18 \text {$\#$1}^4+324 \text {$\#$1}^3+108 \text {$\#$1}^2+216\& ,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})+18 \text {$\#$1}^2 \log (x-\text {$\#$1})+324 \text {$\#$1} \log (x-\text {$\#$1})+108 \log (x-\text {$\#$1})}{\text {$\#$1}^5+12 \text {$\#$1}^3+162 \text {$\#$1}^2+36 \text {$\#$1}}\& \right ]}{1296}-\frac {1}{216 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/216*1/x - 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*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/1296

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(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^2), x)

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maple [C]  time = 0.01, size = 74, normalized size = 0.17 \[ \frac {\left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{4}-18 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}-324 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )-108\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+x \right )}{1296 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5}+15552 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}+209952 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}+46656 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}-\frac {1}{216 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/216/x+1/1296*sum((-_R^4-18*_R^2-324*_R-108)/(_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 \[ -\frac {1}{216 \, x} - \frac {1}{216} \, \int \frac {x^{4} + 18 \, x^{2} + 324 \, x + 108}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.29, size = 340, normalized size = 0.76 \[ \left (\sum _{k=1}^6\ln \left (\frac {5\,\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}{8}-\frac {\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )\,x}{216}-{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^2\,x\,396252-{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^3\,x\,598229670528+{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^4\,x\,82120746212352-{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^5\,x\,6940988288557056+2344464\,{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^2-210297580992\,{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^3-10535082310656\,{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^4-168897381688221696\,{\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )}^5\right )\,\mathrm {root}\left (z^6+\frac {281\,z^4}{118132992}-\frac {50435\,z^3}{9300846726144}-\frac {331\,z^2}{48215589428330496}-\frac {z}{1898054893435658305536}-\frac {1}{1594001683946413330255577088},z,k\right )\right )-\frac {1}{216\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log((5*root(z^6 + (281*z^4)/118132992 - (50435*z^3)/9300846726144 - (331*z^2)/48215589428330496 - z/189
8054893435658305536 - 1/1594001683946413330255577088, z, k))/8 - (root(z^6 + (281*z^4)/118132992 - (50435*z^3)
/9300846726144 - (331*z^2)/48215589428330496 - z/1898054893435658305536 - 1/1594001683946413330255577088, z, k
)*x)/216 - 396252*root(z^6 + (281*z^4)/118132992 - (50435*z^3)/9300846726144 - (331*z^2)/48215589428330496 - z
/1898054893435658305536 - 1/1594001683946413330255577088, z, k)^2*x - 598229670528*root(z^6 + (281*z^4)/118132
992 - (50435*z^3)/9300846726144 - (331*z^2)/48215589428330496 - z/1898054893435658305536 - 1/15940016839464133
30255577088, z, k)^3*x + 82120746212352*root(z^6 + (281*z^4)/118132992 - (50435*z^3)/9300846726144 - (331*z^2)
/48215589428330496 - z/1898054893435658305536 - 1/1594001683946413330255577088, z, k)^4*x - 6940988288557056*r
oot(z^6 + (281*z^4)/118132992 - (50435*z^3)/9300846726144 - (331*z^2)/48215589428330496 - z/189805489343565830
5536 - 1/1594001683946413330255577088, z, k)^5*x + 2344464*root(z^6 + (281*z^4)/118132992 - (50435*z^3)/930084
6726144 - (331*z^2)/48215589428330496 - z/1898054893435658305536 - 1/1594001683946413330255577088, z, k)^2 - 2
10297580992*root(z^6 + (281*z^4)/118132992 - (50435*z^3)/9300846726144 - (331*z^2)/48215589428330496 - z/18980
54893435658305536 - 1/1594001683946413330255577088, z, k)^3 - 10535082310656*root(z^6 + (281*z^4)/118132992 -
(50435*z^3)/9300846726144 - (331*z^2)/48215589428330496 - z/1898054893435658305536 - 1/15940016839464133302555
77088, z, k)^4 - 168897381688221696*root(z^6 + (281*z^4)/118132992 - (50435*z^3)/9300846726144 - (331*z^2)/482
15589428330496 - z/1898054893435658305536 - 1/1594001683946413330255577088, z, k)^5)*root(z^6 + (281*z^4)/1181
32992 - (50435*z^3)/9300846726144 - (331*z^2)/48215589428330496 - z/1898054893435658305536 - 1/159400168394641
3330255577088, z, k), k, 1, 6) - 1/(216*x)

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sympy [A]  time = 0.32, size = 70, normalized size = 0.16 \[ \operatorname {RootSum} {\left (1594001683946413330255577088 t^{6} + 3791612026460331638784 t^{4} - 8643672699589509120 t^{3} - 10942820851968 t^{2} - 839808 t - 1, \left (t \mapsto t \log {\left (- \frac {49875532761902496003293561236914468028416 t^{5}}{12350449784703991795} + \frac {12625489872431620388005975200497664 t^{4}}{12350449784703991795} - \frac {118637692607573771238550798852644864 t^{3}}{12350449784703991795} + \frac {270486324927832147818193778754816 t^{2}}{12350449784703991795} + \frac {273914194897479402961199352 t}{12350449784703991795} + x - \frac {12798926329353908292}{12350449784703991795} \right )} \right )\right )} - \frac {1}{216 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(1594001683946413330255577088*_t**6 + 3791612026460331638784*_t**4 - 8643672699589509120*_t**3 - 109428
20851968*_t**2 - 839808*_t - 1, Lambda(_t, _t*log(-49875532761902496003293561236914468028416*_t**5/12350449784
703991795 + 12625489872431620388005975200497664*_t**4/12350449784703991795 - 118637692607573771238550798852644
864*_t**3/12350449784703991795 + 270486324927832147818193778754816*_t**2/12350449784703991795 + 27391419489747
9402961199352*_t/12350449784703991795 + x - 12798926329353908292/12350449784703991795))) - 1/(216*x)

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