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)^(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))

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Rubi [A]  time = 0.720951, antiderivative size = 377, normalized size of antiderivative = 1., 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 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]

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}{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*}

Mathematica [C]  time = 0.011869, 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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Maple [C]  time = 0.004, size = 53, normalized size = 0.1 \begin{align*}{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \end{align*}

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(x-_R),_R=RootOf(_Z^6+18*_Z^4+324*_Z^3+108*_Z^2+216))

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

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

[Out]

Timed out

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Sympy [A]  time = 0.224718, size = 65, normalized size = 0.17 \begin{align*} \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 )} \end{align*}

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

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)