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

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) - ((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))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqr
t[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))

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Rubi [A]  time = 3.68911, antiderivative size = 448, 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 \[ -\frac{(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\left (3 (-6)^{2/3}+2 \sqrt [3]{-2}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{7776 \sqrt [3]{3}}-\frac{\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{3888 \sqrt [3]{6}}-\frac{1}{216 x}-\frac{\left (27 \sqrt [3]{-6}-(-2)^{2/3}+12\ 3^{2/3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{5832 \sqrt [6]{3} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac{(-1)^{2/3} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{1944 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{5832 \sqrt [6]{6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]

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))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqr
t[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))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.0294103, 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*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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Maple [C]  time = 0.012, size = 74, normalized size = 0.2 \[{\frac{1}{1296}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}-18\,{{\it \_R}}^{2}-324\,{\it \_R}-108 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}}-{\frac{1}{216\,x}} \]

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]

1/1296*sum((-_R^4-18*_R^2-324*_R-108)/(_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/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)*x^2),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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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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^2),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.810258, 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 - 8643
672699589509120*_t**3 - 10942820851968*_t**2 - 839808*_t - 1, Lambda(_t, _t*log(
-49875532761902496003293561236914468028416*_t**5/12350449784703991795 + 12625489
872431620388005975200497664*_t**4/12350449784703991795 - 11863769260757377123855
0798852644864*_t**3/12350449784703991795 + 270486324927832147818193778754816*_t*
*2/12350449784703991795 + 273914194897479402961199352*_t/12350449784703991795 +
x - 12798926329353908292/12350449784703991795))) - 1/(216*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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