Integrand size = 26, antiderivative size = 448 \[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=-\frac {1}{216 x}-\frac {\left (27 \sqrt [3]{-6}-(-2)^{2/3}+12\ 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{5832 \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{1944 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{5832 \sqrt [6]{6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{1296 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\left (3 (-6)^{2/3}+2 \sqrt [3]{-2}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{7776 \sqrt [3]{3}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{3888 \sqrt [3]{6}} \] Output:
-1/216/x-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)-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)-1/3499 2*(2^(1/3)+27*3^(1/3)-6*6^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-1 2+9*2^(1/3)*3^(2/3))^(1/2))*6^(5/6)/(-4+3*2^(1/3)*3^(2/3))^(1/2)-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)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.24 \[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=-\frac {1}{216 x}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {108 \log (x-\text {$\#$1})+324 \log (x-\text {$\#$1}) \text {$\#$1}+18 \log (x-\text {$\#$1}) \text {$\#$1}^2+\log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{1296} \] Input:
Integrate[1/(x^2*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)),x]
Output:
-1/216*1/x - RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (108*L og[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
Time = 2.30 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2466, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^2 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1259712 \int \left (-\frac {(-1)^{2/3} \left (\left (9+\sqrt [3]{-3} 2^{2/3}\right ) x-27 \sqrt [3]{-3} 2^{2/3}-9 (-3)^{2/3} \sqrt [3]{2}+1\right )}{816293376 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {(-1)^{2/3} \left (\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) x+9 \sqrt [3]{-2} 3^{2/3}+27 (-2)^{2/3} \sqrt [3]{3}+1\right )}{2448880128 \sqrt [3]{2} 3^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac {6^{2/3} \left (2^{2/3}-3\ 3^{2/3}\right ) x-54 \sqrt [3]{2} 3^{2/3}-2^{2/3} \sqrt [3]{3}+54}{14693280768 \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac {1}{272097792 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1259712 \left (\frac {(-1)^{2/3} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{7346640384\ 2^{5/6} \sqrt [6]{3} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{2448880128 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{7346640384 \sqrt [6]{6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}-\frac {(-1)^{2/3} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{1632586752 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {(-1)^{2/3} \left ((-2)^{2/3}-3\ 3^{2/3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{4897760256 \sqrt [3]{6}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{4897760256 \sqrt [3]{6}}-\frac {1}{272097792 x}\right )\) |
Input:
Int[1/(x^2*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)),x]
Output:
1259712*(-1/272097792*1/x + ((-1)^(2/3)*(2 + 27*(-2)^(2/3)*3^(1/3) + 12*(- 2)^(1/3)*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^( 1/3)*3^(2/3))]])/(7346640384*2^(5/6)*3^(1/6)*Sqrt[4 + 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))]])/(24488801 28*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))/Sqrt[3 *(-4 + 3*2^(1/3)*3^(2/3))]])/(7346640384*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])/(1632586752*2^(1/3)*3^(2/3)*(1 + (-1)^(1/3))^2) - ((-1)^(2/3)*((- 2)^(2/3) - 3*3^(2/3))*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(4897760256*6 ^(1/3)) - ((2^(2/3) - 3*3^(2/3))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/(4897 760256*6^(1/3)))
Int[(u_.)*(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]}, Sim p[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[Coef f[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.15
| method | result | size |
| risch | \(-\frac {1}{216 x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (633 \textit {\_Z}^{6}+204849 \textit {\_Z}^{4}-5446980 \textit {\_Z}^{3}-80433 \textit {\_Z}^{2}-72 \textit {\_Z} -1\right )}{\sum }\textit {\_R} \ln \left (-462040439801351484393 \textit {\_R}^{5}+1364231865933925308 \textit {\_R}^{4}-149523740969574483417612 \textit {\_R}^{3}+3976310471903162636736042 \textit {\_R}^{2}+46967454543463546461111 \textit {\_R} +24700899569407983590 x -25597852658707816584\right )\right )}{11664}\) | \(69\) |
| default | \(-\frac {1}{216 x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-\textit {\_R}^{4}-18 \textit {\_R}^{2}-324 \textit {\_R} -108\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{1296}\) | \(74\) |
Input:
int(1/x^2/(x^6+18*x^4+324*x^3+108*x^2+216),x,method=_RETURNVERBOSE)
Output:
-1/216/x+1/11664*sum(_R*ln(-462040439801351484393*_R^5+1364231865933925308 *_R^4-149523740969574483417612*_R^3+3976310471903162636736042*_R^2+4696745 4543463546461111*_R+24700899569407983590*x-25597852658707816584),_R=RootOf (633*_Z^6+204849*_Z^4-5446980*_Z^3-80433*_Z^2-72*_Z-1))
Timed out. \[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="fricas")
Output:
Timed out
Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.16 \[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\operatorname {RootSum} {\left (1594001683946413330255577088 t^{6} + 3791612026460331638784 t^{4} - 8643672699589509120 t^{3} - 10942820851968 t^{2} - 839808 t - 1, \left ( t \mapsto t \log {\left (- \frac {49875532761902496003293561236914468028416 t^{5}}{12350449784703991795} + \frac {12625489872431620388005975200497664 t^{4}}{12350449784703991795} - \frac {118637692607573771238550798852644864 t^{3}}{12350449784703991795} + \frac {270486324927832147818193778754816 t^{2}}{12350449784703991795} + \frac {273914194897479402961199352 t}{12350449784703991795} + x - \frac {12798926329353908292}{12350449784703991795} \right )} \right )\right )} - \frac {1}{216 x} \] Input:
integrate(1/x**2/(x**6+18*x**4+324*x**3+108*x**2+216),x)
Output:
RootSum(1594001683946413330255577088*_t**6 + 3791612026460331638784*_t**4 - 8643672699589509120*_t**3 - 10942820851968*_t**2 - 839808*_t - 1, Lambda (_t, _t*log(-49875532761902496003293561236914468028416*_t**5/1235044978470 3991795 + 12625489872431620388005975200497664*_t**4/12350449784703991795 - 118637692607573771238550798852644864*_t**3/12350449784703991795 + 2704863 24927832147818193778754816*_t**2/12350449784703991795 + 273914194897479402 961199352*_t/12350449784703991795 + x - 12798926329353908292/1235044978470 3991795))) - 1/(216*x)
\[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima")
Output:
-1/216/x - 1/216*integrate((x^4 + 18*x^2 + 324*x + 108)/(x^6 + 18*x^4 + 32 4*x^3 + 108*x^2 + 216), x)
\[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="giac")
Output:
integrate(1/((x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)*x^2), x)
Time = 0.26 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx =\text {Too large to display} \] Input:
int(1/(x^2*(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216)),x)
Output:
symsum(log((5*root(z^6 + (281*z^4)/118132992 - (50435*z^3)/9300846726144 - (331*z^2)/48215589428330496 - z/1898054893435658305536 - 1/15940016839464 13330255577088, z, k))/8 - (root(z^6 + (281*z^4)/118132992 - (50435*z^3)/9 300846726144 - (331*z^2)/48215589428330496 - z/1898054893435658305536 - 1/ 1594001683946413330255577088, z, k)*x)/216 - 396252*root(z^6 + (281*z^4)/1 18132992 - (50435*z^3)/9300846726144 - (331*z^2)/48215589428330496 - z/189 8054893435658305536 - 1/1594001683946413330255577088, z, k)^2*x - 59822967 0528*root(z^6 + (281*z^4)/118132992 - (50435*z^3)/9300846726144 - (331*z^2 )/48215589428330496 - z/1898054893435658305536 - 1/15940016839464133302555 77088, z, k)^3*x + 82120746212352*root(z^6 + (281*z^4)/118132992 - (50435* z^3)/9300846726144 - (331*z^2)/48215589428330496 - z/189805489343565830553 6 - 1/1594001683946413330255577088, z, k)^4*x - 6940988288557056*root(z^6 + (281*z^4)/118132992 - (50435*z^3)/9300846726144 - (331*z^2)/482155894283 30496 - z/1898054893435658305536 - 1/1594001683946413330255577088, z, k)^5 *x + 2344464*root(z^6 + (281*z^4)/118132992 - (50435*z^3)/9300846726144 - (331*z^2)/48215589428330496 - z/1898054893435658305536 - 1/159400168394641 3330255577088, z, k)^2 - 210297580992*root(z^6 + (281*z^4)/118132992 - (50 435*z^3)/9300846726144 - (331*z^2)/48215589428330496 - z/18980548934356583 05536 - 1/1594001683946413330255577088, z, k)^3 - 10535082310656*root(z^6 + (281*z^4)/118132992 - (50435*z^3)/9300846726144 - (331*z^2)/482155894...
\[ \int \frac {1}{x^2 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\frac {-\left (\int \frac {x^{4}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \right ) x -18 \left (\int \frac {x^{2}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \right ) x -324 \left (\int \frac {x}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \right ) x -108 \left (\int \frac {1}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \right ) x -1}{216 x} \] Input:
int(1/x^2/(x^6+18*x^4+324*x^3+108*x^2+216),x)
Output:
( - int(x**4/(x**6 + 18*x**4 + 324*x**3 + 108*x**2 + 216),x)*x - 18*int(x* *2/(x**6 + 18*x**4 + 324*x**3 + 108*x**2 + 216),x)*x - 324*int(x/(x**6 + 1 8*x**4 + 324*x**3 + 108*x**2 + 216),x)*x - 108*int(1/(x**6 + 18*x**4 + 324 *x**3 + 108*x**2 + 216),x)*x - 1)/(216*x)