Integrand size = 22, antiderivative size = 383 \[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{324 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac {\left ((-2)^{2/3}-3\ 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{972 \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{972 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3}}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648\ 2^{2/3} \sqrt [3]{3}} \] Output:
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)-1/2916*((-2)^(2/3)-3*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)+1/2916*(2^(2/3)-3*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/3 )+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*3^(5/6)/(-8+6*2^(1/3)*3^(2/3)) ^(1/2)-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)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.16 \[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \] Input:
Integrate[(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^(-1),x]
Output:
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
Time = 1.82 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int \frac {1}{x^6+18 x^4+324 x^3+108 x^2+216} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1259712 \int \left (\frac {\left (-\frac {1}{3}\right )^{2/3} \left (\sqrt [3]{-6} x+2^{2/3} \left (1-3 (-3)^{2/3} \sqrt [3]{2}\right )\right )}{272097792 \left (1+\sqrt [3]{-1}\right )^2 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac {\sqrt [3]{-6} x+2^{2/3} \left ((-1)^{2/3}-3 \sqrt [3]{2} 3^{2/3}\right )}{816293376\ 3^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {\sqrt [3]{2} 3^{2/3} x-2^{2/3} \sqrt [3]{3}+18}{2448880128 \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1259712 \left (\frac {\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1224440064 \sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}+\frac {(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{408146688 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac {\left (9-2^{2/3} \sqrt [3]{3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{1224440064 \sqrt {6 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}-\frac {\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{272097792\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{816293376\ 2^{2/3}}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{816293376\ 2^{2/3} \sqrt [3]{3}}\right )\) |
Input:
Int[(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^(-1),x]
Output:
1259712*(((9 - (-2)^(2/3)*3^(1/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqr t[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(1224440064*Sqrt[6*(4 + 3*(-2)^(1/3)*3^( 2/3))]) + ((-1)^(2/3)*(3*(-3)^(2/3) - 2^(2/3))*ArcTan[(2^(1/6)*(3*(-3)^(1/ 3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(408146688*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))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2 /3))]])/(1224440064*Sqrt[6*(-4 + 3*2^(1/3)*3^(2/3))]) - Log[6 - 3*(-3)^(1/ 3)*2^(2/3)*x + x^2]/(272097792*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])/(816293376*2^(2/3)) + Log[ 6 + 3*2^(2/3)*3^(1/3)*x + x^2]/(816293376*2^(2/3)*3^(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 = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.14
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(53\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(53\) |
Input:
int(1/(x^6+18*x^4+324*x^3+108*x^2+216),x,method=_RETURNVERBOSE)
Output:
1/6*sum(1/(_R^5+12*_R^3+162*_R^2+36*_R)*ln(x-_R),_R=RootOf(_Z^6+18*_Z^4+32 4*_Z^3+108*_Z^2+216))
Timed out. \[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Timed out} \] Input:
integrate(1/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="fricas")
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.17 \[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (34164988081841849499648 t^{6} - 3470494144278528 t^{4} - 86087932019712 t^{3} - 1530550080 t^{2} + 69984 t - 1, \left ( t \mapsto t \log {\left (\frac {185904446699109611410573787136 t^{5}}{57121295165} + \frac {6377301253267917382766592 t^{4}}{57121295165} - \frac {18904636002388564311552 t^{3}}{57121295165} - \frac {469080552915181723968 t^{2}}{57121295165} - \frac {24358640509989936 t}{57121295165} + x + \frac {152427895956}{57121295165} \right )} \right )\right )} \] Input:
integrate(1/(x**6+18*x**4+324*x**3+108*x**2+216),x)
Output:
RootSum(34164988081841849499648*_t**6 - 3470494144278528*_t**4 - 860879320 19712*_t**3 - 1530550080*_t**2 + 69984*_t - 1, Lambda(_t, _t*log(185904446 699109611410573787136*_t**5/57121295165 + 6377301253267917382766592*_t**4/ 57121295165 - 18904636002388564311552*_t**3/57121295165 - 4690805529151817 23968*_t**2/57121295165 - 24358640509989936*_t/57121295165 + x + 152427895 956/57121295165)))
\[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {1}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate(1/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima")
Output:
integrate(1/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
\[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {1}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate(1/(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)
Time = 22.71 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.80 \[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\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 ) \] Input:
int(1/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216),x)
Output:
symsum(log(349920*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2) /111610160713728 + z/488182842961846272 - 1/34164988081841849499648, z, k) ^2*x - 6*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/11161016 0713728 + z/488182842961846272 - 1/34164988081841849499648, z, k)*x - 6122 200320*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/1116101607 13728 + z/488182842961846272 - 1/34164988081841849499648, z, k)^3*x - 2582 63796059136*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/11161 0160713728 + z/488182842961846272 - 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^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/111 610160713728 + z/488182842961846272 - 1/34164988081841849499648, z, k)^2 - 16529940864*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/1116 10160713728 + z/488182842961846272 - 1/34164988081841849499648, z, k)^3 - 33192121254912*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/11 1610160713728 + z/488182842961846272 - 1/34164988081841849499648, z, k)^4 - 168897381688221696*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z ^2)/111610160713728 + z/488182842961846272 - 1/34164988081841849499648, z, k)^5)*root(z^6 - z^4/9844416 - (217*z^3)/86118951168 - (5*z^2)/1116101607 13728 + z/488182842961846272 - 1/34164988081841849499648, z, k), k, 1, ...
\[ \int \frac {1}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int \frac {1}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \] Input:
int(1/(x^6+18*x^4+324*x^3+108*x^2+216),x)
Output:
int(1/(x**6 + 18*x**4 + 324*x**3 + 108*x**2 + 216),x)