Integrand size = 26, antiderivative size = 415 \[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\frac {(-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{216 \sqrt [3]{2} 3^{5/6} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{216 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{216 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\log (x)}{216}-\frac {\left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt {3}\right )\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{46656}-\frac {\left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{23328}-\frac {\left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{23328} \] Output:
1/1296*(-1)^(2/3)*((-2)^(2/3)-2*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x) /(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*2^(2/3)*3^(1/6)/(8+9*I*2^(1/3)*3^(1/6)+ 3*2^(1/3)*3^(2/3))^(1/2)-1/1296*(-1)^(2/3)*((-3)^(1/3)+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)-1/1296*(1-2^(1/3)*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))*2^(5/ 6)*3^(1/6)/(-4+3*2^(1/3)*3^(2/3))^(1/2)+1/216*ln(x)-1/46656*(36+2^(2/3)*3^ (1/3)*(1+I*3^(1/2)))*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)-1/23328*(18-(-2)^(2/ 3)*3^(1/3))*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)-1/23328*(18-2^(2/3)*3^(1/3))* ln(6+3*2^(2/3)*3^(1/3)*x+x^2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\frac {\log (x)}{216}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {108 \log (x-\text {$\#$1})+324 \log (x-\text {$\#$1}) \text {$\#$1}+18 \log (x-\text {$\#$1}) \text {$\#$1}^2+\log (x-\text {$\#$1}) \text {$\#$1}^4}{36+162 \text {$\#$1}+12 \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{1296} \] Input:
Integrate[1/(x*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)),x]
Output:
Log[x]/216 - 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 + 162*#1 + 12*#1^2 + #1^4) & ]/1296
Time = 2.12 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
| \(\displaystyle \int \frac {1}{x \left (x^6+18 x^4+324 x^3+108 x^2+216\right )} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1259712 \int \left (\frac {(-1)^{2/3} \left (6 \left (9+\sqrt [3]{-3} 2^{2/3}\right )-\left (1-3 (-3)^{2/3} \sqrt [3]{2}\right ) x\right )}{816293376 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {1}{272097792 x}-\frac {(-1)^{2/3} \left (6 \left (9-(-2)^{2/3} \sqrt [3]{3}\right )-\left (1+3 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{2448880128 \sqrt [3]{2} 3^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac {\left (18-2^{2/3} \sqrt [3]{3}\right ) x+6 \sqrt [3]{6} \left (9 \sqrt [3]{2}-2 \sqrt [3]{3}\right )}{14693280768 \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1259712 \left (\frac {(-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{272097792\ 6^{5/6} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{272097792 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (1-\sqrt [3]{2} 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{272097792 \sqrt [6]{2} 3^{5/6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}-\frac {\left (36+2^{2/3} \sqrt [3]{3}+i 2^{2/3} 3^{5/6}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{58773123072}-\frac {\left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{29386561536}-\frac {\left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{29386561536}+\frac {\log (x)}{272097792}\right )\) |
Input:
Int[1/(x*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)),x]
Output:
1259712*(((-1)^(2/3)*((-2)^(2/3) - 2*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(272097792*6^(5/6)*Sqrt[4 + 3 *(-2)^(1/3)*3^(2/3)]) - ((-1)^(2/3)*((-3)^(1/3) + 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))]])/(27209 7792*6^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((1 - 2^ (1/3)*3^(2/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^ (1/3)*3^(2/3))]])/(272097792*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + Log[x]/272097792 - ((36 + 2^(2/3)*3^(1/3) + I*2^(2/3)*3^(5/6))*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/58773123072 - ((18 - (-2)^(2/3)*3^(1/3))*Lo g[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/29386561536 - ((18 - 2^(2/3)*3^(1/3)) *Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/29386561536)
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 = 73, normalized size of antiderivative = 0.18
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (136728 \textit {\_Z}^{6}+1230552 \textit {\_Z}^{5}+3682908 \textit {\_Z}^{4}+3630708 \textit {\_Z}^{3}-81810 \textit {\_Z}^{2}+486 \textit {\_Z} -1\right )}{\sum }\textit {\_R} \ln \left (-23672342955240 \textit {\_R}^{5}-213056277916248 \textit {\_R}^{4}-637689647288592 \textit {\_R}^{3}-628763677061560 \textit {\_R}^{2}+14004611129596 \textit {\_R} +2499731391 x -55133083786\right )\right )}{1944}+\frac {\ln \left (x \right )}{216}\) | \(73\) |
| default | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (\textit {\_R}^{5}+18 \textit {\_R}^{3}+324 \textit {\_R}^{2}+108 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{1296}+\frac {\ln \left (x \right )}{216}\) | \(75\) |
Input:
int(1/x/(x^6+18*x^4+324*x^3+108*x^2+216),x,method=_RETURNVERBOSE)
Output:
1/1944*sum(_R*ln(-23672342955240*_R^5-213056277916248*_R^4-637689647288592 *_R^3-628763677061560*_R^2+14004611129596*_R+2499731391*x-55133083786),_R= RootOf(136728*_Z^6+1230552*_Z^5+3682908*_Z^4+3630708*_Z^3-81810*_Z^2+486*_ Z-1))+1/216*ln(x)
Timed out. \[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="fricas")
Output:
Timed out
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.20 \[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\frac {\log {\left (x \right )}}{216} + \operatorname {RootSum} {\left (7379637425677839491923968 t^{6} + 34164988081841849499648 t^{5} + 52598809250685370368 t^{4} + 26673506015311872 t^{3} - 309171116160 t^{2} + 944784 t - 1, \left ( t \mapsto t \log {\left (\frac {8145570099668817936783362115119297360560128 t^{6}}{143425799309052440063} + \frac {977068766770806381087358257564745728 t^{5}}{143425799309052440063} - \frac {116529526608851264288400971539061538816 t^{4}}{143425799309052440063} - \frac {239359794985242202542501440710766592 t^{3}}{143425799309052440063} - \frac {136678312638137094439887341418240 t^{2}}{143425799309052440063} + \frac {1563115569067663795735413696 t}{143425799309052440063} + x - \frac {3164446315075236190044}{143425799309052440063} \right )} \right )\right )} \] Input:
integrate(1/x/(x**6+18*x**4+324*x**3+108*x**2+216),x)
Output:
log(x)/216 + RootSum(7379637425677839491923968*_t**6 + 3416498808184184949 9648*_t**5 + 52598809250685370368*_t**4 + 26673506015311872*_t**3 - 309171 116160*_t**2 + 944784*_t - 1, Lambda(_t, _t*log(81455700996688179367833621 15119297360560128*_t**6/143425799309052440063 + 97706876677080638108735825 7564745728*_t**5/143425799309052440063 - 116529526608851264288400971539061 538816*_t**4/143425799309052440063 - 239359794985242202542501440710766592* _t**3/143425799309052440063 - 136678312638137094439887341418240*_t**2/1434 25799309052440063 + 1563115569067663795735413696*_t/143425799309052440063 + x - 3164446315075236190044/143425799309052440063)))
\[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} x} \,d x } \] Input:
integrate(1/x/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima")
Output:
-1/216*integrate((x^5 + 18*x^3 + 324*x^2 + 108*x)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x) + 1/216*log(x)
\[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} x} \,d x } \] Input:
integrate(1/x/(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), x)
Time = 22.14 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx =\text {Too large to display} \] Input:
int(1/(x*(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216)),x)
Output:
log(x)/216 + symsum(log(7*root(z^6 + z^5/216 + (421*z^4)/59066496 + (10085 3*z^3)/27902540178432 - (505*z^2)/12053897357082624 + z/781092548738954035 2 - 1/7379637425677839491923968, z, k)*x - 5670000*root(z^6 + z^5/216 + (4 21*z^4)/59066496 + (100853*z^3)/27902540178432 - (505*z^2)/120538973570826 24 + z/7810925487389540352 - 1/7379637425677839491923968, z, k)^2*x + 1546 875947520*root(z^6 + z^5/216 + (421*z^4)/59066496 + (100853*z^3)/279025401 78432 - (505*z^2)/12053897357082624 + z/7810925487389540352 - 1/7379637425 677839491923968, z, k)^3*x - 106961147905609728*root(z^6 + z^5/216 + (421* z^4)/59066496 + (100853*z^3)/27902540178432 - (505*z^2)/12053897357082624 + z/7810925487389540352 - 1/7379637425677839491923968, z, k)^4*x - 1405119 95854134018048*root(z^6 + z^5/216 + (421*z^4)/59066496 + (100853*z^3)/2790 2540178432 - (505*z^2)/12053897357082624 + z/7810925487389540352 - 1/73796 37425677839491923968, z, k)^5*x - 45607290567387619000320*root(z^6 + z^5/2 16 + (421*z^4)/59066496 + (100853*z^3)/27902540178432 - (505*z^2)/12053897 357082624 + z/7810925487389540352 - 1/7379637425677839491923968, z, k)^6*x + 839808*root(z^6 + z^5/216 + (421*z^4)/59066496 + (100853*z^3)/279025401 78432 - (505*z^2)/12053897357082624 + z/7810925487389540352 - 1/7379637425 677839491923968, z, k)^2 + 594896472576*root(z^6 + z^5/216 + (421*z^4)/590 66496 + (100853*z^3)/27902540178432 - (505*z^2)/12053897357082624 + z/7810 925487389540352 - 1/7379637425677839491923968, z, k)^3 - 84834301304586...
\[ \int \frac {1}{x \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int \frac {1}{x^{7}+18 x^{5}+324 x^{4}+108 x^{3}+216 x}d x \] Input:
int(1/x/(x^6+18*x^4+324*x^3+108*x^2+216),x)
Output:
int(1/(x**7 + 18*x**5 + 324*x**4 + 108*x**3 + 216*x),x)