Integrand size = 42, antiderivative size = 651 \[ \int \frac {1}{864+2592 x+2376 x^2+2592 x^3+4932 x^4+3132 x^5+166 x^6+12 x^7+9 x^8} \, dx=-\frac {243}{125128 (2+3 x)}-\frac {(-1)^{2/3} \left (274-783 (-2)^{2/3} \sqrt [3]{3}-318 \sqrt [3]{-2} 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{648\ 2^{5/6} \sqrt [6]{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (29-9 (-2)^{2/3} \sqrt [3]{3}\right )^2 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (159 (-6)^{2/3}+783 \sqrt [3]{-3}+137 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{648 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}} \left (29+9 \sqrt [3]{-3} 2^{2/3}\right )^2}+\frac {\left (137 \sqrt [3]{2}-783 \sqrt [3]{3}+159\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{648 \sqrt [6]{6} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (29-9\ 2^{2/3} \sqrt [3]{3}\right )^2 \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {490617 \log (2+3 x)}{489281762}-\frac {\left (83 (-6)^{2/3}+36 \sqrt [3]{-3}-52 \sqrt [3]{2}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{864 \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2 \left (29+9 \sqrt [3]{-3} 2^{2/3}\right )^2}+\frac {(-1)^{2/3} \left (18 \sqrt [3]{-6}-26 (-2)^{2/3}+83\ 3^{2/3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{432 \sqrt [3]{6} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (29-9 (-2)^{2/3} \sqrt [3]{3}\right )^2}-\frac {\left (108-249\ 2^{2/3} \sqrt [3]{3}+52 \sqrt [3]{2} 3^{2/3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{7776 \left (841-522\ 2^{2/3} \sqrt [3]{3}+162 \sqrt [3]{2} 3^{2/3}\right )} \] Output:
-243/(250256+375384*x)-1/3888*(-1)^(2/3)*(274-783*(-2)^(2/3)*3^(1/3)-318*( -2)^(1/3)*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^( 2/3))^(1/2))*2^(1/6)*3^(5/6)/(1-(-1)^(1/3))/(1+(-1)^(1/3))^2/(29-9*(-2)^(2 /3)*3^(1/3))^2/(4+3*(-2)^(1/3)*3^(2/3))^(1/2)-1/3888*(-1)^(2/3)*(159*(-6)^ (2/3)+783*(-3)^(1/3)+137*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)/(29+9*(-3)^(1/3)*2^(2/3))^2+1/3888*(137*2^(1/3)-783*3^(1/3) +159*6^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3) )^(1/2))*6^(5/6)/(1-(-1)^(1/3))/(1+(-1)^(1/3))^2/(29-9*2^(2/3)*3^(1/3))^2/ (-4+3*2^(1/3)*3^(2/3))^(1/2)-490617/489281762*ln(2+3*x)-1/2592*(83*(-6)^(2 /3)+36*(-3)^(1/3)-52*2^(1/3))*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*3^(2/3)/(1+ (-1)^(1/3))^2/(29+9*(-3)^(1/3)*2^(2/3))^2+1/2592*(-1)^(2/3)*(18*(-6)^(1/3) -26*(-2)^(2/3)+83*3^(2/3))*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)*6^(2/3)/(1-(-1 )^(1/3))/(1+(-1)^(1/3))^2/(29-9*(-2)^(2/3)*3^(1/3))^2-(108-249*2^(2/3)*3^( 1/3)+52*2^(1/3)*3^(2/3))*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)/(6539616-4059072*2^ (2/3)*3^(1/3)+1259712*2^(1/3)*3^(2/3))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.28 \[ \int \frac {1}{864+2592 x+2376 x^2+2592 x^3+4932 x^4+3132 x^5+166 x^6+12 x^7+9 x^8} \, dx=-\frac {243}{125128 (2+3 x)}-\frac {490617 \log (2+3 x)}{489281762}+\frac {81 \text {RootSum}\left [125128+64608 \text {$\#$1}-39612 \text {$\#$1}^2+7292 \text {$\#$1}^3+222 \text {$\#$1}^4-12 \text {$\#$1}^5+\text {$\#$1}^6\&,\frac {1141345500 \log (2+3 x-\text {$\#$1})-433960684 \log (2+3 x-\text {$\#$1}) \text {$\#$1}+55417890 \log (2+3 x-\text {$\#$1}) \text {$\#$1}^2+1980564 \log (2+3 x-\text {$\#$1}) \text {$\#$1}^3-112553 \log (2+3 x-\text {$\#$1}) \text {$\#$1}^4+8076 \log (2+3 x-\text {$\#$1}) \text {$\#$1}^5}{10768-13204 \text {$\#$1}+3646 \text {$\#$1}^2+148 \text {$\#$1}^3-10 \text {$\#$1}^4+\text {$\#$1}^5}\&\right ]}{3914254096} \] Input:
Integrate[(864 + 2592*x + 2376*x^2 + 2592*x^3 + 4932*x^4 + 3132*x^5 + 166* x^6 + 12*x^7 + 9*x^8)^(-1),x]
Output:
-243/(125128*(2 + 3*x)) - (490617*Log[2 + 3*x])/489281762 + (81*RootSum[12 5128 + 64608*#1 - 39612*#1^2 + 7292*#1^3 + 222*#1^4 - 12*#1^5 + #1^6 & , ( 1141345500*Log[2 + 3*x - #1] - 433960684*Log[2 + 3*x - #1]*#1 + 55417890*L og[2 + 3*x - #1]*#1^2 + 1980564*Log[2 + 3*x - #1]*#1^3 - 112553*Log[2 + 3* x - #1]*#1^4 + 8076*Log[2 + 3*x - #1]*#1^5)/(10768 - 13204*#1 + 3646*#1^2 + 148*#1^3 - 10*#1^4 + #1^5) & ])/3914254096
Result contains complex when optimal does not.
Time = 2.82 (sec) , antiderivative size = 485, normalized size of antiderivative = 0.75, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2462, 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.
| \(\displaystyle \int \frac {1}{9 x^8+12 x^7+166 x^6+3132 x^5+4932 x^4+2592 x^3+2376 x^2+2592 x+864} \, dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {1962468 x^5-2575233 x^4+37885860 x^3+587214738 x^2-574433916 x+509397788}{1957127048 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )}-\frac {1471851}{489281762 (3 x+2)}+\frac {729}{125128 (3 x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (419298975 \sqrt [3]{-6}-299770033 (-2)^{2/3}-312517050\ 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{475581872664 \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {(-1)^{2/3} \left (156258525 (-6)^{2/3}-419298975 \sqrt [3]{-3}+299770033 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{158527290888 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (299770033 \sqrt [3]{2}+419298975 \sqrt [3]{3}+156258525\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{475581872664 \sqrt [6]{6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}-\frac {(-1)^{2/3} \left (182775057+52986636 (-3)^{2/3} \sqrt [3]{2}-75263272 \sqrt [3]{-3} 2^{2/3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{105684860592 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {\left (317919816+3^{2/3} \left (60925019 (-6)^{2/3}-150526544 \sqrt [3]{-2}\right )\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{1902327490656}+\frac {\left (317919816+6^{2/3} \left (75263272\ 2^{2/3}+60925019\ 3^{2/3}\right )\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{1902327490656}-\frac {243}{125128 (3 x+2)}-\frac {490617 \log (3 x+2)}{489281762}\) |
Input:
Int[(864 + 2592*x + 2376*x^2 + 2592*x^3 + 4932*x^4 + 3132*x^5 + 166*x^6 + 12*x^7 + 9*x^8)^(-1),x]
Output:
-243/(125128*(2 + 3*x)) - ((419298975*(-6)^(1/3) - 299770033*(-2)^(2/3) - 312517050*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^ (1/3)*3^(2/3))]])/(475581872664*3^(1/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3 *2^(1/3)*3^(2/3)]) + ((-1)^(2/3)*(156258525*(-6)^(2/3) - 419298975*(-3)^(1 /3) + 299770033*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))]])/(158527290888*6^(1/6)*(1 + (-1)^(1/3))^2*S qrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((299770033*2^(1/3) + 419298975*3^(1/3) + 156258525*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))]])/(475581872664*6^(1/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) - (490617*Log[2 + 3*x])/489281762 - ((-1)^(2/3)*(182775057 + 52986636*(-3 )^(2/3)*2^(1/3) - 75263272*(-3)^(1/3)*2^(2/3))*Log[6 - 3*(-3)^(1/3)*2^(2/3 )*x + x^2])/(105684860592*2^(1/3)*3^(2/3)*(1 + (-1)^(1/3))^2) + ((31791981 6 + 3^(2/3)*(60925019*(-6)^(2/3) - 150526544*(-2)^(1/3)))*Log[6 + 3*(-2)^( 2/3)*3^(1/3)*x + x^2])/1902327490656 + ((317919816 + 6^(2/3)*(75263272*2^( 2/3) + 60925019*3^(2/3)))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/190232749065 6
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.13
| method | result | size |
| risch | \(-\frac {81}{125128 \left (x +\frac {2}{3}\right )}-\frac {490617 \ln \left (2+3 x \right )}{489281762}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9900753 \textit {\_Z}^{6}-603729730584 \textit {\_Z}^{5}-7778441183375403 \textit {\_Z}^{4}-24286225131714657096 \textit {\_Z}^{3}-249074830842532989021 \textit {\_Z}^{2}+62482523725340232084 \textit {\_Z} -936100721827630814201\right )}{\sum }\textit {\_R} \ln \left (-581639624628989671215509080386163823409178503 \textit {\_R}^{5}+35468189650054705024807095023880440332624355193966 \textit {\_R}^{4}+456906859878617438599177856847948959539967709968421774 \textit {\_R}^{3}+1426057311329693403373942358305451757577328424867159695508 \textit {\_R}^{2}+12496106101584673634988275193178073483743771974241813638447 \textit {\_R} +3711780171170710248762306868718224813196689272625039439882 x -10168834067890754363648654200390349438862480364030103971524\right )\right )}{60812208}\) | \(84\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (1962468 \textit {\_R}^{5}-2575233 \textit {\_R}^{4}+37885860 \textit {\_R}^{3}+587214738 \textit {\_R}^{2}-574433916 \textit {\_R} +509397788\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{11742762288}-\frac {243}{125128 \left (2+3 x \right )}-\frac {490617 \ln \left (2+3 x \right )}{489281762}\) | \(96\) |
Input:
int(1/(9*x^8+12*x^7+166*x^6+3132*x^5+4932*x^4+2592*x^3+2376*x^2+2592*x+864 ),x,method=_RETURNVERBOSE)
Output:
-81/125128/(x+2/3)-490617/489281762*ln(2+3*x)+1/60812208*sum(_R*ln(-581639 624628989671215509080386163823409178503*_R^5+35468189650054705024807095023 880440332624355193966*_R^4+45690685987861743859917785684794895953996770996 8421774*_R^3+1426057311329693403373942358305451757577328424867159695508*_R ^2+12496106101584673634988275193178073483743771974241813638447*_R+37117801 71170710248762306868718224813196689272625039439882*x-101688340678907543636 48654200390349438862480364030103971524),_R=RootOf(9900753*_Z^6-60372973058 4*_Z^5-7778441183375403*_Z^4-24286225131714657096*_Z^3-2490748308425329890 21*_Z^2+62482523725340232084*_Z-936100721827630814201))
Timed out. \[ \int \frac {1}{864+2592 x+2376 x^2+2592 x^3+4932 x^4+3132 x^5+166 x^6+12 x^7+9 x^8} \, dx=\text {Timed out} \] Input:
integrate(1/(9*x^8+12*x^7+166*x^6+3132*x^5+4932*x^4+2592*x^3+2376*x^2+2592 *x+864),x, algorithm="fricas")
Output:
Timed out
Time = 0.77 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.14 \[ \int \frac {1}{864+2592 x+2376 x^2+2592 x^3+4932 x^4+3132 x^5+166 x^6+12 x^7+9 x^8} \, dx=- \frac {490617 \log {\left (x + \frac {2}{3} \right )}}{489281762} + \operatorname {RootSum} {\left (534921778156562570512850938232832 t^{6} - 536381566647968085631001690112 t^{5} - 113640485675194290961317888 t^{4} - 5834582073061472305152 t^{3} - 983985748671744 t^{2} + 4059072 t - 1, \left ( t \mapsto t \log {\left (- \frac {3455750960438682665382725466807609018667495372212774674182519079033868698426615638032740573438585745071222349672612728118706176 t^{6}}{26588246774330856011112805591206001235544173983023377857033288104759495101416203768215908243838411} + \frac {85386916100459442179756844543192772662925680870357731517876453983561313087175523922165948277263533379614391972382900224 t^{5}}{26588246774330856011112805591206001235544173983023377857033288104759495101416203768215908243838411} + \frac {4208788734546251081355806817029177430401212489593741558375531485051007187670489447525037884824679550271713897114626752512 t^{4}}{26588246774330856011112805591206001235544173983023377857033288104759495101416203768215908243838411} + \frac {773743328211408687735928382971741157348346732153516561676592332878344549740957533802268264343331007245717174104207360 t^{3}}{26588246774330856011112805591206001235544173983023377857033288104759495101416203768215908243838411} + \frac {712891082341168850680045281424427993634194541364334319324325766749577944485225815038394351529193544496127289856 t^{2}}{501665033477940679454958596060490589349890075151384487868552605750179152856909505060677514034687} + \frac {187703702755305683110379409050647388786581835024695493483914677097863984075373818733422005751088992341688 t}{916836095666581241762510537627793146053247378724944064035630624302051555221248405800548560132359} + x - \frac {218504985171110848271388000396878344823358304542717602017540598964205204505619164470338751065447302}{79764740322992568033338416773618003706632521949070133571099864314278485304248611304647724731515233} \right )} \right )\right )} - \frac {243}{375384 x + 250256} \] Input:
integrate(1/(9*x**8+12*x**7+166*x**6+3132*x**5+4932*x**4+2592*x**3+2376*x* *2+2592*x+864),x)
Output:
-490617*log(x + 2/3)/489281762 + RootSum(534921778156562570512850938232832 *_t**6 - 536381566647968085631001690112*_t**5 - 11364048567519429096131788 8*_t**4 - 5834582073061472305152*_t**3 - 983985748671744*_t**2 + 4059072*_ t - 1, Lambda(_t, _t*log(-345575096043868266538272546680760901866749537221 27746741825190790338686984266156380327405734385857450712223496726127281187 06176*_t**6/26588246774330856011112805591206001235544173983023377857033288 104759495101416203768215908243838411 + 85386916100459442179756844543192772 66292568087035773151787645398356131308717552392216594827726353337961439197 2382900224*_t**5/265882467743308560111128055912060012355441739830233778570 33288104759495101416203768215908243838411 + 420878873454625108135580681702 91774304012124895937415583755314850510071876704894475250378848246795502717 13897114626752512*_t**4/26588246774330856011112805591206001235544173983023 377857033288104759495101416203768215908243838411 + 77374332821140868773592 83829717411573483467321535165616765923328783445497409575338022682643433310 07245717174104207360*_t**3/26588246774330856011112805591206001235544173983 023377857033288104759495101416203768215908243838411 + 71289108234116885068 00452814244279936341945413643343193243257667495779444852258150383943515291 93544496127289856*_t**2/50166503347794067945495859606049058934989007515138 4487868552605750179152856909505060677514034687 + 1877037027553056831103794 09050647388786581835024695493483914677097863984075373818733422005751088...
\[ \int \frac {1}{864+2592 x+2376 x^2+2592 x^3+4932 x^4+3132 x^5+166 x^6+12 x^7+9 x^8} \, dx=\int { \frac {1}{9 \, x^{8} + 12 \, x^{7} + 166 \, x^{6} + 3132 \, x^{5} + 4932 \, x^{4} + 2592 \, x^{3} + 2376 \, x^{2} + 2592 \, x + 864} \,d x } \] Input:
integrate(1/(9*x^8+12*x^7+166*x^6+3132*x^5+4932*x^4+2592*x^3+2376*x^2+2592 *x+864),x, algorithm="maxima")
Output:
-243/125128/(3*x + 2) + 1/1957127048*integrate((1962468*x^5 - 2575233*x^4 + 37885860*x^3 + 587214738*x^2 - 574433916*x + 509397788)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x) - 490617/489281762*log(3*x + 2)
\[ \int \frac {1}{864+2592 x+2376 x^2+2592 x^3+4932 x^4+3132 x^5+166 x^6+12 x^7+9 x^8} \, dx=\int { \frac {1}{9 \, x^{8} + 12 \, x^{7} + 166 \, x^{6} + 3132 \, x^{5} + 4932 \, x^{4} + 2592 \, x^{3} + 2376 \, x^{2} + 2592 \, x + 864} \,d x } \] Input:
integrate(1/(9*x^8+12*x^7+166*x^6+3132*x^5+4932*x^4+2592*x^3+2376*x^2+2592 *x+864),x, algorithm="giac")
Output:
integrate(1/(9*x^8 + 12*x^7 + 166*x^6 + 3132*x^5 + 4932*x^4 + 2592*x^3 + 2 376*x^2 + 2592*x + 864), x)
Time = 21.94 (sec) , antiderivative size = 480, normalized size of antiderivative = 0.74 \[ \int \frac {1}{864+2592 x+2376 x^2+2592 x^3+4932 x^4+3132 x^5+166 x^6+12 x^7+9 x^8} \, dx=\text {Too large to display} \] Input:
int(1/(2592*x + 2376*x^2 + 2592*x^3 + 4932*x^4 + 3132*x^5 + 166*x^6 + 12*x ^7 + 9*x^8 + 864),x)
Output:
symsum(log(x/15657016384 - (241256*root(z^6 - (490617*z^5)/489281762 - (68 2182427*z^4)/3211128804227328 - (17022113*z^3)/1560608598854481408 - (8036 21*z^2)/436870528728928107429888 + (29*z)/3821743385320663083796660224 - 1 /534921778156562570512850938232832, z, k))/733922643 - (1678771*root(z^6 - (490617*z^5)/489281762 - (682182427*z^4)/3211128804227328 - (17022113*z^3 )/1560608598854481408 - (803621*z^2)/436870528728928107429888 + (29*z)/382 1743385320663083796660224 - 1/534921778156562570512850938232832, z, k)*x)/ 6605303787 + (305207778434812192*root(z^6 - (490617*z^5)/489281762 - (6821 82427*z^4)/3211128804227328 - (17022113*z^3)/1560608598854481408 - (803621 *z^2)/436870528728928107429888 + (29*z)/3821743385320663083796660224 - 1/5 34921778156562570512850938232832, z, k)^2*x)/4815266460723 + (244884659349 9859523392*root(z^6 - (490617*z^5)/489281762 - (682182427*z^4)/32111288042 27328 - (17022113*z^3)/1560608598854481408 - (803621*z^2)/4368705287289281 07429888 + (29*z)/3821743385320663083796660224 - 1/53492177815656257051285 0938232832, z, k)^3*x)/6605303787 + (1533157891304935213946880*root(z^6 - (490617*z^5)/489281762 - (682182427*z^4)/3211128804227328 - (17022113*z^3) /1560608598854481408 - (803621*z^2)/436870528728928107429888 + (29*z)/3821 743385320663083796660224 - 1/534921778156562570512850938232832, z, k)^4*x) /244640881 + (260047433592718410792960*root(z^6 - (490617*z^5)/489281762 - (682182427*z^4)/3211128804227328 - (17022113*z^3)/1560608598854481408 ...
\[ \int \frac {1}{864+2592 x+2376 x^2+2592 x^3+4932 x^4+3132 x^5+166 x^6+12 x^7+9 x^8} \, dx=\int \frac {1}{9 x^{8}+12 x^{7}+166 x^{6}+3132 x^{5}+4932 x^{4}+2592 x^{3}+2376 x^{2}+2592 x +864}d x \] Input:
int(1/(9*x^8+12*x^7+166*x^6+3132*x^5+4932*x^4+2592*x^3+2376*x^2+2592*x+864 ),x)
Output:
int(1/(9*x**8 + 12*x**7 + 166*x**6 + 3132*x**5 + 4932*x**4 + 2592*x**3 + 2 376*x**2 + 2592*x + 864),x)