Is there a general method how to solve integrals including polylogarithms? I have, for example, a following peculiar integral:
$$\int_0^1 \frac{\log (1-q)}{(1-q)^{19}} \big[19122427017242392409179196696957 -2335312823676226739258108857200 \pi ^2-290355604888434431371406862640901 q+32737624286330384009900385643800 \pi ^2 q+2089178147864338015378359942349146 q^2-217908181258375090989657061033200 \pi ^2 q^2-9513735107311961105716228372823832 q^3+923058652929127659169477549327800 \pi ^2 q^3+30860609840601979343569796243664360 q^4-2809234642150007839007261169271200 \pi ^2 q^4-75991613070307679674084684299763488 q^5+6560138082212689647470512926380400 \pi ^2 q^5+147604422934589574241111368769031514 q^6-12225064353406525792601847884728800 \pi ^2 q^6-231661749521735051694587401454488374 q^7+18614430483751971866343908315754000 \pi ^2 q^7+298158588677298781555571530676473698 q^8-23467372646806215931232745831715200 \pi ^2 q^8-317258462849638359532191702700757050 q^9+24645135112158586773581379231285600 \pi ^2 q^9+279929745859987197112521991075600990 q^{10}-21578983058687084626530891022466400 \pi ^2 q^{10}-204525346750399538308156387121441190 q^{11}+15702582554471282285512462526560800 \pi ^2 q^{11}+123049277403587430345156505609140720 q^{12}-9430354354307541344225967691260000 \pi ^2 q^{12}-60342329966544036497302398449157240 q^{13}+4622406501952432638430248014413200 \pi ^2 q^{13}+23738252991464115594141538592264430 q^{14}-1818869679546039116051640669453600 \pi ^2 q^{14}-7312550577333389628853566957018834 q^{15}+560636064795229527948416643193200 \pi ^2 q^{15}+1699354671638585927650255051925577 q^{16}-130382484017845412425019267350800 \pi ^2 q^{16}-280173803150789489806907495704617 q^{17}+21513060264068052978453696664200 \pi ^2 q^{17}+29212691355365195449956724538624 q^{18}-2244803977761215443782290650800 \pi ^2 q^{18}-1448394095369360889798167890490 q^{19}+111380159353759803311707564200 \pi ^2 q^{19}-864839351159251344060440256000 \pi ^2 \log (q)+9614284037777341680385275648000 \pi ^2 q \log (q)-48596947298066064189458694528000 \pi ^2 q^2 \log (q)+149016194597643828944407014096000 \pi ^2 q^3 \log (q)-311515153604740784777334800544000 \pi ^2 q^4 \log (q)+472522854921446523354235979904000 \pi ^2 q^5 \log (q)-540131604846004142202578675328000 \pi ^2 q^6 \log (q)+476835369490723415021630300520000 \pi ^2 q^7 \log (q)-330431763957397218290451028752000 \pi ^2 q^8 \log (q)+181570501304738210584142319096000 \pi ^2 q^9 \log (q)-79469952891751512800087794416000 \pi ^2 q^{10} \log (q)+27618312749376667202083351152000 \pi ^2 q^{11} \log (q)-7469792604527284717674468480000 \pi ^2 q^{12} \log (q)+1441811254291252370952714288000 \pi ^2 q^{13} \log (q)-47548846798971176802583440000 \pi ^2 q^{14} \log (q)-242992965510346357364274216000 \pi ^2 q^{15} \log (q)+315652971889245500921659248000 \pi ^2 q^{16} \log (q)-286658885174869914970791528000 \pi ^2 q^{17} \log (q)+190547414679020492013416496000 \pi ^2 q^{18} \log (q)-91723410310948362363008592000 \pi ^2 q^{19} \log (q)+31251841079492830435486848000 \pi ^2 q^{20} \log (q)-7178196236973842421012672000 \pi ^2 q^{21} \log (q)+1000683546793492378825008000 \pi ^2 q^{22} \log (q)-64152027504348590478672000 \pi ^2 q^{23} \log (q)+116396280 \log (1-q) \left((-1+q)^2 \left(-257586022317616038442244+3541119036337161972270259 q-22985344007557567401249245 q^2+94080232161007231115696965 q^3-272947169011867587813604265 q^4+596855442237497586986976649 q^5-1019309093239565006695750169 q^6+1388155171326881304329317345 q^7-1524633136813377308169137885 q^8+1356305480530438789865344115 q^9-975804898738457785419837691 q^{10}+563741882769447393607104851 q^{11}-257984481217331023678963375 q^{12}+91454555996476460965835495 q^{13}-24229819048292743306257775 q^{14}+4517255786475215854979111 q^{15}-528777243429852450045991 q^{16}+29245925849336575939850 q^{17}\right)+17463600 \left(2552782515875592-28378884658059936 q+143445643657566096 q^2-439857339564018222 q^3+919512319237712508 q^4-1394765491161108528 q^5+1594329068484771696 q^6-1407494920941509015 q^7+975349269884284614 q^8-535949249427887597 q^9+234574786644003962 q^{10}-81522129873123914 q^{11}+22048899523933360 q^{12}-4255854634983666 q^{13}+140351921539830 q^{14}+717252508230687 q^{15}-931726090681386 q^{16}+846143031204471 q^{17}-562446780417522 q^{18}+270743829850494 q^{19}-92247367548336 q^{20}+21188182319304 q^{21}-2953759514706 q^{22}+189360225054 q^{23}\right) \log ^2(q)\right)+2032698075408000 \left(2552782515875592-28378884658059936 q+143445643657566096 q^2-439857339564018222 q^3+919512319237712508 q^4-1394765491161108528 q^5+1594329068484771696 q^6-1407494920941509015 q^7+975349269884284614 q^8-535949249427887597 q^9+234574786644003962 q^{10}-81522129873123914 q^{11}+22048899523933360 q^{12}-4255854634983666 q^{13}+140351921539830 q^{14}+717252508230687 q^{15}-931726090681386 q^{16}+846143031204471 q^{17}-562446780417522 q^{18}+270743829850494 q^{19}-92247367548336 q^{20}+21188182319304 q^{21}-2953759514706 q^{22}+189360225054 q^{23}\right) \log (q) \text{Li}_2(1-q)\big] \mathrm dq$$
This is just a comment, but too long to be written in the comments section.
As I like to proceed when meeting those nasty things, I would suggest you to use Dirac's Elegance.
What I mean is that the integrand function is full of large numbers and patterns which could be written in a shorter way. For example, if you just name $a_0, \ldots, a_{19}$ the first terms and $b_0, \ldots, b_{23}$ the next terms, the first two parts of the integrand function can be written as
$$a_0 + a_1q + a_2q^2 + \ldots + a_{19}q^{19} = \sum_{k = 0}^{19} a_kq^k$$
$$b_0\log(q) + b_1q \log(q) + \ldots + b_{23}q^{23}\log(q) = \log(q)\sum_{k = 0}^{23} b_kq^k$$
The other parts I leave them to you to be written in a compact elegant way.
When you will have finished, please rewrite the whole question by making use of this form so we could go on by investigating the eventual absolute convergence and split into $N$ parts the integrand and make try to get out of this tunnel!