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williamstein
GitHub Repository: williamstein/scratch
 Path: blob/main/2022-06-30-061806.ipynb
Views: 39
Kernel: SageMath 9.6
latex(integrate(sin(x^5),x))
-\frac{{\left({\left(\sqrt{5} - 1\right)} {\left(\Gamma\left(\frac{1}{5}, i \, x^{5}\right) + \Gamma\left(\frac{1}{5}, -i \, x^{5}\right)\right)} + \sqrt{2 \, \sqrt{5} + 10} {\left(i \, \Gamma\left(\frac{1}{5}, i \, x^{5}\right) - i \, \Gamma\left(\frac{1}{5}, -i \, x^{5}\right)\right)}\right)} x}{40 \, {\left(x^{5}\right)}^{\frac{1}{5}}}
−((5−1)(Γ(15,i x5)+Γ(15,−i x5))+2 5+10(i Γ(15,i x5)−i Γ(15,−i x5)))x40 (x5)15-\frac{{\left({\left(\sqrt{5} - 1\right)} {\left(\Gamma\left(\frac{1}{5}, i \, x^{5}\right) + \Gamma\left(\frac{1}{5}, -i \, x^{5}\right)\right)} + \sqrt{2 \, \sqrt{5} + 10} {\left(i \, \Gamma\left(\frac{1}{5}, i \, x^{5}\right) - i \, \Gamma\left(\frac{1}{5}, -i \, x^{5}\right)\right)}\right)} x}{40 \, {\left(x^{5}\right)}^{\frac{1}{5}}}
show(integrate(sin(x^5),x))

−((5−1)(Γ(15,i x5)+Γ(15,−i x5))+2 5+10(i Γ(15,i x5)−i Γ(15,−i x5)))x40 (x5)15\displaystyle -\frac{{\left({\left(\sqrt{5} - 1\right)} {\left(\Gamma\left(\frac{1}{5}, i \, x^{5}\right) + \Gamma\left(\frac{1}{5}, -i \, x^{5}\right)\right)} + \sqrt{2 \, \sqrt{5} + 10} {\left(i \, \Gamma\left(\frac{1}{5}, i \, x^{5}\right) - i \, \Gamma\left(\frac{1}{5}, -i \, x^{5}\right)\right)}\right)} x}{40 \, {\left(x^{5}\right)}^{\frac{1}{5}}}

v = [(0,0,0)]
for i in range(1000):
    v.append([a+random()-.5 for a in v[-1]])
line3d(v, color='red', thickness=3, spin=3)

show(random_matrix(QQ,15)^(-1))

(210706282924403113659792292440315542250129244031−57282946292440319176935029244031−285480292440313748050729244031−21990729229244031781444429244031900315629244031109853125292440316073637929244031−647969029244031849660272924403173260629244031−365170029244031−1425770629244031−35069340292440313568195829244031−5895842329244031273807029244031−186649972924403112155655429244031−934372829244031−385480029244031−6867404129244031−3435120429244031−71065429244031−4868246529244031−276712829244031−936727029244031−557397329244031−4901881292440311068846329244031−3309112558488062−134186129244031−1901367758488062387601372924403128697029244031−754421929244031−2963933558488062−936725629244031192478429244031−2122001558488062440787029244031−5836138829244031−6872323329244031−1247814702924403116028255929244031−41257022158488062766413329244031−1487826355848806248419568329244031−1801265229244031−3130459529244031−45222248558488062−13857796029244031972501829244031−37470649358488062−1837532829244031−5613027229244031−6865958529244031−1432654522924403117868573329244031−49626548558488062−704537929244031−1754228795848806257620843129244031−2190587629244031−3887651929244031−56180731358488062−157682530292440312549722229244031−46204868958488062−2578540229244031157302802924403114640026292440318216948029244031−7638654629244031147401591292440315767602292440315549765529244031−331317566292440318188716292440312241761429244031191440721292440317745895229244031−114684242924403114086063929244031−609422924403111269046292440319803338292440312702861129244031−397617122924403144211809292440311378378292440311583619429244031−1083612922924403114247822924403171436622924403146385098292440312905909029244031−5557118292440314580896929244031619726029244031−1200630829244031−573057629244031−25897370292440312973248029244031−5389097029244031−29334029244031−21776482292440311164354202924403148195629244031−1014664429244031−6508806029244031−3362711829244031262280029244031−4440341229244031108467629244031271247942924403127517787292440315458158429244031−707391272924403118942929758488062−2823493292440317524255958488062−227985867292440319066562292440311145474329244031213730121584880626099601429244031−755379029244031167637105584880621064891829244031−7456362429244031−6048619429244031−1515631382924403119710591029244031−28391653929244031−660697029244031−1086797332924403166706390629244031−2602731629244031−4549053029244031−32804949129244031−181725002292440313149083829244031−26421908529244031−1487664429244031−6809704229244031−7881806129244031−1871359742924403121882457729244031−69166407158488062−150258129244031−2335137055848806277305990929244031−2388977629244031−5280140129244031−80535533558488062−198095502292440312449279829244031−62627721558488062−1637208229244031246108442924403131525499292440317047796529244031−8312254529244031253022121584880621128435292440318874015558488062−285688675292440316167310292440311440039929244031306683117584880627160316529244031−59024002924403122840087558488062314290829244031306030282924403127375260292440314468269229244031−62809578292440316611080329244031−7277578292440312296822529244031−16482949429244031165192429244031128016102924403168661679292440314882027429244031−2242920292440315442674729244031399980229244031−8710702029244031−8463565829244031−1649767502924403121824337629244031−28944526829244031−229450429244031−1046328662924403168206483829244031−2916721229244031−4622735229244031−31859262829244031−187994796292440312316737029244031−26084052829244031−26849366292440318354011829244031808191742924403119316899529244031−2370028842924403134556680329244031−16694362924403112872112829244031−79175645229244031252360742924403150180498292440313969904602924403121185596829244031−2876688229244031314430831292440311837389629244031)\displaystyle \left(\begin{array}{rrrrrrrrrrrrrrr} \frac{21070628}{29244031} & \frac{13659792}{29244031} & \frac{55422501}{29244031} & -\frac{57282946}{29244031} & \frac{91769350}{29244031} & -\frac{285480}{29244031} & \frac{37480507}{29244031} & -\frac{219907292}{29244031} & \frac{7814444}{29244031} & \frac{9003156}{29244031} & \frac{109853125}{29244031} & \frac{60736379}{29244031} & -\frac{6479690}{29244031} & \frac{84966027}{29244031} & \frac{732606}{29244031} \\ -\frac{3651700}{29244031} & -\frac{14257706}{29244031} & -\frac{35069340}{29244031} & \frac{35681958}{29244031} & -\frac{58958423}{29244031} & \frac{2738070}{29244031} & -\frac{18664997}{29244031} & \frac{121556554}{29244031} & -\frac{9343728}{29244031} & -\frac{3854800}{29244031} & -\frac{68674041}{29244031} & -\frac{34351204}{29244031} & -\frac{710654}{29244031} & -\frac{48682465}{29244031} & -\frac{2767128}{29244031} \\ -\frac{9367270}{29244031} & -\frac{5573973}{29244031} & -\frac{4901881}{29244031} & \frac{10688463}{29244031} & -\frac{33091125}{58488062} & -\frac{1341861}{29244031} & -\frac{19013677}{58488062} & \frac{38760137}{29244031} & \frac{286970}{29244031} & -\frac{7544219}{29244031} & -\frac{29639335}{58488062} & -\frac{9367256}{29244031} & \frac{1924784}{29244031} & -\frac{21220015}{58488062} & \frac{4407870}{29244031} \\ -\frac{58361388}{29244031} & -\frac{68723233}{29244031} & -\frac{124781470}{29244031} & \frac{160282559}{29244031} & -\frac{412570221}{58488062} & \frac{7664133}{29244031} & -\frac{148782635}{58488062} & \frac{484195683}{29244031} & -\frac{18012652}{29244031} & -\frac{31304595}{29244031} & -\frac{452222485}{58488062} & -\frac{138577960}{29244031} & \frac{9725018}{29244031} & -\frac{374706493}{58488062} & -\frac{18375328}{29244031} \\ -\frac{56130272}{29244031} & -\frac{68659585}{29244031} & -\frac{143265452}{29244031} & \frac{178685733}{29244031} & -\frac{496265485}{58488062} & -\frac{7045379}{29244031} & -\frac{175422879}{58488062} & \frac{576208431}{29244031} & -\frac{21905876}{29244031} & -\frac{38876519}{29244031} & -\frac{561807313}{58488062} & -\frac{157682530}{29244031} & \frac{25497222}{29244031} & -\frac{462048689}{58488062} & -\frac{25785402}{29244031} \\ \frac{15730280}{29244031} & \frac{14640026}{29244031} & \frac{82169480}{29244031} & -\frac{76386546}{29244031} & \frac{147401591}{29244031} & \frac{5767602}{29244031} & \frac{55497655}{29244031} & -\frac{331317566}{29244031} & \frac{8188716}{29244031} & \frac{22417614}{29244031} & \frac{191440721}{29244031} & \frac{77458952}{29244031} & -\frac{11468424}{29244031} & \frac{140860639}{29244031} & -\frac{60942}{29244031} \\ \frac{11269046}{29244031} & \frac{9803338}{29244031} & \frac{27028611}{29244031} & -\frac{39761712}{29244031} & \frac{44211809}{29244031} & \frac{1378378}{29244031} & \frac{15836194}{29244031} & -\frac{108361292}{29244031} & \frac{1424782}{29244031} & \frac{7143662}{29244031} & \frac{46385098}{29244031} & \frac{29059090}{29244031} & -\frac{5557118}{29244031} & \frac{45808969}{29244031} & \frac{6197260}{29244031} \\ -\frac{12006308}{29244031} & -\frac{5730576}{29244031} & -\frac{25897370}{29244031} & \frac{29732480}{29244031} & -\frac{53890970}{29244031} & -\frac{293340}{29244031} & -\frac{21776482}{29244031} & \frac{116435420}{29244031} & \frac{481956}{29244031} & -\frac{10146644}{29244031} & -\frac{65088060}{29244031} & -\frac{33627118}{29244031} & \frac{2622800}{29244031} & -\frac{44403412}{29244031} & \frac{1084676}{29244031} \\ \frac{27124794}{29244031} & \frac{27517787}{29244031} & \frac{54581584}{29244031} & -\frac{70739127}{29244031} & \frac{189429297}{58488062} & -\frac{2823493}{29244031} & \frac{75242559}{58488062} & -\frac{227985867}{29244031} & \frac{9066562}{29244031} & \frac{11454743}{29244031} & \frac{213730121}{58488062} & \frac{60996014}{29244031} & -\frac{7553790}{29244031} & \frac{167637105}{58488062} & \frac{10648918}{29244031} \\ -\frac{74563624}{29244031} & -\frac{60486194}{29244031} & -\frac{151563138}{29244031} & \frac{197105910}{29244031} & -\frac{283916539}{29244031} & -\frac{6606970}{29244031} & -\frac{108679733}{29244031} & \frac{667063906}{29244031} & -\frac{26027316}{29244031} & -\frac{45490530}{29244031} & -\frac{328049491}{29244031} & -\frac{181725002}{29244031} & \frac{31490838}{29244031} & -\frac{264219085}{29244031} & -\frac{14876644}{29244031} \\ -\frac{68097042}{29244031} & -\frac{78818061}{29244031} & -\frac{187135974}{29244031} & \frac{218824577}{29244031} & -\frac{691664071}{58488062} & -\frac{1502581}{29244031} & -\frac{233513705}{58488062} & \frac{773059909}{29244031} & -\frac{23889776}{29244031} & -\frac{52801401}{29244031} & -\frac{805355335}{58488062} & -\frac{198095502}{29244031} & \frac{24492798}{29244031} & -\frac{626277215}{58488062} & -\frac{16372082}{29244031} \\ \frac{24610844}{29244031} & \frac{31525499}{29244031} & \frac{70477965}{29244031} & -\frac{83122545}{29244031} & \frac{253022121}{58488062} & \frac{1128435}{29244031} & \frac{88740155}{58488062} & -\frac{285688675}{29244031} & \frac{6167310}{29244031} & \frac{14400399}{29244031} & \frac{306683117}{58488062} & \frac{71603165}{29244031} & -\frac{5902400}{29244031} & \frac{228400875}{58488062} & \frac{3142908}{29244031} \\ \frac{30603028}{29244031} & \frac{27375260}{29244031} & \frac{44682692}{29244031} & -\frac{62809578}{29244031} & \frac{66110803}{29244031} & -\frac{7277578}{29244031} & \frac{22968225}{29244031} & -\frac{164829494}{29244031} & \frac{1651924}{29244031} & \frac{12801610}{29244031} & \frac{68661679}{29244031} & \frac{48820274}{29244031} & -\frac{2242920}{29244031} & \frac{54426747}{29244031} & \frac{3999802}{29244031} \\ -\frac{87107020}{29244031} & -\frac{84635658}{29244031} & -\frac{164976750}{29244031} & \frac{218243376}{29244031} & -\frac{289445268}{29244031} & -\frac{2294504}{29244031} & -\frac{104632866}{29244031} & \frac{682064838}{29244031} & -\frac{29167212}{29244031} & -\frac{46227352}{29244031} & -\frac{318592628}{29244031} & -\frac{187994796}{29244031} & \frac{23167370}{29244031} & -\frac{260840528}{29244031} & -\frac{26849366}{29244031} \\ \frac{83540118}{29244031} & \frac{80819174}{29244031} & \frac{193168995}{29244031} & -\frac{237002884}{29244031} & \frac{345566803}{29244031} & -\frac{1669436}{29244031} & \frac{128721128}{29244031} & -\frac{791756452}{29244031} & \frac{25236074}{29244031} & \frac{50180498}{29244031} & \frac{396990460}{29244031} & \frac{211855968}{29244031} & -\frac{28766882}{29244031} & \frac{314430831}{29244031} & \frac{18373896}{29244031} \end{array}\right)