∫ (1 − x)2020xdx [227]

Optimal. Leaf size=23 \[ -\frac {(1-x)^{2021}}{2021}+\frac {(1-x)^{2022}}{2022} \]

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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \begin {gather*} \frac {(1-x)^{2022}}{2022}-\frac {(1-x)^{2021}}{2021} \end {gather*}

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d

*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]

&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

\begin {gather*} \begin {aligned} \text {Integral} &=\int \left ((1-x)^{2020}-(1-x)^{2021}\right ) \, dx\\ &=-\frac {(1-x)^{2021}}{2021}+\frac {(1-x)^{2022}}{2022}\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(11128\) vs. \(2(23)=46\).
time = 0.02, size = 11128, normalized size = 483.83 \begin {gather*} \text {Result too large to show} \end {gather*}

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(10106\) vs. \(2(19)=38\).
time = 6.80, size = 10107, normalized size = 439.43


method	result	size

gosper	\(\text {Expression too large to display}\)	\(10106\)
default	\(\text {Expression too large to display}\)	\(10107\)
risch	\(\text {Expression too large to display}\)	\(10107\)
parallelrisch	\(\text {Expression too large to display}\)	\(10107\)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 10106 vs. \(2 (15) = 30\).
time = 2.62, size = 10106, normalized size = 439.39 \begin {gather*} \text {Too large to display} \end {gather*}

1/2022*x^2022 - 2020/2021*x^2021 + 2019/2*x^2020 - 2038180/3*x^2019 + 685507705/2*x^2018 - 138266870112*x^2017

+ 278745941561035/6*x^2016 - 13373165093095968*x^2015 + 3366693482174180730*x^2014 - 2259050769046992147760/3

*x^2013 + 151506967484463186448138*x^2012 - 27698221479285170723685360*x^2011 + 13918352848288375491988868066/

3*x^2010 - 716973434466946772290161333280*x^2009 + 102834449953653272749024785712470*x^2008 - 4127773796381795

2963643626311862816/3*x^2007 + 1725065037047525925910348951124302995*x^2006 - 20345614934698381583634515569589

3355528*x^2005 + 67954336961055157115435769332338918346705/3*x^2004 - 2387938649240603820483180755599043532230

840*x^2003 + 239032599150157038818882365583996835022772313*x^2002 - 683291587775561350346830150494983456436910

36000/3*x^2001 + 2070580051371710322595432011477108583912261365977*x^2000 - 1799603689949617080081637844544560

51967579363996000*x^1999 + 44945090897709842545903018770867237413834629868000750/3*x^1998 - 119673765355228865

7999009221047362441958082554435826640*x^1997 + 91872606035477133069946098385058726668676396410333163850*x^1996

- 20365089221038364860461105557164855112824957167732656091280/3*x^1995 + 483428305875789720518480870193807402

556007179708207427412550*x^1994 - 33223185208528449708566031169292044668543059434670067575119200*x^1993 + 6618

056825708284424494058802194502943955470785618233551823806130/3*x^1992 - 14168330984944992315661556907483236515

9751152400066575169519360800*x^1991 + 17621857212672637915910494927062643669496400090086408425321559396505/2*x

^1990 - 1593175688015870842174958343408086884084466802056598612815556642435700/3*x^1989 + 62102597379986860750

837045727224787177242246424432593713740407893495825/2*x^1988 - 17628261391368751458545603361475708788554041669

49882699385944631188781540*x^1987 + 58349530411656321466128126930442835872164239998849383966973362576465560642

5/6*x^1986 - 5217287769004881602844605550783515840233337199837209592027677186658756829760*x^1985 + 54479454231

7071048217452589725683577666003679953979611700628832352808384574675/2*x^1984 - 4155105010316363470369118680218

8386365900428229156960691552084059020136098420800/3*x^1983 + 6862846695022287208632201228234657269342964510698

22398649175737920375706771597260*x^1982 - 33159258142694208431400580160346412382904367352290272569444839374856

325440787301600*x^1981 + 4689665312531032901885079475470487545437442238944856864753931415596946731664397417220

/3*x^1980 - 71944537083598867559995458754564932733441509032019128051505486496979940764694848704800*x^1979 + 13

9072024843107780705402031845077308159947349595909730324277836129527455327507705407659100/43*x^1978 - 426271818

247820766649746184002365435930081749751131709578673349952100070121367067635629120/3*x^1977 + 61037166458919525

97625978064311726302575467010042704830898860585684816094929504848447260500*x^1976 - 25648589968398224302894945

3930814576862452231948083118646054803912483168278678210826498756160*x^1975 + 148726516837958557567683275802954

8945380202456735560693973386314041183556179045263649325953362750/141*x^1974 - 42471753124966226272315027135388

7556337666967588520119979077229931100423254663277085869347626000*x^1973 + 167508551250066466837659278061499484

07682165955575943587551682428429558795593704083730408309017030*x^1972 - 19421133501605879757352561249423572857

04479230932630455924561374198235807188773849423679810642603600/3*x^1971 + 245253994485385777510184165040761077

95565624569983127045968938931942467609928105048335642175722600770*x^1970 - 91114149166808287319095374579226028

8041014055912318140921449806736334593989735869641666120045511985600*x^1969 + 996181107013548253662671423156497

66191550146281382513073227650599463042519701580483490702384175388634950/3*x^1968 - 118756832184771006013378257

9617546248169131186934546393497152175140038712256681143811397373961838166297920*x^1967 + 416921199410631247045

88635432001680068686948234595232029842003483354493613505235489103524351070598365583100*x^1966 - 43118418140434

31702808371481379275615951136815593323696241989019901217925493856686652692291654967727428692640/3*x^1965 + 486

69282341048818064483160314590423417580355254067612212045974192124530040492736873323257465234124412330158300*x^

1964 - 1619284346477309911045022289431712708011386742573680951279235099171611387560318575857463878120606538923

403623200*x^1963 + 1588517531232582899057819504815718772136971614804550999524768755825806639797236345464954975

37824209913683068641740/3*x^1962 - 170223063862965036517691321260730218847531427986813824765787744791278453044

0047681486359322893844602835250538849600*x^1961 + 538124384391789517216364764700378495716308332968001976963812

89878409357631286674894258693947248770016518232676054020*x^1960 - 50199304491909934476552632057058959756935320

82123147930077263631604414876259419200957697187279167206713621895919662400/3*x^1959 + 102385638080321174949263

623772645625178023720185120265033712123617268026415416133246729145608800125063372676309620002775/2*x^1958 - 15

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RecursionError} \end {gather*}

Exception raised: RecursionError >> maximum recursion depth exceeded

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 11171 vs. \(2 (12) = 24\).
time = 2.59, size = 11171, normalized size = 485.70 \begin {gather*} \text {Too large to display} \end {gather*}

x**2022/2022 - 2020*x**2021/2021 + 2019*x**2020/2 - 2038180*x**2019/3 + 685507705*x**2018/2 - 138266870112*x**

2017 + 278745941561035*x**2016/6 - 13373165093095968*x**2015 + 3366693482174180730*x**2014 - 22590507690469921

47760*x**2013/3 + 151506967484463186448138*x**2012 - 27698221479285170723685360*x**2011 + 13918352848288375491

988868066*x**2010/3 - 716973434466946772290161333280*x**2009 + 102834449953653272749024785712470*x**2008 - 412

77737963817952963643626311862816*x**2007/3 + 1725065037047525925910348951124302995*x**2006 - 20345614934698381

5836345155695893355528*x**2005 + 67954336961055157115435769332338918346705*x**2004/3 - 23879386492406038204831

80755599043532230840*x**2003 + 239032599150157038818882365583996835022772313*x**2002 - 68329158777556135034683

015049498345643691036000*x**2001/3 + 2070580051371710322595432011477108583912261365977*x**2000 - 1799603689949

61708008163784454456051967579363996000*x**1999 + 44945090897709842545903018770867237413834629868000750*x**1998

/3 - 1196737653552288657999009221047362441958082554435826640*x**1997 + 918726060354771330699460983850587266686

76396410333163850*x**1996 - 20365089221038364860461105557164855112824957167732656091280*x**1995/3 + 4834283058

75789720518480870193807402556007179708207427412550*x**1994 - 3322318520852844970856603116929204466854305943467

0067575119200*x**1993 + 6618056825708284424494058802194502943955470785618233551823806130*x**1992/3 - 141683309

849449923156615569074832365159751152400066575169519360800*x**1991 + 176218572126726379159104949270626436694964

00090086408425321559396505*x**1990/2 - 1593175688015870842174958343408086884084466802056598612815556642435700*

x**1989/3 + 62102597379986860750837045727224787177242246424432593713740407893495825*x**1988/2 - 17628261391368

75145854560336147570878855404166949882699385944631188781540*x**1987 + 5834953041165632146612812693044283587216

42399988493839669733625764655606425*x**1986/6 - 52172877690048816028446055507835158402333371998372095920276771

86658756829760*x**1985 + 544794542317071048217452589725683577666003679953979611700628832352808384574675*x**198

4/2 - 41551050103163634703691186802188386365900428229156960691552084059020136098420800*x**1983/3 + 68628466950

2228720863220122823465726934296451069822398649175737920375706771597260*x**1982 - 33159258142694208431400580160

346412382904367352290272569444839374856325440787301600*x**1981 + 468966531253103290188507947547048754543744223

8944856864753931415596946731664397417220*x**1980/3 - 719445370835988675599954587545649327334415090320191280515

05486496979940764694848704800*x**1979 + 1390720248431077807054020318450773081599473495959097303242778361295274

55327507705407659100*x**1978/43 - 4262718182478207666497461840023654359300817497511317095786733499521000701213

67067635629120*x**1977/3 + 61037166458919525976259780643117263025754670100427048308988605856848160949295048484

47260500*x**1976 - 2564858996839822430289494539308145768624522319480831186460548039124831682786782108264987561

60*x**1975 + 1487265168379585575676832758029548945380202456735560693973386314041183556179045263649325953362750

*x**1974/141 - 42471753124966226272315027135388755633766696758852011997907722993110042325466327708586934762600

0*x**1973 + 16750855125006646683765927806149948407682165955575943587551682428429558795593704083730408309017030

*x**1972 - 194211335016058797573525612494235728570447923093263045592456137419823580718877384942367981064260360

0*x**1971/3 + 245253994485385777510184165040761077955656245699831270459689389319424676099281050483356421757226

00770*x**1970 - 9111414916680828731909537457922602880410140559123181409214498067363345939897358696416661200455

11985600*x**1969 + 9961811070135482536626714231564976619155014628138251307322765059946304251970158048349070238

4175388634950*x**1968/3 - 118756832184771006013378257961754624816913118693454639349715217514003871225668114381

1397373961838166297920*x**1967 + 41692119941063124704588635432001680068686948234595232029842003483354493613505

235489103524351070598365583100*x**1966 - 431184181404343170280837148137927561595113681559332369624198901990121

7925493856686652692291654967727428692640*x**1965/3 + 486692823410488180644831603145904234175803552540676122120

45974192124530040492736873323257465234124412330158300*x**1964 - 1619284346477309911045022289431712708011386742

573680951279235099171611387560318575857463878120606538923403623200*x**1963 + 158851753123258289905781950481571

877213697161480455099952476875582580663979723634546495497537824209913683068641740*x**1962/3 - 1702230638629650

365176913212607302188475314279868138247657877447912784530440047681486359322893844602835250538849600*x**1961 +

53812438439178951721636476470037849571630833296800197696381289878409357631286674894258693947248770016518232676

054020*x**1960 - 501993044919099344765526320570589597569353208212314793007726363160441487625941920095769718727

9167206713621895919662400*x**1959/3 + 102385638080321174949263623772645625178023720185120265033712123617268026

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 10106 vs. \(2 (15) = 30\).
time = 1.99, size = 10106, normalized size = 439.39 \begin {gather*} \text {Too large to display} \end {gather*}