Integrand size = 9, antiderivative size = 23 \[ \int (1-x)^{2020} x \, dx=-\frac {(1-x)^{2021}}{2021}+\frac {(1-x)^{2022}}{2022} \]
Leaf count is larger than twice the leaf count of optimal. \(11128\) vs. \(2(23)=46\).
Time = 0.14 (sec) , antiderivative size = 11128, normalized size of antiderivative = 483.83 \[ \int (1-x)^{2020} x \, dx=\text {Result too large to show} \]
Time = 0.98 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {49, 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.
| \(\displaystyle \int (1-x)^{2020} x \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left ((1-x)^{2020}-(1-x)^{2021}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(1-x)^{2022}}{2022}-\frac {(1-x)^{2021}}{2021}\) |
3.3.27.3.1 Defintions of rubi rules used
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(10105\) vs. \(2(19)=38\).
Time = 9.33 (sec) , antiderivative size = 10106, normalized size of antiderivative = 439.39
| 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 | \(\text {Expression too large to display}\) | \(10107\) |
Exception generated. \[ \int (1-x)^{2020} x \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> System error: Heap exhausted (no more space for allocation).1998848 bytes available, 2179456 requested.PROCEED WITH CAUTION.
Leaf count of result is larger than twice the leaf count of optimal. 11171 vs. \(2 (12) = 24\).
Time = 2.60 (sec) , antiderivative size = 11171, normalized size of antiderivative = 485.70 \[ \int (1-x)^{2020} x \, dx=\text {Too large to display} \]
x**2022/2022 - 2020*x**2021/2021 + 2019*x**2020/2 - 2038180*x**2019/3 + 68 5507705*x**2018/2 - 138266870112*x**2017 + 278745941561035*x**2016/6 - 133 73165093095968*x**2015 + 3366693482174180730*x**2014 - 2259050769046992147 760*x**2013/3 + 151506967484463186448138*x**2012 - 27698221479285170723685 360*x**2011 + 13918352848288375491988868066*x**2010/3 - 716973434466946772 290161333280*x**2009 + 102834449953653272749024785712470*x**2008 - 4127773 7963817952963643626311862816*x**2007/3 + 172506503704752592591034895112430 2995*x**2006 - 203456149346983815836345155695893355528*x**2005 + 679543369 61055157115435769332338918346705*x**2004/3 - 23879386492406038204831807555 99043532230840*x**2003 + 239032599150157038818882365583996835022772313*x** 2002 - 68329158777556135034683015049498345643691036000*x**2001/3 + 2070580 051371710322595432011477108583912261365977*x**2000 - 179960368994961708008 163784454456051967579363996000*x**1999 + 449450908977098425459030187708672 37413834629868000750*x**1998/3 - 11967376535522886579990092210473624419580 82554435826640*x**1997 + 9187260603547713306994609838505872666867639641033 3163850*x**1996 - 20365089221038364860461105557164855112824957167732656091 280*x**1995/3 + 4834283058757897205184808701938074025560071797082074274125 50*x**1994 - 3322318520852844970856603116929204466854305943467006757511920 0*x**1993 + 66180568257082844244940588021945029439554707856182335518238061 30*x**1992/3 - 14168330984944992315661556907483236515975115240006657516...
Leaf count of result is larger than twice the leaf count of optimal. 10106 vs. \(2 (15) = 30\).
Time = 2.40 (sec) , antiderivative size = 10106, normalized size of antiderivative = 439.39 \[ \int (1-x)^{2020} x \, dx=\text {Too large to display} \]
1/2022*x^2022 - 2020/2021*x^2021 + 2019/2*x^2020 - 2038180/3*x^2019 + 6855 07705/2*x^2018 - 138266870112*x^2017 + 278745941561035/6*x^2016 - 13373165 093095968*x^2015 + 3366693482174180730*x^2014 - 2259050769046992147760/3*x ^2013 + 151506967484463186448138*x^2012 - 27698221479285170723685360*x^201 1 + 13918352848288375491988868066/3*x^2010 - 71697343446694677229016133328 0*x^2009 + 102834449953653272749024785712470*x^2008 - 41277737963817952963 643626311862816/3*x^2007 + 1725065037047525925910348951124302995*x^2006 - 203456149346983815836345155695893355528*x^2005 + 6795433696105515711543576 9332338918346705/3*x^2004 - 2387938649240603820483180755599043532230840*x^ 2003 + 239032599150157038818882365583996835022772313*x^2002 - 683291587775 56135034683015049498345643691036000/3*x^2001 + 207058005137171032259543201 1477108583912261365977*x^2000 - 179960368994961708008163784454456051967579 363996000*x^1999 + 44945090897709842545903018770867237413834629868000750/3 *x^1998 - 1196737653552288657999009221047362441958082554435826640*x^1997 + 91872606035477133069946098385058726668676396410333163850*x^1996 - 2036508 9221038364860461105557164855112824957167732656091280/3*x^1995 + 4834283058 75789720518480870193807402556007179708207427412550*x^1994 - 33223185208528 449708566031169292044668543059434670067575119200*x^1993 + 6618056825708284 424494058802194502943955470785618233551823806130/3*x^1992 - 14168330984944 9923156615569074832365159751152400066575169519360800*x^1991 + 176218572...
Leaf count of result is larger than twice the leaf count of optimal. 10106 vs. \(2 (15) = 30\).
Time = 1.10 (sec) , antiderivative size = 10106, normalized size of antiderivative = 439.39 \[ \int (1-x)^{2020} x \, dx=\text {Too large to display} \]
1/2022*x^2022 - 2020/2021*x^2021 + 2019/2*x^2020 - 2038180/3*x^2019 + 6855 07705/2*x^2018 - 138266870112*x^2017 + 278745941561035/6*x^2016 - 13373165 093095968*x^2015 + 3366693482174180730*x^2014 - 2259050769046992147760/3*x ^2013 + 151506967484463186448138*x^2012 - 27698221479285170723685360*x^201 1 + 13918352848288375491988868066/3*x^2010 - 71697343446694677229016133328 0*x^2009 + 102834449953653272749024785712470*x^2008 - 41277737963817952963 643626311862816/3*x^2007 + 1725065037047525925910348951124302995*x^2006 - 203456149346983815836345155695893355528*x^2005 + 6795433696105515711543576 9332338918346705/3*x^2004 - 2387938649240603820483180755599043532230840*x^ 2003 + 239032599150157038818882365583996835022772313*x^2002 - 683291587775 56135034683015049498345643691036000/3*x^2001 + 207058005137171032259543201 1477108583912261365977*x^2000 - 179960368994961708008163784454456051967579 363996000*x^1999 + 44945090897709842545903018770867237413834629868000750/3 *x^1998 - 1196737653552288657999009221047362441958082554435826640*x^1997 + 91872606035477133069946098385058726668676396410333163850*x^1996 - 2036508 9221038364860461105557164855112824957167732656091280/3*x^1995 + 4834283058 75789720518480870193807402556007179708207427412550*x^1994 - 33223185208528 449708566031169292044668543059434670067575119200*x^1993 + 6618056825708284 424494058802194502943955470785618233551823806130/3*x^1992 - 14168330984944 9923156615569074832365159751152400066575169519360800*x^1991 + 176218572...
Time = 31.77 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int (1-x)^{2020} x \, dx=\frac {{\left (x-1\right )}^{2021}}{2021}+\frac {{\left (x-1\right )}^{2022}}{2022} \]