\(\int (1-x)^{2020} x \, dx\) [227]

Optimal result

Integrand size = 9, antiderivative size = 23 \[ \int (1-x)^{2020} x \, dx=-\frac {(1-x)^{2021}}{2021}+\frac {(1-x)^{2022}}{2022} \] Output:

-1/2021*(1-x)^2021+1/2022*(1-x)^2022
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(11128\) vs. \(2(23)=46\).

Time = 0.07 (sec) , antiderivative size = 11128, normalized size of antiderivative = 483.83 \[ \int (1-x)^{2020} x \, dx=\text {Result too large to show} \] Input:

Integrate[(1 - x)^2020*x,x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.16 (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

\(\displaystyle \frac {(1-x)^{2022}}{2022}-\frac {(1-x)^{2021}}{2021}\)

Input:

Int[(1 - x)^2020*x,x]
 

Output:

-1/2021*(1 - x)^2021 + (1 - x)^2022/2022
 

Defintions of rubi rules used

rule 49

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(10105\) vs. \(2(19)=38\).

Time = 8.50 (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	\(\text {Expression too large to display}\)	\(10107\)
orering	\(\text {Expression too large to display}\)	\(10118\)

Input:

int((1-x)^2020*x,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-2)]

Exception generated. \[ \int (1-x)^{2020} x \, dx=\text {Exception raised: RecursionError} \] Input:

integrate((1-x)^2020*x,x, algorithm="fricas")
 

Output:

Exception raised: RecursionError >> maximum recursion depth exceeded
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11171 vs. \(2 (12) = 24\).

Time = 2.73 (sec) , antiderivative size = 11171, normalized size of antiderivative = 485.70 \[ \int (1-x)^{2020} x \, dx=\text {Too large to display} \] Input:

integrate((1-x)**2020*x,x)
 

Output:

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...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10106 vs. \(2 (15) = 30\).

Time = 2.68 (sec) , antiderivative size = 10106, normalized size of antiderivative = 439.39 \[ \int (1-x)^{2020} x \, dx=\text {Too large to display} \] Input:

integrate((1-x)^2020*x,x, algorithm="maxima")
 

Output:

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...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10106 vs. \(2 (15) = 30\).

Time = 1.02 (sec) , antiderivative size = 10106, normalized size of antiderivative = 439.39 \[ \int (1-x)^{2020} x \, dx=\text {Too large to display} \] Input:

integrate((1-x)^2020*x,x, algorithm="giac")
 

Output:

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...
 

Mupad [B] (verification not implemented)

Time = 3.64 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int (1-x)^{2020} x \, dx=\frac {{\left (x-1\right )}^{2021}}{2021}+\frac {{\left (x-1\right )}^{2022}}{2022} \] Input:

int(x*(x - 1)^2020,x)
 

Output:

(x - 1)^2021/2021 + (x - 1)^2022/2022
 

Reduce [B] (verification not implemented)

Time = 1.12 (sec) , antiderivative size = 10105, normalized size of antiderivative = 439.35 \[ \int (1-x)^{2020} x \, dx =\text {Too large to display} \] Input:

int((1-x)^2020*x,x)
                                                                                    
                                                                                    
 

Output:

(x**2*(2021*x**2020 - 4084440*x**2019 + 4125283389*x**2018 - 2776315039720 
*x**2017 + 1400650593594855*x**2016 - 565022310571623744*x**2015 + 1898474 
49640565034695*x**2014 - 54648930972663135585216*x**2013 + 137578649805524 
66934277260*x**2012 - 3077175041260436542039875040*x**2011 + 6191274653604 
94401819230907756*x**2010 - 113187729542682637325852723596320*x**2009 + 18 
958940005707403829914604591574164*x**2008 - 292988469495866824098639726231 
8055360*x**2007 + 420229072026505860264525323872151581140*x**2006 - 562266 
35878366479901239020155209182265664*x**2005 + 7049412721423306890247456395 
509321465553690*x**2004 - 831415822972774178030222697635351753417661936*x* 
*2003 + 92564271908849126485419294939122786981637602570*x**2002 - 97582205 
48453056369459339796886778630807102888080*x**2001 + 9767976331883490331658 
87669429110874440828191726606*x**2000 - 9307483694548319956203360768173436 
9511936319451544000*x**1999 + 84613466978885421082939742884847680980312674 
06133103374*x**1998 - 7354012094038892112304569949493253870355383029538221 
52000*x**1997 + 6122213534667905286327198063081189091220449641321669361550 
0*x**1996 - 4890422945210592613943947219459646819288909951564937002947680* 
x**1995 + 375433913404947956159278073098803854299932684227762881412798700* 
x**1994 - 2774038774279429284813653677911433605135496670585570842536141712 
0*x**1993 + 19755114016857914128893923737739265858638262116047607402391438 
98100*x**1992 - 1357652838736135856529661608641275074403038077435207136...