Integrand size = 9, antiderivative size = 12 \[ \int \sin ^{99}(x) \sin (101 x) \, dx=\frac {1}{100} \sin ^{100}(x) \sin (100 x) \]
Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \sin ^{99}(x) \sin (101 x) \, dx=\frac {1}{100} \sin ^{100}(x) \sin (100 x) \]
Leaf count is larger than twice the leaf count of optimal. \(801\) vs. \(2(12)=24\).
Time = 0.97 (sec) , antiderivative size = 801, normalized size of antiderivative = 66.75, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4854, 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.
| \(\displaystyle \int \sin ^{99}(x) \sin (101 x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (x)^{99} \sin (101 x)dx\) |
| \(\Big \downarrow \) 4854 |
| \(\displaystyle \int \left (-\frac {\cos (2 x)}{633825300114114700748351602688}+\frac {99 \cos (4 x)}{633825300114114700748351602688}-\frac {4851 \cos (6 x)}{633825300114114700748351602688}+\frac {156849 \cos (8 x)}{633825300114114700748351602688}-\frac {470547 \cos (10 x)}{79228162514264337593543950336}+\frac {8940393 \cos (12 x)}{79228162514264337593543950336}-\frac {140066157 \cos (14 x)}{79228162514264337593543950336}+\frac {1860878943 \cos (16 x)}{79228162514264337593543950336}-\frac {42800215689 \cos (18 x)}{158456325028528675187087900672}+\frac {432757736411 \cos (20 x)}{158456325028528675187087900672}-\frac {3894819627699 \cos (22 x)}{158456325028528675187087900672}+\frac {31512631533201 \cos (24 x)}{158456325028528675187087900672}-\frac {115546315621737 \cos (26 x)}{79228162514264337593543950336}+\frac {773271496853163 \cos (28 x)}{79228162514264337593543950336}-\frac {4750096337812287 \cos (30 x)}{79228162514264337593543950336}+\frac {26917212580936293 \cos (32 x)}{79228162514264337593543950336}-\frac {565261464199662153 \cos (34 x)}{316912650057057350374175801344}+\frac {2759805972268938747 \cos (36 x)}{316912650057057350374175801344}-\frac {12572449429225165403 \cos (38 x)}{316912650057057350374175801344}+\frac {53598337040380968297 \cos (40 x)}{316912650057057350374175801344}-\frac {53598337040380968297 \cos (42 x)}{79228162514264337593543950336}+\frac {201631839342385547403 \cos (44 x)}{79228162514264337593543950336}-\frac {714876521304821486247 \cos (46 x)}{79228162514264337593543950336}+\frac {2393282266977011062653 \cos (48 x)}{79228162514264337593543950336}-\frac {15157454357521070063469 \cos (50 x)}{158456325028528675187087900672}+\frac {45472363072563210190407 \cos (52 x)}{158456325028528675187087900672}-\frac {129421341052679905926543 \cos (54 x)}{158456325028528675187087900672}+\frac {349916959142430856764357 \cos (56 x)}{158456325028528675187087900672}-\frac {449893233183125387268459 \cos (58 x)}{79228162514264337593543950336}+\frac {1101462743310410430898641 \cos (60 x)}{79228162514264337593543950336}-\frac {2570079734390957672096829 \cos (62 x)}{79228162514264337593543950336}+\frac {5720500053966970302409071 \cos (64 x)}{79228162514264337593543950336}-\frac {97248500917438495140954207 \cos (66 x)}{633825300114114700748351602688}+\frac {197443926105102399225573693 \cos (68 x)}{633825300114114700748351602688}-\frac {383273503615787010261407757 \cos (70 x)}{633825300114114700748351602688}+\frac {711793649572175876199757263 \cos (72 x)}{633825300114114700748351602688}-\frac {79088183285797319577750807 \cos (74 x)}{39614081257132168796771975168}+\frac {134663663432573814416170293 \cos (76 x)}{39614081257132168796771975168}-\frac {219714398232094118257962057 \cos (78 x)}{39614081257132168796771975168}+\frac {343655853645070287531684243 \cos (80 x)}{39614081257132168796771975168}-\frac {1030967560935210862595052729 \cos (82 x)}{79228162514264337593543950336}+\frac {1483587465736035143734344171 \cos (84 x)}{79228162514264337593543950336}-\frac {2048763643159286627061713379 \cos (86 x)}{79228162514264337593543950336}+\frac {2715802968839054366105061921 \cos (88 x)}{79228162514264337593543950336}-\frac {1728238252897580051157766677 \cos (90 x)}{39614081257132168796771975168}+\frac {2112291197985931173637270383 \cos (92 x)}{39614081257132168796771975168}-\frac {2479646188940006160356795667 \cos (94 x)}{39614081257132168796771975168}+\frac {2796196766251496308487450433 \cos (96 x)}{39614081257132168796771975168}-\frac {12116852653756484003445618543 \cos (98 x)}{158456325028528675187087900672}+\frac {12611418068195524166851562157 \cos (100 x)}{158456325028528675187087900672}-\frac {12611418068195524166851562157 \cos (102 x)}{158456325028528675187087900672}+\frac {12116852653756484003445618543 \cos (104 x)}{158456325028528675187087900672}-\frac {2796196766251496308487450433 \cos (106 x)}{39614081257132168796771975168}+\frac {2479646188940006160356795667 \cos (108 x)}{39614081257132168796771975168}-\frac {2112291197985931173637270383 \cos (110 x)}{39614081257132168796771975168}+\frac {1728238252897580051157766677 \cos (112 x)}{39614081257132168796771975168}-\frac {2715802968839054366105061921 \cos (114 x)}{79228162514264337593543950336}+\frac {2048763643159286627061713379 \cos (116 x)}{79228162514264337593543950336}-\frac {1483587465736035143734344171 \cos (118 x)}{79228162514264337593543950336}+\frac {1030967560935210862595052729 \cos (120 x)}{79228162514264337593543950336}-\frac {343655853645070287531684243 \cos (122 x)}{39614081257132168796771975168}+\frac {219714398232094118257962057 \cos (124 x)}{39614081257132168796771975168}-\frac {134663663432573814416170293 \cos (126 x)}{39614081257132168796771975168}+\frac {79088183285797319577750807 \cos (128 x)}{39614081257132168796771975168}-\frac {711793649572175876199757263 \cos (130 x)}{633825300114114700748351602688}+\frac {383273503615787010261407757 \cos (132 x)}{633825300114114700748351602688}-\frac {197443926105102399225573693 \cos (134 x)}{633825300114114700748351602688}+\frac {97248500917438495140954207 \cos (136 x)}{633825300114114700748351602688}-\frac {5720500053966970302409071 \cos (138 x)}{79228162514264337593543950336}+\frac {2570079734390957672096829 \cos (140 x)}{79228162514264337593543950336}-\frac {1101462743310410430898641 \cos (142 x)}{79228162514264337593543950336}+\frac {449893233183125387268459 \cos (144 x)}{79228162514264337593543950336}-\frac {349916959142430856764357 \cos (146 x)}{158456325028528675187087900672}+\frac {129421341052679905926543 \cos (148 x)}{158456325028528675187087900672}-\frac {45472363072563210190407 \cos (150 x)}{158456325028528675187087900672}+\frac {15157454357521070063469 \cos (152 x)}{158456325028528675187087900672}-\frac {2393282266977011062653 \cos (154 x)}{79228162514264337593543950336}+\frac {714876521304821486247 \cos (156 x)}{79228162514264337593543950336}-\frac {201631839342385547403 \cos (158 x)}{79228162514264337593543950336}+\frac {53598337040380968297 \cos (160 x)}{79228162514264337593543950336}-\frac {53598337040380968297 \cos (162 x)}{316912650057057350374175801344}+\frac {12572449429225165403 \cos (164 x)}{316912650057057350374175801344}-\frac {2759805972268938747 \cos (166 x)}{316912650057057350374175801344}+\frac {565261464199662153 \cos (168 x)}{316912650057057350374175801344}-\frac {26917212580936293 \cos (170 x)}{79228162514264337593543950336}+\frac {4750096337812287 \cos (172 x)}{79228162514264337593543950336}-\frac {773271496853163 \cos (174 x)}{79228162514264337593543950336}+\frac {115546315621737 \cos (176 x)}{79228162514264337593543950336}-\frac {31512631533201 \cos (178 x)}{158456325028528675187087900672}+\frac {3894819627699 \cos (180 x)}{158456325028528675187087900672}-\frac {432757736411 \cos (182 x)}{158456325028528675187087900672}+\frac {42800215689 \cos (184 x)}{158456325028528675187087900672}-\frac {1860878943 \cos (186 x)}{79228162514264337593543950336}+\frac {140066157 \cos (188 x)}{79228162514264337593543950336}-\frac {8940393 \cos (190 x)}{79228162514264337593543950336}+\frac {470547 \cos (192 x)}{79228162514264337593543950336}-\frac {156849 \cos (194 x)}{633825300114114700748351602688}+\frac {4851 \cos (196 x)}{633825300114114700748351602688}-\frac {99 \cos (198 x)}{633825300114114700748351602688}+\frac {\cos (200 x)}{633825300114114700748351602688}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sin (2 x)}{1267650600228229401496703205376}+\frac {99 \sin (4 x)}{2535301200456458802993406410752}-\frac {1617 \sin (6 x)}{1267650600228229401496703205376}+\frac {156849 \sin (8 x)}{5070602400912917605986812821504}-\frac {470547 \sin (10 x)}{792281625142643375935439503360}+\frac {2980131 \sin (12 x)}{316912650057057350374175801344}-\frac {20009451 \sin (14 x)}{158456325028528675187087900672}+\frac {1860878943 \sin (16 x)}{1267650600228229401496703205376}-\frac {4755579521 \sin (18 x)}{316912650057057350374175801344}+\frac {432757736411 \sin (20 x)}{3169126500570573503741758013440}-\frac {354074511609 \sin (22 x)}{316912650057057350374175801344}+\frac {10504210511067 \sin (24 x)}{1267650600228229401496703205376}-\frac {8888178124749 \sin (26 x)}{158456325028528675187087900672}+\frac {110467356693309 \sin (28 x)}{316912650057057350374175801344}-\frac {1583365445937429 \sin (30 x)}{792281625142643375935439503360}+\frac {26917212580936293 \sin (32 x)}{2535301200456458802993406410752}-\frac {33250674364686009 \sin (34 x)}{633825300114114700748351602688}+\frac {306645108029882083 \sin (36 x)}{1267650600228229401496703205376}-\frac {661707864696061337 \sin (38 x)}{633825300114114700748351602688}+\frac {53598337040380968297 \sin (40 x)}{12676506002282294014967032053760}-\frac {2552301763827665157 \sin (42 x)}{158456325028528675187087900672}+\frac {18330167212944140673 \sin (44 x)}{316912650057057350374175801344}-\frac {31081587882818325489 \sin (46 x)}{158456325028528675187087900672}+\frac {797760755659003687551 \sin (48 x)}{1267650600228229401496703205376}-\frac {15157454357521070063469 \sin (50 x)}{7922816251426433759354395033600}+\frac {3497874082504862322339 \sin (52 x)}{633825300114114700748351602688}-\frac {4793383001951107626909 \sin (54 x)}{316912650057057350374175801344}+\frac {49988137020347265252051 \sin (56 x)}{1267650600228229401496703205376}-\frac {15513559764935358181671 \sin (58 x)}{158456325028528675187087900672}+\frac {367154247770136810299547 \sin (60 x)}{1584563250285286751870879006720}-\frac {82905797883579279745059 \sin (62 x)}{158456325028528675187087900672}+\frac {5720500053966970302409071 \sin (64 x)}{5070602400912917605986812821504}-\frac {2946924270225408943665279 \sin (66 x)}{1267650600228229401496703205376}+\frac {11614348594417788189739629 \sin (68 x)}{2535301200456458802993406410752}-\frac {54753357659398144323058251 \sin (70 x)}{6338253001141147007483516026880}+\frac {79088183285797319577750807 \sin (72 x)}{5070602400912917605986812821504}-\frac {2137518467183711339939211 \sin (74 x)}{79228162514264337593543950336}+\frac {7087561233293358653482647 \sin (76 x)}{158456325028528675187087900672}-\frac {5633702518771644057896463 \sin (78 x)}{79228162514264337593543950336}+\frac {343655853645070287531684243 \sin (80 x)}{3169126500570573503741758013440}-\frac {25145550266712460063293969 \sin (82 x)}{158456325028528675187087900672}+\frac {70647022177906435415921151 \sin (84 x)}{316912650057057350374175801344}-\frac {47645666119983409931667753 \sin (86 x)}{158456325028528675187087900672}+\frac {246891178985368578736823811 \sin (88 x)}{633825300114114700748351602688}-\frac {192026472544175561239751853 \sin (90 x)}{396140812571321687967719751680}+\frac {91838747738518746679881321 \sin (92 x)}{158456325028528675187087900672}-\frac {52758429551915024688442461 \sin (94 x)}{79228162514264337593543950336}+\frac {932065588750498769495816811 \sin (96 x)}{1267650600228229401496703205376}-\frac {247282707219520081702971807 \sin (98 x)}{316912650057057350374175801344}+\frac {12611418068195524166851562157 \sin (100 x)}{15845632502852867518708790067200}-\frac {247282707219520081702971807 \sin (102 x)}{316912650057057350374175801344}+\frac {932065588750498769495816811 \sin (104 x)}{1267650600228229401496703205376}-\frac {52758429551915024688442461 \sin (106 x)}{79228162514264337593543950336}+\frac {91838747738518746679881321 \sin (108 x)}{158456325028528675187087900672}-\frac {192026472544175561239751853 \sin (110 x)}{396140812571321687967719751680}+\frac {246891178985368578736823811 \sin (112 x)}{633825300114114700748351602688}-\frac {47645666119983409931667753 \sin (114 x)}{158456325028528675187087900672}+\frac {70647022177906435415921151 \sin (116 x)}{316912650057057350374175801344}-\frac {25145550266712460063293969 \sin (118 x)}{158456325028528675187087900672}+\frac {343655853645070287531684243 \sin (120 x)}{3169126500570573503741758013440}-\frac {5633702518771644057896463 \sin (122 x)}{79228162514264337593543950336}+\frac {7087561233293358653482647 \sin (124 x)}{158456325028528675187087900672}-\frac {2137518467183711339939211 \sin (126 x)}{79228162514264337593543950336}+\frac {79088183285797319577750807 \sin (128 x)}{5070602400912917605986812821504}-\frac {54753357659398144323058251 \sin (130 x)}{6338253001141147007483516026880}+\frac {11614348594417788189739629 \sin (132 x)}{2535301200456458802993406410752}-\frac {2946924270225408943665279 \sin (134 x)}{1267650600228229401496703205376}+\frac {5720500053966970302409071 \sin (136 x)}{5070602400912917605986812821504}-\frac {82905797883579279745059 \sin (138 x)}{158456325028528675187087900672}+\frac {367154247770136810299547 \sin (140 x)}{1584563250285286751870879006720}-\frac {15513559764935358181671 \sin (142 x)}{158456325028528675187087900672}+\frac {49988137020347265252051 \sin (144 x)}{1267650600228229401496703205376}-\frac {4793383001951107626909 \sin (146 x)}{316912650057057350374175801344}+\frac {3497874082504862322339 \sin (148 x)}{633825300114114700748351602688}-\frac {15157454357521070063469 \sin (150 x)}{7922816251426433759354395033600}+\frac {797760755659003687551 \sin (152 x)}{1267650600228229401496703205376}-\frac {31081587882818325489 \sin (154 x)}{158456325028528675187087900672}+\frac {18330167212944140673 \sin (156 x)}{316912650057057350374175801344}-\frac {2552301763827665157 \sin (158 x)}{158456325028528675187087900672}+\frac {53598337040380968297 \sin (160 x)}{12676506002282294014967032053760}-\frac {661707864696061337 \sin (162 x)}{633825300114114700748351602688}+\frac {306645108029882083 \sin (164 x)}{1267650600228229401496703205376}-\frac {33250674364686009 \sin (166 x)}{633825300114114700748351602688}+\frac {26917212580936293 \sin (168 x)}{2535301200456458802993406410752}-\frac {1583365445937429 \sin (170 x)}{792281625142643375935439503360}+\frac {110467356693309 \sin (172 x)}{316912650057057350374175801344}-\frac {8888178124749 \sin (174 x)}{158456325028528675187087900672}+\frac {10504210511067 \sin (176 x)}{1267650600228229401496703205376}-\frac {354074511609 \sin (178 x)}{316912650057057350374175801344}+\frac {432757736411 \sin (180 x)}{3169126500570573503741758013440}-\frac {4755579521 \sin (182 x)}{316912650057057350374175801344}+\frac {1860878943 \sin (184 x)}{1267650600228229401496703205376}-\frac {20009451 \sin (186 x)}{158456325028528675187087900672}+\frac {2980131 \sin (188 x)}{316912650057057350374175801344}-\frac {470547 \sin (190 x)}{792281625142643375935439503360}+\frac {156849 \sin (192 x)}{5070602400912917605986812821504}-\frac {1617 \sin (194 x)}{1267650600228229401496703205376}+\frac {99 \sin (196 x)}{2535301200456458802993406410752}-\frac {\sin (198 x)}{1267650600228229401496703205376}+\frac {\sin (200 x)}{126765060022822940149670320537600}\) |
-1/1267650600228229401496703205376*Sin[2*x] + (99*Sin[4*x])/25353012004564 58802993406410752 - (1617*Sin[6*x])/1267650600228229401496703205376 + (156 849*Sin[8*x])/5070602400912917605986812821504 - (470547*Sin[10*x])/7922816 25142643375935439503360 + (2980131*Sin[12*x])/3169126500570573503741758013 44 - (20009451*Sin[14*x])/158456325028528675187087900672 + (1860878943*Sin [16*x])/1267650600228229401496703205376 - (4755579521*Sin[18*x])/316912650 057057350374175801344 + (432757736411*Sin[20*x])/3169126500570573503741758 013440 - (354074511609*Sin[22*x])/316912650057057350374175801344 + (105042 10511067*Sin[24*x])/1267650600228229401496703205376 - (8888178124749*Sin[2 6*x])/158456325028528675187087900672 + (110467356693309*Sin[28*x])/3169126 50057057350374175801344 - (1583365445937429*Sin[30*x])/7922816251426433759 35439503360 + (26917212580936293*Sin[32*x])/253530120045645880299340641075 2 - (33250674364686009*Sin[34*x])/633825300114114700748351602688 + (306645 108029882083*Sin[36*x])/1267650600228229401496703205376 - (661707864696061 337*Sin[38*x])/633825300114114700748351602688 + (53598337040380968297*Sin[ 40*x])/12676506002282294014967032053760 - (2552301763827665157*Sin[42*x])/ 158456325028528675187087900672 + (18330167212944140673*Sin[44*x])/31691265 0057057350374175801344 - (31081587882818325489*Sin[46*x])/1584563250285286 75187087900672 + (797760755659003687551*Sin[48*x])/12676506002282294014967 03205376 - (15157454357521070063469*Sin[50*x])/792281625142643375935439...
3.1.46.3.1 Defintions of rubi rules used
Int[(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol ] :> Int[ExpandTrigReduce[ActivateTrig[F[a + b*x]^p*G[c + d*x]^q], x], x] / ; FreeQ[{a, b, c, d}, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && IGtQ[p, 0] && IGtQ[q, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(601\) vs. \(2(10)=20\).
Time = 1.16 (sec) , antiderivative size = 602, normalized size of antiderivative = 50.17
\[\text {Expression too large to display}\]
-354074511609/316912650057057350374175801344*sin(178*x)+343655853645070287 531684243/3169126500570573503741758013440*sin(80*x)+1/12676506002282294014 9670320537600*sin(200*x)-1583365445937429/792281625142643375935439503360*s in(170*x)-15157454357521070063469/7922816251426433759354395033600*sin(150* x)-8888178124749/158456325028528675187087900672*sin(174*x)+2980131/3169126 50057057350374175801344*sin(12*x)-20009451/158456325028528675187087900672* sin(14*x)+1860878943/1267650600228229401496703205376*sin(16*x)-4755579521/ 316912650057057350374175801344*sin(18*x)-1583365445937429/7922816251426433 75935439503360*sin(30*x)-354074511609/316912650057057350374175801344*sin(2 2*x)+156849/5070602400912917605986812821504*sin(8*x)-470547/79228162514264 3375935439503360*sin(10*x)-1617/1267650600228229401496703205376*sin(6*x)+4 32757736411/3169126500570573503741758013440*sin(20*x)+99/25353012004564588 02993406410752*sin(4*x)+12611418068195524166851562157/15845632502852867518 708790067200*sin(100*x)-1/1267650600228229401496703205376*sin(2*x)-2472827 07219520081702971807/316912650057057350374175801344*sin(98*x)-192026472544 175561239751853/396140812571321687967719751680*sin(90*x)-54753357659398144 323058251/6338253001141147007483516026880*sin(130*x)+349787408250486232233 9/633825300114114700748351602688*sin(148*x)+26917212580936293/253530120045 6458802993406410752*sin(168*x)-2552301763827665157/15845632502852867518708 7900672*sin(158*x)+99/2535301200456458802993406410752*sin(196*x)-294692...
Timed out. \[ \int \sin ^{99}(x) \sin (101 x) \, dx=\text {Timed out} \]
Timed out. \[ \int \sin ^{99}(x) \sin (101 x) \, dx=\text {Timed out} \]
Exception generated. \[ \int \sin ^{99}(x) \sin (101 x) \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> ECL says: Memory limit reached. Please j ump to an outer pointer, quit program and enlarge thememory limits before executing the program again.
Leaf count of result is larger than twice the leaf count of optimal. 601 vs. \(2 (10) = 20\).
Time = 0.29 (sec) , antiderivative size = 601, normalized size of antiderivative = 50.08 \[ \int \sin ^{99}(x) \sin (101 x) \, dx=\text {Too large to display} \]
1/126765060022822940149670320537600*sin(200*x) - 1/12676506002282294014967 03205376*sin(198*x) + 99/2535301200456458802993406410752*sin(196*x) - 1617 /1267650600228229401496703205376*sin(194*x) + 156849/507060240091291760598 6812821504*sin(192*x) - 470547/792281625142643375935439503360*sin(190*x) + 2980131/316912650057057350374175801344*sin(188*x) - 20009451/158456325028 528675187087900672*sin(186*x) + 1860878943/1267650600228229401496703205376 *sin(184*x) - 4755579521/316912650057057350374175801344*sin(182*x) + 43275 7736411/3169126500570573503741758013440*sin(180*x) - 354074511609/31691265 0057057350374175801344*sin(178*x) + 10504210511067/12676506002282294014967 03205376*sin(176*x) - 8888178124749/158456325028528675187087900672*sin(174 *x) + 110467356693309/316912650057057350374175801344*sin(172*x) - 15833654 45937429/792281625142643375935439503360*sin(170*x) + 26917212580936293/253 5301200456458802993406410752*sin(168*x) - 33250674364686009/63382530011411 4700748351602688*sin(166*x) + 306645108029882083/1267650600228229401496703 205376*sin(164*x) - 661707864696061337/633825300114114700748351602688*sin( 162*x) + 53598337040380968297/12676506002282294014967032053760*sin(160*x) - 2552301763827665157/158456325028528675187087900672*sin(158*x) + 18330167 212944140673/316912650057057350374175801344*sin(156*x) - 31081587882818325 489/158456325028528675187087900672*sin(154*x) + 797760755659003687551/1267 650600228229401496703205376*sin(152*x) - 15157454357521070063469/792281...
Time = 21.21 (sec) , antiderivative size = 601, normalized size of antiderivative = 50.08 \[ \int \sin ^{99}(x) \sin (101 x) \, dx=\text {Too large to display} \]
(99*sin(4*x))/2535301200456458802993406410752 - sin(2*x)/12676506002282294 01496703205376 - (1617*sin(6*x))/1267650600228229401496703205376 + (156849 *sin(8*x))/5070602400912917605986812821504 - (470547*sin(10*x))/7922816251 42643375935439503360 + (2980131*sin(12*x))/316912650057057350374175801344 - (20009451*sin(14*x))/158456325028528675187087900672 + (1860878943*sin(16 *x))/1267650600228229401496703205376 - (4755579521*sin(18*x))/316912650057 057350374175801344 + (432757736411*sin(20*x))/3169126500570573503741758013 440 - (354074511609*sin(22*x))/316912650057057350374175801344 + (105042105 11067*sin(24*x))/1267650600228229401496703205376 - (8888178124749*sin(26*x ))/158456325028528675187087900672 + (110467356693309*sin(28*x))/3169126500 57057350374175801344 - (1583365445937429*sin(30*x))/7922816251426433759354 39503360 + (26917212580936293*sin(32*x))/2535301200456458802993406410752 - (33250674364686009*sin(34*x))/633825300114114700748351602688 + (306645108 029882083*sin(36*x))/1267650600228229401496703205376 - (661707864696061337 *sin(38*x))/633825300114114700748351602688 + (53598337040380968297*sin(40* x))/12676506002282294014967032053760 - (2552301763827665157*sin(42*x))/158 456325028528675187087900672 + (18330167212944140673*sin(44*x))/31691265005 7057350374175801344 - (31081587882818325489*sin(46*x))/1584563250285286751 87087900672 + (797760755659003687551*sin(48*x))/12676506002282294014967032 05376 - (15157454357521070063469*sin(50*x))/792281625142643375935439503...