Integrand size = 11, antiderivative size = 190 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^6} \, dx=-\frac {637 \text {arctanh}(2 \cos (x) \sin (x))}{8192}-\frac {3955 \cot (x)}{1048576}-\frac {2131 \cot ^3(x)}{3145728}-\frac {173 \cot ^5(x)}{2621440}+\frac {158879 \tan (x)}{1048576}+\frac {128051 \tan ^3(x)}{3145728}+\frac {78617 \tan ^5(x)}{5242880}+\frac {4751 \tan ^7(x)}{1048576}+\frac {8501 \tan ^9(x)}{9437184}+\frac {983 \tan ^{11}(x)}{11534336}+\frac {\csc ^5(x) \sec ^{21}(x)}{1310720 \left (1-\tan ^2(x)\right )^5}-\frac {3 \csc ^5(x) \sec ^{19}(x)}{5242880 \left (1-\tan ^2(x)\right )^4}+\frac {61 \csc ^5(x) \sec ^{17}(x)}{15728640 \left (1-\tan ^2(x)\right )^3}-\frac {97 \csc ^5(x) \sec ^{15}(x)}{10485760 \left (1-\tan ^2(x)\right )^2}+\frac {443 \csc ^5(x) \sec ^{13}(x)}{6291456 \left (1-\tan ^2(x)\right )} \] Output:
-637/8192*arctanh(2*cos(x)*sin(x))-3955/1048576*cot(x)-2131/3145728*cot(x) ^3-173/2621440*cot(x)^5+158879/1048576*tan(x)+128051/3145728*tan(x)^3+7861 7/5242880*tan(x)^5+4751/1048576*tan(x)^7+8501/9437184*tan(x)^9+983/1153433 6*tan(x)^11+1/1310720*csc(x)^5*sec(x)^21/(1-tan(x)^2)^5-3/5242880*csc(x)^5 *sec(x)^19/(1-tan(x)^2)^4+61/15728640*csc(x)^5*sec(x)^17/(1-tan(x)^2)^3-97 /10485760*csc(x)^5*sec(x)^15/(1-tan(x)^2)^2+443*csc(x)^5*sec(x)^13/(629145 6-6291456*tan(x)^2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.71 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.31 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^6} \, dx=\frac {1568000 \sqrt {3} \text {arctanh}\left (\frac {-2+\tan \left (\frac {x}{2}\right )}{\sqrt {3}}\right )+2240 i \text {RootSum}\left [i+\text {$\#$1}^3-i \text {$\#$1}^6\&,\frac {2871766 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-1435883 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )-5397130 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}-2698565 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-7271532 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2+3635766 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+5397130 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3+2698565 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2871766 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4-1435883 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{i \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]+\frac {9 \sec ^5(x) (369912+683354970 \cos (2 x)-1397328660 \cos (4 x)-2079944950 \cos (6 x)+460950 \cos (8 x)-433536 \cos (10 x)+1040066515 \cos (12 x)+698762430 \cos (14 x)-341889585 \cos (16 x)-108940 \cos (18 x)-68318421 \cos (20 x)+139743270 \cos (22 x)+207960045 \cos (24 x)+711810 \sin (x)-2080097220 \sin (3 x)-1397410614 \sin (5 x)+683564550 \sin (7 x)+465730 \sin (9 x)+341626245 \sin (11 x)-698684940 \sin (13 x)-1039886629 \sin (15 x)+110220 \sin (17 x)-98700 \sin (19 x)+208013905 \sin (21 x)+139765050 \sin (23 x)-68398425 \sin (25 x))}{(-1+2 \cos (2 x)-2 \cos (4 x)+2 \sin (x)-2 \sin (3 x))^5}}{5314410} \] Input:
Integrate[(Cos[5*x] + Sin[4*x])^(-6),x]
Output:
(1568000*Sqrt[3]*ArcTanh[(-2 + Tan[x/2])/Sqrt[3]] + (2240*I)*RootSum[I + # 1^3 - I*#1^6 & , (2871766*ArcTan[Sin[x]/(Cos[x] - #1)] - (1435883*I)*Log[1 - 2*Cos[x]*#1 + #1^2] - (5397130*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 - 269 8565*Log[1 - 2*Cos[x]*#1 + #1^2]*#1 - 7271532*ArcTan[Sin[x]/(Cos[x] - #1)] *#1^2 + (3635766*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^2 + (5397130*I)*ArcTan[ Sin[x]/(Cos[x] - #1)]*#1^3 + 2698565*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3 + 28 71766*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^4 - (1435883*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^4)/(I*#1^2 + 2*#1^5) & ] + (9*Sec[x]^5*(369912 + 683354970*Cos[ 2*x] - 1397328660*Cos[4*x] - 2079944950*Cos[6*x] + 460950*Cos[8*x] - 43353 6*Cos[10*x] + 1040066515*Cos[12*x] + 698762430*Cos[14*x] - 341889585*Cos[1 6*x] - 108940*Cos[18*x] - 68318421*Cos[20*x] + 139743270*Cos[22*x] + 20796 0045*Cos[24*x] + 711810*Sin[x] - 2080097220*Sin[3*x] - 1397410614*Sin[5*x] + 683564550*Sin[7*x] + 465730*Sin[9*x] + 341626245*Sin[11*x] - 698684940* Sin[13*x] - 1039886629*Sin[15*x] + 110220*Sin[17*x] - 98700*Sin[19*x] + 20 8013905*Sin[21*x] + 139765050*Sin[23*x] - 68398425*Sin[25*x]))/(-1 + 2*Cos [2*x] - 2*Cos[4*x] + 2*Sin[x] - 2*Sin[3*x])^5)/5314410
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.
| \(\displaystyle \int \frac {1}{(\sin (4 x)+\cos (5 x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (4 x)+\cos (5 x))^6}dx\) |
| \(\Big \downarrow \) 4830 |
| \(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^{29}}{\left (-\tan ^{10}\left (\frac {x}{2}\right )-8 \tan ^9\left (\frac {x}{2}\right )+45 \tan ^8\left (\frac {x}{2}\right )+48 \tan ^7\left (\frac {x}{2}\right )-210 \tan ^6\left (\frac {x}{2}\right )+210 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-45 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle 2 \int \left (\frac {65536 \left (209 \tan \left (\frac {x}{2}\right )-56\right )}{19683 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^6}-\frac {331328}{1594323 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}-\frac {1024 \left (2225 \tan ^4\left (\frac {x}{2}\right )+31323 \tan ^3\left (\frac {x}{2}\right )+65593 \tan ^2\left (\frac {x}{2}\right )+71679 \tan \left (\frac {x}{2}\right )-1179151\right )}{2187 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {1043}{\left (\tan \left (\frac {x}{2}\right )-1\right )^2}+\frac {362}{1594323 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {256 \left (1129 \tan \left (\frac {x}{2}\right )+796\right )}{531441 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {4096 \left (1870701 \tan ^5\left (\frac {x}{2}\right )-11849476 \tan ^4\left (\frac {x}{2}\right )+190843194 \tan ^3\left (\frac {x}{2}\right )-2992661684 \tan ^2\left (\frac {x}{2}\right )+44783256989 \tan \left (\frac {x}{2}\right )-655339859632\right )}{729 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {247}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )^3}-\frac {29}{1417176 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {1024 \left (2056 \tan \left (\frac {x}{2}\right )+6137\right )}{531441 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {262144 \left (147054417592 \tan ^5\left (\frac {x}{2}\right )-313397987283 \tan ^4\left (\frac {x}{2}\right )+4461232347428 \tan ^3\left (\frac {x}{2}\right )-63211635234022 \tan ^2\left (\frac {x}{2}\right )+814579158009388 \tan \left (\frac {x}{2}\right )-10419876143279939\right )}{729 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {251}{8 \left (\tan \left (\frac {x}{2}\right )-1\right )^4}+\frac {91}{4251528 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {4096 \left (861 \tan \left (\frac {x}{2}\right )+748\right )}{59049 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {33554432 \left (115025589654795 \tan ^5\left (\frac {x}{2}\right )-74588994561272 \tan ^4\left (\frac {x}{2}\right )+1198889509658190 \tan ^3\left (\frac {x}{2}\right )-18692957643744328 \tan ^2\left (\frac {x}{2}\right )+223768626713731947 \tan \left (\frac {x}{2}\right )-2668294275126091664\right )}{81 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {1}{\left (\tan \left (\frac {x}{2}\right )-1\right )^5}-\frac {1}{531441 \left (\tan \left (\frac {x}{2}\right )+1\right )^5}+\frac {16384 \left (352 \tan \left (\frac {x}{2}\right )+147\right )}{19683 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^5}-\frac {1073741824 \left (331013371196655248 \tan ^5\left (\frac {x}{2}\right )+36445691865972541 \tan ^4\left (\frac {x}{2}\right )+489709579018774756 \tan ^3\left (\frac {x}{2}\right )-18302977809225300198 \tan ^2\left (\frac {x}{2}\right )+210170961771305215764 \tan \left (\frac {x}{2}\right )-2398451564662828263555\right )}{27 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^5}+\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^6}+\frac {1}{1062882 \left (\tan \left (\frac {x}{2}\right )+1\right )^6}-\frac {68719476736 \left (428355383219589218573 \tan ^5\left (\frac {x}{2}\right )+244140164265787302300 \tan ^4\left (\frac {x}{2}\right )-1505468697103927060454 \tan ^3\left (\frac {x}{2}\right )+73370619284291642908 \tan ^2\left (\frac {x}{2}\right )+446436133753324158029 \tan \left (\frac {x}{2}\right )+37476239992016874944\right )}{27 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^6}\right )d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {39200 \log \left (-\tan \left (\frac {x}{2}\right )-\sqrt {3}+2\right )}{177147 \sqrt {3}}-\frac {39200 \log \left (-\tan \left (\frac {x}{2}\right )+\sqrt {3}+2\right )}{177147 \sqrt {3}}+\frac {56297367981513695706973755932672}{27} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )-46018507987867324264331870208 \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )-\frac {593769146402991314229129377742848}{27} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )+\frac {54109245563735799458505634611200}{9} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )+\frac {277586393580403271810907876884480}{27} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )+\frac {2576028603618633336045671809024}{27} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^5}d\tan \left (\frac {x}{2}\right )-\frac {225313928943198447566464221184}{27} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^5}d\tan \left (\frac {x}{2}\right )+\frac {12544214758367384487814234112}{27} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^5}d\tan \left (\frac {x}{2}\right )+\frac {61674715102760000206929920}{9} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^5}d\tan \left (\frac {x}{2}\right )+\frac {3515095745909745367159668736}{27} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^5}d\tan \left (\frac {x}{2}\right )+\frac {89540818047360397610057728}{81} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )-\frac {2501523183490990119845888}{27} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )+\frac {550039209609264833232896}{81} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )-\frac {10836273291558544998400}{27} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )+\frac {41098974609271794171904}{81} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )+\frac {2731585110570466803712}{729} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )-\frac {23722054373774196736}{81} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )+\frac {15799562241882718208}{729} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )-\frac {364128808664563712}{243} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )+\frac {467649734430687232}{729} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )+\frac {2684287389835264}{729} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )-\frac {183424558235648}{729} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )+\frac {12104694431744}{729} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )-\frac {28384034816}{27} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )+\frac {125159366656}{729} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )+\frac {1207450624 \int \frac {1}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )}{2187}-\frac {24466432}{729} \int \frac {\tan \left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )-\frac {67167232 \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )}{2187}-\frac {10691584}{729} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )-\frac {2278400 \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )}{2187}+\frac {1043}{1-\tan \left (\frac {x}{2}\right )}-\frac {362}{1594323 \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {128 \left (907-1018 \tan \left (\frac {x}{2}\right )\right )}{531441 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}-\frac {16640 \left (2-\tan \left (\frac {x}{2}\right )\right )}{1594323 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {1277065216}{729 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )}-\frac {247}{16 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {29}{2834352 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {256 \left (14330-10249 \tan \left (\frac {x}{2}\right )\right )}{1594323 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {863488 \left (2-\tan \left (\frac {x}{2}\right )\right )}{531441 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {9637358311309312}{2187 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {251}{24 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {91}{12754584 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {2048 \left (2357-2470 \tan \left (\frac {x}{2}\right )\right )}{531441 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {14930944 \left (2-\tan \left (\frac {x}{2}\right )\right )}{2657205 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {214423240351762350080}{81 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {1}{4 \left (1-\tan \left (\frac {x}{2}\right )\right )^4}+\frac {1}{2125764 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {2048 \left (646-851 \tan \left (\frac {x}{2}\right )\right )}{59049 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {1482752 \left (2-\tan \left (\frac {x}{2}\right )\right )}{98415 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {44427862619635708585836544}{81 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {1}{10 \left (1-\tan \left (\frac {x}{2}\right )\right )^5}-\frac {1}{5314410 \left (\tan \left (\frac {x}{2}\right )+1\right )^5}+\frac {32768 \left (97-362 \tan \left (\frac {x}{2}\right )\right )}{295245 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^5}+\frac {14718178895949463042601846308864}{405 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^5}\right )\) |
Input:
Int[(Cos[5*x] + Sin[4*x])^(-6),x]
Output:
$Aborted
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Int[(cos[(n_.)*((c_.) + (d_.)*(x_))]*(b_.) + (a_.)*sin[(m_.)*((c_.) + (d_.) *(x_))])^(p_), x_Symbol] :> Simp[2/d Subst[Int[Simplify[TrigExpand[a*Sin[ 2*m*ArcTan[x]] + b*Cos[2*n*ArcTan[x]]]]^p/(1 + x^2), x], x, Tan[(1/2)*(c + d*x)]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && Intege rQ[n]
Time = 2257.52 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.01
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| risch | \(\text {Expression too large to display}\) | \(494\) |
| default | \(\text {Expression too large to display}\) | \(580\) |
Input:
int(1/(cos(5*x)+sin(4*x))^6,x,method=_RETURNVERBOSE)
Output:
0
Result contains complex when optimal does not.
Time = 3.21 (sec) , antiderivative size = 4649, normalized size of antiderivative = 24.47 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^6,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^6} \, dx=\text {Timed out} \] Input:
integrate(1/(cos(5*x)+sin(4*x))**6,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^6} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^6,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (150) = 300\).
Time = 0.19 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.94 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^6,x, algorithm="giac")
Output:
-78400/531441*sqrt(3)*log(abs(-2*sqrt(3) + 2*tan(1/2*x) - 4)/abs(2*sqrt(3) + 2*tan(1/2*x) - 4)) - 2/295245*(570879245*tan(1/2*x)^49 + 23435482320*ta n(1/2*x)^48 + 261760494200*tan(1/2*x)^47 - 1044525945800*tan(1/2*x)^46 - 3 0941381898172*tan(1/2*x)^45 - 8631530312440*tan(1/2*x)^44 + 16079637123166 80*tan(1/2*x)^43 + 633749329566680*tan(1/2*x)^42 - 47684055834309030*tan(1 /2*x)^41 + 26328493931909320*tan(1/2*x)^40 + 748820798895545640*tan(1/2*x) ^39 - 950770122126857720*tan(1/2*x)^38 - 6690753840451590220*tan(1/2*x)^37 + 12335535845921782840*tan(1/2*x)^36 + 35561269336653928280*tan(1/2*x)^35 - 88103415001950478680*tan(1/2*x)^34 - 109819358579340807325*tan(1/2*x)^3 3 + 390657633726427449480*tan(1/2*x)^32 + 155773364304800690480*tan(1/2*x) ^31 - 1129415837133505053648*tan(1/2*x)^30 + 134596391753975528840*tan(1/2 *x)^29 + 2170692932524957437520*tan(1/2*x)^28 - 1057481831338684890160*tan (1/2*x)^27 - 2773487144406606245520*tan(1/2*x)^26 + 2243019453400555433964 *tan(1/2*x)^25 + 2290042927496543814480*tan(1/2*x)^24 - 273669099592372697 0160*tan(1/2*x)^23 - 1100216199124955722480*tan(1/2*x)^22 + 21506065009765 24488840*tan(1/2*x)^21 + 162387425897044906352*tan(1/2*x)^20 - 11220875887 21029229520*tan(1/2*x)^19 + 142875405732537734480*tan(1/2*x)^18 + 38905254 9763681952675*tan(1/2*x)^17 - 105589715296256443680*tan(1/2*x)^16 - 879751 05882599815720*tan(1/2*x)^15 + 34597964386356642840*tan(1/2*x)^14 + 123665 35412686729780*tan(1/2*x)^13 - 6542492768242677720*tan(1/2*x)^12 - 9613...
Time = 36.40 (sec) , antiderivative size = 1424, normalized size of antiderivative = 7.49 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^6} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(4*x))^6,x)
Output:
(78400*3^(1/2)*log(exp(x*1i)*2764668178515143051982344452204134400i - (784 00*3^(1/2)*((78400*3^(1/2)*(exp(x*1i)*638336698300929854077909893928429748 748288i - (78400*3^(1/2)*(267007703491546346947879946801148921045319680*ex p(x*1i) + (78400*3^(1/2)*((78400*3^(1/2)*((78400*3^(1/2)*(exp(x*1i)*130449 1175732155376976616258798333603226112i + 175744772433937284288990442630691 9947277760))/531441 - 8252507670758072195219238877377954861330984960*exp(x *1i) + 5386441821344555767136067829020908665087770624i))/531441 - exp(x*1i )*1832642431326587120631728257003426394824507392i + 2442728074718318432095 3992675402059931451392))/531441 + 1919243696453573764358932070992426027922 22720i))/531441 + 1602316304430981349978723319983041392148480))/531441 - 4 51941199471026873089680245422976663552*exp(x*1i) + 18047225154356337724193 79328408935727104i))/531441 - 1297046851891297022245554723225600000))/5314 41 - (78400*3^(1/2)*log(exp(x*1i)*2764668178515143051982344452204134400i - (78400*3^(1/2)*(451941199471026873089680245422976663552*exp(x*1i) + (7840 0*3^(1/2)*(exp(x*1i)*638336698300929854077909893928429748748288i + (78400* 3^(1/2)*(267007703491546346947879946801148921045319680*exp(x*1i) - (78400* 3^(1/2)*((78400*3^(1/2)*(8252507670758072195219238877377954861330984960*ex p(x*1i) + (78400*3^(1/2)*(exp(x*1i)*13044911757321553769766162587983336032 26112i + 1757447724339372842889904426306919947277760))/531441 - 5386441821 344555767136067829020908665087770624i))/531441 - exp(x*1i)*183264243132...
\[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^6} \, dx=\int \frac {1}{\cos \left (5 x \right )^{6}+6 \cos \left (5 x \right )^{5} \sin \left (4 x \right )+15 \cos \left (5 x \right )^{4} \sin \left (4 x \right )^{2}+20 \cos \left (5 x \right )^{3} \sin \left (4 x \right )^{3}+15 \cos \left (5 x \right )^{2} \sin \left (4 x \right )^{4}+6 \cos \left (5 x \right ) \sin \left (4 x \right )^{5}+\sin \left (4 x \right )^{6}}d x \] Input:
int(1/(cos(5*x)+sin(4*x))^6,x)
Output:
int(1/(cos(5*x)**6 + 6*cos(5*x)**5*sin(4*x) + 15*cos(5*x)**4*sin(4*x)**2 + 20*cos(5*x)**3*sin(4*x)**3 + 15*cos(5*x)**2*sin(4*x)**4 + 6*cos(5*x)*sin( 4*x)**5 + sin(4*x)**6),x)