Integrand size = 11, antiderivative size = 221 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^5} \, dx=-\frac {3889 \text {arctanh}(\cos (x))}{186624}-\frac {332929 \text {arctanh}(2 \cos (x))}{2916}+\frac {82683 \text {arctanh}\left (\sqrt {2} \cos (x)\right )}{512 \sqrt {2}}-\frac {1}{108 (1-2 \cos (x))^4}+\frac {19}{162 (1-2 \cos (x))^3}-\frac {749}{648 (1-2 \cos (x))^2}+\frac {71551}{5832 (1-2 \cos (x))}-\frac {1}{124416 (1-\cos (x))^2}-\frac {209}{373248 (1-\cos (x))}+\frac {1}{124416 (1+\cos (x))^2}+\frac {209}{373248 (1+\cos (x))}+\frac {1}{108 (1+2 \cos (x))^4}-\frac {19}{162 (1+2 \cos (x))^3}+\frac {749}{648 (1+2 \cos (x))^2}-\frac {71551}{5832 (1+2 \cos (x))}-\frac {11643}{512} \cos (x) \sec (2 x)+\frac {681}{256} \cos (x) \sec ^2(2 x)-\frac {21}{64} \cos (x) \sec ^3(2 x)+\frac {1}{32} \cos (x) \sec ^4(2 x) \] Output:
-3889/186624*arctanh(cos(x))-332929/2916*arctanh(2*cos(x))+82683/1024*arct anh(cos(x)*2^(1/2))*2^(1/2)-1/108/(1-2*cos(x))^4+19/162/(1-2*cos(x))^3-749 /648/(1-2*cos(x))^2+71551/(5832-11664*cos(x))-1/124416/(1-cos(x))^2-209/(3 73248-373248*cos(x))+1/124416/(1+cos(x))^2+209/(373248+373248*cos(x))+1/10 8/(1+2*cos(x))^4-19/162/(1+2*cos(x))^3+749/648/(1+2*cos(x))^2-71551/(5832+ 11664*cos(x))-11643/512*cos(x)*sec(2*x)+681/256*cos(x)*sec(2*x)^2-21/64*co s(x)*sec(2*x)^3+1/32*cos(x)*sec(2*x)^4
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.82 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^5} \, dx=\frac {-3070548 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+953225812 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-6069115802624 \log (1-2 \sin (x))-13436928 \left (451639 \log \left (\sec ^2\left (\frac {x}{2}\right )\right )-8 \text {RootSum}\left [107476319-1002557936 \text {$\#$1}-231239168 \text {$\#$1}^2+4096 \text {$\#$1}^3\&,\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (-8477353100119-37436171566600 \sin (x)-6312005469856 \sin (x) \text {$\#$1}+111813632 \sin (x) \text {$\#$1}^2\right )\right ) \text {$\#$1}\&\right ]\right )+\frac {21 \sec ^4(x) (14992301988-135248820940 \cos (2 x)+50511597564 \cos (4 x)+118827142092 \cos (6 x)-118688826492 \cos (8 x)-447277004 \cos (10 x)-7157694992 \cos (12 x)-5002400916 \cos (14 x)+28445689884 \cos (16 x)-10025130752 \cos (18 x)+178131036546 \sin (x)+25679598098 \sin (3 x)-60509728422 \sin (5 x)+19995369696 \sin (7 x)-104947392097 \sin (9 x)+37686703163 \sin (11 x)+29725568451 \sin (13 x)-29658030003 \sin (15 x)+2383629525 \sin (17 x)-8921638271 \sin (19 x))}{(1-2 \cos (2 x)+2 \cos (4 x)+2 \sin (x))^4}}{392073696} \] Input:
Integrate[(Cos[5*x] + Sin[2*x])^(-5),x]
Output:
(-3070548*Log[Cos[x/2] - Sin[x/2]] + 953225812*Log[Cos[x/2] + Sin[x/2]] - 6069115802624*Log[1 - 2*Sin[x]] - 13436928*(451639*Log[Sec[x/2]^2] - 8*Roo tSum[107476319 - 1002557936*#1 - 231239168*#1^2 + 4096*#1^3 & , Log[-(Sec[ x/2]^2*(-8477353100119 - 37436171566600*Sin[x] - 6312005469856*Sin[x]*#1 + 111813632*Sin[x]*#1^2))]*#1 & ]) + (21*Sec[x]^4*(14992301988 - 1352488209 40*Cos[2*x] + 50511597564*Cos[4*x] + 118827142092*Cos[6*x] - 118688826492* Cos[8*x] - 447277004*Cos[10*x] - 7157694992*Cos[12*x] - 5002400916*Cos[14* x] + 28445689884*Cos[16*x] - 10025130752*Cos[18*x] + 178131036546*Sin[x] + 25679598098*Sin[3*x] - 60509728422*Sin[5*x] + 19995369696*Sin[7*x] - 1049 47392097*Sin[9*x] + 37686703163*Sin[11*x] + 29725568451*Sin[13*x] - 296580 30003*Sin[15*x] + 2383629525*Sin[17*x] - 8921638271*Sin[19*x]))/(1 - 2*Cos [2*x] + 2*Cos[4*x] + 2*Sin[x])^4)/392073696
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.
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (5 x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (5 x))^5}dx\) |
| \(\Big \downarrow \) 4829 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (404 \sin ^2(x)+454 \sin (x)+81\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^5}+\frac {64 \left (32518008 \sin ^2(x)+34614094 \sin (x)+3256035\right )}{16807 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {256 \left (644409 \sin ^2(x)+690991 \sin (x)+72711\right )}{2401 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}+\frac {384 \left (30162 \sin ^2(x)+32656 \sin (x)+3919\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}-\frac {256 \left (2607 \sin ^2(x)+2861 \sin (x)+403\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^4}-\frac {1053}{268912 (\sin (x)-1)}+\frac {14179}{11664 (\sin (x)+1)}-\frac {22569152}{729 (2 \sin (x)-1)}+\frac {9}{38416 (\sin (x)-1)^2}+\frac {389}{11664 (\sin (x)+1)^2}-\frac {2904128}{729 (2 \sin (x)-1)^2}-\frac {1}{134456 (\sin (x)-1)^3}+\frac {1}{1944 (\sin (x)+1)^3}-\frac {36224}{81 (2 \sin (x)-1)^3}-\frac {1088}{27 (2 \sin (x)-1)^4}-\frac {64}{27 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^5}d\sin (x)\) |
Input:
Int[(Cos[5*x] + Sin[2*x])^(-5),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[1/d Subst[Int[Simplify[TrigExpand[a*Sin[ m*ArcSin[x]] + b*Cos[n*ArcSin[x]]]]^p/Sqrt[1 - x^2], x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[(p - 1)/2, 0] && IntegerQ[m/2] && Inte gerQ[(n - 1)/2]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.72 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.07
\[\frac {8}{27 \left (2 \sin \left (x \right )-1\right )^{4}}+\frac {544}{81 \left (2 \sin \left (x \right )-1\right )^{3}}+\frac {9056}{81 \left (2 \sin \left (x \right )-1\right )^{2}}+\frac {1452064}{729 \left (2 \sin \left (x \right )-1\right )}-\frac {11284576 \ln \left (2 \sin \left (x \right )-1\right )}{729}+\frac {\frac {9003728896 \sin \left (x \right )^{11}}{2401}+\frac {163775365120 \sin \left (x \right )^{10}}{16807}+\frac {66498560000 \sin \left (x \right )^{9}}{16807}-\frac {149297088512 \sin \left (x \right )^{8}}{16807}-\frac {132678402048 \sin \left (x \right )^{7}}{16807}+\frac {3290523648 \sin \left (x \right )^{6}}{2401}+\frac {7878694912 \sin \left (x \right )^{5}}{2401}+\frac {1619062912 \sin \left (x \right )^{4}}{2401}-\frac {5530391296 \sin \left (x \right )^{3}}{16807}-\frac {2817030592 \sin \left (x \right )^{2}}{16807}-\frac {454497024 \sin \left (x \right )}{16807}-\frac {25720664}{16807}}{\left (8 \sin \left (x \right )^{3}+4 \sin \left (x \right )^{2}-4 \sin \left (x \right )-1\right )^{4}}+\frac {288 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{3}+4 \textit {\_Z}^{2}-4 \textit {\_Z} -1\right )}{\sum }\frac {\left (1806556 \textit {\_R}^{2}+2029861 \textit {\_R} +362232\right ) \ln \left (\sin \left (x \right )-\textit {\_R} \right )}{6 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )}{16807}-\frac {1}{3888 \left (1+\sin \left (x \right )\right )^{2}}-\frac {389}{11664 \left (1+\sin \left (x \right )\right )}+\frac {14179 \ln \left (1+\sin \left (x \right )\right )}{11664}+\frac {1}{268912 \left (\sin \left (x \right )-1\right )^{2}}-\frac {9}{38416 \left (\sin \left (x \right )-1\right )}-\frac {1053 \ln \left (\sin \left (x \right )-1\right )}{268912}\]
Input:
int(1/(cos(5*x)+sin(2*x))^5,x)
Output:
8/27/(2*sin(x)-1)^4+544/81/(2*sin(x)-1)^3+9056/81/(2*sin(x)-1)^2+1452064/7 29/(2*sin(x)-1)-11284576/729*ln(2*sin(x)-1)+524288/16807*(480851/4*sin(x)^ 11+9996055/32*sin(x)^10+2029375/16*sin(x)^9-72898969/256*sin(x)^8-8098047/ 32*sin(x)^7+11246907/256*sin(x)^6+26929133/256*sin(x)^5+88542503/4096*sin( x)^4-21603091/2048*sin(x)^3-44016103/8192*sin(x)^2-1775379/2048*sin(x)-321 5083/65536)/(8*sin(x)^3+4*sin(x)^2-4*sin(x)-1)^4+288/16807*sum((1806556*_R ^2+2029861*_R+362232)/(6*_R^2+2*_R-1)*ln(sin(x)-_R),_R=RootOf(8*_Z^3+4*_Z^ 2-4*_Z-1))-1/3888/(1+sin(x))^2-389/11664/(1+sin(x))+14179/11664*ln(1+sin(x ))+1/268912/(sin(x)-1)^2-9/38416/(sin(x)-1)-1053/268912*ln(sin(x)-1)
Result contains complex when optimal does not.
Time = 1.36 (sec) , antiderivative size = 2056, normalized size of antiderivative = 9.30 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^5,x, algorithm="fricas")
Output:
-1/470684472048*(33126948382087053312*cos(x)^18 - 172570185683432570880*co s(x)^16 + 374537416757797601280*cos(x)^14 - 438603598203625021440*cos(x)^1 2 + 299653580002679476224*cos(x)^10 - 120005887602794516736*cos(x)^8 + 263 21130059384977152*cos(x)^6 - 2457796064465502768*cos(x)^4 + 5832*(65536*co s(x)^20 - 327680*cos(x)^18 + 696320*cos(x)^16 - 825344*cos(x)^14 + 603904* cos(x)^12 - 284800*cos(x)^10 + 86256*cos(x)^8 - 15432*cos(x)^6 + 1241*cos( x)^4 + 8*(4096*cos(x)^16 - 15360*cos(x)^14 + 23040*cos(x)^12 - 17664*cos(x )^10 + 7344*cos(x)^8 - 1600*cos(x)^6 + 145*cos(x)^4)*sin(x))*(40353607*(11 250329208580325376/678223072849*I*sqrt(3) + 93181568782592899694592/678223 072849)^(1/3)*(-I*sqrt(3) + 1) + 1074452790721536*(I*sqrt(3) + 1)/(1125032 9208580325376/678223072849*I*sqrt(3) + 93181568782592899694592/67822307284 9)^(1/3) - 416403931776)*log(58363/23059204*(40353607*(1125032920858032537 6/678223072849*I*sqrt(3) + 93181568782592899694592/678223072849)^(1/3)*(-I *sqrt(3) + 1) + 1074452790721536*(I*sqrt(3) + 1)/(11250329208580325376/678 223072849*I*sqrt(3) + 93181568782592899694592/678223072849)^(1/3) - 416403 931776)^2 + 127611403123292256*(11250329208580325376/678223072849*I*sqrt(3 ) + 93181568782592899694592/678223072849)^(1/3)*(-I*sqrt(3) + 1) + 1165436 410115787637876752384/343*(I*sqrt(3) + 1)/(11250329208580325376/6782230728 49*I*sqrt(3) + 93181568782592899694592/678223072849)^(1/3) + 4687637170241 80224*sin(x) - 1316694816201251589120) + 605165749776*cos(x)^2 - 2916*(...
\[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^5} \, dx=\int \frac {1}{\left (\sin {\left (2 x \right )} + \cos {\left (5 x \right )}\right )^{5}}\, dx \] Input:
integrate(1/(cos(5*x)+sin(2*x))**5,x)
Output:
Integral((sin(2*x) + cos(5*x))**(-5), x)
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^5} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^5,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^5} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^5,x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: 2*(( -89077056*sin(sageVARx)^4-3291925*sin(sageVARx)^3+181375264*sin(sageVARx)^ 2+3343075*sin(sageVARx)-92347900)*1/196036848/(sin(sageVARx)^2-1)^2+(76012 4211200*sin(sageVAR
Time = 22.43 (sec) , antiderivative size = 1217, normalized size of antiderivative = 5.51 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^5} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(2*x))^5,x)
Output:
(14179*log(tan(x/2) + 1))/5832 - (1053*log(tan(x/2) - 1))/134456 - (112845 76*log(tan(x/2)^2 - 4*tan(x/2) + 1))/729 + symsum(log((2199023255552*(3355 8302142494706313979267710332531314776224468449900888064*root(z^3 - (260144 064*z^2)/16807 - (5197260340224*z)/282475249 + 2567380551303168/4747561509 943, z, k)*cos(x) - 120681028458586732072748765655713652467733777601542488 0640*cos(x) - 890863856345561813414462492104570397111221021823603834880*si n(x) - 25611080120468487005099571756628820489567742727129399296000*root(z^ 3 - (260144064*z^2)/16807 - (5197260340224*z)/282475249 + 2567380551303168 /4747561509943, z, k) + 34501468503530603859835169716312084210224049957497 766674432*root(z^3 - (260144064*z^2)/16807 - (5197260340224*z)/282475249 + 2567380551303168/4747561509943, z, k)*sin(x) - 30441449584747752798287027 8435390005304320214291072294846464*root(z^3 - (260144064*z^2)/16807 - (519 7260340224*z)/282475249 + 2567380551303168/4747561509943, z, k)^2 - 391752 94890586797427871989052738884126974312553153808056320*root(z^3 - (26014406 4*z^2)/16807 - (5197260340224*z)/282475249 + 2567380551303168/474756150994 3, z, k)^3 + 158910993749471270999441418708200530860086555354913146082048* root(z^3 - (260144064*z^2)/16807 - (5197260340224*z)/282475249 + 256738055 1303168/4747561509943, z, k)^4 + 82188275993923217656486900130486721710965 619488658442336*root(z^3 - (260144064*z^2)/16807 - (5197260340224*z)/28247 5249 + 2567380551303168/4747561509943, z, k)^5 - 6166384041129098466961...
\[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^5} \, dx=\int \frac {1}{\cos \left (5 x \right )^{5}+5 \cos \left (5 x \right )^{4} \sin \left (2 x \right )+10 \cos \left (5 x \right )^{3} \sin \left (2 x \right )^{2}+10 \cos \left (5 x \right )^{2} \sin \left (2 x \right )^{3}+5 \cos \left (5 x \right ) \sin \left (2 x \right )^{4}+\sin \left (2 x \right )^{5}}d x \] Input:
int(1/(cos(5*x)+sin(2*x))^5,x)
Output:
int(1/(cos(5*x)**5 + 5*cos(5*x)**4*sin(2*x) + 10*cos(5*x)**3*sin(2*x)**2 + 10*cos(5*x)**2*sin(2*x)**3 + 5*cos(5*x)*sin(2*x)**4 + sin(2*x)**5),x)