Integrand size = 11, antiderivative size = 154 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^5} \, dx=-\frac {1063 \text {arctanh}(\cos (x))}{262144}+\frac {3603 \text {arctanh}\left (\sqrt {2} \cos (x)\right )}{16384 \sqrt {2}}-\frac {27761 \sec (x)}{262144}-\frac {13349 \sec ^3(x)}{786432}-\frac {6143 \sec ^5(x)}{1310720}-\frac {635 \sec ^7(x)}{458752}-\frac {1477 \sec ^9(x)}{4718592}+\frac {59 \sec ^9(x) \sec (2 x)}{524288}+\frac {159 \sec ^9(x) \sec ^2(2 x)}{262144}+\frac {149 \sec ^9(x) \sec ^3(2 x)}{393216}+\frac {45 \sec ^9(x) \sec ^4(2 x)}{131072}-\frac {37 \csc ^2(x) \sec ^9(x) \sec ^4(2 x)}{262144}-\frac {\csc ^4(x) \sec ^9(x) \sec ^4(2 x)}{131072} \] Output:
-1063/262144*arctanh(cos(x))+3603/32768*arctanh(cos(x)*2^(1/2))*2^(1/2)-27 761/262144*sec(x)-13349/786432*sec(x)^3-6143/1310720*sec(x)^5-635/458752*s ec(x)^7-1477/4718592*sec(x)^9+59/524288*sec(x)^9*sec(2*x)+159/262144*sec(x )^9*sec(2*x)^2+149/393216*sec(x)^9*sec(2*x)^3+45/131072*sec(x)^9*sec(2*x)^ 4-37/262144*csc(x)^2*sec(x)^9*sec(2*x)^4-1/131072*csc(x)^4*sec(x)^9*sec(2* x)^4
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.46 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.70 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^5} \, dx=\frac {-4147531776 \log \left (\sec ^2\left (\frac {x}{2}\right )\right )-8292605364 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+15796 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-1236992 \log (1-2 \sin (x))+49152 \text {RootSum}\left [851686039+1303482600 \text {$\#$1}-1166493312 \text {$\#$1}^2+13824 \text {$\#$1}^3\&,\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (-22330596672539-51662025250178 \sin (x)-48704028454896 \sin (x) \text {$\#$1}+577204992 \sin (x) \text {$\#$1}^2\right )\right ) \text {$\#$1}\&\right ]+\frac {9 \sec ^4(x) (42765300+302016 \cos (2 x)-412656 \cos (4 x)+302112 \cos (6 x)-27872708 \cos (8 x)+27805588 \cos (10 x)-114192 \cos (12 x)+104480 \cos (14 x)-58960 \cos (16 x)-14277396 \cos (18 x)-41817870 \sin (x)+367056 \sin (3 x)-358512 \sin (5 x)+228432 \sin (7 x)+57060192 \sin (9 x)+97552 \sin (11 x)-123632 \sin (13 x)+84240 \sin (15 x)-6962869 \sin (17 x)+6932549 \sin (19 x))}{(1-2 \cos (2 x)+2 \cos (4 x)-2 \sin (x)+2 \sin (3 x))^4}}{5668704} \] Input:
Integrate[(Cos[5*x] + Sin[4*x])^(-5),x]
Output:
(-4147531776*Log[Sec[x/2]^2] - 8292605364*Log[Cos[x/2] - Sin[x/2]] + 15796 *Log[Cos[x/2] + Sin[x/2]] - 1236992*Log[1 - 2*Sin[x]] + 49152*RootSum[8516 86039 + 1303482600*#1 - 1166493312*#1^2 + 13824*#1^3 & , Log[-(Sec[x/2]^2* (-22330596672539 - 51662025250178*Sin[x] - 48704028454896*Sin[x]*#1 + 5772 04992*Sin[x]*#1^2))]*#1 & ] + (9*Sec[x]^4*(42765300 + 302016*Cos[2*x] - 41 2656*Cos[4*x] + 302112*Cos[6*x] - 27872708*Cos[8*x] + 27805588*Cos[10*x] - 114192*Cos[12*x] + 104480*Cos[14*x] - 58960*Cos[16*x] - 14277396*Cos[18*x ] - 41817870*Sin[x] + 367056*Sin[3*x] - 358512*Sin[5*x] + 228432*Sin[7*x] + 57060192*Sin[9*x] + 97552*Sin[11*x] - 123632*Sin[13*x] + 84240*Sin[15*x] - 6962869*Sin[17*x] + 6932549*Sin[19*x]))/(1 - 2*Cos[2*x] + 2*Cos[4*x] - 2*Sin[x] + 2*Sin[3*x])^4)/5668704
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.
| \(\displaystyle \int \frac {1}{(\sin (4 x)+\cos (5 x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (4 x)+\cos (5 x))^5}dx\) |
| \(\Big \downarrow \) 4829 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^3 \left (16 \sin ^4(x)-8 \sin ^3(x)-12 \sin ^2(x)+4 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {64 \left (196 \sin ^2(x)+190 \sin (x)+37\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^4}+\frac {64 \left (66672 \sin ^2(x)+65510 \sin (x)+14357\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {64 \left (46068 \sin ^2(x)+45130 \sin (x)+9649\right )}{729 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}+\frac {128 \left (1632 \sin ^2(x)+1589 \sin (x)+323\right )}{81 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (264 \sin ^2(x)+248 \sin (x)+35\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^5}-\frac {11703}{16 (\sin (x)-1)}+\frac {3949}{2834352 (\sin (x)+1)}-\frac {77312}{177147 (2 \sin (x)-1)}+\frac {203}{16 (\sin (x)-1)^2}+\frac {209}{2834352 (\sin (x)+1)^2}-\frac {29888}{177147 (2 \sin (x)-1)^2}-\frac {1}{8 (\sin (x)-1)^3}+\frac {1}{472392 (\sin (x)+1)^3}-\frac {1664}{19683 (2 \sin (x)-1)^3}-\frac {128}{6561 (2 \sin (x)-1)^4}-\frac {64}{6561 (2 \sin (x)-1)^5}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^5 \left (1-\sin ^2(x)\right )^3 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^5}d\sin (x)\) |
Input:
Int[(Cos[5*x] + Sin[4*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]]
Time = 594.68 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.01
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| default | \(-\frac {1}{944784 \left (1+\sin \left (x \right )\right )^{2}}-\frac {209}{2834352 \left (1+\sin \left (x \right )\right )}+\frac {3949 \ln \left (1+\sin \left (x \right )\right )}{2834352}+\frac {-\frac {7589920768 \sin \left (x \right )^{11}}{59049}-\frac {7284604928 \sin \left (x \right )^{10}}{59049}+\frac {15777529856 \sin \left (x \right )^{9}}{59049}+\frac {6312200192 \sin \left (x \right )^{8}}{19683}-\frac {2480205824 \sin \left (x \right )^{7}}{19683}-\frac {5225921536 \sin \left (x \right )^{6}}{19683}-\frac {983317504 \sin \left (x \right )^{5}}{19683}+\frac {1190551552 \sin \left (x \right )^{4}}{19683}+\frac {697703168 \sin \left (x \right )^{3}}{19683}+\frac {467236352 \sin \left (x \right )^{2}}{59049}+\frac {48309376 \sin \left (x \right )}{59049}+\frac {1926944}{59049}}{\left (8 \sin \left (x \right )^{3}-6 \sin \left (x \right )-1\right )^{4}}+\frac {128 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{3}-6 \textit {\_Z} -1\right )}{\sum }\frac {\left (1350108 \textit {\_R}^{2}+1268671 \textit {\_R} +179575\right ) \ln \left (\sin \left (x \right )-\textit {\_R} \right )}{4 \textit {\_R}^{2}-1}\right )}{177147}+\frac {1}{16 \left (\sin \left (x \right )-1\right )^{2}}-\frac {203}{16 \left (\sin \left (x \right )-1\right )}-\frac {11703 \ln \left (\sin \left (x \right )-1\right )}{16}+\frac {8}{6561 \left (2 \sin \left (x \right )-1\right )^{4}}+\frac {64}{19683 \left (2 \sin \left (x \right )-1\right )^{3}}+\frac {416}{19683 \left (2 \sin \left (x \right )-1\right )^{2}}+\frac {14944}{177147 \left (2 \sin \left (x \right )-1\right )}-\frac {38656 \ln \left (2 \sin \left (x \right )-1\right )}{177147}\) | \(222\) |
| risch | \(-\frac {i \left (-6932549 \,{\mathrm e}^{i x}+302016 i {\mathrm e}^{18 i x}-228432 \,{\mathrm e}^{13 i x}+358512 \,{\mathrm e}^{15 i x}-367056 \,{\mathrm e}^{17 i x}-58960 i {\mathrm e}^{4 i x}-14277396 i {\mathrm e}^{2 i x}+27805588 i {\mathrm e}^{10 i x}-114192 i {\mathrm e}^{8 i x}+104480 i {\mathrm e}^{6 i x}+302112 i {\mathrm e}^{26 i x}-27872708 i {\mathrm e}^{28 i x}-412656 i {\mathrm e}^{24 i x}+302016 i {\mathrm e}^{22 i x}+85530600 i {\mathrm e}^{20 i x}-84240 \,{\mathrm e}^{5 i x}+6962869 \,{\mathrm e}^{3 i x}-57060192 \,{\mathrm e}^{11 i x}-97552 \,{\mathrm e}^{9 i x}+123632 \,{\mathrm e}^{7 i x}+41817870 \,{\mathrm e}^{19 i x}+302112 i {\mathrm e}^{14 i x}-27872708 i {\mathrm e}^{12 i x}-412656 i {\mathrm e}^{16 i x}-41817870 \,{\mathrm e}^{21 i x}+367056 \,{\mathrm e}^{23 i x}+228432 \,{\mathrm e}^{27 i x}-358512 \,{\mathrm e}^{25 i x}+57060192 \,{\mathrm e}^{29 i x}+104480 i {\mathrm e}^{34 i x}-58960 i {\mathrm e}^{36 i x}-114192 i {\mathrm e}^{32 i x}+27805588 i {\mathrm e}^{30 i x}-14277396 i {\mathrm e}^{38 i x}+6932549 \,{\mathrm e}^{39 i x}-6962869 \,{\mathrm e}^{37 i x}+84240 \,{\mathrm e}^{35 i x}+97552 \,{\mathrm e}^{31 i x}-123632 \,{\mathrm e}^{33 i x}\right )}{78732 \left (-i {\mathrm e}^{9 i x}+{\mathrm e}^{10 i x}+i {\mathrm e}^{i x}+1\right )^{4}}-\frac {11703 \ln \left ({\mathrm e}^{i x}-i\right )}{8}+\frac {3949 \ln \left ({\mathrm e}^{i x}+i\right )}{1417176}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (5559060566555523 \textit {\_Z}^{3}-4067310684152426112 \textit {\_Z}^{2}+39408304152268800 \textit {\_Z} +223264385007616\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (\frac {8365531489507611}{91466123970719744} i \textit {\_R}^{2}-\frac {95640455635235829}{1429158187042496} i \textit {\_R} +\frac {28223399093036}{22330596672539} i\right ) {\mathrm e}^{i x}-1\right )\right )-\frac {38656 \ln \left (-i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{177147}\) | \(411\) |
Input:
int(1/(cos(5*x)+sin(4*x))^5,x,method=_RETURNVERBOSE)
Output:
0
Result contains complex when optimal does not.
Time = 1.43 (sec) , antiderivative size = 2122, normalized size of antiderivative = 13.78 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^5,x, algorithm="fricas")
Output:
-1/502096953744*(1491781604010098688*cos(x)^18 - 6711477101007077376*cos(x )^16 + 12580064525026344960*cos(x)^14 - 12714176602821500928*cos(x)^12 + 7 476036712105789440*cos(x)^10 - 2557995116180228352*cos(x)^8 + 472145715040 915008*cos(x)^6 - 36398752187700384*cos(x)^4 + 8*(65536*cos(x)^20 - 425984 *cos(x)^18 + 1142784*cos(x)^16 - 1665024*cos(x)^14 + 1447680*cos(x)^12 - 7 71968*cos(x)^10 + 247456*cos(x)^8 - 43760*cos(x)^6 + 3281*cos(x)^4 + 16*(8 192*cos(x)^18 - 36864*cos(x)^16 + 69120*cos(x)^14 - 69888*cos(x)^12 + 4115 2*cos(x)^10 - 14136*cos(x)^8 + 2630*cos(x)^6 - 205*cos(x)^4)*sin(x))*(3138 1059609*(2926915967063031808/5559060566555523*I*sqrt(3) + 8063611306999389 0807808/5559060566555523)^(1/3)*(-I*sqrt(3) + 1) + 1866464456667136*(I*sqr t(3) + 1)/(2926915967063031808/5559060566555523*I*sqrt(3) + 80636113069993 890807808/5559060566555523)^(1/3) - 15306725240064)*log(266579/12552423843 6*(31381059609*(2926915967063031808/5559060566555523*I*sqrt(3) + 806361130 69993890807808/5559060566555523)^(1/3)*(-I*sqrt(3) + 1) + 1866464456667136 *(I*sqrt(3) + 1)/(2926915967063031808/5559060566555523*I*sqrt(3) + 8063611 3069993890807808/5559060566555523)^(1/3) - 15306725240064)^2 + 30604945803 27546528*(2926915967063031808/5559060566555523*I*sqrt(3) + 806361130699938 90807808/5559060566555523)^(1/3)*(-I*sqrt(3) + 1) + 1074870842483694059084 5657088/59049*(I*sqrt(3) + 1)/(2926915967063031808/5559060566555523*I*sqrt (3) + 80636113069993890807808/5559060566555523)^(1/3) + 182932247941439...
\[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^5} \, dx=\int \frac {1}{\left (\sin {\left (4 x \right )} + \cos {\left (5 x \right )}\right )^{5}}\, dx \] Input:
integrate(1/(cos(5*x)+sin(4*x))**5,x)
Output:
Integral((sin(4*x) + cos(5*x))**(-5), x)
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^5} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^5,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^5} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^5,x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: 2*(( 777430272*sin(sageVARx)^4-17980525*sin(sageVARx)^3-1572752288*sin(sageVARx )^2+18157675*sin(sageVARx)+795499160)*1/2834352/(sin(sageVARx)^2-1)^2+(-66 34071654400*sin(sag
Time = 23.44 (sec) , antiderivative size = 1216, normalized size of antiderivative = 7.90 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^5} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(4*x))^5,x)
Output:
(3949*log(tan(x/2) + 1))/1417176 - (11703*log(tan(x/2) - 1))/8 - (38656*lo g(tan(x/2)^2 - 4*tan(x/2) + 1))/177147 + symsum(log((8796093022208*(197852 47954114645806985184845140560244969046016*sin(x) - 36163269112055873708639 927590165532859936276480*cos(x) - 8277029935951126161367222769389600356376 239931392*root(z^3 - (59264*z^2)/81 + (74153676800*z)/10460353203 + 223264 385007616/5559060566555523, z, k) + 56113113917200533189968426093082800496 68041211904*root(z^3 - (59264*z^2)/81 + (74153676800*z)/10460353203 + 2232 64385007616/5559060566555523, z, k)*cos(x) - 75124509721717544680531658023 06492880839117897728*root(z^3 - (59264*z^2)/81 + (74153676800*z)/104603532 03 + 223264385007616/5559060566555523, z, k)*sin(x) - 50195909030906274134 83968508646945045343513278939136*root(z^3 - (59264*z^2)/81 + (74153676800* z)/10460353203 + 223264385007616/5559060566555523, z, k)^2 + 4246724871726 1695680059742055928540924381249156153344*root(z^3 - (59264*z^2)/81 + (7415 3676800*z)/10460353203 + 223264385007616/5559060566555523, z, k)^3 + 24130 948804405116934234899534395989301847951601299987456*root(z^3 - (59264*z^2) /81 + (74153676800*z)/10460353203 + 223264385007616/5559060566555523, z, k )^4 + 109091257580509358745927962491452485337567430697523139008*root(z^3 - (59264*z^2)/81 + (74153676800*z)/10460353203 + 223264385007616/5559060566 555523, z, k)^5 - 555954874424632137152452856127196972464769411669929903*r oot(z^3 - (59264*z^2)/81 + (74153676800*z)/10460353203 + 22326438500761...
\[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^5} \, dx=\int \frac {1}{\cos \left (5 x \right )^{5}+5 \cos \left (5 x \right )^{4} \sin \left (4 x \right )+10 \cos \left (5 x \right )^{3} \sin \left (4 x \right )^{2}+10 \cos \left (5 x \right )^{2} \sin \left (4 x \right )^{3}+5 \cos \left (5 x \right ) \sin \left (4 x \right )^{4}+\sin \left (4 x \right )^{5}}d x \] Input:
int(1/(cos(5*x)+sin(4*x))^5,x)
Output:
int(1/(cos(5*x)**5 + 5*cos(5*x)**4*sin(4*x) + 10*cos(5*x)**3*sin(4*x)**2 + 10*cos(5*x)**2*sin(4*x)**3 + 5*cos(5*x)*sin(4*x)**4 + sin(4*x)**5),x)