Integrand size = 11, antiderivative size = 218 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^6} \, dx=-\frac {33243}{128} \text {arctanh}(2 \cos (x) \sin (x))-\frac {715 \cot (x)}{139968}-\frac {11 \cot ^3(x)}{69984}-\frac {\cot ^5(x)}{233280}-\frac {109312 \log \left (\sqrt {3} \cos (x)-\sin (x)\right )}{243 \sqrt {3}}+\frac {109312 \log \left (\sqrt {3} \cos (x)+\sin (x)\right )}{243 \sqrt {3}}+\frac {32 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^5}-\frac {2 \tan (x) \left (76795+61567 \tan ^2(x)\right )}{405 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^4}+\frac {\tan (x) \left (778363-606643 \tan ^2(x)\right )}{2430 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^3}-\frac {\tan (x) \left (91032631-43437157 \tan ^2(x)\right )}{174960 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^2}+\frac {\tan (x) \left (111184863-53226455 \tan ^2(x)\right )}{139968 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )} \] Output:
-33243/128*arctanh(2*cos(x)*sin(x))-715/139968*cot(x)-11/69984*cot(x)^3-1/ 233280*cot(x)^5-109312/729*ln(3^(1/2)*cos(x)-sin(x))*3^(1/2)+109312/729*ln (3^(1/2)*cos(x)+sin(x))*3^(1/2)+32/405*tan(x)*(16627-16465*tan(x)^2)/(3-4* tan(x)^2+tan(x)^4)^5-2/405*tan(x)*(76795+61567*tan(x)^2)/(3-4*tan(x)^2+tan (x)^4)^4+1/2430*tan(x)*(778363-606643*tan(x)^2)/(3-4*tan(x)^2+tan(x)^4)^3- 1/174960*tan(x)*(91032631-43437157*tan(x)^2)/(3-4*tan(x)^2+tan(x)^4)^2+tan (x)*(111184863-53226455*tan(x)^2)/(419904-559872*tan(x)^2+139968*tan(x)^4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.11 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.19 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^6} \, dx=\frac {17013137182720 \sqrt {3} \text {arctanh}\left (\frac {-2+\tan \left (\frac {x}{2}\right )}{\sqrt {3}}\right )-699840 i \text {RootSum}\left [i+\text {$\#$1}-i \text {$\#$1}^2-\text {$\#$1}^3+i \text {$\#$1}^4+\text {$\#$1}^5-i \text {$\#$1}^6\&,\frac {32920146 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-16460073 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )-73970362 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}-36985181 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-92239970 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2+46119985 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+73970362 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3+36985181 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+32920146 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4-16460073 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{i+2 \text {$\#$1}-3 i \text {$\#$1}^2-4 \text {$\#$1}^3+5 i \text {$\#$1}^4+6 \text {$\#$1}^5}\&\right ]+\frac {3 \sec ^5(x) (-1921829222123+617908487785 \cos (2 x)+2410067577070 \cos (4 x)+832867940630 \cos (6 x)+640082095910 \cos (8 x)-1775875210157 \cos (10 x)-442786557220 \cos (12 x)+1152877214505 \cos (14 x)+71920822420 \cos (16 x)-31579364460 \cos (18 x)-153758708336 \cos (20 x)-140817863340 \cos (22 x)+51684808280 \cos (24 x)+128408941550 \sin (x)+2897605989350 \sin (3 x)+785145965921 \sin (5 x)-2882604224865 \sin (7 x)+141794432890 \sin (9 x)+898151545730 \sin (11 x)+627827561740 \sin (13 x)+550510980460 \sin (15 x)-519369889645 \sin (17 x)-95260796820 \sin (19 x)+192101628320 \sin (21 x)+33425806160 \sin (23 x)+45921631644 \sin (25 x))}{(1-2 \cos (2 x)+2 \cos (4 x)+2 \sin (x))^5}}{61261515} \] Input:
Integrate[(Cos[5*x] + Sin[2*x])^(-6),x]
Output:
(17013137182720*Sqrt[3]*ArcTanh[(-2 + Tan[x/2])/Sqrt[3]] - (699840*I)*Root Sum[I + #1 - I*#1^2 - #1^3 + I*#1^4 + #1^5 - I*#1^6 & , (32920146*ArcTan[S in[x]/(Cos[x] - #1)] - (16460073*I)*Log[1 - 2*Cos[x]*#1 + #1^2] - (7397036 2*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 - 36985181*Log[1 - 2*Cos[x]*#1 + #1^2 ]*#1 - 92239970*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^2 + (46119985*I)*Log[1 - 2 *Cos[x]*#1 + #1^2]*#1^2 + (73970362*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 + 36985181*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3 + 32920146*ArcTan[Sin[x]/(Cos[x ] - #1)]*#1^4 - (16460073*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^4)/(I + 2*#1 - (3*I)*#1^2 - 4*#1^3 + (5*I)*#1^4 + 6*#1^5) & ] + (3*Sec[x]^5*(-1921829222 123 + 617908487785*Cos[2*x] + 2410067577070*Cos[4*x] + 832867940630*Cos[6* x] + 640082095910*Cos[8*x] - 1775875210157*Cos[10*x] - 442786557220*Cos[12 *x] + 1152877214505*Cos[14*x] + 71920822420*Cos[16*x] - 31579364460*Cos[18 *x] - 153758708336*Cos[20*x] - 140817863340*Cos[22*x] + 51684808280*Cos[24 *x] + 128408941550*Sin[x] + 2897605989350*Sin[3*x] + 785145965921*Sin[5*x] - 2882604224865*Sin[7*x] + 141794432890*Sin[9*x] + 898151545730*Sin[11*x] + 627827561740*Sin[13*x] + 550510980460*Sin[15*x] - 519369889645*Sin[17*x ] - 95260796820*Sin[19*x] + 192101628320*Sin[21*x] + 33425806160*Sin[23*x] + 45921631644*Sin[25*x]))/(1 - 2*Cos[2*x] + 2*Cos[4*x] + 2*Sin[x])^5)/612 61515
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.
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (5 x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (5 x))^6}dx\) |
| \(\Big \downarrow \) 4830 |
| \(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^{29}}{\left (-\tan ^{10}\left (\frac {x}{2}\right )-4 \tan ^9\left (\frac {x}{2}\right )+45 \tan ^8\left (\frac {x}{2}\right )-8 \tan ^7\left (\frac {x}{2}\right )-210 \tan ^6\left (\frac {x}{2}\right )+210 \tan ^4\left (\frac {x}{2}\right )+8 \tan ^3\left (\frac {x}{2}\right )-45 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle 2 \int \left (\frac {65536 \left (209 \tan \left (\frac {x}{2}\right )-56\right )}{27 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^6}-\frac {768522368}{2187 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {128 \left (322988095 \tan ^4\left (\frac {x}{2}\right )+3725120624 \tan ^3\left (\frac {x}{2}\right )+8635810222 \tan ^2\left (\frac {x}{2}\right )+11201182896 \tan \left (\frac {x}{2}\right )+26470375487\right )}{117649 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {389}{470596 \left (\tan \left (\frac {x}{2}\right )-1\right )^2}+\frac {5093}{8748 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {512 \left (233471 \tan \left (\frac {x}{2}\right )-101068\right )}{729 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {8192 \left (182176425 \tan ^5\left (\frac {x}{2}\right )+2088404808 \tan ^4\left (\frac {x}{2}\right )+3924437826 \tan ^3\left (\frac {x}{2}\right )+6475622760 \tan ^2\left (\frac {x}{2}\right )-17846250263 \tan \left (\frac {x}{2}\right )+347855993696\right )}{16807 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {79}{941192 \left (\tan \left (\frac {x}{2}\right )-1\right )^3}-\frac {53}{1944 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {2048 \left (91204 \tan \left (\frac {x}{2}\right )-61483\right )}{729 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {1048576 \left (3786029776 \tan ^5\left (\frac {x}{2}\right )-12598578709 \tan ^4\left (\frac {x}{2}\right )+64731492616 \tan ^3\left (\frac {x}{2}\right )-846890873498 \tan ^2\left (\frac {x}{2}\right )+8385720705400 \tan \left (\frac {x}{2}\right )-83190365854405\right )}{2401 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {83}{941192 \left (\tan \left (\frac {x}{2}\right )-1\right )^4}+\frac {163}{5832 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {16384 \left (1707 \tan \left (\frac {x}{2}\right )-1082\right )}{81 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {67108864 \left (1830488005075 \tan ^5\left (\frac {x}{2}\right )-3310696109137 \tan ^4\left (\frac {x}{2}\right )+7827681087550 \tan ^3\left (\frac {x}{2}\right )-155814617630357 \tan ^2\left (\frac {x}{2}\right )+1423387288659059 \tan \left (\frac {x}{2}\right )-13167984088413732\right )}{343 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {1}{117649 \left (\tan \left (\frac {x}{2}\right )-1\right )^5}-\frac {1}{729 \left (\tan \left (\frac {x}{2}\right )+1\right )^5}-\frac {16384 \left (656 \tan \left (\frac {x}{2}\right )-417\right )}{27 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^5}-\frac {1073741824 \left (1086933284335984 \tan ^5\left (\frac {x}{2}\right )-1336922294439261 \tan ^4\left (\frac {x}{2}\right )-1927157329105328 \tan ^3\left (\frac {x}{2}\right )-33051456758792506 \tan ^2\left (\frac {x}{2}\right )+279813042087135648 \tan \left (\frac {x}{2}\right )-2475056651618557309\right )}{49 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^5}+\frac {1}{235298 \left (\tan \left (\frac {x}{2}\right )-1\right )^6}+\frac {1}{1458 \left (\tan \left (\frac {x}{2}\right )+1\right )^6}-\frac {68719476736 \left (49035220481070655 \tan ^5\left (\frac {x}{2}\right )-42917855228020264 \tan ^4\left (\frac {x}{2}\right )-256551370056897522 \tan ^3\left (\frac {x}{2}\right )-76734203888959108 \tan ^2\left (\frac {x}{2}\right )+43577028219318607 \tan \left (\frac {x}{2}\right )+5524942465542500\right )}{7 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^6}\right )d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {50613248 \log \left (-\tan \left (\frac {x}{2}\right )-\sqrt {3}+2\right )}{243 \sqrt {3}}-\frac {50613248 \log \left (-\tan \left (\frac {x}{2}\right )+\sqrt {3}+2\right )}{243 \sqrt {3}}+\frac {12339685306688503487471288320}{21} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )-\frac {52789542741040794418907447296}{21} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )-\frac {75599058295242768362013458432}{7} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )-\frac {34721314301791016942519189504}{21} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )+\frac {10891624503367739373535625216}{3} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^6}d\tan \left (\frac {x}{2}\right )+\frac {7977383873745883751998554112}{147} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^5}d\tan \left (\frac {x}{2}\right )-\frac {916513013023649384013758464}{147} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^5}d\tan \left (\frac {x}{2}\right )+\frac {7478674011101471294095360}{49} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^5}d\tan \left (\frac {x}{2}\right )-\frac {24136420632454412789350400}{147} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^5}d\tan \left (\frac {x}{2}\right )+\frac {3949748956388147609993216}{21} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^5}d\tan \left (\frac {x}{2}\right )+\frac {2651556727912908187500544 \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )}{1029}-\frac {41166093934209903099904}{147} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )+\frac {7508334689698602549248}{343} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )-\frac {4769811591860741734400 \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )}{1029}+\frac {3123370576524401967104 \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )}{1029}+\frac {261710142958279327744 \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )}{7203}-\frac {26430805642615717888 \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )}{7203}+\frac {113250126509637632}{343} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )-\frac {306846107722317824 \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )}{7203}+\frac {119030500973084672 \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )}{7203}+\frac {2847646447992832 \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )}{16807}-\frac {139729461968896 \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )}{16807}+\frac {88865644216320 \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )}{16807}+\frac {45083035041792 \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )}{16807}+\frac {7158950363136 \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )}{16807}+\frac {3388208062336 \int \frac {1}{\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )}{117649}+\frac {1433751410688 \int \frac {\tan \left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )}{117649}+\frac {1105383708416 \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )}{117649}+\frac {476815439872 \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )}{117649}+\frac {41342476160 \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )}{117649}+\frac {389}{470596 \left (1-\tan \left (\frac {x}{2}\right )\right )}-\frac {5093}{8748 \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {256 \left (10445-121958 \tan \left (\frac {x}{2}\right )\right )}{729 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {48852352 \left (2-\tan \left (\frac {x}{2}\right )\right )}{2187 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}-\frac {248731545600}{16807 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )}-\frac {79}{1882384 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {53}{3888 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {11930368 \left (2-\tan \left (\frac {x}{2}\right )\right )}{729 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {512 \left (120925 \tan \left (\frac {x}{2}\right )+31762\right )}{2187 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {992484989599744}{7203 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {83}{2823576 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {163}{17496 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {47645696 \left (2-\tan \left (\frac {x}{2}\right )\right )}{3645 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {8192 \left (2332 \tan \left (\frac {x}{2}\right )+457\right )}{729 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {61420985293104742400}{3087 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {1}{470596 \left (1-\tan \left (\frac {x}{2}\right )\right )^4}+\frac {1}{2916 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {1482752 \left (2-\tan \left (\frac {x}{2}\right )\right )}{135 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {2048 \left (895 \tan \left (\frac {x}{2}\right )+178\right )}{81 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {145885715911153761124352}{147 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {1}{1176490 \left (1-\tan \left (\frac {x}{2}\right )\right )^5}-\frac {1}{7290 \left (\tan \left (\frac {x}{2}\right )+1\right )^5}+\frac {32768 \left (97-362 \tan \left (\frac {x}{2}\right )\right )}{405 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^5}+\frac {336967469309356560464478208}{21 \left (\tan ^6\left (\frac {x}{2}\right )+8 \tan ^5\left (\frac {x}{2}\right )-13 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-13 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^5}\right )\) |
Input:
Int[(Cos[5*x] + Sin[2*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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.86 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.68
\[\text {Expression too large to display}\]
Input:
int(1/(cos(5*x)+sin(2*x))^6,x)
Output:
-1/3645/(tan(1/2*x)+1)^5+1/1458/(tan(1/2*x)+1)^4-163/8748/(tan(1/2*x)+1)^3 +53/1944/(tan(1/2*x)+1)^2-5093/4374/(tan(1/2*x)+1)+256/117649*(1139608324* tan(1/2*x)+181317662625834*tan(1/2*x)^8+111701990444896/3*tan(1/2*x)^7-213 633013222/3*tan(1/2*x)^4-56009237441236/15*tan(1/2*x)^5-30410542112408/3*t an(1/2*x)^6+324943703648/3*tan(1/2*x)^3+17681293036*tan(1/2*x)^2+930083521 2690970/3*tan(1/2*x)^12-3568418391473792/3*tan(1/2*x)^11-6092129246546524/ 5*tan(1/2*x)^10-53732817434796*tan(1/2*x)^9+19856315688316396/3*tan(1/2*x) ^13+1262608900769456/3*tan(1/2*x)^14-48148710706309824/5*tan(1/2*x)^15-278 56977579524294/3*tan(1/2*x)^16-2342101723253980/3*tan(1/2*x)^17+1003157840 5774460/3*tan(1/2*x)^18+4606807164518240/3*tan(1/2*x)^19+136727106/5-32260 0338203844*tan(1/2*x)^21-1344782502216014/5*tan(1/2*x)^20-45791476*tan(1/2 *x)^29-1957675846*tan(1/2*x)^28-95242165568/3*tan(1/2*x)^27-650619023804/3 *tan(1/2*x)^26-407268246316/3*tan(1/2*x)^25+17860224325562/3*tan(1/2*x)^24 +69630966616544/3*tan(1/2*x)^23-33601935843720*tan(1/2*x)^22)/(tan(1/2*x)^ 6+8*tan(1/2*x)^5-13*tan(1/2*x)^4-48*tan(1/2*x)^3-13*tan(1/2*x)^2+8*tan(1/2 *x)+1)^5+384/16807*sum((13199839*_R^4+147940724*_R^3+289760846*_R^2+147940 724*_R+13199839)/(3*_R^5+20*_R^4-26*_R^3-72*_R^2-13*_R+4)*ln(tan(1/2*x)-_R ),_R=RootOf(_Z^6+8*_Z^5-13*_Z^4-48*_Z^3-13*_Z^2+8*_Z+1))-1/588245/(tan(1/2 *x)-1)^5-1/235298/(tan(1/2*x)-1)^4-83/1411788/(tan(1/2*x)-1)^3-79/941192/( tan(1/2*x)-1)^2-389/235298/(tan(1/2*x)-1)-256/2187*(1113407*tan(1/2*x)^...
Timed out. \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^6} \, dx=\text {Timed out} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^6,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^6} \, dx=\text {Timed out} \] Input:
integrate(1/(cos(5*x)+sin(2*x))**6,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^6} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(5*x)+sin(2*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. \(2 (188) = 376\).
Time = 0.17 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.56 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^6,x, algorithm="giac")
Output:
-101226496/729*sqrt(3)*log(abs(-2*sqrt(3) + 2*tan(1/2*x) - 4)/abs(2*sqrt(3 ) + 2*tan(1/2*x) - 4)) - 2/20420505*(2348072128025*tan(1/2*x)^49 + 5409679 0137800*tan(1/2*x)^48 + 13549152904760*tan(1/2*x)^47 - 6772739536801060*ta n(1/2*x)^46 - 16732042947484412*tan(1/2*x)^45 + 413382950827430660*tan(1/2 *x)^44 + 860079438673998120*tan(1/2*x)^43 - 15446974901266156660*tan(1/2*x )^42 - 12771099815592707870*tan(1/2*x)^41 + 346843324808711498884*tan(1/2* x)^40 - 128616301300448545880*tan(1/2*x)^39 - 4261744436703269881340*tan(1 /2*x)^38 + 4087003099790949762260*tan(1/2*x)^37 + 32321841562360517280860* tan(1/2*x)^36 - 39300124477699353280392*tan(1/2*x)^35 - 166464525360402634 977420*tan(1/2*x)^34 + 210043995965200725907815*tan(1/2*x)^33 + 6152633010 71302396476740*tan(1/2*x)^32 - 712900843785776889116240*tan(1/2*x)^31 - 16 68740428753947443177832*tan(1/2*x)^30 + 1622442220962425618684040*tan(1/2* x)^29 + 3324257502931363090223400*tan(1/2*x)^28 - 255948656991511101952408 0*tan(1/2*x)^27 - 4839760037792021284833800*tan(1/2*x)^26 + 28751551212638 24099638940*tan(1/2*x)^25 + 5128039883029583065246440*tan(1/2*x)^24 - 2362 769526621783819276080*tan(1/2*x)^23 - 3940075773402854604723320*tan(1/2*x) ^22 + 1464953473634168374848840*tan(1/2*x)^21 + 2183930134431309922808888* tan(1/2*x)^20 - 706814016978668423208080*tan(1/2*x)^19 - 86501756106011011 0030360*tan(1/2*x)^18 + 269100700664240373408375*tan(1/2*x)^17 + 240215839 105178851609040*tan(1/2*x)^16 - 79060948227561609921640*tan(1/2*x)^15 -...
Time = 36.40 (sec) , antiderivative size = 1478, normalized size of antiderivative = 6.78 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^6} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(2*x))^6,x)
Output:
symsum(log(1247391969916038497823910618605161911090410346908745728 - root( z^6 - (114375530107022229504*z^4)/1977326743 + (86370274106142777748767280 98816*z^2)/27368747340080916343 - 57301151339511992506470305855843598336/3 78818692265664781682717625943, z, k)*(585418123911245063659126011685586187 73539640919797530624*exp(x*1i) - root(z^6 - (114375530107022229504*z^4)/19 77326743 + (8637027410614277774876728098816*z^2)/27368747340080916343 - 57 301151339511992506470305855843598336/378818692265664781682717625943, z, k) *(exp(x*1i)*2940635903824389143597992720127636661043972641128448i + root(z ^6 - (114375530107022229504*z^4)/1977326743 + (863702741061427777487672809 8816*z^2)/27368747340080916343 - 57301151339511992506470305855843598336/37 8818692265664781682717625943, z, k)*(1516893653059746760696739190383151125 8935529652210368512*exp(x*1i) - root(z^6 - (114375530107022229504*z^4)/197 7326743 + (8637027410614277774876728098816*z^2)/27368747340080916343 - 573 01151339511992506470305855843598336/378818692265664781682717625943, z, k)* (exp(x*1i)*538997260307260682351482578655848285668872040742912i - root(z^6 - (114375530107022229504*z^4)/1977326743 + (86370274106142777748767280988 16*z^2)/27368747340080916343 - 57301151339511992506470305855843598336/3788 18692265664781682717625943, z, k)*(root(z^6 - (114375530107022229504*z^4)/ 1977326743 + (8637027410614277774876728098816*z^2)/27368747340080916343 - 57301151339511992506470305855843598336/378818692265664781682717625943, ...
\[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^6} \, dx=\int \frac {1}{\cos \left (5 x \right )^{6}+6 \cos \left (5 x \right )^{5} \sin \left (2 x \right )+15 \cos \left (5 x \right )^{4} \sin \left (2 x \right )^{2}+20 \cos \left (5 x \right )^{3} \sin \left (2 x \right )^{3}+15 \cos \left (5 x \right )^{2} \sin \left (2 x \right )^{4}+6 \cos \left (5 x \right ) \sin \left (2 x \right )^{5}+\sin \left (2 x \right )^{6}}d x \] Input:
int(1/(cos(5*x)+sin(2*x))^6,x)
Output:
int(1/(cos(5*x)**6 + 6*cos(5*x)**5*sin(2*x) + 15*cos(5*x)**4*sin(2*x)**2 + 20*cos(5*x)**3*sin(2*x)**3 + 15*cos(5*x)**2*sin(2*x)**4 + 6*cos(5*x)*sin( 2*x)**5 + sin(2*x)**6),x)