Integrand size = 11, antiderivative size = 154 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^4} \, dx=-\frac {103}{8} \text {arctanh}(2 \cos (x) \sin (x))-\frac {43 \cot (x)}{3888}-\frac {\cot ^3(x)}{3888}-\frac {1808 \log \left (\sqrt {3} \cos (x)-\sin (x)\right )}{81 \sqrt {3}}+\frac {1808 \log \left (\sqrt {3} \cos (x)+\sin (x)\right )}{81 \sqrt {3}}+\frac {4 \tan (x) \left (593-539 \tan ^2(x)\right )}{81 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^3}-\frac {2 \tan (x) \left (3502-1237 \tan ^2(x)\right )}{243 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^2}+\frac {\tan (x) \left (38097-18413 \tan ^2(x)\right )}{972 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )} \] Output:
-103/8*arctanh(2*cos(x)*sin(x))-43/3888*cot(x)-1/3888*cot(x)^3-1808/243*ln (3^(1/2)*cos(x)-sin(x))*3^(1/2)+1808/243*ln(3^(1/2)*cos(x)+sin(x))*3^(1/2) +4/81*tan(x)*(593-539*tan(x)^2)/(3-4*tan(x)^2+tan(x)^4)^3-2/243*tan(x)*(35 02-1237*tan(x)^2)/(3-4*tan(x)^2+tan(x)^4)^2+tan(x)*(38097-18413*tan(x)^2)/ (2916-3888*tan(x)^2+972*tan(x)^4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 2.70 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.26 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^4} \, dx=\frac {287392 \text {arctanh}\left (\frac {-2+\tan \left (\frac {x}{2}\right )}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {8}{343} i \text {RootSum}\left [i+\text {$\#$1}-i \text {$\#$1}^2-\text {$\#$1}^3+i \text {$\#$1}^4+\text {$\#$1}^5-i \text {$\#$1}^6\&,\frac {68690 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-34345 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )-154276 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}-77138 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-192440 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2+96220 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+154276 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3+77138 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+68690 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4-34345 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{i+2 \text {$\#$1}-3 i \text {$\#$1}^2-4 \text {$\#$1}^3+5 i \text {$\#$1}^4+6 \text {$\#$1}^5}\&\right ]+\frac {-50806812-46050 \cos (x)+47365479 \cos (2 x)+85384068 \cos (4 x)+138150 \cos (5 x)+17105655 \cos (6 x)+34460531 \cos (8 x)-46050 \cos (9 x)-28082600 \cos (10 x)-15943473 \cos (12 x)+16718128 \cos (14 x)+15350 \cos (15 x)-13261101 \sin (x)+138150 \sin (2 x)+84763014 \sin (3 x)+47550321 \sin (5 x)-15350 \sin (6 x)-67506418 \sin (7 x)-46050 \sin (8 x)+15795675 \sin (9 x)+28666610 \sin (11 x)+46050 \sin (12 x)+10801000 \sin (13 x)+14931819 \sin (15 x)}{388962 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^3 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3 (-1+2 \sin (x))^3 (1-2 \cos (2 x)+2 \sin (x)-2 \sin (3 x))^3} \] Input:
Integrate[(Cos[5*x] + Sin[2*x])^(-4),x]
Output:
(287392*ArcTanh[(-2 + Tan[x/2])/Sqrt[3]])/(81*Sqrt[3]) - ((8*I)/343)*RootS um[I + #1 - I*#1^2 - #1^3 + I*#1^4 + #1^5 - I*#1^6 & , (68690*ArcTan[Sin[x ]/(Cos[x] - #1)] - (34345*I)*Log[1 - 2*Cos[x]*#1 + #1^2] - (154276*I)*ArcT an[Sin[x]/(Cos[x] - #1)]*#1 - 77138*Log[1 - 2*Cos[x]*#1 + #1^2]*#1 - 19244 0*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^2 + (96220*I)*Log[1 - 2*Cos[x]*#1 + #1^2 ]*#1^2 + (154276*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 + 77138*Log[1 - 2*Co s[x]*#1 + #1^2]*#1^3 + 68690*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^4 - (34345*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) & ] + (-50806812 - 46050*Cos[x] + 47365479*Cos[2*x] + 8538 4068*Cos[4*x] + 138150*Cos[5*x] + 17105655*Cos[6*x] + 34460531*Cos[8*x] - 46050*Cos[9*x] - 28082600*Cos[10*x] - 15943473*Cos[12*x] + 16718128*Cos[14 *x] + 15350*Cos[15*x] - 13261101*Sin[x] + 138150*Sin[2*x] + 84763014*Sin[3 *x] + 47550321*Sin[5*x] - 15350*Sin[6*x] - 67506418*Sin[7*x] - 46050*Sin[8 *x] + 15795675*Sin[9*x] + 28666610*Sin[11*x] + 46050*Sin[12*x] + 10801000* Sin[13*x] + 14931819*Sin[15*x])/(388962*(Cos[x/2] - Sin[x/2])^3*(Cos[x/2] + Sin[x/2])^3*(-1 + 2*Sin[x])^3*(1 - 2*Cos[2*x] + 2*Sin[x] - 2*Sin[3*x])^3 )
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.
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (5 x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (5 x))^4}dx\) |
| \(\Big \downarrow \) 4830 |
| \(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^{19}}{\left (-\tan ^{10}\left (\frac {x}{2}\right )-4 \tan ^9\left (\frac {x}{2}\right )+45 \tan ^8\left (\frac {x}{2}\right )-8 \tan ^7\left (\frac {x}{2}\right )-210 \tan ^6\left (\frac {x}{2}\right )+210 \tan ^4\left (\frac {x}{2}\right )+8 \tan ^3\left (\frac {x}{2}\right )-45 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle 2 \int \left (\frac {1024 \left (15 \tan \left (\frac {x}{2}\right )-4\right )}{9 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {362960}{243 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {64 \left (56067 \tan ^4\left (\frac {x}{2}\right )+645324 \tan ^3\left (\frac {x}{2}\right )+1496310 \tan ^2\left (\frac {x}{2}\right )+1796684 \tan \left (\frac {x}{2}\right )+5493443\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 )}+\frac {13}{4802 \left (\tan \left (\frac {x}{2}\right )-1\right )^2}+\frac {79}{486 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {64 \left (301 \tan \left (\frac {x}{2}\right )-124\right )}{27 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {4096 \left (1346 \tan ^5\left (\frac {x}{2}\right )-441294 \tan ^4\left (\frac {x}{2}\right )+97348 \tan ^3\left (\frac {x}{2}\right )-9061157 \tan ^2\left (\frac {x}{2}\right )+82003026 \tan \left (\frac {x}{2}\right )-811301751\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 )^2}+\frac {1}{4802 \left (\tan \left (\frac {x}{2}\right )-1\right )^3}-\frac {1}{162 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {256 \left (112 \tan \left (\frac {x}{2}\right )-79\right )}{27 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {131072 \left (34685044 \tan ^5\left (\frac {x}{2}\right )-55290807 \tan ^4\left (\frac {x}{2}\right )+54522356 \tan ^3\left (\frac {x}{2}\right )-2033210910 \tan ^2\left (\frac {x}{2}\right )+17626524736 \tan \left (\frac {x}{2}\right )-156193140687\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 )^3}+\frac {1}{4802 \left (\tan \left (\frac {x}{2}\right )-1\right )^4}+\frac {1}{162 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {33554432 \left (773578863 \tan ^5\left (\frac {x}{2}\right )-677121297 \tan ^4\left (\frac {x}{2}\right )-4047449272 \tan ^3\left (\frac {x}{2}\right )-1210579237 \tan ^2\left (\frac {x}{2}\right )+687486711 \tan \left (\frac {x}{2}\right )+87163372\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 )^4}\right )d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {71848 \log \left (-\tan \left (\frac {x}{2}\right )-\sqrt {3}+2\right )}{81 \sqrt {3}}-\frac {71848 \log \left (-\tan \left (\frac {x}{2}\right )+\sqrt {3}+2\right )}{81 \sqrt {3}}+\frac {31684615034896384}{7} \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 )-\frac {135548556634226688}{7} \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 {582347685835571200}{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 )^4}d\tan \left (\frac {x}{2}\right )-\frac {89150799707373568}{7} \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 )+\frac {195767082883743744}{7} \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 )+\frac {61435826960728064}{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 )^3}d\tan \left (\frac {x}{2}\right )-\frac {6990132645724160}{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 )^3}d\tan \left (\frac {x}{2}\right )+\frac {157387306303488}{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 )^3}d\tan \left (\frac {x}{2}\right )-\frac {139641253003264}{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 )^3}d\tan \left (\frac {x}{2}\right )+\frac {112665991708672}{147} \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 )+\frac {9969297969152 \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 )}{1029}-\frac {1007724855296 \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 )}{1029}+\frac {36982181888}{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 )^2}d\tan \left (\frac {x}{2}\right )-\frac {191365120}{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 )^2}d\tan \left (\frac {x}{2}\right )+\frac {5532884992 \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 )}{1029}+\frac {351580352 \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 )}{2401}+\frac {114987776 \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 )}{2401}+\frac {95763840 \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 )}{2401}+\frac {41300736 \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 )}{2401}+\frac {3588288 \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 )}{2401}+\frac {13}{4802 \left (1-\tan \left (\frac {x}{2}\right )\right )}-\frac {79}{486 \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {32 \left (53-478 \tan \left (\frac {x}{2}\right )\right )}{81 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {22240 \left (2-\tan \left (\frac {x}{2}\right )\right )}{243 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {2756608}{1029 \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 {1}{9604 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {1}{324 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {16640 \left (2-\tan \left (\frac {x}{2}\right )\right )}{243 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {64 \left (145 \tan \left (\frac {x}{2}\right )+46\right )}{81 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {1136559521792}{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 )^2}+\frac {1}{14406 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {1}{486 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {512 \left (7-26 \tan \left (\frac {x}{2}\right )\right )}{81 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {1442055519731712}{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 )^3}\right )\) |
Input:
Int[(Cos[5*x] + Sin[2*x])^(-4),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 = 113.99 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.01
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| risch | \(\frac {\frac {2764 i}{9}+\frac {4776608 \,{\mathrm e}^{i x}}{27783}-\frac {8072668 i {\mathrm e}^{18 i x}}{9261}-\frac {306952 \,{\mathrm e}^{13 i x}}{343}-\frac {9677488 \,{\mathrm e}^{15 i x}}{9261}+\frac {641100 \,{\mathrm e}^{17 i x}}{343}+\frac {8190460 i {\mathrm e}^{4 i x}}{27783}+\frac {3086000 i {\mathrm e}^{2 i x}}{27783}+\frac {641752 i {\mathrm e}^{10 i x}}{343}-\frac {2755364 i {\mathrm e}^{8 i x}}{3969}-\frac {102292 i {\mathrm e}^{6 i x}}{343}-\frac {8190460 i {\mathrm e}^{26 i x}}{27783}-\frac {3086000 i {\mathrm e}^{28 i x}}{27783}-\frac {641176 i {\mathrm e}^{24 i x}}{1029}+\frac {2755364 i {\mathrm e}^{22 i x}}{3969}+\frac {306300 i {\mathrm e}^{20 i x}}{343}-\frac {8023600 \,{\mathrm e}^{5 i x}}{27783}-\frac {91820 \,{\mathrm e}^{3 i x}}{147}+\frac {1161688 \,{\mathrm e}^{11 i x}}{1323}+\frac {435884 \,{\mathrm e}^{9 i x}}{1323}+\frac {22644568 \,{\mathrm e}^{7 i x}}{27783}+\frac {1161688 \,{\mathrm e}^{19 i x}}{1323}-\frac {5529196 i {\mathrm e}^{14 i x}}{9261}+\frac {8072668 i {\mathrm e}^{12 i x}}{9261}-\frac {3003272 i {\mathrm e}^{16 i x}}{9261}+\frac {29576 \,{\mathrm e}^{21 i x}}{1323}-\frac {2952836 \,{\mathrm e}^{23 i x}}{27783}+\frac {43616 \,{\mathrm e}^{27 i x}}{147}-\frac {8023600 \,{\mathrm e}^{25 i x}}{27783}+\frac {4776608 \,{\mathrm e}^{29 i x}}{27783}}{\left (i {\mathrm e}^{8 i x}+{\mathrm e}^{9 i x}-2 i {\mathrm e}^{6 i x}-{\mathrm e}^{7 i x}+2 i {\mathrm e}^{4 i x}+2 \,{\mathrm e}^{5 i x}-i {\mathrm e}^{2 i x}-2 \,{\mathrm e}^{3 i x}+i+{\mathrm e}^{i x}\right )^{3} \left ({\mathrm e}^{i x}-i\right )^{3}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (558545864083284007 \textit {\_Z}^{6}-585974768513164114355712 \textit {\_Z}^{4}+244298699912207587540992 \textit {\_Z}^{2}-208847362607898689536\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}-\frac {12672570023770820579462826493 \textit {\_R}^{5}}{162436567931301495005603721904128}-\frac {611154271398035239 i \textit {\_R}^{4}}{392380888670297655017472}+\frac {25966583426590654714577002564727 \textit {\_R}^{3}}{317258921740823232432819769344}+\frac {5009110024837374701833 i \textit {\_R}^{2}}{3065475692736700429824}-\frac {27578316087245713379496605195 \textit {\_R}}{826195108700060501127134816}+\frac {2647818291985071967 i}{11974514424752736054}\right )\right )+\frac {143696 \sqrt {3}\, \ln \left ({\mathrm e}^{i x}-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )}{243}-\frac {143696 \sqrt {3}\, \ln \left ({\mathrm e}^{i x}-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )}{243}\) | \(397\) |
| default | \(-\frac {1}{7203 \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{4802 \left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {13}{2401 \left (\tan \left (\frac {x}{2}\right )-1\right )}-\frac {1}{243 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{162 \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {79}{243 \left (\tan \left (\frac {x}{2}\right )+1\right )}-\frac {32 \left (4258 \tan \left (\frac {x}{2}\right )^{5}-37162 \tan \left (\frac {x}{2}\right )^{4}+98648 \tan \left (\frac {x}{2}\right )^{3}-77180 \tan \left (\frac {x}{2}\right )^{2}+22646 \tan \left (\frac {x}{2}\right )-2242\right )}{243 \left (\tan \left (\frac {x}{2}\right )^{2}-4 \tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {287392 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (2 \tan \left (\frac {x}{2}\right )-4\right ) \sqrt {3}}{6}\right )}{243}+\frac {\frac {86144}{343}+\frac {15482048 \tan \left (\frac {x}{2}\right )}{2401}+\frac {17028608 \tan \left (\frac {x}{2}\right )^{2}}{343}+\frac {221038976 \tan \left (\frac {x}{2}\right )^{3}}{7203}-\frac {318767104 \tan \left (\frac {x}{2}\right )^{4}}{343}-\frac {625750656 \tan \left (\frac {x}{2}\right )^{5}}{343}+\frac {4255177216 \tan \left (\frac {x}{2}\right )^{6}}{1029}+\frac {4314996096 \tan \left (\frac {x}{2}\right )^{7}}{343}+\frac {2555392768 \tan \left (\frac {x}{2}\right )^{8}}{2401}-\frac {7370480640 \tan \left (\frac {x}{2}\right )^{9}}{343}-\frac {46677488128 \tan \left (\frac {x}{2}\right )^{10}}{2401}-\frac {543956352 \tan \left (\frac {x}{2}\right )^{11}}{343}+\frac {4420901888 \tan \left (\frac {x}{2}\right )^{12}}{1029}+\frac {452084352 \tan \left (\frac {x}{2}\right )^{13}}{343}-\frac {7772672 \tan \left (\frac {x}{2}\right )^{14}}{49}-\frac {684149120 \tan \left (\frac {x}{2}\right )^{15}}{7203}-\frac {3848576 \tan \left (\frac {x}{2}\right )^{16}}{343}-\frac {1009856 \tan \left (\frac {x}{2}\right )^{17}}{2401}}{\left (\tan \left (\frac {x}{2}\right )^{6}+8 \tan \left (\frac {x}{2}\right )^{5}-13 \tan \left (\frac {x}{2}\right )^{4}-48 \tan \left (\frac {x}{2}\right )^{3}-13 \tan \left (\frac {x}{2}\right )^{2}+8 \tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {32 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+8 \textit {\_Z}^{5}-13 \textit {\_Z}^{4}-48 \textit {\_Z}^{3}-13 \textit {\_Z}^{2}+8 \textit {\_Z} +1\right )}{\sum }\frac {\left (13765 \textit {\_R}^{4}+154276 \textit {\_R}^{3}+302290 \textit {\_R}^{2}+154276 \textit {\_R} +13765\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{3 \textit {\_R}^{5}+20 \textit {\_R}^{4}-26 \textit {\_R}^{3}-72 \textit {\_R}^{2}-13 \textit {\_R} +4}\right )}{343}\) | \(416\) |
Input:
int(1/(cos(5*x)+sin(2*x))^4,x,method=_RETURNVERBOSE)
Output:
0
Result contains complex when optimal does not.
Time = 1.94 (sec) , antiderivative size = 4154, normalized size of antiderivative = 26.97 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^4,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^4} \, dx=\int \frac {1}{\left (\sin {\left (2 x \right )} + \cos {\left (5 x \right )}\right )^{4}}\, dx \] Input:
integrate(1/(cos(5*x)+sin(2*x))**4,x)
Output:
Integral((sin(2*x) + cos(5*x))**(-4), x)
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^4} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^4,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. 398 vs. \(2 (130) = 260\).
Time = 0.16 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.58 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^4,x, algorithm="giac")
Output:
-143696/243*sqrt(3)*log(abs(-2*sqrt(3) + 2*tan(1/2*x) - 4)/abs(2*sqrt(3) + 2*tan(1/2*x) - 4)) - 2/27783*(13636631*tan(1/2*x)^29 + 204766588*tan(1/2* x)^28 - 553680134*tan(1/2*x)^27 - 15018201276*tan(1/2*x)^26 + 18647592429* tan(1/2*x)^25 + 420080085120*tan(1/2*x)^24 - 718319708124*tan(1/2*x)^23 - 4206791944584*tan(1/2*x)^22 + 7146174304719*tan(1/2*x)^21 + 22393376841796 *tan(1/2*x)^20 - 30960170007130*tan(1/2*x)^19 - 70996981497940*tan(1/2*x)^ 18 + 67072099281005*tan(1/2*x)^17 + 131929900367416*tan(1/2*x)^16 - 811159 14524552*tan(1/2*x)^15 - 144156490971824*tan(1/2*x)^14 + 59297186514317*ta n(1/2*x)^13 + 92704275150980*tan(1/2*x)^12 - 28321773645658*tan(1/2*x)^11 - 34467255328676*tan(1/2*x)^10 + 9389108574703*tan(1/2*x)^9 + 706790571412 8*tan(1/2*x)^8 - 2026230242268*tan(1/2*x)^7 - 693687479112*tan(1/2*x)^6 + 227742142605*tan(1/2*x)^5 + 20915657532*tan(1/2*x)^4 - 8078777862*tan(1/2* x)^3 - 455196620*tan(1/2*x)^2 + 104666039*tan(1/2*x) + 7585784)/(tan(1/2*x )^10 + 4*tan(1/2*x)^9 - 45*tan(1/2*x)^8 + 8*tan(1/2*x)^7 + 210*tan(1/2*x)^ 6 - 210*tan(1/2*x)^4 - 8*tan(1/2*x)^3 + 45*tan(1/2*x)^2 - 4*tan(1/2*x) - 1 )^3 - 0.0292685064921866*log(tan(1/2*x) + 8.87524547497000) + 0.6450220540 46647*log(tan(1/2*x) + 1.59149088298000) - 0.645022054029154*log(tan(1/2*x ) + 0.628341645367000) + 0.0292685064952187*log(tan(1/2*x) + 0.11267293990 0000) - 1024.25938420408*log(tan(1/2*x) - 0.349915133947000) + 1024.259384 20408*log(tan(1/2*x) - 2.85783580927000)
Time = 23.05 (sec) , antiderivative size = 864, normalized size of antiderivative = 5.61 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^4} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(2*x))^4,x)
Output:
((209332078*tan(x/2))/27783 - (910393240*tan(x/2)^2)/27783 - (5385851908*t an(x/2)^3)/9261 + (663989128*tan(x/2)^4)/441 + (151828095070*tan(x/2)^5)/9 261 - (17128085904*tan(x/2)^6)/343 - (1350820161512*tan(x/2)^7)/9261 + (67 3133877536*tan(x/2)^8)/1323 + (18778217149406*tan(x/2)^9)/27783 - (6893451 0657352*tan(x/2)^10)/27783 - (56643547291316*tan(x/2)^11)/27783 + (1854085 50301960*tan(x/2)^12)/27783 + (118594373028634*tan(x/2)^13)/27783 - (28831 2981943648*tan(x/2)^14)/27783 - (162231829049104*tan(x/2)^15)/27783 + (263 859800734832*tan(x/2)^16)/27783 + (19163456937430*tan(x/2)^17)/3969 - (141 993962995880*tan(x/2)^18)/27783 - (8845762859180*tan(x/2)^19)/3969 + (4478 6753683592*tan(x/2)^20)/27783 + (529346244794*tan(x/2)^21)/1029 - (9348426 54352*tan(x/2)^22)/3087 - (478879805416*tan(x/2)^23)/9261 + (280053390080* tan(x/2)^24)/9261 + (12431728286*tan(x/2)^25)/9261 - (10012134184*tan(x/2) ^26)/9261 - (158194324*tan(x/2)^27)/3969 + (409533176*tan(x/2)^28)/27783 + (27273262*tan(x/2)^29)/27783 + 15171568/27783)/(12*tan(x/2) - 87*tan(x/2) ^2 - 992*tan(x/2)^3 + 4737*tan(x/2)^4 + 27564*tan(x/2)^5 - 146823*tan(x/2) ^6 - 172416*tan(x/2)^7 + 1486293*tan(x/2)^8 + 295676*tan(x/2)^9 - 7495683* tan(x/2)^10 + 347808*tan(x/2)^11 + 21580005*tan(x/2)^12 - 1024932*tan(x/2) ^13 - 36395235*tan(x/2)^14 + 36395235*tan(x/2)^16 + 1024932*tan(x/2)^17 - 21580005*tan(x/2)^18 - 347808*tan(x/2)^19 + 7495683*tan(x/2)^20 - 295676*t an(x/2)^21 - 1486293*tan(x/2)^22 + 172416*tan(x/2)^23 + 146823*tan(x/2)...
\[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^4} \, dx=\int \frac {1}{\cos \left (5 x \right )^{4}+4 \cos \left (5 x \right )^{3} \sin \left (2 x \right )+6 \cos \left (5 x \right )^{2} \sin \left (2 x \right )^{2}+4 \cos \left (5 x \right ) \sin \left (2 x \right )^{3}+\sin \left (2 x \right )^{4}}d x \] Input:
int(1/(cos(5*x)+sin(2*x))^4,x)
Output:
int(1/(cos(5*x)**4 + 4*cos(5*x)**3*sin(2*x) + 6*cos(5*x)**2*sin(2*x)**2 + 4*cos(5*x)*sin(2*x)**3 + sin(2*x)**4),x)