Integrand size = 11, antiderivative size = 122 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^4} \, dx=-\frac {9}{128} \text {arctanh}(2 \cos (x) \sin (x))-\frac {37 \cot (x)}{4096}-\frac {11 \cot ^3(x)}{12288}+\frac {289 \tan (x)}{2048}+\frac {199 \tan ^3(x)}{6144}+\frac {163 \tan ^5(x)}{20480}+\frac {29 \tan ^7(x)}{28672}+\frac {\csc ^3(x) \sec ^{13}(x)}{12288 \left (1-\tan ^2(x)\right )^3}-\frac {\csc ^3(x) \sec ^{11}(x)}{12288 \left (1-\tan ^2(x)\right )^2}+\frac {5 \csc ^3(x) \sec ^9(x)}{6144 \left (1-\tan ^2(x)\right )} \] Output:
-9/128*arctanh(2*cos(x)*sin(x))-37/4096*cot(x)-11/12288*cot(x)^3+289/2048* tan(x)+199/6144*tan(x)^3+163/20480*tan(x)^5+29/28672*tan(x)^7+1/12288*csc( x)^3*sec(x)^13/(1-tan(x)^2)^3-1/12288*csc(x)^3*sec(x)^11/(1-tan(x)^2)^2+5* csc(x)^3*sec(x)^9/(6144-6144*tan(x)^2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.04 (sec) , antiderivative size = 515, normalized size of antiderivative = 4.22 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^4} \, dx=\frac {316872 i+13696 \sqrt {3} \text {arctanh}\left (\frac {-2+\tan \left (\frac {x}{2}\right )}{\sqrt {3}}\right )+32 i \text {RootSum}\left [i+\text {$\#$1}^3-i \text {$\#$1}^6\&,\frac {57926 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right )-28963 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right )-108818 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}-54409 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-146610 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^2+73305 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+108818 i \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^3+54409 \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+57926 \arctan \left (\frac {\sin (x)}{\cos (x)-\text {$\#$1}}\right ) \text {$\#$1}^4-28963 i \log \left (1-2 \cos (x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{i \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ]+\frac {6561}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}+\frac {13122 \sin \left (\frac {x}{2}\right )}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^3}+\frac {3201768 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}+\frac {2 \sin \left (\frac {x}{2}\right )}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3}-\frac {1}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}+\frac {168 \sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}-\frac {1088 \cos (x)}{(1-2 \sin (x))^2}+\frac {8256 \cos (x)}{1-2 \sin (x)}-\frac {384 \cos (x)}{(-1+2 \sin (x))^3}+\frac {10368 (35 \cos (x)-23 \cos (3 x)+43 \sin (2 x))}{(1+2 \sin (3 x))^3}+\frac {1728 (-723 \cos (x)+427 \cos (3 x)-821 \sin (2 x))}{(1+2 \sin (3 x))^2}+\frac {192 (21739 \cos (x)-12987 \cos (3 x)+24878 \sin (2 x))}{1+2 \sin (3 x)}}{78732} \] Input:
Integrate[(Cos[5*x] + Sin[4*x])^(-4),x]
Output:
(316872*I + 13696*Sqrt[3]*ArcTanh[(-2 + Tan[x/2])/Sqrt[3]] + (32*I)*RootSu m[I + #1^3 - I*#1^6 & , (57926*ArcTan[Sin[x]/(Cos[x] - #1)] - (28963*I)*Lo g[1 - 2*Cos[x]*#1 + #1^2] - (108818*I)*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 - 5 4409*Log[1 - 2*Cos[x]*#1 + #1^2]*#1 - 146610*ArcTan[Sin[x]/(Cos[x] - #1)]* #1^2 + (73305*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^2 + (108818*I)*ArcTan[Sin[ x]/(Cos[x] - #1)]*#1^3 + 54409*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3 + 57926*Ar cTan[Sin[x]/(Cos[x] - #1)]*#1^4 - (28963*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1 ^4)/(I*#1^2 + 2*#1^5) & ] + 6561/(Cos[x/2] - Sin[x/2])^2 + (13122*Sin[x/2] )/(Cos[x/2] - Sin[x/2])^3 + (3201768*Sin[x/2])/(Cos[x/2] - Sin[x/2]) + (2* Sin[x/2])/(Cos[x/2] + Sin[x/2])^3 - (Cos[x/2] + Sin[x/2])^(-2) + (168*Sin[ x/2])/(Cos[x/2] + Sin[x/2]) - (1088*Cos[x])/(1 - 2*Sin[x])^2 + (8256*Cos[x ])/(1 - 2*Sin[x]) - (384*Cos[x])/(-1 + 2*Sin[x])^3 + (10368*(35*Cos[x] - 2 3*Cos[3*x] + 43*Sin[2*x]))/(1 + 2*Sin[3*x])^3 + (1728*(-723*Cos[x] + 427*C os[3*x] - 821*Sin[2*x]))/(1 + 2*Sin[3*x])^2 + (192*(21739*Cos[x] - 12987*C os[3*x] + 24878*Sin[2*x]))/(1 + 2*Sin[3*x]))/78732
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.
| \(\displaystyle \int \frac {1}{(\sin (4 x)+\cos (5 x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (4 x)+\cos (5 x))^4}dx\) |
| \(\Big \downarrow \) 4830 |
| \(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^{19}}{\left (-\tan ^{10}\left (\frac {x}{2}\right )-8 \tan ^9\left (\frac {x}{2}\right )+45 \tan ^8\left (\frac {x}{2}\right )+48 \tan ^7\left (\frac {x}{2}\right )-210 \tan ^6\left (\frac {x}{2}\right )+210 \tan ^4\left (\frac {x}{2}\right )-48 \tan ^3\left (\frac {x}{2}\right )-45 \tan ^2\left (\frac {x}{2}\right )+8 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle 2 \int \left (\frac {1024 \left (15 \tan \left (\frac {x}{2}\right )-4\right )}{729 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {1520}{19683 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}-\frac {16 \left (295 \tan ^4\left (\frac {x}{2}\right )+4660 \tan ^3\left (\frac {x}{2}\right )+2606 \tan ^2\left (\frac {x}{2}\right )+123892 \tan \left (\frac {x}{2}\right )-2018009\right )}{243 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {41}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^2}+\frac {43}{39366 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {64 \left (19 \tan \left (\frac {x}{2}\right )+20\right )}{2187 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {1024 \left (54609 \tan ^5\left (\frac {x}{2}\right )-162772 \tan ^4\left (\frac {x}{2}\right )+2211554 \tan ^3\left (\frac {x}{2}\right )-30039908 \tan ^2\left (\frac {x}{2}\right )+379640689 \tan \left (\frac {x}{2}\right )-4752012176\right )}{27 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^3}-\frac {1}{13122 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {256 \left (32 \tan \left (\frac {x}{2}\right )+43\right )}{2187 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {65536 \left (922476208 \tan ^5\left (\frac {x}{2}\right )-317683387 \tan ^4\left (\frac {x}{2}\right )+5917846028 \tan ^3\left (\frac {x}{2}\right )-101576018038 \tan ^2\left (\frac {x}{2}\right )+1163642157596 \tan \left (\frac {x}{2}\right )-13290300057595\right )}{27 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^4}+\frac {1}{13122 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {4194304 \left (791196215943 \tan ^5\left (\frac {x}{2}\right )+450940461332 \tan ^4\left (\frac {x}{2}\right )-2780684410738 \tan ^3\left (\frac {x}{2}\right )+135519614356 \tan ^2\left (\frac {x}{2}\right )+824592367751 \tan \left (\frac {x}{2}\right )+69220699520\right )}{9 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}\right )d\tan \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {856 \log \left (-\tan \left (\frac {x}{2}\right )-\sqrt {3}+2\right )}{6561 \sqrt {3}}-\frac {856 \log \left (-\tan \left (\frac {x}{2}\right )+\sqrt {3}+2\right )}{6561 \sqrt {3}}+\frac {6346702249749643264}{9} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )-\frac {140073613112901632}{9} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )-\frac {66938759526863601664}{9} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )+\frac {18300070653325213696}{9} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )+\frac {31293793152419233792}{9} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )+\frac {290371338458693632}{9} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )-\frac {76199997039443968}{27} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )+\frac {5447777902788608}{27} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )-\frac {88973718585344}{9} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )+\frac {625373706125312}{27} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}d\tan \left (\frac {x}{2}\right )+\frac {4866172307456}{27} \int \frac {1}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )-\frac {388696145920}{27} \int \frac {\tan \left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )+\frac {29642473472}{27} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )-\frac {2152792064}{27} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )+\frac {725874688}{27} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}d\tan \left (\frac {x}{2}\right )+\frac {32288144}{243} \int \frac {1}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )-\frac {1982272}{243} \int \frac {\tan \left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )-\frac {41696}{243} \int \frac {\tan ^2\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )-\frac {74560}{243} \int \frac {\tan ^3\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )-\frac {4720}{243} \int \frac {\tan ^4\left (\frac {x}{2}\right )}{\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1}d\tan \left (\frac {x}{2}\right )+\frac {41}{2 \left (1-\tan \left (\frac {x}{2}\right )\right )}-\frac {43}{39366 \left (\tan \left (\frac {x}{2}\right )+1\right )}+\frac {32 \left (59-58 \tan \left (\frac {x}{2}\right )\right )}{6561 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}-\frac {1952 \left (2-\tan \left (\frac {x}{2}\right )\right )}{19683 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {9319936}{27 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )}-\frac {1}{4 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {1}{26244 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {64 \left (118-107 \tan \left (\frac {x}{2}\right )\right )}{6561 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {16640 \left (2-\tan \left (\frac {x}{2}\right )\right )}{19683 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {15113850191872}{81 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {1}{6 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {1}{39366 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {512 \left (7-26 \tan \left (\frac {x}{2}\right )\right )}{6561 \left (\tan ^2\left (\frac {x}{2}\right )-4 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {553086242219098112}{27 \left (\tan ^6\left (\frac {x}{2}\right )+12 \tan ^5\left (\frac {x}{2}\right )+3 \tan ^4\left (\frac {x}{2}\right )-40 \tan ^3\left (\frac {x}{2}\right )+3 \tan ^2\left (\frac {x}{2}\right )+12 \tan \left (\frac {x}{2}\right )+1\right )^3}\right )\) |
Input:
Int[(Cos[5*x] + Sin[4*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 = 114.89 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.02
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| risch | \(\frac {\frac {652 i}{9}-\frac {106756 \,{\mathrm e}^{i x}}{729}-\frac {159380 i {\mathrm e}^{18 i x}}{729}+\frac {3232 \,{\mathrm e}^{13 i x}}{2187}-\frac {1184 \,{\mathrm e}^{15 i x}}{729}+\frac {3232 \,{\mathrm e}^{17 i x}}{2187}+\frac {1148 i {\mathrm e}^{4 i x}}{2187}-\frac {155564 i {\mathrm e}^{2 i x}}{2187}+\frac {959704 i {\mathrm e}^{10 i x}}{2187}+\frac {1436 i {\mathrm e}^{8 i x}}{2187}-\frac {484 i {\mathrm e}^{6 i x}}{729}-\frac {1148 i {\mathrm e}^{26 i x}}{2187}-\frac {319744 i {\mathrm e}^{28 i x}}{2187}+\frac {484 i {\mathrm e}^{24 i x}}{729}-\frac {1436 i {\mathrm e}^{22 i x}}{2187}+\frac {466220 i {\mathrm e}^{20 i x}}{2187}+\frac {1340 \,{\mathrm e}^{5 i x}}{2187}-\frac {292 \,{\mathrm e}^{3 i x}}{729}-\frac {467392 \,{\mathrm e}^{11 i x}}{2187}+\frac {158872 \,{\mathrm e}^{9 i x}}{729}-\frac {1484 \,{\mathrm e}^{7 i x}}{2187}+\frac {958532 \,{\mathrm e}^{19 i x}}{2187}-\frac {3472 i {\mathrm e}^{14 i x}}{2187}+\frac {944 i {\mathrm e}^{12 i x}}{729}+\frac {3472 i {\mathrm e}^{16 i x}}{2187}+\frac {436 \,{\mathrm e}^{21 i x}}{729}-\frac {1484 \,{\mathrm e}^{23 i x}}{2187}-\frac {53104 \,{\mathrm e}^{27 i x}}{729}+\frac {1340 \,{\mathrm e}^{25 i x}}{2187}+\frac {51680 \,{\mathrm e}^{29 i x}}{729}}{\left (-2 i {\mathrm e}^{8 i x}+{\mathrm e}^{9 i x}+2 i {\mathrm e}^{6 i x}-2 \,{\mathrm e}^{7 i x}-2 i {\mathrm e}^{4 i x}+2 \,{\mathrm e}^{5 i x}+2 i {\mathrm e}^{2 i x}-2 \,{\mathrm e}^{3 i x}-i+2 \,{\mathrm e}^{i x}\right )^{3} \left ({\mathrm e}^{i x}+i\right )^{3}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (19383245667680019896796723 \textit {\_Z}^{6}-206513670890742663320201910528 \textit {\_Z}^{4}+26359922556168117138358272 \textit {\_Z}^{2}-104871273876793851904\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}-\frac {2457091464413307694166177088059575611 \textit {\_R}^{5}}{13686056932668822222033510619676672}+\frac {159294169191699354553721397 i \textit {\_R}^{4}}{342129116498324864769654784}+\frac {204519000560237383493058642014976865935 \textit {\_R}^{3}}{106922319786475173609636801716224}-\frac {13259044400524987285398012561 i \textit {\_R}^{2}}{2672883722643163006012928}-\frac {67848663825112919755054083469605 \textit {\_R}}{208832655832959323456321878352}+\frac {8211146036386220090123 i}{10440952041574855492238}\right )\right )+\frac {1712 \sqrt {3}\, \ln \left ({\mathrm e}^{i x}-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )}{19683}-\frac {1712 \sqrt {3}\, \ln \left ({\mathrm e}^{i x}-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )}{19683}\) | \(399\) |
| default | \(-\frac {32 \left (-\frac {6736}{27}-\frac {249910 \tan \left (\frac {x}{2}\right )}{27}-\frac {3238904 \tan \left (\frac {x}{2}\right )^{2}}{27}-\frac {15729764 \tan \left (\frac {x}{2}\right )^{3}}{27}-\frac {6320408 \tan \left (\frac {x}{2}\right )^{4}}{27}+\frac {109380812 \tan \left (\frac {x}{2}\right )^{5}}{27}+\frac {122921288 \tan \left (\frac {x}{2}\right )^{6}}{27}-\frac {282708628 \tan \left (\frac {x}{2}\right )^{7}}{27}-\frac {318537272 \tan \left (\frac {x}{2}\right )^{8}}{27}+\frac {336395840 \tan \left (\frac {x}{2}\right )^{9}}{27}+\frac {277919192 \tan \left (\frac {x}{2}\right )^{10}}{27}-\frac {134384492 \tan \left (\frac {x}{2}\right )^{11}}{27}-\frac {112170632 \tan \left (\frac {x}{2}\right )^{12}}{27}+\frac {7987252 \tan \left (\frac {x}{2}\right )^{13}}{27}+\frac {16747736 \tan \left (\frac {x}{2}\right )^{14}}{27}+\frac {3443300 \tan \left (\frac {x}{2}\right )^{15}}{27}+\frac {268328 \tan \left (\frac {x}{2}\right )^{16}}{27}+\frac {7414 \tan \left (\frac {x}{2}\right )^{17}}{27}\right )}{243 \left (\tan \left (\frac {x}{2}\right )^{6}+12 \tan \left (\frac {x}{2}\right )^{5}+3 \tan \left (\frac {x}{2}\right )^{4}-40 \tan \left (\frac {x}{2}\right )^{3}+3 \tan \left (\frac {x}{2}\right )^{2}+12 \tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {16 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+12 \textit {\_Z}^{5}+3 \textit {\_Z}^{4}-40 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )}{\sum }\frac {\left (15379 \textit {\_R}^{4}+217636 \textit {\_R}^{3}+494166 \textit {\_R}^{2}+217636 \textit {\_R} +15379\right ) \ln \left (\tan \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5}+10 \textit {\_R}^{4}+2 \textit {\_R}^{3}-20 \textit {\_R}^{2}+\textit {\_R} +2}\right )}{19683}-\frac {1}{3 \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {41}{\tan \left (\frac {x}{2}\right )-1}-\frac {32 \left (226 \tan \left (\frac {x}{2}\right )^{5}-1918 \tan \left (\frac {x}{2}\right )^{4}+5192 \tan \left (\frac {x}{2}\right )^{3}-4100 \tan \left (\frac {x}{2}\right )^{2}+1190 \tan \left (\frac {x}{2}\right )-118\right )}{19683 \left (\tan \left (\frac {x}{2}\right )^{2}-4 \tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {3424 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (2 \tan \left (\frac {x}{2}\right )-4\right ) \sqrt {3}}{6}\right )}{19683}-\frac {1}{19683 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{13122 \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {43}{19683 \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(412\) |
Input:
int(1/(cos(5*x)+sin(4*x))^4,x,method=_RETURNVERBOSE)
Output:
0
Result contains complex when optimal does not.
Time = 1.78 (sec) , antiderivative size = 4179, normalized size of antiderivative = 34.25 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^4,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^4} \, dx=\int \frac {1}{\left (\sin {\left (4 x \right )} + \cos {\left (5 x \right )}\right )^{4}}\, dx \] Input:
integrate(1/(cos(5*x)+sin(4*x))**4,x)
Output:
Integral((sin(4*x) + cos(5*x))**(-4), x)
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^4} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(5*x)+sin(4*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. \(2 (96) = 192\).
Time = 0.16 (sec) , antiderivative size = 398, normalized size of antiderivative = 3.26 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^4,x, algorithm="giac")
Output:
-1712/19683*sqrt(3)*log(abs(-2*sqrt(3) + 2*tan(1/2*x) - 4)/abs(2*sqrt(3) + 2*tan(1/2*x) - 4)) - 2/2187*(84779*tan(1/2*x)^29 + 2089080*tan(1/2*x)^28 + 6842722*tan(1/2*x)^27 - 141493912*tan(1/2*x)^26 - 496870231*tan(1/2*x)^2 5 + 4630724320*tan(1/2*x)^24 + 5060118068*tan(1/2*x)^23 - 57259181968*tan( 1/2*x)^22 - 8100002429*tan(1/2*x)^21 + 334520168968*tan(1/2*x)^20 - 116212 308802*tan(1/2*x)^19 - 1023875673160*tan(1/2*x)^18 + 669078512041*tan(1/2* x)^17 + 1689393065296*tan(1/2*x)^16 - 1481034238120*tan(1/2*x)^15 - 150989 3990240*tan(1/2*x)^14 + 1663359582505*tan(1/2*x)^13 + 688806280328*tan(1/2 *x)^12 - 1009432595266*tan(1/2*x)^11 - 124324145576*tan(1/2*x)^10 + 329728 991491*tan(1/2*x)^9 - 6204582592*tan(1/2*x)^8 - 56361572812*tan(1/2*x)^7 + 4829210608*tan(1/2*x)^6 + 4545990185*tan(1/2*x)^5 - 483937288*tan(1/2*x)^ 4 - 137922206*tan(1/2*x)^3 + 6693768*tan(1/2*x)^2 + 2023595*tan(1/2*x) + 8 0784)/(tan(1/2*x)^10 + 8*tan(1/2*x)^9 - 45*tan(1/2*x)^8 - 48*tan(1/2*x)^7 + 210*tan(1/2*x)^6 - 210*tan(1/2*x)^4 + 48*tan(1/2*x)^3 + 45*tan(1/2*x)^2 - 8*tan(1/2*x) - 1)^3 - 0.0111144805997257*log(tan(1/2*x) + 11.43005230280 00) - 0.00202751790850175*log(tan(1/2*x) + 2.14450692051000) + 0.002027517 90841030*log(tan(1/2*x) + 0.466307658155000) + 0.0111144805996952*log(tan( 1/2*x) + 0.0874886635259000) + 103.219357306813*log(tan(1/2*x) - 0.7002075 38210000) - 103.219357306813*log(tan(1/2*x) - 1.42814800674000)
Time = 22.55 (sec) , antiderivative size = 864, normalized size of antiderivative = 7.08 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^4} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(4*x))^4,x)
Output:
(1712*3^(1/2)*log(tan(x/2) + 3^(1/2) - 2))/19683 + symsum(log((93041736942 57979384182831645624107008*root(z^6 - (1375889754368*z^4)/129140163 + (226 79855871164416*z^2)/16677181699666569 - 104871273876793851904/193832456676 80019896796723, z, k)^2*tan(x/2))/1853020188851841 - (70659383725811536301 63032781740750929920*tan(x/2))/239299329230617529590083 - (248738930312971 440627939967033696845824*root(z^6 - (1375889754368*z^4)/129140163 + (22679 855871164416*z^2)/16677181699666569 - 104871273876793851904/19383245667680 019896796723, z, k))/109418989131512359209 + (1400219463908049721827252837 2334592*root(z^6 - (1375889754368*z^4)/129140163 + (22679855871164416*z^2) /16677181699666569 - 104871273876793851904/19383245667680019896796723, z, k)^3*tan(x/2))/31381059609 - (51396907458973907160183630462976*root(z^6 - (1375889754368*z^4)/129140163 + (22679855871164416*z^2)/16677181699666569 - 104871273876793851904/19383245667680019896796723, z, k)^4*tan(x/2))/1434 8907 - (30581771712029100737225031680*root(z^6 - (1375889754368*z^4)/12914 0163 + (22679855871164416*z^2)/16677181699666569 - 104871273876793851904/1 9383245667680019896796723, z, k)^5*tan(x/2))/729 + 38676433403547551268864 *root(z^6 - (1375889754368*z^4)/129140163 + (22679855871164416*z^2)/166771 81699666569 - 104871273876793851904/19383245667680019896796723, z, k)^6*ta n(x/2) + 4319334531957283356672*root(z^6 - (1375889754368*z^4)/129140163 + (22679855871164416*z^2)/16677181699666569 - 104871273876793851904/1938...
\[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^4} \, dx=\int \frac {1}{\cos \left (5 x \right )^{4}+4 \cos \left (5 x \right )^{3} \sin \left (4 x \right )+6 \cos \left (5 x \right )^{2} \sin \left (4 x \right )^{2}+4 \cos \left (5 x \right ) \sin \left (4 x \right )^{3}+\sin \left (4 x \right )^{4}}d x \] Input:
int(1/(cos(5*x)+sin(4*x))^4,x)
Output:
int(1/(cos(5*x)**4 + 4*cos(5*x)**3*sin(4*x) + 6*cos(5*x)**2*sin(4*x)**2 + 4*cos(5*x)*sin(4*x)**3 + sin(4*x)**4),x)