Integrand size = 11, antiderivative size = 127 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^3} \, dx=-\frac {7}{144} \text {arctanh}(\cos (x))-\frac {55}{9} \text {arctanh}(2 \cos (x))+\frac {139 \text {arctanh}\left (\sqrt {2} \cos (x)\right )}{16 \sqrt {2}}-\frac {1}{18 (1-2 \cos (x))^2}+\frac {35}{54 (1-2 \cos (x))}-\frac {1}{864 (1-\cos (x))}+\frac {1}{864 (1+\cos (x))}+\frac {1}{18 (1+2 \cos (x))^2}-\frac {35}{54 (1+2 \cos (x))}-\frac {19}{16} \cos (x) \sec (2 x)+\frac {1}{8} \cos (x) \sec ^2(2 x) \] Output:
-7/144*arctanh(cos(x))-55/9*arctanh(2*cos(x))+139/32*arctanh(cos(x)*2^(1/2 ))*2^(1/2)-1/18/(1-2*cos(x))^2+35/(54-108*cos(x))-1/(864-864*cos(x))+1/(86 4+864*cos(x))+1/18/(1+2*cos(x))^2-35/(54+108*cos(x))-19/16*cos(x)*sec(2*x) +1/8*cos(x)*sec(2*x)^2
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.92 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.92 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^3} \, dx=-\frac {19}{686} \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\frac {13}{18} \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-\frac {640}{9} \log (1-2 \sin (x))+\frac {1}{343} \left (-24272 \log \left (\sec ^2\left (\frac {x}{2}\right )\right )+256 \text {RootSum}\left [13033-167008 \text {$\#$1}-388352 \text {$\#$1}^2+4096 \text {$\#$1}^3\&,\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (-2999947-12270846 \sin (x)-17960480 \sin (x) \text {$\#$1}+190464 \sin (x) \text {$\#$1}^2\right )\right ) \text {$\#$1}\&\right ]\right )-\frac {1}{108 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}+\frac {1}{1372-1372 \sin (x)}+\frac {4}{9 (1-2 \sin (x))^2}+\frac {248}{-27+54 \sin (x)}+\frac {4 (-57+40 \cos (2 x)-92 \sin (x))}{49 (1-2 \cos (2 x)+2 \sin (x)-2 \sin (3 x))^2}+\frac {48 (-158+118 \cos (2 x)-263 \sin (x))}{343 (-1+2 \cos (2 x)-2 \sin (x)+2 \sin (3 x))} \] Input:
Integrate[(Cos[5*x] + Sin[2*x])^(-3),x]
Output:
(-19*Log[Cos[x/2] - Sin[x/2]])/686 + (13*Log[Cos[x/2] + Sin[x/2]])/18 - (6 40*Log[1 - 2*Sin[x]])/9 + (-24272*Log[Sec[x/2]^2] + 256*RootSum[13033 - 16 7008*#1 - 388352*#1^2 + 4096*#1^3 & , Log[-(Sec[x/2]^2*(-2999947 - 1227084 6*Sin[x] - 17960480*Sin[x]*#1 + 190464*Sin[x]*#1^2))]*#1 & ])/343 - 1/(108 *(Cos[x/2] + Sin[x/2])^2) + (1372 - 1372*Sin[x])^(-1) + 4/(9*(1 - 2*Sin[x] )^2) + 248/(-27 + 54*Sin[x]) + (4*(-57 + 40*Cos[2*x] - 92*Sin[x]))/(49*(1 - 2*Cos[2*x] + 2*Sin[x] - 2*Sin[3*x])^2) + (48*(-158 + 118*Cos[2*x] - 263* Sin[x]))/(343*(-1 + 2*Cos[2*x] - 2*Sin[x] + 2*Sin[3*x]))
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.
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (5 x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (5 x))^3}dx\) |
| \(\Big \downarrow \) 4829 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {64 \left (15 \sin ^2(x)+17 \sin (x)+3\right )}{7 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^3}+\frac {128 \left (1517 \sin ^2(x)+1627 \sin (x)+153\right )}{343 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )}-\frac {64 \left (243 \sin ^2(x)+265 \sin (x)+31\right )}{49 \left (8 \sin ^3(x)+4 \sin ^2(x)-4 \sin (x)-1\right )^2}-\frac {19}{1372 (\sin (x)-1)}+\frac {13}{36 (\sin (x)+1)}-\frac {1280}{9 (2 \sin (x)-1)}+\frac {1}{1372 (\sin (x)-1)^2}+\frac {1}{108 (\sin (x)+1)^2}-\frac {496}{27 (2 \sin (x)-1)^2}-\frac {16}{9 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-12 \sin ^2(x)+2 \sin (x)+1\right )^3}d\sin (x)\) |
Input:
Int[(Cos[5*x] + Sin[2*x])^(-3),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 = 15.36 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.02
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| default | \(\frac {\frac {90624 \sin \left (x \right )^{5}}{343}+\frac {146304 \sin \left (x \right )^{4}}{343}+\frac {20544 \sin \left (x \right )^{3}}{343}-\frac {56384 \sin \left (x \right )^{2}}{343}-\frac {22880 \sin \left (x \right )}{343}-\frac {2396}{343}}{\left (8 \sin \left (x \right )^{3}+4 \sin \left (x \right )^{2}-4 \sin \left (x \right )-1\right )^{2}}+\frac {16 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{3}+4 \textit {\_Z}^{2}-4 \textit {\_Z} -1\right )}{\sum }\frac {\left (3034 \textit {\_R}^{2}+3431 \textit {\_R} +612\right ) \ln \left (\sin \left (x \right )-\textit {\_R} \right )}{6 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )}{343}+\frac {4}{9 \left (2 \sin \left (x \right )-1\right )^{2}}+\frac {248}{27 \left (2 \sin \left (x \right )-1\right )}-\frac {640 \ln \left (2 \sin \left (x \right )-1\right )}{9}-\frac {1}{1372 \left (\sin \left (x \right )-1\right )}-\frac {19 \ln \left (\sin \left (x \right )-1\right )}{1372}-\frac {1}{108 \left (1+\sin \left (x \right )\right )}+\frac {13 \ln \left (1+\sin \left (x \right )\right )}{36}\) | \(164\) |
| risch | \(\frac {i \left (2912 i {\mathrm e}^{18 i x}+2561 \,{\mathrm e}^{19 i x}+2912 i {\mathrm e}^{2 i x}-667 \,{\mathrm e}^{17 i x}-1292 i {\mathrm e}^{10 i x}+2765 \,{\mathrm e}^{15 i x}+54 i {\mathrm e}^{14 i x}+96 \,{\mathrm e}^{13 i x}+5368 i {\mathrm e}^{12 i x}-5932 \,{\mathrm e}^{11 i x}+5368 i {\mathrm e}^{8 i x}+5932 \,{\mathrm e}^{9 i x}-3062 i {\mathrm e}^{4 i x}-96 \,{\mathrm e}^{7 i x}+54 i {\mathrm e}^{6 i x}-2765 \,{\mathrm e}^{5 i x}-3062 i {\mathrm e}^{16 i x}+667 \,{\mathrm e}^{3 i x}-2561 \,{\mathrm e}^{i x}\right )}{147 \left ({\mathrm e}^{i x}-i\right )^{2} \left (i {\mathrm e}^{8 i x}+{\mathrm e}^{9 i x}-2 i {\mathrm e}^{6 i x}-{\mathrm e}^{7 i x}+2 i {\mathrm e}^{4 i x}+2 \,{\mathrm e}^{5 i x}-i {\mathrm e}^{2 i x}-2 \,{\mathrm e}^{3 i x}+i+{\mathrm e}^{i x}\right )^{2}}+\frac {13 \ln \left ({\mathrm e}^{i x}+i\right )}{18}-\frac {19 \ln \left ({\mathrm e}^{i x}-i\right )}{686}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (40353607 \textit {\_Z}^{3}-2855576528 \textit {\_Z}^{2}-916539904 \textit {\_Z} +53383168\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (\frac {34471157}{767986432} i \textit {\_R}^{2}-\frac {19299581}{5999894} i \textit {\_R} +\frac {1820638}{2999947} i\right ) {\mathrm e}^{i x}-1\right )\right )-\frac {640 \ln \left (-i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{9}\) | \(314\) |
Input:
int(1/(cos(5*x)+sin(2*x))^3,x,method=_RETURNVERBOSE)
Output:
0
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 1468, normalized size of antiderivative = 11.56 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^3,x, algorithm="fricas")
Output:
-1/605052*(767090688*cos(x)^8 - 1735832448*cos(x)^6 + 1262229024*cos(x)^4 + 2*(256*cos(x)^10 - 640*cos(x)^8 + 560*cos(x)^6 - 204*cos(x)^4 + 29*cos(x )^2 + 4*(16*cos(x)^6 - 20*cos(x)^4 + 5*cos(x)^2)*sin(x))*(151263*(61438914 56/51883209*I*sqrt(3) + 2084357543936/155649627)^(1/3)*(-I*sqrt(3) + 1) + 85306624*(I*sqrt(3) + 1)/(6143891456/51883209*I*sqrt(3) + 2084357543936/15 5649627)^(1/3) - 7135968)*log(293/777924*(151263*(6143891456/51883209*I*sq rt(3) + 2084357543936/155649627)^(1/3)*(-I*sqrt(3) + 1) + 85306624*(I*sqrt (3) + 1)/(6143891456/51883209*I*sqrt(3) + 2084357543936/155649627)^(1/3) - 7135968)^2 + 1235173184*(6143891456/51883209*I*sqrt(3) + 2084357543936/15 5649627)^(1/3)*(-I*sqrt(3) + 1) + 307196660006912/441*(I*sqrt(3) + 1)/(614 3891456/51883209*I*sqrt(3) + 2084357543936/155649627)^(1/3) + 1535972864*s in(x) - 173412965888/3) - 283995768*cos(x)^2 - (5480423424*cos(x)^10 - 137 01058560*cos(x)^8 + 11988426240*cos(x)^6 - 4367212416*cos(x)^4 + (256*cos( x)^10 - 640*cos(x)^8 + 560*cos(x)^6 - 204*cos(x)^4 + 29*cos(x)^2 + 4*(16*c os(x)^6 - 20*cos(x)^4 + 5*cos(x)^2)*sin(x))*(151263*(6143891456/51883209*I *sqrt(3) + 2084357543936/155649627)^(1/3)*(-I*sqrt(3) + 1) + 85306624*(I*s qrt(3) + 1)/(6143891456/51883209*I*sqrt(3) + 2084357543936/155649627)^(1/3 ) - 7135968) + 620829216*cos(x)^2 + 85631616*(16*cos(x)^6 - 20*cos(x)^4 + 5*cos(x)^2)*sin(x) + 882*(256*cos(x)^10 - 640*cos(x)^8 + 560*cos(x)^6 - 20 4*cos(x)^4 + 29*cos(x)^2 + 4*(16*cos(x)^6 - 20*cos(x)^4 + 5*cos(x)^2)*s...
\[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^3} \, dx=\int \frac {1}{\left (\sin {\left (2 x \right )} + \cos {\left (5 x \right )}\right )^{3}}\, dx \] Input:
integrate(1/(cos(5*x)+sin(2*x))**3,x)
Output:
Integral((sin(2*x) + cos(5*x))**(-3), x)
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^3} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(1/(cos(5*x)+sin(2*x))^3,x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: 2*(( -3216*sin(sageVARx)^2-185*sin(sageVARx)+3374)*1/37044/(sin(sageVARx)^2-1)+ (154368*sin(sageVARx)^8+10337792*sin(sageVARx)^7+5628928*sin(sageVARx)^6-1 2202432*sin(sageVAR
Time = 23.95 (sec) , antiderivative size = 3812, normalized size of antiderivative = 30.02 \[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^3} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(2*x))^3,x)
Output:
log((tan(x/2) + 1)^(13/18)) - (640*log(tan(x/2)^2 - 4*tan(x/2) + 1))/9 + l og(1/(tan(x/2) - 1)^(19/686)) - (1151*tan(x/2))/(21*(8*tan(x/2) - 74*tan(x /2)^2 - 344*tan(x/2)^3 + 2509*tan(x/2)^4 + 960*tan(x/2)^5 - 19256*tan(x/2) ^6 + 1664*tan(x/2)^7 + 63026*tan(x/2)^8 - 2288*tan(x/2)^9 - 92412*tan(x/2) ^10 - 2288*tan(x/2)^11 + 63026*tan(x/2)^12 + 1664*tan(x/2)^13 - 19256*tan( x/2)^14 + 960*tan(x/2)^15 + 2509*tan(x/2)^16 - 344*tan(x/2)^17 - 74*tan(x/ 2)^18 + 8*tan(x/2)^19 + tan(x/2)^20 + 1)) + symsum(log(-(35184372088832*(9 076554944856055150128136192*root(z^3 - (24272*z^2)/343 - (2672128*z)/11764 9 + 53383168/40353607, z, k) + 6298178645246991627368529920*cos(x) + 17602 02297405634484348387328*sin(x) - 20217616625110063036134064128*root(z^3 - (24272*z^2)/343 - (2672128*z)/117649 + 53383168/40353607, z, k)*cos(x) - 7 8034534172559312435727040512*root(z^3 - (24272*z^2)/343 - (2672128*z)/1176 49 + 53383168/40353607, z, k)*sin(x) + 2121876823628170823527417036800*roo t(z^3 - (24272*z^2)/343 - (2672128*z)/117649 + 53383168/40353607, z, k)^2 + 1290119019714658581791140861952*root(z^3 - (24272*z^2)/343 - (2672128*z) /117649 + 53383168/40353607, z, k)^3 - 12111407380848387785076052472512*ro ot(z^3 - (24272*z^2)/343 - (2672128*z)/117649 + 53383168/40353607, z, k)^4 - 1401724705173810769795036656432*root(z^3 - (24272*z^2)/343 - (2672128*z )/117649 + 53383168/40353607, z, k)^5 + 225542914364791881678819969663*roo t(z^3 - (24272*z^2)/343 - (2672128*z)/117649 + 53383168/40353607, z, k)...
\[ \int \frac {1}{(\cos (5 x)+\sin (2 x))^3} \, dx=\int \frac {1}{\cos \left (5 x \right )^{3}+3 \cos \left (5 x \right )^{2} \sin \left (2 x \right )+3 \cos \left (5 x \right ) \sin \left (2 x \right )^{2}+\sin \left (2 x \right )^{3}}d x \] Input:
int(1/(cos(5*x)+sin(2*x))^3,x)
Output:
int(1/(cos(5*x)**3 + 3*cos(5*x)**2*sin(2*x) + 3*cos(5*x)*sin(2*x)**2 + sin (2*x)**3),x)