Integrand size = 21, antiderivative size = 39 \[ \int (\cos (3 x)+\sin (2 x) (\cos (3 x)-\sin (2019 x))) \, dx=\frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x)+\frac {1}{3} \sin (3 x)-\frac {\sin (2017 x)}{4034}+\frac {\sin (2021 x)}{4042} \] Output:
1/2*cos(x)-1/10*cos(5*x)+1/3*sin(3*x)-1/4034*sin(2017*x)+1/4042*sin(2021*x )
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int (\cos (3 x)+\sin (2 x) (\cos (3 x)-\sin (2019 x))) \, dx=\frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x)+\frac {1}{3} \sin (3 x)-\frac {\sin (2017 x)}{4034}+\frac {\sin (2021 x)}{4042} \] Input:
Integrate[Cos[3*x] + Sin[2*x]*(Cos[3*x] - Sin[2019*x]),x]
Output:
Cos[x]/2 - Cos[5*x]/10 + Sin[3*x]/3 - Sin[2017*x]/4034 + Sin[2021*x]/4042
Time = 0.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2009}
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.
\(\displaystyle \int (\cos (3 x)+\sin (2 x) (\cos (3 x)-\sin (2019 x))) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \sin (3 x)-\frac {\sin (2017 x)}{4034}+\frac {\sin (2021 x)}{4042}+\frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x)\) |
Input:
Int[Cos[3*x] + Sin[2*x]*(Cos[3*x] - Sin[2019*x]),x]
Output:
Cos[x]/2 - Cos[5*x]/10 + Sin[3*x]/3 - Sin[2017*x]/4034 + Sin[2021*x]/4042
Time = 2.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {\cos \left (x \right )}{2}-\frac {\cos \left (5 x \right )}{10}+\frac {\sin \left (3 x \right )}{3}-\frac {\sin \left (2017 x \right )}{4034}+\frac {\sin \left (2021 x \right )}{4042}\) | \(30\) |
risch | \(\frac {\cos \left (x \right )}{2}-\frac {\cos \left (5 x \right )}{10}+\frac {\sin \left (3 x \right )}{3}-\frac {\sin \left (2017 x \right )}{4034}+\frac {\sin \left (2021 x \right )}{4042}\) | \(30\) |
parts | \(\frac {\cos \left (x \right )}{2}-\frac {\cos \left (5 x \right )}{10}+\frac {\sin \left (3 x \right )}{3}-\frac {\sin \left (2017 x \right )}{4034}+\frac {\sin \left (2021 x \right )}{4042}\) | \(30\) |
parallelrisch | \(\frac {\frac {2 \left (-8152714+\left (-10095 \tan \left (x \right )-8152714\right ) \tan \left (\frac {2019 x}{2}\right )^{2}+10 \left (\tan \left (x \right )^{2}-1\right ) \tan \left (\frac {2019 x}{2}\right )+10095 \tan \left (x \right )\right ) \tan \left (\frac {3 x}{2}\right )^{2}}{20381785}+\frac {2 \left (\tan \left (x \right )^{2}+\frac {18 \tan \left (x \right )}{5}+1\right ) \left (1+\tan \left (\frac {2019 x}{2}\right )^{2}\right ) \tan \left (\frac {3 x}{2}\right )}{3}+\frac {2 \left (-8152714 \tan \left (x \right )^{2}-10095 \tan \left (x \right )\right ) \tan \left (\frac {2019 x}{2}\right )^{2}}{20381785}+\frac {4 \left (\tan \left (x \right )^{2}-1\right ) \tan \left (\frac {2019 x}{2}\right )}{4076357}-\frac {4 \tan \left (x \right )^{2}}{5}+\frac {4038 \tan \left (x \right )}{4076357}}{\left (1+\tan \left (x \right )^{2}\right ) \left (1+\tan \left (\frac {3 x}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {2019 x}{2}\right )^{2}\right )}\) | \(136\) |
orering | \(\frac {\sin \left (3 x \right )}{3}-\frac {19940000842106 \cos \left (2 x \right ) \left (\cos \left (3 x \right )-\sin \left (2019 x \right )\right )}{16616686391449}+\frac {29910131706719 \sin \left (2 x \right ) \left (-3 \sin \left (3 x \right )-2019 \cos \left (2019 x \right )\right )}{83083431957245}-\frac {240018283468988 \cos \left (2 x \right ) \left (-9 \cos \left (3 x \right )+4076361 \sin \left (2019 x \right )\right )}{415417159786225}+\frac {9232018743254 \sin \left (2 x \right ) \left (27 \sin \left (3 x \right )+8230172859 \cos \left (2019 x \right )\right )}{415417159786225}-\frac {166167434602412 \cos \left (2 x \right ) \left (81 \cos \left (3 x \right )-16616719002321 \sin \left (2019 x \right )\right )}{3738754438076025}-\frac {5538762302338 \sin \left (2 x \right ) \left (-243 \sin \left (3 x \right )-33549155665686099 \cos \left (2019 x \right )\right )}{1246251479358675}-\frac {114138038 \cos \left (2 x \right ) \left (-729 \cos \left (3 x \right )+67735745289020233881 \sin \left (2019 x \right )\right )}{3738754438076025}-\frac {8152621 \sin \left (2 x \right ) \left (2187 \sin \left (3 x \right )+136758469738531852205739 \cos \left (2019 x \right )\right )}{3738754438076025}-\frac {2 \cos \left (2 x \right ) \left (6561 \cos \left (3 x \right )-276115350402095809603387041 \sin \left (2019 x \right )\right )}{415417159786225}-\frac {\sin \left (2 x \right ) \left (-19683 \sin \left (3 x \right )-557476892461831439589238435779 \cos \left (2019 x \right )\right )}{3738754438076025}\) | \(196\) |
Input:
int(cos(3*x)+sin(2*x)*(cos(3*x)-sin(2019*x)),x,method=_RETURNVERBOSE)
Output:
1/2*cos(x)-1/10*cos(5*x)+1/3*sin(3*x)-1/4034*sin(2017*x)+1/4042*sin(2021*x )
Timed out. \[ \int (\cos (3 x)+\sin (2 x) (\cos (3 x)-\sin (2019 x))) \, dx=\text {Timed out} \] Input:
integrate(cos(3*x)+sin(2*x)*(cos(3*x)-sin(2019*x)),x, algorithm="fricas")
Output:
Timed out
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.54 \[ \int (\cos (3 x)+\sin (2 x) (\cos (3 x)-\sin (2019 x))) \, dx=\frac {3 \sin {\left (2 x \right )} \sin {\left (3 x \right )}}{5} + \frac {2019 \sin {\left (2 x \right )} \cos {\left (2019 x \right )}}{4076357} + \frac {\sin {\left (3 x \right )}}{3} - \frac {2 \sin {\left (2019 x \right )} \cos {\left (2 x \right )}}{4076357} + \frac {2 \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{5} \] Input:
integrate(cos(3*x)+sin(2*x)*(cos(3*x)-sin(2019*x)),x)
Output:
3*sin(2*x)*sin(3*x)/5 + 2019*sin(2*x)*cos(2019*x)/4076357 + sin(3*x)/3 - 2 *sin(2019*x)*cos(2*x)/4076357 + 2*cos(2*x)*cos(3*x)/5
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int (\cos (3 x)+\sin (2 x) (\cos (3 x)-\sin (2019 x))) \, dx=-\frac {1}{10} \, \cos \left (5 \, x\right ) + \frac {1}{2} \, \cos \left (x\right ) + \frac {1}{4042} \, \sin \left (2021 \, x\right ) - \frac {1}{4034} \, \sin \left (2017 \, x\right ) + \frac {1}{3} \, \sin \left (3 \, x\right ) \] Input:
integrate(cos(3*x)+sin(2*x)*(cos(3*x)-sin(2019*x)),x, algorithm="maxima")
Output:
-1/10*cos(5*x) + 1/2*cos(x) + 1/4042*sin(2021*x) - 1/4034*sin(2017*x) + 1/ 3*sin(3*x)
Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int (\cos (3 x)+\sin (2 x) (\cos (3 x)-\sin (2019 x))) \, dx=-\frac {1}{10} \, \cos \left (5 \, x\right ) + \frac {1}{2} \, \cos \left (x\right ) + \frac {1}{4042} \, \sin \left (2021 \, x\right ) - \frac {1}{4034} \, \sin \left (2017 \, x\right ) + \frac {1}{3} \, \sin \left (3 \, x\right ) \] Input:
integrate(cos(3*x)+sin(2*x)*(cos(3*x)-sin(2019*x)),x, algorithm="giac")
Output:
-1/10*cos(5*x) + 1/2*cos(x) + 1/4042*sin(2021*x) - 1/4034*sin(2017*x) + 1/ 3*sin(3*x)
Time = 0.50 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int (\cos (3 x)+\sin (2 x) (\cos (3 x)-\sin (2019 x))) \, dx=\frac {\sin \left (3\,x\right )}{3}-\frac {\cos \left (5\,x\right )}{10}-\frac {\sin \left (2017\,x\right )}{4034}+\frac {\sin \left (2021\,x\right )}{4042}+\frac {\cos \left (x\right )}{2} \] Input:
int(cos(3*x) + sin(2*x)*(cos(3*x) - sin(2019*x)),x)
Output:
sin(3*x)/3 - cos(5*x)/10 - sin(2017*x)/4034 + sin(2021*x)/4042 + cos(x)/2
Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.21 \[ \int (\cos (3 x)+\sin (2 x) (\cos (3 x)-\sin (2019 x))) \, dx=\frac {2019 \cos \left (2019 x \right ) \sin \left (2 x \right )}{4076357}+\frac {2 \cos \left (3 x \right ) \cos \left (2 x \right )}{5}-\frac {2 \cos \left (2 x \right ) \sin \left (2019 x \right )}{4076357}+\frac {3 \sin \left (3 x \right ) \sin \left (2 x \right )}{5}+\frac {\sin \left (3 x \right )}{3} \] Input:
int(cos(3*x)+sin(2*x)*(cos(3*x)-sin(2019*x)),x)
Output:
(30285*cos(2019*x)*sin(2*x) + 24458142*cos(3*x)*cos(2*x) - 30*cos(2*x)*sin (2019*x) + 36687213*sin(3*x)*sin(2*x) + 20381785*sin(3*x))/61145355