1,1,14,0,1.000963," ","integrate(3/(5-4*cos(x)),x, algorithm=""giac"")","x - 2 \, \arctan\left(\frac{\sin\left(x\right)}{\cos\left(x\right) - 2}\right)"," ",0,"x - 2*arctan(sin(x)/(cos(x) - 2))","A",0
2,1,18,0,1.231309," ","integrate((1+cos(x)+2*sin(x))/(3+cos(x)^2+2*sin(x)-2*cos(x)*sin(x)),x, algorithm=""giac"")","-\arctan\left(-\tan\left(\frac{1}{2} \, x\right)^{2} - \tan\left(\frac{1}{2} \, x\right)\right)"," ",0,"-arctan(-tan(1/2*x)^2 - tan(1/2*x))","A",0
3,1,36,0,1.258451," ","integrate((2+cos(x)+5*sin(x))/(4*cos(x)-2*sin(x)+cos(x)*sin(x)-2*sin(x)^2),x, algorithm=""giac"")","\log\left(\tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 2\right) - \log\left({\left| 2 \, \tan\left(\frac{1}{2} \, x\right) - 1 \right|}\right) - \log\left({\left| \tan\left(\frac{1}{2} \, x\right) + 1 \right|}\right)"," ",0,"log(tan(1/2*x)^2 + tan(1/2*x) + 2) - log(abs(2*tan(1/2*x) - 1)) - log(abs(tan(1/2*x) + 1))","A",0
4,1,26,0,1.019062," ","integrate((3+7*cos(x)+2*sin(x))/(1+4*cos(x)+3*cos(x)^2-5*sin(x)-cos(x)*sin(x)),x, algorithm=""giac"")","\log\left(\tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 2\right) - \log\left({\left| 2 \, \tan\left(\frac{1}{2} \, x\right) - 1 \right|}\right)"," ",0,"log(tan(1/2*x)^2 + tan(1/2*x) + 2) - log(abs(2*tan(1/2*x) - 1))","A",0
5,1,18,0,0.963435," ","integrate((-1+4*cos(x)+5*cos(x)^2)/(-1-4*cos(x)-3*cos(x)^2+4*cos(x)^3),x, algorithm=""giac"")","-2 \, \arctan\left(-\tan\left(\frac{1}{2} \, x\right)^{3} + 2 \, \tan\left(\frac{1}{2} \, x\right)\right)"," ",0,"-2*arctan(-tan(1/2*x)^3 + 2*tan(1/2*x))","A",0
6,1,16,0,1.089164," ","integrate((-5+2*cos(x)+7*cos(x)^2)/(-1+2*cos(x)-9*cos(x)^2+4*cos(x)^3),x, algorithm=""giac"")","-2 \, \arctan\left(-2 \, \tan\left(\frac{1}{2} \, x\right)^{3} + \tan\left(\frac{1}{2} \, x\right)\right)"," ",0,"-2*arctan(-2*tan(1/2*x)^3 + tan(1/2*x))","A",0
7,1,25,0,1.138909," ","integrate(3/(5+4*sin(x)),x, algorithm=""giac"")","x + 2 \, \arctan\left(-\frac{2 \, \cos\left(x\right) + \sin\left(x\right) + 2}{\cos\left(x\right) - 2 \, \sin\left(x\right) - 4}\right)"," ",0,"x + 2*arctan(-(2*cos(x) + sin(x) + 2)/(cos(x) - 2*sin(x) - 4))","A",0
8,1,45,0,0.846991," ","integrate(2/(1+cos(x)^2),x, algorithm=""giac"")","\sqrt{2} {\left(x + \arctan\left(-\frac{\sqrt{2} \sin\left(2 \, x\right) - \sin\left(2 \, x\right)}{\sqrt{2} \cos\left(2 \, x\right) + \sqrt{2} - \cos\left(2 \, x\right) + 1}\right)\right)}"," ",0,"sqrt(2)*(x + arctan(-(sqrt(2)*sin(2*x) - sin(2*x))/(sqrt(2)*cos(2*x) + sqrt(2) - cos(2*x) + 1)))","A",0
9,1,72,0,0.909043," ","integrate(1/(p+q*cos(x)+r*sin(x)),x, algorithm=""giac"")","-\frac{2 \, {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(-2 \, p + 2 \, q\right) + \arctan\left(-\frac{p \tan\left(\frac{1}{2} \, x\right) - q \tan\left(\frac{1}{2} \, x\right) + r}{\sqrt{p^{2} - q^{2} - r^{2}}}\right)\right)}}{\sqrt{p^{2} - q^{2} - r^{2}}}"," ",0,"-2*(pi*floor(1/2*x/pi + 1/2)*sgn(-2*p + 2*q) + arctan(-(p*tan(1/2*x) - q*tan(1/2*x) + r)/sqrt(p^2 - q^2 - r^2)))/sqrt(p^2 - q^2 - r^2)","A",0
