1,1,88,0,0.157150," ","integrate((A+B*cos(x))/(a+b*sin(x)),x, algorithm=""giac"")","\frac{2 \, {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(a\right) + \arctan\left(\frac{a \tan\left(\frac{1}{2} \, x\right) + b}{\sqrt{a^{2} - b^{2}}}\right)\right)} A}{\sqrt{a^{2} - b^{2}}} + \frac{B \log\left(a \tan\left(\frac{1}{2} \, x\right)^{2} + 2 \, b \tan\left(\frac{1}{2} \, x\right) + a\right)}{b} - \frac{B \log\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)}{b}"," ",0,"2*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))*A/sqrt(a^2 - b^2) + B*log(a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)/b - B*log(tan(1/2*x)^2 + 1)/b","A",0
2,1,43,0,0.156547," ","integrate((A+B*cos(x))/(1+sin(x)),x, algorithm=""giac"")","-B \log\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right) + 2 \, B \log\left({\left| \tan\left(\frac{1}{2} \, x\right) + 1 \right|}\right) - \frac{2 \, {\left(B \tan\left(\frac{1}{2} \, x\right) + A + B\right)}}{\tan\left(\frac{1}{2} \, x\right) + 1}"," ",0,"-B*log(tan(1/2*x)^2 + 1) + 2*B*log(abs(tan(1/2*x) + 1)) - 2*(B*tan(1/2*x) + A + B)/(tan(1/2*x) + 1)","B",0
3,1,46,0,0.133631," ","integrate((A+B*cos(x))/(1-sin(x)),x, algorithm=""giac"")","B \log\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right) - 2 \, B \log\left({\left| \tan\left(\frac{1}{2} \, x\right) - 1 \right|}\right) + \frac{2 \, {\left(B \tan\left(\frac{1}{2} \, x\right) - A - B\right)}}{\tan\left(\frac{1}{2} \, x\right) - 1}"," ",0,"B*log(tan(1/2*x)^2 + 1) - 2*B*log(abs(tan(1/2*x) - 1)) + 2*(B*tan(1/2*x) - A - B)/(tan(1/2*x) - 1)","B",0
4,1,88,0,0.152599," ","integrate((b+c+cos(x))/(a+b*sin(x)),x, algorithm=""giac"")","\frac{2 \, {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(a\right) + \arctan\left(\frac{a \tan\left(\frac{1}{2} \, x\right) + b}{\sqrt{a^{2} - b^{2}}}\right)\right)} {\left(b + c\right)}}{\sqrt{a^{2} - b^{2}}} + \frac{\log\left(a \tan\left(\frac{1}{2} \, x\right)^{2} + 2 \, b \tan\left(\frac{1}{2} \, x\right) + a\right)}{b} - \frac{\log\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)}{b}"," ",0,"2*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))*(b + c)/sqrt(a^2 - b^2) + log(a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)/b - log(tan(1/2*x)^2 + 1)/b","A",0
5,1,90,0,0.155443," ","integrate((b+c+cos(x))/(a-b*sin(x)),x, algorithm=""giac"")","\frac{2 \, {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(a\right) + \arctan\left(\frac{a \tan\left(\frac{1}{2} \, x\right) - b}{\sqrt{a^{2} - b^{2}}}\right)\right)} {\left(b + c\right)}}{\sqrt{a^{2} - b^{2}}} - \frac{\log\left(a \tan\left(\frac{1}{2} \, x\right)^{2} - 2 \, b \tan\left(\frac{1}{2} \, x\right) + a\right)}{b} + \frac{\log\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)}{b}"," ",0,"2*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) - b)/sqrt(a^2 - b^2)))*(b + c)/sqrt(a^2 - b^2) - log(a*tan(1/2*x)^2 - 2*b*tan(1/2*x) + a)/b + log(tan(1/2*x)^2 + 1)/b","A",0
6,1,116,0,0.157643," ","integrate((A+B*tan(x))/(a+b*sin(x)),x, algorithm=""giac"")","\frac{B a \log\left(a \tan\left(\frac{1}{2} \, x\right)^{2} + 2 \, b \tan\left(\frac{1}{2} \, x\right) + a\right)}{a^{2} - b^{2}} + \frac{2 \, {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(a\right) + \arctan\left(\frac{a \tan\left(\frac{1}{2} \, x\right) + b}{\sqrt{a^{2} - b^{2}}}\right)\right)} A}{\sqrt{a^{2} - b^{2}}} - \frac{B \log\left({\left| \tan\left(\frac{1}{2} \, x\right) + 1 \right|}\right)}{a - b} - \frac{B \log\left({\left| \tan\left(\frac{1}{2} \, x\right) - 1 \right|}\right)}{a + b}"," ",0,"B*a*log(a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)/(a^2 - b^2) + 2*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))*A/sqrt(a^2 - b^2) - B*log(abs(tan(1/2*x) + 1))/(a - b) - B*log(abs(tan(1/2*x) - 1))/(a + b)","A",0
7,1,85,0,0.145839," ","integrate((A+B*cot(x))/(a+b*sin(x)),x, algorithm=""giac"")","\frac{2 \, {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(a\right) + \arctan\left(\frac{a \tan\left(\frac{1}{2} \, x\right) + b}{\sqrt{a^{2} - b^{2}}}\right)\right)} A}{\sqrt{a^{2} - b^{2}}} - \frac{B \log\left(a \tan\left(\frac{1}{2} \, x\right)^{2} + 2 \, b \tan\left(\frac{1}{2} \, x\right) + a\right)}{a} + \frac{B \log\left({\left| \tan\left(\frac{1}{2} \, x\right) \right|}\right)}{a}"," ",0,"2*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))*A/sqrt(a^2 - b^2) - B*log(a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)/a + B*log(abs(tan(1/2*x)))/a","A",0
8,1,116,0,0.168525," ","integrate((A+B*sec(x))/(a+b*sin(x)),x, algorithm=""giac"")","-\frac{B b \log\left(a \tan\left(\frac{1}{2} \, x\right)^{2} + 2 \, b \tan\left(\frac{1}{2} \, x\right) + a\right)}{a^{2} - b^{2}} + \frac{2 \, {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(a\right) + \arctan\left(\frac{a \tan\left(\frac{1}{2} \, x\right) + b}{\sqrt{a^{2} - b^{2}}}\right)\right)} A}{\sqrt{a^{2} - b^{2}}} + \frac{B \log\left({\left| \tan\left(\frac{1}{2} \, x\right) + 1 \right|}\right)}{a - b} - \frac{B \log\left({\left| \tan\left(\frac{1}{2} \, x\right) - 1 \right|}\right)}{a + b}"," ",0,"-B*b*log(a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)/(a^2 - b^2) + 2*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))*A/sqrt(a^2 - b^2) + B*log(abs(tan(1/2*x) + 1))/(a - b) - B*log(abs(tan(1/2*x) - 1))/(a + b)","A",0
9,1,71,0,0.167495," ","integrate((A+B*csc(x))/(a+b*sin(x)),x, algorithm=""giac"")","\frac{B \log\left({\left| \tan\left(\frac{1}{2} \, x\right) \right|}\right)}{a} + \frac{2 \, {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(a\right) + \arctan\left(\frac{a \tan\left(\frac{1}{2} \, x\right) + b}{\sqrt{a^{2} - b^{2}}}\right)\right)} {\left(A a - B b\right)}}{\sqrt{a^{2} - b^{2}} a}"," ",0,"B*log(abs(tan(1/2*x)))/a + 2*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))*(A*a - B*b)/(sqrt(a^2 - b^2)*a)","A",0
