1,1,246,0,0.570278," ","integrate((A+B*cos(x))/(a+b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} A b \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(x\right) \sin\left(x\right) + b \cos\left(x\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) - {\left(B a^{2} - B b^{2}\right)} \log\left(-b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)}}, -\frac{2 \, \sqrt{a^{2} - b^{2}} A b \arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right) - {\left(B a^{2} - B b^{2}\right)} \log\left(-b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)}}\right]"," ",0,"[-1/2*(sqrt(-a^2 + b^2)*A*b*log(((2*a^2 - b^2)*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2 + 2*(a*cos(x)*sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - (B*a^2 - B*b^2)*log(-b^2*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2))/(a^2*b - b^3), -1/2*(2*sqrt(a^2 - b^2)*A*b*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (B*a^2 - B*b^2)*log(-b^2*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2))/(a^2*b - b^3)]","A",0
2,1,38,0,0.598160," ","integrate((A+B*cos(x))/(1+sin(x)),x, algorithm=""fricas"")","-\frac{A \cos\left(x\right) - {\left(B \cos\left(x\right) + B \sin\left(x\right) + B\right)} \log\left(\sin\left(x\right) + 1\right) - A \sin\left(x\right) + A}{\cos\left(x\right) + \sin\left(x\right) + 1}"," ",0,"-(A*cos(x) - (B*cos(x) + B*sin(x) + B)*log(sin(x) + 1) - A*sin(x) + A)/(cos(x) + sin(x) + 1)","A",0
3,1,41,0,0.602711," ","integrate((A+B*cos(x))/(1-sin(x)),x, algorithm=""fricas"")","\frac{A \cos\left(x\right) - {\left(B \cos\left(x\right) - B \sin\left(x\right) + B\right)} \log\left(-\sin\left(x\right) + 1\right) + A \sin\left(x\right) + A}{\cos\left(x\right) - \sin\left(x\right) + 1}"," ",0,"(A*cos(x) - (B*cos(x) - B*sin(x) + B)*log(-sin(x) + 1) + A*sin(x) + A)/(cos(x) - sin(x) + 1)","A",0
4,1,250,0,0.602987," ","integrate((b+c+cos(x))/(a+b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} {\left(b^{2} + b c\right)} \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(x\right) \sin\left(x\right) + b \cos\left(x\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) - {\left(a^{2} - b^{2}\right)} \log\left(-b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)}}, -\frac{2 \, \sqrt{a^{2} - b^{2}} {\left(b^{2} + b c\right)} \arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right) - {\left(a^{2} - b^{2}\right)} \log\left(-b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)}}\right]"," ",0,"[-1/2*(sqrt(-a^2 + b^2)*(b^2 + b*c)*log(((2*a^2 - b^2)*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2 + 2*(a*cos(x)*sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - (a^2 - b^2)*log(-b^2*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2))/(a^2*b - b^3), -1/2*(2*sqrt(a^2 - b^2)*(b^2 + b*c)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (a^2 - b^2)*log(-b^2*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2))/(a^2*b - b^3)]","A",0
5,1,252,0,0.847061," ","integrate((b+c+cos(x))/(a-b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} {\left(b^{2} + b c\right)} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(x\right) \sin\left(x\right) - b \cos\left(x\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) + {\left(a^{2} - b^{2}\right)} \log\left(-b^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)}}, -\frac{2 \, \sqrt{a^{2} - b^{2}} {\left(b^{2} + b c\right)} \arctan\left(-\frac{a \sin\left(x\right) - b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right) + {\left(a^{2} - b^{2}\right)} \log\left(-b^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)}}\right]"," ",0,"[-1/2*(sqrt(-a^2 + b^2)*(b^2 + b*c)*log(-((2*a^2 - b^2)*cos(x)^2 + 2*a*b*sin(x) - a^2 - b^2 + 2*(a*cos(x)*sin(x) - b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 + 2*a*b*sin(x) - a^2 - b^2)) + (a^2 - b^2)*log(-b^2*cos(x)^2 - 2*a*b*sin(x) + a^2 + b^2))/(a^2*b - b^3), -1/2*(2*sqrt(a^2 - b^2)*(b^2 + b*c)*arctan(-(a*sin(x) - b)/(sqrt(a^2 - b^2)*cos(x))) + (a^2 - b^2)*log(-b^2*cos(x)^2 - 2*a*b*sin(x) + a^2 + b^2))/(a^2*b - b^3)]","A",0
6,1,281,0,7.206171," ","integrate((A+B*tan(x))/(a+b*sin(x)),x, algorithm=""fricas"")","\left[\frac{B a \log\left(-b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right) - \sqrt{-a^{2} + b^{2}} A \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(x\right) \sin\left(x\right) + b \cos\left(x\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) - {\left(B a + B b\right)} \log\left(\sin\left(x\right) + 1\right) - {\left(B a - B b\right)} \log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a^{2} - b^{2}\right)}}, \frac{B a \log\left(-b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right) - 2 \, \sqrt{a^{2} - b^{2}} A \arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right) - {\left(B a + B b\right)} \log\left(\sin\left(x\right) + 1\right) - {\left(B a - B b\right)} \log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a^{2} - b^{2}\right)}}\right]"," ",0,"[1/2*(B*a*log(-b^2*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2) - sqrt(-a^2 + b^2)*A*log(((2*a^2 - b^2)*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2 + 2*(a*cos(x)*sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - (B*a + B*b)*log(sin(x) + 1) - (B*a - B*b)*log(-sin(x) + 1))/(a^2 - b^2), 1/2*(B*a*log(-b^2*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2) - 2*sqrt(a^2 - b^2)*A*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (B*a + B*b)*log(sin(x) + 1) - (B*a - B*b)*log(-sin(x) + 1))/(a^2 - b^2)]","A",0
7,1,280,0,0.731917," ","integrate((A+B*cot(x))/(a+b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} A a \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(x\right) \sin\left(x\right) + b \cos\left(x\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) + {\left(B a^{2} - B b^{2}\right)} \log\left(-b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right) - 2 \, {\left(B a^{2} - B b^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(x\right)\right)}{2 \, {\left(a^{3} - a b^{2}\right)}}, -\frac{2 \, \sqrt{a^{2} - b^{2}} A a \arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right) + {\left(B a^{2} - B b^{2}\right)} \log\left(-b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right) - 2 \, {\left(B a^{2} - B b^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(x\right)\right)}{2 \, {\left(a^{3} - a b^{2}\right)}}\right]"," ",0,"[-1/2*(sqrt(-a^2 + b^2)*A*a*log(((2*a^2 - b^2)*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2 + 2*(a*cos(x)*sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) + (B*a^2 - B*b^2)*log(-b^2*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2) - 2*(B*a^2 - B*b^2)*log(1/2*sin(x)))/(a^3 - a*b^2), -1/2*(2*sqrt(a^2 - b^2)*A*a*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + (B*a^2 - B*b^2)*log(-b^2*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2) - 2*(B*a^2 - B*b^2)*log(1/2*sin(x)))/(a^3 - a*b^2)]","A",0
8,1,278,0,10.313765," ","integrate((A+B*sec(x))/(a+b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{B b \log\left(-b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right) + \sqrt{-a^{2} + b^{2}} A \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(x\right) \sin\left(x\right) + b \cos\left(x\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) - {\left(B a + B b\right)} \log\left(\sin\left(x\right) + 1\right) + {\left(B a - B b\right)} \log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a^{2} - b^{2}\right)}}, -\frac{B b \log\left(-b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2}\right) + 2 \, \sqrt{a^{2} - b^{2}} A \arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right) - {\left(B a + B b\right)} \log\left(\sin\left(x\right) + 1\right) + {\left(B a - B b\right)} \log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a^{2} - b^{2}\right)}}\right]"," ",0,"[-1/2*(B*b*log(-b^2*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2) + sqrt(-a^2 + b^2)*A*log(((2*a^2 - b^2)*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2 + 2*(a*cos(x)*sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - (B*a + B*b)*log(sin(x) + 1) + (B*a - B*b)*log(-sin(x) + 1))/(a^2 - b^2), -1/2*(B*b*log(-b^2*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2) + 2*sqrt(a^2 - b^2)*A*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (B*a + B*b)*log(sin(x) + 1) + (B*a - B*b)*log(-sin(x) + 1))/(a^2 - b^2)]","A",0
9,1,265,0,2.046347," ","integrate((A+B*csc(x))/(a+b*sin(x)),x, algorithm=""fricas"")","\left[\frac{{\left(A a - B b\right)} \sqrt{-a^{2} + b^{2}} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(x\right) \sin\left(x\right) + b \cos\left(x\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) - {\left(B a^{2} - B b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(B a^{2} - B b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{3} - a b^{2}\right)}}, -\frac{2 \, {\left(A a - B b\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right) + {\left(B a^{2} - B b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(B a^{2} - B b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{3} - a b^{2}\right)}}\right]"," ",0,"[1/2*((A*a - B*b)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2 - 2*(a*cos(x)*sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - (B*a^2 - B*b^2)*log(1/2*cos(x) + 1/2) + (B*a^2 - B*b^2)*log(-1/2*cos(x) + 1/2))/(a^3 - a*b^2), -1/2*(2*(A*a - B*b)*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + (B*a^2 - B*b^2)*log(1/2*cos(x) + 1/2) - (B*a^2 - B*b^2)*log(-1/2*cos(x) + 1/2))/(a^3 - a*b^2)]","A",0
