1,1,227,0,0.770894," ","integrate((A+B*sin(x))/(a+b*cos(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} A b \log\left(\frac{2 \, a b \cos\left(x\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(x\right) + b\right)} \sin\left(x\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}}\right) + {\left(B a^{2} - B b^{2}\right)} \log\left(b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)}}, \frac{2 \, \sqrt{a^{2} - b^{2}} A b \arctan\left(-\frac{a \cos\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(x\right)}\right) - {\left(B a^{2} - B b^{2}\right)} \log\left(b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)}}\right]"," ",0,"[-1/2*(sqrt(-a^2 + b^2)*A*b*log((2*a*b*cos(x) + (2*a^2 - b^2)*cos(x)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(x) + b)*sin(x) - a^2 + 2*b^2)/(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2)) + (B*a^2 - B*b^2)*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2))/(a^2*b - b^3), 1/2*(2*sqrt(a^2 - b^2)*A*b*arctan(-(a*cos(x) + b)/(sqrt(a^2 - b^2)*sin(x))) - (B*a^2 - B*b^2)*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2))/(a^2*b - b^3)]","A",0
2,1,28,0,0.645754," ","integrate((A+B*sin(x))/(1+cos(x)),x, algorithm=""fricas"")","-\frac{{\left(B \cos\left(x\right) + B\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - A \sin\left(x\right)}{\cos\left(x\right) + 1}"," ",0,"-((B*cos(x) + B)*log(1/2*cos(x) + 1/2) - A*sin(x))/(cos(x) + 1)","A",0
3,1,25,0,0.462426," ","integrate((A+B*sin(x))/(1-cos(x)),x, algorithm=""fricas"")","\frac{B \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) - A \cos\left(x\right) - A}{\sin\left(x\right)}"," ",0,"(B*log(-1/2*cos(x) + 1/2)*sin(x) - A*cos(x) - A)/sin(x)","A",0
4,1,231,0,0.541057," ","integrate((b+c+sin(x))/(a+b*cos(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} {\left(b^{2} + b c\right)} \log\left(\frac{2 \, a b \cos\left(x\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(x\right) + b\right)} \sin\left(x\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}}\right) + {\left(a^{2} - b^{2}\right)} \log\left(b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{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 \cos\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(x\right)}\right) - {\left(a^{2} - b^{2}\right)} \log\left(b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{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*b*cos(x) + (2*a^2 - b^2)*cos(x)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(x) + b)*sin(x) - a^2 + 2*b^2)/(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2)) + (a^2 - b^2)*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2))/(a^2*b - b^3), 1/2*(2*sqrt(a^2 - b^2)*(b^2 + b*c)*arctan(-(a*cos(x) + b)/(sqrt(a^2 - b^2)*sin(x))) - (a^2 - b^2)*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2))/(a^2*b - b^3)]","A",0
5,1,235,0,0.564471," ","integrate((b+c+sin(x))/(a-b*cos(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} {\left(b^{2} + b c\right)} \log\left(-\frac{2 \, a b \cos\left(x\right) - {\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(x\right) - b\right)} \sin\left(x\right) + a^{2} - 2 \, b^{2}}{b^{2} \cos\left(x\right)^{2} - 2 \, a b \cos\left(x\right) + a^{2}}\right) - {\left(a^{2} - b^{2}\right)} \log\left(b^{2} \cos\left(x\right)^{2} - 2 \, a b \cos\left(x\right) + a^{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 \cos\left(x\right) - b}{\sqrt{a^{2} - b^{2}} \sin\left(x\right)}\right) + {\left(a^{2} - b^{2}\right)} \log\left(b^{2} \cos\left(x\right)^{2} - 2 \, a b \cos\left(x\right) + a^{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*b*cos(x) - (2*a^2 - b^2)*cos(x)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(x) - b)*sin(x) + a^2 - 2*b^2)/(b^2*cos(x)^2 - 2*a*b*cos(x) + a^2)) - (a^2 - b^2)*log(b^2*cos(x)^2 - 2*a*b*cos(x) + a^2))/(a^2*b - b^3), 1/2*(2*sqrt(a^2 - b^2)*(b^2 + b*c)*arctan(-(a*cos(x) - b)/(sqrt(a^2 - b^2)*sin(x))) + (a^2 - b^2)*log(b^2*cos(x)^2 - 2*a*b*cos(x) + a^2))/(a^2*b - b^3)]","A",0
6,1,263,0,0.792063," ","integrate((A+B*tan(x))/(a+b*cos(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} A a \log\left(\frac{2 \, a b \cos\left(x\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(x\right) + b\right)} \sin\left(x\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}}\right) - {\left(B a^{2} - B b^{2}\right)} \log\left(b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}\right) + 2 \, {\left(B a^{2} - B b^{2}\right)} \log\left(-\cos\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 \cos\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(x\right)}\right) + {\left(B a^{2} - B b^{2}\right)} \log\left(b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}\right) - 2 \, {\left(B a^{2} - B b^{2}\right)} \log\left(-\cos\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*b*cos(x) + (2*a^2 - b^2)*cos(x)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(x) + b)*sin(x) - a^2 + 2*b^2)/(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2)) - (B*a^2 - B*b^2)*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2) + 2*(B*a^2 - B*b^2)*log(-cos(x)))/(a^3 - a*b^2), 1/2*(2*sqrt(a^2 - b^2)*A*a*arctan(-(a*cos(x) + b)/(sqrt(a^2 - b^2)*sin(x))) + (B*a^2 - B*b^2)*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2) - 2*(B*a^2 - B*b^2)*log(-cos(x)))/(a^3 - a*b^2)]","A",0
7,1,266,0,7.234782," ","integrate((A+B*cot(x))/(a+b*cos(x)),x, algorithm=""fricas"")","\left[-\frac{B a \log\left(b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}\right) + \sqrt{-a^{2} + b^{2}} A \log\left(\frac{2 \, a b \cos\left(x\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(x\right) + b\right)} \sin\left(x\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}}\right) - {\left(B a + B b\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(B a - B b\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{2} - b^{2}\right)}}, -\frac{B a \log\left(b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}\right) - 2 \, \sqrt{a^{2} - b^{2}} A \arctan\left(-\frac{a \cos\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(x\right)}\right) - {\left(B a + B b\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(B a - B b\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{2} - b^{2}\right)}}\right]"," ",0,"[-1/2*(B*a*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2) + sqrt(-a^2 + b^2)*A*log((2*a*b*cos(x) + (2*a^2 - b^2)*cos(x)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(x) + b)*sin(x) - a^2 + 2*b^2)/(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2)) - (B*a + B*b)*log(1/2*cos(x) + 1/2) - (B*a - B*b)*log(-1/2*cos(x) + 1/2))/(a^2 - b^2), -1/2*(B*a*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2) - 2*sqrt(a^2 - b^2)*A*arctan(-(a*cos(x) + b)/(sqrt(a^2 - b^2)*sin(x))) - (B*a + B*b)*log(1/2*cos(x) + 1/2) - (B*a - B*b)*log(-1/2*cos(x) + 1/2))/(a^2 - b^2)]","A",0
8,1,265,0,8.187143," ","integrate((A+B*csc(x))/(a+b*cos(x)),x, algorithm=""fricas"")","\left[\frac{B b \log\left(b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}\right) - \sqrt{-a^{2} + b^{2}} A \log\left(\frac{2 \, a b \cos\left(x\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(x\right) + b\right)} \sin\left(x\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}}\right) - {\left(B a + B b\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(B a - B b\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{2} - b^{2}\right)}}, \frac{B b \log\left(b^{2} \cos\left(x\right)^{2} + 2 \, a b \cos\left(x\right) + a^{2}\right) + 2 \, \sqrt{a^{2} - b^{2}} A \arctan\left(-\frac{a \cos\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(x\right)}\right) - {\left(B a + B b\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(B a - B b\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{2} - b^{2}\right)}}\right]"," ",0,"[1/2*(B*b*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2) - sqrt(-a^2 + b^2)*A*log((2*a*b*cos(x) + (2*a^2 - b^2)*cos(x)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(x) + b)*sin(x) - a^2 + 2*b^2)/(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2)) - (B*a + B*b)*log(1/2*cos(x) + 1/2) + (B*a - B*b)*log(-1/2*cos(x) + 1/2))/(a^2 - b^2), 1/2*(B*b*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2) + 2*sqrt(a^2 - b^2)*A*arctan(-(a*cos(x) + b)/(sqrt(a^2 - b^2)*sin(x))) - (B*a + B*b)*log(1/2*cos(x) + 1/2) + (B*a - B*b)*log(-1/2*cos(x) + 1/2))/(a^2 - b^2)]","A",0
9,-1,0,0,0.000000," ","integrate((c+d*sec(f*x+e))^4/(a+b*cos(f*x+e)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
10,1,747,0,111.796053," ","integrate((c+d*sec(f*x+e))^3/(a+b*cos(f*x+e)),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{3} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a b^{2} c d^{2} - b^{3} d^{3}\right)} \sqrt{-a^{2} + b^{2}} \cos\left(f x + e\right)^{2} \log\left(\frac{2 \, a b \cos\left(f x + e\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(f x + e\right) + b\right)} \sin\left(f x + e\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(f x + e\right)^{2} + 2 \, a b \cos\left(f x + e\right) + a^{2}}\right) + {\left(6 \, {\left(a^{4} - a^{2} b^{2}\right)} c^{2} d - 6 \, {\left(a^{3} b - a b^{3}\right)} c d^{2} + {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4}\right)} d^{3}\right)} \cos\left(f x + e\right)^{2} \log\left(\sin\left(f x + e\right) + 1\right) - {\left(6 \, {\left(a^{4} - a^{2} b^{2}\right)} c^{2} d - 6 \, {\left(a^{3} b - a b^{3}\right)} c d^{2} + {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4}\right)} d^{3}\right)} \cos\left(f x + e\right)^{2} \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, {\left({\left(a^{4} - a^{2} b^{2}\right)} d^{3} + 2 \, {\left(3 \, {\left(a^{4} - a^{2} b^{2}\right)} c d^{2} - {\left(a^{3} b - a b^{3}\right)} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{4 \, {\left(a^{5} - a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2}}, \frac{4 \, {\left(a^{3} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a b^{2} c d^{2} - b^{3} d^{3}\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \cos\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{2} + {\left(6 \, {\left(a^{4} - a^{2} b^{2}\right)} c^{2} d - 6 \, {\left(a^{3} b - a b^{3}\right)} c d^{2} + {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4}\right)} d^{3}\right)} \cos\left(f x + e\right)^{2} \log\left(\sin\left(f x + e\right) + 1\right) - {\left(6 \, {\left(a^{4} - a^{2} b^{2}\right)} c^{2} d - 6 \, {\left(a^{3} b - a b^{3}\right)} c d^{2} + {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4}\right)} d^{3}\right)} \cos\left(f x + e\right)^{2} \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, {\left({\left(a^{4} - a^{2} b^{2}\right)} d^{3} + 2 \, {\left(3 \, {\left(a^{4} - a^{2} b^{2}\right)} c d^{2} - {\left(a^{3} b - a b^{3}\right)} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{4 \, {\left(a^{5} - a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2}}\right]"," ",0,"[1/4*(2*(a^3*c^3 - 3*a^2*b*c^2*d + 3*a*b^2*c*d^2 - b^3*d^3)*sqrt(-a^2 + b^2)*cos(f*x + e)^2*log((2*a*b*cos(f*x + e) + (2*a^2 - b^2)*cos(f*x + e)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(f*x + e) + b)*sin(f*x + e) - a^2 + 2*b^2)/(b^2*cos(f*x + e)^2 + 2*a*b*cos(f*x + e) + a^2)) + (6*(a^4 - a^2*b^2)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 + (a^4 + a^2*b^2 - 2*b^4)*d^3)*cos(f*x + e)^2*log(sin(f*x + e) + 1) - (6*(a^4 - a^2*b^2)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 + (a^4 + a^2*b^2 - 2*b^4)*d^3)*cos(f*x + e)^2*log(-sin(f*x + e) + 1) + 2*((a^4 - a^2*b^2)*d^3 + 2*(3*(a^4 - a^2*b^2)*c*d^2 - (a^3*b - a*b^3)*d^3)*cos(f*x + e))*sin(f*x + e))/((a^5 - a^3*b^2)*f*cos(f*x + e)^2), 1/4*(4*(a^3*c^3 - 3*a^2*b*c^2*d + 3*a*b^2*c*d^2 - b^3*d^3)*sqrt(a^2 - b^2)*arctan(-(a*cos(f*x + e) + b)/(sqrt(a^2 - b^2)*sin(f*x + e)))*cos(f*x + e)^2 + (6*(a^4 - a^2*b^2)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 + (a^4 + a^2*b^2 - 2*b^4)*d^3)*cos(f*x + e)^2*log(sin(f*x + e) + 1) - (6*(a^4 - a^2*b^2)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 + (a^4 + a^2*b^2 - 2*b^4)*d^3)*cos(f*x + e)^2*log(-sin(f*x + e) + 1) + 2*((a^4 - a^2*b^2)*d^3 + 2*(3*(a^4 - a^2*b^2)*c*d^2 - (a^3*b - a*b^3)*d^3)*cos(f*x + e))*sin(f*x + e))/((a^5 - a^3*b^2)*f*cos(f*x + e)^2)]","B",0
11,1,505,0,12.845299," ","integrate((c+d*sec(f*x+e))^2/(a+b*cos(f*x+e)),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{3} - a b^{2}\right)} d^{2} \sin\left(f x + e\right) - {\left(a^{2} c^{2} - 2 \, a b c d + b^{2} d^{2}\right)} \sqrt{-a^{2} + b^{2}} \cos\left(f x + e\right) \log\left(\frac{2 \, a b \cos\left(f x + e\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(f x + e\right) + b\right)} \sin\left(f x + e\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(f x + e\right)^{2} + 2 \, a b \cos\left(f x + e\right) + a^{2}}\right) + {\left(2 \, {\left(a^{3} - a b^{2}\right)} c d - {\left(a^{2} b - b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right) \log\left(\sin\left(f x + e\right) + 1\right) - {\left(2 \, {\left(a^{3} - a b^{2}\right)} c d - {\left(a^{2} b - b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right) \log\left(-\sin\left(f x + e\right) + 1\right)}{2 \, {\left(a^{4} - a^{2} b^{2}\right)} f \cos\left(f x + e\right)}, \frac{2 \, {\left(a^{3} - a b^{2}\right)} d^{2} \sin\left(f x + e\right) + 2 \, {\left(a^{2} c^{2} - 2 \, a b c d + b^{2} d^{2}\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \cos\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right) + {\left(2 \, {\left(a^{3} - a b^{2}\right)} c d - {\left(a^{2} b - b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right) \log\left(\sin\left(f x + e\right) + 1\right) - {\left(2 \, {\left(a^{3} - a b^{2}\right)} c d - {\left(a^{2} b - b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right) \log\left(-\sin\left(f x + e\right) + 1\right)}{2 \, {\left(a^{4} - a^{2} b^{2}\right)} f \cos\left(f x + e\right)}\right]"," ",0,"[1/2*(2*(a^3 - a*b^2)*d^2*sin(f*x + e) - (a^2*c^2 - 2*a*b*c*d + b^2*d^2)*sqrt(-a^2 + b^2)*cos(f*x + e)*log((2*a*b*cos(f*x + e) + (2*a^2 - b^2)*cos(f*x + e)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(f*x + e) + b)*sin(f*x + e) - a^2 + 2*b^2)/(b^2*cos(f*x + e)^2 + 2*a*b*cos(f*x + e) + a^2)) + (2*(a^3 - a*b^2)*c*d - (a^2*b - b^3)*d^2)*cos(f*x + e)*log(sin(f*x + e) + 1) - (2*(a^3 - a*b^2)*c*d - (a^2*b - b^3)*d^2)*cos(f*x + e)*log(-sin(f*x + e) + 1))/((a^4 - a^2*b^2)*f*cos(f*x + e)), 1/2*(2*(a^3 - a*b^2)*d^2*sin(f*x + e) + 2*(a^2*c^2 - 2*a*b*c*d + b^2*d^2)*sqrt(a^2 - b^2)*arctan(-(a*cos(f*x + e) + b)/(sqrt(a^2 - b^2)*sin(f*x + e)))*cos(f*x + e) + (2*(a^3 - a*b^2)*c*d - (a^2*b - b^3)*d^2)*cos(f*x + e)*log(sin(f*x + e) + 1) - (2*(a^3 - a*b^2)*c*d - (a^2*b - b^3)*d^2)*cos(f*x + e)*log(-sin(f*x + e) + 1))/((a^4 - a^2*b^2)*f*cos(f*x + e))]","B",0
12,1,296,0,1.633657," ","integrate((c+d*sec(f*x+e))/(a+b*cos(f*x+e)),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} - b^{2}\right)} d \log\left(\sin\left(f x + e\right) + 1\right) - {\left(a^{2} - b^{2}\right)} d \log\left(-\sin\left(f x + e\right) + 1\right) + \sqrt{-a^{2} + b^{2}} {\left(a c - b d\right)} \log\left(\frac{2 \, a b \cos\left(f x + e\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(f x + e\right) + b\right)} \sin\left(f x + e\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(f x + e\right)^{2} + 2 \, a b \cos\left(f x + e\right) + a^{2}}\right)}{2 \, {\left(a^{3} - a b^{2}\right)} f}, \frac{{\left(a^{2} - b^{2}\right)} d \log\left(\sin\left(f x + e\right) + 1\right) - {\left(a^{2} - b^{2}\right)} d \log\left(-\sin\left(f x + e\right) + 1\right) + 2 \, \sqrt{a^{2} - b^{2}} {\left(a c - b d\right)} \arctan\left(-\frac{a \cos\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(f x + e\right)}\right)}{2 \, {\left(a^{3} - a b^{2}\right)} f}\right]"," ",0,"[1/2*((a^2 - b^2)*d*log(sin(f*x + e) + 1) - (a^2 - b^2)*d*log(-sin(f*x + e) + 1) + sqrt(-a^2 + b^2)*(a*c - b*d)*log((2*a*b*cos(f*x + e) + (2*a^2 - b^2)*cos(f*x + e)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(f*x + e) + b)*sin(f*x + e) - a^2 + 2*b^2)/(b^2*cos(f*x + e)^2 + 2*a*b*cos(f*x + e) + a^2)))/((a^3 - a*b^2)*f), 1/2*((a^2 - b^2)*d*log(sin(f*x + e) + 1) - (a^2 - b^2)*d*log(-sin(f*x + e) + 1) + 2*sqrt(a^2 - b^2)*(a*c - b*d)*arctan(-(a*cos(f*x + e) + b)/(sqrt(a^2 - b^2)*sin(f*x + e))))/((a^3 - a*b^2)*f)]","A",0
13,1,1022,0,5.920675," ","integrate(1/(a+b*cos(f*x+e))/(c+d*sec(f*x+e)),x, algorithm=""fricas"")","\left[-\frac{{\left(a^{2} - b^{2}\right)} \sqrt{c^{2} - d^{2}} d \log\left(\frac{2 \, c d \cos\left(f x + e\right) - {\left(c^{2} - 2 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{c^{2} - d^{2}} {\left(d \cos\left(f x + e\right) + c\right)} \sin\left(f x + e\right) + 2 \, c^{2} - d^{2}}{c^{2} \cos\left(f x + e\right)^{2} + 2 \, c d \cos\left(f x + e\right) + d^{2}}\right) - {\left(a c^{2} - a d^{2}\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{2 \, a b \cos\left(f x + e\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(f x + e\right) + b\right)} \sin\left(f x + e\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(f x + e\right)^{2} + 2 \, a b \cos\left(f x + e\right) + a^{2}}\right)}{2 \, {\left({\left(a^{3} - a b^{2}\right)} c^{3} - {\left(a^{2} b - b^{3}\right)} c^{2} d - {\left(a^{3} - a b^{2}\right)} c d^{2} + {\left(a^{2} b - b^{3}\right)} d^{3}\right)} f}, -\frac{2 \, {\left(a^{2} - b^{2}\right)} \sqrt{-c^{2} + d^{2}} d \arctan\left(-\frac{\sqrt{-c^{2} + d^{2}} {\left(d \cos\left(f x + e\right) + c\right)}}{{\left(c^{2} - d^{2}\right)} \sin\left(f x + e\right)}\right) - {\left(a c^{2} - a d^{2}\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{2 \, a b \cos\left(f x + e\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(f x + e\right) + b\right)} \sin\left(f x + e\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(f x + e\right)^{2} + 2 \, a b \cos\left(f x + e\right) + a^{2}}\right)}{2 \, {\left({\left(a^{3} - a b^{2}\right)} c^{3} - {\left(a^{2} b - b^{3}\right)} c^{2} d - {\left(a^{3} - a b^{2}\right)} c d^{2} + {\left(a^{2} b - b^{3}\right)} d^{3}\right)} f}, -\frac{{\left(a^{2} - b^{2}\right)} \sqrt{c^{2} - d^{2}} d \log\left(\frac{2 \, c d \cos\left(f x + e\right) - {\left(c^{2} - 2 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, \sqrt{c^{2} - d^{2}} {\left(d \cos\left(f x + e\right) + c\right)} \sin\left(f x + e\right) + 2 \, c^{2} - d^{2}}{c^{2} \cos\left(f x + e\right)^{2} + 2 \, c d \cos\left(f x + e\right) + d^{2}}\right) - 2 \, {\left(a c^{2} - a d^{2}\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \cos\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(f x + e\right)}\right)}{2 \, {\left({\left(a^{3} - a b^{2}\right)} c^{3} - {\left(a^{2} b - b^{3}\right)} c^{2} d - {\left(a^{3} - a b^{2}\right)} c d^{2} + {\left(a^{2} b - b^{3}\right)} d^{3}\right)} f}, -\frac{{\left(a^{2} - b^{2}\right)} \sqrt{-c^{2} + d^{2}} d \arctan\left(-\frac{\sqrt{-c^{2} + d^{2}} {\left(d \cos\left(f x + e\right) + c\right)}}{{\left(c^{2} - d^{2}\right)} \sin\left(f x + e\right)}\right) - {\left(a c^{2} - a d^{2}\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \cos\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(f x + e\right)}\right)}{{\left({\left(a^{3} - a b^{2}\right)} c^{3} - {\left(a^{2} b - b^{3}\right)} c^{2} d - {\left(a^{3} - a b^{2}\right)} c d^{2} + {\left(a^{2} b - b^{3}\right)} d^{3}\right)} f}\right]"," ",0,"[-1/2*((a^2 - b^2)*sqrt(c^2 - d^2)*d*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 + 2*sqrt(c^2 - d^2)*(d*cos(f*x + e) + c)*sin(f*x + e) + 2*c^2 - d^2)/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) - (a*c^2 - a*d^2)*sqrt(-a^2 + b^2)*log((2*a*b*cos(f*x + e) + (2*a^2 - b^2)*cos(f*x + e)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(f*x + e) + b)*sin(f*x + e) - a^2 + 2*b^2)/(b^2*cos(f*x + e)^2 + 2*a*b*cos(f*x + e) + a^2)))/(((a^3 - a*b^2)*c^3 - (a^2*b - b^3)*c^2*d - (a^3 - a*b^2)*c*d^2 + (a^2*b - b^3)*d^3)*f), -1/2*(2*(a^2 - b^2)*sqrt(-c^2 + d^2)*d*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) - (a*c^2 - a*d^2)*sqrt(-a^2 + b^2)*log((2*a*b*cos(f*x + e) + (2*a^2 - b^2)*cos(f*x + e)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(f*x + e) + b)*sin(f*x + e) - a^2 + 2*b^2)/(b^2*cos(f*x + e)^2 + 2*a*b*cos(f*x + e) + a^2)))/(((a^3 - a*b^2)*c^3 - (a^2*b - b^3)*c^2*d - (a^3 - a*b^2)*c*d^2 + (a^2*b - b^3)*d^3)*f), -1/2*((a^2 - b^2)*sqrt(c^2 - d^2)*d*log((2*c*d*cos(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 + 2*sqrt(c^2 - d^2)*(d*cos(f*x + e) + c)*sin(f*x + e) + 2*c^2 - d^2)/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) - 2*(a*c^2 - a*d^2)*sqrt(a^2 - b^2)*arctan(-(a*cos(f*x + e) + b)/(sqrt(a^2 - b^2)*sin(f*x + e))))/(((a^3 - a*b^2)*c^3 - (a^2*b - b^3)*c^2*d - (a^3 - a*b^2)*c*d^2 + (a^2*b - b^3)*d^3)*f), -((a^2 - b^2)*sqrt(-c^2 + d^2)*d*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) - (a*c^2 - a*d^2)*sqrt(a^2 - b^2)*arctan(-(a*cos(f*x + e) + b)/(sqrt(a^2 - b^2)*sin(f*x + e))))/(((a^3 - a*b^2)*c^3 - (a^2*b - b^3)*c^2*d - (a^3 - a*b^2)*c*d^2 + (a^2*b - b^3)*d^3)*f)]","A",0
14,-1,0,0,0.000000," ","integrate(1/(a+b*cos(f*x+e))/(c+d*sec(f*x+e))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
15,-1,0,0,0.000000," ","integrate(1/(a+b*cos(f*x+e))/(c+d*sec(f*x+e))^3,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
16,0,0,0,2.062537," ","integrate((c+d*sec(f*x+e))^(1/2)/(a+b*cos(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{d \sec\left(f x + e\right) + c}}{b \cos\left(f x + e\right) + a}, x\right)"," ",0,"integral(sqrt(d*sec(f*x + e) + c)/(b*cos(f*x + e) + a), x)","F",0
17,0,0,0,2.459524," ","integrate(1/(a+b*cos(f*x+e))/(c+d*sec(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{d \sec\left(f x + e\right) + c}}{b c \cos\left(f x + e\right) + a c + {\left(b d \cos\left(f x + e\right) + a d\right)} \sec\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(d*sec(f*x + e) + c)/(b*c*cos(f*x + e) + a*c + (b*d*cos(f*x + e) + a*d)*sec(f*x + e)), x)","F",0
18,1,326,0,0.878828," ","integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d)),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(B a^{2} - B b^{2}\right)} e x + {\left(B a - A b\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{2 \, a b \cos\left(e x + d\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(e x + d\right)^{2} + 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(e x + d\right) + b\right)} \sin\left(e x + d\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(e x + d\right)^{2} + 2 \, a b \cos\left(e x + d\right) + a^{2}}\right) - {\left(C a^{2} - C b^{2}\right)} \log\left(b^{2} \cos\left(e x + d\right)^{2} + 2 \, a b \cos\left(e x + d\right) + a^{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)} e}, \frac{2 \, {\left(B a^{2} - B b^{2}\right)} e x - 2 \, {\left(B a - A b\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \cos\left(e x + d\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(e x + d\right)}\right) - {\left(C a^{2} - C b^{2}\right)} \log\left(b^{2} \cos\left(e x + d\right)^{2} + 2 \, a b \cos\left(e x + d\right) + a^{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)} e}\right]"," ",0,"[1/2*(2*(B*a^2 - B*b^2)*e*x + (B*a - A*b)*sqrt(-a^2 + b^2)*log((2*a*b*cos(e*x + d) + (2*a^2 - b^2)*cos(e*x + d)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(e*x + d) + b)*sin(e*x + d) - a^2 + 2*b^2)/(b^2*cos(e*x + d)^2 + 2*a*b*cos(e*x + d) + a^2)) - (C*a^2 - C*b^2)*log(b^2*cos(e*x + d)^2 + 2*a*b*cos(e*x + d) + a^2))/((a^2*b - b^3)*e), 1/2*(2*(B*a^2 - B*b^2)*e*x - 2*(B*a - A*b)*sqrt(a^2 - b^2)*arctan(-(a*cos(e*x + d) + b)/(sqrt(a^2 - b^2)*sin(e*x + d))) - (C*a^2 - C*b^2)*log(b^2*cos(e*x + d)^2 + 2*a*b*cos(e*x + d) + a^2))/((a^2*b - b^3)*e)]","A",0
19,1,444,0,1.250083," ","integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))^2,x, algorithm=""fricas"")","\left[\frac{2 \, C a^{4} - 4 \, C a^{2} b^{2} + 2 \, C b^{4} - {\left(A a^{2} b - B a b^{2} + {\left(A a b^{2} - B b^{3}\right)} \cos\left(e x + d\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{2 \, a b \cos\left(e x + d\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(e x + d\right)^{2} + 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(e x + d\right) + b\right)} \sin\left(e x + d\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(e x + d\right)^{2} + 2 \, a b \cos\left(e x + d\right) + a^{2}}\right) + 2 \, {\left(B a^{3} b - A a^{2} b^{2} - B a b^{3} + A b^{4}\right)} \sin\left(e x + d\right)}{2 \, {\left({\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} e \cos\left(e x + d\right) + {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} e\right)}}, \frac{C a^{4} - 2 \, C a^{2} b^{2} + C b^{4} + {\left(A a^{2} b - B a b^{2} + {\left(A a b^{2} - B b^{3}\right)} \cos\left(e x + d\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \cos\left(e x + d\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(e x + d\right)}\right) + {\left(B a^{3} b - A a^{2} b^{2} - B a b^{3} + A b^{4}\right)} \sin\left(e x + d\right)}{{\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} e \cos\left(e x + d\right) + {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} e}\right]"," ",0,"[1/2*(2*C*a^4 - 4*C*a^2*b^2 + 2*C*b^4 - (A*a^2*b - B*a*b^2 + (A*a*b^2 - B*b^3)*cos(e*x + d))*sqrt(-a^2 + b^2)*log((2*a*b*cos(e*x + d) + (2*a^2 - b^2)*cos(e*x + d)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(e*x + d) + b)*sin(e*x + d) - a^2 + 2*b^2)/(b^2*cos(e*x + d)^2 + 2*a*b*cos(e*x + d) + a^2)) + 2*(B*a^3*b - A*a^2*b^2 - B*a*b^3 + A*b^4)*sin(e*x + d))/((a^4*b^2 - 2*a^2*b^4 + b^6)*e*cos(e*x + d) + (a^5*b - 2*a^3*b^3 + a*b^5)*e), (C*a^4 - 2*C*a^2*b^2 + C*b^4 + (A*a^2*b - B*a*b^2 + (A*a*b^2 - B*b^3)*cos(e*x + d))*sqrt(a^2 - b^2)*arctan(-(a*cos(e*x + d) + b)/(sqrt(a^2 - b^2)*sin(e*x + d))) + (B*a^3*b - A*a^2*b^2 - B*a*b^3 + A*b^4)*sin(e*x + d))/((a^4*b^2 - 2*a^2*b^4 + b^6)*e*cos(e*x + d) + (a^5*b - 2*a^3*b^3 + a*b^5)*e)]","A",0
20,1,830,0,1.810557," ","integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))^3,x, algorithm=""fricas"")","\left[\frac{2 \, C a^{6} - 6 \, C a^{4} b^{2} + 6 \, C a^{2} b^{4} - 2 \, C b^{6} - {\left(2 \, A a^{4} b - 3 \, B a^{3} b^{2} + A a^{2} b^{3} + {\left(2 \, A a^{2} b^{3} - 3 \, B a b^{4} + A b^{5}\right)} \cos\left(e x + d\right)^{2} + 2 \, {\left(2 \, A a^{3} b^{2} - 3 \, B a^{2} b^{3} + A a b^{4}\right)} \cos\left(e x + d\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{2 \, a b \cos\left(e x + d\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(e x + d\right)^{2} + 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(e x + d\right) + b\right)} \sin\left(e x + d\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(e x + d\right)^{2} + 2 \, a b \cos\left(e x + d\right) + a^{2}}\right) + 2 \, {\left(2 \, B a^{5} b - 4 \, A a^{4} b^{2} - B a^{3} b^{3} + 5 \, A a^{2} b^{4} - B a b^{5} - A b^{6} + {\left(B a^{4} b^{2} - 3 \, A a^{3} b^{3} + B a^{2} b^{4} + 3 \, A a b^{5} - 2 \, B b^{6}\right)} \cos\left(e x + d\right)\right)} \sin\left(e x + d\right)}{4 \, {\left({\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} e \cos\left(e x + d\right)^{2} + 2 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} e \cos\left(e x + d\right) + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} e\right)}}, \frac{C a^{6} - 3 \, C a^{4} b^{2} + 3 \, C a^{2} b^{4} - C b^{6} + {\left(2 \, A a^{4} b - 3 \, B a^{3} b^{2} + A a^{2} b^{3} + {\left(2 \, A a^{2} b^{3} - 3 \, B a b^{4} + A b^{5}\right)} \cos\left(e x + d\right)^{2} + 2 \, {\left(2 \, A a^{3} b^{2} - 3 \, B a^{2} b^{3} + A a b^{4}\right)} \cos\left(e x + d\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \cos\left(e x + d\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(e x + d\right)}\right) + {\left(2 \, B a^{5} b - 4 \, A a^{4} b^{2} - B a^{3} b^{3} + 5 \, A a^{2} b^{4} - B a b^{5} - A b^{6} + {\left(B a^{4} b^{2} - 3 \, A a^{3} b^{3} + B a^{2} b^{4} + 3 \, A a b^{5} - 2 \, B b^{6}\right)} \cos\left(e x + d\right)\right)} \sin\left(e x + d\right)}{2 \, {\left({\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} e \cos\left(e x + d\right)^{2} + 2 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} e \cos\left(e x + d\right) + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} e\right)}}\right]"," ",0,"[1/4*(2*C*a^6 - 6*C*a^4*b^2 + 6*C*a^2*b^4 - 2*C*b^6 - (2*A*a^4*b - 3*B*a^3*b^2 + A*a^2*b^3 + (2*A*a^2*b^3 - 3*B*a*b^4 + A*b^5)*cos(e*x + d)^2 + 2*(2*A*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4)*cos(e*x + d))*sqrt(-a^2 + b^2)*log((2*a*b*cos(e*x + d) + (2*a^2 - b^2)*cos(e*x + d)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(e*x + d) + b)*sin(e*x + d) - a^2 + 2*b^2)/(b^2*cos(e*x + d)^2 + 2*a*b*cos(e*x + d) + a^2)) + 2*(2*B*a^5*b - 4*A*a^4*b^2 - B*a^3*b^3 + 5*A*a^2*b^4 - B*a*b^5 - A*b^6 + (B*a^4*b^2 - 3*A*a^3*b^3 + B*a^2*b^4 + 3*A*a*b^5 - 2*B*b^6)*cos(e*x + d))*sin(e*x + d))/((a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*e*cos(e*x + d)^2 + 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*e*cos(e*x + d) + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*e), 1/2*(C*a^6 - 3*C*a^4*b^2 + 3*C*a^2*b^4 - C*b^6 + (2*A*a^4*b - 3*B*a^3*b^2 + A*a^2*b^3 + (2*A*a^2*b^3 - 3*B*a*b^4 + A*b^5)*cos(e*x + d)^2 + 2*(2*A*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4)*cos(e*x + d))*sqrt(a^2 - b^2)*arctan(-(a*cos(e*x + d) + b)/(sqrt(a^2 - b^2)*sin(e*x + d))) + (2*B*a^5*b - 4*A*a^4*b^2 - B*a^3*b^3 + 5*A*a^2*b^4 - B*a*b^5 - A*b^6 + (B*a^4*b^2 - 3*A*a^3*b^3 + B*a^2*b^4 + 3*A*a*b^5 - 2*B*b^6)*cos(e*x + d))*sin(e*x + d))/((a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*e*cos(e*x + d)^2 + 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*e*cos(e*x + d) + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*e)]","B",0
21,1,1334,0,1.191729," ","integrate((A+B*cos(e*x+d)+C*sin(e*x+d))/(a+b*cos(e*x+d))^4,x, algorithm=""fricas"")","\left[\frac{4 \, C a^{8} - 16 \, C a^{6} b^{2} + 24 \, C a^{4} b^{4} - 16 \, C a^{2} b^{6} + 4 \, C b^{8} - 3 \, {\left(2 \, A a^{6} b - 4 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} - B a^{3} b^{4} + {\left(2 \, A a^{3} b^{4} - 4 \, B a^{2} b^{5} + 3 \, A a b^{6} - B b^{7}\right)} \cos\left(e x + d\right)^{3} + 3 \, {\left(2 \, A a^{4} b^{3} - 4 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5} - B a b^{6}\right)} \cos\left(e x + d\right)^{2} + 3 \, {\left(2 \, A a^{5} b^{2} - 4 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4} - B a^{2} b^{5}\right)} \cos\left(e x + d\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{2 \, a b \cos\left(e x + d\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(e x + d\right)^{2} + 2 \, \sqrt{-a^{2} + b^{2}} {\left(a \cos\left(e x + d\right) + b\right)} \sin\left(e x + d\right) - a^{2} + 2 \, b^{2}}{b^{2} \cos\left(e x + d\right)^{2} + 2 \, a b \cos\left(e x + d\right) + a^{2}}\right) + 2 \, {\left(6 \, B a^{7} b - 18 \, A a^{6} b^{2} + 4 \, B a^{5} b^{3} + 23 \, A a^{4} b^{4} - 11 \, B a^{3} b^{5} - 7 \, A a^{2} b^{6} + B a b^{7} + 2 \, A b^{8} + {\left(2 \, B a^{5} b^{3} - 11 \, A a^{4} b^{4} + 11 \, B a^{3} b^{5} + 7 \, A a^{2} b^{6} - 13 \, B a b^{7} + 4 \, A b^{8}\right)} \cos\left(e x + d\right)^{2} + 3 \, {\left(2 \, B a^{6} b^{2} - 9 \, A a^{5} b^{3} + 7 \, B a^{4} b^{4} + 8 \, A a^{3} b^{5} - 10 \, B a^{2} b^{6} + A a b^{7} + B b^{8}\right)} \cos\left(e x + d\right)\right)} \sin\left(e x + d\right)}{12 \, {\left({\left(a^{8} b^{4} - 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} - 4 \, a^{2} b^{10} + b^{12}\right)} e \cos\left(e x + d\right)^{3} + 3 \, {\left(a^{9} b^{3} - 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} - 4 \, a^{3} b^{9} + a b^{11}\right)} e \cos\left(e x + d\right)^{2} + 3 \, {\left(a^{10} b^{2} - 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} - 4 \, a^{4} b^{8} + a^{2} b^{10}\right)} e \cos\left(e x + d\right) + {\left(a^{11} b - 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 4 \, a^{5} b^{7} + a^{3} b^{9}\right)} e\right)}}, \frac{2 \, C a^{8} - 8 \, C a^{6} b^{2} + 12 \, C a^{4} b^{4} - 8 \, C a^{2} b^{6} + 2 \, C b^{8} + 3 \, {\left(2 \, A a^{6} b - 4 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} - B a^{3} b^{4} + {\left(2 \, A a^{3} b^{4} - 4 \, B a^{2} b^{5} + 3 \, A a b^{6} - B b^{7}\right)} \cos\left(e x + d\right)^{3} + 3 \, {\left(2 \, A a^{4} b^{3} - 4 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5} - B a b^{6}\right)} \cos\left(e x + d\right)^{2} + 3 \, {\left(2 \, A a^{5} b^{2} - 4 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4} - B a^{2} b^{5}\right)} \cos\left(e x + d\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \cos\left(e x + d\right) + b}{\sqrt{a^{2} - b^{2}} \sin\left(e x + d\right)}\right) + {\left(6 \, B a^{7} b - 18 \, A a^{6} b^{2} + 4 \, B a^{5} b^{3} + 23 \, A a^{4} b^{4} - 11 \, B a^{3} b^{5} - 7 \, A a^{2} b^{6} + B a b^{7} + 2 \, A b^{8} + {\left(2 \, B a^{5} b^{3} - 11 \, A a^{4} b^{4} + 11 \, B a^{3} b^{5} + 7 \, A a^{2} b^{6} - 13 \, B a b^{7} + 4 \, A b^{8}\right)} \cos\left(e x + d\right)^{2} + 3 \, {\left(2 \, B a^{6} b^{2} - 9 \, A a^{5} b^{3} + 7 \, B a^{4} b^{4} + 8 \, A a^{3} b^{5} - 10 \, B a^{2} b^{6} + A a b^{7} + B b^{8}\right)} \cos\left(e x + d\right)\right)} \sin\left(e x + d\right)}{6 \, {\left({\left(a^{8} b^{4} - 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} - 4 \, a^{2} b^{10} + b^{12}\right)} e \cos\left(e x + d\right)^{3} + 3 \, {\left(a^{9} b^{3} - 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} - 4 \, a^{3} b^{9} + a b^{11}\right)} e \cos\left(e x + d\right)^{2} + 3 \, {\left(a^{10} b^{2} - 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} - 4 \, a^{4} b^{8} + a^{2} b^{10}\right)} e \cos\left(e x + d\right) + {\left(a^{11} b - 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 4 \, a^{5} b^{7} + a^{3} b^{9}\right)} e\right)}}\right]"," ",0,"[1/12*(4*C*a^8 - 16*C*a^6*b^2 + 24*C*a^4*b^4 - 16*C*a^2*b^6 + 4*C*b^8 - 3*(2*A*a^6*b - 4*B*a^5*b^2 + 3*A*a^4*b^3 - B*a^3*b^4 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cos(e*x + d)^3 + 3*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cos(e*x + d)^2 + 3*(2*A*a^5*b^2 - 4*B*a^4*b^3 + 3*A*a^3*b^4 - B*a^2*b^5)*cos(e*x + d))*sqrt(-a^2 + b^2)*log((2*a*b*cos(e*x + d) + (2*a^2 - b^2)*cos(e*x + d)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(e*x + d) + b)*sin(e*x + d) - a^2 + 2*b^2)/(b^2*cos(e*x + d)^2 + 2*a*b*cos(e*x + d) + a^2)) + 2*(6*B*a^7*b - 18*A*a^6*b^2 + 4*B*a^5*b^3 + 23*A*a^4*b^4 - 11*B*a^3*b^5 - 7*A*a^2*b^6 + B*a*b^7 + 2*A*b^8 + (2*B*a^5*b^3 - 11*A*a^4*b^4 + 11*B*a^3*b^5 + 7*A*a^2*b^6 - 13*B*a*b^7 + 4*A*b^8)*cos(e*x + d)^2 + 3*(2*B*a^6*b^2 - 9*A*a^5*b^3 + 7*B*a^4*b^4 + 8*A*a^3*b^5 - 10*B*a^2*b^6 + A*a*b^7 + B*b^8)*cos(e*x + d))*sin(e*x + d))/((a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*cos(e*x + d)^3 + 3*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e*cos(e*x + d)^2 + 3*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10)*e*cos(e*x + d) + (a^11*b - 4*a^9*b^3 + 6*a^7*b^5 - 4*a^5*b^7 + a^3*b^9)*e), 1/6*(2*C*a^8 - 8*C*a^6*b^2 + 12*C*a^4*b^4 - 8*C*a^2*b^6 + 2*C*b^8 + 3*(2*A*a^6*b - 4*B*a^5*b^2 + 3*A*a^4*b^3 - B*a^3*b^4 + (2*A*a^3*b^4 - 4*B*a^2*b^5 + 3*A*a*b^6 - B*b^7)*cos(e*x + d)^3 + 3*(2*A*a^4*b^3 - 4*B*a^3*b^4 + 3*A*a^2*b^5 - B*a*b^6)*cos(e*x + d)^2 + 3*(2*A*a^5*b^2 - 4*B*a^4*b^3 + 3*A*a^3*b^4 - B*a^2*b^5)*cos(e*x + d))*sqrt(a^2 - b^2)*arctan(-(a*cos(e*x + d) + b)/(sqrt(a^2 - b^2)*sin(e*x + d))) + (6*B*a^7*b - 18*A*a^6*b^2 + 4*B*a^5*b^3 + 23*A*a^4*b^4 - 11*B*a^3*b^5 - 7*A*a^2*b^6 + B*a*b^7 + 2*A*b^8 + (2*B*a^5*b^3 - 11*A*a^4*b^4 + 11*B*a^3*b^5 + 7*A*a^2*b^6 - 13*B*a*b^7 + 4*A*b^8)*cos(e*x + d)^2 + 3*(2*B*a^6*b^2 - 9*A*a^5*b^3 + 7*B*a^4*b^4 + 8*A*a^3*b^5 - 10*B*a^2*b^6 + A*a*b^7 + B*b^8)*cos(e*x + d))*sin(e*x + d))/((a^8*b^4 - 4*a^6*b^6 + 6*a^4*b^8 - 4*a^2*b^10 + b^12)*e*cos(e*x + d)^3 + 3*(a^9*b^3 - 4*a^7*b^5 + 6*a^5*b^7 - 4*a^3*b^9 + a*b^11)*e*cos(e*x + d)^2 + 3*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10)*e*cos(e*x + d) + (a^11*b - 4*a^9*b^3 + 6*a^7*b^5 - 4*a^5*b^7 + a^3*b^9)*e)]","B",0
