1,1,83,0,1.225359," ","integrate(sin(f*x+e)^3*(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{4 \, a^{2} \cos\left(f x + e\right)^{5} - 20 \, a^{2} \cos\left(f x + e\right)^{3} - 15 \, a^{2} f x + 40 \, a^{2} \cos\left(f x + e\right) - 5 \, {\left(2 \, a^{2} \cos\left(f x + e\right)^{3} - 5 \, a^{2} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{20 \, f}"," ",0,"-1/20*(4*a^2*cos(f*x + e)^5 - 20*a^2*cos(f*x + e)^3 - 15*a^2*f*x + 40*a^2*cos(f*x + e) - 5*(2*a^2*cos(f*x + e)^3 - 5*a^2*cos(f*x + e))*sin(f*x + e))/f","A",0
2,1,96,0,1.461401," ","integrate(sin(f*x+e)^3*(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{144 \, a^{3} \cos\left(f x + e\right)^{5} - 560 \, a^{3} \cos\left(f x + e\right)^{3} - 345 \, a^{3} f x + 960 \, a^{3} \cos\left(f x + e\right) + 5 \, {\left(8 \, a^{3} \cos\left(f x + e\right)^{5} - 62 \, a^{3} \cos\left(f x + e\right)^{3} + 123 \, a^{3} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{240 \, f}"," ",0,"-1/240*(144*a^3*cos(f*x + e)^5 - 560*a^3*cos(f*x + e)^3 - 345*a^3*f*x + 960*a^3*cos(f*x + e) + 5*(8*a^3*cos(f*x + e)^5 - 62*a^3*cos(f*x + e)^3 + 123*a^3*cos(f*x + e))*sin(f*x + e))/f","A",0
3,1,70,0,1.360439," ","integrate(sin(x)^4/(a+a*sin(x)),x, algorithm=""fricas"")","\frac{2 \, \cos\left(x\right)^{4} - \cos\left(x\right)^{3} - 3 \, {\left(3 \, x + 5\right)} \cos\left(x\right) - 12 \, \cos\left(x\right)^{2} + {\left(2 \, \cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} - 9 \, x - 9 \, \cos\left(x\right) + 6\right)} \sin\left(x\right) - 9 \, x - 6}{6 \, {\left(a \cos\left(x\right) + a \sin\left(x\right) + a\right)}}"," ",0,"1/6*(2*cos(x)^4 - cos(x)^3 - 3*(3*x + 5)*cos(x) - 12*cos(x)^2 + (2*cos(x)^3 + 3*cos(x)^2 - 9*x - 9*cos(x) + 6)*sin(x) - 9*x - 6)/(a*cos(x) + a*sin(x) + a)","A",0
4,1,53,0,1.329066," ","integrate(sin(x)^3/(a+a*sin(x)),x, algorithm=""fricas"")","\frac{\cos\left(x\right)^{3} + 3 \, {\left(x + 1\right)} \cos\left(x\right) + 2 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{2} - 3 \, x - \cos\left(x\right) + 2\right)} \sin\left(x\right) + 3 \, x + 2}{2 \, {\left(a \cos\left(x\right) + a \sin\left(x\right) + a\right)}}"," ",0,"1/2*(cos(x)^3 + 3*(x + 1)*cos(x) + 2*cos(x)^2 - (cos(x)^2 - 3*x - cos(x) + 2)*sin(x) + 3*x + 2)/(a*cos(x) + a*sin(x) + a)","A",0
5,1,35,0,1.782405," ","integrate(sin(x)^2/(a+a*sin(x)),x, algorithm=""fricas"")","-\frac{{\left(x + 2\right)} \cos\left(x\right) + \cos\left(x\right)^{2} + {\left(x + \cos\left(x\right) - 1\right)} \sin\left(x\right) + x + 1}{a \cos\left(x\right) + a \sin\left(x\right) + a}"," ",0,"-((x + 2)*cos(x) + cos(x)^2 + (x + cos(x) - 1)*sin(x) + x + 1)/(a*cos(x) + a*sin(x) + a)","A",0
6,1,28,0,0.979789," ","integrate(sin(x)/(a+a*sin(x)),x, algorithm=""fricas"")","\frac{{\left(x + 1\right)} \cos\left(x\right) + {\left(x - 1\right)} \sin\left(x\right) + x + 1}{a \cos\left(x\right) + a \sin\left(x\right) + a}"," ",0,"((x + 1)*cos(x) + (x - 1)*sin(x) + x + 1)/(a*cos(x) + a*sin(x) + a)","A",0
7,1,22,0,1.174938," ","integrate(1/(a+a*sin(x)),x, algorithm=""fricas"")","-\frac{\cos\left(x\right) - \sin\left(x\right) + 1}{a \cos\left(x\right) + a \sin\left(x\right) + a}"," ",0,"-(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)","A",0
8,1,53,0,1.322759," ","integrate(csc(x)/(a+a*sin(x)),x, algorithm=""fricas"")","-\frac{{\left(\cos\left(x\right) + \sin\left(x\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(\cos\left(x\right) + \sin\left(x\right) + 1\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 2 \, \cos\left(x\right) + 2 \, \sin\left(x\right) - 2}{2 \, {\left(a \cos\left(x\right) + a \sin\left(x\right) + a\right)}}"," ",0,"-1/2*((cos(x) + sin(x) + 1)*log(1/2*cos(x) + 1/2) - (cos(x) + sin(x) + 1)*log(-1/2*cos(x) + 1/2) - 2*cos(x) + 2*sin(x) - 2)/(a*cos(x) + a*sin(x) + a)","B",0
9,1,91,0,1.245251," ","integrate(csc(x)^2/(a+a*sin(x)),x, algorithm=""fricas"")","\frac{4 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 2 \, {\left(2 \, \cos\left(x\right) + 1\right)} \sin\left(x\right) + 2 \, \cos\left(x\right) - 2}{2 \, {\left(a \cos\left(x\right)^{2} - {\left(a \cos\left(x\right) + a\right)} \sin\left(x\right) - a\right)}}"," ",0,"1/2*(4*cos(x)^2 + (cos(x)^2 - (cos(x) + 1)*sin(x) - 1)*log(1/2*cos(x) + 1/2) - (cos(x)^2 - (cos(x) + 1)*sin(x) - 1)*log(-1/2*cos(x) + 1/2) + 2*(2*cos(x) + 1)*sin(x) + 2*cos(x) - 2)/(a*cos(x)^2 - (a*cos(x) + a)*sin(x) - a)","B",0
10,1,134,0,1.386248," ","integrate(csc(x)^3/(a+a*sin(x)),x, algorithm=""fricas"")","\frac{8 \, \cos\left(x\right)^{3} + 6 \, \cos\left(x\right)^{2} - 3 \, {\left(\cos\left(x\right)^{3} + \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 1\right)} \sin\left(x\right) - \cos\left(x\right) - 1\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 3 \, {\left(\cos\left(x\right)^{3} + \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 1\right)} \sin\left(x\right) - \cos\left(x\right) - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 2 \, {\left(4 \, \cos\left(x\right)^{2} + \cos\left(x\right) - 2\right)} \sin\left(x\right) - 6 \, \cos\left(x\right) - 4}{4 \, {\left(a \cos\left(x\right)^{3} + a \cos\left(x\right)^{2} - a \cos\left(x\right) + {\left(a \cos\left(x\right)^{2} - a\right)} \sin\left(x\right) - a\right)}}"," ",0,"1/4*(8*cos(x)^3 + 6*cos(x)^2 - 3*(cos(x)^3 + cos(x)^2 + (cos(x)^2 - 1)*sin(x) - cos(x) - 1)*log(1/2*cos(x) + 1/2) + 3*(cos(x)^3 + cos(x)^2 + (cos(x)^2 - 1)*sin(x) - cos(x) - 1)*log(-1/2*cos(x) + 1/2) - 2*(4*cos(x)^2 + cos(x) - 2)*sin(x) - 6*cos(x) - 4)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) + (a*cos(x)^2 - a)*sin(x) - a)","B",0
11,1,168,0,1.244769," ","integrate(csc(x)^4/(a+a*sin(x)),x, algorithm=""fricas"")","\frac{32 \, \cos\left(x\right)^{4} + 14 \, \cos\left(x\right)^{3} - 48 \, \cos\left(x\right)^{2} + 9 \, {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{3} + \cos\left(x\right)^{2} - \cos\left(x\right) - 1\right)} \sin\left(x\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 9 \, {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{3} + \cos\left(x\right)^{2} - \cos\left(x\right) - 1\right)} \sin\left(x\right) + 1\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 2 \, {\left(16 \, \cos\left(x\right)^{3} + 9 \, \cos\left(x\right)^{2} - 15 \, \cos\left(x\right) - 6\right)} \sin\left(x\right) - 18 \, \cos\left(x\right) + 12}{12 \, {\left(a \cos\left(x\right)^{4} - 2 \, a \cos\left(x\right)^{2} - {\left(a \cos\left(x\right)^{3} + a \cos\left(x\right)^{2} - a \cos\left(x\right) - a\right)} \sin\left(x\right) + a\right)}}"," ",0,"1/12*(32*cos(x)^4 + 14*cos(x)^3 - 48*cos(x)^2 + 9*(cos(x)^4 - 2*cos(x)^2 - (cos(x)^3 + cos(x)^2 - cos(x) - 1)*sin(x) + 1)*log(1/2*cos(x) + 1/2) - 9*(cos(x)^4 - 2*cos(x)^2 - (cos(x)^3 + cos(x)^2 - cos(x) - 1)*sin(x) + 1)*log(-1/2*cos(x) + 1/2) + 2*(16*cos(x)^3 + 9*cos(x)^2 - 15*cos(x) - 6)*sin(x) - 18*cos(x) + 12)/(a*cos(x)^4 - 2*a*cos(x)^2 - (a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a)*sin(x) + a)","B",0
12,1,105,0,1.236719," ","integrate(sin(x)^4/(a+a*sin(x))^2,x, algorithm=""fricas"")","-\frac{3 \, \cos\left(x\right)^{4} - {\left(21 \, x - 31\right)} \cos\left(x\right)^{2} - 6 \, \cos\left(x\right)^{3} + {\left(21 \, x + 38\right)} \cos\left(x\right) + {\left(3 \, \cos\left(x\right)^{3} + {\left(21 \, x + 40\right)} \cos\left(x\right) + 9 \, \cos\left(x\right)^{2} + 42 \, x + 2\right)} \sin\left(x\right) + 42 \, x - 2}{6 \, {\left(a^{2} \cos\left(x\right)^{2} - a^{2} \cos\left(x\right) - 2 \, a^{2} - {\left(a^{2} \cos\left(x\right) + 2 \, a^{2}\right)} \sin\left(x\right)\right)}}"," ",0,"-1/6*(3*cos(x)^4 - (21*x - 31)*cos(x)^2 - 6*cos(x)^3 + (21*x + 38)*cos(x) + (3*cos(x)^3 + (21*x + 40)*cos(x) + 9*cos(x)^2 + 42*x + 2)*sin(x) + 42*x - 2)/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))","A",0
13,1,95,0,1.320202," ","integrate(sin(x)^3/(a+a*sin(x))^2,x, algorithm=""fricas"")","-\frac{{\left(6 \, x - 11\right)} \cos\left(x\right)^{2} + 3 \, \cos\left(x\right)^{3} - {\left(6 \, x + 13\right)} \cos\left(x\right) - {\left(2 \, {\left(3 \, x + 7\right)} \cos\left(x\right) + 3 \, \cos\left(x\right)^{2} + 12 \, x + 1\right)} \sin\left(x\right) - 12 \, x + 1}{3 \, {\left(a^{2} \cos\left(x\right)^{2} - a^{2} \cos\left(x\right) - 2 \, a^{2} - {\left(a^{2} \cos\left(x\right) + 2 \, a^{2}\right)} \sin\left(x\right)\right)}}"," ",0,"-1/3*((6*x - 11)*cos(x)^2 + 3*cos(x)^3 - (6*x + 13)*cos(x) - (2*(3*x + 7)*cos(x) + 3*cos(x)^2 + 12*x + 1)*sin(x) - 12*x + 1)/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))","B",0
14,1,82,0,1.403275," ","integrate(sin(x)^2/(a+a*sin(x))^2,x, algorithm=""fricas"")","\frac{{\left(3 \, x - 5\right)} \cos\left(x\right)^{2} - {\left(3 \, x + 4\right)} \cos\left(x\right) - {\left({\left(3 \, x + 5\right)} \cos\left(x\right) + 6 \, x + 1\right)} \sin\left(x\right) - 6 \, x + 1}{3 \, {\left(a^{2} \cos\left(x\right)^{2} - a^{2} \cos\left(x\right) - 2 \, a^{2} - {\left(a^{2} \cos\left(x\right) + 2 \, a^{2}\right)} \sin\left(x\right)\right)}}"," ",0,"1/3*((3*x - 5)*cos(x)^2 - (3*x + 4)*cos(x) - ((3*x + 5)*cos(x) + 6*x + 1)*sin(x) - 6*x + 1)/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))","B",0
15,1,60,0,1.259494," ","integrate(sin(x)/(a+a*sin(x))^2,x, algorithm=""fricas"")","\frac{2 \, \cos\left(x\right)^{2} + {\left(2 \, \cos\left(x\right) + 1\right)} \sin\left(x\right) + \cos\left(x\right) - 1}{3 \, {\left(a^{2} \cos\left(x\right)^{2} - a^{2} \cos\left(x\right) - 2 \, a^{2} - {\left(a^{2} \cos\left(x\right) + 2 \, a^{2}\right)} \sin\left(x\right)\right)}}"," ",0,"1/3*(2*cos(x)^2 + (2*cos(x) + 1)*sin(x) + cos(x) - 1)/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))","B",0
16,1,58,0,1.257189," ","integrate(1/(a+a*sin(x))^2,x, algorithm=""fricas"")","\frac{\cos\left(x\right)^{2} + {\left(\cos\left(x\right) - 1\right)} \sin\left(x\right) + 2 \, \cos\left(x\right) + 1}{3 \, {\left(a^{2} \cos\left(x\right)^{2} - a^{2} \cos\left(x\right) - 2 \, a^{2} - {\left(a^{2} \cos\left(x\right) + 2 \, a^{2}\right)} \sin\left(x\right)\right)}}"," ",0,"1/3*(cos(x)^2 + (cos(x) - 1)*sin(x) + 2*cos(x) + 1)/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))","A",0
17,1,117,0,1.452185," ","integrate(csc(x)/(a+a*sin(x))^2,x, algorithm=""fricas"")","-\frac{8 \, \cos\left(x\right)^{2} + 3 \, {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 2\right)} \sin\left(x\right) - \cos\left(x\right) - 2\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 3 \, {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 2\right)} \sin\left(x\right) - \cos\left(x\right) - 2\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 2 \, {\left(4 \, \cos\left(x\right) - 1\right)} \sin\left(x\right) + 10 \, \cos\left(x\right) + 2}{6 \, {\left(a^{2} \cos\left(x\right)^{2} - a^{2} \cos\left(x\right) - 2 \, a^{2} - {\left(a^{2} \cos\left(x\right) + 2 \, a^{2}\right)} \sin\left(x\right)\right)}}"," ",0,"-1/6*(8*cos(x)^2 + 3*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*log(1/2*cos(x) + 1/2) - 3*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*log(-1/2*cos(x) + 1/2) + 2*(4*cos(x) - 1)*sin(x) + 10*cos(x) + 2)/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))","B",0
18,1,168,0,1.308316," ","integrate(csc(x)^2/(a+a*sin(x))^2,x, algorithm=""fricas"")","-\frac{10 \, \cos\left(x\right)^{3} - 4 \, \cos\left(x\right)^{2} - 3 \, {\left(\cos\left(x\right)^{3} + 2 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - \cos\left(x\right) - 2\right)} \sin\left(x\right) - \cos\left(x\right) - 2\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 3 \, {\left(\cos\left(x\right)^{3} + 2 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - \cos\left(x\right) - 2\right)} \sin\left(x\right) - \cos\left(x\right) - 2\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(10 \, \cos\left(x\right)^{2} + 14 \, \cos\left(x\right) + 1\right)} \sin\left(x\right) - 13 \, \cos\left(x\right) + 1}{3 \, {\left(a^{2} \cos\left(x\right)^{3} + 2 \, a^{2} \cos\left(x\right)^{2} - a^{2} \cos\left(x\right) - 2 \, a^{2} + {\left(a^{2} \cos\left(x\right)^{2} - a^{2} \cos\left(x\right) - 2 \, a^{2}\right)} \sin\left(x\right)\right)}}"," ",0,"-1/3*(10*cos(x)^3 - 4*cos(x)^2 - 3*(cos(x)^3 + 2*cos(x)^2 + (cos(x)^2 - cos(x) - 2)*sin(x) - cos(x) - 2)*log(1/2*cos(x) + 1/2) + 3*(cos(x)^3 + 2*cos(x)^2 + (cos(x)^2 - cos(x) - 2)*sin(x) - cos(x) - 2)*log(-1/2*cos(x) + 1/2) - (10*cos(x)^2 + 14*cos(x) + 1)*sin(x) - 13*cos(x) + 1)/(a^2*cos(x)^3 + 2*a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 + (a^2*cos(x)^2 - a^2*cos(x) - 2*a^2)*sin(x))","B",0
19,1,220,0,0.935365," ","integrate(csc(x)^3/(a+a*sin(x))^2,x, algorithm=""fricas"")","-\frac{64 \, \cos\left(x\right)^{4} + 86 \, \cos\left(x\right)^{3} - 54 \, \cos\left(x\right)^{2} + 21 \, {\left(\cos\left(x\right)^{4} - \cos\left(x\right)^{3} - 3 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{3} + 2 \, \cos\left(x\right)^{2} - \cos\left(x\right) - 2\right)} \sin\left(x\right) + \cos\left(x\right) + 2\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 21 \, {\left(\cos\left(x\right)^{4} - \cos\left(x\right)^{3} - 3 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{3} + 2 \, \cos\left(x\right)^{2} - \cos\left(x\right) - 2\right)} \sin\left(x\right) + \cos\left(x\right) + 2\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 2 \, {\left(32 \, \cos\left(x\right)^{3} - 11 \, \cos\left(x\right)^{2} - 38 \, \cos\left(x\right) + 2\right)} \sin\left(x\right) - 80 \, \cos\left(x\right) - 4}{12 \, {\left(a^{2} \cos\left(x\right)^{4} - a^{2} \cos\left(x\right)^{3} - 3 \, a^{2} \cos\left(x\right)^{2} + a^{2} \cos\left(x\right) + 2 \, a^{2} - {\left(a^{2} \cos\left(x\right)^{3} + 2 \, a^{2} \cos\left(x\right)^{2} - a^{2} \cos\left(x\right) - 2 \, a^{2}\right)} \sin\left(x\right)\right)}}"," ",0,"-1/12*(64*cos(x)^4 + 86*cos(x)^3 - 54*cos(x)^2 + 21*(cos(x)^4 - cos(x)^3 - 3*cos(x)^2 - (cos(x)^3 + 2*cos(x)^2 - cos(x) - 2)*sin(x) + cos(x) + 2)*log(1/2*cos(x) + 1/2) - 21*(cos(x)^4 - cos(x)^3 - 3*cos(x)^2 - (cos(x)^3 + 2*cos(x)^2 - cos(x) - 2)*sin(x) + cos(x) + 2)*log(-1/2*cos(x) + 1/2) + 2*(32*cos(x)^3 - 11*cos(x)^2 - 38*cos(x) + 2)*sin(x) - 80*cos(x) - 4)/(a^2*cos(x)^4 - a^2*cos(x)^3 - 3*a^2*cos(x)^2 + a^2*cos(x) + 2*a^2 - (a^2*cos(x)^3 + 2*a^2*cos(x)^2 - a^2*cos(x) - 2*a^2)*sin(x))","B",0
20,1,266,0,1.417581," ","integrate(csc(x)^4/(a+a*sin(x))^2,x, algorithm=""fricas"")","-\frac{48 \, \cos\left(x\right)^{5} - 18 \, \cos\left(x\right)^{4} - 108 \, \cos\left(x\right)^{3} + 22 \, \cos\left(x\right)^{2} - 15 \, {\left(\cos\left(x\right)^{5} + 2 \, \cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{3} - 4 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{4} - \cos\left(x\right)^{3} - 3 \, \cos\left(x\right)^{2} + \cos\left(x\right) + 2\right)} \sin\left(x\right) + \cos\left(x\right) + 2\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 15 \, {\left(\cos\left(x\right)^{5} + 2 \, \cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{3} - 4 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{4} - \cos\left(x\right)^{3} - 3 \, \cos\left(x\right)^{2} + \cos\left(x\right) + 2\right)} \sin\left(x\right) + \cos\left(x\right) + 2\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 2 \, {\left(24 \, \cos\left(x\right)^{4} + 33 \, \cos\left(x\right)^{3} - 21 \, \cos\left(x\right)^{2} - 32 \, \cos\left(x\right) - 1\right)} \sin\left(x\right) + 62 \, \cos\left(x\right) - 2}{6 \, {\left(a^{2} \cos\left(x\right)^{5} + 2 \, a^{2} \cos\left(x\right)^{4} - 2 \, a^{2} \cos\left(x\right)^{3} - 4 \, a^{2} \cos\left(x\right)^{2} + a^{2} \cos\left(x\right) + 2 \, a^{2} + {\left(a^{2} \cos\left(x\right)^{4} - a^{2} \cos\left(x\right)^{3} - 3 \, a^{2} \cos\left(x\right)^{2} + a^{2} \cos\left(x\right) + 2 \, a^{2}\right)} \sin\left(x\right)\right)}}"," ",0,"-1/6*(48*cos(x)^5 - 18*cos(x)^4 - 108*cos(x)^3 + 22*cos(x)^2 - 15*(cos(x)^5 + 2*cos(x)^4 - 2*cos(x)^3 - 4*cos(x)^2 + (cos(x)^4 - cos(x)^3 - 3*cos(x)^2 + cos(x) + 2)*sin(x) + cos(x) + 2)*log(1/2*cos(x) + 1/2) + 15*(cos(x)^5 + 2*cos(x)^4 - 2*cos(x)^3 - 4*cos(x)^2 + (cos(x)^4 - cos(x)^3 - 3*cos(x)^2 + cos(x) + 2)*sin(x) + cos(x) + 2)*log(-1/2*cos(x) + 1/2) - 2*(24*cos(x)^4 + 33*cos(x)^3 - 21*cos(x)^2 - 32*cos(x) - 1)*sin(x) + 62*cos(x) - 2)/(a^2*cos(x)^5 + 2*a^2*cos(x)^4 - 2*a^2*cos(x)^3 - 4*a^2*cos(x)^2 + a^2*cos(x) + 2*a^2 + (a^2*cos(x)^4 - a^2*cos(x)^3 - 3*a^2*cos(x)^2 + a^2*cos(x) + 2*a^2)*sin(x))","B",0
21,1,158,0,1.246021," ","integrate(sin(x)^6/(a+a*sin(x))^3,x, algorithm=""fricas"")","\frac{10 \, \cos\left(x\right)^{6} - 15 \, \cos\left(x\right)^{5} - {\left(345 \, x + 839\right)} \cos\left(x\right)^{3} - 140 \, \cos\left(x\right)^{4} - {\left(1035 \, x - 668\right)} \cos\left(x\right)^{2} + 6 \, {\left(115 \, x + 233\right)} \cos\left(x\right) + {\left(10 \, \cos\left(x\right)^{5} + 25 \, \cos\left(x\right)^{4} - {\left(345 \, x - 724\right)} \cos\left(x\right)^{2} - 115 \, \cos\left(x\right)^{3} + 6 \, {\left(115 \, x + 232\right)} \cos\left(x\right) + 1380 \, x - 6\right)} \sin\left(x\right) + 1380 \, x + 6}{30 \, {\left(a^{3} \cos\left(x\right)^{3} + 3 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3} + {\left(a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3}\right)} \sin\left(x\right)\right)}}"," ",0,"1/30*(10*cos(x)^6 - 15*cos(x)^5 - (345*x + 839)*cos(x)^3 - 140*cos(x)^4 - (1035*x - 668)*cos(x)^2 + 6*(115*x + 233)*cos(x) + (10*cos(x)^5 + 25*cos(x)^4 - (345*x - 724)*cos(x)^2 - 115*cos(x)^3 + 6*(115*x + 232)*cos(x) + 1380*x - 6)*sin(x) + 1380*x + 6)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))","A",0
22,1,145,0,1.310512," ","integrate(sin(x)^5/(a+a*sin(x))^3,x, algorithm=""fricas"")","\frac{15 \, \cos\left(x\right)^{5} + {\left(195 \, x + 449\right)} \cos\left(x\right)^{3} + 60 \, \cos\left(x\right)^{4} + {\left(585 \, x - 358\right)} \cos\left(x\right)^{2} - 6 \, {\left(65 \, x + 128\right)} \cos\left(x\right) - {\left(15 \, \cos\left(x\right)^{4} - {\left(195 \, x - 404\right)} \cos\left(x\right)^{2} - 45 \, \cos\left(x\right)^{3} + 6 \, {\left(65 \, x + 127\right)} \cos\left(x\right) + 780 \, x - 6\right)} \sin\left(x\right) - 780 \, x - 6}{30 \, {\left(a^{3} \cos\left(x\right)^{3} + 3 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3} + {\left(a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3}\right)} \sin\left(x\right)\right)}}"," ",0,"1/30*(15*cos(x)^5 + (195*x + 449)*cos(x)^3 + 60*cos(x)^4 + (585*x - 358)*cos(x)^2 - 6*(65*x + 128)*cos(x) - (15*cos(x)^4 - (195*x - 404)*cos(x)^2 - 45*cos(x)^3 + 6*(65*x + 127)*cos(x) + 780*x - 6)*sin(x) - 780*x - 6)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))","A",0
23,1,132,0,1.099442," ","integrate(sin(x)^4/(a+a*sin(x))^3,x, algorithm=""fricas"")","-\frac{3 \, {\left(5 \, x + 13\right)} \cos\left(x\right)^{3} + 5 \, \cos\left(x\right)^{4} + {\left(45 \, x - 28\right)} \cos\left(x\right)^{2} - 3 \, {\left(10 \, x + 21\right)} \cos\left(x\right) + {\left({\left(15 \, x - 34\right)} \cos\left(x\right)^{2} + 5 \, \cos\left(x\right)^{3} - 2 \, {\left(15 \, x + 31\right)} \cos\left(x\right) - 60 \, x + 1\right)} \sin\left(x\right) - 60 \, x - 1}{5 \, {\left(a^{3} \cos\left(x\right)^{3} + 3 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3} + {\left(a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3}\right)} \sin\left(x\right)\right)}}"," ",0,"-1/5*(3*(5*x + 13)*cos(x)^3 + 5*cos(x)^4 + (45*x - 28)*cos(x)^2 - 3*(10*x + 21)*cos(x) + ((15*x - 34)*cos(x)^2 + 5*cos(x)^3 - 2*(15*x + 31)*cos(x) - 60*x + 1)*sin(x) - 60*x - 1)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))","B",0
24,1,119,0,0.879084," ","integrate(sin(x)^3/(a+a*sin(x))^3,x, algorithm=""fricas"")","\frac{{\left(15 \, x + 32\right)} \cos\left(x\right)^{3} + {\left(45 \, x - 19\right)} \cos\left(x\right)^{2} - 6 \, {\left(5 \, x + 9\right)} \cos\left(x\right) + {\left({\left(15 \, x - 32\right)} \cos\left(x\right)^{2} - 3 \, {\left(10 \, x + 17\right)} \cos\left(x\right) - 60 \, x + 3\right)} \sin\left(x\right) - 60 \, x - 3}{15 \, {\left(a^{3} \cos\left(x\right)^{3} + 3 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3} + {\left(a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3}\right)} \sin\left(x\right)\right)}}"," ",0,"1/15*((15*x + 32)*cos(x)^3 + (45*x - 19)*cos(x)^2 - 6*(5*x + 9)*cos(x) + ((15*x - 32)*cos(x)^2 - 3*(10*x + 17)*cos(x) - 60*x + 3)*sin(x) - 60*x - 3)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))","B",0
25,1,90,0,1.302948," ","integrate(sin(x)^2/(a+a*sin(x))^3,x, algorithm=""fricas"")","-\frac{7 \, \cos\left(x\right)^{3} + \cos\left(x\right)^{2} - {\left(7 \, \cos\left(x\right)^{2} + 6 \, \cos\left(x\right) - 3\right)} \sin\left(x\right) - 9 \, \cos\left(x\right) - 3}{15 \, {\left(a^{3} \cos\left(x\right)^{3} + 3 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3} + {\left(a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3}\right)} \sin\left(x\right)\right)}}"," ",0,"-1/15*(7*cos(x)^3 + cos(x)^2 - (7*cos(x)^2 + 6*cos(x) - 3)*sin(x) - 9*cos(x) - 3)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))","B",0
26,1,88,0,0.943844," ","integrate(sin(x)/(a+a*sin(x))^3,x, algorithm=""fricas"")","-\frac{\cos\left(x\right)^{3} - 2 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{2} + 3 \, \cos\left(x\right) + 1\right)} \sin\left(x\right) - 2 \, \cos\left(x\right) + 1}{5 \, {\left(a^{3} \cos\left(x\right)^{3} + 3 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3} + {\left(a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3}\right)} \sin\left(x\right)\right)}}"," ",0,"-1/5*(cos(x)^3 - 2*cos(x)^2 - (cos(x)^2 + 3*cos(x) + 1)*sin(x) - 2*cos(x) + 1)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))","A",0
27,1,92,0,1.189863," ","integrate(1/(a+a*sin(x))^3,x, algorithm=""fricas"")","-\frac{2 \, \cos\left(x\right)^{3} - 4 \, \cos\left(x\right)^{2} - {\left(2 \, \cos\left(x\right)^{2} + 6 \, \cos\left(x\right) - 3\right)} \sin\left(x\right) - 9 \, \cos\left(x\right) - 3}{15 \, {\left(a^{3} \cos\left(x\right)^{3} + 3 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3} + {\left(a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3}\right)} \sin\left(x\right)\right)}}"," ",0,"-1/15*(2*cos(x)^3 - 4*cos(x)^2 - (2*cos(x)^2 + 6*cos(x) - 3)*sin(x) - 9*cos(x) - 3)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))","B",0
28,1,168,0,1.480648," ","integrate(csc(x)/(a+a*sin(x))^3,x, algorithm=""fricas"")","\frac{44 \, \cos\left(x\right)^{3} - 58 \, \cos\left(x\right)^{2} - 15 \, {\left(\cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 2 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 2 \, \cos\left(x\right) - 4\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 15 \, {\left(\cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 2 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 2 \, \cos\left(x\right) - 4\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 2 \, {\left(22 \, \cos\left(x\right)^{2} + 51 \, \cos\left(x\right) - 3\right)} \sin\left(x\right) - 108 \, \cos\left(x\right) - 6}{30 \, {\left(a^{3} \cos\left(x\right)^{3} + 3 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3} + {\left(a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3}\right)} \sin\left(x\right)\right)}}"," ",0,"1/30*(44*cos(x)^3 - 58*cos(x)^2 - 15*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*log(1/2*cos(x) + 1/2) + 15*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*log(-1/2*cos(x) + 1/2) - 2*(22*cos(x)^2 + 51*cos(x) - 3)*sin(x) - 108*cos(x) - 6)/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))","B",0
29,1,225,0,1.092957," ","integrate(csc(x)^2/(a+a*sin(x))^3,x, algorithm=""fricas"")","\frac{48 \, \cos\left(x\right)^{4} + 114 \, \cos\left(x\right)^{3} - 60 \, \cos\left(x\right)^{2} + 15 \, {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{3} - 5 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} - 2 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) + 2 \, \cos\left(x\right) + 4\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 15 \, {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{3} - 5 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} - 2 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) + 2 \, \cos\left(x\right) + 4\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 2 \, {\left(24 \, \cos\left(x\right)^{3} - 33 \, \cos\left(x\right)^{2} - 63 \, \cos\left(x\right) - 1\right)} \sin\left(x\right) - 124 \, \cos\left(x\right) + 2}{10 \, {\left(a^{3} \cos\left(x\right)^{4} - 2 \, a^{3} \cos\left(x\right)^{3} - 5 \, a^{3} \cos\left(x\right)^{2} + 2 \, a^{3} \cos\left(x\right) + 4 \, a^{3} - {\left(a^{3} \cos\left(x\right)^{3} + 3 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3}\right)} \sin\left(x\right)\right)}}"," ",0,"1/10*(48*cos(x)^4 + 114*cos(x)^3 - 60*cos(x)^2 + 15*(cos(x)^4 - 2*cos(x)^3 - 5*cos(x)^2 - (cos(x)^3 + 3*cos(x)^2 - 2*cos(x) - 4)*sin(x) + 2*cos(x) + 4)*log(1/2*cos(x) + 1/2) - 15*(cos(x)^4 - 2*cos(x)^3 - 5*cos(x)^2 - (cos(x)^3 + 3*cos(x)^2 - 2*cos(x) - 4)*sin(x) + 2*cos(x) + 4)*log(-1/2*cos(x) + 1/2) + 2*(24*cos(x)^3 - 33*cos(x)^2 - 63*cos(x) - 1)*sin(x) - 124*cos(x) + 2)/(a^3*cos(x)^4 - 2*a^3*cos(x)^3 - 5*a^3*cos(x)^2 + 2*a^3*cos(x) + 4*a^3 - (a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))","B",0
30,1,276,0,1.461088," ","integrate(csc(x)^3/(a+a*sin(x))^3,x, algorithm=""fricas"")","\frac{608 \, \cos\left(x\right)^{5} - 826 \, \cos\left(x\right)^{4} - 2174 \, \cos\left(x\right)^{3} + 784 \, \cos\left(x\right)^{2} - 195 \, {\left(\cos\left(x\right)^{5} + 3 \, \cos\left(x\right)^{4} - 3 \, \cos\left(x\right)^{3} - 7 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{3} - 5 \, \cos\left(x\right)^{2} + 2 \, \cos\left(x\right) + 4\right)} \sin\left(x\right) + 2 \, \cos\left(x\right) + 4\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 195 \, {\left(\cos\left(x\right)^{5} + 3 \, \cos\left(x\right)^{4} - 3 \, \cos\left(x\right)^{3} - 7 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{3} - 5 \, \cos\left(x\right)^{2} + 2 \, \cos\left(x\right) + 4\right)} \sin\left(x\right) + 2 \, \cos\left(x\right) + 4\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 2 \, {\left(304 \, \cos\left(x\right)^{4} + 717 \, \cos\left(x\right)^{3} - 370 \, \cos\left(x\right)^{2} - 762 \, \cos\left(x\right) + 6\right)} \sin\left(x\right) + 1536 \, \cos\left(x\right) + 12}{60 \, {\left(a^{3} \cos\left(x\right)^{5} + 3 \, a^{3} \cos\left(x\right)^{4} - 3 \, a^{3} \cos\left(x\right)^{3} - 7 \, a^{3} \cos\left(x\right)^{2} + 2 \, a^{3} \cos\left(x\right) + 4 \, a^{3} + {\left(a^{3} \cos\left(x\right)^{4} - 2 \, a^{3} \cos\left(x\right)^{3} - 5 \, a^{3} \cos\left(x\right)^{2} + 2 \, a^{3} \cos\left(x\right) + 4 \, a^{3}\right)} \sin\left(x\right)\right)}}"," ",0,"1/60*(608*cos(x)^5 - 826*cos(x)^4 - 2174*cos(x)^3 + 784*cos(x)^2 - 195*(cos(x)^5 + 3*cos(x)^4 - 3*cos(x)^3 - 7*cos(x)^2 + (cos(x)^4 - 2*cos(x)^3 - 5*cos(x)^2 + 2*cos(x) + 4)*sin(x) + 2*cos(x) + 4)*log(1/2*cos(x) + 1/2) + 195*(cos(x)^5 + 3*cos(x)^4 - 3*cos(x)^3 - 7*cos(x)^2 + (cos(x)^4 - 2*cos(x)^3 - 5*cos(x)^2 + 2*cos(x) + 4)*sin(x) + 2*cos(x) + 4)*log(-1/2*cos(x) + 1/2) - 2*(304*cos(x)^4 + 717*cos(x)^3 - 370*cos(x)^2 - 762*cos(x) + 6)*sin(x) + 1536*cos(x) + 12)/(a^3*cos(x)^5 + 3*a^3*cos(x)^4 - 3*a^3*cos(x)^3 - 7*a^3*cos(x)^2 + 2*a^3*cos(x) + 4*a^3 + (a^3*cos(x)^4 - 2*a^3*cos(x)^3 - 5*a^3*cos(x)^2 + 2*a^3*cos(x) + 4*a^3)*sin(x))","B",0
31,1,333,0,0.804945," ","integrate(csc(x)^4/(a+a*sin(x))^3,x, algorithm=""fricas"")","\frac{1088 \, \cos\left(x\right)^{6} + 2574 \, \cos\left(x\right)^{5} - 2428 \, \cos\left(x\right)^{4} - 5338 \, \cos\left(x\right)^{3} + 1372 \, \cos\left(x\right)^{2} + 345 \, {\left(\cos\left(x\right)^{6} - 2 \, \cos\left(x\right)^{5} - 6 \, \cos\left(x\right)^{4} + 4 \, \cos\left(x\right)^{3} + 9 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{5} + 3 \, \cos\left(x\right)^{4} - 3 \, \cos\left(x\right)^{3} - 7 \, \cos\left(x\right)^{2} + 2 \, \cos\left(x\right) + 4\right)} \sin\left(x\right) - 2 \, \cos\left(x\right) - 4\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 345 \, {\left(\cos\left(x\right)^{6} - 2 \, \cos\left(x\right)^{5} - 6 \, \cos\left(x\right)^{4} + 4 \, \cos\left(x\right)^{3} + 9 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{5} + 3 \, \cos\left(x\right)^{4} - 3 \, \cos\left(x\right)^{3} - 7 \, \cos\left(x\right)^{2} + 2 \, \cos\left(x\right) + 4\right)} \sin\left(x\right) - 2 \, \cos\left(x\right) - 4\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 2 \, {\left(544 \, \cos\left(x\right)^{5} - 743 \, \cos\left(x\right)^{4} - 1957 \, \cos\left(x\right)^{3} + 712 \, \cos\left(x\right)^{2} + 1398 \, \cos\left(x\right) + 6\right)} \sin\left(x\right) + 2784 \, \cos\left(x\right) - 12}{60 \, {\left(a^{3} \cos\left(x\right)^{6} - 2 \, a^{3} \cos\left(x\right)^{5} - 6 \, a^{3} \cos\left(x\right)^{4} + 4 \, a^{3} \cos\left(x\right)^{3} + 9 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3} - {\left(a^{3} \cos\left(x\right)^{5} + 3 \, a^{3} \cos\left(x\right)^{4} - 3 \, a^{3} \cos\left(x\right)^{3} - 7 \, a^{3} \cos\left(x\right)^{2} + 2 \, a^{3} \cos\left(x\right) + 4 \, a^{3}\right)} \sin\left(x\right)\right)}}"," ",0,"1/60*(1088*cos(x)^6 + 2574*cos(x)^5 - 2428*cos(x)^4 - 5338*cos(x)^3 + 1372*cos(x)^2 + 345*(cos(x)^6 - 2*cos(x)^5 - 6*cos(x)^4 + 4*cos(x)^3 + 9*cos(x)^2 - (cos(x)^5 + 3*cos(x)^4 - 3*cos(x)^3 - 7*cos(x)^2 + 2*cos(x) + 4)*sin(x) - 2*cos(x) - 4)*log(1/2*cos(x) + 1/2) - 345*(cos(x)^6 - 2*cos(x)^5 - 6*cos(x)^4 + 4*cos(x)^3 + 9*cos(x)^2 - (cos(x)^5 + 3*cos(x)^4 - 3*cos(x)^3 - 7*cos(x)^2 + 2*cos(x) + 4)*sin(x) - 2*cos(x) - 4)*log(-1/2*cos(x) + 1/2) + 2*(544*cos(x)^5 - 743*cos(x)^4 - 1957*cos(x)^3 + 712*cos(x)^2 + 1398*cos(x) + 6)*sin(x) + 2784*cos(x) - 12)/(a^3*cos(x)^6 - 2*a^3*cos(x)^5 - 6*a^3*cos(x)^4 + 4*a^3*cos(x)^3 + 9*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 - (a^3*cos(x)^5 + 3*a^3*cos(x)^4 - 3*a^3*cos(x)^3 - 7*a^3*cos(x)^2 + 2*a^3*cos(x) + 4*a^3)*sin(x))","B",0
32,1,132,0,0.962791," ","integrate(sin(d*x+c)^4*(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(35 \, \cos\left(d x + c\right)^{5} - 5 \, \cos\left(d x + c\right)^{4} - 118 \, \cos\left(d x + c\right)^{3} + 26 \, \cos\left(d x + c\right)^{2} - {\left(35 \, \cos\left(d x + c\right)^{4} + 40 \, \cos\left(d x + c\right)^{3} - 78 \, \cos\left(d x + c\right)^{2} - 104 \, \cos\left(d x + c\right) + 107\right)} \sin\left(d x + c\right) + 211 \, \cos\left(d x + c\right) + 107\right)} \sqrt{a \sin\left(d x + c\right) + a}}{315 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"-2/315*(35*cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 118*cos(d*x + c)^3 + 26*cos(d*x + c)^2 - (35*cos(d*x + c)^4 + 40*cos(d*x + c)^3 - 78*cos(d*x + c)^2 - 104*cos(d*x + c) + 107)*sin(d*x + c) + 211*cos(d*x + c) + 107)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
33,1,111,0,1.257224," ","integrate(sin(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{2 \, {\left(5 \, \cos\left(d x + c\right)^{4} + 6 \, \cos\left(d x + c\right)^{3} - 12 \, \cos\left(d x + c\right)^{2} + {\left(5 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} - 13 \, \cos\left(d x + c\right) + 9\right)} \sin\left(d x + c\right) - 22 \, \cos\left(d x + c\right) - 9\right)} \sqrt{a \sin\left(d x + c\right) + a}}{35 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"2/35*(5*cos(d*x + c)^4 + 6*cos(d*x + c)^3 - 12*cos(d*x + c)^2 + (5*cos(d*x + c)^3 - cos(d*x + c)^2 - 13*cos(d*x + c) + 9)*sin(d*x + c) - 22*cos(d*x + c) - 9)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
34,1,92,0,1.137305," ","integrate(sin(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} - {\left(3 \, \cos\left(d x + c\right)^{2} + 4 \, \cos\left(d x + c\right) - 7\right)} \sin\left(d x + c\right) - 11 \, \cos\left(d x + c\right) - 7\right)} \sqrt{a \sin\left(d x + c\right) + a}}{15 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"2/15*(3*cos(d*x + c)^3 - cos(d*x + c)^2 - (3*cos(d*x + c)^2 + 4*cos(d*x + c) - 7)*sin(d*x + c) - 11*cos(d*x + c) - 7)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
35,1,67,0,1.304845," ","integrate(sin(d*x+c)*(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) - 1\right)} \sin\left(d x + c\right) + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a \sin\left(d x + c\right) + a}}{3 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"-2/3*(cos(d*x + c)^2 + (cos(d*x + c) - 1)*sin(d*x + c) + 2*cos(d*x + c) + 1)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
36,1,50,0,1.212451," ","integrate((a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d}"," ",0,"-2*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","B",0
37,1,219,0,1.349930," ","integrate(csc(d*x+c)*(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right)}{2 \, d}, -\frac{\sqrt{-a} \arctan\left(\frac{\sqrt{a \sin\left(d x + c\right) + a} \sqrt{-a} {\left(\sin\left(d x + c\right) - 2\right)}}{2 \, a \cos\left(d x + c\right)}\right)}{d}\right]"," ",0,"[1/2*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1))/d, -sqrt(-a)*arctan(1/2*sqrt(a*sin(d*x + c) + a)*sqrt(-a)*(sin(d*x + c) - 2)/(a*cos(d*x + c)))/d]","A",0
38,1,258,0,1.338485," ","integrate(csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{{\left(\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 1\right)} \sin\left(d x + c\right) - 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{4 \, {\left(d \cos\left(d x + c\right)^{2} - {\left(d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right) - d\right)}}"," ",0,"1/4*((cos(d*x + c)^2 - (cos(d*x + c) + 1)*sin(d*x + c) - 1)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1))/(d*cos(d*x + c)^2 - (d*cos(d*x + c) + d)*sin(d*x + c) - d)","B",0
39,1,319,0,1.372850," ","integrate(csc(d*x+c)^3*(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{3 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, {\left(3 \, \cos\left(d x + c\right)^{2} + {\left(3 \, \cos\left(d x + c\right) + 1\right)} \sin\left(d x + c\right) + 2 \, \cos\left(d x + c\right) - 1\right)} \sqrt{a \sin\left(d x + c\right) + a}}{16 \, {\left(d \cos\left(d x + c\right)^{3} + d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) + {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right) - d\right)}}"," ",0,"1/16*(3*(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(3*cos(d*x + c)^2 + (3*cos(d*x + c) + 1)*sin(d*x + c) + 2*cos(d*x + c) - 1)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c) + (d*cos(d*x + c)^2 - d)*sin(d*x + c) - d)","B",0
40,1,361,0,1.313164," ","integrate(csc(d*x+c)^4*(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{15 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 1\right)} \sin\left(d x + c\right) + 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, {\left(15 \, \cos\left(d x + c\right)^{3} + 5 \, \cos\left(d x + c\right)^{2} - {\left(15 \, \cos\left(d x + c\right)^{2} + 10 \, \cos\left(d x + c\right) - 13\right)} \sin\left(d x + c\right) - 23 \, \cos\left(d x + c\right) - 13\right)} \sqrt{a \sin\left(d x + c\right) + a}}{96 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} - {\left(d \cos\left(d x + c\right)^{3} + d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right) + d\right)}}"," ",0,"1/96*(15*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 - (cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sin(d*x + c) + 1)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(15*cos(d*x + c)^3 + 5*cos(d*x + c)^2 - (15*cos(d*x + c)^2 + 10*cos(d*x + c) - 13)*sin(d*x + c) - 23*cos(d*x + c) - 13)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 - (d*cos(d*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c) - d)*sin(d*x + c) + d)","B",0
41,1,223,0,1.339623," ","integrate(csc(d*x+c)*(a-a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{-a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right)}{2 \, d}, \frac{\sqrt{-a} \arctan\left(\frac{\sqrt{-a \sin\left(d x + c\right) + a} \sqrt{-a} {\left(\sin\left(d x + c\right) + 2\right)}}{2 \, a \cos\left(d x + c\right)}\right)}{d}\right]"," ",0,"[1/2*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 - (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(-a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) - (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 - (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1))/d, sqrt(-a)*arctan(1/2*sqrt(-a*sin(d*x + c) + a)*sqrt(-a)*(sin(d*x + c) + 2)/(a*cos(d*x + c)))/d]","A",0
42,1,223,0,1.401307," ","integrate(csc(d*x+c)*(-a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) - a} \sqrt{-a} - 9 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right)}{2 \, d}, -\frac{\sqrt{a} \arctan\left(\frac{\sqrt{a \sin\left(d x + c\right) - a} {\left(\sin\left(d x + c\right) + 2\right)}}{2 \, \sqrt{a} \cos\left(d x + c\right)}\right)}{d}\right]"," ",0,"[1/2*sqrt(-a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 + 4*(cos(d*x + c)^2 - (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) - a)*sqrt(-a) - 9*a*cos(d*x + c) - (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 - (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1))/d, -sqrt(a)*arctan(1/2*sqrt(a*sin(d*x + c) - a)*(sin(d*x + c) + 2)/(sqrt(a)*cos(d*x + c)))/d]","A",0
43,1,221,0,1.392802," ","integrate(csc(d*x+c)*(-a-a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{-a \sin\left(d x + c\right) - a} \sqrt{-a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right)}{2 \, d}, \frac{\sqrt{a} \arctan\left(\frac{\sqrt{-a \sin\left(d x + c\right) - a} {\left(\sin\left(d x + c\right) - 2\right)}}{2 \, \sqrt{a} \cos\left(d x + c\right)}\right)}{d}\right]"," ",0,"[1/2*sqrt(-a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 + 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(-a*sin(d*x + c) - a)*sqrt(-a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1))/d, sqrt(a)*arctan(1/2*sqrt(-a*sin(d*x + c) - a)*(sin(d*x + c) - 2)/(sqrt(a)*cos(d*x + c)))/d]","A",0
44,1,145,0,1.281955," ","integrate(sin(d*x+c)^3*(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(35 \, a \cos\left(d x + c\right)^{5} - 50 \, a \cos\left(d x + c\right)^{4} - 172 \, a \cos\left(d x + c\right)^{3} + 134 \, a \cos\left(d x + c\right)^{2} + 409 \, a \cos\left(d x + c\right) - {\left(35 \, a \cos\left(d x + c\right)^{4} + 85 \, a \cos\left(d x + c\right)^{3} - 87 \, a \cos\left(d x + c\right)^{2} - 221 \, a \cos\left(d x + c\right) + 188 \, a\right)} \sin\left(d x + c\right) + 188 \, a\right)} \sqrt{a \sin\left(d x + c\right) + a}}{315 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"-2/315*(35*a*cos(d*x + c)^5 - 50*a*cos(d*x + c)^4 - 172*a*cos(d*x + c)^3 + 134*a*cos(d*x + c)^2 + 409*a*cos(d*x + c) - (35*a*cos(d*x + c)^4 + 85*a*cos(d*x + c)^3 - 87*a*cos(d*x + c)^2 - 221*a*cos(d*x + c) + 188*a)*sin(d*x + c) + 188*a)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
45,1,122,0,1.141689," ","integrate(sin(d*x+c)^2*(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{2 \, {\left(15 \, a \cos\left(d x + c\right)^{4} + 39 \, a \cos\left(d x + c\right)^{3} - 43 \, a \cos\left(d x + c\right)^{2} - 143 \, a \cos\left(d x + c\right) + {\left(15 \, a \cos\left(d x + c\right)^{3} - 24 \, a \cos\left(d x + c\right)^{2} - 67 \, a \cos\left(d x + c\right) + 76 \, a\right)} \sin\left(d x + c\right) - 76 \, a\right)} \sqrt{a \sin\left(d x + c\right) + a}}{105 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"2/105*(15*a*cos(d*x + c)^4 + 39*a*cos(d*x + c)^3 - 43*a*cos(d*x + c)^2 - 143*a*cos(d*x + c) + (15*a*cos(d*x + c)^3 - 24*a*cos(d*x + c)^2 - 67*a*cos(d*x + c) + 76*a)*sin(d*x + c) - 76*a)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
46,1,99,0,1.447875," ","integrate(sin(d*x+c)*(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{2 \, {\left(a \cos\left(d x + c\right)^{3} - 2 \, a \cos\left(d x + c\right)^{2} - 7 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right)^{2} + 3 \, a \cos\left(d x + c\right) - 4 \, a\right)} \sin\left(d x + c\right) - 4 \, a\right)} \sqrt{a \sin\left(d x + c\right) + a}}{5 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"2/5*(a*cos(d*x + c)^3 - 2*a*cos(d*x + c)^2 - 7*a*cos(d*x + c) - (a*cos(d*x + c)^2 + 3*a*cos(d*x + c) - 4*a)*sin(d*x + c) - 4*a)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
47,1,76,0,1.184485," ","integrate((a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(a \cos\left(d x + c\right)^{2} + 5 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right) - 4 \, a\right)} \sin\left(d x + c\right) + 4 \, a\right)} \sqrt{a \sin\left(d x + c\right) + a}}{3 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"-2/3*(a*cos(d*x + c)^2 + 5*a*cos(d*x + c) + (a*cos(d*x + c) - 4*a)*sin(d*x + c) + 4*a)*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
48,1,239,0,1.348415," ","integrate(csc(d*x+c)*(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{{\left(a \cos\left(d x + c\right) + a \sin\left(d x + c\right) + a\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) - 4 \, {\left(a \cos\left(d x + c\right) - a \sin\left(d x + c\right) + a\right)} \sqrt{a \sin\left(d x + c\right) + a}}{2 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"1/2*((a*cos(d*x + c) + a*sin(d*x + c) + a)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) - 4*(a*cos(d*x + c) - a*sin(d*x + c) + a)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c) + d*sin(d*x + c) + d)","B",0
49,1,268,0,1.259161," ","integrate(csc(d*x+c)^2*(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{3 \, {\left(a \cos\left(d x + c\right)^{2} - {\left(a \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) - a\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, {\left(a \cos\left(d x + c\right) - a \sin\left(d x + c\right) + a\right)} \sqrt{a \sin\left(d x + c\right) + a}}{4 \, {\left(d \cos\left(d x + c\right)^{2} - {\left(d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right) - d\right)}}"," ",0,"1/4*(3*(a*cos(d*x + c)^2 - (a*cos(d*x + c) + a)*sin(d*x + c) - a)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(a*cos(d*x + c) - a*sin(d*x + c) + a)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^2 - (d*cos(d*x + c) + d)*sin(d*x + c) - d)","B",0
50,1,337,0,1.064916," ","integrate(csc(d*x+c)^3*(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{7 \, {\left(a \cos\left(d x + c\right)^{3} + a \cos\left(d x + c\right)^{2} - a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} - a\right)} \sin\left(d x + c\right) - a\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, {\left(7 \, a \cos\left(d x + c\right)^{2} + 2 \, a \cos\left(d x + c\right) + {\left(7 \, a \cos\left(d x + c\right) + 5 \, a\right)} \sin\left(d x + c\right) - 5 \, a\right)} \sqrt{a \sin\left(d x + c\right) + a}}{16 \, {\left(d \cos\left(d x + c\right)^{3} + d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) + {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right) - d\right)}}"," ",0,"1/16*(7*(a*cos(d*x + c)^3 + a*cos(d*x + c)^2 - a*cos(d*x + c) + (a*cos(d*x + c)^2 - a)*sin(d*x + c) - a)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(7*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) + (7*a*cos(d*x + c) + 5*a)*sin(d*x + c) - 5*a)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c) + (d*cos(d*x + c)^2 - d)*sin(d*x + c) - d)","B",0
51,1,380,0,0.907046," ","integrate(csc(d*x+c)^4*(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{33 \, {\left(a \cos\left(d x + c\right)^{4} - 2 \, a \cos\left(d x + c\right)^{2} - {\left(a \cos\left(d x + c\right)^{3} + a \cos\left(d x + c\right)^{2} - a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) + a\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, {\left(33 \, a \cos\left(d x + c\right)^{3} + 11 \, a \cos\left(d x + c\right)^{2} - 41 \, a \cos\left(d x + c\right) - {\left(33 \, a \cos\left(d x + c\right)^{2} + 22 \, a \cos\left(d x + c\right) - 19 \, a\right)} \sin\left(d x + c\right) - 19 \, a\right)} \sqrt{a \sin\left(d x + c\right) + a}}{96 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} - {\left(d \cos\left(d x + c\right)^{3} + d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right) + d\right)}}"," ",0,"1/96*(33*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 - (a*cos(d*x + c)^3 + a*cos(d*x + c)^2 - a*cos(d*x + c) - a)*sin(d*x + c) + a)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(33*a*cos(d*x + c)^3 + 11*a*cos(d*x + c)^2 - 41*a*cos(d*x + c) - (33*a*cos(d*x + c)^2 + 22*a*cos(d*x + c) - 19*a)*sin(d*x + c) - 19*a)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 - (d*cos(d*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c) - d)*sin(d*x + c) + d)","B",0
52,1,192,0,1.151048," ","integrate(sin(d*x+c)^3*(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(63 \, a^{2} \cos\left(d x + c\right)^{6} + 224 \, a^{2} \cos\left(d x + c\right)^{5} - 320 \, a^{2} \cos\left(d x + c\right)^{4} - 874 \, a^{2} \cos\left(d x + c\right)^{3} + 593 \, a^{2} \cos\left(d x + c\right)^{2} + 1786 \, a^{2} \cos\left(d x + c\right) + 800 \, a^{2} + {\left(63 \, a^{2} \cos\left(d x + c\right)^{5} - 161 \, a^{2} \cos\left(d x + c\right)^{4} - 481 \, a^{2} \cos\left(d x + c\right)^{3} + 393 \, a^{2} \cos\left(d x + c\right)^{2} + 986 \, a^{2} \cos\left(d x + c\right) - 800 \, a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a \sin\left(d x + c\right) + a}}{693 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"-2/693*(63*a^2*cos(d*x + c)^6 + 224*a^2*cos(d*x + c)^5 - 320*a^2*cos(d*x + c)^4 - 874*a^2*cos(d*x + c)^3 + 593*a^2*cos(d*x + c)^2 + 1786*a^2*cos(d*x + c) + 800*a^2 + (63*a^2*cos(d*x + c)^5 - 161*a^2*cos(d*x + c)^4 - 481*a^2*cos(d*x + c)^3 + 393*a^2*cos(d*x + c)^2 + 986*a^2*cos(d*x + c) - 800*a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
53,1,167,0,1.217450," ","integrate(sin(d*x+c)^2*(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(35 \, a^{2} \cos\left(d x + c\right)^{5} - 95 \, a^{2} \cos\left(d x + c\right)^{4} - 289 \, a^{2} \cos\left(d x + c\right)^{3} + 263 \, a^{2} \cos\left(d x + c\right)^{2} + 838 \, a^{2} \cos\left(d x + c\right) + 416 \, a^{2} - {\left(35 \, a^{2} \cos\left(d x + c\right)^{4} + 130 \, a^{2} \cos\left(d x + c\right)^{3} - 159 \, a^{2} \cos\left(d x + c\right)^{2} - 422 \, a^{2} \cos\left(d x + c\right) + 416 \, a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a \sin\left(d x + c\right) + a}}{315 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"-2/315*(35*a^2*cos(d*x + c)^5 - 95*a^2*cos(d*x + c)^4 - 289*a^2*cos(d*x + c)^3 + 263*a^2*cos(d*x + c)^2 + 838*a^2*cos(d*x + c) + 416*a^2 - (35*a^2*cos(d*x + c)^4 + 130*a^2*cos(d*x + c)^3 - 159*a^2*cos(d*x + c)^2 - 422*a^2*cos(d*x + c) + 416*a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
54,1,140,0,1.223720," ","integrate(sin(d*x+c)*(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, a^{2} \cos\left(d x + c\right)^{4} + 12 \, a^{2} \cos\left(d x + c\right)^{3} - 17 \, a^{2} \cos\left(d x + c\right)^{2} - 58 \, a^{2} \cos\left(d x + c\right) - 32 \, a^{2} + {\left(3 \, a^{2} \cos\left(d x + c\right)^{3} - 9 \, a^{2} \cos\left(d x + c\right)^{2} - 26 \, a^{2} \cos\left(d x + c\right) + 32 \, a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a \sin\left(d x + c\right) + a}}{21 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"2/21*(3*a^2*cos(d*x + c)^4 + 12*a^2*cos(d*x + c)^3 - 17*a^2*cos(d*x + c)^2 - 58*a^2*cos(d*x + c) - 32*a^2 + (3*a^2*cos(d*x + c)^3 - 9*a^2*cos(d*x + c)^2 - 26*a^2*cos(d*x + c) + 32*a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
55,1,115,0,1.314395," ","integrate((a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, a^{2} \cos\left(d x + c\right)^{3} - 11 \, a^{2} \cos\left(d x + c\right)^{2} - 46 \, a^{2} \cos\left(d x + c\right) - 32 \, a^{2} - {\left(3 \, a^{2} \cos\left(d x + c\right)^{2} + 14 \, a^{2} \cos\left(d x + c\right) - 32 \, a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a \sin\left(d x + c\right) + a}}{15 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"2/15*(3*a^2*cos(d*x + c)^3 - 11*a^2*cos(d*x + c)^2 - 46*a^2*cos(d*x + c) - 32*a^2 - (3*a^2*cos(d*x + c)^2 + 14*a^2*cos(d*x + c) - 32*a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a)/(d*cos(d*x + c) + d*sin(d*x + c) + d)","A",0
56,1,279,0,1.223079," ","integrate(csc(d*x+c)*(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{3 \, {\left(a^{2} \cos\left(d x + c\right) + a^{2} \sin\left(d x + c\right) + a^{2}\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) - 4 \, {\left(a^{2} \cos\left(d x + c\right)^{2} + 8 \, a^{2} \cos\left(d x + c\right) + 7 \, a^{2} + {\left(a^{2} \cos\left(d x + c\right) - 7 \, a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a \sin\left(d x + c\right) + a}}{6 \, {\left(d \cos\left(d x + c\right) + d \sin\left(d x + c\right) + d\right)}}"," ",0,"1/6*(3*(a^2*cos(d*x + c) + a^2*sin(d*x + c) + a^2)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) - 4*(a^2*cos(d*x + c)^2 + 8*a^2*cos(d*x + c) + 7*a^2 + (a^2*cos(d*x + c) - 7*a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c) + d*sin(d*x + c) + d)","B",0
57,1,308,0,1.425600," ","integrate(csc(d*x+c)^2*(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{5 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - a^{2} - {\left(a^{2} \cos\left(d x + c\right) + a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, {\left(2 \, a^{2} \cos\left(d x + c\right)^{2} + a^{2} \cos\left(d x + c\right) - a^{2} + {\left(2 \, a^{2} \cos\left(d x + c\right) + a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a \sin\left(d x + c\right) + a}}{4 \, {\left(d \cos\left(d x + c\right)^{2} - {\left(d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right) - d\right)}}"," ",0,"1/4*(5*(a^2*cos(d*x + c)^2 - a^2 - (a^2*cos(d*x + c) + a^2)*sin(d*x + c))*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c) - a^2 + (2*a^2*cos(d*x + c) + a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^2 - (d*cos(d*x + c) + d)*sin(d*x + c) - d)","B",0
58,1,359,0,1.362854," ","integrate(csc(d*x+c)^3*(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{19 \, {\left(a^{2} \cos\left(d x + c\right)^{3} + a^{2} \cos\left(d x + c\right)^{2} - a^{2} \cos\left(d x + c\right) - a^{2} + {\left(a^{2} \cos\left(d x + c\right)^{2} - a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, {\left(11 \, a^{2} \cos\left(d x + c\right)^{2} + 2 \, a^{2} \cos\left(d x + c\right) - 9 \, a^{2} + {\left(11 \, a^{2} \cos\left(d x + c\right) + 9 \, a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a \sin\left(d x + c\right) + a}}{16 \, {\left(d \cos\left(d x + c\right)^{3} + d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) + {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right) - d\right)}}"," ",0,"1/16*(19*(a^2*cos(d*x + c)^3 + a^2*cos(d*x + c)^2 - a^2*cos(d*x + c) - a^2 + (a^2*cos(d*x + c)^2 - a^2)*sin(d*x + c))*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(11*a^2*cos(d*x + c)^2 + 2*a^2*cos(d*x + c) - 9*a^2 + (11*a^2*cos(d*x + c) + 9*a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c) + (d*cos(d*x + c)^2 - d)*sin(d*x + c) - d)","B",0
59,1,408,0,1.407084," ","integrate(csc(d*x+c)^4*(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{75 \, {\left(a^{2} \cos\left(d x + c\right)^{4} - 2 \, a^{2} \cos\left(d x + c\right)^{2} + a^{2} - {\left(a^{2} \cos\left(d x + c\right)^{3} + a^{2} \cos\left(d x + c\right)^{2} - a^{2} \cos\left(d x + c\right) - a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, {\left(75 \, a^{2} \cos\left(d x + c\right)^{3} + 41 \, a^{2} \cos\left(d x + c\right)^{2} - 83 \, a^{2} \cos\left(d x + c\right) - 49 \, a^{2} - {\left(75 \, a^{2} \cos\left(d x + c\right)^{2} + 34 \, a^{2} \cos\left(d x + c\right) - 49 \, a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a \sin\left(d x + c\right) + a}}{96 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} - {\left(d \cos\left(d x + c\right)^{3} + d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right) + d\right)}}"," ",0,"1/96*(75*(a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^2 + a^2 - (a^2*cos(d*x + c)^3 + a^2*cos(d*x + c)^2 - a^2*cos(d*x + c) - a^2)*sin(d*x + c))*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(75*a^2*cos(d*x + c)^3 + 41*a^2*cos(d*x + c)^2 - 83*a^2*cos(d*x + c) - 49*a^2 - (75*a^2*cos(d*x + c)^2 + 34*a^2*cos(d*x + c) - 49*a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 - (d*cos(d*x + c)^3 + d*cos(d*x + c)^2 - d*cos(d*x + c) - d)*sin(d*x + c) + d)","B",0
60,1,473,0,1.466280," ","integrate(csc(d*x+c)^5*(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{489 \, {\left(a^{2} \cos\left(d x + c\right)^{5} + a^{2} \cos\left(d x + c\right)^{4} - 2 \, a^{2} \cos\left(d x + c\right)^{3} - 2 \, a^{2} \cos\left(d x + c\right)^{2} + a^{2} \cos\left(d x + c\right) + a^{2} + {\left(a^{2} \cos\left(d x + c\right)^{4} - 2 \, a^{2} \cos\left(d x + c\right)^{2} + a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, {\left(489 \, a^{2} \cos\left(d x + c\right)^{4} + 326 \, a^{2} \cos\left(d x + c\right)^{3} - 836 \, a^{2} \cos\left(d x + c\right)^{2} - 374 \, a^{2} \cos\left(d x + c\right) + 299 \, a^{2} + {\left(489 \, a^{2} \cos\left(d x + c\right)^{3} + 163 \, a^{2} \cos\left(d x + c\right)^{2} - 673 \, a^{2} \cos\left(d x + c\right) - 299 \, a^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a \sin\left(d x + c\right) + a}}{768 \, {\left(d \cos\left(d x + c\right)^{5} + d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{3} - 2 \, d \cos\left(d x + c\right)^{2} + d \cos\left(d x + c\right) + {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} + d\right)} \sin\left(d x + c\right) + d\right)}}"," ",0,"1/768*(489*(a^2*cos(d*x + c)^5 + a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c) + a^2 + (a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^2 + a^2)*sin(d*x + c))*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(489*a^2*cos(d*x + c)^4 + 326*a^2*cos(d*x + c)^3 - 836*a^2*cos(d*x + c)^2 - 374*a^2*cos(d*x + c) + 299*a^2 + (489*a^2*cos(d*x + c)^3 + 163*a^2*cos(d*x + c)^2 - 673*a^2*cos(d*x + c) - 299*a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^5 + d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 - 2*d*cos(d*x + c)^2 + d*cos(d*x + c) + (d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c) + d)","B",0
61,1,234,0,1.446711," ","integrate(sin(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{\frac{15 \, \sqrt{2} {\left(a \cos\left(d x + c\right) + a \sin\left(d x + c\right) + a\right)} \log\left(-\frac{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) + \frac{2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(d x + c\right) + 2}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right)}{\sqrt{a}} + 4 \, {\left(3 \, \cos\left(d x + c\right)^{3} + 4 \, \cos\left(d x + c\right)^{2} - {\left(3 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 17\right)} \sin\left(d x + c\right) - 16 \, \cos\left(d x + c\right) - 17\right)} \sqrt{a \sin\left(d x + c\right) + a}}{30 \, {\left(a d \cos\left(d x + c\right) + a d \sin\left(d x + c\right) + a d\right)}}"," ",0,"1/30*(15*sqrt(2)*(a*cos(d*x + c) + a*sin(d*x + c) + a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/sqrt(a) + 4*(3*cos(d*x + c)^3 + 4*cos(d*x + c)^2 - (3*cos(d*x + c)^2 - cos(d*x + c) - 17)*sin(d*x + c) - 16*cos(d*x + c) - 17)*sqrt(a*sin(d*x + c) + a))/(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d)","A",0
62,1,209,0,1.316233," ","integrate(sin(d*x+c)^2/(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{\frac{3 \, \sqrt{2} {\left(a \cos\left(d x + c\right) + a \sin\left(d x + c\right) + a\right)} \log\left(-\frac{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) - \frac{2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(d x + c\right) + 2}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right)}{\sqrt{a}} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2\right)} \sqrt{a \sin\left(d x + c\right) + a}}{6 \, {\left(a d \cos\left(d x + c\right) + a d \sin\left(d x + c\right) + a d\right)}}"," ",0,"1/6*(3*sqrt(2)*(a*cos(d*x + c) + a*sin(d*x + c) + a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c) - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/sqrt(a) - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a*sin(d*x + c) + a))/(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d)","B",0
63,1,191,0,1.115177," ","integrate(sin(d*x+c)/(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{\frac{\sqrt{2} {\left(a \cos\left(d x + c\right) + a \sin\left(d x + c\right) + a\right)} \log\left(-\frac{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) + \frac{2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(d x + c\right) + 2}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right)}{\sqrt{a}} - 4 \, \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{2 \, {\left(a d \cos\left(d x + c\right) + a d \sin\left(d x + c\right) + a d\right)}}"," ",0,"1/2*(sqrt(2)*(a*cos(d*x + c) + a*sin(d*x + c) + a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/sqrt(a) - 4*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1))/(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d)","B",0
64,1,167,0,1.027477," ","integrate(1/(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} \log\left(-\frac{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) - \frac{2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(d x + c\right) + 2}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right)}{2 \, \sqrt{a} d}, \frac{\sqrt{2} \sqrt{-\frac{1}{a}} \arctan\left(\frac{\sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{-\frac{1}{a}}}{\cos\left(d x + c\right)}\right)}{d}\right]"," ",0,"[1/2*sqrt(2)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c) - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/(sqrt(a)*d), sqrt(2)*sqrt(-1/a)*arctan(sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(-1/a)/cos(d*x + c))/d]","A",0
65,1,290,0,1.281690," ","integrate(csc(d*x+c)/(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{2} \sqrt{a} \log\left(-\frac{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) + \frac{2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(d x + c\right) + 2}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right)}{2 \, a d}"," ",0,"1/2*(sqrt(2)*sqrt(a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)))/(a*d)","B",0
66,1,412,0,1.249809," ","integrate(csc(d*x+c)^2/(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{{\left(\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 1\right)} \sin\left(d x + c\right) - 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + \frac{2 \, \sqrt{2} {\left(a \cos\left(d x + c\right)^{2} - {\left(a \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) - a\right)} \log\left(-\frac{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) - \frac{2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(d x + c\right) + 2}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right)}{\sqrt{a}} + 4 \, \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{4 \, {\left(a d \cos\left(d x + c\right)^{2} - a d - {\left(a d \cos\left(d x + c\right) + a d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/4*((cos(d*x + c)^2 - (cos(d*x + c) + 1)*sin(d*x + c) - 1)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 + 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 2*sqrt(2)*(a*cos(d*x + c)^2 - (a*cos(d*x + c) + a)*sin(d*x + c) - a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c) - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/sqrt(a) + 4*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1))/(a*d*cos(d*x + c)^2 - a*d - (a*d*cos(d*x + c) + a*d)*sin(d*x + c))","B",0
67,1,492,0,0.928352," ","integrate(csc(d*x+c)^3/(a+a*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{7 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + \frac{8 \, \sqrt{2} {\left(a \cos\left(d x + c\right)^{3} + a \cos\left(d x + c\right)^{2} - a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} - a\right)} \sin\left(d x + c\right) - a\right)} \log\left(-\frac{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) + \frac{2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(d x + c\right) + 2}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right)}{\sqrt{a}} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a}}{16 \, {\left(a d \cos\left(d x + c\right)^{3} + a d \cos\left(d x + c\right)^{2} - a d \cos\left(d x + c\right) - a d + {\left(a d \cos\left(d x + c\right)^{2} - a d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/16*(7*(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 8*sqrt(2)*(a*cos(d*x + c)^3 + a*cos(d*x + c)^2 - a*cos(d*x + c) + (a*cos(d*x + c)^2 - a)*sin(d*x + c) - a)*log(-(cos(d*x + c)^2 - (cos(d*x + c) - 2)*sin(d*x + c) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/sqrt(a) - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a))/(a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)^2 - a*d*cos(d*x + c) - a*d + (a*d*cos(d*x + c)^2 - a*d)*sin(d*x + c))","B",0
68,1,314,0,1.587022," ","integrate(sin(d*x+c)^4/(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{75 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) - 4 \, {\left(4 \, \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{3} - 48 \, \cos\left(d x + c\right)^{2} + {\left(4 \, \cos\left(d x + c\right)^{3} + 8 \, \cos\left(d x + c\right)^{2} - 40 \, \cos\left(d x + c\right) + 5\right)} \sin\left(d x + c\right) - 45 \, \cos\left(d x + c\right) - 5\right)} \sqrt{a \sin\left(d x + c\right) + a}}{40 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d \cos\left(d x + c\right) - 2 \, a^{2} d - {\left(a^{2} d \cos\left(d x + c\right) + 2 \, a^{2} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/40*(75*sqrt(2)*(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) - 4*(4*cos(d*x + c)^4 - 4*cos(d*x + c)^3 - 48*cos(d*x + c)^2 + (4*cos(d*x + c)^3 + 8*cos(d*x + c)^2 - 40*cos(d*x + c) + 5)*sin(d*x + c) - 45*cos(d*x + c) - 5)*sqrt(a*sin(d*x + c) + a))/(a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2*d - (a^2*d*cos(d*x + c) + 2*a^2*d)*sin(d*x + c))","B",0
69,1,295,0,1.463490," ","integrate(sin(d*x+c)^3/(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{33 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) - 4 \, {\left(4 \, \cos\left(d x + c\right)^{3} + 16 \, \cos\left(d x + c\right)^{2} - {\left(4 \, \cos\left(d x + c\right)^{2} - 12 \, \cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) + 15 \, \cos\left(d x + c\right) + 3\right)} \sqrt{a \sin\left(d x + c\right) + a}}{24 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d \cos\left(d x + c\right) - 2 \, a^{2} d - {\left(a^{2} d \cos\left(d x + c\right) + 2 \, a^{2} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/24*(33*sqrt(2)*(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) - 4*(4*cos(d*x + c)^3 + 16*cos(d*x + c)^2 - (4*cos(d*x + c)^2 - 12*cos(d*x + c) + 3)*sin(d*x + c) + 15*cos(d*x + c) + 3)*sqrt(a*sin(d*x + c) + a))/(a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2*d - (a^2*d*cos(d*x + c) + 2*a^2*d)*sin(d*x + c))","B",0
70,1,274,0,1.406998," ","integrate(sin(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{7 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + 4 \, {\left(4 \, \cos\left(d x + c\right)^{2} + {\left(4 \, \cos\left(d x + c\right) - 1\right)} \sin\left(d x + c\right) + 5 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a \sin\left(d x + c\right) + a}}{8 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d \cos\left(d x + c\right) - 2 \, a^{2} d - {\left(a^{2} d \cos\left(d x + c\right) + 2 \, a^{2} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/8*(7*sqrt(2)*(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 4*(4*cos(d*x + c)^2 + (4*cos(d*x + c) - 1)*sin(d*x + c) + 5*cos(d*x + c) + 1)*sqrt(a*sin(d*x + c) + a))/(a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2*d - (a^2*d*cos(d*x + c) + 2*a^2*d)*sin(d*x + c))","B",0
71,1,253,0,1.439131," ","integrate(sin(d*x+c)/(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{3 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) - 4 \, \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{8 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d \cos\left(d x + c\right) - 2 \, a^{2} d - {\left(a^{2} d \cos\left(d x + c\right) + 2 \, a^{2} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/8*(3*sqrt(2)*(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) - 4*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1))/(a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2*d - (a^2*d*cos(d*x + c) + 2*a^2*d)*sin(d*x + c))","B",0
72,1,252,0,1.338381," ","integrate(1/(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{2} {\left(\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + 4 \, \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{8 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d \cos\left(d x + c\right) - 2 \, a^{2} d - {\left(a^{2} d \cos\left(d x + c\right) + 2 \, a^{2} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/8*(sqrt(2)*(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 4*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1))/(a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2*d - (a^2*d*cos(d*x + c) + 2*a^2*d)*sin(d*x + c))","B",0
73,1,453,0,1.375730," ","integrate(csc(d*x+c)/(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{5 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + 4 \, {\left(\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) - 4 \, \sqrt{a \sin\left(d x + c\right) + a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)}}{8 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d \cos\left(d x + c\right) - 2 \, a^{2} d - {\left(a^{2} d \cos\left(d x + c\right) + 2 \, a^{2} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/8*(5*sqrt(2)*(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 4*(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) - 4*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) - sin(d*x + c) + 1))/(a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2*d - (a^2*d*cos(d*x + c) + 2*a^2*d)*sin(d*x + c))","B",0
74,1,539,0,1.275480," ","integrate(csc(d*x+c)^2/(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{9 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 2 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + 6 \, {\left(\cos\left(d x + c\right)^{3} + 2 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, {\left(3 \, \cos\left(d x + c\right)^{2} + {\left(3 \, \cos\left(d x + c\right) + 1\right)} \sin\left(d x + c\right) + 2 \, \cos\left(d x + c\right) - 1\right)} \sqrt{a \sin\left(d x + c\right) + a}}{8 \, {\left(a^{2} d \cos\left(d x + c\right)^{3} + 2 \, a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d \cos\left(d x + c\right) - 2 \, a^{2} d + {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d \cos\left(d x + c\right) - 2 \, a^{2} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/8*(9*sqrt(2)*(cos(d*x + c)^3 + 2*cos(d*x + c)^2 + (cos(d*x + c)^2 - cos(d*x + c) - 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 6*(cos(d*x + c)^3 + 2*cos(d*x + c)^2 + (cos(d*x + c)^2 - cos(d*x + c) - 2)*sin(d*x + c) - cos(d*x + c) - 2)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 + 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(3*cos(d*x + c)^2 + (3*cos(d*x + c) + 1)*sin(d*x + c) + 2*cos(d*x + c) - 1)*sqrt(a*sin(d*x + c) + a))/(a^2*d*cos(d*x + c)^3 + 2*a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2*d + (a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2*d)*sin(d*x + c))","B",0
75,1,626,0,1.713513," ","integrate(csc(d*x+c)^3/(a+a*sin(d*x+c))^(3/2),x, algorithm=""fricas"")","\frac{26 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{4} - \cos\left(d x + c\right)^{3} - 3 \, \cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right)^{3} + 2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) + \cos\left(d x + c\right) + 2\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + 19 \, {\left(\cos\left(d x + c\right)^{4} - \cos\left(d x + c\right)^{3} - 3 \, \cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right)^{3} + 2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) + \cos\left(d x + c\right) + 2\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) - 4 \, {\left(7 \, \cos\left(d x + c\right)^{3} + 4 \, \cos\left(d x + c\right)^{2} - {\left(7 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) - 2\right)} \sin\left(d x + c\right) - 5 \, \cos\left(d x + c\right) - 2\right)} \sqrt{a \sin\left(d x + c\right) + a}}{16 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - a^{2} d \cos\left(d x + c\right)^{3} - 3 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d \cos\left(d x + c\right) + 2 \, a^{2} d - {\left(a^{2} d \cos\left(d x + c\right)^{3} + 2 \, a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d \cos\left(d x + c\right) - 2 \, a^{2} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/16*(26*sqrt(2)*(cos(d*x + c)^4 - cos(d*x + c)^3 - 3*cos(d*x + c)^2 - (cos(d*x + c)^3 + 2*cos(d*x + c)^2 - cos(d*x + c) - 2)*sin(d*x + c) + cos(d*x + c) + 2)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 19*(cos(d*x + c)^4 - cos(d*x + c)^3 - 3*cos(d*x + c)^2 - (cos(d*x + c)^3 + 2*cos(d*x + c)^2 - cos(d*x + c) - 2)*sin(d*x + c) + cos(d*x + c) + 2)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) - 4*(7*cos(d*x + c)^3 + 4*cos(d*x + c)^2 - (7*cos(d*x + c)^2 + 3*cos(d*x + c) - 2)*sin(d*x + c) - 5*cos(d*x + c) - 2)*sqrt(a*sin(d*x + c) + a))/(a^2*d*cos(d*x + c)^4 - a^2*d*cos(d*x + c)^3 - 3*a^2*d*cos(d*x + c)^2 + a^2*d*cos(d*x + c) + 2*a^2*d - (a^2*d*cos(d*x + c)^3 + 2*a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2*d)*sin(d*x + c))","B",0
76,1,381,0,1.366037," ","integrate(sin(d*x+c)^5/(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{4245 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 4\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + 4 \, {\left(96 \, \cos\left(d x + c\right)^{5} + 256 \, \cos\left(d x + c\right)^{4} - 1760 \, \cos\left(d x + c\right)^{3} + 2475 \, \cos\left(d x + c\right)^{2} - {\left(96 \, \cos\left(d x + c\right)^{4} - 160 \, \cos\left(d x + c\right)^{3} - 1920 \, \cos\left(d x + c\right)^{2} - 4395 \, \cos\left(d x + c\right) - 60\right)} \sin\left(d x + c\right) + 4335 \, \cos\left(d x + c\right) - 60\right)} \sqrt{a \sin\left(d x + c\right) + a}}{960 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d + {\left(a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/960*(4245*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) - 2*cos(d*x + c) - 4)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 4*(96*cos(d*x + c)^5 + 256*cos(d*x + c)^4 - 1760*cos(d*x + c)^3 + 2475*cos(d*x + c)^2 - (96*cos(d*x + c)^4 - 160*cos(d*x + c)^3 - 1920*cos(d*x + c)^2 - 4395*cos(d*x + c) - 60)*sin(d*x + c) + 4335*cos(d*x + c) - 60)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d + (a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))","B",0
77,1,360,0,1.110326," ","integrate(sin(d*x+c)^4/(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{489 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 4\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) - 4 \, {\left(32 \, \cos\left(d x + c\right)^{4} - 160 \, \cos\left(d x + c\right)^{3} + 279 \, \cos\left(d x + c\right)^{2} + {\left(32 \, \cos\left(d x + c\right)^{3} + 192 \, \cos\left(d x + c\right)^{2} + 471 \, \cos\left(d x + c\right) + 12\right)} \sin\left(d x + c\right) + 459 \, \cos\left(d x + c\right) - 12\right)} \sqrt{a \sin\left(d x + c\right) + a}}{192 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d + {\left(a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/192*(489*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) - 2*cos(d*x + c) - 4)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) - 4*(32*cos(d*x + c)^4 - 160*cos(d*x + c)^3 + 279*cos(d*x + c)^2 + (32*cos(d*x + c)^3 + 192*cos(d*x + c)^2 + 471*cos(d*x + c) + 12)*sin(d*x + c) + 459*cos(d*x + c) - 12)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d + (a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))","B",0
78,1,341,0,1.509679," ","integrate(sin(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{75 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 4\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) - 4 \, {\left(32 \, \cos\left(d x + c\right)^{3} - 53 \, \cos\left(d x + c\right)^{2} - {\left(32 \, \cos\left(d x + c\right)^{2} + 85 \, \cos\left(d x + c\right) + 4\right)} \sin\left(d x + c\right) - 81 \, \cos\left(d x + c\right) + 4\right)} \sqrt{a \sin\left(d x + c\right) + a}}{64 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d + {\left(a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/64*(75*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) - 2*cos(d*x + c) - 4)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) - 4*(32*cos(d*x + c)^3 - 53*cos(d*x + c)^2 - (32*cos(d*x + c)^2 + 85*cos(d*x + c) + 4)*sin(d*x + c) - 81*cos(d*x + c) + 4)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d + (a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))","B",0
79,1,320,0,1.456049," ","integrate(sin(d*x+c)^2/(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{19 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 4\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) - 4 \, {\left(13 \, \cos\left(d x + c\right)^{2} + {\left(13 \, \cos\left(d x + c\right) + 4\right)} \sin\left(d x + c\right) + 9 \, \cos\left(d x + c\right) - 4\right)} \sqrt{a \sin\left(d x + c\right) + a}}{64 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d + {\left(a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/64*(19*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) - 2*cos(d*x + c) - 4)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) - 4*(13*cos(d*x + c)^2 + (13*cos(d*x + c) + 4)*sin(d*x + c) + 9*cos(d*x + c) - 4)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d + (a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))","B",0
80,1,318,0,1.350266," ","integrate(sin(d*x+c)/(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{5 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 4\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + 4 \, {\left(5 \, \cos\left(d x + c\right)^{2} + {\left(5 \, \cos\left(d x + c\right) + 4\right)} \sin\left(d x + c\right) + \cos\left(d x + c\right) - 4\right)} \sqrt{a \sin\left(d x + c\right) + a}}{64 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d + {\left(a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/64*(5*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) - 2*cos(d*x + c) - 4)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 4*(5*cos(d*x + c)^2 + (5*cos(d*x + c) + 4)*sin(d*x + c) + cos(d*x + c) - 4)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d + (a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))","B",0
81,1,320,0,1.415635," ","integrate(1/(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{3 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 4\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + 4 \, {\left(3 \, \cos\left(d x + c\right)^{2} + {\left(3 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) + 7 \, \cos\left(d x + c\right) + 4\right)} \sqrt{a \sin\left(d x + c\right) + a}}{64 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d + {\left(a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/64*(3*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) - 2*cos(d*x + c) - 4)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 4*(3*cos(d*x + c)^2 + (3*cos(d*x + c) - 4)*sin(d*x + c) + 7*cos(d*x + c) + 4)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d + (a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))","B",0
82,1,539,0,1.524349," ","integrate(csc(d*x+c)/(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{43 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 4\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + 32 \, {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 4\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) - 4 \, {\left(11 \, \cos\left(d x + c\right)^{2} + {\left(11 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) + 15 \, \cos\left(d x + c\right) + 4\right)} \sqrt{a \sin\left(d x + c\right) + a}}{64 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d + {\left(a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/64*(43*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) - 2*cos(d*x + c) - 4)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 32*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) - 2*cos(d*x + c) - 4)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) - 4*(11*cos(d*x + c)^2 + (11*cos(d*x + c) - 4)*sin(d*x + c) + 15*cos(d*x + c) + 4)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d + (a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))","B",0
83,1,631,0,1.530017," ","integrate(csc(d*x+c)^2/(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{115 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{3} - 5 \, \cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) + 2 \, \cos\left(d x + c\right) + 4\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + 80 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{3} - 5 \, \cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) + 2 \, \cos\left(d x + c\right) + 4\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) + 4 \, {\left(35 \, \cos\left(d x + c\right)^{3} - 20 \, \cos\left(d x + c\right)^{2} - {\left(35 \, \cos\left(d x + c\right)^{2} + 55 \, \cos\left(d x + c\right) + 4\right)} \sin\left(d x + c\right) - 51 \, \cos\left(d x + c\right) + 4\right)} \sqrt{a \sin\left(d x + c\right) + a}}{64 \, {\left(a^{3} d \cos\left(d x + c\right)^{4} - 2 \, a^{3} d \cos\left(d x + c\right)^{3} - 5 \, a^{3} d \cos\left(d x + c\right)^{2} + 2 \, a^{3} d \cos\left(d x + c\right) + 4 \, a^{3} d - {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/64*(115*sqrt(2)*(cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 5*cos(d*x + c)^2 - (cos(d*x + c)^3 + 3*cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) + 2*cos(d*x + c) + 4)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 80*(cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 5*cos(d*x + c)^2 - (cos(d*x + c)^3 + 3*cos(d*x + c)^2 - 2*cos(d*x + c) - 4)*sin(d*x + c) + 2*cos(d*x + c) + 4)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 + 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(35*cos(d*x + c)^3 - 20*cos(d*x + c)^2 - (35*cos(d*x + c)^2 + 55*cos(d*x + c) + 4)*sin(d*x + c) - 51*cos(d*x + c) + 4)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^3 - 5*a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + 4*a^3*d - (a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))","B",0
84,1,715,0,1.344525," ","integrate(csc(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x, algorithm=""fricas"")","\frac{219 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{4} - 3 \, \cos\left(d x + c\right)^{3} - 7 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{3} - 5 \, \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 4\right)} \sin\left(d x + c\right) + 2 \, \cos\left(d x + c\right) + 4\right)} \sqrt{a} \log\left(-\frac{a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} {\left(\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1\right)} + 3 \, a \cos\left(d x + c\right) - {\left(a \cos\left(d x + c\right) - 2 \, a\right)} \sin\left(d x + c\right) + 2 \, a}{\cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 2}\right) + 156 \, {\left(\cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{4} - 3 \, \cos\left(d x + c\right)^{3} - 7 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{3} - 5 \, \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 4\right)} \sin\left(d x + c\right) + 2 \, \cos\left(d x + c\right) + 4\right)} \sqrt{a} \log\left(\frac{a \cos\left(d x + c\right)^{3} - 7 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right) - 3\right)} \sqrt{a \sin\left(d x + c\right) + a} \sqrt{a} - 9 \, a \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right)^{2} + 8 \, a \cos\left(d x + c\right) - a\right)} \sin\left(d x + c\right) - a}{\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - \cos\left(d x + c\right) - 1}\right) - 4 \, {\left(63 \, \cos\left(d x + c\right)^{4} + 95 \, \cos\left(d x + c\right)^{3} - 51 \, \cos\left(d x + c\right)^{2} + {\left(63 \, \cos\left(d x + c\right)^{3} - 32 \, \cos\left(d x + c\right)^{2} - 83 \, \cos\left(d x + c\right) + 4\right)} \sin\left(d x + c\right) - 87 \, \cos\left(d x + c\right) - 4\right)} \sqrt{a \sin\left(d x + c\right) + a}}{64 \, {\left(a^{3} d \cos\left(d x + c\right)^{5} + 3 \, a^{3} d \cos\left(d x + c\right)^{4} - 3 \, a^{3} d \cos\left(d x + c\right)^{3} - 7 \, a^{3} d \cos\left(d x + c\right)^{2} + 2 \, a^{3} d \cos\left(d x + c\right) + 4 \, a^{3} d + {\left(a^{3} d \cos\left(d x + c\right)^{4} - 2 \, a^{3} d \cos\left(d x + c\right)^{3} - 5 \, a^{3} d \cos\left(d x + c\right)^{2} + 2 \, a^{3} d \cos\left(d x + c\right) + 4 \, a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/64*(219*sqrt(2)*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 - 3*cos(d*x + c)^3 - 7*cos(d*x + c)^2 + (cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 5*cos(d*x + c)^2 + 2*cos(d*x + c) + 4)*sin(d*x + c) + 2*cos(d*x + c) + 4)*sqrt(a)*log(-(a*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 156*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 - 3*cos(d*x + c)^3 - 7*cos(d*x + c)^2 + (cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 5*cos(d*x + c)^2 + 2*cos(d*x + c) + 4)*sin(d*x + c) + 2*cos(d*x + c) + 4)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) - 4*(63*cos(d*x + c)^4 + 95*cos(d*x + c)^3 - 51*cos(d*x + c)^2 + (63*cos(d*x + c)^3 - 32*cos(d*x + c)^2 - 83*cos(d*x + c) + 4)*sin(d*x + c) - 87*cos(d*x + c) - 4)*sqrt(a*sin(d*x + c) + a))/(a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 - 3*a^3*d*cos(d*x + c)^3 - 7*a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + 4*a^3*d + (a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^3 - 5*a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + 4*a^3*d)*sin(d*x + c))","B",0
85,1,330,0,1.684302," ","integrate((a+a*sin(f*x+e))^(1/2)/sin(f*x+e)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} \log\left(\frac{128 \, a \cos\left(f x + e\right)^{5} - 128 \, a \cos\left(f x + e\right)^{4} - 416 \, a \cos\left(f x + e\right)^{3} + 128 \, a \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, \cos\left(f x + e\right)^{4} - 24 \, \cos\left(f x + e\right)^{3} - 66 \, \cos\left(f x + e\right)^{2} + {\left(16 \, \cos\left(f x + e\right)^{3} + 40 \, \cos\left(f x + e\right)^{2} - 26 \, \cos\left(f x + e\right) - 51\right)} \sin\left(f x + e\right) + 25 \, \cos\left(f x + e\right) + 51\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-a} \sqrt{\sin\left(f x + e\right)} + 289 \, a \cos\left(f x + e\right) + {\left(128 \, a \cos\left(f x + e\right)^{4} + 256 \, a \cos\left(f x + e\right)^{3} - 160 \, a \cos\left(f x + e\right)^{2} - 288 \, a \cos\left(f x + e\right) + a\right)} \sin\left(f x + e\right) + a}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right)}{4 \, f}, \frac{\sqrt{a} \arctan\left(\frac{{\left(8 \, \cos\left(f x + e\right)^{2} + 8 \, \sin\left(f x + e\right) - 9\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} \sqrt{\sin\left(f x + e\right)}}{4 \, {\left(2 \, a \cos\left(f x + e\right)^{3} + a \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a \cos\left(f x + e\right)\right)}}\right)}{2 \, f}\right]"," ",0,"[1/4*sqrt(-a)*log((128*a*cos(f*x + e)^5 - 128*a*cos(f*x + e)^4 - 416*a*cos(f*x + e)^3 + 128*a*cos(f*x + e)^2 - 8*(16*cos(f*x + e)^4 - 24*cos(f*x + e)^3 - 66*cos(f*x + e)^2 + (16*cos(f*x + e)^3 + 40*cos(f*x + e)^2 - 26*cos(f*x + e) - 51)*sin(f*x + e) + 25*cos(f*x + e) + 51)*sqrt(a*sin(f*x + e) + a)*sqrt(-a)*sqrt(sin(f*x + e)) + 289*a*cos(f*x + e) + (128*a*cos(f*x + e)^4 + 256*a*cos(f*x + e)^3 - 160*a*cos(f*x + e)^2 - 288*a*cos(f*x + e) + a)*sin(f*x + e) + a)/(cos(f*x + e) + sin(f*x + e) + 1))/f, 1/2*sqrt(a)*arctan(1/4*(8*cos(f*x + e)^2 + 8*sin(f*x + e) - 9)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*sqrt(sin(f*x + e))/(2*a*cos(f*x + e)^3 + a*cos(f*x + e)*sin(f*x + e) - 2*a*cos(f*x + e)))/f]","B",0
86,1,341,0,1.709177," ","integrate((a-a*sin(f*x+e))^(1/2)/(-sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} \log\left(\frac{128 \, a \cos\left(f x + e\right)^{5} - 128 \, a \cos\left(f x + e\right)^{4} - 416 \, a \cos\left(f x + e\right)^{3} + 128 \, a \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, \cos\left(f x + e\right)^{4} - 24 \, \cos\left(f x + e\right)^{3} - 66 \, \cos\left(f x + e\right)^{2} - {\left(16 \, \cos\left(f x + e\right)^{3} + 40 \, \cos\left(f x + e\right)^{2} - 26 \, \cos\left(f x + e\right) - 51\right)} \sin\left(f x + e\right) + 25 \, \cos\left(f x + e\right) + 51\right)} \sqrt{-a \sin\left(f x + e\right) + a} \sqrt{-a} \sqrt{-\sin\left(f x + e\right)} + 289 \, a \cos\left(f x + e\right) - {\left(128 \, a \cos\left(f x + e\right)^{4} + 256 \, a \cos\left(f x + e\right)^{3} - 160 \, a \cos\left(f x + e\right)^{2} - 288 \, a \cos\left(f x + e\right) + a\right)} \sin\left(f x + e\right) + a}{\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1}\right)}{4 \, f}, -\frac{\sqrt{a} \arctan\left(\frac{{\left(8 \, \cos\left(f x + e\right)^{2} - 8 \, \sin\left(f x + e\right) - 9\right)} \sqrt{-a \sin\left(f x + e\right) + a} \sqrt{a} \sqrt{-\sin\left(f x + e\right)}}{4 \, {\left(2 \, a \cos\left(f x + e\right)^{3} - a \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a \cos\left(f x + e\right)\right)}}\right)}{2 \, f}\right]"," ",0,"[1/4*sqrt(-a)*log((128*a*cos(f*x + e)^5 - 128*a*cos(f*x + e)^4 - 416*a*cos(f*x + e)^3 + 128*a*cos(f*x + e)^2 + 8*(16*cos(f*x + e)^4 - 24*cos(f*x + e)^3 - 66*cos(f*x + e)^2 - (16*cos(f*x + e)^3 + 40*cos(f*x + e)^2 - 26*cos(f*x + e) - 51)*sin(f*x + e) + 25*cos(f*x + e) + 51)*sqrt(-a*sin(f*x + e) + a)*sqrt(-a)*sqrt(-sin(f*x + e)) + 289*a*cos(f*x + e) - (128*a*cos(f*x + e)^4 + 256*a*cos(f*x + e)^3 - 160*a*cos(f*x + e)^2 - 288*a*cos(f*x + e) + a)*sin(f*x + e) + a)/(cos(f*x + e) - sin(f*x + e) + 1))/f, -1/2*sqrt(a)*arctan(1/4*(8*cos(f*x + e)^2 - 8*sin(f*x + e) - 9)*sqrt(-a*sin(f*x + e) + a)*sqrt(a)*sqrt(-sin(f*x + e))/(2*a*cos(f*x + e)^3 - a*cos(f*x + e)*sin(f*x + e) - 2*a*cos(f*x + e)))/f]","B",0
87,1,28,0,1.274456," ","integrate(1/sin(x)^(1/2)/(1+sin(x))^(1/2),x, algorithm=""fricas"")","2 \, \sqrt{2} \arctan\left(\frac{\sqrt{2} \sqrt{\sin\left(x\right) + 1} \sqrt{\sin\left(x\right)}}{\cos\left(x\right) + \sin\left(x\right) + 1}\right)"," ",0,"2*sqrt(2)*arctan(sqrt(2)*sqrt(sin(x) + 1)*sqrt(sin(x))/(cos(x) + sin(x) + 1))","A",0
88,1,163,0,1.089202," ","integrate(1/sin(x)^(1/2)/(a+a*sin(x))^(1/2),x, algorithm=""fricas"")","\left[\frac{1}{4} \, \sqrt{2} \sqrt{-\frac{1}{a}} \log\left(\frac{17 \, \cos\left(x\right)^{3} - 4 \, \sqrt{2} {\left(3 \, \cos\left(x\right)^{2} + {\left(3 \, \cos\left(x\right) + 4\right)} \sin\left(x\right) - \cos\left(x\right) - 4\right)} \sqrt{a \sin\left(x\right) + a} \sqrt{-\frac{1}{a}} \sqrt{\sin\left(x\right)} + 3 \, \cos\left(x\right)^{2} + {\left(17 \, \cos\left(x\right)^{2} + 14 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 18 \, \cos\left(x\right) - 4}{\cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 2 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 2 \, \cos\left(x\right) - 4}\right), \frac{\sqrt{2} \arctan\left(\frac{\sqrt{2} \sqrt{a \sin\left(x\right) + a} {\left(3 \, \sin\left(x\right) - 1\right)}}{4 \, \sqrt{a} \cos\left(x\right) \sqrt{\sin\left(x\right)}}\right)}{2 \, \sqrt{a}}\right]"," ",0,"[1/4*sqrt(2)*sqrt(-1/a)*log((17*cos(x)^3 - 4*sqrt(2)*(3*cos(x)^2 + (3*cos(x) + 4)*sin(x) - cos(x) - 4)*sqrt(a*sin(x) + a)*sqrt(-1/a)*sqrt(sin(x)) + 3*cos(x)^2 + (17*cos(x)^2 + 14*cos(x) - 4)*sin(x) - 18*cos(x) - 4)/(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)), 1/2*sqrt(2)*arctan(1/4*sqrt(2)*sqrt(a*sin(x) + a)*(3*sin(x) - 1)/(sqrt(a)*cos(x)*sqrt(sin(x))))/sqrt(a)]","A",0
89,1,31,0,0.790115," ","integrate(1/(1-sin(x))^(1/2)/sin(x)^(1/2),x, algorithm=""fricas"")","\sqrt{2} \log\left(\frac{\sqrt{2} \sqrt{-\sin\left(x\right) + 1} \sqrt{\sin\left(x\right)} + \cos\left(x\right)}{\sin\left(x\right) - 1}\right)"," ",0,"sqrt(2)*log((sqrt(2)*sqrt(-sin(x) + 1)*sqrt(sin(x)) + cos(x))/(sin(x) - 1))","A",0
90,1,168,0,1.264255," ","integrate(1/sin(x)^(1/2)/(a-a*sin(x))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} \log\left(\frac{17 \, \cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} + \frac{4 \, \sqrt{2} {\left(3 \, \cos\left(x\right)^{2} - {\left(3 \, \cos\left(x\right) + 4\right)} \sin\left(x\right) - \cos\left(x\right) - 4\right)} \sqrt{-a \sin\left(x\right) + a} \sqrt{\sin\left(x\right)}}{\sqrt{a}} - {\left(17 \, \cos\left(x\right)^{2} + 14 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 18 \, \cos\left(x\right) - 4}{\cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{2} - 2 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 2 \, \cos\left(x\right) - 4}\right)}{4 \, \sqrt{a}}, -\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{a}} \arctan\left(\frac{\sqrt{2} \sqrt{-a \sin\left(x\right) + a} \sqrt{-\frac{1}{a}} {\left(3 \, \sin\left(x\right) + 1\right)}}{4 \, \cos\left(x\right) \sqrt{\sin\left(x\right)}}\right)\right]"," ",0,"[1/4*sqrt(2)*log((17*cos(x)^3 + 3*cos(x)^2 + 4*sqrt(2)*(3*cos(x)^2 - (3*cos(x) + 4)*sin(x) - cos(x) - 4)*sqrt(-a*sin(x) + a)*sqrt(sin(x))/sqrt(a) - (17*cos(x)^2 + 14*cos(x) - 4)*sin(x) - 18*cos(x) - 4)/(cos(x)^3 + 3*cos(x)^2 - (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4))/sqrt(a), -1/2*sqrt(2)*sqrt(-1/a)*arctan(1/4*sqrt(2)*sqrt(-a*sin(x) + a)*sqrt(-1/a)*(3*sin(x) + 1)/(cos(x)*sqrt(sin(x))))]","A",0
91,0,0,0,1.356098," ","integrate(sin(d*x+c)^(1/3)/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sin\left(d x + c\right)^{\frac{1}{3}}}{a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \sin\left(d x + c\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-sin(d*x + c)^(1/3)/(a^2*cos(d*x + c)^2 - 2*a^2*sin(d*x + c) - 2*a^2), x)","F",0
92,0,0,0,1.294544," ","integrate(sin(d*x+c)^3*(a+a*sin(d*x+c))^(2/3),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(d x + c\right)^{2} - 1\right)} {\left(a \sin\left(d x + c\right) + a\right)}^{\frac{2}{3}} \sin\left(d x + c\right), x\right)"," ",0,"integral(-(cos(d*x + c)^2 - 1)*(a*sin(d*x + c) + a)^(2/3)*sin(d*x + c), x)","F",0
93,0,0,0,1.808342," ","integrate(sin(d*x+c)^2*(a+a*sin(d*x+c))^(2/3),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(d x + c\right)^{2} - 1\right)} {\left(a \sin\left(d x + c\right) + a\right)}^{\frac{2}{3}}, x\right)"," ",0,"integral(-(cos(d*x + c)^2 - 1)*(a*sin(d*x + c) + a)^(2/3), x)","F",0
94,0,0,0,0.832108," ","integrate(sin(d*x+c)*(a+a*sin(d*x+c))^(2/3),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(d x + c\right) + a\right)}^{\frac{2}{3}} \sin\left(d x + c\right), x\right)"," ",0,"integral((a*sin(d*x + c) + a)^(2/3)*sin(d*x + c), x)","F",0
95,0,0,0,1.219994," ","integrate((a+a*sin(d*x+c))^(2/3),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(d x + c\right) + a\right)}^{\frac{2}{3}}, x\right)"," ",0,"integral((a*sin(d*x + c) + a)^(2/3), x)","F",0
96,-1,0,0,0.000000," ","integrate(csc(d*x+c)*(a+a*sin(d*x+c))^(2/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
97,-1,0,0,0.000000," ","integrate(csc(d*x+c)^2*(a+a*sin(d*x+c))^(2/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
98,0,0,0,2.232162," ","integrate(sin(d*x+c)^3*(a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \cos\left(d x + c\right)^{4} - 2 \, a \cos\left(d x + c\right)^{2} - {\left(a \cos\left(d x + c\right)^{2} - a\right)} \sin\left(d x + c\right) + a\right)} {\left(a \sin\left(d x + c\right) + a\right)}^{\frac{1}{3}}, x\right)"," ",0,"integral((a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 - (a*cos(d*x + c)^2 - a)*sin(d*x + c) + a)*(a*sin(d*x + c) + a)^(1/3), x)","F",0
99,0,0,0,2.193175," ","integrate(sin(d*x+c)^2*(a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a \cos\left(d x + c\right)^{2} + {\left(a \cos\left(d x + c\right)^{2} - a\right)} \sin\left(d x + c\right) - a\right)} {\left(a \sin\left(d x + c\right) + a\right)}^{\frac{1}{3}}, x\right)"," ",0,"integral(-(a*cos(d*x + c)^2 + (a*cos(d*x + c)^2 - a)*sin(d*x + c) - a)*(a*sin(d*x + c) + a)^(1/3), x)","F",0
100,0,0,0,1.439637," ","integrate(sin(d*x+c)*(a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a \cos\left(d x + c\right)^{2} - a \sin\left(d x + c\right) - a\right)} {\left(a \sin\left(d x + c\right) + a\right)}^{\frac{1}{3}}, x\right)"," ",0,"integral(-(a*cos(d*x + c)^2 - a*sin(d*x + c) - a)*(a*sin(d*x + c) + a)^(1/3), x)","F",0
101,0,0,0,1.636862," ","integrate((a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(d x + c\right) + a\right)}^{\frac{4}{3}}, x\right)"," ",0,"integral((a*sin(d*x + c) + a)^(4/3), x)","F",0
102,-1,0,0,0.000000," ","integrate(csc(d*x+c)*(a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
103,-1,0,0,0.000000," ","integrate(csc(d*x+c)^2*(a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
104,0,0,0,1.026829," ","integrate(sin(d*x+c)^3/(a+a*sin(d*x+c))^(1/3),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right)}{{\left(a \sin\left(d x + c\right) + a\right)}^{\frac{1}{3}}}, x\right)"," ",0,"integral(-(cos(d*x + c)^2 - 1)*sin(d*x + c)/(a*sin(d*x + c) + a)^(1/3), x)","F",0
105,0,0,0,0.847521," ","integrate(sin(d*x+c)^2/(a+a*sin(d*x+c))^(1/3),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\cos\left(d x + c\right)^{2} - 1}{{\left(a \sin\left(d x + c\right) + a\right)}^{\frac{1}{3}}}, x\right)"," ",0,"integral(-(cos(d*x + c)^2 - 1)/(a*sin(d*x + c) + a)^(1/3), x)","F",0
106,0,0,0,1.207630," ","integrate(sin(d*x+c)/(a+a*sin(d*x+c))^(1/3),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sin\left(d x + c\right)}{{\left(a \sin\left(d x + c\right) + a\right)}^{\frac{1}{3}}}, x\right)"," ",0,"integral(sin(d*x + c)/(a*sin(d*x + c) + a)^(1/3), x)","F",0
107,0,0,0,1.380245," ","integrate(1/(a+a*sin(d*x+c))^(1/3),x, algorithm=""fricas"")","{\rm integral}\left(\frac{1}{{\left(a \sin\left(d x + c\right) + a\right)}^{\frac{1}{3}}}, x\right)"," ",0,"integral((a*sin(d*x + c) + a)^(-1/3), x)","F",0
108,-1,0,0,0.000000," ","integrate(csc(d*x+c)/(a+a*sin(d*x+c))^(1/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
109,-1,0,0,0.000000," ","integrate(csc(d*x+c)^2/(a+a*sin(d*x+c))^(1/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
110,0,0,0,1.686069," ","integrate(sin(d*x+c)^3/(a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(\cos\left(d x + c\right)^{2} - 1\right)} {\left(a \sin\left(d x + c\right) + a\right)}^{\frac{2}{3}} \sin\left(d x + c\right)}{a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \sin\left(d x + c\right) - 2 \, a^{2}}, x\right)"," ",0,"integral((cos(d*x + c)^2 - 1)*(a*sin(d*x + c) + a)^(2/3)*sin(d*x + c)/(a^2*cos(d*x + c)^2 - 2*a^2*sin(d*x + c) - 2*a^2), x)","F",0
111,0,0,0,1.844243," ","integrate(sin(d*x+c)^2/(a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(\cos\left(d x + c\right)^{2} - 1\right)} {\left(a \sin\left(d x + c\right) + a\right)}^{\frac{2}{3}}}{a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \sin\left(d x + c\right) - 2 \, a^{2}}, x\right)"," ",0,"integral((cos(d*x + c)^2 - 1)*(a*sin(d*x + c) + a)^(2/3)/(a^2*cos(d*x + c)^2 - 2*a^2*sin(d*x + c) - 2*a^2), x)","F",0
112,0,0,0,1.980121," ","integrate(sin(d*x+c)/(a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(d x + c\right) + a\right)}^{\frac{2}{3}} \sin\left(d x + c\right)}{a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \sin\left(d x + c\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-(a*sin(d*x + c) + a)^(2/3)*sin(d*x + c)/(a^2*cos(d*x + c)^2 - 2*a^2*sin(d*x + c) - 2*a^2), x)","F",0
113,0,0,0,1.754985," ","integrate(1/(a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(d x + c\right) + a\right)}^{\frac{2}{3}}}{a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \sin\left(d x + c\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-(a*sin(d*x + c) + a)^(2/3)/(a^2*cos(d*x + c)^2 - 2*a^2*sin(d*x + c) - 2*a^2), x)","F",0
114,-1,0,0,0.000000," ","integrate(csc(d*x+c)/(a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
115,-1,0,0,0.000000," ","integrate(csc(d*x+c)^2/(a+a*sin(d*x+c))^(4/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
116,0,0,0,1.556041," ","integrate(sin(f*x+e)^n*(1+sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\sin\left(f x + e\right)^{n} {\left(\sin\left(f x + e\right) + 1\right)}^{\frac{3}{2}}, x\right)"," ",0,"integral(sin(f*x + e)^n*(sin(f*x + e) + 1)^(3/2), x)","F",0
117,0,0,0,1.614769," ","integrate(sin(f*x+e)^n*(1+sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sin\left(f x + e\right)^{n} \sqrt{\sin\left(f x + e\right) + 1}, x\right)"," ",0,"integral(sin(f*x + e)^n*sqrt(sin(f*x + e) + 1), x)","F",0
118,0,0,0,1.207042," ","integrate(sin(f*x+e)^n/(1+sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sin\left(f x + e\right)^{n}}{\sqrt{\sin\left(f x + e\right) + 1}}, x\right)"," ",0,"integral(sin(f*x + e)^n/sqrt(sin(f*x + e) + 1), x)","F",0
119,0,0,0,1.516534," ","integrate(sin(f*x+e)^n/(1+sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sin\left(f x + e\right)^{n} \sqrt{\sin\left(f x + e\right) + 1}}{\cos\left(f x + e\right)^{2} - 2 \, \sin\left(f x + e\right) - 2}, x\right)"," ",0,"integral(-sin(f*x + e)^n*sqrt(sin(f*x + e) + 1)/(cos(f*x + e)^2 - 2*sin(f*x + e) - 2), x)","F",0
120,0,0,0,1.292226," ","integrate(sin(f*x+e)^n*(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{\frac{3}{2}} \sin\left(f x + e\right)^{n}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^(3/2)*sin(f*x + e)^n, x)","F",0
121,0,0,0,1.074432," ","integrate(sin(f*x+e)^n*(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{a \sin\left(f x + e\right) + a} \sin\left(f x + e\right)^{n}, x\right)"," ",0,"integral(sqrt(a*sin(f*x + e) + a)*sin(f*x + e)^n, x)","F",0
122,0,0,0,1.139547," ","integrate(sin(f*x+e)^n/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sin\left(f x + e\right)^{n}}{\sqrt{a \sin\left(f x + e\right) + a}}, x\right)"," ",0,"integral(sin(f*x + e)^n/sqrt(a*sin(f*x + e) + a), x)","F",0
123,0,0,0,1.300960," ","integrate(sin(f*x+e)^n/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{a \sin\left(f x + e\right) + a} \sin\left(f x + e\right)^{n}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-sqrt(a*sin(f*x + e) + a)*sin(f*x + e)^n/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
124,0,0,0,1.391718," ","integrate((d*sin(f*x+e))^n*(1+sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\left(d \sin\left(f x + e\right)\right)^{n} {\left(\sin\left(f x + e\right) + 1\right)}^{\frac{3}{2}}, x\right)"," ",0,"integral((d*sin(f*x + e))^n*(sin(f*x + e) + 1)^(3/2), x)","F",0
125,0,0,0,1.478202," ","integrate((d*sin(f*x+e))^n*(1+sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\left(d \sin\left(f x + e\right)\right)^{n} \sqrt{\sin\left(f x + e\right) + 1}, x\right)"," ",0,"integral((d*sin(f*x + e))^n*sqrt(sin(f*x + e) + 1), x)","F",0
126,0,0,0,1.323478," ","integrate((d*sin(f*x+e))^n/(1+sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(d \sin\left(f x + e\right)\right)^{n}}{\sqrt{\sin\left(f x + e\right) + 1}}, x\right)"," ",0,"integral((d*sin(f*x + e))^n/sqrt(sin(f*x + e) + 1), x)","F",0
127,0,0,0,0.968093," ","integrate((d*sin(f*x+e))^n/(1+sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(d \sin\left(f x + e\right)\right)^{n} \sqrt{\sin\left(f x + e\right) + 1}}{\cos\left(f x + e\right)^{2} - 2 \, \sin\left(f x + e\right) - 2}, x\right)"," ",0,"integral(-(d*sin(f*x + e))^n*sqrt(sin(f*x + e) + 1)/(cos(f*x + e)^2 - 2*sin(f*x + e) - 2), x)","F",0
128,0,0,0,1.293209," ","integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{\frac{3}{2}} \left(d \sin\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^(3/2)*(d*sin(f*x + e))^n, x)","F",0
129,0,0,0,1.408181," ","integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{a \sin\left(f x + e\right) + a} \left(d \sin\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral(sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e))^n, x)","F",0
130,0,0,0,1.332601," ","integrate((d*sin(f*x+e))^n/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(d \sin\left(f x + e\right)\right)^{n}}{\sqrt{a \sin\left(f x + e\right) + a}}, x\right)"," ",0,"integral((d*sin(f*x + e))^n/sqrt(a*sin(f*x + e) + a), x)","F",0
131,0,0,0,1.289967," ","integrate((d*sin(f*x+e))^n/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{a \sin\left(f x + e\right) + a} \left(d \sin\left(f x + e\right)\right)^{n}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e))^n/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
132,0,0,0,1.206164," ","integrate(sin(f*x+e)^n*(1+sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(\sin\left(f x + e\right) + 1\right)}^{m} \sin\left(f x + e\right)^{n}, x\right)"," ",0,"integral((sin(f*x + e) + 1)^m*sin(f*x + e)^n, x)","F",0
133,0,0,0,1.245758," ","integrate((1-sin(f*x+e))^m*(-sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left(\left(-\sin\left(f x + e\right)\right)^{n} {\left(-\sin\left(f x + e\right) + 1\right)}^{m}, x\right)"," ",0,"integral((-sin(f*x + e))^n*(-sin(f*x + e) + 1)^m, x)","F",0
134,0,0,0,1.306605," ","integrate((d*sin(f*x+e))^n*(1+sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left(\left(d \sin\left(f x + e\right)\right)^{n} {\left(\sin\left(f x + e\right) + 1\right)}^{m}, x\right)"," ",0,"integral((d*sin(f*x + e))^n*(sin(f*x + e) + 1)^m, x)","F",0
135,0,0,0,1.605543," ","integrate((1-sin(f*x+e))^m*(d*sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left(\left(d \sin\left(f x + e\right)\right)^{n} {\left(-\sin\left(f x + e\right) + 1\right)}^{m}, x\right)"," ",0,"integral((d*sin(f*x + e))^n*(-sin(f*x + e) + 1)^m, x)","F",0
136,0,0,0,1.469859," ","integrate(sin(f*x+e)^n*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \sin\left(f x + e\right)^{n}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*sin(f*x + e)^n, x)","F",0
137,0,0,0,1.437900," ","integrate((-sin(f*x+e))^n*(a-a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(-a \sin\left(f x + e\right) + a\right)}^{m} \left(-\sin\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral((-a*sin(f*x + e) + a)^m*(-sin(f*x + e))^n, x)","F",0
138,0,0,0,1.467040," ","integrate((d*sin(f*x+e))^n*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \left(d \sin\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(d*sin(f*x + e))^n, x)","F",0
139,0,0,0,1.410652," ","integrate((d*sin(f*x+e))^n*(a-a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(-a \sin\left(f x + e\right) + a\right)}^{m} \left(d \sin\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral((-a*sin(f*x + e) + a)^m*(d*sin(f*x + e))^n, x)","F",0
140,0,0,0,1.477422," ","integrate(sin(d*x+c)^4*(a+a*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1\right)} {\left(a \sin\left(d x + c\right) + a\right)}^{n}, x\right)"," ",0,"integral((cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*(a*sin(d*x + c) + a)^n, x)","F",0
141,0,0,0,1.143969," ","integrate(sin(d*x+c)^3*(a+a*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(d x + c\right)^{2} - 1\right)} {\left(a \sin\left(d x + c\right) + a\right)}^{n} \sin\left(d x + c\right), x\right)"," ",0,"integral(-(cos(d*x + c)^2 - 1)*(a*sin(d*x + c) + a)^n*sin(d*x + c), x)","F",0
142,0,0,0,1.272251," ","integrate(sin(d*x+c)^2*(a+a*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(d x + c\right)^{2} - 1\right)} {\left(a \sin\left(d x + c\right) + a\right)}^{n}, x\right)"," ",0,"integral(-(cos(d*x + c)^2 - 1)*(a*sin(d*x + c) + a)^n, x)","F",0
143,0,0,0,1.477179," ","integrate(sin(d*x+c)*(a+a*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(d x + c\right) + a\right)}^{n} \sin\left(d x + c\right), x\right)"," ",0,"integral((a*sin(d*x + c) + a)^n*sin(d*x + c), x)","F",0
144,0,0,0,1.330325," ","integrate((a+a*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(d x + c\right) + a\right)}^{n}, x\right)"," ",0,"integral((a*sin(d*x + c) + a)^n, x)","F",0
145,0,0,0,1.426765," ","integrate(csc(d*x+c)*(a+a*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(d x + c\right) + a\right)}^{n} \csc\left(d x + c\right), x\right)"," ",0,"integral((a*sin(d*x + c) + a)^n*csc(d*x + c), x)","F",0
146,0,0,0,1.424917," ","integrate(csc(d*x+c)^2*(a+a*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(d x + c\right) + a\right)}^{n} \csc\left(d x + c\right)^{2}, x\right)"," ",0,"integral((a*sin(d*x + c) + a)^n*csc(d*x + c)^2, x)","F",0
147,0,0,0,1.398095," ","integrate((1+sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(\sin\left(d x + c\right) + 1\right)}^{n}, x\right)"," ",0,"integral((sin(d*x + c) + 1)^n, x)","F",0
148,0,0,0,1.348995," ","integrate((1-sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(-\sin\left(d x + c\right) + 1\right)}^{n}, x\right)"," ",0,"integral((-sin(d*x + c) + 1)^n, x)","F",0
149,1,60,0,1.338415," ","integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\frac{8 \, a \cos\left(f x + e\right)^{3} + 9 \, b f x - 24 \, a \cos\left(f x + e\right) + 3 \, {\left(2 \, b \cos\left(f x + e\right)^{3} - 5 \, b \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, f}"," ",0,"1/24*(8*a*cos(f*x + e)^3 + 9*b*f*x - 24*a*cos(f*x + e) + 3*(2*b*cos(f*x + e)^3 - 5*b*cos(f*x + e))*sin(f*x + e))/f","A",0
150,1,46,0,1.187543," ","integrate(sin(f*x+e)^2*(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\frac{2 \, b \cos\left(f x + e\right)^{3} + 3 \, a f x - 3 \, a \cos\left(f x + e\right) \sin\left(f x + e\right) - 6 \, b \cos\left(f x + e\right)}{6 \, f}"," ",0,"1/6*(2*b*cos(f*x + e)^3 + 3*a*f*x - 3*a*cos(f*x + e)*sin(f*x + e) - 6*b*cos(f*x + e))/f","A",0
151,1,34,0,1.130282," ","integrate(sin(f*x+e)*(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\frac{b f x - b \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a \cos\left(f x + e\right)}{2 \, f}"," ",0,"1/2*(b*f*x - b*cos(f*x + e)*sin(f*x + e) - 2*a*cos(f*x + e))/f","A",0
152,1,18,0,1.286705," ","integrate(a+b*sin(f*x+e),x, algorithm=""fricas"")","\frac{a f x - b \cos\left(f x + e\right)}{f}"," ",0,"(a*f*x - b*cos(f*x + e))/f","A",0
153,1,38,0,1.350607," ","integrate(csc(f*x+e)*(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\frac{2 \, b f x - a \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + a \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{2 \, f}"," ",0,"1/2*(2*b*f*x - a*log(1/2*cos(f*x + e) + 1/2) + a*log(-1/2*cos(f*x + e) + 1/2))/f","B",0
154,1,62,0,1.396037," ","integrate(csc(f*x+e)^2*(a+b*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{b \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) - b \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) + 2 \, a \cos\left(f x + e\right)}{2 \, f \sin\left(f x + e\right)}"," ",0,"-1/2*(b*log(1/2*cos(f*x + e) + 1/2)*sin(f*x + e) - b*log(-1/2*cos(f*x + e) + 1/2)*sin(f*x + e) + 2*a*cos(f*x + e))/(f*sin(f*x + e))","B",0
155,1,96,0,1.355128," ","integrate(csc(f*x+e)^3*(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\frac{4 \, b \cos\left(f x + e\right) \sin\left(f x + e\right) + 2 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right)^{2} - a\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left(a \cos\left(f x + e\right)^{2} - a\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}"," ",0,"1/4*(4*b*cos(f*x + e)*sin(f*x + e) + 2*a*cos(f*x + e) - (a*cos(f*x + e)^2 - a)*log(1/2*cos(f*x + e) + 1/2) + (a*cos(f*x + e)^2 - a)*log(-1/2*cos(f*x + e) + 1/2))/(f*cos(f*x + e)^2 - f)","B",0
156,1,128,0,1.472560," ","integrate(csc(f*x+e)^4*(a+b*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{8 \, a \cos\left(f x + e\right)^{3} - 6 \, b \cos\left(f x + e\right) \sin\left(f x + e\right) + 3 \, {\left(b \cos\left(f x + e\right)^{2} - b\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) - 3 \, {\left(b \cos\left(f x + e\right)^{2} - b\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) - 12 \, a \cos\left(f x + e\right)}{12 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)} \sin\left(f x + e\right)}"," ",0,"-1/12*(8*a*cos(f*x + e)^3 - 6*b*cos(f*x + e)*sin(f*x + e) + 3*(b*cos(f*x + e)^2 - b)*log(1/2*cos(f*x + e) + 1/2)*sin(f*x + e) - 3*(b*cos(f*x + e)^2 - b)*log(-1/2*cos(f*x + e) + 1/2)*sin(f*x + e) - 12*a*cos(f*x + e))/((f*cos(f*x + e)^2 - f)*sin(f*x + e))","B",0
157,1,90,0,1.433949," ","integrate(sin(f*x+e)^3*(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{12 \, b^{2} \cos\left(f x + e\right)^{5} - 45 \, a b f x - 20 \, {\left(a^{2} + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + 60 \, {\left(a^{2} + b^{2}\right)} \cos\left(f x + e\right) - 15 \, {\left(2 \, a b \cos\left(f x + e\right)^{3} - 5 \, a b \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{60 \, f}"," ",0,"-1/60*(12*b^2*cos(f*x + e)^5 - 45*a*b*f*x - 20*(a^2 + 2*b^2)*cos(f*x + e)^3 + 60*(a^2 + b^2)*cos(f*x + e) - 15*(2*a*b*cos(f*x + e)^3 - 5*a*b*cos(f*x + e))*sin(f*x + e))/f","A",0
158,1,84,0,1.735326," ","integrate(sin(f*x+e)^2*(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{16 \, a b \cos\left(f x + e\right)^{3} + 3 \, {\left(4 \, a^{2} + 3 \, b^{2}\right)} f x - 48 \, a b \cos\left(f x + e\right) + 3 \, {\left(2 \, b^{2} \cos\left(f x + e\right)^{3} - {\left(4 \, a^{2} + 5 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, f}"," ",0,"1/24*(16*a*b*cos(f*x + e)^3 + 3*(4*a^2 + 3*b^2)*f*x - 48*a*b*cos(f*x + e) + 3*(2*b^2*cos(f*x + e)^3 - (4*a^2 + 5*b^2)*cos(f*x + e))*sin(f*x + e))/f","A",0
159,1,55,0,1.179372," ","integrate(sin(f*x+e)*(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{b^{2} \cos\left(f x + e\right)^{3} + 3 \, a b f x - 3 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right) - 3 \, {\left(a^{2} + b^{2}\right)} \cos\left(f x + e\right)}{3 \, f}"," ",0,"1/3*(b^2*cos(f*x + e)^3 + 3*a*b*f*x - 3*a*b*cos(f*x + e)*sin(f*x + e) - 3*(a^2 + b^2)*cos(f*x + e))/f","A",0
160,1,45,0,1.206631," ","integrate((a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{b^{2} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(2 \, a^{2} + b^{2}\right)} f x + 4 \, a b \cos\left(f x + e\right)}{2 \, f}"," ",0,"-1/2*(b^2*cos(f*x + e)*sin(f*x + e) - (2*a^2 + b^2)*f*x + 4*a*b*cos(f*x + e))/f","A",0
161,1,54,0,1.070027," ","integrate(csc(f*x+e)*(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{4 \, a b f x - 2 \, b^{2} \cos\left(f x + e\right) - a^{2} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + a^{2} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{2 \, f}"," ",0,"1/2*(4*a*b*f*x - 2*b^2*cos(f*x + e) - a^2*log(1/2*cos(f*x + e) + 1/2) + a^2*log(-1/2*cos(f*x + e) + 1/2))/f","A",0
162,1,77,0,1.375913," ","integrate(csc(f*x+e)^2*(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{b^{2} f x \sin\left(f x + e\right) - a b \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) + a b \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) - a^{2} \cos\left(f x + e\right)}{f \sin\left(f x + e\right)}"," ",0,"(b^2*f*x*sin(f*x + e) - a*b*log(1/2*cos(f*x + e) + 1/2)*sin(f*x + e) + a*b*log(-1/2*cos(f*x + e) + 1/2)*sin(f*x + e) - a^2*cos(f*x + e))/(f*sin(f*x + e))","B",0
163,1,129,0,1.482793," ","integrate(csc(f*x+e)^3*(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{8 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right) + 2 \, a^{2} \cos\left(f x + e\right) - {\left({\left(a^{2} + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + {\left({\left(a^{2} + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}"," ",0,"1/4*(8*a*b*cos(f*x + e)*sin(f*x + e) + 2*a^2*cos(f*x + e) - ((a^2 + 2*b^2)*cos(f*x + e)^2 - a^2 - 2*b^2)*log(1/2*cos(f*x + e) + 1/2) + ((a^2 + 2*b^2)*cos(f*x + e)^2 - a^2 - 2*b^2)*log(-1/2*cos(f*x + e) + 1/2))/(f*cos(f*x + e)^2 - f)","B",0
164,1,149,0,1.345649," ","integrate(csc(f*x+e)^4*(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{2 \, {\left(2 \, a^{2} + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 6 \, a b \cos\left(f x + e\right) \sin\left(f x + e\right) + 3 \, {\left(a b \cos\left(f x + e\right)^{2} - a b\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) - 3 \, {\left(a b \cos\left(f x + e\right)^{2} - a b\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) - 6 \, {\left(a^{2} + b^{2}\right)} \cos\left(f x + e\right)}{6 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)} \sin\left(f x + e\right)}"," ",0,"-1/6*(2*(2*a^2 + 3*b^2)*cos(f*x + e)^3 - 6*a*b*cos(f*x + e)*sin(f*x + e) + 3*(a*b*cos(f*x + e)^2 - a*b)*log(1/2*cos(f*x + e) + 1/2)*sin(f*x + e) - 3*(a*b*cos(f*x + e)^2 - a*b)*log(-1/2*cos(f*x + e) + 1/2)*sin(f*x + e) - 6*(a^2 + b^2)*cos(f*x + e))/((f*cos(f*x + e)^2 - f)*sin(f*x + e))","A",0
165,1,229,0,1.377370," ","integrate(csc(f*x+e)^5*(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{6 \, {\left(3 \, a^{2} + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{3} - 6 \, {\left(5 \, a^{2} + 4 \, b^{2}\right)} \cos\left(f x + e\right) - 3 \, {\left({\left(3 \, a^{2} + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 4 \, b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(3 \, a^{2} + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 4 \, b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 32 \, {\left(2 \, a b \cos\left(f x + e\right)^{3} - 3 \, a b \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, {\left(f \cos\left(f x + e\right)^{4} - 2 \, f \cos\left(f x + e\right)^{2} + f\right)}}"," ",0,"1/48*(6*(3*a^2 + 4*b^2)*cos(f*x + e)^3 - 6*(5*a^2 + 4*b^2)*cos(f*x + e) - 3*((3*a^2 + 4*b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 4*b^2)*cos(f*x + e)^2 + 3*a^2 + 4*b^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((3*a^2 + 4*b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 4*b^2)*cos(f*x + e)^2 + 3*a^2 + 4*b^2)*log(-1/2*cos(f*x + e) + 1/2) + 32*(2*a*b*cos(f*x + e)^3 - 3*a*b*cos(f*x + e))*sin(f*x + e))/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)","B",0
166,1,138,0,1.228113," ","integrate(sin(f*x+e)^3*(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{144 \, a b^{2} \cos\left(f x + e\right)^{5} - 80 \, {\left(a^{3} + 6 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(18 \, a^{2} b + 5 \, b^{3}\right)} f x + 240 \, {\left(a^{3} + 3 \, a b^{2}\right)} \cos\left(f x + e\right) + 5 \, {\left(8 \, b^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(18 \, a^{2} b + 13 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(30 \, a^{2} b + 11 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{240 \, f}"," ",0,"-1/240*(144*a*b^2*cos(f*x + e)^5 - 80*(a^3 + 6*a*b^2)*cos(f*x + e)^3 - 15*(18*a^2*b + 5*b^3)*f*x + 240*(a^3 + 3*a*b^2)*cos(f*x + e) + 5*(8*b^3*cos(f*x + e)^5 - 2*(18*a^2*b + 13*b^3)*cos(f*x + e)^3 + 3*(30*a^2*b + 11*b^3)*cos(f*x + e))*sin(f*x + e))/f","A",0
167,1,118,0,1.384507," ","integrate(sin(f*x+e)^2*(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{24 \, b^{3} \cos\left(f x + e\right)^{5} - 40 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(4 \, a^{3} + 9 \, a b^{2}\right)} f x + 120 \, {\left(3 \, a^{2} b + b^{3}\right)} \cos\left(f x + e\right) - 15 \, {\left(6 \, a b^{2} \cos\left(f x + e\right)^{3} - {\left(4 \, a^{3} + 15 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{120 \, f}"," ",0,"-1/120*(24*b^3*cos(f*x + e)^5 - 40*(3*a^2*b + 2*b^3)*cos(f*x + e)^3 - 15*(4*a^3 + 9*a*b^2)*f*x + 120*(3*a^2*b + b^3)*cos(f*x + e) - 15*(6*a*b^2*cos(f*x + e)^3 - (4*a^3 + 15*a*b^2)*cos(f*x + e))*sin(f*x + e))/f","A",0
168,1,93,0,1.216177," ","integrate(sin(f*x+e)*(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{8 \, a b^{2} \cos\left(f x + e\right)^{3} + 3 \, {\left(4 \, a^{2} b + b^{3}\right)} f x - 8 \, {\left(a^{3} + 3 \, a b^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, b^{3} \cos\left(f x + e\right)^{3} - {\left(12 \, a^{2} b + 5 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, f}"," ",0,"1/8*(8*a*b^2*cos(f*x + e)^3 + 3*(4*a^2*b + b^3)*f*x - 8*(a^3 + 3*a*b^2)*cos(f*x + e) + (2*b^3*cos(f*x + e)^3 - (12*a^2*b + 5*b^3)*cos(f*x + e))*sin(f*x + e))/f","A",0
169,1,71,0,1.517434," ","integrate((a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{2 \, b^{3} \cos\left(f x + e\right)^{3} - 9 \, a b^{2} \cos\left(f x + e\right) \sin\left(f x + e\right) + 3 \, {\left(2 \, a^{3} + 3 \, a b^{2}\right)} f x - 6 \, {\left(3 \, a^{2} b + b^{3}\right)} \cos\left(f x + e\right)}{6 \, f}"," ",0,"1/6*(2*b^3*cos(f*x + e)^3 - 9*a*b^2*cos(f*x + e)*sin(f*x + e) + 3*(2*a^3 + 3*a*b^2)*f*x - 6*(3*a^2*b + b^3)*cos(f*x + e))/f","A",0
170,1,79,0,1.333084," ","integrate(csc(f*x+e)*(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{b^{3} \cos\left(f x + e\right) \sin\left(f x + e\right) + 6 \, a b^{2} \cos\left(f x + e\right) + a^{3} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - a^{3} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - {\left(6 \, a^{2} b + b^{3}\right)} f x}{2 \, f}"," ",0,"-1/2*(b^3*cos(f*x + e)*sin(f*x + e) + 6*a*b^2*cos(f*x + e) + a^3*log(1/2*cos(f*x + e) + 1/2) - a^3*log(-1/2*cos(f*x + e) + 1/2) - (6*a^2*b + b^3)*f*x)/f","A",0
171,1,99,0,1.286279," ","integrate(csc(f*x+e)^2*(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{3 \, a^{2} b \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) - 3 \, a^{2} b \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) + 2 \, a^{3} \cos\left(f x + e\right) - 2 \, {\left(3 \, a b^{2} f x - b^{3} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, f \sin\left(f x + e\right)}"," ",0,"-1/2*(3*a^2*b*log(1/2*cos(f*x + e) + 1/2)*sin(f*x + e) - 3*a^2*b*log(-1/2*cos(f*x + e) + 1/2)*sin(f*x + e) + 2*a^3*cos(f*x + e) - 2*(3*a*b^2*f*x - b^3*cos(f*x + e))*sin(f*x + e))/(f*sin(f*x + e))","A",0
172,1,155,0,1.291051," ","integrate(csc(f*x+e)^3*(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{4 \, b^{3} f x \cos\left(f x + e\right)^{2} - 4 \, b^{3} f x + 12 \, a^{2} b \cos\left(f x + e\right) \sin\left(f x + e\right) + 2 \, a^{3} \cos\left(f x + e\right) + {\left(a^{3} + 6 \, a b^{2} - {\left(a^{3} + 6 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) - {\left(a^{3} + 6 \, a b^{2} - {\left(a^{3} + 6 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right)}{4 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}"," ",0,"1/4*(4*b^3*f*x*cos(f*x + e)^2 - 4*b^3*f*x + 12*a^2*b*cos(f*x + e)*sin(f*x + e) + 2*a^3*cos(f*x + e) + (a^3 + 6*a*b^2 - (a^3 + 6*a*b^2)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) - (a^3 + 6*a*b^2 - (a^3 + 6*a*b^2)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2))/(f*cos(f*x + e)^2 - f)","B",0
173,1,191,0,1.041262," ","integrate(csc(f*x+e)^4*(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{18 \, a^{2} b \cos\left(f x + e\right) \sin\left(f x + e\right) - 4 \, {\left(2 \, a^{3} + 9 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, a^{2} b + 2 \, b^{3} - {\left(3 \, a^{2} b + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) - 3 \, {\left(3 \, a^{2} b + 2 \, b^{3} - {\left(3 \, a^{2} b + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) \sin\left(f x + e\right) + 12 \, {\left(a^{3} + 3 \, a b^{2}\right)} \cos\left(f x + e\right)}{12 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)} \sin\left(f x + e\right)}"," ",0,"1/12*(18*a^2*b*cos(f*x + e)*sin(f*x + e) - 4*(2*a^3 + 9*a*b^2)*cos(f*x + e)^3 + 3*(3*a^2*b + 2*b^3 - (3*a^2*b + 2*b^3)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2)*sin(f*x + e) - 3*(3*a^2*b + 2*b^3 - (3*a^2*b + 2*b^3)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2)*sin(f*x + e) + 12*(a^3 + 3*a*b^2)*cos(f*x + e))/((f*cos(f*x + e)^2 - f)*sin(f*x + e))","A",0
174,1,238,0,1.324295," ","integrate(csc(f*x+e)^5*(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{6 \, {\left(a^{3} + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, a^{3} + 12 \, a b^{2}\right)} \cos\left(f x + e\right) - 3 \, {\left({\left(a^{3} + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{3} + 4 \, a b^{2} - 2 \, {\left(a^{3} + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 3 \, {\left({\left(a^{3} + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + a^{3} + 4 \, a b^{2} - 2 \, {\left(a^{3} + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(f x + e\right) + \frac{1}{2}\right) + 16 \, {\left({\left(2 \, a^{2} b + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, a^{2} b + b^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{16 \, {\left(f \cos\left(f x + e\right)^{4} - 2 \, f \cos\left(f x + e\right)^{2} + f\right)}}"," ",0,"1/16*(6*(a^3 + 4*a*b^2)*cos(f*x + e)^3 - 2*(5*a^3 + 12*a*b^2)*cos(f*x + e) - 3*((a^3 + 4*a*b^2)*cos(f*x + e)^4 + a^3 + 4*a*b^2 - 2*(a^3 + 4*a*b^2)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^3 + 4*a*b^2)*cos(f*x + e)^4 + a^3 + 4*a*b^2 - 2*(a^3 + 4*a*b^2)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2) + 16*((2*a^2*b + b^3)*cos(f*x + e)^3 - (3*a^2*b + b^3)*cos(f*x + e))*sin(f*x + e))/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)","A",0
175,1,106,0,1.269586," ","integrate((a+b*sin(f*x+e))^4,x, algorithm=""fricas"")","\frac{32 \, a b^{3} \cos\left(f x + e\right)^{3} + 3 \, {\left(8 \, a^{4} + 24 \, a^{2} b^{2} + 3 \, b^{4}\right)} f x - 96 \, {\left(a^{3} b + a b^{3}\right)} \cos\left(f x + e\right) + 3 \, {\left(2 \, b^{4} \cos\left(f x + e\right)^{3} - {\left(24 \, a^{2} b^{2} + 5 \, b^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, f}"," ",0,"1/24*(32*a*b^3*cos(f*x + e)^3 + 3*(8*a^4 + 24*a^2*b^2 + 3*b^4)*f*x - 96*(a^3*b + a*b^3)*cos(f*x + e) + 3*(2*b^4*cos(f*x + e)^3 - (24*a^2*b^2 + 5*b^4)*cos(f*x + e))*sin(f*x + e))/f","A",0
176,1,333,0,1.143731," ","integrate(sin(x)^4/(a+b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{3 \, \sqrt{-a^{2} + b^{2}} a^{4} \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) - 2 \, {\left(a^{2} b^{3} - b^{5}\right)} \cos\left(x\right)^{3} - 3 \, {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right) \sin\left(x\right) + 3 \, {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} x + 6 \, {\left(a^{4} b - b^{5}\right)} \cos\left(x\right)}{6 \, {\left(a^{2} b^{4} - b^{6}\right)}}, -\frac{6 \, \sqrt{a^{2} - b^{2}} a^{4} \arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right) - 2 \, {\left(a^{2} b^{3} - b^{5}\right)} \cos\left(x\right)^{3} - 3 \, {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right) \sin\left(x\right) + 3 \, {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} x + 6 \, {\left(a^{4} b - b^{5}\right)} \cos\left(x\right)}{6 \, {\left(a^{2} b^{4} - b^{6}\right)}}\right]"," ",0,"[-1/6*(3*sqrt(-a^2 + b^2)*a^4*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)) - 2*(a^2*b^3 - b^5)*cos(x)^3 - 3*(a^3*b^2 - a*b^4)*cos(x)*sin(x) + 3*(2*a^5 - a^3*b^2 - a*b^4)*x + 6*(a^4*b - b^5)*cos(x))/(a^2*b^4 - b^6), -1/6*(6*sqrt(a^2 - b^2)*a^4*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - 2*(a^2*b^3 - b^5)*cos(x)^3 - 3*(a^3*b^2 - a*b^4)*cos(x)*sin(x) + 3*(2*a^5 - a^3*b^2 - a*b^4)*x + 6*(a^4*b - b^5)*cos(x))/(a^2*b^4 - b^6)]","A",0
177,1,291,0,1.280773," ","integrate(sin(x)^3/(a+b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} a^{3} \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} - b^{4}\right)} \cos\left(x\right) \sin\left(x\right) - {\left(2 \, a^{4} - a^{2} b^{2} - b^{4}\right)} x - 2 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(x\right)}{2 \, {\left(a^{2} b^{3} - b^{5}\right)}}, \frac{2 \, \sqrt{a^{2} - b^{2}} a^{3} \arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right) - {\left(a^{2} b^{2} - b^{4}\right)} \cos\left(x\right) \sin\left(x\right) + {\left(2 \, a^{4} - a^{2} b^{2} - b^{4}\right)} x + 2 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(x\right)}{2 \, {\left(a^{2} b^{3} - b^{5}\right)}}\right]"," ",0,"[-1/2*(sqrt(-a^2 + b^2)*a^3*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 - b^4)*cos(x)*sin(x) - (2*a^4 - a^2*b^2 - b^4)*x - 2*(a^3*b - a*b^3)*cos(x))/(a^2*b^3 - b^5), 1/2*(2*sqrt(a^2 - b^2)*a^3*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (a^2*b^2 - b^4)*cos(x)*sin(x) + (2*a^4 - a^2*b^2 - b^4)*x + 2*(a^3*b - a*b^3)*cos(x))/(a^2*b^3 - b^5)]","A",0
178,1,231,0,1.434438," ","integrate(sin(x)^2/(a+b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} a^{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) + 2 \, {\left(a^{3} - a b^{2}\right)} x + 2 \, {\left(a^{2} b - b^{3}\right)} \cos\left(x\right)}{2 \, {\left(a^{2} b^{2} - b^{4}\right)}}, -\frac{\sqrt{a^{2} - b^{2}} a^{2} \arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right) + {\left(a^{3} - a b^{2}\right)} x + {\left(a^{2} b - b^{3}\right)} \cos\left(x\right)}{a^{2} b^{2} - b^{4}}\right]"," ",0,"[-1/2*(sqrt(-a^2 + b^2)*a^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)) + 2*(a^3 - a*b^2)*x + 2*(a^2*b - b^3)*cos(x))/(a^2*b^2 - b^4), -(sqrt(a^2 - b^2)*a^2*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + (a^3 - a*b^2)*x + (a^2*b - b^3)*cos(x))/(a^2*b^2 - b^4)]","A",0
179,1,192,0,1.397978," ","integrate(sin(x)/(a+b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{\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) - 2 \, {\left(a^{2} - b^{2}\right)} x}{2 \, {\left(a^{2} b - b^{3}\right)}}, \frac{\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(a^{2} - b^{2}\right)} x}{a^{2} b - b^{3}}\right]"," ",0,"[-1/2*(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)) - 2*(a^2 - b^2)*x)/(a^2*b - b^3), (sqrt(a^2 - b^2)*a*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + (a^2 - b^2)*x)/(a^2*b - b^3)]","A",0
180,1,148,0,0.938840," ","integrate(1/(a+b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{\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)}{2 \, {\left(a^{2} - b^{2}\right)}}, -\frac{\arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right)}{\sqrt{a^{2} - b^{2}}}\right]"," ",0,"[-1/2*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))/(a^2 - b^2), -arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x)))/sqrt(a^2 - b^2)]","A",0
181,1,239,0,1.492001," ","integrate(csc(x)/(a+b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} 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(a^{2} - b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(a^{2} - 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 \, \sqrt{a^{2} - b^{2}} b \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(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(a^{2} - 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*(sqrt(-a^2 + b^2)*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)) + (a^2 - b^2)*log(1/2*cos(x) + 1/2) - (a^2 - b^2)*log(-1/2*cos(x) + 1/2))/(a^3 - a*b^2), 1/2*(2*sqrt(a^2 - b^2)*b*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (a^2 - b^2)*log(1/2*cos(x) + 1/2) + (a^2 - b^2)*log(-1/2*cos(x) + 1/2))/(a^3 - a*b^2)]","A",0
182,1,302,0,1.779779," ","integrate(csc(x)^2/(a+b*sin(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{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) \sin\left(x\right) - {\left(a^{2} b - b^{3}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) + {\left(a^{2} b - b^{3}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) + 2 \, {\left(a^{3} - a b^{2}\right)} \cos\left(x\right)}{2 \, {\left(a^{4} - a^{2} b^{2}\right)} \sin\left(x\right)}, -\frac{2 \, \sqrt{a^{2} - b^{2}} b^{2} \arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right) \sin\left(x\right) - {\left(a^{2} b - b^{3}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) + {\left(a^{2} b - b^{3}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) + 2 \, {\left(a^{3} - a b^{2}\right)} \cos\left(x\right)}{2 \, {\left(a^{4} - a^{2} b^{2}\right)} \sin\left(x\right)}\right]"," ",0,"[-1/2*(sqrt(-a^2 + b^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))*sin(x) - (a^2*b - b^3)*log(1/2*cos(x) + 1/2)*sin(x) + (a^2*b - b^3)*log(-1/2*cos(x) + 1/2)*sin(x) + 2*(a^3 - a*b^2)*cos(x))/((a^4 - a^2*b^2)*sin(x)), -1/2*(2*sqrt(a^2 - b^2)*b^2*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x)))*sin(x) - (a^2*b - b^3)*log(1/2*cos(x) + 1/2)*sin(x) + (a^2*b - b^3)*log(-1/2*cos(x) + 1/2)*sin(x) + 2*(a^3 - a*b^2)*cos(x))/((a^4 - a^2*b^2)*sin(x))]","B",0
183,1,490,0,2.209891," ","integrate(csc(x)^3/(a+b*sin(x)),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(x\right) \sin\left(x\right) + 2 \, {\left(b^{3} \cos\left(x\right)^{2} - b^{3}\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) - 2 \, {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(x\right) - {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4} - {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4} - {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{4 \, {\left(a^{5} - a^{3} b^{2} - {\left(a^{5} - a^{3} b^{2}\right)} \cos\left(x\right)^{2}\right)}}, \frac{4 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(x\right) \sin\left(x\right) - 4 \, {\left(b^{3} \cos\left(x\right)^{2} - b^{3}\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) - 2 \, {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(x\right) - {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4} - {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4} - {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{4 \, {\left(a^{5} - a^{3} b^{2} - {\left(a^{5} - a^{3} b^{2}\right)} \cos\left(x\right)^{2}\right)}}\right]"," ",0,"[1/4*(4*(a^3*b - a*b^3)*cos(x)*sin(x) + 2*(b^3*cos(x)^2 - b^3)*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)) - 2*(a^4 - a^2*b^2)*cos(x) - (a^4 + a^2*b^2 - 2*b^4 - (a^4 + a^2*b^2 - 2*b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2) + (a^4 + a^2*b^2 - 2*b^4 - (a^4 + a^2*b^2 - 2*b^4)*cos(x)^2)*log(-1/2*cos(x) + 1/2))/(a^5 - a^3*b^2 - (a^5 - a^3*b^2)*cos(x)^2), 1/4*(4*(a^3*b - a*b^3)*cos(x)*sin(x) - 4*(b^3*cos(x)^2 - b^3)*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - 2*(a^4 - a^2*b^2)*cos(x) - (a^4 + a^2*b^2 - 2*b^4 - (a^4 + a^2*b^2 - 2*b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2) + (a^4 + a^2*b^2 - 2*b^4 - (a^4 + a^2*b^2 - 2*b^4)*cos(x)^2)*log(-1/2*cos(x) + 1/2))/(a^5 - a^3*b^2 - (a^5 - a^3*b^2)*cos(x)^2)]","B",0
184,1,577,0,1.740962," ","integrate(csc(x)^4/(a+b*sin(x)),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(2 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4}\right)} \cos\left(x\right)^{3} + 6 \, {\left(b^{4} \cos\left(x\right)^{2} - b^{4}\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) \sin\left(x\right) + 6 \, {\left(a^{4} b - a^{2} b^{3}\right)} \cos\left(x\right) \sin\left(x\right) + 3 \, {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5} - {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) - 3 \, {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5} - {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) - 12 \, {\left(a^{5} - a b^{4}\right)} \cos\left(x\right)}{12 \, {\left(a^{6} - a^{4} b^{2} - {\left(a^{6} - a^{4} b^{2}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}, \frac{4 \, {\left(2 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4}\right)} \cos\left(x\right)^{3} + 12 \, {\left(b^{4} \cos\left(x\right)^{2} - b^{4}\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) \sin\left(x\right) + 6 \, {\left(a^{4} b - a^{2} b^{3}\right)} \cos\left(x\right) \sin\left(x\right) + 3 \, {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5} - {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) - 3 \, {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5} - {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) - 12 \, {\left(a^{5} - a b^{4}\right)} \cos\left(x\right)}{12 \, {\left(a^{6} - a^{4} b^{2} - {\left(a^{6} - a^{4} b^{2}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}\right]"," ",0,"[1/12*(4*(2*a^5 + a^3*b^2 - 3*a*b^4)*cos(x)^3 + 6*(b^4*cos(x)^2 - b^4)*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))*sin(x) + 6*(a^4*b - a^2*b^3)*cos(x)*sin(x) + 3*(a^4*b + a^2*b^3 - 2*b^5 - (a^4*b + a^2*b^3 - 2*b^5)*cos(x)^2)*log(1/2*cos(x) + 1/2)*sin(x) - 3*(a^4*b + a^2*b^3 - 2*b^5 - (a^4*b + a^2*b^3 - 2*b^5)*cos(x)^2)*log(-1/2*cos(x) + 1/2)*sin(x) - 12*(a^5 - a*b^4)*cos(x))/((a^6 - a^4*b^2 - (a^6 - a^4*b^2)*cos(x)^2)*sin(x)), 1/12*(4*(2*a^5 + a^3*b^2 - 3*a*b^4)*cos(x)^3 + 12*(b^4*cos(x)^2 - b^4)*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x)))*sin(x) + 6*(a^4*b - a^2*b^3)*cos(x)*sin(x) + 3*(a^4*b + a^2*b^3 - 2*b^5 - (a^4*b + a^2*b^3 - 2*b^5)*cos(x)^2)*log(1/2*cos(x) + 1/2)*sin(x) - 3*(a^4*b + a^2*b^3 - 2*b^5 - (a^4*b + a^2*b^3 - 2*b^5)*cos(x)^2)*log(-1/2*cos(x) + 1/2)*sin(x) - 12*(a^5 - a*b^4)*cos(x))/((a^6 - a^4*b^2 - (a^6 - a^4*b^2)*cos(x)^2)*sin(x))]","B",0
185,1,580,0,1.646889," ","integrate(sin(x)^4/(a+b*sin(x))^2,x, algorithm=""fricas"")","\left[\frac{{\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} \cos\left(x\right)^{3} - {\left(3 \, a^{6} - 4 \, a^{4} b^{2} + {\left(3 \, a^{5} b - 4 \, a^{3} b^{3}\right)} \sin\left(x\right)\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(6 \, a^{7} - 11 \, a^{5} b^{2} + 4 \, a^{3} b^{4} + a b^{6}\right)} x + {\left(6 \, a^{6} b - 11 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - b^{7}\right)} \cos\left(x\right) + {\left({\left(6 \, a^{6} b - 11 \, a^{4} b^{3} + 4 \, a^{2} b^{5} + b^{7}\right)} x + 3 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{2 \, {\left(a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8} + {\left(a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right)} \sin\left(x\right)\right)}}, \frac{{\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} \cos\left(x\right)^{3} + 2 \, {\left(3 \, a^{6} - 4 \, a^{4} b^{2} + {\left(3 \, a^{5} b - 4 \, a^{3} b^{3}\right)} \sin\left(x\right)\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(6 \, a^{7} - 11 \, a^{5} b^{2} + 4 \, a^{3} b^{4} + a b^{6}\right)} x + {\left(6 \, a^{6} b - 11 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - b^{7}\right)} \cos\left(x\right) + {\left({\left(6 \, a^{6} b - 11 \, a^{4} b^{3} + 4 \, a^{2} b^{5} + b^{7}\right)} x + 3 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{2 \, {\left(a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8} + {\left(a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[1/2*((a^4*b^3 - 2*a^2*b^5 + b^7)*cos(x)^3 - (3*a^6 - 4*a^4*b^2 + (3*a^5*b - 4*a^3*b^3)*sin(x))*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)) + (6*a^7 - 11*a^5*b^2 + 4*a^3*b^4 + a*b^6)*x + (6*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 - b^7)*cos(x) + ((6*a^6*b - 11*a^4*b^3 + 4*a^2*b^5 + b^7)*x + 3*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(x))*sin(x))/(a^5*b^4 - 2*a^3*b^6 + a*b^8 + (a^4*b^5 - 2*a^2*b^7 + b^9)*sin(x)), 1/2*((a^4*b^3 - 2*a^2*b^5 + b^7)*cos(x)^3 + 2*(3*a^6 - 4*a^4*b^2 + (3*a^5*b - 4*a^3*b^3)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + (6*a^7 - 11*a^5*b^2 + 4*a^3*b^4 + a*b^6)*x + (6*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 - b^7)*cos(x) + ((6*a^6*b - 11*a^4*b^3 + 4*a^2*b^5 + b^7)*x + 3*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(x))*sin(x))/(a^5*b^4 - 2*a^3*b^6 + a*b^8 + (a^4*b^5 - 2*a^2*b^7 + b^9)*sin(x))]","A",0
186,1,483,0,0.847394," ","integrate(sin(x)^3/(a+b*sin(x))^2,x, algorithm=""fricas"")","\left[-\frac{{\left(2 \, a^{5} - 3 \, a^{3} b^{2} + {\left(2 \, a^{4} b - 3 \, a^{2} b^{3}\right)} \sin\left(x\right)\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) + 4 \, {\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} x + 2 \, {\left(2 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right)} \cos\left(x\right) + 2 \, {\left(2 \, {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} x + {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{2 \, {\left(a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7} + {\left(a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right)} \sin\left(x\right)\right)}}, -\frac{{\left(2 \, a^{5} - 3 \, a^{3} b^{2} + {\left(2 \, a^{4} b - 3 \, a^{2} b^{3}\right)} \sin\left(x\right)\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) + 2 \, {\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} x + {\left(2 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right)} \cos\left(x\right) + {\left(2 \, {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} x + {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7} + {\left(a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right)} \sin\left(x\right)}\right]"," ",0,"[-1/2*((2*a^5 - 3*a^3*b^2 + (2*a^4*b - 3*a^2*b^3)*sin(x))*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)) + 4*(a^6 - 2*a^4*b^2 + a^2*b^4)*x + 2*(2*a^5*b - 3*a^3*b^3 + a*b^5)*cos(x) + 2*(2*(a^5*b - 2*a^3*b^3 + a*b^5)*x + (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(x))*sin(x))/(a^5*b^3 - 2*a^3*b^5 + a*b^7 + (a^4*b^4 - 2*a^2*b^6 + b^8)*sin(x)), -((2*a^5 - 3*a^3*b^2 + (2*a^4*b - 3*a^2*b^3)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + 2*(a^6 - 2*a^4*b^2 + a^2*b^4)*x + (2*a^5*b - 3*a^3*b^3 + a*b^5)*cos(x) + (2*(a^5*b - 2*a^3*b^3 + a*b^5)*x + (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(x))*sin(x))/(a^5*b^3 - 2*a^3*b^5 + a*b^7 + (a^4*b^4 - 2*a^2*b^6 + b^8)*sin(x))]","A",0
187,1,403,0,0.712706," ","integrate(sin(x)^2/(a+b*sin(x))^2,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} x \sin\left(x\right) - {\left(a^{4} - 2 \, a^{2} b^{2} + {\left(a^{3} b - 2 \, a b^{3}\right)} \sin\left(x\right)\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) + 2 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} x + 2 \, {\left(a^{4} b - a^{2} b^{3}\right)} \cos\left(x\right)}{2 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6} + {\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} \sin\left(x\right)\right)}}, \frac{{\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} x \sin\left(x\right) + {\left(a^{4} - 2 \, a^{2} b^{2} + {\left(a^{3} b - 2 \, a b^{3}\right)} \sin\left(x\right)\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(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} x + {\left(a^{4} b - a^{2} b^{3}\right)} \cos\left(x\right)}{a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6} + {\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} \sin\left(x\right)}\right]"," ",0,"[1/2*(2*(a^4*b - 2*a^2*b^3 + b^5)*x*sin(x) - (a^4 - 2*a^2*b^2 + (a^3*b - 2*a*b^3)*sin(x))*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)) + 2*(a^5 - 2*a^3*b^2 + a*b^4)*x + 2*(a^4*b - a^2*b^3)*cos(x))/(a^5*b^2 - 2*a^3*b^4 + a*b^6 + (a^4*b^3 - 2*a^2*b^5 + b^7)*sin(x)), ((a^4*b - 2*a^2*b^3 + b^5)*x*sin(x) + (a^4 - 2*a^2*b^2 + (a^3*b - 2*a*b^3)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + (a^5 - 2*a^3*b^2 + a*b^4)*x + (a^4*b - a^2*b^3)*cos(x))/(a^5*b^2 - 2*a^3*b^4 + a*b^6 + (a^4*b^3 - 2*a^2*b^5 + b^7)*sin(x))]","B",0
188,1,266,0,1.293515," ","integrate(sin(x)/(a+b*sin(x))^2,x, algorithm=""fricas"")","\left[\frac{{\left(b^{2} \sin\left(x\right) + a 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) - 2 \, {\left(a^{3} - a b^{2}\right)} \cos\left(x\right)}{2 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} \sin\left(x\right)\right)}}, \frac{{\left(b^{2} \sin\left(x\right) + a 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(a^{3} - a b^{2}\right)} \cos\left(x\right)}{a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} \sin\left(x\right)}\right]"," ",0,"[1/2*((b^2*sin(x) + a*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)) - 2*(a^3 - a*b^2)*cos(x))/(a^5 - 2*a^3*b^2 + a*b^4 + (a^4*b - 2*a^2*b^3 + b^5)*sin(x)), ((b^2*sin(x) + a*b)*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (a^3 - a*b^2)*cos(x))/(a^5 - 2*a^3*b^2 + a*b^4 + (a^4*b - 2*a^2*b^3 + b^5)*sin(x))]","A",0
189,1,268,0,1.015961," ","integrate(1/(a+b*sin(x))^2,x, algorithm=""fricas"")","\left[\frac{{\left(a b \sin\left(x\right) + a^{2}\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) + 2 \, {\left(a^{2} b - b^{3}\right)} \cos\left(x\right)}{2 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} \sin\left(x\right)\right)}}, -\frac{{\left(a b \sin\left(x\right) + a^{2}\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(a^{2} b - b^{3}\right)} \cos\left(x\right)}{a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} \sin\left(x\right)}\right]"," ",0,"[1/2*((a*b*sin(x) + a^2)*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)) + 2*(a^2*b - b^3)*cos(x))/(a^5 - 2*a^3*b^2 + a*b^4 + (a^4*b - 2*a^2*b^3 + b^5)*sin(x)), -((a*b*sin(x) + a^2)*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (a^2*b - b^3)*cos(x))/(a^5 - 2*a^3*b^2 + a*b^4 + (a^4*b - 2*a^2*b^3 + b^5)*sin(x))]","A",0
190,1,511,0,1.762687," ","integrate(csc(x)/(a+b*sin(x))^2,x, algorithm=""fricas"")","\left[-\frac{{\left(2 \, a^{3} b - a b^{3} + {\left(2 \, a^{2} b^{2} - b^{4}\right)} \sin\left(x\right)\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) + 2 \, {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right) + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4} + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \sin\left(x\right)\right)}}, \frac{2 \, {\left(2 \, a^{3} b - a b^{3} + {\left(2 \, a^{2} b^{2} - b^{4}\right)} \sin\left(x\right)\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) - 2 \, {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right) - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4} + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[-1/2*((2*a^3*b - a*b^3 + (2*a^2*b^2 - b^4)*sin(x))*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)) + 2*(a^3*b^2 - a*b^4)*cos(x) + (a^5 - 2*a^3*b^2 + a*b^4 + (a^4*b - 2*a^2*b^3 + b^5)*sin(x))*log(1/2*cos(x) + 1/2) - (a^5 - 2*a^3*b^2 + a*b^4 + (a^4*b - 2*a^2*b^3 + b^5)*sin(x))*log(-1/2*cos(x) + 1/2))/(a^7 - 2*a^5*b^2 + a^3*b^4 + (a^6*b - 2*a^4*b^3 + a^2*b^5)*sin(x)), 1/2*(2*(2*a^3*b - a*b^3 + (2*a^2*b^2 - b^4)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - 2*(a^3*b^2 - a*b^4)*cos(x) - (a^5 - 2*a^3*b^2 + a*b^4 + (a^4*b - 2*a^2*b^3 + b^5)*sin(x))*log(1/2*cos(x) + 1/2) + (a^5 - 2*a^3*b^2 + a*b^4 + (a^4*b - 2*a^2*b^3 + b^5)*sin(x))*log(-1/2*cos(x) + 1/2))/(a^7 - 2*a^5*b^2 + a^3*b^4 + (a^6*b - 2*a^4*b^3 + a^2*b^5)*sin(x))]","B",0
191,1,784,0,1.913743," ","integrate(csc(x)^2/(a+b*sin(x))^2,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right)} \cos\left(x\right) \sin\left(x\right) + {\left(3 \, a^{2} b^{3} - 2 \, b^{5} - {\left(3 \, a^{2} b^{3} - 2 \, b^{5}\right)} \cos\left(x\right)^{2} + {\left(3 \, a^{3} b^{2} - 2 \, a b^{4}\right)} \sin\left(x\right)\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) + 2 \, {\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} \cos\left(x\right) - 2 \, {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6} - {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} \cos\left(x\right)^{2} + {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 2 \, {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6} - {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} \cos\left(x\right)^{2} + {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5} - {\left(a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right)} \cos\left(x\right)^{2} + {\left(a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right)} \sin\left(x\right)\right)}}, -\frac{{\left(a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right)} \cos\left(x\right) \sin\left(x\right) + {\left(3 \, a^{2} b^{3} - 2 \, b^{5} - {\left(3 \, a^{2} b^{3} - 2 \, b^{5}\right)} \cos\left(x\right)^{2} + {\left(3 \, a^{3} b^{2} - 2 \, a b^{4}\right)} \sin\left(x\right)\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(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} \cos\left(x\right) - {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6} - {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} \cos\left(x\right)^{2} + {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6} - {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} \cos\left(x\right)^{2} + {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5} - {\left(a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right)} \cos\left(x\right)^{2} + {\left(a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right)} \sin\left(x\right)}\right]"," ",0,"[-1/2*(2*(a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(x)*sin(x) + (3*a^2*b^3 - 2*b^5 - (3*a^2*b^3 - 2*b^5)*cos(x)^2 + (3*a^3*b^2 - 2*a*b^4)*sin(x))*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)) + 2*(a^6 - 2*a^4*b^2 + a^2*b^4)*cos(x) - 2*(a^4*b^2 - 2*a^2*b^4 + b^6 - (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(x)^2 + (a^5*b - 2*a^3*b^3 + a*b^5)*sin(x))*log(1/2*cos(x) + 1/2) + 2*(a^4*b^2 - 2*a^2*b^4 + b^6 - (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(x)^2 + (a^5*b - 2*a^3*b^3 + a*b^5)*sin(x))*log(-1/2*cos(x) + 1/2))/(a^7*b - 2*a^5*b^3 + a^3*b^5 - (a^7*b - 2*a^5*b^3 + a^3*b^5)*cos(x)^2 + (a^8 - 2*a^6*b^2 + a^4*b^4)*sin(x)), -((a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(x)*sin(x) + (3*a^2*b^3 - 2*b^5 - (3*a^2*b^3 - 2*b^5)*cos(x)^2 + (3*a^3*b^2 - 2*a*b^4)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + (a^6 - 2*a^4*b^2 + a^2*b^4)*cos(x) - (a^4*b^2 - 2*a^2*b^4 + b^6 - (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(x)^2 + (a^5*b - 2*a^3*b^3 + a*b^5)*sin(x))*log(1/2*cos(x) + 1/2) + (a^4*b^2 - 2*a^2*b^4 + b^6 - (a^4*b^2 - 2*a^2*b^4 + b^6)*cos(x)^2 + (a^5*b - 2*a^3*b^3 + a*b^5)*sin(x))*log(-1/2*cos(x) + 1/2))/(a^7*b - 2*a^5*b^3 + a^3*b^5 - (a^7*b - 2*a^5*b^3 + a^3*b^5)*cos(x)^2 + (a^8 - 2*a^6*b^2 + a^4*b^4)*sin(x))]","B",0
192,1,1174,0,2.682258," ","integrate(csc(x)^3/(a+b*sin(x))^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(2 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 3 \, a b^{6}\right)} \cos\left(x\right)^{3} - 6 \, {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(x\right) \sin\left(x\right) + 2 \, {\left(4 \, a^{3} b^{3} - 3 \, a b^{5} - {\left(4 \, a^{3} b^{3} - 3 \, a b^{5}\right)} \cos\left(x\right)^{2} + {\left(4 \, a^{2} b^{4} - 3 \, b^{6} - {\left(4 \, a^{2} b^{4} - 3 \, b^{6}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\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) + 2 \, {\left(a^{7} - 6 \, a^{5} b^{2} + 11 \, a^{3} b^{4} - 6 \, a b^{6}\right)} \cos\left(x\right) + {\left(a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6} - {\left(a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right)} \cos\left(x\right)^{2} + {\left(a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7} - {\left(a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6} - {\left(a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right)} \cos\left(x\right)^{2} + {\left(a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7} - {\left(a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{4 \, {\left(a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4} - {\left(a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right)} \cos\left(x\right)^{2} + {\left(a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5} - {\left(a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)}}, -\frac{4 \, {\left(2 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 3 \, a b^{6}\right)} \cos\left(x\right)^{3} - 6 \, {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(x\right) \sin\left(x\right) - 4 \, {\left(4 \, a^{3} b^{3} - 3 \, a b^{5} - {\left(4 \, a^{3} b^{3} - 3 \, a b^{5}\right)} \cos\left(x\right)^{2} + {\left(4 \, a^{2} b^{4} - 3 \, b^{6} - {\left(4 \, a^{2} b^{4} - 3 \, b^{6}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\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) + 2 \, {\left(a^{7} - 6 \, a^{5} b^{2} + 11 \, a^{3} b^{4} - 6 \, a b^{6}\right)} \cos\left(x\right) + {\left(a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6} - {\left(a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right)} \cos\left(x\right)^{2} + {\left(a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7} - {\left(a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6} - {\left(a^{7} + 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right)} \cos\left(x\right)^{2} + {\left(a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7} - {\left(a^{6} b + 4 \, a^{4} b^{3} - 11 \, a^{2} b^{5} + 6 \, b^{7}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{4 \, {\left(a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4} - {\left(a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right)} \cos\left(x\right)^{2} + {\left(a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5} - {\left(a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[-1/4*(4*(2*a^5*b^2 - 5*a^3*b^4 + 3*a*b^6)*cos(x)^3 - 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(x)*sin(x) + 2*(4*a^3*b^3 - 3*a*b^5 - (4*a^3*b^3 - 3*a*b^5)*cos(x)^2 + (4*a^2*b^4 - 3*b^6 - (4*a^2*b^4 - 3*b^6)*cos(x)^2)*sin(x))*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)) + 2*(a^7 - 6*a^5*b^2 + 11*a^3*b^4 - 6*a*b^6)*cos(x) + (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6 - (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6)*cos(x)^2 + (a^6*b + 4*a^4*b^3 - 11*a^2*b^5 + 6*b^7 - (a^6*b + 4*a^4*b^3 - 11*a^2*b^5 + 6*b^7)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) - (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6 - (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6)*cos(x)^2 + (a^6*b + 4*a^4*b^3 - 11*a^2*b^5 + 6*b^7 - (a^6*b + 4*a^4*b^3 - 11*a^2*b^5 + 6*b^7)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2))/(a^9 - 2*a^7*b^2 + a^5*b^4 - (a^9 - 2*a^7*b^2 + a^5*b^4)*cos(x)^2 + (a^8*b - 2*a^6*b^3 + a^4*b^5 - (a^8*b - 2*a^6*b^3 + a^4*b^5)*cos(x)^2)*sin(x)), -1/4*(4*(2*a^5*b^2 - 5*a^3*b^4 + 3*a*b^6)*cos(x)^3 - 6*(a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(x)*sin(x) - 4*(4*a^3*b^3 - 3*a*b^5 - (4*a^3*b^3 - 3*a*b^5)*cos(x)^2 + (4*a^2*b^4 - 3*b^6 - (4*a^2*b^4 - 3*b^6)*cos(x)^2)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + 2*(a^7 - 6*a^5*b^2 + 11*a^3*b^4 - 6*a*b^6)*cos(x) + (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6 - (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6)*cos(x)^2 + (a^6*b + 4*a^4*b^3 - 11*a^2*b^5 + 6*b^7 - (a^6*b + 4*a^4*b^3 - 11*a^2*b^5 + 6*b^7)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) - (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6 - (a^7 + 4*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6)*cos(x)^2 + (a^6*b + 4*a^4*b^3 - 11*a^2*b^5 + 6*b^7 - (a^6*b + 4*a^4*b^3 - 11*a^2*b^5 + 6*b^7)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2))/(a^9 - 2*a^7*b^2 + a^5*b^4 - (a^9 - 2*a^7*b^2 + a^5*b^4)*cos(x)^2 + (a^8*b - 2*a^6*b^3 + a^4*b^5 - (a^8*b - 2*a^6*b^3 + a^4*b^5)*cos(x)^2)*sin(x))]","B",0
193,1,1090,0,1.551326," ","integrate(sin(x)^5/(a+b*sin(x))^3,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(12 \, a^{8} b^{2} - 35 \, a^{6} b^{4} + 33 \, a^{4} b^{6} - 9 \, a^{2} b^{8} - b^{10}\right)} x \cos\left(x\right)^{2} + 8 \, {\left(a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right)} \cos\left(x\right)^{3} + {\left(12 \, a^{9} - 17 \, a^{7} b^{2} - 9 \, a^{5} b^{4} + 20 \, a^{3} b^{6} - {\left(12 \, a^{7} b^{2} - 29 \, a^{5} b^{4} + 20 \, a^{3} b^{6}\right)} \cos\left(x\right)^{2} + 2 \, {\left(12 \, a^{8} b - 29 \, a^{6} b^{3} + 20 \, a^{4} b^{5}\right)} \sin\left(x\right)\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) - 2 \, {\left(12 \, a^{10} - 23 \, a^{8} b^{2} - 2 \, a^{6} b^{4} + 24 \, a^{4} b^{6} - 10 \, a^{2} b^{8} - b^{10}\right)} x - 2 \, {\left(12 \, a^{9} b - 29 \, a^{7} b^{3} + 15 \, a^{5} b^{5} + 6 \, a^{3} b^{7} - 4 \, a b^{9}\right)} \cos\left(x\right) - 2 \, {\left({\left(a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right)} \cos\left(x\right)^{3} + 2 \, {\left(12 \, a^{9} b - 35 \, a^{7} b^{3} + 33 \, a^{5} b^{5} - 9 \, a^{3} b^{7} - a b^{9}\right)} x + {\left(18 \, a^{8} b^{2} - 51 \, a^{6} b^{4} + 46 \, a^{4} b^{6} - 14 \, a^{2} b^{8} + b^{10}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{4 \, {\left(a^{8} b^{5} - 2 \, a^{6} b^{7} + 2 \, a^{2} b^{11} - b^{13} - {\left(a^{6} b^{7} - 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} - b^{13}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b^{6} - 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} - a b^{12}\right)} \sin\left(x\right)\right)}}, -\frac{{\left(12 \, a^{8} b^{2} - 35 \, a^{6} b^{4} + 33 \, a^{4} b^{6} - 9 \, a^{2} b^{8} - b^{10}\right)} x \cos\left(x\right)^{2} + 4 \, {\left(a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right)} \cos\left(x\right)^{3} - {\left(12 \, a^{9} - 17 \, a^{7} b^{2} - 9 \, a^{5} b^{4} + 20 \, a^{3} b^{6} - {\left(12 \, a^{7} b^{2} - 29 \, a^{5} b^{4} + 20 \, a^{3} b^{6}\right)} \cos\left(x\right)^{2} + 2 \, {\left(12 \, a^{8} b - 29 \, a^{6} b^{3} + 20 \, a^{4} b^{5}\right)} \sin\left(x\right)\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(12 \, a^{10} - 23 \, a^{8} b^{2} - 2 \, a^{6} b^{4} + 24 \, a^{4} b^{6} - 10 \, a^{2} b^{8} - b^{10}\right)} x - {\left(12 \, a^{9} b - 29 \, a^{7} b^{3} + 15 \, a^{5} b^{5} + 6 \, a^{3} b^{7} - 4 \, a b^{9}\right)} \cos\left(x\right) - {\left({\left(a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right)} \cos\left(x\right)^{3} + 2 \, {\left(12 \, a^{9} b - 35 \, a^{7} b^{3} + 33 \, a^{5} b^{5} - 9 \, a^{3} b^{7} - a b^{9}\right)} x + {\left(18 \, a^{8} b^{2} - 51 \, a^{6} b^{4} + 46 \, a^{4} b^{6} - 14 \, a^{2} b^{8} + b^{10}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{2 \, {\left(a^{8} b^{5} - 2 \, a^{6} b^{7} + 2 \, a^{2} b^{11} - b^{13} - {\left(a^{6} b^{7} - 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} - b^{13}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b^{6} - 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} - a b^{12}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[-1/4*(2*(12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b^8 - b^10)*x*cos(x)^2 + 8*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*cos(x)^3 + (12*a^9 - 17*a^7*b^2 - 9*a^5*b^4 + 20*a^3*b^6 - (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*cos(x)^2 + 2*(12*a^8*b - 29*a^6*b^3 + 20*a^4*b^5)*sin(x))*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)) - 2*(12*a^10 - 23*a^8*b^2 - 2*a^6*b^4 + 24*a^4*b^6 - 10*a^2*b^8 - b^10)*x - 2*(12*a^9*b - 29*a^7*b^3 + 15*a^5*b^5 + 6*a^3*b^7 - 4*a*b^9)*cos(x) - 2*((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cos(x)^3 + 2*(12*a^9*b - 35*a^7*b^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*x + (18*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8 + b^10)*cos(x))*sin(x))/(a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^13 - (a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*cos(x)^2 + 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*sin(x)), -1/2*((12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b^8 - b^10)*x*cos(x)^2 + 4*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*cos(x)^3 - (12*a^9 - 17*a^7*b^2 - 9*a^5*b^4 + 20*a^3*b^6 - (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*cos(x)^2 + 2*(12*a^8*b - 29*a^6*b^3 + 20*a^4*b^5)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (12*a^10 - 23*a^8*b^2 - 2*a^6*b^4 + 24*a^4*b^6 - 10*a^2*b^8 - b^10)*x - (12*a^9*b - 29*a^7*b^3 + 15*a^5*b^5 + 6*a^3*b^7 - 4*a*b^9)*cos(x) - ((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*cos(x)^3 + 2*(12*a^9*b - 35*a^7*b^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*x + (18*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8 + b^10)*cos(x))*sin(x))/(a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^13 - (a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*cos(x)^2 + 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*sin(x))]","B",0
194,1,945,0,1.371251," ","integrate(sin(x)^4/(a+b*sin(x))^3,x, algorithm=""fricas"")","\left[\frac{12 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} x \cos\left(x\right)^{2} + 4 \, {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} \cos\left(x\right)^{3} - 3 \, {\left(2 \, a^{8} - 3 \, a^{6} b^{2} - a^{4} b^{4} + 4 \, a^{2} b^{6} - {\left(2 \, a^{6} b^{2} - 5 \, a^{4} b^{4} + 4 \, a^{2} b^{6}\right)} \cos\left(x\right)^{2} + 2 \, {\left(2 \, a^{7} b - 5 \, a^{5} b^{3} + 4 \, a^{3} b^{5}\right)} \sin\left(x\right)\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) - 12 \, {\left(a^{9} - 2 \, a^{7} b^{2} + 2 \, a^{3} b^{6} - a b^{8}\right)} x - 2 \, {\left(6 \, a^{8} b - 15 \, a^{6} b^{3} + 7 \, a^{4} b^{5} + 4 \, a^{2} b^{7} - 2 \, b^{9}\right)} \cos\left(x\right) - 2 \, {\left(12 \, {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} x + {\left(9 \, a^{7} b^{2} - 25 \, a^{5} b^{4} + 20 \, a^{3} b^{6} - 4 \, a b^{8}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{4 \, {\left(a^{8} b^{4} - 2 \, a^{6} b^{6} + 2 \, a^{2} b^{10} - b^{12} - {\left(a^{6} b^{6} - 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} - b^{12}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b^{5} - 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} - a b^{11}\right)} \sin\left(x\right)\right)}}, \frac{6 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} x \cos\left(x\right)^{2} + 2 \, {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} \cos\left(x\right)^{3} - 3 \, {\left(2 \, a^{8} - 3 \, a^{6} b^{2} - a^{4} b^{4} + 4 \, a^{2} b^{6} - {\left(2 \, a^{6} b^{2} - 5 \, a^{4} b^{4} + 4 \, a^{2} b^{6}\right)} \cos\left(x\right)^{2} + 2 \, {\left(2 \, a^{7} b - 5 \, a^{5} b^{3} + 4 \, a^{3} b^{5}\right)} \sin\left(x\right)\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) - 6 \, {\left(a^{9} - 2 \, a^{7} b^{2} + 2 \, a^{3} b^{6} - a b^{8}\right)} x - {\left(6 \, a^{8} b - 15 \, a^{6} b^{3} + 7 \, a^{4} b^{5} + 4 \, a^{2} b^{7} - 2 \, b^{9}\right)} \cos\left(x\right) - {\left(12 \, {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} x + {\left(9 \, a^{7} b^{2} - 25 \, a^{5} b^{4} + 20 \, a^{3} b^{6} - 4 \, a b^{8}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{2 \, {\left(a^{8} b^{4} - 2 \, a^{6} b^{6} + 2 \, a^{2} b^{10} - b^{12} - {\left(a^{6} b^{6} - 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} - b^{12}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b^{5} - 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} - a b^{11}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[1/4*(12*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*x*cos(x)^2 + 4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*cos(x)^3 - 3*(2*a^8 - 3*a^6*b^2 - a^4*b^4 + 4*a^2*b^6 - (2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b^6)*cos(x)^2 + 2*(2*a^7*b - 5*a^5*b^3 + 4*a^3*b^5)*sin(x))*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)) - 12*(a^9 - 2*a^7*b^2 + 2*a^3*b^6 - a*b^8)*x - 2*(6*a^8*b - 15*a^6*b^3 + 7*a^4*b^5 + 4*a^2*b^7 - 2*b^9)*cos(x) - 2*(12*(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*x + (9*a^7*b^2 - 25*a^5*b^4 + 20*a^3*b^6 - 4*a*b^8)*cos(x))*sin(x))/(a^8*b^4 - 2*a^6*b^6 + 2*a^2*b^10 - b^12 - (a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*cos(x)^2 + 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*sin(x)), 1/2*(6*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*x*cos(x)^2 + 2*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*cos(x)^3 - 3*(2*a^8 - 3*a^6*b^2 - a^4*b^4 + 4*a^2*b^6 - (2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b^6)*cos(x)^2 + 2*(2*a^7*b - 5*a^5*b^3 + 4*a^3*b^5)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - 6*(a^9 - 2*a^7*b^2 + 2*a^3*b^6 - a*b^8)*x - (6*a^8*b - 15*a^6*b^3 + 7*a^4*b^5 + 4*a^2*b^7 - 2*b^9)*cos(x) - (12*(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*x + (9*a^7*b^2 - 25*a^5*b^4 + 20*a^3*b^6 - 4*a*b^8)*cos(x))*sin(x))/(a^8*b^4 - 2*a^6*b^6 + 2*a^2*b^10 - b^12 - (a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*cos(x)^2 + 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*sin(x))]","B",0
195,1,819,0,1.097664," ","integrate(sin(x)^3/(a+b*sin(x))^3,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} x \cos\left(x\right)^{2} + {\left(2 \, a^{7} - 3 \, a^{5} b^{2} + a^{3} b^{4} + 6 \, a b^{6} - {\left(2 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 6 \, a b^{6}\right)} \cos\left(x\right)^{2} + 2 \, {\left(2 \, a^{6} b - 5 \, a^{4} b^{3} + 6 \, a^{2} b^{5}\right)} \sin\left(x\right)\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) - 4 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} x - 2 \, {\left(2 \, a^{7} b - 7 \, a^{5} b^{3} + 5 \, a^{3} b^{5}\right)} \cos\left(x\right) - 2 \, {\left(4 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} x + 3 \, {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 2 \, a^{2} b^{6}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{4 \, {\left(a^{8} b^{3} - 2 \, a^{6} b^{5} + 2 \, a^{2} b^{9} - b^{11} - {\left(a^{6} b^{5} - 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} - b^{11}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b^{4} - 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} - a b^{10}\right)} \sin\left(x\right)\right)}}, -\frac{2 \, {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} x \cos\left(x\right)^{2} - {\left(2 \, a^{7} - 3 \, a^{5} b^{2} + a^{3} b^{4} + 6 \, a b^{6} - {\left(2 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 6 \, a b^{6}\right)} \cos\left(x\right)^{2} + 2 \, {\left(2 \, a^{6} b - 5 \, a^{4} b^{3} + 6 \, a^{2} b^{5}\right)} \sin\left(x\right)\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) - 2 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} x - {\left(2 \, a^{7} b - 7 \, a^{5} b^{3} + 5 \, a^{3} b^{5}\right)} \cos\left(x\right) - {\left(4 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} x + 3 \, {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 2 \, a^{2} b^{6}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{2 \, {\left(a^{8} b^{3} - 2 \, a^{6} b^{5} + 2 \, a^{2} b^{9} - b^{11} - {\left(a^{6} b^{5} - 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} - b^{11}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b^{4} - 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} - a b^{10}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[-1/4*(4*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*x*cos(x)^2 + (2*a^7 - 3*a^5*b^2 + a^3*b^4 + 6*a*b^6 - (2*a^5*b^2 - 5*a^3*b^4 + 6*a*b^6)*cos(x)^2 + 2*(2*a^6*b - 5*a^4*b^3 + 6*a^2*b^5)*sin(x))*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)) - 4*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*x - 2*(2*a^7*b - 7*a^5*b^3 + 5*a^3*b^5)*cos(x) - 2*(4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*x + 3*(a^6*b^2 - 3*a^4*b^4 + 2*a^2*b^6)*cos(x))*sin(x))/(a^8*b^3 - 2*a^6*b^5 + 2*a^2*b^9 - b^11 - (a^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*cos(x)^2 + 2*(a^7*b^4 - 3*a^5*b^6 + 3*a^3*b^8 - a*b^10)*sin(x)), -1/2*(2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*x*cos(x)^2 - (2*a^7 - 3*a^5*b^2 + a^3*b^4 + 6*a*b^6 - (2*a^5*b^2 - 5*a^3*b^4 + 6*a*b^6)*cos(x)^2 + 2*(2*a^6*b - 5*a^4*b^3 + 6*a^2*b^5)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - 2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*x - (2*a^7*b - 7*a^5*b^3 + 5*a^3*b^5)*cos(x) - (4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*x + 3*(a^6*b^2 - 3*a^4*b^4 + 2*a^2*b^6)*cos(x))*sin(x))/(a^8*b^3 - 2*a^6*b^5 + 2*a^2*b^9 - b^11 - (a^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*cos(x)^2 + 2*(a^7*b^4 - 3*a^5*b^6 + 3*a^3*b^8 - a*b^10)*sin(x))]","B",0
196,1,516,0,1.275303," ","integrate(sin(x)^2/(a+b*sin(x))^3,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a^{5} - 5 \, a^{3} b^{2} + 4 \, a b^{4}\right)} \cos\left(x\right) \sin\left(x\right) + {\left(a^{4} + 3 \, a^{2} b^{2} + 2 \, b^{4} - {\left(a^{2} b^{2} + 2 \, b^{4}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{3} b + 2 \, a b^{3}\right)} \sin\left(x\right)\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) - 6 \, {\left(a^{4} b - a^{2} b^{3}\right)} \cos\left(x\right)}{4 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} \sin\left(x\right)\right)}}, -\frac{{\left(a^{5} - 5 \, a^{3} b^{2} + 4 \, a b^{4}\right)} \cos\left(x\right) \sin\left(x\right) + {\left(a^{4} + 3 \, a^{2} b^{2} + 2 \, b^{4} - {\left(a^{2} b^{2} + 2 \, b^{4}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{3} b + 2 \, a b^{3}\right)} \sin\left(x\right)\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) - 3 \, {\left(a^{4} b - a^{2} b^{3}\right)} \cos\left(x\right)}{2 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[-1/4*(2*(a^5 - 5*a^3*b^2 + 4*a*b^4)*cos(x)*sin(x) + (a^4 + 3*a^2*b^2 + 2*b^4 - (a^2*b^2 + 2*b^4)*cos(x)^2 + 2*(a^3*b + 2*a*b^3)*sin(x))*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)) - 6*(a^4*b - a^2*b^3)*cos(x))/(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x)), -1/2*((a^5 - 5*a^3*b^2 + 4*a*b^4)*cos(x)*sin(x) + (a^4 + 3*a^2*b^2 + 2*b^4 - (a^2*b^2 + 2*b^4)*cos(x)^2 + 2*(a^3*b + 2*a*b^3)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - 3*(a^4*b - a^2*b^3)*cos(x))/(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x))]","B",0
197,1,490,0,1.336584," ","integrate(sin(x)/(a+b*sin(x))^3,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} \cos\left(x\right) \sin\left(x\right) - 3 \, {\left(a b^{3} \cos\left(x\right)^{2} - 2 \, a^{2} b^{2} \sin\left(x\right) - a^{3} b - a b^{3}\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) + 2 \, {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right)}{4 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} \sin\left(x\right)\right)}}, -\frac{{\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} \cos\left(x\right) \sin\left(x\right) + 3 \, {\left(a b^{3} \cos\left(x\right)^{2} - 2 \, a^{2} b^{2} \sin\left(x\right) - a^{3} b - a b^{3}\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(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right)}{2 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[-1/4*(2*(a^4*b + a^2*b^3 - 2*b^5)*cos(x)*sin(x) - 3*(a*b^3*cos(x)^2 - 2*a^2*b^2*sin(x) - a^3*b - a*b^3)*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)) + 2*(2*a^5 - a^3*b^2 - a*b^4)*cos(x))/(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x)), -1/2*((a^4*b + a^2*b^3 - 2*b^5)*cos(x)*sin(x) + 3*(a*b^3*cos(x)^2 - 2*a^2*b^2*sin(x) - a^3*b - a*b^3)*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + (2*a^5 - a^3*b^2 - a*b^4)*cos(x))/(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x))]","B",0
198,1,516,0,1.296633," ","integrate(1/(a+b*sin(x))^3,x, algorithm=""fricas"")","\left[\frac{6 \, {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right) \sin\left(x\right) - {\left(2 \, a^{4} + 3 \, a^{2} b^{2} + b^{4} - {\left(2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(x\right)^{2} + 2 \, {\left(2 \, a^{3} b + a b^{3}\right)} \sin\left(x\right)\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) + 2 \, {\left(4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right)} \cos\left(x\right)}{4 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} \sin\left(x\right)\right)}}, \frac{3 \, {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right) \sin\left(x\right) - {\left(2 \, a^{4} + 3 \, a^{2} b^{2} + b^{4} - {\left(2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(x\right)^{2} + 2 \, {\left(2 \, a^{3} b + a b^{3}\right)} \sin\left(x\right)\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(4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right)} \cos\left(x\right)}{2 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[1/4*(6*(a^3*b^2 - a*b^4)*cos(x)*sin(x) - (2*a^4 + 3*a^2*b^2 + b^4 - (2*a^2*b^2 + b^4)*cos(x)^2 + 2*(2*a^3*b + a*b^3)*sin(x))*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)) + 2*(4*a^4*b - 5*a^2*b^3 + b^5)*cos(x))/(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x)), 1/2*(3*(a^3*b^2 - a*b^4)*cos(x)*sin(x) - (2*a^4 + 3*a^2*b^2 + b^4 - (2*a^2*b^2 + b^4)*cos(x)^2 + 2*(2*a^3*b + a*b^3)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + (4*a^4*b - 5*a^2*b^3 + b^5)*cos(x))/(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x))]","B",0
199,1,1027,0,3.164161," ","integrate(csc(x)/(a+b*sin(x))^3,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(5 \, a^{5} b^{3} - 7 \, a^{3} b^{5} + 2 \, a b^{7}\right)} \cos\left(x\right) \sin\left(x\right) + {\left(6 \, a^{6} b + a^{4} b^{3} - 3 \, a^{2} b^{5} + 2 \, b^{7} - {\left(6 \, a^{4} b^{3} - 5 \, a^{2} b^{5} + 2 \, b^{7}\right)} \cos\left(x\right)^{2} + 2 \, {\left(6 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 2 \, a b^{6}\right)} \sin\left(x\right)\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) + 6 \, {\left(2 \, a^{6} b^{2} - 3 \, a^{4} b^{4} + a^{2} b^{6}\right)} \cos\left(x\right) + 2 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 2 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{4 \, {\left(a^{11} - 2 \, a^{9} b^{2} + 2 \, a^{5} b^{6} - a^{3} b^{8} - {\left(a^{9} b^{2} - 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} - a^{3} b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{10} b - 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} - a^{4} b^{7}\right)} \sin\left(x\right)\right)}}, -\frac{{\left(5 \, a^{5} b^{3} - 7 \, a^{3} b^{5} + 2 \, a b^{7}\right)} \cos\left(x\right) \sin\left(x\right) - {\left(6 \, a^{6} b + a^{4} b^{3} - 3 \, a^{2} b^{5} + 2 \, b^{7} - {\left(6 \, a^{4} b^{3} - 5 \, a^{2} b^{5} + 2 \, b^{7}\right)} \cos\left(x\right)^{2} + 2 \, {\left(6 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 2 \, a b^{6}\right)} \sin\left(x\right)\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) + 3 \, {\left(2 \, a^{6} b^{2} - 3 \, a^{4} b^{4} + a^{2} b^{6}\right)} \cos\left(x\right) + {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{11} - 2 \, a^{9} b^{2} + 2 \, a^{5} b^{6} - a^{3} b^{8} - {\left(a^{9} b^{2} - 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} - a^{3} b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{10} b - 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} - a^{4} b^{7}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[-1/4*(2*(5*a^5*b^3 - 7*a^3*b^5 + 2*a*b^7)*cos(x)*sin(x) + (6*a^6*b + a^4*b^3 - 3*a^2*b^5 + 2*b^7 - (6*a^4*b^3 - 5*a^2*b^5 + 2*b^7)*cos(x)^2 + 2*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*sin(x))*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)) + 6*(2*a^6*b^2 - 3*a^4*b^4 + a^2*b^6)*cos(x) + 2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x))*log(1/2*cos(x) + 1/2) - 2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x))*log(-1/2*cos(x) + 1/2))/(a^11 - 2*a^9*b^2 + 2*a^5*b^6 - a^3*b^8 - (a^9*b^2 - 3*a^7*b^4 + 3*a^5*b^6 - a^3*b^8)*cos(x)^2 + 2*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*sin(x)), -1/2*((5*a^5*b^3 - 7*a^3*b^5 + 2*a*b^7)*cos(x)*sin(x) - (6*a^6*b + a^4*b^3 - 3*a^2*b^5 + 2*b^7 - (6*a^4*b^3 - 5*a^2*b^5 + 2*b^7)*cos(x)^2 + 2*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + 3*(2*a^6*b^2 - 3*a^4*b^4 + a^2*b^6)*cos(x) + (a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x))*log(1/2*cos(x) + 1/2) - (a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(x))*log(-1/2*cos(x) + 1/2))/(a^11 - 2*a^9*b^2 + 2*a^5*b^6 - a^3*b^8 - (a^9*b^2 - 3*a^7*b^4 + 3*a^5*b^6 - a^3*b^8)*cos(x)^2 + 2*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*sin(x))]","B",0
200,1,1436,0,2.968029," ","integrate(csc(x)^2/(a+b*sin(x))^3,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(2 \, a^{7} b^{2} - 13 \, a^{5} b^{4} + 17 \, a^{3} b^{6} - 6 \, a b^{8}\right)} \cos\left(x\right)^{3} - 2 \, {\left(4 \, a^{8} b - 20 \, a^{6} b^{3} + 25 \, a^{4} b^{5} - 9 \, a^{2} b^{7}\right)} \cos\left(x\right) \sin\left(x\right) - 3 \, {\left(8 \, a^{5} b^{3} - 10 \, a^{3} b^{5} + 4 \, a b^{7} - 2 \, {\left(4 \, a^{5} b^{3} - 5 \, a^{3} b^{5} + 2 \, a b^{7}\right)} \cos\left(x\right)^{2} + {\left(4 \, a^{6} b^{2} - a^{4} b^{4} - 3 \, a^{2} b^{6} + 2 \, b^{8} - {\left(4 \, a^{4} b^{4} - 5 \, a^{2} b^{6} + 2 \, b^{8}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\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) - 2 \, {\left(2 \, a^{9} - 4 \, a^{7} b^{2} - 7 \, a^{5} b^{4} + 15 \, a^{3} b^{6} - 6 \, a b^{8}\right)} \cos\left(x\right) + 6 \, {\left(2 \, a^{7} b^{2} - 6 \, a^{5} b^{4} + 6 \, a^{3} b^{6} - 2 \, a b^{8} - 2 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} \cos\left(x\right)^{2} + {\left(a^{8} b - 2 \, a^{6} b^{3} + 2 \, a^{2} b^{7} - b^{9} - {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 6 \, {\left(2 \, a^{7} b^{2} - 6 \, a^{5} b^{4} + 6 \, a^{3} b^{6} - 2 \, a b^{8} - 2 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} \cos\left(x\right)^{2} + {\left(a^{8} b - 2 \, a^{6} b^{3} + 2 \, a^{2} b^{7} - b^{9} - {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{4 \, {\left(2 \, a^{11} b - 6 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 2 \, a^{5} b^{7} - 2 \, {\left(a^{11} b - 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} - a^{5} b^{7}\right)} \cos\left(x\right)^{2} + {\left(a^{12} - 2 \, a^{10} b^{2} + 2 \, a^{6} b^{6} - a^{4} b^{8} - {\left(a^{10} b^{2} - 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} - a^{4} b^{8}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)}}, \frac{{\left(2 \, a^{7} b^{2} - 13 \, a^{5} b^{4} + 17 \, a^{3} b^{6} - 6 \, a b^{8}\right)} \cos\left(x\right)^{3} - {\left(4 \, a^{8} b - 20 \, a^{6} b^{3} + 25 \, a^{4} b^{5} - 9 \, a^{2} b^{7}\right)} \cos\left(x\right) \sin\left(x\right) - 3 \, {\left(8 \, a^{5} b^{3} - 10 \, a^{3} b^{5} + 4 \, a b^{7} - 2 \, {\left(4 \, a^{5} b^{3} - 5 \, a^{3} b^{5} + 2 \, a b^{7}\right)} \cos\left(x\right)^{2} + {\left(4 \, a^{6} b^{2} - a^{4} b^{4} - 3 \, a^{2} b^{6} + 2 \, b^{8} - {\left(4 \, a^{4} b^{4} - 5 \, a^{2} b^{6} + 2 \, b^{8}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\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(2 \, a^{9} - 4 \, a^{7} b^{2} - 7 \, a^{5} b^{4} + 15 \, a^{3} b^{6} - 6 \, a b^{8}\right)} \cos\left(x\right) + 3 \, {\left(2 \, a^{7} b^{2} - 6 \, a^{5} b^{4} + 6 \, a^{3} b^{6} - 2 \, a b^{8} - 2 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} \cos\left(x\right)^{2} + {\left(a^{8} b - 2 \, a^{6} b^{3} + 2 \, a^{2} b^{7} - b^{9} - {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 3 \, {\left(2 \, a^{7} b^{2} - 6 \, a^{5} b^{4} + 6 \, a^{3} b^{6} - 2 \, a b^{8} - 2 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} \cos\left(x\right)^{2} + {\left(a^{8} b - 2 \, a^{6} b^{3} + 2 \, a^{2} b^{7} - b^{9} - {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(2 \, a^{11} b - 6 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 2 \, a^{5} b^{7} - 2 \, {\left(a^{11} b - 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} - a^{5} b^{7}\right)} \cos\left(x\right)^{2} + {\left(a^{12} - 2 \, a^{10} b^{2} + 2 \, a^{6} b^{6} - a^{4} b^{8} - {\left(a^{10} b^{2} - 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} - a^{4} b^{8}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[1/4*(2*(2*a^7*b^2 - 13*a^5*b^4 + 17*a^3*b^6 - 6*a*b^8)*cos(x)^3 - 2*(4*a^8*b - 20*a^6*b^3 + 25*a^4*b^5 - 9*a^2*b^7)*cos(x)*sin(x) - 3*(8*a^5*b^3 - 10*a^3*b^5 + 4*a*b^7 - 2*(4*a^5*b^3 - 5*a^3*b^5 + 2*a*b^7)*cos(x)^2 + (4*a^6*b^2 - a^4*b^4 - 3*a^2*b^6 + 2*b^8 - (4*a^4*b^4 - 5*a^2*b^6 + 2*b^8)*cos(x)^2)*sin(x))*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)) - 2*(2*a^9 - 4*a^7*b^2 - 7*a^5*b^4 + 15*a^3*b^6 - 6*a*b^8)*cos(x) + 6*(2*a^7*b^2 - 6*a^5*b^4 + 6*a^3*b^6 - 2*a*b^8 - 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*cos(x)^2 + (a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9 - (a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) - 6*(2*a^7*b^2 - 6*a^5*b^4 + 6*a^3*b^6 - 2*a*b^8 - 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*cos(x)^2 + (a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9 - (a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2))/(2*a^11*b - 6*a^9*b^3 + 6*a^7*b^5 - 2*a^5*b^7 - 2*(a^11*b - 3*a^9*b^3 + 3*a^7*b^5 - a^5*b^7)*cos(x)^2 + (a^12 - 2*a^10*b^2 + 2*a^6*b^6 - a^4*b^8 - (a^10*b^2 - 3*a^8*b^4 + 3*a^6*b^6 - a^4*b^8)*cos(x)^2)*sin(x)), 1/2*((2*a^7*b^2 - 13*a^5*b^4 + 17*a^3*b^6 - 6*a*b^8)*cos(x)^3 - (4*a^8*b - 20*a^6*b^3 + 25*a^4*b^5 - 9*a^2*b^7)*cos(x)*sin(x) - 3*(8*a^5*b^3 - 10*a^3*b^5 + 4*a*b^7 - 2*(4*a^5*b^3 - 5*a^3*b^5 + 2*a*b^7)*cos(x)^2 + (4*a^6*b^2 - a^4*b^4 - 3*a^2*b^6 + 2*b^8 - (4*a^4*b^4 - 5*a^2*b^6 + 2*b^8)*cos(x)^2)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (2*a^9 - 4*a^7*b^2 - 7*a^5*b^4 + 15*a^3*b^6 - 6*a*b^8)*cos(x) + 3*(2*a^7*b^2 - 6*a^5*b^4 + 6*a^3*b^6 - 2*a*b^8 - 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*cos(x)^2 + (a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9 - (a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) - 3*(2*a^7*b^2 - 6*a^5*b^4 + 6*a^3*b^6 - 2*a*b^8 - 2*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*cos(x)^2 + (a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9 - (a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2))/(2*a^11*b - 6*a^9*b^3 + 6*a^7*b^5 - 2*a^5*b^7 - 2*(a^11*b - 3*a^9*b^3 + 3*a^7*b^5 - a^5*b^7)*cos(x)^2 + (a^12 - 2*a^10*b^2 + 2*a^6*b^6 - a^4*b^8 - (a^10*b^2 - 3*a^8*b^4 + 3*a^6*b^6 - a^4*b^8)*cos(x)^2)*sin(x))]","B",0
201,1,2005,0,5.751288," ","integrate(csc(x)^3/(a+b*sin(x))^3,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(11 \, a^{8} b^{2} - 43 \, a^{6} b^{4} + 50 \, a^{4} b^{6} - 18 \, a^{2} b^{8}\right)} \cos\left(x\right)^{3} + {\left(20 \, a^{6} b^{3} - 9 \, a^{4} b^{5} - 17 \, a^{2} b^{7} + 12 \, b^{9} + {\left(20 \, a^{4} b^{5} - 29 \, a^{2} b^{7} + 12 \, b^{9}\right)} \cos\left(x\right)^{4} - {\left(20 \, a^{6} b^{3} + 11 \, a^{4} b^{5} - 46 \, a^{2} b^{7} + 24 \, b^{9}\right)} \cos\left(x\right)^{2} + 2 \, {\left(20 \, a^{5} b^{4} - 29 \, a^{3} b^{6} + 12 \, a b^{8} - {\left(20 \, a^{5} b^{4} - 29 \, a^{3} b^{6} + 12 \, a b^{8}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\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) + 2 \, {\left(a^{10} - 14 \, a^{8} b^{2} + 46 \, a^{6} b^{4} - 51 \, a^{4} b^{6} + 18 \, a^{2} b^{8}\right)} \cos\left(x\right) + {\left(a^{10} + 10 \, a^{8} b^{2} - 24 \, a^{6} b^{4} + 2 \, a^{4} b^{6} + 23 \, a^{2} b^{8} - 12 \, b^{10} + {\left(a^{8} b^{2} + 9 \, a^{6} b^{4} - 33 \, a^{4} b^{6} + 35 \, a^{2} b^{8} - 12 \, b^{10}\right)} \cos\left(x\right)^{4} - {\left(a^{10} + 11 \, a^{8} b^{2} - 15 \, a^{6} b^{4} - 31 \, a^{4} b^{6} + 58 \, a^{2} b^{8} - 24 \, b^{10}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{9} b + 9 \, a^{7} b^{3} - 33 \, a^{5} b^{5} + 35 \, a^{3} b^{7} - 12 \, a b^{9} - {\left(a^{9} b + 9 \, a^{7} b^{3} - 33 \, a^{5} b^{5} + 35 \, a^{3} b^{7} - 12 \, a b^{9}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(a^{10} + 10 \, a^{8} b^{2} - 24 \, a^{6} b^{4} + 2 \, a^{4} b^{6} + 23 \, a^{2} b^{8} - 12 \, b^{10} + {\left(a^{8} b^{2} + 9 \, a^{6} b^{4} - 33 \, a^{4} b^{6} + 35 \, a^{2} b^{8} - 12 \, b^{10}\right)} \cos\left(x\right)^{4} - {\left(a^{10} + 11 \, a^{8} b^{2} - 15 \, a^{6} b^{4} - 31 \, a^{4} b^{6} + 58 \, a^{2} b^{8} - 24 \, b^{10}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{9} b + 9 \, a^{7} b^{3} - 33 \, a^{5} b^{5} + 35 \, a^{3} b^{7} - 12 \, a b^{9} - {\left(a^{9} b + 9 \, a^{7} b^{3} - 33 \, a^{5} b^{5} + 35 \, a^{3} b^{7} - 12 \, a b^{9}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 2 \, {\left(3 \, {\left(2 \, a^{7} b^{3} - 9 \, a^{5} b^{5} + 11 \, a^{3} b^{7} - 4 \, a b^{9}\right)} \cos\left(x\right)^{3} - {\left(4 \, a^{9} b - 6 \, a^{7} b^{3} - 15 \, a^{5} b^{5} + 29 \, a^{3} b^{7} - 12 \, a b^{9}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{4 \, {\left(a^{13} - 2 \, a^{11} b^{2} + 2 \, a^{7} b^{6} - a^{5} b^{8} + {\left(a^{11} b^{2} - 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} - a^{5} b^{8}\right)} \cos\left(x\right)^{4} - {\left(a^{13} - a^{11} b^{2} - 3 \, a^{9} b^{4} + 5 \, a^{7} b^{6} - 2 \, a^{5} b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{12} b - 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} - a^{6} b^{7} - {\left(a^{12} b - 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} - a^{6} b^{7}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)}}, -\frac{2 \, {\left(11 \, a^{8} b^{2} - 43 \, a^{6} b^{4} + 50 \, a^{4} b^{6} - 18 \, a^{2} b^{8}\right)} \cos\left(x\right)^{3} - 2 \, {\left(20 \, a^{6} b^{3} - 9 \, a^{4} b^{5} - 17 \, a^{2} b^{7} + 12 \, b^{9} + {\left(20 \, a^{4} b^{5} - 29 \, a^{2} b^{7} + 12 \, b^{9}\right)} \cos\left(x\right)^{4} - {\left(20 \, a^{6} b^{3} + 11 \, a^{4} b^{5} - 46 \, a^{2} b^{7} + 24 \, b^{9}\right)} \cos\left(x\right)^{2} + 2 \, {\left(20 \, a^{5} b^{4} - 29 \, a^{3} b^{6} + 12 \, a b^{8} - {\left(20 \, a^{5} b^{4} - 29 \, a^{3} b^{6} + 12 \, a b^{8}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\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) + 2 \, {\left(a^{10} - 14 \, a^{8} b^{2} + 46 \, a^{6} b^{4} - 51 \, a^{4} b^{6} + 18 \, a^{2} b^{8}\right)} \cos\left(x\right) + {\left(a^{10} + 10 \, a^{8} b^{2} - 24 \, a^{6} b^{4} + 2 \, a^{4} b^{6} + 23 \, a^{2} b^{8} - 12 \, b^{10} + {\left(a^{8} b^{2} + 9 \, a^{6} b^{4} - 33 \, a^{4} b^{6} + 35 \, a^{2} b^{8} - 12 \, b^{10}\right)} \cos\left(x\right)^{4} - {\left(a^{10} + 11 \, a^{8} b^{2} - 15 \, a^{6} b^{4} - 31 \, a^{4} b^{6} + 58 \, a^{2} b^{8} - 24 \, b^{10}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{9} b + 9 \, a^{7} b^{3} - 33 \, a^{5} b^{5} + 35 \, a^{3} b^{7} - 12 \, a b^{9} - {\left(a^{9} b + 9 \, a^{7} b^{3} - 33 \, a^{5} b^{5} + 35 \, a^{3} b^{7} - 12 \, a b^{9}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(a^{10} + 10 \, a^{8} b^{2} - 24 \, a^{6} b^{4} + 2 \, a^{4} b^{6} + 23 \, a^{2} b^{8} - 12 \, b^{10} + {\left(a^{8} b^{2} + 9 \, a^{6} b^{4} - 33 \, a^{4} b^{6} + 35 \, a^{2} b^{8} - 12 \, b^{10}\right)} \cos\left(x\right)^{4} - {\left(a^{10} + 11 \, a^{8} b^{2} - 15 \, a^{6} b^{4} - 31 \, a^{4} b^{6} + 58 \, a^{2} b^{8} - 24 \, b^{10}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{9} b + 9 \, a^{7} b^{3} - 33 \, a^{5} b^{5} + 35 \, a^{3} b^{7} - 12 \, a b^{9} - {\left(a^{9} b + 9 \, a^{7} b^{3} - 33 \, a^{5} b^{5} + 35 \, a^{3} b^{7} - 12 \, a b^{9}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 2 \, {\left(3 \, {\left(2 \, a^{7} b^{3} - 9 \, a^{5} b^{5} + 11 \, a^{3} b^{7} - 4 \, a b^{9}\right)} \cos\left(x\right)^{3} - {\left(4 \, a^{9} b - 6 \, a^{7} b^{3} - 15 \, a^{5} b^{5} + 29 \, a^{3} b^{7} - 12 \, a b^{9}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{4 \, {\left(a^{13} - 2 \, a^{11} b^{2} + 2 \, a^{7} b^{6} - a^{5} b^{8} + {\left(a^{11} b^{2} - 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} - a^{5} b^{8}\right)} \cos\left(x\right)^{4} - {\left(a^{13} - a^{11} b^{2} - 3 \, a^{9} b^{4} + 5 \, a^{7} b^{6} - 2 \, a^{5} b^{8}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{12} b - 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} - a^{6} b^{7} - {\left(a^{12} b - 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} - a^{6} b^{7}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[-1/4*(2*(11*a^8*b^2 - 43*a^6*b^4 + 50*a^4*b^6 - 18*a^2*b^8)*cos(x)^3 + (20*a^6*b^3 - 9*a^4*b^5 - 17*a^2*b^7 + 12*b^9 + (20*a^4*b^5 - 29*a^2*b^7 + 12*b^9)*cos(x)^4 - (20*a^6*b^3 + 11*a^4*b^5 - 46*a^2*b^7 + 24*b^9)*cos(x)^2 + 2*(20*a^5*b^4 - 29*a^3*b^6 + 12*a*b^8 - (20*a^5*b^4 - 29*a^3*b^6 + 12*a*b^8)*cos(x)^2)*sin(x))*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)) + 2*(a^10 - 14*a^8*b^2 + 46*a^6*b^4 - 51*a^4*b^6 + 18*a^2*b^8)*cos(x) + (a^10 + 10*a^8*b^2 - 24*a^6*b^4 + 2*a^4*b^6 + 23*a^2*b^8 - 12*b^10 + (a^8*b^2 + 9*a^6*b^4 - 33*a^4*b^6 + 35*a^2*b^8 - 12*b^10)*cos(x)^4 - (a^10 + 11*a^8*b^2 - 15*a^6*b^4 - 31*a^4*b^6 + 58*a^2*b^8 - 24*b^10)*cos(x)^2 + 2*(a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9 - (a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) - (a^10 + 10*a^8*b^2 - 24*a^6*b^4 + 2*a^4*b^6 + 23*a^2*b^8 - 12*b^10 + (a^8*b^2 + 9*a^6*b^4 - 33*a^4*b^6 + 35*a^2*b^8 - 12*b^10)*cos(x)^4 - (a^10 + 11*a^8*b^2 - 15*a^6*b^4 - 31*a^4*b^6 + 58*a^2*b^8 - 24*b^10)*cos(x)^2 + 2*(a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9 - (a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2) + 2*(3*(2*a^7*b^3 - 9*a^5*b^5 + 11*a^3*b^7 - 4*a*b^9)*cos(x)^3 - (4*a^9*b - 6*a^7*b^3 - 15*a^5*b^5 + 29*a^3*b^7 - 12*a*b^9)*cos(x))*sin(x))/(a^13 - 2*a^11*b^2 + 2*a^7*b^6 - a^5*b^8 + (a^11*b^2 - 3*a^9*b^4 + 3*a^7*b^6 - a^5*b^8)*cos(x)^4 - (a^13 - a^11*b^2 - 3*a^9*b^4 + 5*a^7*b^6 - 2*a^5*b^8)*cos(x)^2 + 2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7 - (a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*cos(x)^2)*sin(x)), -1/4*(2*(11*a^8*b^2 - 43*a^6*b^4 + 50*a^4*b^6 - 18*a^2*b^8)*cos(x)^3 - 2*(20*a^6*b^3 - 9*a^4*b^5 - 17*a^2*b^7 + 12*b^9 + (20*a^4*b^5 - 29*a^2*b^7 + 12*b^9)*cos(x)^4 - (20*a^6*b^3 + 11*a^4*b^5 - 46*a^2*b^7 + 24*b^9)*cos(x)^2 + 2*(20*a^5*b^4 - 29*a^3*b^6 + 12*a*b^8 - (20*a^5*b^4 - 29*a^3*b^6 + 12*a*b^8)*cos(x)^2)*sin(x))*sqrt(a^2 - b^2)*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) + 2*(a^10 - 14*a^8*b^2 + 46*a^6*b^4 - 51*a^4*b^6 + 18*a^2*b^8)*cos(x) + (a^10 + 10*a^8*b^2 - 24*a^6*b^4 + 2*a^4*b^6 + 23*a^2*b^8 - 12*b^10 + (a^8*b^2 + 9*a^6*b^4 - 33*a^4*b^6 + 35*a^2*b^8 - 12*b^10)*cos(x)^4 - (a^10 + 11*a^8*b^2 - 15*a^6*b^4 - 31*a^4*b^6 + 58*a^2*b^8 - 24*b^10)*cos(x)^2 + 2*(a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9 - (a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9)*cos(x)^2)*sin(x))*log(1/2*cos(x) + 1/2) - (a^10 + 10*a^8*b^2 - 24*a^6*b^4 + 2*a^4*b^6 + 23*a^2*b^8 - 12*b^10 + (a^8*b^2 + 9*a^6*b^4 - 33*a^4*b^6 + 35*a^2*b^8 - 12*b^10)*cos(x)^4 - (a^10 + 11*a^8*b^2 - 15*a^6*b^4 - 31*a^4*b^6 + 58*a^2*b^8 - 24*b^10)*cos(x)^2 + 2*(a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9 - (a^9*b + 9*a^7*b^3 - 33*a^5*b^5 + 35*a^3*b^7 - 12*a*b^9)*cos(x)^2)*sin(x))*log(-1/2*cos(x) + 1/2) + 2*(3*(2*a^7*b^3 - 9*a^5*b^5 + 11*a^3*b^7 - 4*a*b^9)*cos(x)^3 - (4*a^9*b - 6*a^7*b^3 - 15*a^5*b^5 + 29*a^3*b^7 - 12*a*b^9)*cos(x))*sin(x))/(a^13 - 2*a^11*b^2 + 2*a^7*b^6 - a^5*b^8 + (a^11*b^2 - 3*a^9*b^4 + 3*a^7*b^6 - a^5*b^8)*cos(x)^4 - (a^13 - a^11*b^2 - 3*a^9*b^4 + 5*a^7*b^6 - 2*a^5*b^8)*cos(x)^2 + 2*(a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7 - (a^12*b - 3*a^10*b^3 + 3*a^8*b^5 - a^6*b^7)*cos(x)^2)*sin(x))]","B",0
202,1,965,0,1.292011," ","integrate(1/(a+b*sin(d*x+c))^4,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(11 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - 4 \, b^{7}\right)} \cos\left(d x + c\right)^{3} - 6 \, {\left(9 \, a^{5} b^{2} - 8 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) - 3 \, {\left(2 \, a^{6} + 9 \, a^{4} b^{2} + 9 \, a^{2} b^{4} - 3 \, {\left(2 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(6 \, a^{5} b + 11 \, a^{3} b^{3} + 3 \, a b^{5} - {\left(2 \, a^{3} b^{3} + 3 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, a b \sin\left(d x + c\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(d x + c\right) \sin\left(d x + c\right) + b \cos\left(d x + c\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(d x + c\right)^{2} - 2 \, a b \sin\left(d x + c\right) - a^{2} - b^{2}}\right) - 12 \, {\left(3 \, a^{6} b - 2 \, a^{4} b^{3} - b^{7}\right)} \cos\left(d x + c\right)}{12 \, {\left(3 \, {\left(a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{11} - a^{9} b^{2} - 6 \, a^{7} b^{4} + 14 \, a^{5} b^{6} - 11 \, a^{3} b^{8} + 3 \, a b^{10}\right)} d + {\left({\left(a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right)} d \cos\left(d x + c\right)^{2} - {\left(3 \, a^{10} b - 11 \, a^{8} b^{3} + 14 \, a^{6} b^{5} - 6 \, a^{4} b^{7} - a^{2} b^{9} + b^{11}\right)} d\right)} \sin\left(d x + c\right)\right)}}, \frac{{\left(11 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - 4 \, b^{7}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(9 \, a^{5} b^{2} - 8 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) + 3 \, {\left(2 \, a^{6} + 9 \, a^{4} b^{2} + 9 \, a^{2} b^{4} - 3 \, {\left(2 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(6 \, a^{5} b + 11 \, a^{3} b^{3} + 3 \, a b^{5} - {\left(2 \, a^{3} b^{3} + 3 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(d x + c\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(d x + c\right)}\right) - 6 \, {\left(3 \, a^{6} b - 2 \, a^{4} b^{3} - b^{7}\right)} \cos\left(d x + c\right)}{6 \, {\left(3 \, {\left(a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{11} - a^{9} b^{2} - 6 \, a^{7} b^{4} + 14 \, a^{5} b^{6} - 11 \, a^{3} b^{8} + 3 \, a b^{10}\right)} d + {\left({\left(a^{8} b^{3} - 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} - 4 \, a^{2} b^{9} + b^{11}\right)} d \cos\left(d x + c\right)^{2} - {\left(3 \, a^{10} b - 11 \, a^{8} b^{3} + 14 \, a^{6} b^{5} - 6 \, a^{4} b^{7} - a^{2} b^{9} + b^{11}\right)} d\right)} \sin\left(d x + c\right)\right)}}\right]"," ",0,"[1/12*(2*(11*a^4*b^3 - 7*a^2*b^5 - 4*b^7)*cos(d*x + c)^3 - 6*(9*a^5*b^2 - 8*a^3*b^4 - a*b^6)*cos(d*x + c)*sin(d*x + c) - 3*(2*a^6 + 9*a^4*b^2 + 9*a^2*b^4 - 3*(2*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c)^2 + (6*a^5*b + 11*a^3*b^3 + 3*a*b^5 - (2*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 12*(3*a^6*b - 2*a^4*b^3 - b^7)*cos(d*x + c))/(3*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cos(d*x + c)^2 - (a^11 - a^9*b^2 - 6*a^7*b^4 + 14*a^5*b^6 - 11*a^3*b^8 + 3*a*b^10)*d + ((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cos(d*x + c)^2 - (3*a^10*b - 11*a^8*b^3 + 14*a^6*b^5 - 6*a^4*b^7 - a^2*b^9 + b^11)*d)*sin(d*x + c)), 1/6*((11*a^4*b^3 - 7*a^2*b^5 - 4*b^7)*cos(d*x + c)^3 - 3*(9*a^5*b^2 - 8*a^3*b^4 - a*b^6)*cos(d*x + c)*sin(d*x + c) + 3*(2*a^6 + 9*a^4*b^2 + 9*a^2*b^4 - 3*(2*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c)^2 + (6*a^5*b + 11*a^3*b^3 + 3*a*b^5 - (2*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 6*(3*a^6*b - 2*a^4*b^3 - b^7)*cos(d*x + c))/(3*(a^9*b^2 - 4*a^7*b^4 + 6*a^5*b^6 - 4*a^3*b^8 + a*b^10)*d*cos(d*x + c)^2 - (a^11 - a^9*b^2 - 6*a^7*b^4 + 14*a^5*b^6 - 11*a^3*b^8 + 3*a*b^10)*d + ((a^8*b^3 - 4*a^6*b^5 + 6*a^4*b^7 - 4*a^2*b^9 + b^11)*d*cos(d*x + c)^2 - (3*a^10*b - 11*a^8*b^3 + 14*a^6*b^5 - 6*a^4*b^7 - a^2*b^9 + b^11)*d)*sin(d*x + c))]","B",0
203,0,0,0,0.915589," ","integrate(sin(f*x+e)*(a+b*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sin\left(f x + e\right) + a} \sin\left(f x + e\right), x\right)"," ",0,"integral(sqrt(b*sin(f*x + e) + a)*sin(f*x + e), x)","F",0
204,0,0,0,0.804295," ","integrate((a+b*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral(sqrt(b*sin(f*x + e) + a), x)","F",0
205,-1,0,0,0.000000," ","integrate(csc(f*x+e)*(a+b*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
206,0,0,0,1.605762," ","integrate(csc(f*x+e)^2*(a+b*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sin\left(f x + e\right) + a} \csc\left(f x + e\right)^{2}, x\right)"," ",0,"integral(sqrt(b*sin(f*x + e) + a)*csc(f*x + e)^2, x)","F",0
207,0,0,0,1.008154," ","integrate(sin(f*x+e)/(a+b*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sin\left(f x + e\right)}{\sqrt{b \sin\left(f x + e\right) + a}}, x\right)"," ",0,"integral(sin(f*x + e)/sqrt(b*sin(f*x + e) + a), x)","F",0
208,0,0,0,1.329030," ","integrate(1/(a+b*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{1}{\sqrt{b \sin\left(f x + e\right) + a}}, x\right)"," ",0,"integral(1/sqrt(b*sin(f*x + e) + a), x)","F",0
209,-2,0,0,0.000000," ","integrate(csc(f*x+e)/(a+b*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   failed of mode Union(SparseUnivariatePolynomial(SimpleAlgebraicExtension(InnerPrimeField(7),SparseUnivariatePolynomial(InnerPrimeField(7)),?^2+2)),failed) cannot be coerced to mode SparseUnivariatePolynomial(SimpleAlgebraicExtension(InnerPrimeField(7),SparseUnivariatePolynomial(InnerPrimeField(7)),?^2+2))","F(-2)",0
210,-1,0,0,0.000000," ","integrate(csc(f*x+e)^2/(a+b*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
211,0,0,0,5.385025," ","integrate(sin(d*x+c)^(1/2)*(a+b*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sin\left(d x + c\right) + a} \sqrt{\sin\left(d x + c\right)}, x\right)"," ",0,"integral(sqrt(b*sin(d*x + c) + a)*sqrt(sin(d*x + c)), x)","F",0
212,0,0,0,1.168602," ","integrate(1/sin(d*x+c)^(1/2)/(a+b*sin(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{b \sin\left(d x + c\right) + a} \sqrt{\sin\left(d x + c\right)}}{b \cos\left(d x + c\right)^{2} - a \sin\left(d x + c\right) - b}, x\right)"," ",0,"integral(-sqrt(b*sin(d*x + c) + a)*sqrt(sin(d*x + c))/(b*cos(d*x + c)^2 - a*sin(d*x + c) - b), x)","F",0
213,0,0,0,1.490483," ","integrate((d*sin(f*x+e))^m*(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)\right)} \left(d \sin\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(-(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))*(d*sin(f*x + e))^m, x)","F",0
214,0,0,0,1.189477," ","integrate((d*sin(f*x+e))^m*(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}\right)} \left(d \sin\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*(d*sin(f*x + e))^m, x)","F",0
215,0,0,0,1.696969," ","integrate((d*sin(f*x+e))^m*(a+b*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right) + a\right)} \left(d \sin\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)*(d*sin(f*x + e))^m, x)","F",0
216,0,0,0,1.633018," ","integrate((d*sin(f*x+e))^m/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(d \sin\left(f x + e\right)\right)^{m}}{b \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral((d*sin(f*x + e))^m/(b*sin(f*x + e) + a), x)","F",0
217,0,0,0,1.398125," ","integrate((d*sin(f*x+e))^m/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(d \sin\left(f x + e\right)\right)^{m}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}, x\right)"," ",0,"integral(-(d*sin(f*x + e))^m/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2), x)","F",0
218,0,0,0,1.566372," ","integrate((d*sin(f*x+e))^m/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(d \sin\left(f x + e\right)\right)^{m}}{3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(d*sin(f*x + e))^m/(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e)), x)","F",0
219,0,0,0,1.349340," ","integrate(sin(d*x+c)^(-1-a^2/(a^2+b^2))*(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b^{2} \cos\left(d x + c\right)^{2} - 2 \, a b \sin\left(d x + c\right) - a^{2} - b^{2}\right)} \sin\left(d x + c\right)^{-\frac{2 \, a^{2} + b^{2}}{a^{2} + b^{2}}}, x\right)"," ",0,"integral(-(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)*sin(d*x + c)^(-(2*a^2 + b^2)/(a^2 + b^2)), x)","F",0
220,0,0,0,1.480803," ","integrate((1+2*sin(d*x+c))^2/sin(d*x+c)^(6/5),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(4 \, \cos\left(d x + c\right)^{2} - 4 \, \sin\left(d x + c\right) - 5\right)} \sin\left(d x + c\right)^{\frac{4}{5}}}{\cos\left(d x + c\right)^{2} - 1}, x\right)"," ",0,"integral((4*cos(d*x + c)^2 - 4*sin(d*x + c) - 5)*sin(d*x + c)^(4/5)/(cos(d*x + c)^2 - 1), x)","F",0
221,0,0,0,1.533321," ","integrate(sin(d*x+c)^m*(a+b*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right) + a\right)}^{n} \sin\left(d x + c\right)^{m}, x\right)"," ",0,"integral((b*sin(d*x + c) + a)^n*sin(d*x + c)^m, x)","F",0
222,0,0,0,1.429717," ","integrate(sin(d*x+c)^3*(a+b*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(d x + c\right)^{2} - 1\right)} {\left(b \sin\left(d x + c\right) + a\right)}^{n} \sin\left(d x + c\right), x\right)"," ",0,"integral(-(cos(d*x + c)^2 - 1)*(b*sin(d*x + c) + a)^n*sin(d*x + c), x)","F",0
223,0,0,0,1.462282," ","integrate(sin(d*x+c)^2*(a+b*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(d x + c\right)^{2} - 1\right)} {\left(b \sin\left(d x + c\right) + a\right)}^{n}, x\right)"," ",0,"integral(-(cos(d*x + c)^2 - 1)*(b*sin(d*x + c) + a)^n, x)","F",0
224,0,0,0,1.403263," ","integrate(sin(d*x+c)*(a+b*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right) + a\right)}^{n} \sin\left(d x + c\right), x\right)"," ",0,"integral((b*sin(d*x + c) + a)^n*sin(d*x + c), x)","F",0
225,0,0,0,1.302129," ","integrate((a+b*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right) + a\right)}^{n}, x\right)"," ",0,"integral((b*sin(d*x + c) + a)^n, x)","F",0
226,0,0,0,1.383350," ","integrate(csc(d*x+c)*(a+b*sin(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right) + a\right)}^{n} \csc\left(d x + c\right), x\right)"," ",0,"integral((b*sin(d*x + c) + a)^n*csc(d*x + c), x)","F",0
227,1,77,0,1.380632," ","integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))^4,x, algorithm=""fricas"")","-\frac{24 \, a c^{4} \cos\left(f x + e\right)^{5} - 160 \, a c^{4} \cos\left(f x + e\right)^{3} - 105 \, a c^{4} f x + 15 \, {\left(6 \, a c^{4} \cos\left(f x + e\right)^{3} - 7 \, a c^{4} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{120 \, f}"," ",0,"-1/120*(24*a*c^4*cos(f*x + e)^5 - 160*a*c^4*cos(f*x + e)^3 - 105*a*c^4*f*x + 15*(6*a*c^4*cos(f*x + e)^3 - 7*a*c^4*cos(f*x + e))*sin(f*x + e))/f","A",0
228,1,63,0,1.309141," ","integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{16 \, a c^{3} \cos\left(f x + e\right)^{3} + 15 \, a c^{3} f x - 3 \, {\left(2 \, a c^{3} \cos\left(f x + e\right)^{3} - 5 \, a c^{3} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, f}"," ",0,"1/24*(16*a*c^3*cos(f*x + e)^3 + 15*a*c^3*f*x - 3*(2*a*c^3*cos(f*x + e)^3 - 5*a*c^3*cos(f*x + e))*sin(f*x + e))/f","A",0
229,1,46,0,1.312923," ","integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{2 \, a c^{2} \cos\left(f x + e\right)^{3} + 3 \, a c^{2} f x + 3 \, a c^{2} \cos\left(f x + e\right) \sin\left(f x + e\right)}{6 \, f}"," ",0,"1/6*(2*a*c^2*cos(f*x + e)^3 + 3*a*c^2*f*x + 3*a*c^2*cos(f*x + e)*sin(f*x + e))/f","A",0
230,1,26,0,1.437881," ","integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e)),x, algorithm=""fricas"")","\frac{a c f x + a c \cos\left(f x + e\right) \sin\left(f x + e\right)}{2 \, f}"," ",0,"1/2*(a*c*f*x + a*c*cos(f*x + e)*sin(f*x + e))/f","A",0
231,1,66,0,1.452151," ","integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{a f x + {\left(a f x - 2 \, a\right)} \cos\left(f x + e\right) - {\left(a f x + 2 \, a\right)} \sin\left(f x + e\right) - 2 \, a}{c f \cos\left(f x + e\right) - c f \sin\left(f x + e\right) + c f}"," ",0,"-(a*f*x + (a*f*x - 2*a)*cos(f*x + e) - (a*f*x + 2*a)*sin(f*x + e) - 2*a)/(c*f*cos(f*x + e) - c*f*sin(f*x + e) + c*f)","A",0
232,1,104,0,1.229444," ","integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{a \cos\left(f x + e\right)^{2} - a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) + 2 \, a\right)} \sin\left(f x + e\right) - 2 \, a}{3 \, {\left(c^{2} f \cos\left(f x + e\right)^{2} - c^{2} f \cos\left(f x + e\right) - 2 \, c^{2} f + {\left(c^{2} f \cos\left(f x + e\right) + 2 \, c^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/3*(a*cos(f*x + e)^2 - a*cos(f*x + e) - (a*cos(f*x + e) + 2*a)*sin(f*x + e) - 2*a)/(c^2*f*cos(f*x + e)^2 - c^2*f*cos(f*x + e) - 2*c^2*f + (c^2*f*cos(f*x + e) + 2*c^2*f)*sin(f*x + e))","B",0
233,1,154,0,1.478726," ","integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{a \cos\left(f x + e\right)^{3} - 2 \, a \cos\left(f x + e\right)^{2} + 3 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 3 \, a \cos\left(f x + e\right) + 6 \, a\right)} \sin\left(f x + e\right) + 6 \, a}{15 \, {\left(c^{3} f \cos\left(f x + e\right)^{3} + 3 \, c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f - {\left(c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/15*(a*cos(f*x + e)^3 - 2*a*cos(f*x + e)^2 + 3*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 3*a*cos(f*x + e) + 6*a)*sin(f*x + e) + 6*a)/(c^3*f*cos(f*x + e)^3 + 3*c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f - (c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f)*sin(f*x + e))","B",0
234,1,206,0,1.387537," ","integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^4,x, algorithm=""fricas"")","\frac{2 \, a \cos\left(f x + e\right)^{4} + 8 \, a \cos\left(f x + e\right)^{3} - 9 \, a \cos\left(f x + e\right)^{2} + 15 \, a \cos\left(f x + e\right) - {\left(2 \, a \cos\left(f x + e\right)^{3} - 6 \, a \cos\left(f x + e\right)^{2} - 15 \, a \cos\left(f x + e\right) - 30 \, a\right)} \sin\left(f x + e\right) + 30 \, a}{105 \, {\left(c^{4} f \cos\left(f x + e\right)^{4} - 3 \, c^{4} f \cos\left(f x + e\right)^{3} - 8 \, c^{4} f \cos\left(f x + e\right)^{2} + 4 \, c^{4} f \cos\left(f x + e\right) + 8 \, c^{4} f + {\left(c^{4} f \cos\left(f x + e\right)^{3} + 4 \, c^{4} f \cos\left(f x + e\right)^{2} - 4 \, c^{4} f \cos\left(f x + e\right) - 8 \, c^{4} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/105*(2*a*cos(f*x + e)^4 + 8*a*cos(f*x + e)^3 - 9*a*cos(f*x + e)^2 + 15*a*cos(f*x + e) - (2*a*cos(f*x + e)^3 - 6*a*cos(f*x + e)^2 - 15*a*cos(f*x + e) - 30*a)*sin(f*x + e) + 30*a)/(c^4*f*cos(f*x + e)^4 - 3*c^4*f*cos(f*x + e)^3 - 8*c^4*f*cos(f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^4*f + (c^4*f*cos(f*x + e)^3 + 4*c^4*f*cos(f*x + e)^2 - 4*c^4*f*cos(f*x + e) - 8*c^4*f)*sin(f*x + e))","B",0
235,1,256,0,1.384371," ","integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm=""fricas"")","-\frac{2 \, a \cos\left(f x + e\right)^{5} - 8 \, a \cos\left(f x + e\right)^{4} - 25 \, a \cos\left(f x + e\right)^{3} + 20 \, a \cos\left(f x + e\right)^{2} - 35 \, a \cos\left(f x + e\right) + {\left(2 \, a \cos\left(f x + e\right)^{4} + 10 \, a \cos\left(f x + e\right)^{3} - 15 \, a \cos\left(f x + e\right)^{2} - 35 \, a \cos\left(f x + e\right) - 70 \, a\right)} \sin\left(f x + e\right) - 70 \, a}{315 \, {\left(c^{5} f \cos\left(f x + e\right)^{5} + 5 \, c^{5} f \cos\left(f x + e\right)^{4} - 8 \, c^{5} f \cos\left(f x + e\right)^{3} - 20 \, c^{5} f \cos\left(f x + e\right)^{2} + 8 \, c^{5} f \cos\left(f x + e\right) + 16 \, c^{5} f - {\left(c^{5} f \cos\left(f x + e\right)^{4} - 4 \, c^{5} f \cos\left(f x + e\right)^{3} - 12 \, c^{5} f \cos\left(f x + e\right)^{2} + 8 \, c^{5} f \cos\left(f x + e\right) + 16 \, c^{5} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/315*(2*a*cos(f*x + e)^5 - 8*a*cos(f*x + e)^4 - 25*a*cos(f*x + e)^3 + 20*a*cos(f*x + e)^2 - 35*a*cos(f*x + e) + (2*a*cos(f*x + e)^4 + 10*a*cos(f*x + e)^3 - 15*a*cos(f*x + e)^2 - 35*a*cos(f*x + e) - 70*a)*sin(f*x + e) - 70*a)/(c^5*f*cos(f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f)*sin(f*x + e))","B",0
236,1,103,0,1.270191," ","integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^5,x, algorithm=""fricas"")","-\frac{80 \, a^{2} c^{5} \cos\left(f x + e\right)^{7} - 448 \, a^{2} c^{5} \cos\left(f x + e\right)^{5} - 315 \, a^{2} c^{5} f x + 35 \, {\left(8 \, a^{2} c^{5} \cos\left(f x + e\right)^{5} - 6 \, a^{2} c^{5} \cos\left(f x + e\right)^{3} - 9 \, a^{2} c^{5} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{560 \, f}"," ",0,"-1/560*(80*a^2*c^5*cos(f*x + e)^7 - 448*a^2*c^5*cos(f*x + e)^5 - 315*a^2*c^5*f*x + 35*(8*a^2*c^5*cos(f*x + e)^5 - 6*a^2*c^5*cos(f*x + e)^3 - 9*a^2*c^5*cos(f*x + e))*sin(f*x + e))/f","A",0
237,1,87,0,0.860895," ","integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^4,x, algorithm=""fricas"")","\frac{96 \, a^{2} c^{4} \cos\left(f x + e\right)^{5} + 105 \, a^{2} c^{4} f x - 5 \, {\left(8 \, a^{2} c^{4} \cos\left(f x + e\right)^{5} - 14 \, a^{2} c^{4} \cos\left(f x + e\right)^{3} - 21 \, a^{2} c^{4} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{240 \, f}"," ",0,"1/240*(96*a^2*c^4*cos(f*x + e)^5 + 105*a^2*c^4*f*x - 5*(8*a^2*c^4*cos(f*x + e)^5 - 14*a^2*c^4*cos(f*x + e)^3 - 21*a^2*c^4*cos(f*x + e))*sin(f*x + e))/f","A",0
238,1,71,0,0.976027," ","integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{8 \, a^{2} c^{3} \cos\left(f x + e\right)^{5} + 15 \, a^{2} c^{3} f x + 5 \, {\left(2 \, a^{2} c^{3} \cos\left(f x + e\right)^{3} + 3 \, a^{2} c^{3} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{40 \, f}"," ",0,"1/40*(8*a^2*c^3*cos(f*x + e)^5 + 15*a^2*c^3*f*x + 5*(2*a^2*c^3*cos(f*x + e)^3 + 3*a^2*c^3*cos(f*x + e))*sin(f*x + e))/f","A",0
239,1,54,0,1.267776," ","integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{3 \, a^{2} c^{2} f x + {\left(2 \, a^{2} c^{2} \cos\left(f x + e\right)^{3} + 3 \, a^{2} c^{2} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, f}"," ",0,"1/8*(3*a^2*c^2*f*x + (2*a^2*c^2*cos(f*x + e)^3 + 3*a^2*c^2*cos(f*x + e))*sin(f*x + e))/f","A",0
240,1,46,0,1.440531," ","integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{2 \, a^{2} c \cos\left(f x + e\right)^{3} - 3 \, a^{2} c f x - 3 \, a^{2} c \cos\left(f x + e\right) \sin\left(f x + e\right)}{6 \, f}"," ",0,"-1/6*(2*a^2*c*cos(f*x + e)^3 - 3*a^2*c*f*x - 3*a^2*c*cos(f*x + e)*sin(f*x + e))/f","A",0
241,1,105,0,1.431098," ","integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{3 \, a^{2} f x - a^{2} \cos\left(f x + e\right)^{2} - 4 \, a^{2} + {\left(3 \, a^{2} f x - 5 \, a^{2}\right)} \cos\left(f x + e\right) - {\left(3 \, a^{2} f x - a^{2} \cos\left(f x + e\right) + 4 \, a^{2}\right)} \sin\left(f x + e\right)}{c f \cos\left(f x + e\right) - c f \sin\left(f x + e\right) + c f}"," ",0,"-(3*a^2*f*x - a^2*cos(f*x + e)^2 - 4*a^2 + (3*a^2*f*x - 5*a^2)*cos(f*x + e) - (3*a^2*f*x - a^2*cos(f*x + e) + 4*a^2)*sin(f*x + e))/(c*f*cos(f*x + e) - c*f*sin(f*x + e) + c*f)","A",0
242,1,158,0,0.920011," ","integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{6 \, a^{2} f x - {\left(3 \, a^{2} f x + 8 \, a^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, a^{2} + {\left(3 \, a^{2} f x - 4 \, a^{2}\right)} \cos\left(f x + e\right) - {\left(6 \, a^{2} f x - 4 \, a^{2} + {\left(3 \, a^{2} f x - 8 \, a^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{3 \, {\left(c^{2} f \cos\left(f x + e\right)^{2} - c^{2} f \cos\left(f x + e\right) - 2 \, c^{2} f + {\left(c^{2} f \cos\left(f x + e\right) + 2 \, c^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/3*(6*a^2*f*x - (3*a^2*f*x + 8*a^2)*cos(f*x + e)^2 + 4*a^2 + (3*a^2*f*x - 4*a^2)*cos(f*x + e) - (6*a^2*f*x - 4*a^2 + (3*a^2*f*x - 8*a^2)*cos(f*x + e))*sin(f*x + e))/(c^2*f*cos(f*x + e)^2 - c^2*f*cos(f*x + e) - 2*c^2*f + (c^2*f*cos(f*x + e) + 2*c^2*f)*sin(f*x + e))","B",0
243,1,168,0,1.094642," ","integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{a^{2} \cos\left(f x + e\right)^{3} + 3 \, a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \cos\left(f x + e\right) - 4 \, a^{2} + {\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \cos\left(f x + e\right) - 4 \, a^{2}\right)} \sin\left(f x + e\right)}{5 \, {\left(c^{3} f \cos\left(f x + e\right)^{3} + 3 \, c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f - {\left(c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/5*(a^2*cos(f*x + e)^3 + 3*a^2*cos(f*x + e)^2 - 2*a^2*cos(f*x + e) - 4*a^2 + (a^2*cos(f*x + e)^2 - 2*a^2*cos(f*x + e) - 4*a^2)*sin(f*x + e))/(c^3*f*cos(f*x + e)^3 + 3*c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f - (c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f)*sin(f*x + e))","B",0
244,1,222,0,1.228659," ","integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^4,x, algorithm=""fricas"")","-\frac{a^{2} \cos\left(f x + e\right)^{4} + 4 \, a^{2} \cos\left(f x + e\right)^{3} + 13 \, a^{2} \cos\left(f x + e\right)^{2} - 10 \, a^{2} \cos\left(f x + e\right) - 20 \, a^{2} - {\left(a^{2} \cos\left(f x + e\right)^{3} - 3 \, a^{2} \cos\left(f x + e\right)^{2} + 10 \, a^{2} \cos\left(f x + e\right) + 20 \, a^{2}\right)} \sin\left(f x + e\right)}{35 \, {\left(c^{4} f \cos\left(f x + e\right)^{4} - 3 \, c^{4} f \cos\left(f x + e\right)^{3} - 8 \, c^{4} f \cos\left(f x + e\right)^{2} + 4 \, c^{4} f \cos\left(f x + e\right) + 8 \, c^{4} f + {\left(c^{4} f \cos\left(f x + e\right)^{3} + 4 \, c^{4} f \cos\left(f x + e\right)^{2} - 4 \, c^{4} f \cos\left(f x + e\right) - 8 \, c^{4} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/35*(a^2*cos(f*x + e)^4 + 4*a^2*cos(f*x + e)^3 + 13*a^2*cos(f*x + e)^2 - 10*a^2*cos(f*x + e) - 20*a^2 - (a^2*cos(f*x + e)^3 - 3*a^2*cos(f*x + e)^2 + 10*a^2*cos(f*x + e) + 20*a^2)*sin(f*x + e))/(c^4*f*cos(f*x + e)^4 - 3*c^4*f*cos(f*x + e)^3 - 8*c^4*f*cos(f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^4*f + (c^4*f*cos(f*x + e)^3 + 4*c^4*f*cos(f*x + e)^2 - 4*c^4*f*cos(f*x + e) - 8*c^4*f)*sin(f*x + e))","B",0
245,1,278,0,1.676574," ","integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm=""fricas"")","\frac{2 \, a^{2} \cos\left(f x + e\right)^{5} - 8 \, a^{2} \cos\left(f x + e\right)^{4} - 25 \, a^{2} \cos\left(f x + e\right)^{3} - 85 \, a^{2} \cos\left(f x + e\right)^{2} + 70 \, a^{2} \cos\left(f x + e\right) + 140 \, a^{2} + {\left(2 \, a^{2} \cos\left(f x + e\right)^{4} + 10 \, a^{2} \cos\left(f x + e\right)^{3} - 15 \, a^{2} \cos\left(f x + e\right)^{2} + 70 \, a^{2} \cos\left(f x + e\right) + 140 \, a^{2}\right)} \sin\left(f x + e\right)}{315 \, {\left(c^{5} f \cos\left(f x + e\right)^{5} + 5 \, c^{5} f \cos\left(f x + e\right)^{4} - 8 \, c^{5} f \cos\left(f x + e\right)^{3} - 20 \, c^{5} f \cos\left(f x + e\right)^{2} + 8 \, c^{5} f \cos\left(f x + e\right) + 16 \, c^{5} f - {\left(c^{5} f \cos\left(f x + e\right)^{4} - 4 \, c^{5} f \cos\left(f x + e\right)^{3} - 12 \, c^{5} f \cos\left(f x + e\right)^{2} + 8 \, c^{5} f \cos\left(f x + e\right) + 16 \, c^{5} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/315*(2*a^2*cos(f*x + e)^5 - 8*a^2*cos(f*x + e)^4 - 25*a^2*cos(f*x + e)^3 - 85*a^2*cos(f*x + e)^2 + 70*a^2*cos(f*x + e) + 140*a^2 + (2*a^2*cos(f*x + e)^4 + 10*a^2*cos(f*x + e)^3 - 15*a^2*cos(f*x + e)^2 + 70*a^2*cos(f*x + e) + 140*a^2)*sin(f*x + e))/(c^5*f*cos(f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f)*sin(f*x + e))","B",0
246,1,332,0,1.782583," ","integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^6,x, algorithm=""fricas"")","-\frac{2 \, a^{2} \cos\left(f x + e\right)^{6} + 12 \, a^{2} \cos\left(f x + e\right)^{5} - 25 \, a^{2} \cos\left(f x + e\right)^{4} - 70 \, a^{2} \cos\left(f x + e\right)^{3} - 245 \, a^{2} \cos\left(f x + e\right)^{2} + 210 \, a^{2} \cos\left(f x + e\right) + 420 \, a^{2} - {\left(2 \, a^{2} \cos\left(f x + e\right)^{5} - 10 \, a^{2} \cos\left(f x + e\right)^{4} - 35 \, a^{2} \cos\left(f x + e\right)^{3} + 35 \, a^{2} \cos\left(f x + e\right)^{2} - 210 \, a^{2} \cos\left(f x + e\right) - 420 \, a^{2}\right)} \sin\left(f x + e\right)}{1155 \, {\left(c^{6} f \cos\left(f x + e\right)^{6} - 5 \, c^{6} f \cos\left(f x + e\right)^{5} - 18 \, c^{6} f \cos\left(f x + e\right)^{4} + 20 \, c^{6} f \cos\left(f x + e\right)^{3} + 48 \, c^{6} f \cos\left(f x + e\right)^{2} - 16 \, c^{6} f \cos\left(f x + e\right) - 32 \, c^{6} f + {\left(c^{6} f \cos\left(f x + e\right)^{5} + 6 \, c^{6} f \cos\left(f x + e\right)^{4} - 12 \, c^{6} f \cos\left(f x + e\right)^{3} - 32 \, c^{6} f \cos\left(f x + e\right)^{2} + 16 \, c^{6} f \cos\left(f x + e\right) + 32 \, c^{6} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/1155*(2*a^2*cos(f*x + e)^6 + 12*a^2*cos(f*x + e)^5 - 25*a^2*cos(f*x + e)^4 - 70*a^2*cos(f*x + e)^3 - 245*a^2*cos(f*x + e)^2 + 210*a^2*cos(f*x + e) + 420*a^2 - (2*a^2*cos(f*x + e)^5 - 10*a^2*cos(f*x + e)^4 - 35*a^2*cos(f*x + e)^3 + 35*a^2*cos(f*x + e)^2 - 210*a^2*cos(f*x + e) - 420*a^2)*sin(f*x + e))/(c^6*f*cos(f*x + e)^6 - 5*c^6*f*cos(f*x + e)^5 - 18*c^6*f*cos(f*x + e)^4 + 20*c^6*f*cos(f*x + e)^3 + 48*c^6*f*cos(f*x + e)^2 - 16*c^6*f*cos(f*x + e) - 32*c^6*f + (c^6*f*cos(f*x + e)^5 + 6*c^6*f*cos(f*x + e)^4 - 12*c^6*f*cos(f*x + e)^3 - 32*c^6*f*cos(f*x + e)^2 + 16*c^6*f*cos(f*x + e) + 32*c^6*f)*sin(f*x + e))","B",0
247,1,119,0,1.353194," ","integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^6,x, algorithm=""fricas"")","-\frac{896 \, a^{3} c^{6} \cos\left(f x + e\right)^{9} - 4608 \, a^{3} c^{6} \cos\left(f x + e\right)^{7} - 3465 \, a^{3} c^{6} f x + 21 \, {\left(144 \, a^{3} c^{6} \cos\left(f x + e\right)^{7} - 88 \, a^{3} c^{6} \cos\left(f x + e\right)^{5} - 110 \, a^{3} c^{6} \cos\left(f x + e\right)^{3} - 165 \, a^{3} c^{6} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8064 \, f}"," ",0,"-1/8064*(896*a^3*c^6*cos(f*x + e)^9 - 4608*a^3*c^6*cos(f*x + e)^7 - 3465*a^3*c^6*f*x + 21*(144*a^3*c^6*cos(f*x + e)^7 - 88*a^3*c^6*cos(f*x + e)^5 - 110*a^3*c^6*cos(f*x + e)^3 - 165*a^3*c^6*cos(f*x + e))*sin(f*x + e))/f","A",0
248,1,103,0,0.786148," ","integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^5,x, algorithm=""fricas"")","\frac{256 \, a^{3} c^{5} \cos\left(f x + e\right)^{7} + 315 \, a^{3} c^{5} f x - 7 \, {\left(16 \, a^{3} c^{5} \cos\left(f x + e\right)^{7} - 24 \, a^{3} c^{5} \cos\left(f x + e\right)^{5} - 30 \, a^{3} c^{5} \cos\left(f x + e\right)^{3} - 45 \, a^{3} c^{5} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{896 \, f}"," ",0,"1/896*(256*a^3*c^5*cos(f*x + e)^7 + 315*a^3*c^5*f*x - 7*(16*a^3*c^5*cos(f*x + e)^7 - 24*a^3*c^5*cos(f*x + e)^5 - 30*a^3*c^5*cos(f*x + e)^3 - 45*a^3*c^5*cos(f*x + e))*sin(f*x + e))/f","A",0
249,1,87,0,0.842448," ","integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^4,x, algorithm=""fricas"")","\frac{48 \, a^{3} c^{4} \cos\left(f x + e\right)^{7} + 105 \, a^{3} c^{4} f x + 7 \, {\left(8 \, a^{3} c^{4} \cos\left(f x + e\right)^{5} + 10 \, a^{3} c^{4} \cos\left(f x + e\right)^{3} + 15 \, a^{3} c^{4} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{336 \, f}"," ",0,"1/336*(48*a^3*c^4*cos(f*x + e)^7 + 105*a^3*c^4*f*x + 7*(8*a^3*c^4*cos(f*x + e)^5 + 10*a^3*c^4*cos(f*x + e)^3 + 15*a^3*c^4*cos(f*x + e))*sin(f*x + e))/f","A",0
250,1,70,0,1.221035," ","integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{15 \, a^{3} c^{3} f x + {\left(8 \, a^{3} c^{3} \cos\left(f x + e\right)^{5} + 10 \, a^{3} c^{3} \cos\left(f x + e\right)^{3} + 15 \, a^{3} c^{3} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, f}"," ",0,"1/48*(15*a^3*c^3*f*x + (8*a^3*c^3*cos(f*x + e)^5 + 10*a^3*c^3*cos(f*x + e)^3 + 15*a^3*c^3*cos(f*x + e))*sin(f*x + e))/f","A",0
251,1,71,0,1.366671," ","integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{8 \, a^{3} c^{2} \cos\left(f x + e\right)^{5} - 15 \, a^{3} c^{2} f x - 5 \, {\left(2 \, a^{3} c^{2} \cos\left(f x + e\right)^{3} + 3 \, a^{3} c^{2} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{40 \, f}"," ",0,"-1/40*(8*a^3*c^2*cos(f*x + e)^5 - 15*a^3*c^2*f*x - 5*(2*a^3*c^2*cos(f*x + e)^3 + 3*a^3*c^2*cos(f*x + e))*sin(f*x + e))/f","A",0
252,1,63,0,1.375632," ","integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{16 \, a^{3} c \cos\left(f x + e\right)^{3} - 15 \, a^{3} c f x + 3 \, {\left(2 \, a^{3} c \cos\left(f x + e\right)^{3} - 5 \, a^{3} c \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, f}"," ",0,"-1/24*(16*a^3*c*cos(f*x + e)^3 - 15*a^3*c*f*x + 3*(2*a^3*c*cos(f*x + e)^3 - 5*a^3*c*cos(f*x + e))*sin(f*x + e))/f","A",0
253,1,129,0,1.366132," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e)),x, algorithm=""fricas"")","\frac{a^{3} \cos\left(f x + e\right)^{3} - 15 \, a^{3} f x + 8 \, a^{3} \cos\left(f x + e\right)^{2} + 16 \, a^{3} - {\left(15 \, a^{3} f x - 23 \, a^{3}\right)} \cos\left(f x + e\right) + {\left(15 \, a^{3} f x + a^{3} \cos\left(f x + e\right)^{2} - 7 \, a^{3} \cos\left(f x + e\right) + 16 \, a^{3}\right)} \sin\left(f x + e\right)}{2 \, {\left(c f \cos\left(f x + e\right) - c f \sin\left(f x + e\right) + c f\right)}}"," ",0,"1/2*(a^3*cos(f*x + e)^3 - 15*a^3*f*x + 8*a^3*cos(f*x + e)^2 + 16*a^3 - (15*a^3*f*x - 23*a^3)*cos(f*x + e) + (15*a^3*f*x + a^3*cos(f*x + e)^2 - 7*a^3*cos(f*x + e) + 16*a^3)*sin(f*x + e))/(c*f*cos(f*x + e) - c*f*sin(f*x + e) + c*f)","A",0
254,1,184,0,1.361188," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{3 \, a^{3} \cos\left(f x + e\right)^{3} + 30 \, a^{3} f x + 8 \, a^{3} - {\left(15 \, a^{3} f x + 31 \, a^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(15 \, a^{3} f x - 26 \, a^{3}\right)} \cos\left(f x + e\right) - {\left(30 \, a^{3} f x - 3 \, a^{3} \cos\left(f x + e\right)^{2} - 8 \, a^{3} + {\left(15 \, a^{3} f x - 34 \, a^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{3 \, {\left(c^{2} f \cos\left(f x + e\right)^{2} - c^{2} f \cos\left(f x + e\right) - 2 \, c^{2} f + {\left(c^{2} f \cos\left(f x + e\right) + 2 \, c^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/3*(3*a^3*cos(f*x + e)^3 + 30*a^3*f*x + 8*a^3 - (15*a^3*f*x + 31*a^3)*cos(f*x + e)^2 + (15*a^3*f*x - 26*a^3)*cos(f*x + e) - (30*a^3*f*x - 3*a^3*cos(f*x + e)^2 - 8*a^3 + (15*a^3*f*x - 34*a^3)*cos(f*x + e))*sin(f*x + e))/(c^2*f*cos(f*x + e)^2 - c^2*f*cos(f*x + e) - 2*c^2*f + (c^2*f*cos(f*x + e) + 2*c^2*f)*sin(f*x + e))","B",0
255,1,235,0,1.386762," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{60 \, a^{3} f x - {\left(15 \, a^{3} f x - 46 \, a^{3}\right)} \cos\left(f x + e\right)^{3} - 24 \, a^{3} - {\left(45 \, a^{3} f x + 2 \, a^{3}\right)} \cos\left(f x + e\right)^{2} + 6 \, {\left(5 \, a^{3} f x - 12 \, a^{3}\right)} \cos\left(f x + e\right) - {\left(60 \, a^{3} f x + 24 \, a^{3} - {\left(15 \, a^{3} f x + 46 \, a^{3}\right)} \cos\left(f x + e\right)^{2} + 6 \, {\left(5 \, a^{3} f x - 8 \, a^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{15 \, {\left(c^{3} f \cos\left(f x + e\right)^{3} + 3 \, c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f - {\left(c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/15*(60*a^3*f*x - (15*a^3*f*x - 46*a^3)*cos(f*x + e)^3 - 24*a^3 - (45*a^3*f*x + 2*a^3)*cos(f*x + e)^2 + 6*(5*a^3*f*x - 12*a^3)*cos(f*x + e) - (60*a^3*f*x + 24*a^3 - (15*a^3*f*x + 46*a^3)*cos(f*x + e)^2 + 6*(5*a^3*f*x - 8*a^3)*cos(f*x + e))*sin(f*x + e))/(c^3*f*cos(f*x + e)^3 + 3*c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f - (c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f)*sin(f*x + e))","B",0
256,1,222,0,1.282177," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algorithm=""fricas"")","\frac{a^{3} \cos\left(f x + e\right)^{4} - 3 \, a^{3} \cos\left(f x + e\right)^{3} - 8 \, a^{3} \cos\left(f x + e\right)^{2} + 4 \, a^{3} \cos\left(f x + e\right) + 8 \, a^{3} - {\left(a^{3} \cos\left(f x + e\right)^{3} + 4 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} \cos\left(f x + e\right) - 8 \, a^{3}\right)} \sin\left(f x + e\right)}{7 \, {\left(c^{4} f \cos\left(f x + e\right)^{4} - 3 \, c^{4} f \cos\left(f x + e\right)^{3} - 8 \, c^{4} f \cos\left(f x + e\right)^{2} + 4 \, c^{4} f \cos\left(f x + e\right) + 8 \, c^{4} f + {\left(c^{4} f \cos\left(f x + e\right)^{3} + 4 \, c^{4} f \cos\left(f x + e\right)^{2} - 4 \, c^{4} f \cos\left(f x + e\right) - 8 \, c^{4} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/7*(a^3*cos(f*x + e)^4 - 3*a^3*cos(f*x + e)^3 - 8*a^3*cos(f*x + e)^2 + 4*a^3*cos(f*x + e) + 8*a^3 - (a^3*cos(f*x + e)^3 + 4*a^3*cos(f*x + e)^2 - 4*a^3*cos(f*x + e) - 8*a^3)*sin(f*x + e))/(c^4*f*cos(f*x + e)^4 - 3*c^4*f*cos(f*x + e)^3 - 8*c^4*f*cos(f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^4*f + (c^4*f*cos(f*x + e)^3 + 4*c^4*f*cos(f*x + e)^2 - 4*c^4*f*cos(f*x + e) - 8*c^4*f)*sin(f*x + e))","B",0
257,1,276,0,1.377798," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x, algorithm=""fricas"")","-\frac{a^{3} \cos\left(f x + e\right)^{5} - 4 \, a^{3} \cos\left(f x + e\right)^{4} + 19 \, a^{3} \cos\left(f x + e\right)^{3} + 52 \, a^{3} \cos\left(f x + e\right)^{2} - 28 \, a^{3} \cos\left(f x + e\right) - 56 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{4} + 5 \, a^{3} \cos\left(f x + e\right)^{3} + 24 \, a^{3} \cos\left(f x + e\right)^{2} - 28 \, a^{3} \cos\left(f x + e\right) - 56 \, a^{3}\right)} \sin\left(f x + e\right)}{63 \, {\left(c^{5} f \cos\left(f x + e\right)^{5} + 5 \, c^{5} f \cos\left(f x + e\right)^{4} - 8 \, c^{5} f \cos\left(f x + e\right)^{3} - 20 \, c^{5} f \cos\left(f x + e\right)^{2} + 8 \, c^{5} f \cos\left(f x + e\right) + 16 \, c^{5} f - {\left(c^{5} f \cos\left(f x + e\right)^{4} - 4 \, c^{5} f \cos\left(f x + e\right)^{3} - 12 \, c^{5} f \cos\left(f x + e\right)^{2} + 8 \, c^{5} f \cos\left(f x + e\right) + 16 \, c^{5} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/63*(a^3*cos(f*x + e)^5 - 4*a^3*cos(f*x + e)^4 + 19*a^3*cos(f*x + e)^3 + 52*a^3*cos(f*x + e)^2 - 28*a^3*cos(f*x + e) - 56*a^3 + (a^3*cos(f*x + e)^4 + 5*a^3*cos(f*x + e)^3 + 24*a^3*cos(f*x + e)^2 - 28*a^3*cos(f*x + e) - 56*a^3)*sin(f*x + e))/(c^5*f*cos(f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f)*sin(f*x + e))","B",0
258,1,332,0,1.390899," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm=""fricas"")","\frac{2 \, a^{3} \cos\left(f x + e\right)^{6} + 12 \, a^{3} \cos\left(f x + e\right)^{5} - 25 \, a^{3} \cos\left(f x + e\right)^{4} + 161 \, a^{3} \cos\left(f x + e\right)^{3} + 448 \, a^{3} \cos\left(f x + e\right)^{2} - 252 \, a^{3} \cos\left(f x + e\right) - 504 \, a^{3} - {\left(2 \, a^{3} \cos\left(f x + e\right)^{5} - 10 \, a^{3} \cos\left(f x + e\right)^{4} - 35 \, a^{3} \cos\left(f x + e\right)^{3} - 196 \, a^{3} \cos\left(f x + e\right)^{2} + 252 \, a^{3} \cos\left(f x + e\right) + 504 \, a^{3}\right)} \sin\left(f x + e\right)}{693 \, {\left(c^{6} f \cos\left(f x + e\right)^{6} - 5 \, c^{6} f \cos\left(f x + e\right)^{5} - 18 \, c^{6} f \cos\left(f x + e\right)^{4} + 20 \, c^{6} f \cos\left(f x + e\right)^{3} + 48 \, c^{6} f \cos\left(f x + e\right)^{2} - 16 \, c^{6} f \cos\left(f x + e\right) - 32 \, c^{6} f + {\left(c^{6} f \cos\left(f x + e\right)^{5} + 6 \, c^{6} f \cos\left(f x + e\right)^{4} - 12 \, c^{6} f \cos\left(f x + e\right)^{3} - 32 \, c^{6} f \cos\left(f x + e\right)^{2} + 16 \, c^{6} f \cos\left(f x + e\right) + 32 \, c^{6} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/693*(2*a^3*cos(f*x + e)^6 + 12*a^3*cos(f*x + e)^5 - 25*a^3*cos(f*x + e)^4 + 161*a^3*cos(f*x + e)^3 + 448*a^3*cos(f*x + e)^2 - 252*a^3*cos(f*x + e) - 504*a^3 - (2*a^3*cos(f*x + e)^5 - 10*a^3*cos(f*x + e)^4 - 35*a^3*cos(f*x + e)^3 - 196*a^3*cos(f*x + e)^2 + 252*a^3*cos(f*x + e) + 504*a^3)*sin(f*x + e))/(c^6*f*cos(f*x + e)^6 - 5*c^6*f*cos(f*x + e)^5 - 18*c^6*f*cos(f*x + e)^4 + 20*c^6*f*cos(f*x + e)^3 + 48*c^6*f*cos(f*x + e)^2 - 16*c^6*f*cos(f*x + e) - 32*c^6*f + (c^6*f*cos(f*x + e)^5 + 6*c^6*f*cos(f*x + e)^4 - 12*c^6*f*cos(f*x + e)^3 - 32*c^6*f*cos(f*x + e)^2 + 16*c^6*f*cos(f*x + e) + 32*c^6*f)*sin(f*x + e))","B",0
259,1,386,0,1.365700," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^7,x, algorithm=""fricas"")","-\frac{2 \, a^{3} \cos\left(f x + e\right)^{7} - 12 \, a^{3} \cos\left(f x + e\right)^{6} - 49 \, a^{3} \cos\left(f x + e\right)^{5} + 70 \, a^{3} \cos\left(f x + e\right)^{4} - 567 \, a^{3} \cos\left(f x + e\right)^{3} - 1596 \, a^{3} \cos\left(f x + e\right)^{2} + 924 \, a^{3} \cos\left(f x + e\right) + 1848 \, a^{3} + {\left(2 \, a^{3} \cos\left(f x + e\right)^{6} + 14 \, a^{3} \cos\left(f x + e\right)^{5} - 35 \, a^{3} \cos\left(f x + e\right)^{4} - 105 \, a^{3} \cos\left(f x + e\right)^{3} - 672 \, a^{3} \cos\left(f x + e\right)^{2} + 924 \, a^{3} \cos\left(f x + e\right) + 1848 \, a^{3}\right)} \sin\left(f x + e\right)}{3003 \, {\left(c^{7} f \cos\left(f x + e\right)^{7} + 7 \, c^{7} f \cos\left(f x + e\right)^{6} - 18 \, c^{7} f \cos\left(f x + e\right)^{5} - 56 \, c^{7} f \cos\left(f x + e\right)^{4} + 48 \, c^{7} f \cos\left(f x + e\right)^{3} + 112 \, c^{7} f \cos\left(f x + e\right)^{2} - 32 \, c^{7} f \cos\left(f x + e\right) - 64 \, c^{7} f - {\left(c^{7} f \cos\left(f x + e\right)^{6} - 6 \, c^{7} f \cos\left(f x + e\right)^{5} - 24 \, c^{7} f \cos\left(f x + e\right)^{4} + 32 \, c^{7} f \cos\left(f x + e\right)^{3} + 80 \, c^{7} f \cos\left(f x + e\right)^{2} - 32 \, c^{7} f \cos\left(f x + e\right) - 64 \, c^{7} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/3003*(2*a^3*cos(f*x + e)^7 - 12*a^3*cos(f*x + e)^6 - 49*a^3*cos(f*x + e)^5 + 70*a^3*cos(f*x + e)^4 - 567*a^3*cos(f*x + e)^3 - 1596*a^3*cos(f*x + e)^2 + 924*a^3*cos(f*x + e) + 1848*a^3 + (2*a^3*cos(f*x + e)^6 + 14*a^3*cos(f*x + e)^5 - 35*a^3*cos(f*x + e)^4 - 105*a^3*cos(f*x + e)^3 - 672*a^3*cos(f*x + e)^2 + 924*a^3*cos(f*x + e) + 1848*a^3)*sin(f*x + e))/(c^7*f*cos(f*x + e)^7 + 7*c^7*f*cos(f*x + e)^6 - 18*c^7*f*cos(f*x + e)^5 - 56*c^7*f*cos(f*x + e)^4 + 48*c^7*f*cos(f*x + e)^3 + 112*c^7*f*cos(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^7*f - (c^7*f*cos(f*x + e)^6 - 6*c^7*f*cos(f*x + e)^5 - 24*c^7*f*cos(f*x + e)^4 + 32*c^7*f*cos(f*x + e)^3 + 80*c^7*f*cos(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^7*f)*sin(f*x + e))","B",0
260,1,440,0,0.937953," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^8,x, algorithm=""fricas"")","\frac{8 \, a^{3} \cos\left(f x + e\right)^{8} + 64 \, a^{3} \cos\left(f x + e\right)^{7} - 196 \, a^{3} \cos\left(f x + e\right)^{6} - 672 \, a^{3} \cos\left(f x + e\right)^{5} + 735 \, a^{3} \cos\left(f x + e\right)^{4} - 7161 \, a^{3} \cos\left(f x + e\right)^{3} - 20328 \, a^{3} \cos\left(f x + e\right)^{2} + 12012 \, a^{3} \cos\left(f x + e\right) + 24024 \, a^{3} - {\left(8 \, a^{3} \cos\left(f x + e\right)^{7} - 56 \, a^{3} \cos\left(f x + e\right)^{6} - 252 \, a^{3} \cos\left(f x + e\right)^{5} + 420 \, a^{3} \cos\left(f x + e\right)^{4} + 1155 \, a^{3} \cos\left(f x + e\right)^{3} + 8316 \, a^{3} \cos\left(f x + e\right)^{2} - 12012 \, a^{3} \cos\left(f x + e\right) - 24024 \, a^{3}\right)} \sin\left(f x + e\right)}{45045 \, {\left(c^{8} f \cos\left(f x + e\right)^{8} - 7 \, c^{8} f \cos\left(f x + e\right)^{7} - 32 \, c^{8} f \cos\left(f x + e\right)^{6} + 56 \, c^{8} f \cos\left(f x + e\right)^{5} + 160 \, c^{8} f \cos\left(f x + e\right)^{4} - 112 \, c^{8} f \cos\left(f x + e\right)^{3} - 256 \, c^{8} f \cos\left(f x + e\right)^{2} + 64 \, c^{8} f \cos\left(f x + e\right) + 128 \, c^{8} f + {\left(c^{8} f \cos\left(f x + e\right)^{7} + 8 \, c^{8} f \cos\left(f x + e\right)^{6} - 24 \, c^{8} f \cos\left(f x + e\right)^{5} - 80 \, c^{8} f \cos\left(f x + e\right)^{4} + 80 \, c^{8} f \cos\left(f x + e\right)^{3} + 192 \, c^{8} f \cos\left(f x + e\right)^{2} - 64 \, c^{8} f \cos\left(f x + e\right) - 128 \, c^{8} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/45045*(8*a^3*cos(f*x + e)^8 + 64*a^3*cos(f*x + e)^7 - 196*a^3*cos(f*x + e)^6 - 672*a^3*cos(f*x + e)^5 + 735*a^3*cos(f*x + e)^4 - 7161*a^3*cos(f*x + e)^3 - 20328*a^3*cos(f*x + e)^2 + 12012*a^3*cos(f*x + e) + 24024*a^3 - (8*a^3*cos(f*x + e)^7 - 56*a^3*cos(f*x + e)^6 - 252*a^3*cos(f*x + e)^5 + 420*a^3*cos(f*x + e)^4 + 1155*a^3*cos(f*x + e)^3 + 8316*a^3*cos(f*x + e)^2 - 12012*a^3*cos(f*x + e) - 24024*a^3)*sin(f*x + e))/(c^8*f*cos(f*x + e)^8 - 7*c^8*f*cos(f*x + e)^7 - 32*c^8*f*cos(f*x + e)^6 + 56*c^8*f*cos(f*x + e)^5 + 160*c^8*f*cos(f*x + e)^4 - 112*c^8*f*cos(f*x + e)^3 - 256*c^8*f*cos(f*x + e)^2 + 64*c^8*f*cos(f*x + e) + 128*c^8*f + (c^8*f*cos(f*x + e)^7 + 8*c^8*f*cos(f*x + e)^6 - 24*c^8*f*cos(f*x + e)^5 - 80*c^8*f*cos(f*x + e)^4 + 80*c^8*f*cos(f*x + e)^3 + 192*c^8*f*cos(f*x + e)^2 - 64*c^8*f*cos(f*x + e) - 128*c^8*f)*sin(f*x + e))","B",0
261,1,156,0,1.237217," ","integrate((c-c*sin(f*x+e))^4/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","\frac{2 \, c^{4} \cos\left(f x + e\right)^{4} - 13 \, c^{4} \cos\left(f x + e\right)^{3} - 105 \, c^{4} f x - 72 \, c^{4} \cos\left(f x + e\right)^{2} - 96 \, c^{4} - 3 \, {\left(35 \, c^{4} f x + 51 \, c^{4}\right)} \cos\left(f x + e\right) + {\left(2 \, c^{4} \cos\left(f x + e\right)^{3} - 105 \, c^{4} f x + 15 \, c^{4} \cos\left(f x + e\right)^{2} - 57 \, c^{4} \cos\left(f x + e\right) + 96 \, c^{4}\right)} \sin\left(f x + e\right)}{6 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}"," ",0,"1/6*(2*c^4*cos(f*x + e)^4 - 13*c^4*cos(f*x + e)^3 - 105*c^4*f*x - 72*c^4*cos(f*x + e)^2 - 96*c^4 - 3*(35*c^4*f*x + 51*c^4)*cos(f*x + e) + (2*c^4*cos(f*x + e)^3 - 105*c^4*f*x + 15*c^4*cos(f*x + e)^2 - 57*c^4*cos(f*x + e) + 96*c^4)*sin(f*x + e))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","A",0
262,1,128,0,1.448938," ","integrate((c-c*sin(f*x+e))^3/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{c^{3} \cos\left(f x + e\right)^{3} + 15 \, c^{3} f x + 8 \, c^{3} \cos\left(f x + e\right)^{2} + 16 \, c^{3} + {\left(15 \, c^{3} f x + 23 \, c^{3}\right)} \cos\left(f x + e\right) + {\left(15 \, c^{3} f x - c^{3} \cos\left(f x + e\right)^{2} + 7 \, c^{3} \cos\left(f x + e\right) - 16 \, c^{3}\right)} \sin\left(f x + e\right)}{2 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}"," ",0,"-1/2*(c^3*cos(f*x + e)^3 + 15*c^3*f*x + 8*c^3*cos(f*x + e)^2 + 16*c^3 + (15*c^3*f*x + 23*c^3)*cos(f*x + e) + (15*c^3*f*x - c^3*cos(f*x + e)^2 + 7*c^3*cos(f*x + e) - 16*c^3)*sin(f*x + e))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","A",0
263,1,101,0,1.001282," ","integrate((c-c*sin(f*x+e))^2/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{3 \, c^{2} f x + c^{2} \cos\left(f x + e\right)^{2} + 4 \, c^{2} + {\left(3 \, c^{2} f x + 5 \, c^{2}\right)} \cos\left(f x + e\right) + {\left(3 \, c^{2} f x + c^{2} \cos\left(f x + e\right) - 4 \, c^{2}\right)} \sin\left(f x + e\right)}{a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f}"," ",0,"-(3*c^2*f*x + c^2*cos(f*x + e)^2 + 4*c^2 + (3*c^2*f*x + 5*c^2)*cos(f*x + e) + (3*c^2*f*x + c^2*cos(f*x + e) - 4*c^2)*sin(f*x + e))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","A",0
264,1,64,0,1.332397," ","integrate((c-c*sin(f*x+e))/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{c f x + {\left(c f x + 2 \, c\right)} \cos\left(f x + e\right) + {\left(c f x - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f}"," ",0,"-(c*f*x + (c*f*x + 2*c)*cos(f*x + e) + (c*f*x - 2*c)*sin(f*x + e) + 2*c)/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","A",0
265,1,24,0,1.402479," ","integrate(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e)),x, algorithm=""fricas"")","\frac{\sin\left(f x + e\right)}{a c f \cos\left(f x + e\right)}"," ",0,"sin(f*x + e)/(a*c*f*cos(f*x + e))","A",0
266,1,56,0,0.965577," ","integrate(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{2 \, \cos\left(f x + e\right)^{2} + 2 \, \sin\left(f x + e\right) - 1}{3 \, {\left(a c^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) - a c^{2} f \cos\left(f x + e\right)\right)}}"," ",0,"-1/3*(2*cos(f*x + e)^2 + 2*sin(f*x + e) - 1)/(a*c^2*f*cos(f*x + e)*sin(f*x + e) - a*c^2*f*cos(f*x + e))","A",0
267,1,83,0,1.281193," ","integrate(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{4 \, \cos\left(f x + e\right)^{2} - {\left(2 \, \cos\left(f x + e\right)^{2} - 3\right)} \sin\left(f x + e\right) - 2}{5 \, {\left(a c^{3} f \cos\left(f x + e\right)^{3} + 2 \, a c^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a c^{3} f \cos\left(f x + e\right)\right)}}"," ",0,"-1/5*(4*cos(f*x + e)^2 - (2*cos(f*x + e)^2 - 3)*sin(f*x + e) - 2)/(a*c^3*f*cos(f*x + e)^3 + 2*a*c^3*f*cos(f*x + e)*sin(f*x + e) - 2*a*c^3*f*cos(f*x + e))","A",0
268,1,111,0,1.322474," ","integrate(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^4,x, algorithm=""fricas"")","\frac{8 \, \cos\left(f x + e\right)^{4} - 36 \, \cos\left(f x + e\right)^{2} + 4 \, {\left(6 \, \cos\left(f x + e\right)^{2} - 5\right)} \sin\left(f x + e\right) + 15}{35 \, {\left(3 \, a c^{4} f \cos\left(f x + e\right)^{3} - 4 \, a c^{4} f \cos\left(f x + e\right) - {\left(a c^{4} f \cos\left(f x + e\right)^{3} - 4 \, a c^{4} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/35*(8*cos(f*x + e)^4 - 36*cos(f*x + e)^2 + 4*(6*cos(f*x + e)^2 - 5)*sin(f*x + e) + 15)/(3*a*c^4*f*cos(f*x + e)^3 - 4*a*c^4*f*cos(f*x + e) - (a*c^4*f*cos(f*x + e)^3 - 4*a*c^4*f*cos(f*x + e))*sin(f*x + e))","A",0
269,1,238,0,1.472556," ","integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{2 \, c^{5} \cos\left(f x + e\right)^{5} + 19 \, c^{5} \cos\left(f x + e\right)^{4} - 106 \, c^{5} \cos\left(f x + e\right)^{3} + 630 \, c^{5} f x - 64 \, c^{5} - 7 \, {\left(45 \, c^{5} f x - 77 \, c^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(315 \, c^{5} f x + 598 \, c^{5}\right)} \cos\left(f x + e\right) - {\left(2 \, c^{5} \cos\left(f x + e\right)^{4} - 17 \, c^{5} \cos\left(f x + e\right)^{3} - 630 \, c^{5} f x - 123 \, c^{5} \cos\left(f x + e\right)^{2} - 64 \, c^{5} - {\left(315 \, c^{5} f x + 662 \, c^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{6 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/6*(2*c^5*cos(f*x + e)^5 + 19*c^5*cos(f*x + e)^4 - 106*c^5*cos(f*x + e)^3 + 630*c^5*f*x - 64*c^5 - 7*(45*c^5*f*x - 77*c^5)*cos(f*x + e)^2 + (315*c^5*f*x + 598*c^5)*cos(f*x + e) - (2*c^5*cos(f*x + e)^4 - 17*c^5*cos(f*x + e)^3 - 630*c^5*f*x - 123*c^5*cos(f*x + e)^2 - 64*c^5 - (315*c^5*f*x + 662*c^5)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","A",0
270,1,210,0,1.684708," ","integrate((c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{3 \, c^{4} \cos\left(f x + e\right)^{4} - 30 \, c^{4} \cos\left(f x + e\right)^{3} + 210 \, c^{4} f x - 32 \, c^{4} - {\left(105 \, c^{4} f x - 193 \, c^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(105 \, c^{4} f x + 194 \, c^{4}\right)} \cos\left(f x + e\right) + {\left(3 \, c^{4} \cos\left(f x + e\right)^{3} + 210 \, c^{4} f x + 33 \, c^{4} \cos\left(f x + e\right)^{2} + 32 \, c^{4} + {\left(105 \, c^{4} f x + 226 \, c^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{6 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/6*(3*c^4*cos(f*x + e)^4 - 30*c^4*cos(f*x + e)^3 + 210*c^4*f*x - 32*c^4 - (105*c^4*f*x - 193*c^4)*cos(f*x + e)^2 + (105*c^4*f*x + 194*c^4)*cos(f*x + e) + (3*c^4*cos(f*x + e)^3 + 210*c^4*f*x + 33*c^4*cos(f*x + e)^2 + 32*c^4 + (105*c^4*f*x + 226*c^4)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","A",0
271,1,185,0,1.533710," ","integrate((c-c*sin(f*x+e))^3/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{3 \, c^{3} \cos\left(f x + e\right)^{3} - 30 \, c^{3} f x + 8 \, c^{3} + {\left(15 \, c^{3} f x - 31 \, c^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(15 \, c^{3} f x + 26 \, c^{3}\right)} \cos\left(f x + e\right) - {\left(30 \, c^{3} f x + 3 \, c^{3} \cos\left(f x + e\right)^{2} + 8 \, c^{3} + {\left(15 \, c^{3} f x + 34 \, c^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{3 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/3*(3*c^3*cos(f*x + e)^3 - 30*c^3*f*x + 8*c^3 + (15*c^3*f*x - 31*c^3)*cos(f*x + e)^2 - (15*c^3*f*x + 26*c^3)*cos(f*x + e) - (30*c^3*f*x + 3*c^3*cos(f*x + e)^2 + 8*c^3 + (15*c^3*f*x + 34*c^3)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","B",0
272,1,158,0,0.839495," ","integrate((c-c*sin(f*x+e))^2/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{6 \, c^{2} f x - {\left(3 \, c^{2} f x - 8 \, c^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, c^{2} + {\left(3 \, c^{2} f x + 4 \, c^{2}\right)} \cos\left(f x + e\right) + {\left(6 \, c^{2} f x + 4 \, c^{2} + {\left(3 \, c^{2} f x + 8 \, c^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{3 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/3*(6*c^2*f*x - (3*c^2*f*x - 8*c^2)*cos(f*x + e)^2 - 4*c^2 + (3*c^2*f*x + 4*c^2)*cos(f*x + e) + (6*c^2*f*x + 4*c^2 + (3*c^2*f*x + 8*c^2)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","B",0
273,1,104,0,1.507811," ","integrate((c-c*sin(f*x+e))/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{c \cos\left(f x + e\right)^{2} - c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) + 2 \, c\right)} \sin\left(f x + e\right) - 2 \, c}{3 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/3*(c*cos(f*x + e)^2 - c*cos(f*x + e) + (c*cos(f*x + e) + 2*c)*sin(f*x + e) - 2*c)/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","B",0
274,1,55,0,1.274336," ","integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{2 \, \cos\left(f x + e\right)^{2} - 2 \, \sin\left(f x + e\right) - 1}{3 \, {\left(a^{2} c f \cos\left(f x + e\right) \sin\left(f x + e\right) + a^{2} c f \cos\left(f x + e\right)\right)}}"," ",0,"-1/3*(2*cos(f*x + e)^2 - 2*sin(f*x + e) - 1)/(a^2*c*f*cos(f*x + e)*sin(f*x + e) + a^2*c*f*cos(f*x + e))","A",0
275,1,37,0,1.651179," ","integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{{\left(2 \, \cos\left(f x + e\right)^{2} + 1\right)} \sin\left(f x + e\right)}{3 \, a^{2} c^{2} f \cos\left(f x + e\right)^{3}}"," ",0,"1/3*(2*cos(f*x + e)^2 + 1)*sin(f*x + e)/(a^2*c^2*f*cos(f*x + e)^3)","A",0
276,1,86,0,1.520889," ","integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{8 \, \cos\left(f x + e\right)^{4} - 4 \, \cos\left(f x + e\right)^{2} + 4 \, {\left(2 \, \cos\left(f x + e\right)^{2} + 1\right)} \sin\left(f x + e\right) - 1}{15 \, {\left(a^{2} c^{3} f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) - a^{2} c^{3} f \cos\left(f x + e\right)^{3}\right)}}"," ",0,"-1/15*(8*cos(f*x + e)^4 - 4*cos(f*x + e)^2 + 4*(2*cos(f*x + e)^2 + 1)*sin(f*x + e) - 1)/(a^2*c^3*f*cos(f*x + e)^3*sin(f*x + e) - a^2*c^3*f*cos(f*x + e)^3)","A",0
277,1,113,0,1.209504," ","integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^4,x, algorithm=""fricas"")","-\frac{16 \, \cos\left(f x + e\right)^{4} - 8 \, \cos\left(f x + e\right)^{2} - {\left(8 \, \cos\left(f x + e\right)^{4} - 12 \, \cos\left(f x + e\right)^{2} - 5\right)} \sin\left(f x + e\right) - 2}{21 \, {\left(a^{2} c^{4} f \cos\left(f x + e\right)^{5} + 2 \, a^{2} c^{4} f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) - 2 \, a^{2} c^{4} f \cos\left(f x + e\right)^{3}\right)}}"," ",0,"-1/21*(16*cos(f*x + e)^4 - 8*cos(f*x + e)^2 - (8*cos(f*x + e)^4 - 12*cos(f*x + e)^2 - 5)*sin(f*x + e) - 2)/(a^2*c^4*f*cos(f*x + e)^5 + 2*a^2*c^4*f*cos(f*x + e)^3*sin(f*x + e) - 2*a^2*c^4*f*cos(f*x + e)^3)","A",0
278,1,143,0,1.369914," ","integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm=""fricas"")","\frac{16 \, \cos\left(f x + e\right)^{6} - 72 \, \cos\left(f x + e\right)^{4} + 30 \, \cos\left(f x + e\right)^{2} + 2 \, {\left(24 \, \cos\left(f x + e\right)^{4} - 20 \, \cos\left(f x + e\right)^{2} - 7\right)} \sin\left(f x + e\right) + 7}{63 \, {\left(3 \, a^{2} c^{5} f \cos\left(f x + e\right)^{5} - 4 \, a^{2} c^{5} f \cos\left(f x + e\right)^{3} - {\left(a^{2} c^{5} f \cos\left(f x + e\right)^{5} - 4 \, a^{2} c^{5} f \cos\left(f x + e\right)^{3}\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/63*(16*cos(f*x + e)^6 - 72*cos(f*x + e)^4 + 30*cos(f*x + e)^2 + 2*(24*cos(f*x + e)^4 - 20*cos(f*x + e)^2 - 7)*sin(f*x + e) + 7)/(3*a^2*c^5*f*cos(f*x + e)^5 - 4*a^2*c^5*f*cos(f*x + e)^3 - (a^2*c^5*f*cos(f*x + e)^5 - 4*a^2*c^5*f*cos(f*x + e)^3)*sin(f*x + e))","A",0
279,1,285,0,1.108331," ","integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{5 \, c^{5} \cos\left(f x + e\right)^{5} + 70 \, c^{5} \cos\left(f x + e\right)^{4} - 1260 \, c^{5} f x - 64 \, c^{5} + 7 \, {\left(45 \, c^{5} f x + 113 \, c^{5}\right)} \cos\left(f x + e\right)^{3} + {\left(945 \, c^{5} f x - 502 \, c^{5}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(315 \, c^{5} f x + 646 \, c^{5}\right)} \cos\left(f x + e\right) - {\left(5 \, c^{5} \cos\left(f x + e\right)^{4} - 65 \, c^{5} \cos\left(f x + e\right)^{3} + 1260 \, c^{5} f x - 64 \, c^{5} - 3 \, {\left(105 \, c^{5} f x - 242 \, c^{5}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(315 \, c^{5} f x + 614 \, c^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{10 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/10*(5*c^5*cos(f*x + e)^5 + 70*c^5*cos(f*x + e)^4 - 1260*c^5*f*x - 64*c^5 + 7*(45*c^5*f*x + 113*c^5)*cos(f*x + e)^3 + (945*c^5*f*x - 502*c^5)*cos(f*x + e)^2 - 2*(315*c^5*f*x + 646*c^5)*cos(f*x + e) - (5*c^5*cos(f*x + e)^4 - 65*c^5*cos(f*x + e)^3 + 1260*c^5*f*x - 64*c^5 - 3*(105*c^5*f*x - 242*c^5)*cos(f*x + e)^2 + 2*(315*c^5*f*x + 614*c^5)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","A",0
280,1,256,0,1.170866," ","integrate((c-c*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{15 \, c^{4} \cos\left(f x + e\right)^{4} - 420 \, c^{4} f x - 48 \, c^{4} + {\left(105 \, c^{4} f x + 277 \, c^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(315 \, c^{4} f x - 134 \, c^{4}\right)} \cos\left(f x + e\right)^{2} - 6 \, {\left(35 \, c^{4} f x + 74 \, c^{4}\right)} \cos\left(f x + e\right) + {\left(15 \, c^{4} \cos\left(f x + e\right)^{3} - 420 \, c^{4} f x + 48 \, c^{4} + {\left(105 \, c^{4} f x - 262 \, c^{4}\right)} \cos\left(f x + e\right)^{2} - 6 \, {\left(35 \, c^{4} f x + 66 \, c^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/15*(15*c^4*cos(f*x + e)^4 - 420*c^4*f*x - 48*c^4 + (105*c^4*f*x + 277*c^4)*cos(f*x + e)^3 + (315*c^4*f*x - 134*c^4)*cos(f*x + e)^2 - 6*(35*c^4*f*x + 74*c^4)*cos(f*x + e) + (15*c^4*cos(f*x + e)^3 - 420*c^4*f*x + 48*c^4 + (105*c^4*f*x - 262*c^4)*cos(f*x + e)^2 - 6*(35*c^4*f*x + 66*c^4)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
281,1,233,0,1.545169," ","integrate((c-c*sin(f*x+e))^3/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{60 \, c^{3} f x - {\left(15 \, c^{3} f x + 46 \, c^{3}\right)} \cos\left(f x + e\right)^{3} + 24 \, c^{3} - {\left(45 \, c^{3} f x - 2 \, c^{3}\right)} \cos\left(f x + e\right)^{2} + 6 \, {\left(5 \, c^{3} f x + 12 \, c^{3}\right)} \cos\left(f x + e\right) + {\left(60 \, c^{3} f x - 24 \, c^{3} - {\left(15 \, c^{3} f x - 46 \, c^{3}\right)} \cos\left(f x + e\right)^{2} + 6 \, {\left(5 \, c^{3} f x + 8 \, c^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/15*(60*c^3*f*x - (15*c^3*f*x + 46*c^3)*cos(f*x + e)^3 + 24*c^3 - (45*c^3*f*x - 2*c^3)*cos(f*x + e)^2 + 6*(5*c^3*f*x + 12*c^3)*cos(f*x + e) + (60*c^3*f*x - 24*c^3 - (15*c^3*f*x - 46*c^3)*cos(f*x + e)^2 + 6*(5*c^3*f*x + 8*c^3)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
282,1,168,0,1.603222," ","integrate((c-c*sin(f*x+e))^2/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{c^{2} \cos\left(f x + e\right)^{3} + 3 \, c^{2} \cos\left(f x + e\right)^{2} - 2 \, c^{2} \cos\left(f x + e\right) - 4 \, c^{2} - {\left(c^{2} \cos\left(f x + e\right)^{2} - 2 \, c^{2} \cos\left(f x + e\right) - 4 \, c^{2}\right)} \sin\left(f x + e\right)}{5 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/5*(c^2*cos(f*x + e)^3 + 3*c^2*cos(f*x + e)^2 - 2*c^2*cos(f*x + e) - 4*c^2 - (c^2*cos(f*x + e)^2 - 2*c^2*cos(f*x + e) - 4*c^2)*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
283,1,154,0,1.241642," ","integrate((c-c*sin(f*x+e))/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{c \cos\left(f x + e\right)^{3} - 2 \, c \cos\left(f x + e\right)^{2} + 3 \, c \cos\left(f x + e\right) - {\left(c \cos\left(f x + e\right)^{2} + 3 \, c \cos\left(f x + e\right) + 6 \, c\right)} \sin\left(f x + e\right) + 6 \, c}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/15*(c*cos(f*x + e)^3 - 2*c*cos(f*x + e)^2 + 3*c*cos(f*x + e) - (c*cos(f*x + e)^2 + 3*c*cos(f*x + e) + 6*c)*sin(f*x + e) + 6*c)/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
284,1,82,0,1.515601," ","integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e)),x, algorithm=""fricas"")","\frac{4 \, \cos\left(f x + e\right)^{2} + {\left(2 \, \cos\left(f x + e\right)^{2} - 3\right)} \sin\left(f x + e\right) - 2}{5 \, {\left(a^{3} c f \cos\left(f x + e\right)^{3} - 2 \, a^{3} c f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{3} c f \cos\left(f x + e\right)\right)}}"," ",0,"1/5*(4*cos(f*x + e)^2 + (2*cos(f*x + e)^2 - 3)*sin(f*x + e) - 2)/(a^3*c*f*cos(f*x + e)^3 - 2*a^3*c*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*c*f*cos(f*x + e))","A",0
285,1,85,0,0.835434," ","integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{8 \, \cos\left(f x + e\right)^{4} - 4 \, \cos\left(f x + e\right)^{2} - 4 \, {\left(2 \, \cos\left(f x + e\right)^{2} + 1\right)} \sin\left(f x + e\right) - 1}{15 \, {\left(a^{3} c^{2} f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) + a^{3} c^{2} f \cos\left(f x + e\right)^{3}\right)}}"," ",0,"-1/15*(8*cos(f*x + e)^4 - 4*cos(f*x + e)^2 - 4*(2*cos(f*x + e)^2 + 1)*sin(f*x + e) - 1)/(a^3*c^2*f*cos(f*x + e)^3*sin(f*x + e) + a^3*c^2*f*cos(f*x + e)^3)","A",0
286,1,47,0,1.623198," ","integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{{\left(8 \, \cos\left(f x + e\right)^{4} + 4 \, \cos\left(f x + e\right)^{2} + 3\right)} \sin\left(f x + e\right)}{15 \, a^{3} c^{3} f \cos\left(f x + e\right)^{5}}"," ",0,"1/15*(8*cos(f*x + e)^4 + 4*cos(f*x + e)^2 + 3)*sin(f*x + e)/(a^3*c^3*f*cos(f*x + e)^5)","A",0
287,1,106,0,1.169676," ","integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algorithm=""fricas"")","-\frac{16 \, \cos\left(f x + e\right)^{6} - 8 \, \cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 2 \, {\left(8 \, \cos\left(f x + e\right)^{4} + 4 \, \cos\left(f x + e\right)^{2} + 3\right)} \sin\left(f x + e\right) - 1}{35 \, {\left(a^{3} c^{4} f \cos\left(f x + e\right)^{5} \sin\left(f x + e\right) - a^{3} c^{4} f \cos\left(f x + e\right)^{5}\right)}}"," ",0,"-1/35*(16*cos(f*x + e)^6 - 8*cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 2*(8*cos(f*x + e)^4 + 4*cos(f*x + e)^2 + 3)*sin(f*x + e) - 1)/(a^3*c^4*f*cos(f*x + e)^5*sin(f*x + e) - a^3*c^4*f*cos(f*x + e)^5)","A",0
288,1,133,0,1.638922," ","integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^5,x, algorithm=""fricas"")","-\frac{32 \, \cos\left(f x + e\right)^{6} - 16 \, \cos\left(f x + e\right)^{4} - 4 \, \cos\left(f x + e\right)^{2} - {\left(16 \, \cos\left(f x + e\right)^{6} - 24 \, \cos\left(f x + e\right)^{4} - 10 \, \cos\left(f x + e\right)^{2} - 7\right)} \sin\left(f x + e\right) - 2}{45 \, {\left(a^{3} c^{5} f \cos\left(f x + e\right)^{7} + 2 \, a^{3} c^{5} f \cos\left(f x + e\right)^{5} \sin\left(f x + e\right) - 2 \, a^{3} c^{5} f \cos\left(f x + e\right)^{5}\right)}}"," ",0,"-1/45*(32*cos(f*x + e)^6 - 16*cos(f*x + e)^4 - 4*cos(f*x + e)^2 - (16*cos(f*x + e)^6 - 24*cos(f*x + e)^4 - 10*cos(f*x + e)^2 - 7)*sin(f*x + e) - 2)/(a^3*c^5*f*cos(f*x + e)^7 + 2*a^3*c^5*f*cos(f*x + e)^5*sin(f*x + e) - 2*a^3*c^5*f*cos(f*x + e)^5)","A",0
289,1,163,0,1.735238," ","integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algorithm=""fricas"")","\frac{128 \, \cos\left(f x + e\right)^{8} - 576 \, \cos\left(f x + e\right)^{6} + 240 \, \cos\left(f x + e\right)^{4} + 56 \, \cos\left(f x + e\right)^{2} + 8 \, {\left(48 \, \cos\left(f x + e\right)^{6} - 40 \, \cos\left(f x + e\right)^{4} - 14 \, \cos\left(f x + e\right)^{2} - 9\right)} \sin\left(f x + e\right) + 27}{495 \, {\left(3 \, a^{3} c^{6} f \cos\left(f x + e\right)^{7} - 4 \, a^{3} c^{6} f \cos\left(f x + e\right)^{5} - {\left(a^{3} c^{6} f \cos\left(f x + e\right)^{7} - 4 \, a^{3} c^{6} f \cos\left(f x + e\right)^{5}\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/495*(128*cos(f*x + e)^8 - 576*cos(f*x + e)^6 + 240*cos(f*x + e)^4 + 56*cos(f*x + e)^2 + 8*(48*cos(f*x + e)^6 - 40*cos(f*x + e)^4 - 14*cos(f*x + e)^2 - 9)*sin(f*x + e) + 27)/(3*a^3*c^6*f*cos(f*x + e)^7 - 4*a^3*c^6*f*cos(f*x + e)^5 - (a^3*c^6*f*cos(f*x + e)^7 - 4*a^3*c^6*f*cos(f*x + e)^5)*sin(f*x + e))","A",0
290,1,179,0,1.459660," ","integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(35 \, a c^{3} \cos\left(f x + e\right)^{5} - 95 \, a c^{3} \cos\left(f x + e\right)^{4} - 226 \, a c^{3} \cos\left(f x + e\right)^{3} + 32 \, a c^{3} \cos\left(f x + e\right)^{2} - 128 \, a c^{3} \cos\left(f x + e\right) - 256 \, a c^{3} + {\left(35 \, a c^{3} \cos\left(f x + e\right)^{4} + 130 \, a c^{3} \cos\left(f x + e\right)^{3} - 96 \, a c^{3} \cos\left(f x + e\right)^{2} - 128 \, a c^{3} \cos\left(f x + e\right) - 256 \, a c^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{315 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"-2/315*(35*a*c^3*cos(f*x + e)^5 - 95*a*c^3*cos(f*x + e)^4 - 226*a*c^3*cos(f*x + e)^3 + 32*a*c^3*cos(f*x + e)^2 - 128*a*c^3*cos(f*x + e) - 256*a*c^3 + (35*a*c^3*cos(f*x + e)^4 + 130*a*c^3*cos(f*x + e)^3 - 96*a*c^3*cos(f*x + e)^2 - 128*a*c^3*cos(f*x + e) - 256*a*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","A",0
291,1,152,0,1.209541," ","integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{2 \, {\left(15 \, a c^{2} \cos\left(f x + e\right)^{4} + 39 \, a c^{2} \cos\left(f x + e\right)^{3} - 8 \, a c^{2} \cos\left(f x + e\right)^{2} + 32 \, a c^{2} \cos\left(f x + e\right) + 64 \, a c^{2} - {\left(15 \, a c^{2} \cos\left(f x + e\right)^{3} - 24 \, a c^{2} \cos\left(f x + e\right)^{2} - 32 \, a c^{2} \cos\left(f x + e\right) - 64 \, a c^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{105 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/105*(15*a*c^2*cos(f*x + e)^4 + 39*a*c^2*cos(f*x + e)^3 - 8*a*c^2*cos(f*x + e)^2 + 32*a*c^2*cos(f*x + e) + 64*a*c^2 - (15*a*c^2*cos(f*x + e)^3 - 24*a*c^2*cos(f*x + e)^2 - 32*a*c^2*cos(f*x + e) - 64*a*c^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","A",0
292,1,109,0,1.464373," ","integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, a c \cos\left(f x + e\right)^{3} - a c \cos\left(f x + e\right)^{2} + 4 \, a c \cos\left(f x + e\right) + 8 \, a c + {\left(3 \, a c \cos\left(f x + e\right)^{2} + 4 \, a c \cos\left(f x + e\right) + 8 \, a c\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{15 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/15*(3*a*c*cos(f*x + e)^3 - a*c*cos(f*x + e)^2 + 4*a*c*cos(f*x + e) + 8*a*c + (3*a*c*cos(f*x + e)^2 + 4*a*c*cos(f*x + e) + 8*a*c)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","A",0
293,1,79,0,1.124408," ","integrate((a+a*sin(f*x+e))*(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) + 2 \, a\right)} \sin\left(f x + e\right) - 2 \, a\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"-2/3*(a*cos(f*x + e)^2 - a*cos(f*x + e) - (a*cos(f*x + e) + 2*a)*sin(f*x + e) - 2*a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","B",0
294,1,196,0,1.453837," ","integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{\frac{\sqrt{2} {\left(a c \cos\left(f x + e\right) - a c \sin\left(f x + e\right) + a c\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) + \frac{2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)}}{\sqrt{c}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{c}} - 2 \, {\left(a \cos\left(f x + e\right) + a \sin\left(f x + e\right) + a\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{c f \cos\left(f x + e\right) - c f \sin\left(f x + e\right) + c f}"," ",0,"(sqrt(2)*(a*c*cos(f*x + e) - a*c*sin(f*x + e) + a*c)*log(-(cos(f*x + e)^2 + (cos(f*x + e) - 2)*sin(f*x + e) + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*(cos(f*x + e) + sin(f*x + e) + 1)/sqrt(c) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(c) - 2*(a*cos(f*x + e) + a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(c*f*cos(f*x + e) - c*f*sin(f*x + e) + c*f)","B",0
295,1,255,0,1.560200," ","integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{\frac{\sqrt{2} {\left(a c \cos\left(f x + e\right)^{2} - a c \cos\left(f x + e\right) - 2 \, a c + {\left(a c \cos\left(f x + e\right) + 2 \, a c\right)} \sin\left(f x + e\right)\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) - \frac{2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)}}{\sqrt{c}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{c}} - 4 \, {\left(a \cos\left(f x + e\right) + a \sin\left(f x + e\right) + a\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{4 \, {\left(c^{2} f \cos\left(f x + e\right)^{2} - c^{2} f \cos\left(f x + e\right) - 2 \, c^{2} f + {\left(c^{2} f \cos\left(f x + e\right) + 2 \, c^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/4*(sqrt(2)*(a*c*cos(f*x + e)^2 - a*c*cos(f*x + e) - 2*a*c + (a*c*cos(f*x + e) + 2*a*c)*sin(f*x + e))*log(-(cos(f*x + e)^2 + (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*(cos(f*x + e) + sin(f*x + e) + 1)/sqrt(c) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(c) - 4*(a*cos(f*x + e) + a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(c^2*f*cos(f*x + e)^2 - c^2*f*cos(f*x + e) - 2*c^2*f + (c^2*f*cos(f*x + e) + 2*c^2*f)*sin(f*x + e))","B",0
296,1,336,0,1.662766," ","integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{\sqrt{2} {\left(a \cos\left(f x + e\right)^{3} + 3 \, a \cos\left(f x + e\right)^{2} - 2 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right)^{2} - 2 \, a \cos\left(f x + e\right) - 4 \, a\right)} \sin\left(f x + e\right) - 4 \, a\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left(a \cos\left(f x + e\right)^{2} - 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) + 4 \, a\right)} \sin\left(f x + e\right) - 4 \, a\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{32 \, {\left(c^{3} f \cos\left(f x + e\right)^{3} + 3 \, c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f - {\left(c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/32*(sqrt(2)*(a*cos(f*x + e)^3 + 3*a*cos(f*x + e)^2 - 2*a*cos(f*x + e) - (a*cos(f*x + e)^2 - 2*a*cos(f*x + e) - 4*a)*sin(f*x + e) - 4*a)*sqrt(c)*log(-(c*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(a*cos(f*x + e)^2 - 3*a*cos(f*x + e) - (a*cos(f*x + e) + 4*a)*sin(f*x + e) - 4*a)*sqrt(-c*sin(f*x + e) + c))/(c^3*f*cos(f*x + e)^3 + 3*c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f - (c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f)*sin(f*x + e))","B",0
297,1,408,0,1.333987," ","integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","\frac{3 \, \sqrt{2} {\left(a \cos\left(f x + e\right)^{4} - 3 \, a \cos\left(f x + e\right)^{3} - 8 \, a \cos\left(f x + e\right)^{2} + 4 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{3} + 4 \, a \cos\left(f x + e\right)^{2} - 4 \, a \cos\left(f x + e\right) - 8 \, a\right)} \sin\left(f x + e\right) + 8 \, a\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left(3 \, a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} + 22 \, a \cos\left(f x + e\right) + {\left(3 \, a \cos\left(f x + e\right)^{2} + 10 \, a \cos\left(f x + e\right) + 32 \, a\right)} \sin\left(f x + e\right) + 32 \, a\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{384 \, {\left(c^{4} f \cos\left(f x + e\right)^{4} - 3 \, c^{4} f \cos\left(f x + e\right)^{3} - 8 \, c^{4} f \cos\left(f x + e\right)^{2} + 4 \, c^{4} f \cos\left(f x + e\right) + 8 \, c^{4} f + {\left(c^{4} f \cos\left(f x + e\right)^{3} + 4 \, c^{4} f \cos\left(f x + e\right)^{2} - 4 \, c^{4} f \cos\left(f x + e\right) - 8 \, c^{4} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/384*(3*sqrt(2)*(a*cos(f*x + e)^4 - 3*a*cos(f*x + e)^3 - 8*a*cos(f*x + e)^2 + 4*a*cos(f*x + e) + (a*cos(f*x + e)^3 + 4*a*cos(f*x + e)^2 - 4*a*cos(f*x + e) - 8*a)*sin(f*x + e) + 8*a)*sqrt(c)*log(-(c*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(3*a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 22*a*cos(f*x + e) + (3*a*cos(f*x + e)^2 + 10*a*cos(f*x + e) + 32*a)*sin(f*x + e) + 32*a)*sqrt(-c*sin(f*x + e) + c))/(c^4*f*cos(f*x + e)^4 - 3*c^4*f*cos(f*x + e)^3 - 8*c^4*f*cos(f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^4*f + (c^4*f*cos(f*x + e)^3 + 4*c^4*f*cos(f*x + e)^2 - 4*c^4*f*cos(f*x + e) - 8*c^4*f)*sin(f*x + e))","B",0
298,1,234,0,1.253747," ","integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","\frac{2 \, {\left(105 \, a^{2} c^{3} \cos\left(f x + e\right)^{6} + 245 \, a^{2} c^{3} \cos\left(f x + e\right)^{5} - 20 \, a^{2} c^{3} \cos\left(f x + e\right)^{4} + 32 \, a^{2} c^{3} \cos\left(f x + e\right)^{3} - 64 \, a^{2} c^{3} \cos\left(f x + e\right)^{2} + 256 \, a^{2} c^{3} \cos\left(f x + e\right) + 512 \, a^{2} c^{3} - {\left(105 \, a^{2} c^{3} \cos\left(f x + e\right)^{5} - 140 \, a^{2} c^{3} \cos\left(f x + e\right)^{4} - 160 \, a^{2} c^{3} \cos\left(f x + e\right)^{3} - 192 \, a^{2} c^{3} \cos\left(f x + e\right)^{2} - 256 \, a^{2} c^{3} \cos\left(f x + e\right) - 512 \, a^{2} c^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{1155 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/1155*(105*a^2*c^3*cos(f*x + e)^6 + 245*a^2*c^3*cos(f*x + e)^5 - 20*a^2*c^3*cos(f*x + e)^4 + 32*a^2*c^3*cos(f*x + e)^3 - 64*a^2*c^3*cos(f*x + e)^2 + 256*a^2*c^3*cos(f*x + e) + 512*a^2*c^3 - (105*a^2*c^3*cos(f*x + e)^5 - 140*a^2*c^3*cos(f*x + e)^4 - 160*a^2*c^3*cos(f*x + e)^3 - 192*a^2*c^3*cos(f*x + e)^2 - 256*a^2*c^3*cos(f*x + e) - 512*a^2*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","A",0
299,1,201,0,1.301293," ","integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{2 \, {\left(35 \, a^{2} c^{2} \cos\left(f x + e\right)^{5} - 5 \, a^{2} c^{2} \cos\left(f x + e\right)^{4} + 8 \, a^{2} c^{2} \cos\left(f x + e\right)^{3} - 16 \, a^{2} c^{2} \cos\left(f x + e\right)^{2} + 64 \, a^{2} c^{2} \cos\left(f x + e\right) + 128 \, a^{2} c^{2} + {\left(35 \, a^{2} c^{2} \cos\left(f x + e\right)^{4} + 40 \, a^{2} c^{2} \cos\left(f x + e\right)^{3} + 48 \, a^{2} c^{2} \cos\left(f x + e\right)^{2} + 64 \, a^{2} c^{2} \cos\left(f x + e\right) + 128 \, a^{2} c^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{315 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/315*(35*a^2*c^2*cos(f*x + e)^5 - 5*a^2*c^2*cos(f*x + e)^4 + 8*a^2*c^2*cos(f*x + e)^3 - 16*a^2*c^2*cos(f*x + e)^2 + 64*a^2*c^2*cos(f*x + e) + 128*a^2*c^2 + (35*a^2*c^2*cos(f*x + e)^4 + 40*a^2*c^2*cos(f*x + e)^3 + 48*a^2*c^2*cos(f*x + e)^2 + 64*a^2*c^2*cos(f*x + e) + 128*a^2*c^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","B",0
300,1,152,0,0.943504," ","integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(5 \, a^{2} c \cos\left(f x + e\right)^{4} - a^{2} c \cos\left(f x + e\right)^{3} + 2 \, a^{2} c \cos\left(f x + e\right)^{2} - 8 \, a^{2} c \cos\left(f x + e\right) - 16 \, a^{2} c - {\left(5 \, a^{2} c \cos\left(f x + e\right)^{3} + 6 \, a^{2} c \cos\left(f x + e\right)^{2} + 8 \, a^{2} c \cos\left(f x + e\right) + 16 \, a^{2} c\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{35 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"-2/35*(5*a^2*c*cos(f*x + e)^4 - a^2*c*cos(f*x + e)^3 + 2*a^2*c*cos(f*x + e)^2 - 8*a^2*c*cos(f*x + e) - 16*a^2*c - (5*a^2*c*cos(f*x + e)^3 + 6*a^2*c*cos(f*x + e)^2 + 8*a^2*c*cos(f*x + e) + 16*a^2*c)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","B",0
301,1,114,0,1.682871," ","integrate((a+a*sin(f*x+e))^2*(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(a^{2} \cos\left(f x + e\right)^{3} + 3 \, a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \cos\left(f x + e\right) - 4 \, a^{2} + {\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \cos\left(f x + e\right) - 4 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{5 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"-2/5*(a^2*cos(f*x + e)^3 + 3*a^2*cos(f*x + e)^2 - 2*a^2*cos(f*x + e) - 4*a^2 + (a^2*cos(f*x + e)^2 - 2*a^2*cos(f*x + e) - 4*a^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","B",0
302,1,238,0,1.534763," ","integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{2 \, {\left(\frac{3 \, \sqrt{2} {\left(a^{2} c \cos\left(f x + e\right) - a^{2} c \sin\left(f x + e\right) + a^{2} c\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) + \frac{2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)}}{\sqrt{c}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{c}} + {\left(a^{2} \cos\left(f x + e\right)^{2} - 7 \, a^{2} \cos\left(f x + e\right) - 8 \, a^{2} - {\left(a^{2} \cos\left(f x + e\right) + 8 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}\right)}}{3 \, {\left(c f \cos\left(f x + e\right) - c f \sin\left(f x + e\right) + c f\right)}}"," ",0,"2/3*(3*sqrt(2)*(a^2*c*cos(f*x + e) - a^2*c*sin(f*x + e) + a^2*c)*log(-(cos(f*x + e)^2 + (cos(f*x + e) - 2)*sin(f*x + e) + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*(cos(f*x + e) + sin(f*x + e) + 1)/sqrt(c) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(c) + (a^2*cos(f*x + e)^2 - 7*a^2*cos(f*x + e) - 8*a^2 - (a^2*cos(f*x + e) + 8*a^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c*f*cos(f*x + e) - c*f*sin(f*x + e) + c*f)","B",0
303,1,299,0,1.039378," ","integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{\frac{3 \, \sqrt{2} {\left(a^{2} c \cos\left(f x + e\right)^{2} - a^{2} c \cos\left(f x + e\right) - 2 \, a^{2} c + {\left(a^{2} c \cos\left(f x + e\right) + 2 \, a^{2} c\right)} \sin\left(f x + e\right)\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) - \frac{2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)}}{\sqrt{c}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{c}} - 4 \, {\left(a^{2} \cos\left(f x + e\right)^{2} + 2 \, a^{2} \cos\left(f x + e\right) + a^{2} - {\left(a^{2} \cos\left(f x + e\right) - a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{2 \, {\left(c^{2} f \cos\left(f x + e\right)^{2} - c^{2} f \cos\left(f x + e\right) - 2 \, c^{2} f + {\left(c^{2} f \cos\left(f x + e\right) + 2 \, c^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/2*(3*sqrt(2)*(a^2*c*cos(f*x + e)^2 - a^2*c*cos(f*x + e) - 2*a^2*c + (a^2*c*cos(f*x + e) + 2*a^2*c)*sin(f*x + e))*log(-(cos(f*x + e)^2 + (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*(cos(f*x + e) + sin(f*x + e) + 1)/sqrt(c) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(c) - 4*(a^2*cos(f*x + e)^2 + 2*a^2*cos(f*x + e) + a^2 - (a^2*cos(f*x + e) - a^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^2*f*cos(f*x + e)^2 - c^2*f*cos(f*x + e) - 2*c^2*f + (c^2*f*cos(f*x + e) + 2*c^2*f)*sin(f*x + e))","B",0
304,1,362,0,1.429390," ","integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{3 \, \sqrt{2} {\left(a^{2} \cos\left(f x + e\right)^{3} + 3 \, a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \cos\left(f x + e\right) - 4 \, a^{2} - {\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \cos\left(f x + e\right) - 4 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left(5 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2} \cos\left(f x + e\right) - 4 \, a^{2} - {\left(5 \, a^{2} \cos\left(f x + e\right) + 4 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{16 \, {\left(c^{3} f \cos\left(f x + e\right)^{3} + 3 \, c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f - {\left(c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/16*(3*sqrt(2)*(a^2*cos(f*x + e)^3 + 3*a^2*cos(f*x + e)^2 - 2*a^2*cos(f*x + e) - 4*a^2 - (a^2*cos(f*x + e)^2 - 2*a^2*cos(f*x + e) - 4*a^2)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(5*a^2*cos(f*x + e)^2 + a^2*cos(f*x + e) - 4*a^2 - (5*a^2*cos(f*x + e) + 4*a^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^3*f*cos(f*x + e)^3 + 3*c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f - (c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f)*sin(f*x + e))","B",0
305,1,440,0,1.477985," ","integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","\frac{3 \, \sqrt{2} {\left(a^{2} \cos\left(f x + e\right)^{4} - 3 \, a^{2} \cos\left(f x + e\right)^{3} - 8 \, a^{2} \cos\left(f x + e\right)^{2} + 4 \, a^{2} \cos\left(f x + e\right) + 8 \, a^{2} + {\left(a^{2} \cos\left(f x + e\right)^{3} + 4 \, a^{2} \cos\left(f x + e\right)^{2} - 4 \, a^{2} \cos\left(f x + e\right) - 8 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, {\left(3 \, a^{2} \cos\left(f x + e\right)^{3} + 25 \, a^{2} \cos\left(f x + e\right)^{2} - 10 \, a^{2} \cos\left(f x + e\right) - 32 \, a^{2} + {\left(3 \, a^{2} \cos\left(f x + e\right)^{2} - 22 \, a^{2} \cos\left(f x + e\right) - 32 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{192 \, {\left(c^{4} f \cos\left(f x + e\right)^{4} - 3 \, c^{4} f \cos\left(f x + e\right)^{3} - 8 \, c^{4} f \cos\left(f x + e\right)^{2} + 4 \, c^{4} f \cos\left(f x + e\right) + 8 \, c^{4} f + {\left(c^{4} f \cos\left(f x + e\right)^{3} + 4 \, c^{4} f \cos\left(f x + e\right)^{2} - 4 \, c^{4} f \cos\left(f x + e\right) - 8 \, c^{4} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/192*(3*sqrt(2)*(a^2*cos(f*x + e)^4 - 3*a^2*cos(f*x + e)^3 - 8*a^2*cos(f*x + e)^2 + 4*a^2*cos(f*x + e) + 8*a^2 + (a^2*cos(f*x + e)^3 + 4*a^2*cos(f*x + e)^2 - 4*a^2*cos(f*x + e) - 8*a^2)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(3*a^2*cos(f*x + e)^3 + 25*a^2*cos(f*x + e)^2 - 10*a^2*cos(f*x + e) - 32*a^2 + (3*a^2*cos(f*x + e)^2 - 22*a^2*cos(f*x + e) - 32*a^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^4*f*cos(f*x + e)^4 - 3*c^4*f*cos(f*x + e)^3 - 8*c^4*f*cos(f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^4*f + (c^4*f*cos(f*x + e)^3 + 4*c^4*f*cos(f*x + e)^2 - 4*c^4*f*cos(f*x + e) - 8*c^4*f)*sin(f*x + e))","B",0
306,1,523,0,1.108735," ","integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^(9/2),x, algorithm=""fricas"")","\frac{3 \, \sqrt{2} {\left(a^{2} \cos\left(f x + e\right)^{5} + 5 \, a^{2} \cos\left(f x + e\right)^{4} - 8 \, a^{2} \cos\left(f x + e\right)^{3} - 20 \, a^{2} \cos\left(f x + e\right)^{2} + 8 \, a^{2} \cos\left(f x + e\right) + 16 \, a^{2} - {\left(a^{2} \cos\left(f x + e\right)^{4} - 4 \, a^{2} \cos\left(f x + e\right)^{3} - 12 \, a^{2} \cos\left(f x + e\right)^{2} + 8 \, a^{2} \cos\left(f x + e\right) + 16 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, {\left(3 \, a^{2} \cos\left(f x + e\right)^{4} + 13 \, a^{2} \cos\left(f x + e\right)^{3} + 86 \, a^{2} \cos\left(f x + e\right)^{2} - 52 \, a^{2} \cos\left(f x + e\right) - 128 \, a^{2} - {\left(3 \, a^{2} \cos\left(f x + e\right)^{3} - 10 \, a^{2} \cos\left(f x + e\right)^{2} + 76 \, a^{2} \cos\left(f x + e\right) + 128 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{1024 \, {\left(c^{5} f \cos\left(f x + e\right)^{5} + 5 \, c^{5} f \cos\left(f x + e\right)^{4} - 8 \, c^{5} f \cos\left(f x + e\right)^{3} - 20 \, c^{5} f \cos\left(f x + e\right)^{2} + 8 \, c^{5} f \cos\left(f x + e\right) + 16 \, c^{5} f - {\left(c^{5} f \cos\left(f x + e\right)^{4} - 4 \, c^{5} f \cos\left(f x + e\right)^{3} - 12 \, c^{5} f \cos\left(f x + e\right)^{2} + 8 \, c^{5} f \cos\left(f x + e\right) + 16 \, c^{5} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/1024*(3*sqrt(2)*(a^2*cos(f*x + e)^5 + 5*a^2*cos(f*x + e)^4 - 8*a^2*cos(f*x + e)^3 - 20*a^2*cos(f*x + e)^2 + 8*a^2*cos(f*x + e) + 16*a^2 - (a^2*cos(f*x + e)^4 - 4*a^2*cos(f*x + e)^3 - 12*a^2*cos(f*x + e)^2 + 8*a^2*cos(f*x + e) + 16*a^2)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(3*a^2*cos(f*x + e)^4 + 13*a^2*cos(f*x + e)^3 + 86*a^2*cos(f*x + e)^2 - 52*a^2*cos(f*x + e) - 128*a^2 - (3*a^2*cos(f*x + e)^3 - 10*a^2*cos(f*x + e)^2 + 76*a^2*cos(f*x + e) + 128*a^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^5*f*cos(f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f)*sin(f*x + e))","B",0
307,1,265,0,1.547160," ","integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","\frac{2 \, {\left(231 \, a^{3} c^{3} \cos\left(f x + e\right)^{7} - 21 \, a^{3} c^{3} \cos\left(f x + e\right)^{6} + 28 \, a^{3} c^{3} \cos\left(f x + e\right)^{5} - 40 \, a^{3} c^{3} \cos\left(f x + e\right)^{4} + 64 \, a^{3} c^{3} \cos\left(f x + e\right)^{3} - 128 \, a^{3} c^{3} \cos\left(f x + e\right)^{2} + 512 \, a^{3} c^{3} \cos\left(f x + e\right) + 1024 \, a^{3} c^{3} + {\left(231 \, a^{3} c^{3} \cos\left(f x + e\right)^{6} + 252 \, a^{3} c^{3} \cos\left(f x + e\right)^{5} + 280 \, a^{3} c^{3} \cos\left(f x + e\right)^{4} + 320 \, a^{3} c^{3} \cos\left(f x + e\right)^{3} + 384 \, a^{3} c^{3} \cos\left(f x + e\right)^{2} + 512 \, a^{3} c^{3} \cos\left(f x + e\right) + 1024 \, a^{3} c^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{3003 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/3003*(231*a^3*c^3*cos(f*x + e)^7 - 21*a^3*c^3*cos(f*x + e)^6 + 28*a^3*c^3*cos(f*x + e)^5 - 40*a^3*c^3*cos(f*x + e)^4 + 64*a^3*c^3*cos(f*x + e)^3 - 128*a^3*c^3*cos(f*x + e)^2 + 512*a^3*c^3*cos(f*x + e) + 1024*a^3*c^3 + (231*a^3*c^3*cos(f*x + e)^6 + 252*a^3*c^3*cos(f*x + e)^5 + 280*a^3*c^3*cos(f*x + e)^4 + 320*a^3*c^3*cos(f*x + e)^3 + 384*a^3*c^3*cos(f*x + e)^2 + 512*a^3*c^3*cos(f*x + e) + 1024*a^3*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","B",0
308,1,234,0,1.336231," ","integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(63 \, a^{3} c^{2} \cos\left(f x + e\right)^{6} - 7 \, a^{3} c^{2} \cos\left(f x + e\right)^{5} + 10 \, a^{3} c^{2} \cos\left(f x + e\right)^{4} - 16 \, a^{3} c^{2} \cos\left(f x + e\right)^{3} + 32 \, a^{3} c^{2} \cos\left(f x + e\right)^{2} - 128 \, a^{3} c^{2} \cos\left(f x + e\right) - 256 \, a^{3} c^{2} - {\left(63 \, a^{3} c^{2} \cos\left(f x + e\right)^{5} + 70 \, a^{3} c^{2} \cos\left(f x + e\right)^{4} + 80 \, a^{3} c^{2} \cos\left(f x + e\right)^{3} + 96 \, a^{3} c^{2} \cos\left(f x + e\right)^{2} + 128 \, a^{3} c^{2} \cos\left(f x + e\right) + 256 \, a^{3} c^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{693 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"-2/693*(63*a^3*c^2*cos(f*x + e)^6 - 7*a^3*c^2*cos(f*x + e)^5 + 10*a^3*c^2*cos(f*x + e)^4 - 16*a^3*c^2*cos(f*x + e)^3 + 32*a^3*c^2*cos(f*x + e)^2 - 128*a^3*c^2*cos(f*x + e) - 256*a^3*c^2 - (63*a^3*c^2*cos(f*x + e)^5 + 70*a^3*c^2*cos(f*x + e)^4 + 80*a^3*c^2*cos(f*x + e)^3 + 96*a^3*c^2*cos(f*x + e)^2 + 128*a^3*c^2*cos(f*x + e) + 256*a^3*c^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","B",0
309,1,179,0,1.006214," ","integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(7 \, a^{3} c \cos\left(f x + e\right)^{5} + 17 \, a^{3} c \cos\left(f x + e\right)^{4} - 2 \, a^{3} c \cos\left(f x + e\right)^{3} + 4 \, a^{3} c \cos\left(f x + e\right)^{2} - 16 \, a^{3} c \cos\left(f x + e\right) - 32 \, a^{3} c + {\left(7 \, a^{3} c \cos\left(f x + e\right)^{4} - 10 \, a^{3} c \cos\left(f x + e\right)^{3} - 12 \, a^{3} c \cos\left(f x + e\right)^{2} - 16 \, a^{3} c \cos\left(f x + e\right) - 32 \, a^{3} c\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{63 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"-2/63*(7*a^3*c*cos(f*x + e)^5 + 17*a^3*c*cos(f*x + e)^4 - 2*a^3*c*cos(f*x + e)^3 + 4*a^3*c*cos(f*x + e)^2 - 16*a^3*c*cos(f*x + e) - 32*a^3*c + (7*a^3*c*cos(f*x + e)^4 - 10*a^3*c*cos(f*x + e)^3 - 12*a^3*c*cos(f*x + e)^2 - 16*a^3*c*cos(f*x + e) - 32*a^3*c)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","B",0
310,1,141,0,1.254312," ","integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{2 \, {\left(a^{3} \cos\left(f x + e\right)^{4} - 3 \, a^{3} \cos\left(f x + e\right)^{3} - 8 \, a^{3} \cos\left(f x + e\right)^{2} + 4 \, a^{3} \cos\left(f x + e\right) + 8 \, a^{3} - {\left(a^{3} \cos\left(f x + e\right)^{3} + 4 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} \cos\left(f x + e\right) - 8 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{7 \, {\left(f \cos\left(f x + e\right) - f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/7*(a^3*cos(f*x + e)^4 - 3*a^3*cos(f*x + e)^3 - 8*a^3*cos(f*x + e)^2 + 4*a^3*cos(f*x + e) + 8*a^3 - (a^3*cos(f*x + e)^3 + 4*a^3*cos(f*x + e)^2 - 4*a^3*cos(f*x + e) - 8*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)","B",0
311,1,265,0,1.401699," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{2 \, {\left(\frac{30 \, \sqrt{2} {\left(a^{3} c \cos\left(f x + e\right) - a^{3} c \sin\left(f x + e\right) + a^{3} c\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) + \frac{2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)}}{\sqrt{c}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{c}} + {\left(3 \, a^{3} \cos\left(f x + e\right)^{3} + 19 \, a^{3} \cos\left(f x + e\right)^{2} - 76 \, a^{3} \cos\left(f x + e\right) - 92 \, a^{3} + {\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 16 \, a^{3} \cos\left(f x + e\right) - 92 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}\right)}}{15 \, {\left(c f \cos\left(f x + e\right) - c f \sin\left(f x + e\right) + c f\right)}}"," ",0,"2/15*(30*sqrt(2)*(a^3*c*cos(f*x + e) - a^3*c*sin(f*x + e) + a^3*c)*log(-(cos(f*x + e)^2 + (cos(f*x + e) - 2)*sin(f*x + e) + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*(cos(f*x + e) + sin(f*x + e) + 1)/sqrt(c) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(c) + (3*a^3*cos(f*x + e)^3 + 19*a^3*cos(f*x + e)^2 - 76*a^3*cos(f*x + e) - 92*a^3 + (3*a^3*cos(f*x + e)^2 - 16*a^3*cos(f*x + e) - 92*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c*f*cos(f*x + e) - c*f*sin(f*x + e) + c*f)","B",0
312,1,326,0,0.843627," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{\frac{15 \, \sqrt{2} {\left(a^{3} c \cos\left(f x + e\right)^{2} - a^{3} c \cos\left(f x + e\right) - 2 \, a^{3} c + {\left(a^{3} c \cos\left(f x + e\right) + 2 \, a^{3} c\right)} \sin\left(f x + e\right)\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) - \frac{2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)}}{\sqrt{c}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{c}} - 2 \, {\left(a^{3} \cos\left(f x + e\right)^{3} + 13 \, a^{3} \cos\left(f x + e\right)^{2} + 18 \, a^{3} \cos\left(f x + e\right) + 6 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 12 \, a^{3} \cos\left(f x + e\right) + 6 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, {\left(c^{2} f \cos\left(f x + e\right)^{2} - c^{2} f \cos\left(f x + e\right) - 2 \, c^{2} f + {\left(c^{2} f \cos\left(f x + e\right) + 2 \, c^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/3*(15*sqrt(2)*(a^3*c*cos(f*x + e)^2 - a^3*c*cos(f*x + e) - 2*a^3*c + (a^3*c*cos(f*x + e) + 2*a^3*c)*sin(f*x + e))*log(-(cos(f*x + e)^2 + (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*(cos(f*x + e) + sin(f*x + e) + 1)/sqrt(c) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(c) - 2*(a^3*cos(f*x + e)^3 + 13*a^3*cos(f*x + e)^2 + 18*a^3*cos(f*x + e) + 6*a^3 + (a^3*cos(f*x + e)^2 - 12*a^3*cos(f*x + e) + 6*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^2*f*cos(f*x + e)^2 - c^2*f*cos(f*x + e) - 2*c^2*f + (c^2*f*cos(f*x + e) + 2*c^2*f)*sin(f*x + e))","B",0
313,1,388,0,0.927347," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{15 \, \sqrt{2} {\left(a^{3} \cos\left(f x + e\right)^{3} + 3 \, a^{3} \cos\left(f x + e\right)^{2} - 2 \, a^{3} \cos\left(f x + e\right) - 4 \, a^{3} - {\left(a^{3} \cos\left(f x + e\right)^{2} - 2 \, a^{3} \cos\left(f x + e\right) - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, {\left(4 \, a^{3} \cos\left(f x + e\right)^{3} - 13 \, a^{3} \cos\left(f x + e\right)^{2} - 13 \, a^{3} \cos\left(f x + e\right) + 4 \, a^{3} + {\left(4 \, a^{3} \cos\left(f x + e\right)^{2} + 17 \, a^{3} \cos\left(f x + e\right) + 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{8 \, {\left(c^{3} f \cos\left(f x + e\right)^{3} + 3 \, c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f - {\left(c^{3} f \cos\left(f x + e\right)^{2} - 2 \, c^{3} f \cos\left(f x + e\right) - 4 \, c^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/8*(15*sqrt(2)*(a^3*cos(f*x + e)^3 + 3*a^3*cos(f*x + e)^2 - 2*a^3*cos(f*x + e) - 4*a^3 - (a^3*cos(f*x + e)^2 - 2*a^3*cos(f*x + e) - 4*a^3)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(4*a^3*cos(f*x + e)^3 - 13*a^3*cos(f*x + e)^2 - 13*a^3*cos(f*x + e) + 4*a^3 + (4*a^3*cos(f*x + e)^2 + 17*a^3*cos(f*x + e) + 4*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^3*f*cos(f*x + e)^3 + 3*c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f - (c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f)*sin(f*x + e))","B",0
314,1,440,0,1.083736," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","\frac{15 \, \sqrt{2} {\left(a^{3} \cos\left(f x + e\right)^{4} - 3 \, a^{3} \cos\left(f x + e\right)^{3} - 8 \, a^{3} \cos\left(f x + e\right)^{2} + 4 \, a^{3} \cos\left(f x + e\right) + 8 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{3} + 4 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} \cos\left(f x + e\right) - 8 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, {\left(33 \, a^{3} \cos\left(f x + e\right)^{3} + 19 \, a^{3} \cos\left(f x + e\right)^{2} - 46 \, a^{3} \cos\left(f x + e\right) - 32 \, a^{3} + {\left(33 \, a^{3} \cos\left(f x + e\right)^{2} + 14 \, a^{3} \cos\left(f x + e\right) - 32 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{96 \, {\left(c^{4} f \cos\left(f x + e\right)^{4} - 3 \, c^{4} f \cos\left(f x + e\right)^{3} - 8 \, c^{4} f \cos\left(f x + e\right)^{2} + 4 \, c^{4} f \cos\left(f x + e\right) + 8 \, c^{4} f + {\left(c^{4} f \cos\left(f x + e\right)^{3} + 4 \, c^{4} f \cos\left(f x + e\right)^{2} - 4 \, c^{4} f \cos\left(f x + e\right) - 8 \, c^{4} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/96*(15*sqrt(2)*(a^3*cos(f*x + e)^4 - 3*a^3*cos(f*x + e)^3 - 8*a^3*cos(f*x + e)^2 + 4*a^3*cos(f*x + e) + 8*a^3 + (a^3*cos(f*x + e)^3 + 4*a^3*cos(f*x + e)^2 - 4*a^3*cos(f*x + e) - 8*a^3)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(33*a^3*cos(f*x + e)^3 + 19*a^3*cos(f*x + e)^2 - 46*a^3*cos(f*x + e) - 32*a^3 + (33*a^3*cos(f*x + e)^2 + 14*a^3*cos(f*x + e) - 32*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^4*f*cos(f*x + e)^4 - 3*c^4*f*cos(f*x + e)^3 - 8*c^4*f*cos(f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^4*f + (c^4*f*cos(f*x + e)^3 + 4*c^4*f*cos(f*x + e)^2 - 4*c^4*f*cos(f*x + e) - 8*c^4*f)*sin(f*x + e))","B",0
315,1,523,0,1.365600," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(9/2),x, algorithm=""fricas"")","\frac{15 \, \sqrt{2} {\left(a^{3} \cos\left(f x + e\right)^{5} + 5 \, a^{3} \cos\left(f x + e\right)^{4} - 8 \, a^{3} \cos\left(f x + e\right)^{3} - 20 \, a^{3} \cos\left(f x + e\right)^{2} + 8 \, a^{3} \cos\left(f x + e\right) + 16 \, a^{3} - {\left(a^{3} \cos\left(f x + e\right)^{4} - 4 \, a^{3} \cos\left(f x + e\right)^{3} - 12 \, a^{3} \cos\left(f x + e\right)^{2} + 8 \, a^{3} \cos\left(f x + e\right) + 16 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left(15 \, a^{3} \cos\left(f x + e\right)^{4} - 191 \, a^{3} \cos\left(f x + e\right)^{3} - 338 \, a^{3} \cos\left(f x + e\right)^{2} + 252 \, a^{3} \cos\left(f x + e\right) + 384 \, a^{3} - {\left(15 \, a^{3} \cos\left(f x + e\right)^{3} + 206 \, a^{3} \cos\left(f x + e\right)^{2} - 132 \, a^{3} \cos\left(f x + e\right) - 384 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{1536 \, {\left(c^{5} f \cos\left(f x + e\right)^{5} + 5 \, c^{5} f \cos\left(f x + e\right)^{4} - 8 \, c^{5} f \cos\left(f x + e\right)^{3} - 20 \, c^{5} f \cos\left(f x + e\right)^{2} + 8 \, c^{5} f \cos\left(f x + e\right) + 16 \, c^{5} f - {\left(c^{5} f \cos\left(f x + e\right)^{4} - 4 \, c^{5} f \cos\left(f x + e\right)^{3} - 12 \, c^{5} f \cos\left(f x + e\right)^{2} + 8 \, c^{5} f \cos\left(f x + e\right) + 16 \, c^{5} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/1536*(15*sqrt(2)*(a^3*cos(f*x + e)^5 + 5*a^3*cos(f*x + e)^4 - 8*a^3*cos(f*x + e)^3 - 20*a^3*cos(f*x + e)^2 + 8*a^3*cos(f*x + e) + 16*a^3 - (a^3*cos(f*x + e)^4 - 4*a^3*cos(f*x + e)^3 - 12*a^3*cos(f*x + e)^2 + 8*a^3*cos(f*x + e) + 16*a^3)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(15*a^3*cos(f*x + e)^4 - 191*a^3*cos(f*x + e)^3 - 338*a^3*cos(f*x + e)^2 + 252*a^3*cos(f*x + e) + 384*a^3 - (15*a^3*cos(f*x + e)^3 + 206*a^3*cos(f*x + e)^2 - 132*a^3*cos(f*x + e) - 384*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^5*f*cos(f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f)*sin(f*x + e))","B",0
316,1,600,0,1.571333," ","integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(11/2),x, algorithm=""fricas"")","\frac{15 \, \sqrt{2} {\left(a^{3} \cos\left(f x + e\right)^{6} - 5 \, a^{3} \cos\left(f x + e\right)^{5} - 18 \, a^{3} \cos\left(f x + e\right)^{4} + 20 \, a^{3} \cos\left(f x + e\right)^{3} + 48 \, a^{3} \cos\left(f x + e\right)^{2} - 16 \, a^{3} \cos\left(f x + e\right) - 32 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{5} + 6 \, a^{3} \cos\left(f x + e\right)^{4} - 12 \, a^{3} \cos\left(f x + e\right)^{3} - 32 \, a^{3} \cos\left(f x + e\right)^{2} + 16 \, a^{3} \cos\left(f x + e\right) + 32 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left(15 \, a^{3} \cos\left(f x + e\right)^{5} - 65 \, a^{3} \cos\left(f x + e\right)^{4} + 812 \, a^{3} \cos\left(f x + e\right)^{3} + 1796 \, a^{3} \cos\left(f x + e\right)^{2} - 1144 \, a^{3} \cos\left(f x + e\right) - 2048 \, a^{3} + {\left(15 \, a^{3} \cos\left(f x + e\right)^{4} + 80 \, a^{3} \cos\left(f x + e\right)^{3} + 892 \, a^{3} \cos\left(f x + e\right)^{2} - 904 \, a^{3} \cos\left(f x + e\right) - 2048 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{10240 \, {\left(c^{6} f \cos\left(f x + e\right)^{6} - 5 \, c^{6} f \cos\left(f x + e\right)^{5} - 18 \, c^{6} f \cos\left(f x + e\right)^{4} + 20 \, c^{6} f \cos\left(f x + e\right)^{3} + 48 \, c^{6} f \cos\left(f x + e\right)^{2} - 16 \, c^{6} f \cos\left(f x + e\right) - 32 \, c^{6} f + {\left(c^{6} f \cos\left(f x + e\right)^{5} + 6 \, c^{6} f \cos\left(f x + e\right)^{4} - 12 \, c^{6} f \cos\left(f x + e\right)^{3} - 32 \, c^{6} f \cos\left(f x + e\right)^{2} + 16 \, c^{6} f \cos\left(f x + e\right) + 32 \, c^{6} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/10240*(15*sqrt(2)*(a^3*cos(f*x + e)^6 - 5*a^3*cos(f*x + e)^5 - 18*a^3*cos(f*x + e)^4 + 20*a^3*cos(f*x + e)^3 + 48*a^3*cos(f*x + e)^2 - 16*a^3*cos(f*x + e) - 32*a^3 + (a^3*cos(f*x + e)^5 + 6*a^3*cos(f*x + e)^4 - 12*a^3*cos(f*x + e)^3 - 32*a^3*cos(f*x + e)^2 + 16*a^3*cos(f*x + e) + 32*a^3)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(15*a^3*cos(f*x + e)^5 - 65*a^3*cos(f*x + e)^4 + 812*a^3*cos(f*x + e)^3 + 1796*a^3*cos(f*x + e)^2 - 1144*a^3*cos(f*x + e) - 2048*a^3 + (15*a^3*cos(f*x + e)^4 + 80*a^3*cos(f*x + e)^3 + 892*a^3*cos(f*x + e)^2 - 904*a^3*cos(f*x + e) - 2048*a^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^6*f*cos(f*x + e)^6 - 5*c^6*f*cos(f*x + e)^5 - 18*c^6*f*cos(f*x + e)^4 + 20*c^6*f*cos(f*x + e)^3 + 48*c^6*f*cos(f*x + e)^2 - 16*c^6*f*cos(f*x + e) - 32*c^6*f + (c^6*f*cos(f*x + e)^5 + 6*c^6*f*cos(f*x + e)^4 - 12*c^6*f*cos(f*x + e)^3 - 32*c^6*f*cos(f*x + e)^2 + 16*c^6*f*cos(f*x + e) + 32*c^6*f)*sin(f*x + e))","B",0
317,1,74,0,1.608252," ","integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{2 \, {\left(7 \, c^{3} \cos\left(f x + e\right)^{2} + 84 \, c^{3} - {\left(c^{3} \cos\left(f x + e\right)^{2} - 44 \, c^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{5 \, a f \cos\left(f x + e\right)}"," ",0,"-2/5*(7*c^3*cos(f*x + e)^2 + 84*c^3 - (c^3*cos(f*x + e)^2 - 44*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(a*f*cos(f*x + e))","A",0
318,1,58,0,1.573360," ","integrate((c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{2 \, {\left(c^{2} \cos\left(f x + e\right)^{2} + 10 \, c^{2} \sin\left(f x + e\right) + 22 \, c^{2}\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, a f \cos\left(f x + e\right)}"," ",0,"-2/3*(c^2*cos(f*x + e)^2 + 10*c^2*sin(f*x + e) + 22*c^2)*sqrt(-c*sin(f*x + e) + c)/(a*f*cos(f*x + e))","A",0
319,1,41,0,1.314035," ","integrate((c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{2 \, {\left(c \sin\left(f x + e\right) + 3 \, c\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{a f \cos\left(f x + e\right)}"," ",0,"-2*(c*sin(f*x + e) + 3*c)*sqrt(-c*sin(f*x + e) + c)/(a*f*cos(f*x + e))","A",0
320,1,29,0,0.880499," ","integrate((c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-c \sin\left(f x + e\right) + c}}{a f \cos\left(f x + e\right)}"," ",0,"-2*sqrt(-c*sin(f*x + e) + c)/(a*f*cos(f*x + e))","A",0
321,1,154,0,1.590751," ","integrate(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{2} \sqrt{c} \cos\left(f x + e\right) \log\left(-\frac{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) + \frac{2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)}}{\sqrt{c}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, \sqrt{-c \sin\left(f x + e\right) + c}}{4 \, a c f \cos\left(f x + e\right)}"," ",0,"1/4*(sqrt(2)*sqrt(c)*cos(f*x + e)*log(-(cos(f*x + e)^2 + (cos(f*x + e) - 2)*sin(f*x + e) + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*(cos(f*x + e) + sin(f*x + e) + 1)/sqrt(c) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*sqrt(-c*sin(f*x + e) + c))/(a*c*f*cos(f*x + e))","B",0
322,1,207,0,0.975046," ","integrate(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{3 \, \sqrt{2} {\left(\cos\left(f x + e\right) \sin\left(f x + e\right) - \cos\left(f x + e\right)\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, \sqrt{-c \sin\left(f x + e\right) + c} {\left(3 \, \sin\left(f x + e\right) - 1\right)}}{16 \, {\left(a c^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) - a c^{2} f \cos\left(f x + e\right)\right)}}"," ",0,"1/16*(3*sqrt(2)*(cos(f*x + e)*sin(f*x + e) - cos(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*sqrt(-c*sin(f*x + e) + c)*(3*sin(f*x + e) - 1))/(a*c^2*f*cos(f*x + e)*sin(f*x + e) - a*c^2*f*cos(f*x + e))","B",0
323,1,241,0,1.658488," ","integrate(1/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{15 \, \sqrt{2} {\left(\cos\left(f x + e\right)^{3} + 2 \, \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right)\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, {\left(15 \, \cos\left(f x + e\right)^{2} + 20 \, \sin\left(f x + e\right) - 12\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{128 \, {\left(a c^{3} f \cos\left(f x + e\right)^{3} + 2 \, a c^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a c^{3} f \cos\left(f x + e\right)\right)}}"," ",0,"1/128*(15*sqrt(2)*(cos(f*x + e)^3 + 2*cos(f*x + e)*sin(f*x + e) - 2*cos(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(15*cos(f*x + e)^2 + 20*sin(f*x + e) - 12)*sqrt(-c*sin(f*x + e) + c))/(a*c^3*f*cos(f*x + e)^3 + 2*a*c^3*f*cos(f*x + e)*sin(f*x + e) - 2*a*c^3*f*cos(f*x + e))","A",0
324,1,104,0,1.561378," ","integrate((c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, c^{4} \cos\left(f x + e\right)^{4} - 264 \, c^{4} \cos\left(f x + e\right)^{2} + 984 \, c^{4} + 28 \, {\left(c^{4} \cos\left(f x + e\right)^{2} + 38 \, c^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{15 \, {\left(a^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) + a^{2} f \cos\left(f x + e\right)\right)}}"," ",0,"2/15*(3*c^4*cos(f*x + e)^4 - 264*c^4*cos(f*x + e)^2 + 984*c^4 + 28*(c^4*cos(f*x + e)^2 + 38*c^4)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(a^2*f*cos(f*x + e)*sin(f*x + e) + a^2*f*cos(f*x + e))","A",0
325,1,91,0,1.378766," ","integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{2 \, {\left(15 \, c^{3} \cos\left(f x + e\right)^{2} - 60 \, c^{3} - {\left(c^{3} \cos\left(f x + e\right)^{2} + 68 \, c^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, {\left(a^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) + a^{2} f \cos\left(f x + e\right)\right)}}"," ",0,"-2/3*(15*c^3*cos(f*x + e)^2 - 60*c^3 - (c^3*cos(f*x + e)^2 + 68*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(a^2*f*cos(f*x + e)*sin(f*x + e) + a^2*f*cos(f*x + e))","A",0
326,1,76,0,1.466453," ","integrate((c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{2 \, {\left(3 \, c^{2} \cos\left(f x + e\right)^{2} - 18 \, c^{2} \sin\left(f x + e\right) - 14 \, c^{2}\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, {\left(a^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) + a^{2} f \cos\left(f x + e\right)\right)}}"," ",0,"-2/3*(3*c^2*cos(f*x + e)^2 - 18*c^2*sin(f*x + e) - 14*c^2)*sqrt(-c*sin(f*x + e) + c)/(a^2*f*cos(f*x + e)*sin(f*x + e) + a^2*f*cos(f*x + e))","A",0
327,1,57,0,1.467415," ","integrate((c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, c \sin\left(f x + e\right) + c\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, {\left(a^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) + a^{2} f \cos\left(f x + e\right)\right)}}"," ",0,"2/3*(3*c*sin(f*x + e) + c)*sqrt(-c*sin(f*x + e) + c)/(a^2*f*cos(f*x + e)*sin(f*x + e) + a^2*f*cos(f*x + e))","A",0
328,1,46,0,1.426649," ","integrate((c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{2 \, \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, {\left(a^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) + a^{2} f \cos\left(f x + e\right)\right)}}"," ",0,"-2/3*sqrt(-c*sin(f*x + e) + c)/(a^2*f*cos(f*x + e)*sin(f*x + e) + a^2*f*cos(f*x + e))","A",0
329,1,204,0,1.578825," ","integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{3 \, \sqrt{2} {\left(\cos\left(f x + e\right) \sin\left(f x + e\right) + \cos\left(f x + e\right)\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, \sqrt{-c \sin\left(f x + e\right) + c} {\left(3 \, \sin\left(f x + e\right) + 5\right)}}{24 \, {\left(a^{2} c f \cos\left(f x + e\right) \sin\left(f x + e\right) + a^{2} c f \cos\left(f x + e\right)\right)}}"," ",0,"1/24*(3*sqrt(2)*(cos(f*x + e)*sin(f*x + e) + cos(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*sqrt(-c*sin(f*x + e) + c)*(3*sin(f*x + e) + 5))/(a^2*c*f*cos(f*x + e)*sin(f*x + e) + a^2*c*f*cos(f*x + e))","A",0
330,1,186,0,1.439848," ","integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{15 \, \sqrt{2} \sqrt{c} \cos\left(f x + e\right)^{3} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, {\left(15 \, \cos\left(f x + e\right)^{2} - 10 \, \sin\left(f x + e\right) - 2\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{96 \, a^{2} c^{2} f \cos\left(f x + e\right)^{3}}"," ",0,"1/96*(15*sqrt(2)*sqrt(c)*cos(f*x + e)^3*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(15*cos(f*x + e)^2 - 10*sin(f*x + e) - 2)*sqrt(-c*sin(f*x + e) + c))/(a^2*c^2*f*cos(f*x + e)^3)","A",0
331,1,241,0,1.639359," ","integrate(1/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{105 \, \sqrt{2} {\left(\cos\left(f x + e\right)^{3} \sin\left(f x + e\right) - \cos\left(f x + e\right)^{3}\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left(35 \, \cos\left(f x + e\right)^{2} - 7 \, {\left(15 \, \cos\left(f x + e\right)^{2} + 8\right)} \sin\left(f x + e\right) + 8\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{768 \, {\left(a^{2} c^{3} f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) - a^{2} c^{3} f \cos\left(f x + e\right)^{3}\right)}}"," ",0,"1/768*(105*sqrt(2)*(cos(f*x + e)^3*sin(f*x + e) - cos(f*x + e)^3)*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(35*cos(f*x + e)^2 - 7*(15*cos(f*x + e)^2 + 8)*sin(f*x + e) + 8)*sqrt(-c*sin(f*x + e) + c))/(a^2*c^3*f*cos(f*x + e)^3*sin(f*x + e) - a^2*c^3*f*cos(f*x + e)^3)","A",0
332,1,119,0,1.658998," ","integrate((c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{2 \, {\left(5 \, c^{4} \cos\left(f x + e\right)^{4} + 680 \, c^{4} \cos\left(f x + e\right)^{2} - 1048 \, c^{4} + 100 \, {\left(c^{4} \cos\left(f x + e\right)^{2} - 10 \, c^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} - 2 \, a^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{3} f \cos\left(f x + e\right)\right)}}"," ",0,"-2/15*(5*c^4*cos(f*x + e)^4 + 680*c^4*cos(f*x + e)^2 - 1048*c^4 + 100*(c^4*cos(f*x + e)^2 - 10*c^4)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(a^3*f*cos(f*x + e)^3 - 2*a^3*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*f*cos(f*x + e))","A",0
333,1,106,0,0.935215," ","integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{2 \, {\left(45 \, c^{3} \cos\left(f x + e\right)^{2} - 68 \, c^{3} + 5 \, {\left(c^{3} \cos\left(f x + e\right)^{2} - 12 \, c^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{5 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} - 2 \, a^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{3} f \cos\left(f x + e\right)\right)}}"," ",0,"-2/5*(45*c^3*cos(f*x + e)^2 - 68*c^3 + 5*(c^3*cos(f*x + e)^2 - 12*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(a^3*f*cos(f*x + e)^3 - 2*a^3*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*f*cos(f*x + e))","A",0
334,1,91,0,1.523772," ","integrate((c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{2 \, {\left(15 \, c^{2} \cos\left(f x + e\right)^{2} - 10 \, c^{2} \sin\left(f x + e\right) - 22 \, c^{2}\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} - 2 \, a^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{3} f \cos\left(f x + e\right)\right)}}"," ",0,"-2/15*(15*c^2*cos(f*x + e)^2 - 10*c^2*sin(f*x + e) - 22*c^2)*sqrt(-c*sin(f*x + e) + c)/(a^3*f*cos(f*x + e)^3 - 2*a^3*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*f*cos(f*x + e))","A",0
335,1,74,0,1.361893," ","integrate((c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{2 \, {\left(5 \, c \sin\left(f x + e\right) - c\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} - 2 \, a^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{3} f \cos\left(f x + e\right)\right)}}"," ",0,"-2/15*(5*c*sin(f*x + e) - c)*sqrt(-c*sin(f*x + e) + c)/(a^3*f*cos(f*x + e)^3 - 2*a^3*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*f*cos(f*x + e))","A",0
336,1,61,0,1.439903," ","integrate((c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{2 \, \sqrt{-c \sin\left(f x + e\right) + c}}{5 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} - 2 \, a^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{3} f \cos\left(f x + e\right)\right)}}"," ",0,"2/5*sqrt(-c*sin(f*x + e) + c)/(a^3*f*cos(f*x + e)^3 - 2*a^3*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*f*cos(f*x + e))","A",0
337,1,241,0,1.604135," ","integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{15 \, \sqrt{2} {\left(\cos\left(f x + e\right)^{3} - 2 \, \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right)\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, {\left(15 \, \cos\left(f x + e\right)^{2} - 40 \, \sin\left(f x + e\right) - 52\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{240 \, {\left(a^{3} c f \cos\left(f x + e\right)^{3} - 2 \, a^{3} c f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{3} c f \cos\left(f x + e\right)\right)}}"," ",0,"1/240*(15*sqrt(2)*(cos(f*x + e)^3 - 2*cos(f*x + e)*sin(f*x + e) - 2*cos(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(15*cos(f*x + e)^2 - 40*sin(f*x + e) - 52)*sqrt(-c*sin(f*x + e) + c))/(a^3*c*f*cos(f*x + e)^3 - 2*a^3*c*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*c*f*cos(f*x + e))","A",0
338,1,238,0,1.803100," ","integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{105 \, \sqrt{2} {\left(\cos\left(f x + e\right)^{3} \sin\left(f x + e\right) + \cos\left(f x + e\right)^{3}\right)} \sqrt{c} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, {\left(175 \, \cos\left(f x + e\right)^{2} + 21 \, {\left(5 \, \cos\left(f x + e\right)^{2} - 4\right)} \sin\left(f x + e\right) - 36\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{960 \, {\left(a^{3} c^{2} f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) + a^{3} c^{2} f \cos\left(f x + e\right)^{3}\right)}}"," ",0,"1/960*(105*sqrt(2)*(cos(f*x + e)^3*sin(f*x + e) + cos(f*x + e)^3)*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(175*cos(f*x + e)^2 + 21*(5*cos(f*x + e)^2 - 4)*sin(f*x + e) - 36)*sqrt(-c*sin(f*x + e) + c))/(a^3*c^2*f*cos(f*x + e)^3*sin(f*x + e) + a^3*c^2*f*cos(f*x + e)^3)","A",0
339,1,208,0,1.486943," ","integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{315 \, \sqrt{2} \sqrt{c} \cos\left(f x + e\right)^{5} \log\left(-\frac{c \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{-c \sin\left(f x + e\right) + c} \sqrt{c} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)} + 3 \, c \cos\left(f x + e\right) + {\left(c \cos\left(f x + e\right) - 2 \, c\right)} \sin\left(f x + e\right) + 2 \, c}{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, {\left(315 \, \cos\left(f x + e\right)^{4} - 42 \, \cos\left(f x + e\right)^{2} - 6 \, {\left(35 \, \cos\left(f x + e\right)^{2} + 24\right)} \sin\left(f x + e\right) - 16\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{2560 \, a^{3} c^{3} f \cos\left(f x + e\right)^{5}}"," ",0,"1/2560*(315*sqrt(2)*sqrt(c)*cos(f*x + e)^5*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(315*cos(f*x + e)^4 - 42*cos(f*x + e)^2 - 6*(35*cos(f*x + e)^2 + 24)*sin(f*x + e) - 16)*sqrt(-c*sin(f*x + e) + c))/(a^3*c^3*f*cos(f*x + e)^5)","A",0
340,1,95,0,1.274737," ","integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","-\frac{{\left(c^{3} \cos\left(f x + e\right)^{4} - 8 \, c^{3} \cos\left(f x + e\right)^{2} + 7 \, c^{3} + 4 \, {\left(c^{3} \cos\left(f x + e\right)^{2} - 2 \, c^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{4 \, f \cos\left(f x + e\right)}"," ",0,"-1/4*(c^3*cos(f*x + e)^4 - 8*c^3*cos(f*x + e)^2 + 7*c^3 + 4*(c^3*cos(f*x + e)^2 - 2*c^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","B",0
341,1,83,0,0.834998," ","integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{{\left(3 \, c^{2} \cos\left(f x + e\right)^{2} - 3 \, c^{2} - {\left(c^{2} \cos\left(f x + e\right)^{2} - 4 \, c^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, f \cos\left(f x + e\right)}"," ",0,"1/3*(3*c^2*cos(f*x + e)^2 - 3*c^2 - (c^2*cos(f*x + e)^2 - 4*c^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","B",0
342,1,61,0,1.048165," ","integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{{\left(c \cos\left(f x + e\right)^{2} + 2 \, c \sin\left(f x + e\right) - c\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{2 \, f \cos\left(f x + e\right)}"," ",0,"1/2*(c*cos(f*x + e)^2 + 2*c*sin(f*x + e) - c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","A",0
343,1,43,0,1.018635," ","integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{f \cos\left(f x + e\right)}"," ",0,"sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e)/(f*cos(f*x + e))","A",0
344,0,0,0,2.321215," ","integrate((a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{c \sin\left(f x + e\right) - c}, x\right)"," ",0,"integral(-sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c*sin(f*x + e) - c), x)","F",0
345,1,59,0,1.035800," ","integrate((a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{c^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) - c^{2} f \cos\left(f x + e\right)}"," ",0,"-sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^2*f*cos(f*x + e)*sin(f*x + e) - c^2*f*cos(f*x + e))","A",0
346,1,73,0,1.243182," ","integrate((a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{2 \, {\left(c^{3} f \cos\left(f x + e\right)^{3} + 2 \, c^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, c^{3} f \cos\left(f x + e\right)\right)}}"," ",0,"-1/2*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^3*f*cos(f*x + e)^3 + 2*c^3*f*cos(f*x + e)*sin(f*x + e) - 2*c^3*f*cos(f*x + e))","A",0
347,1,90,0,1.631960," ","integrate((a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, {\left(3 \, c^{4} f \cos\left(f x + e\right)^{3} - 4 \, c^{4} f \cos\left(f x + e\right) - {\left(c^{4} f \cos\left(f x + e\right)^{3} - 4 \, c^{4} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/3*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(3*c^4*f*cos(f*x + e)^3 - 4*c^4*f*cos(f*x + e) - (c^4*f*cos(f*x + e)^3 - 4*c^4*f*cos(f*x + e))*sin(f*x + e))","B",0
348,1,101,0,1.555645," ","integrate((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","\frac{{\left(5 \, a c^{3} \cos\left(f x + e\right)^{4} - 5 \, a c^{3} - 2 \, {\left(a c^{3} \cos\left(f x + e\right)^{4} - 2 \, a c^{3} \cos\left(f x + e\right)^{2} - 4 \, a c^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{10 \, f \cos\left(f x + e\right)}"," ",0,"1/10*(5*a*c^3*cos(f*x + e)^4 - 5*a*c^3 - 2*(a*c^3*cos(f*x + e)^4 - 2*a*c^3*cos(f*x + e)^2 - 4*a*c^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","A",0
349,1,87,0,1.840380," ","integrate((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{{\left(3 \, a c^{2} \cos\left(f x + e\right)^{4} - 3 \, a c^{2} + 4 \, {\left(a c^{2} \cos\left(f x + e\right)^{2} + 2 \, a c^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{12 \, f \cos\left(f x + e\right)}"," ",0,"1/12*(3*a*c^2*cos(f*x + e)^4 - 3*a*c^2 + 4*(a*c^2*cos(f*x + e)^2 + 2*a*c^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","A",0
350,1,60,0,1.658523," ","integrate((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{{\left(a c \cos\left(f x + e\right)^{2} + 2 \, a c\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{3 \, f \cos\left(f x + e\right)}"," ",0,"1/3*(a*c*cos(f*x + e)^2 + 2*a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e)/(f*cos(f*x + e))","A",0
351,1,61,0,1.707010," ","integrate((a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","-\frac{{\left(a \cos\left(f x + e\right)^{2} - 2 \, a \sin\left(f x + e\right) - a\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{2 \, f \cos\left(f x + e\right)}"," ",0,"-1/2*(a*cos(f*x + e)^2 - 2*a*sin(f*x + e) - a)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","A",0
352,0,0,0,2.296043," ","integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{\frac{3}{2}} \sqrt{-c \sin\left(f x + e\right) + c}}{c \sin\left(f x + e\right) - c}, x\right)"," ",0,"integral(-(a*sin(f*x + e) + a)^(3/2)*sqrt(-c*sin(f*x + e) + c)/(c*sin(f*x + e) - c), x)","F",0
353,0,0,0,2.727228," ","integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{\frac{3}{2}} \sqrt{-c \sin\left(f x + e\right) + c}}{c^{2} \cos\left(f x + e\right)^{2} + 2 \, c^{2} \sin\left(f x + e\right) - 2 \, c^{2}}, x\right)"," ",0,"integral(-(a*sin(f*x + e) + a)^(3/2)*sqrt(-c*sin(f*x + e) + c)/(c^2*cos(f*x + e)^2 + 2*c^2*sin(f*x + e) - 2*c^2), x)","F",0
354,1,80,0,1.628967," ","integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} a \sin\left(f x + e\right)}{c^{3} f \cos\left(f x + e\right)^{3} + 2 \, c^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, c^{3} f \cos\left(f x + e\right)}"," ",0,"-sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*a*sin(f*x + e)/(c^3*f*cos(f*x + e)^3 + 2*c^3*f*cos(f*x + e)*sin(f*x + e) - 2*c^3*f*cos(f*x + e))","B",0
355,1,101,0,1.181103," ","integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, a \sin\left(f x + e\right) + a\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{6 \, {\left(3 \, c^{4} f \cos\left(f x + e\right)^{3} - 4 \, c^{4} f \cos\left(f x + e\right) - {\left(c^{4} f \cos\left(f x + e\right)^{3} - 4 \, c^{4} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/6*(3*a*sin(f*x + e) + a)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(3*c^4*f*cos(f*x + e)^3 - 4*c^4*f*cos(f*x + e) - (c^4*f*cos(f*x + e)^3 - 4*c^4*f*cos(f*x + e))*sin(f*x + e))","A",0
356,1,114,0,1.521412," ","integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm=""fricas"")","\frac{{\left(2 \, a \sin\left(f x + e\right) + a\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{6 \, {\left(c^{5} f \cos\left(f x + e\right)^{5} - 8 \, c^{5} f \cos\left(f x + e\right)^{3} + 8 \, c^{5} f \cos\left(f x + e\right) + 4 \, {\left(c^{5} f \cos\left(f x + e\right)^{3} - 2 \, c^{5} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/6*(2*a*sin(f*x + e) + a)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^5*f*cos(f*x + e)^5 - 8*c^5*f*cos(f*x + e)^3 + 8*c^5*f*cos(f*x + e) + 4*(c^5*f*cos(f*x + e)^3 - 2*c^5*f*cos(f*x + e))*sin(f*x + e))","A",0
357,1,131,0,1.508822," ","integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm=""fricas"")","\frac{{\left(5 \, a \sin\left(f x + e\right) + 3 \, a\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{20 \, {\left(5 \, c^{6} f \cos\left(f x + e\right)^{5} - 20 \, c^{6} f \cos\left(f x + e\right)^{3} + 16 \, c^{6} f \cos\left(f x + e\right) - {\left(c^{6} f \cos\left(f x + e\right)^{5} - 12 \, c^{6} f \cos\left(f x + e\right)^{3} + 16 \, c^{6} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/20*(5*a*sin(f*x + e) + 3*a)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(5*c^6*f*cos(f*x + e)^5 - 20*c^6*f*cos(f*x + e)^3 + 16*c^6*f*cos(f*x + e) - (c^6*f*cos(f*x + e)^5 - 12*c^6*f*cos(f*x + e)^3 + 16*c^6*f*cos(f*x + e))*sin(f*x + e))","A",0
358,1,112,0,1.368895," ","integrate((a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","\frac{{\left(5 \, a^{2} c^{3} \cos\left(f x + e\right)^{6} - 5 \, a^{2} c^{3} + 2 \, {\left(3 \, a^{2} c^{3} \cos\left(f x + e\right)^{4} + 4 \, a^{2} c^{3} \cos\left(f x + e\right)^{2} + 8 \, a^{2} c^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{30 \, f \cos\left(f x + e\right)}"," ",0,"1/30*(5*a^2*c^3*cos(f*x + e)^6 - 5*a^2*c^3 + 2*(3*a^2*c^3*cos(f*x + e)^4 + 4*a^2*c^3*cos(f*x + e)^2 + 8*a^2*c^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","A",0
359,1,85,0,0.733582," ","integrate((a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{{\left(3 \, a^{2} c^{2} \cos\left(f x + e\right)^{4} + 4 \, a^{2} c^{2} \cos\left(f x + e\right)^{2} + 8 \, a^{2} c^{2}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{15 \, f \cos\left(f x + e\right)}"," ",0,"1/15*(3*a^2*c^2*cos(f*x + e)^4 + 4*a^2*c^2*cos(f*x + e)^2 + 8*a^2*c^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e)/(f*cos(f*x + e))","A",0
360,1,87,0,1.270588," ","integrate((a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, a^{2} c \cos\left(f x + e\right)^{4} - 3 \, a^{2} c - 4 \, {\left(a^{2} c \cos\left(f x + e\right)^{2} + 2 \, a^{2} c\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{12 \, f \cos\left(f x + e\right)}"," ",0,"-1/12*(3*a^2*c*cos(f*x + e)^4 - 3*a^2*c - 4*(a^2*c*cos(f*x + e)^2 + 2*a^2*c)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","A",0
361,1,82,0,0.946478," ","integrate((a+a*sin(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, a^{2} \cos\left(f x + e\right)^{2} - 3 \, a^{2} + {\left(a^{2} \cos\left(f x + e\right)^{2} - 4 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, f \cos\left(f x + e\right)}"," ",0,"-1/3*(3*a^2*cos(f*x + e)^2 - 3*a^2 + (a^2*cos(f*x + e)^2 - 4*a^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","B",0
362,0,0,0,2.289881," ","integrate((a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{c \sin\left(f x + e\right) - c}, x\right)"," ",0,"integral((a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c*sin(f*x + e) - c), x)","F",0
363,0,0,0,2.637467," ","integrate((a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{c^{2} \cos\left(f x + e\right)^{2} + 2 \, c^{2} \sin\left(f x + e\right) - 2 \, c^{2}}, x\right)"," ",0,"integral((a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^2*cos(f*x + e)^2 + 2*c^2*sin(f*x + e) - 2*c^2), x)","F",0
364,0,0,0,4.470828," ","integrate((a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3} - {\left(c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(3*c^3*cos(f*x + e)^2 - 4*c^3 - (c^3*cos(f*x + e)^2 - 4*c^3)*sin(f*x + e)), x)","F",0
365,1,109,0,0.900863," ","integrate((a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","\frac{{\left(3 \, a^{2} \cos\left(f x + e\right)^{2} - 4 \, a^{2}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, {\left(3 \, c^{4} f \cos\left(f x + e\right)^{3} - 4 \, c^{4} f \cos\left(f x + e\right) - {\left(c^{4} f \cos\left(f x + e\right)^{3} - 4 \, c^{4} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/3*(3*a^2*cos(f*x + e)^2 - 4*a^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(3*c^4*f*cos(f*x + e)^3 - 4*c^4*f*cos(f*x + e) - (c^4*f*cos(f*x + e)^3 - 4*c^4*f*cos(f*x + e))*sin(f*x + e))","B",0
366,1,133,0,1.267221," ","integrate((a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 4 \, a^{2}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{6 \, {\left(c^{5} f \cos\left(f x + e\right)^{5} - 8 \, c^{5} f \cos\left(f x + e\right)^{3} + 8 \, c^{5} f \cos\left(f x + e\right) + 4 \, {\left(c^{5} f \cos\left(f x + e\right)^{3} - 2 \, c^{5} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/6*(3*a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 4*a^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^5*f*cos(f*x + e)^5 - 8*c^5*f*cos(f*x + e)^3 + 8*c^5*f*cos(f*x + e) + 4*(c^5*f*cos(f*x + e)^3 - 2*c^5*f*cos(f*x + e))*sin(f*x + e))","A",0
367,1,148,0,1.578139," ","integrate((a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm=""fricas"")","-\frac{{\left(5 \, a^{2} \cos\left(f x + e\right)^{2} - 5 \, a^{2} \sin\left(f x + e\right) - 7 \, a^{2}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{15 \, {\left(5 \, c^{6} f \cos\left(f x + e\right)^{5} - 20 \, c^{6} f \cos\left(f x + e\right)^{3} + 16 \, c^{6} f \cos\left(f x + e\right) - {\left(c^{6} f \cos\left(f x + e\right)^{5} - 12 \, c^{6} f \cos\left(f x + e\right)^{3} + 16 \, c^{6} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/15*(5*a^2*cos(f*x + e)^2 - 5*a^2*sin(f*x + e) - 7*a^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(5*c^6*f*cos(f*x + e)^5 - 20*c^6*f*cos(f*x + e)^3 + 16*c^6*f*cos(f*x + e) - (c^6*f*cos(f*x + e)^5 - 12*c^6*f*cos(f*x + e)^3 + 16*c^6*f*cos(f*x + e))*sin(f*x + e))","A",0
368,1,162,0,1.243891," ","integrate((a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm=""fricas"")","\frac{{\left(15 \, a^{2} \cos\left(f x + e\right)^{2} - 18 \, a^{2} \sin\left(f x + e\right) - 22 \, a^{2}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{60 \, {\left(c^{7} f \cos\left(f x + e\right)^{7} - 18 \, c^{7} f \cos\left(f x + e\right)^{5} + 48 \, c^{7} f \cos\left(f x + e\right)^{3} - 32 \, c^{7} f \cos\left(f x + e\right) + 2 \, {\left(3 \, c^{7} f \cos\left(f x + e\right)^{5} - 16 \, c^{7} f \cos\left(f x + e\right)^{3} + 16 \, c^{7} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/60*(15*a^2*cos(f*x + e)^2 - 18*a^2*sin(f*x + e) - 22*a^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^7*f*cos(f*x + e)^7 - 18*c^7*f*cos(f*x + e)^5 + 48*c^7*f*cos(f*x + e)^3 - 32*c^7*f*cos(f*x + e) + 2*(3*c^7*f*cos(f*x + e)^5 - 16*c^7*f*cos(f*x + e)^3 + 16*c^7*f*cos(f*x + e))*sin(f*x + e))","A",0
369,1,128,0,1.960910," ","integrate((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(9/2),x, algorithm=""fricas"")","\frac{{\left(35 \, a^{3} c^{4} \cos\left(f x + e\right)^{8} - 35 \, a^{3} c^{4} + 8 \, {\left(5 \, a^{3} c^{4} \cos\left(f x + e\right)^{6} + 6 \, a^{3} c^{4} \cos\left(f x + e\right)^{4} + 8 \, a^{3} c^{4} \cos\left(f x + e\right)^{2} + 16 \, a^{3} c^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{280 \, f \cos\left(f x + e\right)}"," ",0,"1/280*(35*a^3*c^4*cos(f*x + e)^8 - 35*a^3*c^4 + 8*(5*a^3*c^4*cos(f*x + e)^6 + 6*a^3*c^4*cos(f*x + e)^4 + 8*a^3*c^4*cos(f*x + e)^2 + 16*a^3*c^4)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","A",0
370,1,101,0,1.799423," ","integrate((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","\frac{{\left(5 \, a^{3} c^{3} \cos\left(f x + e\right)^{6} + 6 \, a^{3} c^{3} \cos\left(f x + e\right)^{4} + 8 \, a^{3} c^{3} \cos\left(f x + e\right)^{2} + 16 \, a^{3} c^{3}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{35 \, f \cos\left(f x + e\right)}"," ",0,"1/35*(5*a^3*c^3*cos(f*x + e)^6 + 6*a^3*c^3*cos(f*x + e)^4 + 8*a^3*c^3*cos(f*x + e)^2 + 16*a^3*c^3)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e)/(f*cos(f*x + e))","A",0
371,1,112,0,1.670782," ","integrate((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","-\frac{{\left(5 \, a^{3} c^{2} \cos\left(f x + e\right)^{6} - 5 \, a^{3} c^{2} - 2 \, {\left(3 \, a^{3} c^{2} \cos\left(f x + e\right)^{4} + 4 \, a^{3} c^{2} \cos\left(f x + e\right)^{2} + 8 \, a^{3} c^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{30 \, f \cos\left(f x + e\right)}"," ",0,"-1/30*(5*a^3*c^2*cos(f*x + e)^6 - 5*a^3*c^2 - 2*(3*a^3*c^2*cos(f*x + e)^4 + 4*a^3*c^2*cos(f*x + e)^2 + 8*a^3*c^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","A",0
372,1,101,0,1.437890," ","integrate((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","-\frac{{\left(5 \, a^{3} c \cos\left(f x + e\right)^{4} - 5 \, a^{3} c + 2 \, {\left(a^{3} c \cos\left(f x + e\right)^{4} - 2 \, a^{3} c \cos\left(f x + e\right)^{2} - 4 \, a^{3} c\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{10 \, f \cos\left(f x + e\right)}"," ",0,"-1/10*(5*a^3*c*cos(f*x + e)^4 - 5*a^3*c + 2*(a^3*c*cos(f*x + e)^4 - 2*a^3*c*cos(f*x + e)^2 - 4*a^3*c)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","A",0
373,1,95,0,1.787372," ","integrate((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{{\left(a^{3} \cos\left(f x + e\right)^{4} - 8 \, a^{3} \cos\left(f x + e\right)^{2} + 7 \, a^{3} - 4 \, {\left(a^{3} \cos\left(f x + e\right)^{2} - 2 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{4 \, f \cos\left(f x + e\right)}"," ",0,"1/4*(a^3*cos(f*x + e)^4 - 8*a^3*cos(f*x + e)^2 + 7*a^3 - 4*(a^3*cos(f*x + e)^2 - 2*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))","B",0
374,0,0,0,1.595733," ","integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{c \sin\left(f x + e\right) - c}, x\right)"," ",0,"integral((3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c*sin(f*x + e) - c), x)","F",0
375,0,0,0,3.043451," ","integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{c^{2} \cos\left(f x + e\right)^{2} + 2 \, c^{2} \sin\left(f x + e\right) - 2 \, c^{2}}, x\right)"," ",0,"integral((3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^2*cos(f*x + e)^2 + 2*c^2*sin(f*x + e) - 2*c^2), x)","F",0
376,0,0,0,5.189378," ","integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3} - {\left(c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(3*c^3*cos(f*x + e)^2 - 4*c^3 - (c^3*cos(f*x + e)^2 - 4*c^3)*sin(f*x + e)), x)","F",0
377,0,0,0,7.209905," ","integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{c^{4} \cos\left(f x + e\right)^{4} - 8 \, c^{4} \cos\left(f x + e\right)^{2} + 8 \, c^{4} + 4 \, {\left(c^{4} \cos\left(f x + e\right)^{2} - 2 \, c^{4}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^4*cos(f*x + e)^4 - 8*c^4*cos(f*x + e)^2 + 8*c^4 + 4*(c^4*cos(f*x + e)^2 - 2*c^4)*sin(f*x + e)), x)","F",0
378,1,127,0,1.168775," ","integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm=""fricas"")","-\frac{{\left(a^{3} \cos\left(f x + e\right)^{2} - 2 \, a^{3}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{c^{5} f \cos\left(f x + e\right)^{5} - 8 \, c^{5} f \cos\left(f x + e\right)^{3} + 8 \, c^{5} f \cos\left(f x + e\right) + 4 \, {\left(c^{5} f \cos\left(f x + e\right)^{3} - 2 \, c^{5} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"-(a^3*cos(f*x + e)^2 - 2*a^3)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e)/(c^5*f*cos(f*x + e)^5 - 8*c^5*f*cos(f*x + e)^3 + 8*c^5*f*cos(f*x + e) + 4*(c^5*f*cos(f*x + e)^3 - 2*c^5*f*cos(f*x + e))*sin(f*x + e))","B",0
379,1,163,0,1.489776," ","integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm=""fricas"")","-\frac{{\left(5 \, a^{3} \cos\left(f x + e\right)^{2} - 6 \, a^{3} + 5 \, {\left(a^{3} \cos\left(f x + e\right)^{2} - 2 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{10 \, {\left(5 \, c^{6} f \cos\left(f x + e\right)^{5} - 20 \, c^{6} f \cos\left(f x + e\right)^{3} + 16 \, c^{6} f \cos\left(f x + e\right) - {\left(c^{6} f \cos\left(f x + e\right)^{5} - 12 \, c^{6} f \cos\left(f x + e\right)^{3} + 16 \, c^{6} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/10*(5*a^3*cos(f*x + e)^2 - 6*a^3 + 5*(a^3*cos(f*x + e)^2 - 2*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(5*c^6*f*cos(f*x + e)^5 - 20*c^6*f*cos(f*x + e)^3 + 16*c^6*f*cos(f*x + e) - (c^6*f*cos(f*x + e)^5 - 12*c^6*f*cos(f*x + e)^3 + 16*c^6*f*cos(f*x + e))*sin(f*x + e))","B",0
380,1,178,0,1.699965," ","integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm=""fricas"")","\frac{{\left(15 \, a^{3} \cos\left(f x + e\right)^{2} - 18 \, a^{3} + 2 \, {\left(5 \, a^{3} \cos\left(f x + e\right)^{2} - 11 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{30 \, {\left(c^{7} f \cos\left(f x + e\right)^{7} - 18 \, c^{7} f \cos\left(f x + e\right)^{5} + 48 \, c^{7} f \cos\left(f x + e\right)^{3} - 32 \, c^{7} f \cos\left(f x + e\right) + 2 \, {\left(3 \, c^{7} f \cos\left(f x + e\right)^{5} - 16 \, c^{7} f \cos\left(f x + e\right)^{3} + 16 \, c^{7} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/30*(15*a^3*cos(f*x + e)^2 - 18*a^3 + 2*(5*a^3*cos(f*x + e)^2 - 11*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^7*f*cos(f*x + e)^7 - 18*c^7*f*cos(f*x + e)^5 + 48*c^7*f*cos(f*x + e)^3 - 32*c^7*f*cos(f*x + e) + 2*(3*c^7*f*cos(f*x + e)^5 - 16*c^7*f*cos(f*x + e)^3 + 16*c^7*f*cos(f*x + e))*sin(f*x + e))","A",0
381,1,192,0,1.107233," ","integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, algorithm=""fricas"")","\frac{{\left(63 \, a^{3} \cos\left(f x + e\right)^{2} - 76 \, a^{3} + 7 \, {\left(5 \, a^{3} \cos\left(f x + e\right)^{2} - 12 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{140 \, {\left(7 \, c^{8} f \cos\left(f x + e\right)^{7} - 56 \, c^{8} f \cos\left(f x + e\right)^{5} + 112 \, c^{8} f \cos\left(f x + e\right)^{3} - 64 \, c^{8} f \cos\left(f x + e\right) - {\left(c^{8} f \cos\left(f x + e\right)^{7} - 24 \, c^{8} f \cos\left(f x + e\right)^{5} + 80 \, c^{8} f \cos\left(f x + e\right)^{3} - 64 \, c^{8} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/140*(63*a^3*cos(f*x + e)^2 - 76*a^3 + 7*(5*a^3*cos(f*x + e)^2 - 12*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(7*c^8*f*cos(f*x + e)^7 - 56*c^8*f*cos(f*x + e)^5 + 112*c^8*f*cos(f*x + e)^3 - 64*c^8*f*cos(f*x + e) - (c^8*f*cos(f*x + e)^7 - 24*c^8*f*cos(f*x + e)^5 + 80*c^8*f*cos(f*x + e)^3 - 64*c^8*f*cos(f*x + e))*sin(f*x + e))","A",0
382,1,204,0,1.947977," ","integrate((a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(17/2),x, algorithm=""fricas"")","-\frac{{\left(14 \, a^{3} \cos\left(f x + e\right)^{2} - 17 \, a^{3} + {\left(7 \, a^{3} \cos\left(f x + e\right)^{2} - 18 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{35 \, {\left(c^{9} f \cos\left(f x + e\right)^{9} - 32 \, c^{9} f \cos\left(f x + e\right)^{7} + 160 \, c^{9} f \cos\left(f x + e\right)^{5} - 256 \, c^{9} f \cos\left(f x + e\right)^{3} + 128 \, c^{9} f \cos\left(f x + e\right) + 8 \, {\left(c^{9} f \cos\left(f x + e\right)^{7} - 10 \, c^{9} f \cos\left(f x + e\right)^{5} + 24 \, c^{9} f \cos\left(f x + e\right)^{3} - 16 \, c^{9} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/35*(14*a^3*cos(f*x + e)^2 - 17*a^3 + (7*a^3*cos(f*x + e)^2 - 18*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^9*f*cos(f*x + e)^9 - 32*c^9*f*cos(f*x + e)^7 + 160*c^9*f*cos(f*x + e)^5 - 256*c^9*f*cos(f*x + e)^3 + 128*c^9*f*cos(f*x + e) + 8*(c^9*f*cos(f*x + e)^7 - 10*c^9*f*cos(f*x + e)^5 + 24*c^9*f*cos(f*x + e)^3 - 16*c^9*f*cos(f*x + e))*sin(f*x + e))","A",0
383,0,0,0,2.027837," ","integrate((c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(c^{2} \cos\left(f x + e\right)^{2} + 2 \, c^{2} \sin\left(f x + e\right) - 2 \, c^{2}\right)} \sqrt{-c \sin\left(f x + e\right) + c}}{\sqrt{a \sin\left(f x + e\right) + a}}, x\right)"," ",0,"integral(-(c^2*cos(f*x + e)^2 + 2*c^2*sin(f*x + e) - 2*c^2)*sqrt(-c*sin(f*x + e) + c)/sqrt(a*sin(f*x + e) + a), x)","F",0
384,0,0,0,2.025324," ","integrate((c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(-c \sin\left(f x + e\right) + c\right)}^{\frac{3}{2}}}{\sqrt{a \sin\left(f x + e\right) + a}}, x\right)"," ",0,"integral((-c*sin(f*x + e) + c)^(3/2)/sqrt(a*sin(f*x + e) + a), x)","F",0
385,0,0,0,1.778641," ","integrate((c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-c \sin\left(f x + e\right) + c}}{\sqrt{a \sin\left(f x + e\right) + a}}, x\right)"," ",0,"integral(sqrt(-c*sin(f*x + e) + c)/sqrt(a*sin(f*x + e) + a), x)","F",0
386,1,160,0,1.405970," ","integrate(1/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a c} \log\left(-\frac{a c \cos\left(f x + e\right)^{3} - 2 \, a c \cos\left(f x + e\right) - 2 \, \sqrt{a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3}}\right)}{2 \, a c f}, -\frac{\sqrt{-a c} \arctan\left(\frac{\sqrt{-a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a c \cos\left(f x + e\right) \sin\left(f x + e\right)}\right)}{a c f}\right]"," ",0,"[1/2*sqrt(a*c)*log(-(a*c*cos(f*x + e)^3 - 2*a*c*cos(f*x + e) - 2*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/cos(f*x + e)^3)/(a*c*f), -sqrt(-a*c)*arctan(sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a*c*cos(f*x + e)*sin(f*x + e)))/(a*c*f)]","A",0
387,1,311,0,1.152163," ","integrate(1/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a c} {\left(\cos\left(f x + e\right) \sin\left(f x + e\right) - \cos\left(f x + e\right)\right)} \log\left(-\frac{a c \cos\left(f x + e\right)^{3} - 2 \, a c \cos\left(f x + e\right) - 2 \, \sqrt{a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3}}\right) - 2 \, \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{4 \, {\left(a c^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) - a c^{2} f \cos\left(f x + e\right)\right)}}, -\frac{\sqrt{-a c} {\left(\cos\left(f x + e\right) \sin\left(f x + e\right) - \cos\left(f x + e\right)\right)} \arctan\left(\frac{\sqrt{-a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a c \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{2 \, {\left(a c^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) - a c^{2} f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[1/4*(sqrt(a*c)*(cos(f*x + e)*sin(f*x + e) - cos(f*x + e))*log(-(a*c*cos(f*x + e)^3 - 2*a*c*cos(f*x + e) - 2*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/cos(f*x + e)^3) - 2*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(a*c^2*f*cos(f*x + e)*sin(f*x + e) - a*c^2*f*cos(f*x + e)), -1/2*(sqrt(-a*c)*(cos(f*x + e)*sin(f*x + e) - cos(f*x + e))*arctan(sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a*c*cos(f*x + e)*sin(f*x + e))) + sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(a*c^2*f*cos(f*x + e)*sin(f*x + e) - a*c^2*f*cos(f*x + e))]","A",0
388,1,376,0,1.392053," ","integrate(1/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[\frac{{\left(\cos\left(f x + e\right)^{3} + 2 \, \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right)\right)} \sqrt{a c} \log\left(-\frac{a c \cos\left(f x + e\right)^{3} - 2 \, a c \cos\left(f x + e\right) - 2 \, \sqrt{a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3}}\right) + 2 \, \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} {\left(\sin\left(f x + e\right) - 2\right)}}{8 \, {\left(a c^{3} f \cos\left(f x + e\right)^{3} + 2 \, a c^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a c^{3} f \cos\left(f x + e\right)\right)}}, -\frac{{\left(\cos\left(f x + e\right)^{3} + 2 \, \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right)\right)} \sqrt{-a c} \arctan\left(\frac{\sqrt{-a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a c \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} {\left(\sin\left(f x + e\right) - 2\right)}}{4 \, {\left(a c^{3} f \cos\left(f x + e\right)^{3} + 2 \, a c^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a c^{3} f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[1/8*((cos(f*x + e)^3 + 2*cos(f*x + e)*sin(f*x + e) - 2*cos(f*x + e))*sqrt(a*c)*log(-(a*c*cos(f*x + e)^3 - 2*a*c*cos(f*x + e) - 2*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/cos(f*x + e)^3) + 2*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*(sin(f*x + e) - 2))/(a*c^3*f*cos(f*x + e)^3 + 2*a*c^3*f*cos(f*x + e)*sin(f*x + e) - 2*a*c^3*f*cos(f*x + e)), -1/4*((cos(f*x + e)^3 + 2*cos(f*x + e)*sin(f*x + e) - 2*cos(f*x + e))*sqrt(-a*c)*arctan(sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a*c*cos(f*x + e)*sin(f*x + e))) - sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*(sin(f*x + e) - 2))/(a*c^3*f*cos(f*x + e)^3 + 2*a*c^3*f*cos(f*x + e)*sin(f*x + e) - 2*a*c^3*f*cos(f*x + e))]","A",0
389,0,0,0,2.616234," ","integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(3 \, c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3} - {\left(c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral((3*c^3*cos(f*x + e)^2 - 4*c^3 - (c^3*cos(f*x + e)^2 - 4*c^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
390,0,0,0,2.180062," ","integrate((c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(c^{2} \cos\left(f x + e\right)^{2} + 2 \, c^{2} \sin\left(f x + e\right) - 2 \, c^{2}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral((c^2*cos(f*x + e)^2 + 2*c^2*sin(f*x + e) - 2*c^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
391,0,0,0,2.271548," ","integrate((c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(-c \sin\left(f x + e\right) + c\right)}^{\frac{3}{2}}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-sqrt(a*sin(f*x + e) + a)*(-c*sin(f*x + e) + c)^(3/2)/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
392,1,58,0,0.719520," ","integrate((c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) + a^{2} f \cos\left(f x + e\right)}"," ",0,"-sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a^2*f*cos(f*x + e)*sin(f*x + e) + a^2*f*cos(f*x + e))","A",0
393,1,305,0,1.548261," ","integrate(1/(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a c} {\left(\cos\left(f x + e\right) \sin\left(f x + e\right) + \cos\left(f x + e\right)\right)} \log\left(-\frac{a c \cos\left(f x + e\right)^{3} - 2 \, a c \cos\left(f x + e\right) - 2 \, \sqrt{a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3}}\right) - 2 \, \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{4 \, {\left(a^{2} c f \cos\left(f x + e\right) \sin\left(f x + e\right) + a^{2} c f \cos\left(f x + e\right)\right)}}, -\frac{\sqrt{-a c} {\left(\cos\left(f x + e\right) \sin\left(f x + e\right) + \cos\left(f x + e\right)\right)} \arctan\left(\frac{\sqrt{-a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a c \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{2 \, {\left(a^{2} c f \cos\left(f x + e\right) \sin\left(f x + e\right) + a^{2} c f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[1/4*(sqrt(a*c)*(cos(f*x + e)*sin(f*x + e) + cos(f*x + e))*log(-(a*c*cos(f*x + e)^3 - 2*a*c*cos(f*x + e) - 2*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/cos(f*x + e)^3) - 2*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(a^2*c*f*cos(f*x + e)*sin(f*x + e) + a^2*c*f*cos(f*x + e)), -1/2*(sqrt(-a*c)*(cos(f*x + e)*sin(f*x + e) + cos(f*x + e))*arctan(sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a*c*cos(f*x + e)*sin(f*x + e))) + sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(a^2*c*f*cos(f*x + e)*sin(f*x + e) + a^2*c*f*cos(f*x + e))]","A",0
394,1,262,0,1.317901," ","integrate(1/(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a c} \cos\left(f x + e\right)^{3} \log\left(-\frac{a c \cos\left(f x + e\right)^{3} - 2 \, a c \cos\left(f x + e\right) - 2 \, \sqrt{a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3}}\right) + 2 \, \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{4 \, a^{2} c^{2} f \cos\left(f x + e\right)^{3}}, -\frac{\sqrt{-a c} \arctan\left(\frac{\sqrt{-a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a c \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{3} - \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{2 \, a^{2} c^{2} f \cos\left(f x + e\right)^{3}}\right]"," ",0,"[1/4*(sqrt(a*c)*cos(f*x + e)^3*log(-(a*c*cos(f*x + e)^3 - 2*a*c*cos(f*x + e) - 2*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/cos(f*x + e)^3) + 2*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/(a^2*c^2*f*cos(f*x + e)^3), -1/2*(sqrt(-a*c)*arctan(sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a*c*cos(f*x + e)*sin(f*x + e)))*cos(f*x + e)^3 - sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/(a^2*c^2*f*cos(f*x + e)^3)]","A",0
395,1,377,0,1.265239," ","integrate(1/(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(\cos\left(f x + e\right)^{3} \sin\left(f x + e\right) - \cos\left(f x + e\right)^{3}\right)} \sqrt{a c} \log\left(-\frac{a c \cos\left(f x + e\right)^{3} - 2 \, a c \cos\left(f x + e\right) - 2 \, \sqrt{a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3}}\right) - 2 \, {\left(3 \, \cos\left(f x + e\right)^{2} + 3 \, \sin\left(f x + e\right) - 1\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{16 \, {\left(a^{2} c^{3} f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) - a^{2} c^{3} f \cos\left(f x + e\right)^{3}\right)}}, -\frac{3 \, {\left(\cos\left(f x + e\right)^{3} \sin\left(f x + e\right) - \cos\left(f x + e\right)^{3}\right)} \sqrt{-a c} \arctan\left(\frac{\sqrt{-a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a c \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + {\left(3 \, \cos\left(f x + e\right)^{2} + 3 \, \sin\left(f x + e\right) - 1\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{8 \, {\left(a^{2} c^{3} f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) - a^{2} c^{3} f \cos\left(f x + e\right)^{3}\right)}}\right]"," ",0,"[1/16*(3*(cos(f*x + e)^3*sin(f*x + e) - cos(f*x + e)^3)*sqrt(a*c)*log(-(a*c*cos(f*x + e)^3 - 2*a*c*cos(f*x + e) - 2*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/cos(f*x + e)^3) - 2*(3*cos(f*x + e)^2 + 3*sin(f*x + e) - 1)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(a^2*c^3*f*cos(f*x + e)^3*sin(f*x + e) - a^2*c^3*f*cos(f*x + e)^3), -1/8*(3*(cos(f*x + e)^3*sin(f*x + e) - cos(f*x + e)^3)*sqrt(-a*c)*arctan(sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a*c*cos(f*x + e)*sin(f*x + e))) + (3*cos(f*x + e)^2 + 3*sin(f*x + e) - 1)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(a^2*c^3*f*cos(f*x + e)^3*sin(f*x + e) - a^2*c^3*f*cos(f*x + e)^3)]","A",0
396,0,0,0,4.530431," ","integrate((c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(c^{4} \cos\left(f x + e\right)^{4} - 8 \, c^{4} \cos\left(f x + e\right)^{2} + 8 \, c^{4} + 4 \, {\left(c^{4} \cos\left(f x + e\right)^{2} - 2 \, c^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(c^4*cos(f*x + e)^4 - 8*c^4*cos(f*x + e)^2 + 8*c^4 + 4*(c^4*cos(f*x + e)^2 - 2*c^4)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e)), x)","F",0
397,0,0,0,3.461911," ","integrate((c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(3 \, c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3} - {\left(c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((3*c^3*cos(f*x + e)^2 - 4*c^3 - (c^3*cos(f*x + e)^2 - 4*c^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e)), x)","F",0
398,0,0,0,3.161544," ","integrate((c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(c^{2} \cos\left(f x + e\right)^{2} + 2 \, c^{2} \sin\left(f x + e\right) - 2 \, c^{2}\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((c^2*cos(f*x + e)^2 + 2*c^2*sin(f*x + e) - 2*c^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e)), x)","F",0
399,1,80,0,0.841297," ","integrate((c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} c \sin\left(f x + e\right)}{a^{3} f \cos\left(f x + e\right)^{3} - 2 \, a^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{3} f \cos\left(f x + e\right)}"," ",0,"-sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*c*sin(f*x + e)/(a^3*f*cos(f*x + e)^3 - 2*a^3*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*f*cos(f*x + e))","B",0
400,1,73,0,0.824686," ","integrate((c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{\sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{2 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} - 2 \, a^{3} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{3} f \cos\left(f x + e\right)\right)}}"," ",0,"1/2*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a^3*f*cos(f*x + e)^3 - 2*a^3*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*f*cos(f*x + e))","A",0
401,1,376,0,0.893794," ","integrate(1/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(\cos\left(f x + e\right)^{3} - 2 \, \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right)\right)} \sqrt{a c} \log\left(-\frac{a c \cos\left(f x + e\right)^{3} - 2 \, a c \cos\left(f x + e\right) - 2 \, \sqrt{a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3}}\right) + 2 \, \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} {\left(\sin\left(f x + e\right) + 2\right)}}{8 \, {\left(a^{3} c f \cos\left(f x + e\right)^{3} - 2 \, a^{3} c f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{3} c f \cos\left(f x + e\right)\right)}}, -\frac{{\left(\cos\left(f x + e\right)^{3} - 2 \, \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right)\right)} \sqrt{-a c} \arctan\left(\frac{\sqrt{-a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a c \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) - \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} {\left(\sin\left(f x + e\right) + 2\right)}}{4 \, {\left(a^{3} c f \cos\left(f x + e\right)^{3} - 2 \, a^{3} c f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{3} c f \cos\left(f x + e\right)\right)}}\right]"," ",0,"[1/8*((cos(f*x + e)^3 - 2*cos(f*x + e)*sin(f*x + e) - 2*cos(f*x + e))*sqrt(a*c)*log(-(a*c*cos(f*x + e)^3 - 2*a*c*cos(f*x + e) - 2*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/cos(f*x + e)^3) + 2*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*(sin(f*x + e) + 2))/(a^3*c*f*cos(f*x + e)^3 - 2*a^3*c*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*c*f*cos(f*x + e)), -1/4*((cos(f*x + e)^3 - 2*cos(f*x + e)*sin(f*x + e) - 2*cos(f*x + e))*sqrt(-a*c)*arctan(sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a*c*cos(f*x + e)*sin(f*x + e))) - sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*(sin(f*x + e) + 2))/(a^3*c*f*cos(f*x + e)^3 - 2*a^3*c*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*c*f*cos(f*x + e))]","A",0
402,1,371,0,1.008996," ","integrate(1/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(\cos\left(f x + e\right)^{3} \sin\left(f x + e\right) + \cos\left(f x + e\right)^{3}\right)} \sqrt{a c} \log\left(-\frac{a c \cos\left(f x + e\right)^{3} - 2 \, a c \cos\left(f x + e\right) - 2 \, \sqrt{a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3}}\right) - 2 \, {\left(3 \, \cos\left(f x + e\right)^{2} - 3 \, \sin\left(f x + e\right) - 1\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{16 \, {\left(a^{3} c^{2} f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) + a^{3} c^{2} f \cos\left(f x + e\right)^{3}\right)}}, -\frac{3 \, {\left(\cos\left(f x + e\right)^{3} \sin\left(f x + e\right) + \cos\left(f x + e\right)^{3}\right)} \sqrt{-a c} \arctan\left(\frac{\sqrt{-a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a c \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) + {\left(3 \, \cos\left(f x + e\right)^{2} - 3 \, \sin\left(f x + e\right) - 1\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{8 \, {\left(a^{3} c^{2} f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) + a^{3} c^{2} f \cos\left(f x + e\right)^{3}\right)}}\right]"," ",0,"[1/16*(3*(cos(f*x + e)^3*sin(f*x + e) + cos(f*x + e)^3)*sqrt(a*c)*log(-(a*c*cos(f*x + e)^3 - 2*a*c*cos(f*x + e) - 2*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/cos(f*x + e)^3) - 2*(3*cos(f*x + e)^2 - 3*sin(f*x + e) - 1)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(a^3*c^2*f*cos(f*x + e)^3*sin(f*x + e) + a^3*c^2*f*cos(f*x + e)^3), -1/8*(3*(cos(f*x + e)^3*sin(f*x + e) + cos(f*x + e)^3)*sqrt(-a*c)*arctan(sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a*c*cos(f*x + e)*sin(f*x + e))) + (3*cos(f*x + e)^2 - 3*sin(f*x + e) - 1)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(a^3*c^2*f*cos(f*x + e)^3*sin(f*x + e) + a^3*c^2*f*cos(f*x + e)^3)]","A",0
403,1,288,0,1.417705," ","integrate(1/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a c} \cos\left(f x + e\right)^{5} \log\left(-\frac{a c \cos\left(f x + e\right)^{3} - 2 \, a c \cos\left(f x + e\right) - 2 \, \sqrt{a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3}}\right) + 2 \, {\left(3 \, \cos\left(f x + e\right)^{2} + 2\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{16 \, a^{3} c^{3} f \cos\left(f x + e\right)^{5}}, -\frac{3 \, \sqrt{-a c} \arctan\left(\frac{\sqrt{-a c} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c}}{a c \cos\left(f x + e\right) \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{5} - {\left(3 \, \cos\left(f x + e\right)^{2} + 2\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-c \sin\left(f x + e\right) + c} \sin\left(f x + e\right)}{8 \, a^{3} c^{3} f \cos\left(f x + e\right)^{5}}\right]"," ",0,"[1/16*(3*sqrt(a*c)*cos(f*x + e)^5*log(-(a*c*cos(f*x + e)^3 - 2*a*c*cos(f*x + e) - 2*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/cos(f*x + e)^3) + 2*(3*cos(f*x + e)^2 + 2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/(a^3*c^3*f*cos(f*x + e)^5), -1/8*(3*sqrt(-a*c)*arctan(sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a*c*cos(f*x + e)*sin(f*x + e)))*cos(f*x + e)^5 - (3*cos(f*x + e)^2 + 2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e))/(a^3*c^3*f*cos(f*x + e)^5)]","A",0
404,0,0,0,1.148908," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-c \sin\left(f x + e\right) + c\right)}^{n}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^n, x)","F",0
405,0,0,0,1.184023," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(3 \, c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3} - {\left(c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3}\right)} \sin\left(f x + e\right)\right)} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(-(3*c^3*cos(f*x + e)^2 - 4*c^3 - (c^3*cos(f*x + e)^2 - 4*c^3)*sin(f*x + e))*(a*sin(f*x + e) + a)^m, x)","F",0
406,0,0,0,1.225249," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(c^{2} \cos\left(f x + e\right)^{2} + 2 \, c^{2} \sin\left(f x + e\right) - 2 \, c^{2}\right)} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(-(c^2*cos(f*x + e)^2 + 2*c^2*sin(f*x + e) - 2*c^2)*(a*sin(f*x + e) + a)^m, x)","F",0
407,0,0,0,1.192927," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(c \sin\left(f x + e\right) - c\right)} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(-(c*sin(f*x + e) - c)*(a*sin(f*x + e) + a)^m, x)","F",0
408,0,0,0,0.890718," ","integrate((a+a*sin(f*x+e))^m/(c-c*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{m}}{c \sin\left(f x + e\right) - c}, x\right)"," ",0,"integral(-(a*sin(f*x + e) + a)^m/(c*sin(f*x + e) - c), x)","F",0
409,0,0,0,1.160771," ","integrate((a+a*sin(f*x+e))^m/(c-c*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{m}}{c^{2} \cos\left(f x + e\right)^{2} + 2 \, c^{2} \sin\left(f x + e\right) - 2 \, c^{2}}, x\right)"," ",0,"integral(-(a*sin(f*x + e) + a)^m/(c^2*cos(f*x + e)^2 + 2*c^2*sin(f*x + e) - 2*c^2), x)","F",0
410,0,0,0,1.011374," ","integrate((a+a*sin(f*x+e))^m/(c-c*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{m}}{3 \, c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3} - {\left(c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(a*sin(f*x + e) + a)^m/(3*c^3*cos(f*x + e)^2 - 4*c^3 - (c^3*cos(f*x + e)^2 - 4*c^3)*sin(f*x + e)), x)","F",0
411,1,267,0,1.250554," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","-\frac{2 \, {\left({\left(4 \, c^{2} m^{2} + 8 \, c^{2} m + 3 \, c^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(4 \, c^{2} m^{2} + 24 \, c^{2} m + 11 \, c^{2}\right)} \cos\left(f x + e\right)^{2} - 32 \, c^{2} - 2 \, {\left(4 \, c^{2} m^{2} + 16 \, c^{2} m + 23 \, c^{2}\right)} \cos\left(f x + e\right) + {\left({\left(4 \, c^{2} m^{2} + 8 \, c^{2} m + 3 \, c^{2}\right)} \cos\left(f x + e\right)^{2} - 32 \, c^{2} + 2 \, {\left(4 \, c^{2} m^{2} + 16 \, c^{2} m + 7 \, c^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-c \sin\left(f x + e\right) + c} {\left(a \sin\left(f x + e\right) + a\right)}^{m}}{8 \, f m^{3} + 36 \, f m^{2} + 46 \, f m + {\left(8 \, f m^{3} + 36 \, f m^{2} + 46 \, f m + 15 \, f\right)} \cos\left(f x + e\right) - {\left(8 \, f m^{3} + 36 \, f m^{2} + 46 \, f m + 15 \, f\right)} \sin\left(f x + e\right) + 15 \, f}"," ",0,"-2*((4*c^2*m^2 + 8*c^2*m + 3*c^2)*cos(f*x + e)^3 - (4*c^2*m^2 + 24*c^2*m + 11*c^2)*cos(f*x + e)^2 - 32*c^2 - 2*(4*c^2*m^2 + 16*c^2*m + 23*c^2)*cos(f*x + e) + ((4*c^2*m^2 + 8*c^2*m + 3*c^2)*cos(f*x + e)^2 - 32*c^2 + 2*(4*c^2*m^2 + 16*c^2*m + 7*c^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(8*f*m^3 + 36*f*m^2 + 46*f*m + (8*f*m^3 + 36*f*m^2 + 46*f*m + 15*f)*cos(f*x + e) - (8*f*m^3 + 36*f*m^2 + 46*f*m + 15*f)*sin(f*x + e) + 15*f)","A",0
412,1,145,0,1.164142," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{2 \, {\left({\left(2 \, c m + c\right)} \cos\left(f x + e\right)^{2} + {\left(2 \, c m + 5 \, c\right)} \cos\left(f x + e\right) - {\left({\left(2 \, c m + c\right)} \cos\left(f x + e\right) - 4 \, c\right)} \sin\left(f x + e\right) + 4 \, c\right)} \sqrt{-c \sin\left(f x + e\right) + c} {\left(a \sin\left(f x + e\right) + a\right)}^{m}}{4 \, f m^{2} + 8 \, f m + {\left(4 \, f m^{2} + 8 \, f m + 3 \, f\right)} \cos\left(f x + e\right) - {\left(4 \, f m^{2} + 8 \, f m + 3 \, f\right)} \sin\left(f x + e\right) + 3 \, f}"," ",0,"2*((2*c*m + c)*cos(f*x + e)^2 + (2*c*m + 5*c)*cos(f*x + e) - ((2*c*m + c)*cos(f*x + e) - 4*c)*sin(f*x + e) + 4*c)*sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(4*f*m^2 + 8*f*m + (4*f*m^2 + 8*f*m + 3*f)*cos(f*x + e) - (4*f*m^2 + 8*f*m + 3*f)*sin(f*x + e) + 3*f)","A",0
413,1,76,0,0.941016," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{2 \, \sqrt{-c \sin\left(f x + e\right) + c} {\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1\right)}}{2 \, f m + {\left(2 \, f m + f\right)} \cos\left(f x + e\right) - {\left(2 \, f m + f\right)} \sin\left(f x + e\right) + f}"," ",0,"2*sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m*(cos(f*x + e) + sin(f*x + e) + 1)/(2*f*m + (2*f*m + f)*cos(f*x + e) - (2*f*m + f)*sin(f*x + e) + f)","A",0
414,0,0,0,1.041114," ","integrate((a+a*sin(f*x+e))^m/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-c \sin\left(f x + e\right) + c} {\left(a \sin\left(f x + e\right) + a\right)}^{m}}{c \sin\left(f x + e\right) - c}, x\right)"," ",0,"integral(-sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(c*sin(f*x + e) - c), x)","F",0
415,0,0,0,1.222643," ","integrate((a+a*sin(f*x+e))^m/(c-c*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-c \sin\left(f x + e\right) + c} {\left(a \sin\left(f x + e\right) + a\right)}^{m}}{c^{2} \cos\left(f x + e\right)^{2} + 2 \, c^{2} \sin\left(f x + e\right) - 2 \, c^{2}}, x\right)"," ",0,"integral(-sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(c^2*cos(f*x + e)^2 + 2*c^2*sin(f*x + e) - 2*c^2), x)","F",0
416,0,0,0,1.009408," ","integrate((a+a*sin(f*x+e))^m/(c-c*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-c \sin\left(f x + e\right) + c} {\left(a \sin\left(f x + e\right) + a\right)}^{m}}{3 \, c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3} - {\left(c^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(3*c^3*cos(f*x + e)^2 - 4*c^3 - (c^3*cos(f*x + e)^2 - 4*c^3)*sin(f*x + e)), x)","F",0
417,0,0,0,0.719974," ","integrate((a+a*sin(f*x+e))^m/(c-c*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-c \sin\left(f x + e\right) + c} {\left(a \sin\left(f x + e\right) + a\right)}^{m}}{c \sin\left(f x + e\right) - c}, x\right)"," ",0,"integral(-sqrt(-c*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(c*sin(f*x + e) - c), x)","F",0
418,0,0,0,1.010301," ","integrate((c+c*sin(f*x+e))^m/(a-a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-a \sin\left(f x + e\right) + a} {\left(c \sin\left(f x + e\right) + c\right)}^{m}}{a \sin\left(f x + e\right) - a}, x\right)"," ",0,"integral(-sqrt(-a*sin(f*x + e) + a)*(c*sin(f*x + e) + c)^m/(a*sin(f*x + e) - a), x)","F",0
419,1,101,0,1.224300," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-3-m),x, algorithm=""fricas"")","-\frac{{\left(2 \, \cos\left(f x + e\right)^{3} + 2 \, {\left(2 \, m + 3\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(4 \, m^{2} + 12 \, m + 9\right)} \cos\left(f x + e\right)\right)} {\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-c \sin\left(f x + e\right) + c\right)}^{-m - 3}}{8 \, f m^{3} + 36 \, f m^{2} + 46 \, f m + 15 \, f}"," ",0,"-(2*cos(f*x + e)^3 + 2*(2*m + 3)*cos(f*x + e)*sin(f*x + e) - (4*m^2 + 12*m + 9)*cos(f*x + e))*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(-m - 3)/(8*f*m^3 + 36*f*m^2 + 46*f*m + 15*f)","A",0
420,1,72,0,0.825845," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-2-m),x, algorithm=""fricas"")","\frac{{\left(2 \, {\left(m + 1\right)} \cos\left(f x + e\right) - \cos\left(f x + e\right) \sin\left(f x + e\right)\right)} {\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-c \sin\left(f x + e\right) + c\right)}^{-m - 2}}{4 \, f m^{2} + 8 \, f m + 3 \, f}"," ",0,"(2*(m + 1)*cos(f*x + e) - cos(f*x + e)*sin(f*x + e))*(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(-m - 2)/(4*f*m^2 + 8*f*m + 3*f)","A",0
421,1,44,0,1.075490," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(-1-m),x, algorithm=""fricas"")","\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-c \sin\left(f x + e\right) + c\right)}^{-m - 1} \cos\left(f x + e\right)}{2 \, f m + f}"," ",0,"(a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(-m - 1)*cos(f*x + e)/(2*f*m + f)","A",0
422,0,0,0,1.223317," ","integrate((a+a*sin(f*x+e))^m/((c-c*sin(f*x+e))^m),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{m}}{{\left(-c \sin\left(f x + e\right) + c\right)}^{m}}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m/(-c*sin(f*x + e) + c)^m, x)","F",0
423,0,0,0,1.128014," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(1-m),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-c \sin\left(f x + e\right) + c\right)}^{-m + 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(-m + 1), x)","F",0
424,0,0,0,0.860192," ","integrate((a+a*sin(f*x+e))^m*(c-c*sin(f*x+e))^(2-m),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-c \sin\left(f x + e\right) + c\right)}^{-m + 2}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(-c*sin(f*x + e) + c)^(-m + 2), x)","F",0
425,1,204,0,1.274309," ","integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^4,x, algorithm=""fricas"")","-\frac{24 \, a d^{4} \cos\left(f x + e\right)^{5} - 80 \, {\left(3 \, a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(8 \, a c^{4} + 16 \, a c^{3} d + 24 \, a c^{2} d^{2} + 12 \, a c d^{3} + 3 \, a d^{4}\right)} f x + 120 \, {\left(a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right) - 15 \, {\left(2 \, {\left(4 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(16 \, a c^{3} d + 24 \, a c^{2} d^{2} + 20 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{120 \, f}"," ",0,"-1/120*(24*a*d^4*cos(f*x + e)^5 - 80*(3*a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^3 - 15*(8*a*c^4 + 16*a*c^3*d + 24*a*c^2*d^2 + 12*a*c*d^3 + 3*a*d^4)*f*x + 120*(a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*cos(f*x + e) - 15*(2*(4*a*c*d^3 + a*d^4)*cos(f*x + e)^3 - (16*a*c^3*d + 24*a*c^2*d^2 + 20*a*c*d^3 + 5*a*d^4)*cos(f*x + e))*sin(f*x + e))/f","A",0
426,1,145,0,1.105922," ","integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{8 \, {\left(3 \, a c d^{2} + a d^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right)} f x - 24 \, {\left(a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3}\right)} \cos\left(f x + e\right) + 3 \, {\left(2 \, a d^{3} \cos\left(f x + e\right)^{3} - {\left(12 \, a c^{2} d + 12 \, a c d^{2} + 5 \, a d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, f}"," ",0,"1/24*(8*(3*a*c*d^2 + a*d^3)*cos(f*x + e)^3 + 3*(8*a*c^3 + 12*a*c^2*d + 12*a*c*d^2 + 3*a*d^3)*f*x - 24*(a*c^3 + 3*a*c^2*d + 3*a*c*d^2 + a*d^3)*cos(f*x + e) + 3*(2*a*d^3*cos(f*x + e)^3 - (12*a*c^2*d + 12*a*c*d^2 + 5*a*d^3)*cos(f*x + e))*sin(f*x + e))/f","A",0
427,1,90,0,1.020626," ","integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{2 \, a d^{2} \cos\left(f x + e\right)^{3} + 3 \, {\left(2 \, a c^{2} + 2 \, a c d + a d^{2}\right)} f x - 3 \, {\left(2 \, a c d + a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 6 \, {\left(a c^{2} + 2 \, a c d + a d^{2}\right)} \cos\left(f x + e\right)}{6 \, f}"," ",0,"1/6*(2*a*d^2*cos(f*x + e)^3 + 3*(2*a*c^2 + 2*a*c*d + a*d^2)*f*x - 3*(2*a*c*d + a*d^2)*cos(f*x + e)*sin(f*x + e) - 6*(a*c^2 + 2*a*c*d + a*d^2)*cos(f*x + e))/f","A",0
428,1,48,0,1.112190," ","integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{a d \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(2 \, a c + a d\right)} f x + 2 \, {\left(a c + a d\right)} \cos\left(f x + e\right)}{2 \, f}"," ",0,"-1/2*(a*d*cos(f*x + e)*sin(f*x + e) - (2*a*c + a*d)*f*x + 2*(a*c + a*d)*cos(f*x + e))/f","A",0
429,1,18,0,1.145957," ","integrate(a+a*sin(f*x+e),x, algorithm=""fricas"")","\frac{a f x - a \cos\left(f x + e\right)}{f}"," ",0,"(a*f*x - a*cos(f*x + e))/f","A",0
430,1,228,0,0.910337," ","integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[\frac{2 \, a f x + a \sqrt{-\frac{c - d}{c + d}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left({\left(c^{2} + c d\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{c + d}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right)}{2 \, d f}, \frac{a f x + a \sqrt{\frac{c - d}{c + d}} \arctan\left(-\frac{{\left(c \sin\left(f x + e\right) + d\right)} \sqrt{\frac{c - d}{c + d}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right)}{d f}\right]"," ",0,"[1/2*(2*a*f*x + a*sqrt(-(c - d)/(c + d))*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*((c^2 + c*d)*cos(f*x + e)*sin(f*x + e) + (c*d + d^2)*cos(f*x + e))*sqrt(-(c - d)/(c + d)))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)))/(d*f), (a*f*x + a*sqrt((c - d)/(c + d))*arctan(-(c*sin(f*x + e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(f*x + e))))/(d*f)]","A",0
431,1,362,0,0.995166," ","integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[-\frac{{\left(a d \sin\left(f x + e\right) + a c\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(a c^{2} - a d^{2}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(c^{3} d + c^{2} d^{2} - c d^{3} - d^{4}\right)} f \sin\left(f x + e\right) + {\left(c^{4} + c^{3} d - c^{2} d^{2} - c d^{3}\right)} f\right)}}, -\frac{{\left(a d \sin\left(f x + e\right) + a c\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(a c^{2} - a d^{2}\right)} \cos\left(f x + e\right)}{{\left(c^{3} d + c^{2} d^{2} - c d^{3} - d^{4}\right)} f \sin\left(f x + e\right) + {\left(c^{4} + c^{3} d - c^{2} d^{2} - c d^{3}\right)} f}\right]"," ",0,"[-1/2*((a*d*sin(f*x + e) + a*c)*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(a*c^2 - a*d^2)*cos(f*x + e))/((c^3*d + c^2*d^2 - c*d^3 - d^4)*f*sin(f*x + e) + (c^4 + c^3*d - c^2*d^2 - c*d^3)*f), -((a*d*sin(f*x + e) + a*c)*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (a*c^2 - a*d^2)*cos(f*x + e))/((c^3*d + c^2*d^2 - c*d^3 - d^4)*f*sin(f*x + e) + (c^4 + c^3*d - c^2*d^2 - c*d^3)*f)]","A",0
432,1,803,0,0.779415," ","integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} - a c d^{3} + 2 \, a d^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(2 \, a c^{3} - a c^{2} d + 2 \, a c d^{2} - a d^{3} - {\left(2 \, a c d^{2} - a d^{3}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, a c^{2} d - a c d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(2 \, a c^{4} - 2 \, a c^{3} d - 3 \, a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)}{4 \, {\left({\left(c^{5} d^{2} + c^{4} d^{3} - 2 \, c^{3} d^{4} - 2 \, c^{2} d^{5} + c d^{6} + d^{7}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{6} d + c^{5} d^{2} - 2 \, c^{4} d^{3} - 2 \, c^{3} d^{4} + c^{2} d^{5} + c d^{6}\right)} f \sin\left(f x + e\right) - {\left(c^{7} + c^{6} d - c^{5} d^{2} - c^{4} d^{3} - c^{3} d^{4} - c^{2} d^{5} + c d^{6} + d^{7}\right)} f\right)}}, \frac{{\left(a c^{3} d - 2 \, a c^{2} d^{2} - a c d^{3} + 2 \, a d^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(2 \, a c^{3} - a c^{2} d + 2 \, a c d^{2} - a d^{3} - {\left(2 \, a c d^{2} - a d^{3}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, a c^{2} d - a c d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(2 \, a c^{4} - 2 \, a c^{3} d - 3 \, a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(c^{5} d^{2} + c^{4} d^{3} - 2 \, c^{3} d^{4} - 2 \, c^{2} d^{5} + c d^{6} + d^{7}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{6} d + c^{5} d^{2} - 2 \, c^{4} d^{3} - 2 \, c^{3} d^{4} + c^{2} d^{5} + c d^{6}\right)} f \sin\left(f x + e\right) - {\left(c^{7} + c^{6} d - c^{5} d^{2} - c^{4} d^{3} - c^{3} d^{4} - c^{2} d^{5} + c d^{6} + d^{7}\right)} f\right)}}\right]"," ",0,"[1/4*(2*(a*c^3*d - 2*a*c^2*d^2 - a*c*d^3 + 2*a*d^4)*cos(f*x + e)*sin(f*x + e) + (2*a*c^3 - a*c^2*d + 2*a*c*d^2 - a*d^3 - (2*a*c*d^2 - a*d^3)*cos(f*x + e)^2 + 2*(2*a*c^2*d - a*c*d^2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(2*a*c^4 - 2*a*c^3*d - 3*a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e))/((c^5*d^2 + c^4*d^3 - 2*c^3*d^4 - 2*c^2*d^5 + c*d^6 + d^7)*f*cos(f*x + e)^2 - 2*(c^6*d + c^5*d^2 - 2*c^4*d^3 - 2*c^3*d^4 + c^2*d^5 + c*d^6)*f*sin(f*x + e) - (c^7 + c^6*d - c^5*d^2 - c^4*d^3 - c^3*d^4 - c^2*d^5 + c*d^6 + d^7)*f), 1/2*((a*c^3*d - 2*a*c^2*d^2 - a*c*d^3 + 2*a*d^4)*cos(f*x + e)*sin(f*x + e) + (2*a*c^3 - a*c^2*d + 2*a*c*d^2 - a*d^3 - (2*a*c*d^2 - a*d^3)*cos(f*x + e)^2 + 2*(2*a*c^2*d - a*c*d^2)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (2*a*c^4 - 2*a*c^3*d - 3*a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e))/((c^5*d^2 + c^4*d^3 - 2*c^3*d^4 - 2*c^2*d^5 + c*d^6 + d^7)*f*cos(f*x + e)^2 - 2*(c^6*d + c^5*d^2 - 2*c^4*d^3 - 2*c^3*d^4 + c^2*d^5 + c*d^6)*f*sin(f*x + e) - (c^7 + c^6*d - c^5*d^2 - c^4*d^3 - c^3*d^4 - c^2*d^5 + c*d^6 + d^7)*f)]","B",0
433,1,1344,0,1.518624," ","integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^4,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(2 \, a c^{4} d^{2} - 9 \, a c^{3} d^{3} + 2 \, a c^{2} d^{4} + 9 \, a c d^{5} - 4 \, a d^{6}\right)} \cos\left(f x + e\right)^{3} - 6 \, {\left(2 \, a c^{5} d - 7 \, a c^{4} d^{2} + 8 \, a c^{2} d^{4} - 2 \, a c d^{5} - a d^{6}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 3 \, {\left(2 \, a c^{5} - 2 \, a c^{4} d + 7 \, a c^{3} d^{2} - 6 \, a c^{2} d^{3} + 3 \, a c d^{4} - 3 \, {\left(2 \, a c^{3} d^{2} - 2 \, a c^{2} d^{3} + a c d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(6 \, a c^{4} d - 6 \, a c^{3} d^{2} + 5 \, a c^{2} d^{3} - 2 \, a c d^{4} + a d^{5} - {\left(2 \, a c^{2} d^{3} - 2 \, a c d^{4} + a d^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 12 \, {\left(a c^{6} - 2 \, a c^{5} d - a c^{4} d^{2} + a c^{3} d^{3} + a c^{2} d^{4} + a c d^{5} - a d^{6}\right)} \cos\left(f x + e\right)}{12 \, {\left(3 \, {\left(c^{8} d^{2} + c^{7} d^{3} - 3 \, c^{6} d^{4} - 3 \, c^{5} d^{5} + 3 \, c^{4} d^{6} + 3 \, c^{3} d^{7} - c^{2} d^{8} - c d^{9}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{10} + c^{9} d - 6 \, c^{6} d^{4} - 6 \, c^{5} d^{5} + 8 \, c^{4} d^{6} + 8 \, c^{3} d^{7} - 3 \, c^{2} d^{8} - 3 \, c d^{9}\right)} f + {\left({\left(c^{7} d^{3} + c^{6} d^{4} - 3 \, c^{5} d^{5} - 3 \, c^{4} d^{6} + 3 \, c^{3} d^{7} + 3 \, c^{2} d^{8} - c d^{9} - d^{10}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{9} d + 3 \, c^{8} d^{2} - 8 \, c^{7} d^{3} - 8 \, c^{6} d^{4} + 6 \, c^{5} d^{5} + 6 \, c^{4} d^{6} - c d^{9} - d^{10}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left(2 \, a c^{4} d^{2} - 9 \, a c^{3} d^{3} + 2 \, a c^{2} d^{4} + 9 \, a c d^{5} - 4 \, a d^{6}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(2 \, a c^{5} d - 7 \, a c^{4} d^{2} + 8 \, a c^{2} d^{4} - 2 \, a c d^{5} - a d^{6}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 3 \, {\left(2 \, a c^{5} - 2 \, a c^{4} d + 7 \, a c^{3} d^{2} - 6 \, a c^{2} d^{3} + 3 \, a c d^{4} - 3 \, {\left(2 \, a c^{3} d^{2} - 2 \, a c^{2} d^{3} + a c d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(6 \, a c^{4} d - 6 \, a c^{3} d^{2} + 5 \, a c^{2} d^{3} - 2 \, a c d^{4} + a d^{5} - {\left(2 \, a c^{2} d^{3} - 2 \, a c d^{4} + a d^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) - 6 \, {\left(a c^{6} - 2 \, a c^{5} d - a c^{4} d^{2} + a c^{3} d^{3} + a c^{2} d^{4} + a c d^{5} - a d^{6}\right)} \cos\left(f x + e\right)}{6 \, {\left(3 \, {\left(c^{8} d^{2} + c^{7} d^{3} - 3 \, c^{6} d^{4} - 3 \, c^{5} d^{5} + 3 \, c^{4} d^{6} + 3 \, c^{3} d^{7} - c^{2} d^{8} - c d^{9}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{10} + c^{9} d - 6 \, c^{6} d^{4} - 6 \, c^{5} d^{5} + 8 \, c^{4} d^{6} + 8 \, c^{3} d^{7} - 3 \, c^{2} d^{8} - 3 \, c d^{9}\right)} f + {\left({\left(c^{7} d^{3} + c^{6} d^{4} - 3 \, c^{5} d^{5} - 3 \, c^{4} d^{6} + 3 \, c^{3} d^{7} + 3 \, c^{2} d^{8} - c d^{9} - d^{10}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{9} d + 3 \, c^{8} d^{2} - 8 \, c^{7} d^{3} - 8 \, c^{6} d^{4} + 6 \, c^{5} d^{5} + 6 \, c^{4} d^{6} - c d^{9} - d^{10}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/12*(2*(2*a*c^4*d^2 - 9*a*c^3*d^3 + 2*a*c^2*d^4 + 9*a*c*d^5 - 4*a*d^6)*cos(f*x + e)^3 - 6*(2*a*c^5*d - 7*a*c^4*d^2 + 8*a*c^2*d^4 - 2*a*c*d^5 - a*d^6)*cos(f*x + e)*sin(f*x + e) - 3*(2*a*c^5 - 2*a*c^4*d + 7*a*c^3*d^2 - 6*a*c^2*d^3 + 3*a*c*d^4 - 3*(2*a*c^3*d^2 - 2*a*c^2*d^3 + a*c*d^4)*cos(f*x + e)^2 + (6*a*c^4*d - 6*a*c^3*d^2 + 5*a*c^2*d^3 - 2*a*c*d^4 + a*d^5 - (2*a*c^2*d^3 - 2*a*c*d^4 + a*d^5)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 12*(a*c^6 - 2*a*c^5*d - a*c^4*d^2 + a*c^3*d^3 + a*c^2*d^4 + a*c*d^5 - a*d^6)*cos(f*x + e))/(3*(c^8*d^2 + c^7*d^3 - 3*c^6*d^4 - 3*c^5*d^5 + 3*c^4*d^6 + 3*c^3*d^7 - c^2*d^8 - c*d^9)*f*cos(f*x + e)^2 - (c^10 + c^9*d - 6*c^6*d^4 - 6*c^5*d^5 + 8*c^4*d^6 + 8*c^3*d^7 - 3*c^2*d^8 - 3*c*d^9)*f + ((c^7*d^3 + c^6*d^4 - 3*c^5*d^5 - 3*c^4*d^6 + 3*c^3*d^7 + 3*c^2*d^8 - c*d^9 - d^10)*f*cos(f*x + e)^2 - (3*c^9*d + 3*c^8*d^2 - 8*c^7*d^3 - 8*c^6*d^4 + 6*c^5*d^5 + 6*c^4*d^6 - c*d^9 - d^10)*f)*sin(f*x + e)), -1/6*((2*a*c^4*d^2 - 9*a*c^3*d^3 + 2*a*c^2*d^4 + 9*a*c*d^5 - 4*a*d^6)*cos(f*x + e)^3 - 3*(2*a*c^5*d - 7*a*c^4*d^2 + 8*a*c^2*d^4 - 2*a*c*d^5 - a*d^6)*cos(f*x + e)*sin(f*x + e) - 3*(2*a*c^5 - 2*a*c^4*d + 7*a*c^3*d^2 - 6*a*c^2*d^3 + 3*a*c*d^4 - 3*(2*a*c^3*d^2 - 2*a*c^2*d^3 + a*c*d^4)*cos(f*x + e)^2 + (6*a*c^4*d - 6*a*c^3*d^2 + 5*a*c^2*d^3 - 2*a*c*d^4 + a*d^5 - (2*a*c^2*d^3 - 2*a*c*d^4 + a*d^5)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - 6*(a*c^6 - 2*a*c^5*d - a*c^4*d^2 + a*c^3*d^3 + a*c^2*d^4 + a*c*d^5 - a*d^6)*cos(f*x + e))/(3*(c^8*d^2 + c^7*d^3 - 3*c^6*d^4 - 3*c^5*d^5 + 3*c^4*d^6 + 3*c^3*d^7 - c^2*d^8 - c*d^9)*f*cos(f*x + e)^2 - (c^10 + c^9*d - 6*c^6*d^4 - 6*c^5*d^5 + 8*c^4*d^6 + 8*c^3*d^7 - 3*c^2*d^8 - 3*c*d^9)*f + ((c^7*d^3 + c^6*d^4 - 3*c^5*d^5 - 3*c^4*d^6 + 3*c^3*d^7 + 3*c^2*d^8 - c*d^9 - d^10)*f*cos(f*x + e)^2 - (3*c^9*d + 3*c^8*d^2 - 8*c^7*d^3 - 8*c^6*d^4 + 6*c^5*d^5 + 6*c^4*d^6 - c*d^9 - d^10)*f)*sin(f*x + e))]","B",0
434,1,299,0,1.096295," ","integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^4,x, algorithm=""fricas"")","-\frac{96 \, {\left(2 \, a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right)^{5} - 320 \, {\left(a^{2} c^{3} d + 3 \, a^{2} c^{2} d^{2} + 3 \, a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(24 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 84 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 11 \, a^{2} d^{4}\right)} f x + 480 \, {\left(a^{2} c^{4} + 4 \, a^{2} c^{3} d + 6 \, a^{2} c^{2} d^{2} + 4 \, a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right) + 5 \, {\left(8 \, a^{2} d^{4} \cos\left(f x + e\right)^{5} - 2 \, {\left(36 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(8 \, a^{2} c^{4} + 64 \, a^{2} c^{3} d + 108 \, a^{2} c^{2} d^{2} + 80 \, a^{2} c d^{3} + 21 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{240 \, f}"," ",0,"-1/240*(96*(2*a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^5 - 320*(a^2*c^3*d + 3*a^2*c^2*d^2 + 3*a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^3 - 15*(24*a^2*c^4 + 64*a^2*c^3*d + 84*a^2*c^2*d^2 + 48*a^2*c*d^3 + 11*a^2*d^4)*f*x + 480*(a^2*c^4 + 4*a^2*c^3*d + 6*a^2*c^2*d^2 + 4*a^2*c*d^3 + a^2*d^4)*cos(f*x + e) + 5*(8*a^2*d^4*cos(f*x + e)^5 - 2*(36*a^2*c^2*d^2 + 48*a^2*c*d^3 + 19*a^2*d^4)*cos(f*x + e)^3 + 3*(8*a^2*c^4 + 64*a^2*c^3*d + 108*a^2*c^2*d^2 + 80*a^2*c*d^3 + 21*a^2*d^4)*cos(f*x + e))*sin(f*x + e))/f","A",0
435,1,217,0,1.223176," ","integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{8 \, a^{2} d^{3} \cos\left(f x + e\right)^{5} - 40 \, {\left(a^{2} c^{2} d + 2 \, a^{2} c d^{2} + a^{2} d^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(4 \, a^{2} c^{3} + 8 \, a^{2} c^{2} d + 7 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right)} f x + 80 \, {\left(a^{2} c^{3} + 3 \, a^{2} c^{2} d + 3 \, a^{2} c d^{2} + a^{2} d^{3}\right)} \cos\left(f x + e\right) - 5 \, {\left(2 \, {\left(3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(4 \, a^{2} c^{3} + 24 \, a^{2} c^{2} d + 27 \, a^{2} c d^{2} + 10 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{40 \, f}"," ",0,"-1/40*(8*a^2*d^3*cos(f*x + e)^5 - 40*(a^2*c^2*d + 2*a^2*c*d^2 + a^2*d^3)*cos(f*x + e)^3 - 15*(4*a^2*c^3 + 8*a^2*c^2*d + 7*a^2*c*d^2 + 2*a^2*d^3)*f*x + 80*(a^2*c^3 + 3*a^2*c^2*d + 3*a^2*c*d^2 + a^2*d^3)*cos(f*x + e) - 5*(2*(3*a^2*c*d^2 + 2*a^2*d^3)*cos(f*x + e)^3 - (4*a^2*c^3 + 24*a^2*c^2*d + 27*a^2*c*d^2 + 10*a^2*d^3)*cos(f*x + e))*sin(f*x + e))/f","A",0
436,1,145,0,1.196408," ","integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{16 \, {\left(a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(12 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right)} f x - 48 \, {\left(a^{2} c^{2} + 2 \, a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right) + 3 \, {\left(2 \, a^{2} d^{2} \cos\left(f x + e\right)^{3} - {\left(4 \, a^{2} c^{2} + 16 \, a^{2} c d + 9 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, f}"," ",0,"1/24*(16*(a^2*c*d + a^2*d^2)*cos(f*x + e)^3 + 3*(12*a^2*c^2 + 16*a^2*c*d + 7*a^2*d^2)*f*x - 48*(a^2*c^2 + 2*a^2*c*d + a^2*d^2)*cos(f*x + e) + 3*(2*a^2*d^2*cos(f*x + e)^3 - (4*a^2*c^2 + 16*a^2*c*d + 9*a^2*d^2)*cos(f*x + e))*sin(f*x + e))/f","A",0
437,1,82,0,1.298669," ","integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\frac{2 \, a^{2} d \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, a^{2} c + 2 \, a^{2} d\right)} f x - 3 \, {\left(a^{2} c + 2 \, a^{2} d\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 12 \, {\left(a^{2} c + a^{2} d\right)} \cos\left(f x + e\right)}{6 \, f}"," ",0,"1/6*(2*a^2*d*cos(f*x + e)^3 + 3*(3*a^2*c + 2*a^2*d)*f*x - 3*(a^2*c + 2*a^2*d)*cos(f*x + e)*sin(f*x + e) - 12*(a^2*c + a^2*d)*cos(f*x + e))/f","A",0
438,1,41,0,0.815422," ","integrate((a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{3 \, a^{2} f x - a^{2} \cos\left(f x + e\right) \sin\left(f x + e\right) - 4 \, a^{2} \cos\left(f x + e\right)}{2 \, f}"," ",0,"1/2*(3*a^2*f*x - a^2*cos(f*x + e)*sin(f*x + e) - 4*a^2*cos(f*x + e))/f","A",0
439,1,296,0,1.454975," ","integrate((a+a*sin(f*x+e))^2/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[-\frac{2 \, a^{2} d \cos\left(f x + e\right) + 2 \, {\left(a^{2} c - 2 \, a^{2} d\right)} f x + {\left(a^{2} c - a^{2} d\right)} \sqrt{-\frac{c - d}{c + d}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left({\left(c^{2} + c d\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{c + d}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right)}{2 \, d^{2} f}, -\frac{a^{2} d \cos\left(f x + e\right) + {\left(a^{2} c - 2 \, a^{2} d\right)} f x + {\left(a^{2} c - a^{2} d\right)} \sqrt{\frac{c - d}{c + d}} \arctan\left(-\frac{{\left(c \sin\left(f x + e\right) + d\right)} \sqrt{\frac{c - d}{c + d}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right)}{d^{2} f}\right]"," ",0,"[-1/2*(2*a^2*d*cos(f*x + e) + 2*(a^2*c - 2*a^2*d)*f*x + (a^2*c - a^2*d)*sqrt(-(c - d)/(c + d))*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*((c^2 + c*d)*cos(f*x + e)*sin(f*x + e) + (c*d + d^2)*cos(f*x + e))*sqrt(-(c - d)/(c + d)))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)))/(d^2*f), -(a^2*d*cos(f*x + e) + (a^2*c - 2*a^2*d)*f*x + (a^2*c - a^2*d)*sqrt((c - d)/(c + d))*arctan(-(c*sin(f*x + e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(f*x + e))))/(d^2*f)]","A",0
440,1,476,0,1.396787," ","integrate((a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{2} c d + a^{2} d^{2}\right)} f x \sin\left(f x + e\right) + 2 \, {\left(a^{2} c^{2} + a^{2} c d\right)} f x + {\left(a^{2} c^{2} + 2 \, a^{2} c d + {\left(a^{2} c d + 2 \, a^{2} d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{c + d}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left({\left(c^{2} + c d\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{c + d}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(a^{2} c d - a^{2} d^{2}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(c d^{3} + d^{4}\right)} f \sin\left(f x + e\right) + {\left(c^{2} d^{2} + c d^{3}\right)} f\right)}}, \frac{{\left(a^{2} c d + a^{2} d^{2}\right)} f x \sin\left(f x + e\right) + {\left(a^{2} c^{2} + a^{2} c d\right)} f x + {\left(a^{2} c^{2} + 2 \, a^{2} c d + {\left(a^{2} c d + 2 \, a^{2} d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{c + d}} \arctan\left(-\frac{{\left(c \sin\left(f x + e\right) + d\right)} \sqrt{\frac{c - d}{c + d}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) + {\left(a^{2} c d - a^{2} d^{2}\right)} \cos\left(f x + e\right)}{{\left(c d^{3} + d^{4}\right)} f \sin\left(f x + e\right) + {\left(c^{2} d^{2} + c d^{3}\right)} f}\right]"," ",0,"[1/2*(2*(a^2*c*d + a^2*d^2)*f*x*sin(f*x + e) + 2*(a^2*c^2 + a^2*c*d)*f*x + (a^2*c^2 + 2*a^2*c*d + (a^2*c*d + 2*a^2*d^2)*sin(f*x + e))*sqrt(-(c - d)/(c + d))*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*((c^2 + c*d)*cos(f*x + e)*sin(f*x + e) + (c*d + d^2)*cos(f*x + e))*sqrt(-(c - d)/(c + d)))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(a^2*c*d - a^2*d^2)*cos(f*x + e))/((c*d^3 + d^4)*f*sin(f*x + e) + (c^2*d^2 + c*d^3)*f), ((a^2*c*d + a^2*d^2)*f*x*sin(f*x + e) + (a^2*c^2 + a^2*c*d)*f*x + (a^2*c^2 + 2*a^2*c*d + (a^2*c*d + 2*a^2*d^2)*sin(f*x + e))*sqrt((c - d)/(c + d))*arctan(-(c*sin(f*x + e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(f*x + e))) + (a^2*c*d - a^2*d^2)*cos(f*x + e))/((c*d^3 + d^4)*f*sin(f*x + e) + (c^2*d^2 + c*d^3)*f)]","A",0
441,1,679,0,1.068266," ","integrate((a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{2} c^{3} + 4 \, a^{2} c^{2} d - a^{2} c d^{2} - 4 \, a^{2} d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 3 \, {\left(a^{2} d^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} c d \sin\left(f x + e\right) - a^{2} c^{2} - a^{2} d^{2}\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(4 \, a^{2} c^{3} + a^{2} c^{2} d - 4 \, a^{2} c d^{2} - a^{2} d^{3}\right)} \cos\left(f x + e\right)}{4 \, {\left({\left(c^{4} d^{2} + 2 \, c^{3} d^{3} - 2 \, c d^{5} - d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{5} d + 2 \, c^{4} d^{2} - 2 \, c^{2} d^{4} - c d^{5}\right)} f \sin\left(f x + e\right) - {\left(c^{6} + 2 \, c^{5} d + c^{4} d^{2} - c^{2} d^{4} - 2 \, c d^{5} - d^{6}\right)} f\right)}}, \frac{{\left(a^{2} c^{3} + 4 \, a^{2} c^{2} d - a^{2} c d^{2} - 4 \, a^{2} d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 3 \, {\left(a^{2} d^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} c d \sin\left(f x + e\right) - a^{2} c^{2} - a^{2} d^{2}\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(4 \, a^{2} c^{3} + a^{2} c^{2} d - 4 \, a^{2} c d^{2} - a^{2} d^{3}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(c^{4} d^{2} + 2 \, c^{3} d^{3} - 2 \, c d^{5} - d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{5} d + 2 \, c^{4} d^{2} - 2 \, c^{2} d^{4} - c d^{5}\right)} f \sin\left(f x + e\right) - {\left(c^{6} + 2 \, c^{5} d + c^{4} d^{2} - c^{2} d^{4} - 2 \, c d^{5} - d^{6}\right)} f\right)}}\right]"," ",0,"[1/4*(2*(a^2*c^3 + 4*a^2*c^2*d - a^2*c*d^2 - 4*a^2*d^3)*cos(f*x + e)*sin(f*x + e) - 3*(a^2*d^2*cos(f*x + e)^2 - 2*a^2*c*d*sin(f*x + e) - a^2*c^2 - a^2*d^2)*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(4*a^2*c^3 + a^2*c^2*d - 4*a^2*c*d^2 - a^2*d^3)*cos(f*x + e))/((c^4*d^2 + 2*c^3*d^3 - 2*c*d^5 - d^6)*f*cos(f*x + e)^2 - 2*(c^5*d + 2*c^4*d^2 - 2*c^2*d^4 - c*d^5)*f*sin(f*x + e) - (c^6 + 2*c^5*d + c^4*d^2 - c^2*d^4 - 2*c*d^5 - d^6)*f), 1/2*((a^2*c^3 + 4*a^2*c^2*d - a^2*c*d^2 - 4*a^2*d^3)*cos(f*x + e)*sin(f*x + e) - 3*(a^2*d^2*cos(f*x + e)^2 - 2*a^2*c*d*sin(f*x + e) - a^2*c^2 - a^2*d^2)*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (4*a^2*c^3 + a^2*c^2*d - 4*a^2*c*d^2 - a^2*d^3)*cos(f*x + e))/((c^4*d^2 + 2*c^3*d^3 - 2*c*d^5 - d^6)*f*cos(f*x + e)^2 - 2*(c^5*d + 2*c^4*d^2 - 2*c^2*d^4 - c*d^5)*f*sin(f*x + e) - (c^6 + 2*c^5*d + c^4*d^2 - c^2*d^4 - 2*c*d^5 - d^6)*f)]","B",0
442,1,1366,0,1.294647," ","integrate((a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^4,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a^{2} c^{4} d + 6 \, a^{2} c^{3} d^{2} - 11 \, a^{2} c^{2} d^{3} - 6 \, a^{2} c d^{4} + 10 \, a^{2} d^{5}\right)} \cos\left(f x + e\right)^{3} - 6 \, {\left(a^{2} c^{5} + 6 \, a^{2} c^{4} d - 8 \, a^{2} c^{3} d^{2} - 8 \, a^{2} c^{2} d^{3} + 7 \, a^{2} c d^{4} + 2 \, a^{2} d^{5}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 3 \, {\left(3 \, a^{2} c^{4} - 2 \, a^{2} c^{3} d + 9 \, a^{2} c^{2} d^{2} - 6 \, a^{2} c d^{3} - 3 \, {\left(3 \, a^{2} c^{2} d^{2} - 2 \, a^{2} c d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(9 \, a^{2} c^{3} d - 6 \, a^{2} c^{2} d^{2} + 3 \, a^{2} c d^{3} - 2 \, a^{2} d^{4} - {\left(3 \, a^{2} c d^{3} - 2 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 12 \, {\left(2 \, a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} - a^{2} c^{2} d^{3} + 2 \, a^{2} d^{5}\right)} \cos\left(f x + e\right)}{12 \, {\left(3 \, {\left(c^{7} d^{2} + 2 \, c^{6} d^{3} - c^{5} d^{4} - 4 \, c^{4} d^{5} - c^{3} d^{6} + 2 \, c^{2} d^{7} + c d^{8}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{9} + 2 \, c^{8} d + 2 \, c^{7} d^{2} + 2 \, c^{6} d^{3} - 4 \, c^{5} d^{4} - 10 \, c^{4} d^{5} - 2 \, c^{3} d^{6} + 6 \, c^{2} d^{7} + 3 \, c d^{8}\right)} f + {\left({\left(c^{6} d^{3} + 2 \, c^{5} d^{4} - c^{4} d^{5} - 4 \, c^{3} d^{6} - c^{2} d^{7} + 2 \, c d^{8} + d^{9}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{8} d + 6 \, c^{7} d^{2} - 2 \, c^{6} d^{3} - 10 \, c^{5} d^{4} - 4 \, c^{4} d^{5} + 2 \, c^{3} d^{6} + 2 \, c^{2} d^{7} + 2 \, c d^{8} + d^{9}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left(a^{2} c^{4} d + 6 \, a^{2} c^{3} d^{2} - 11 \, a^{2} c^{2} d^{3} - 6 \, a^{2} c d^{4} + 10 \, a^{2} d^{5}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(a^{2} c^{5} + 6 \, a^{2} c^{4} d - 8 \, a^{2} c^{3} d^{2} - 8 \, a^{2} c^{2} d^{3} + 7 \, a^{2} c d^{4} + 2 \, a^{2} d^{5}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 3 \, {\left(3 \, a^{2} c^{4} - 2 \, a^{2} c^{3} d + 9 \, a^{2} c^{2} d^{2} - 6 \, a^{2} c d^{3} - 3 \, {\left(3 \, a^{2} c^{2} d^{2} - 2 \, a^{2} c d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(9 \, a^{2} c^{3} d - 6 \, a^{2} c^{2} d^{2} + 3 \, a^{2} c d^{3} - 2 \, a^{2} d^{4} - {\left(3 \, a^{2} c d^{3} - 2 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) - 6 \, {\left(2 \, a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} - a^{2} c^{2} d^{3} + 2 \, a^{2} d^{5}\right)} \cos\left(f x + e\right)}{6 \, {\left(3 \, {\left(c^{7} d^{2} + 2 \, c^{6} d^{3} - c^{5} d^{4} - 4 \, c^{4} d^{5} - c^{3} d^{6} + 2 \, c^{2} d^{7} + c d^{8}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{9} + 2 \, c^{8} d + 2 \, c^{7} d^{2} + 2 \, c^{6} d^{3} - 4 \, c^{5} d^{4} - 10 \, c^{4} d^{5} - 2 \, c^{3} d^{6} + 6 \, c^{2} d^{7} + 3 \, c d^{8}\right)} f + {\left({\left(c^{6} d^{3} + 2 \, c^{5} d^{4} - c^{4} d^{5} - 4 \, c^{3} d^{6} - c^{2} d^{7} + 2 \, c d^{8} + d^{9}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{8} d + 6 \, c^{7} d^{2} - 2 \, c^{6} d^{3} - 10 \, c^{5} d^{4} - 4 \, c^{4} d^{5} + 2 \, c^{3} d^{6} + 2 \, c^{2} d^{7} + 2 \, c d^{8} + d^{9}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/12*(2*(a^2*c^4*d + 6*a^2*c^3*d^2 - 11*a^2*c^2*d^3 - 6*a^2*c*d^4 + 10*a^2*d^5)*cos(f*x + e)^3 - 6*(a^2*c^5 + 6*a^2*c^4*d - 8*a^2*c^3*d^2 - 8*a^2*c^2*d^3 + 7*a^2*c*d^4 + 2*a^2*d^5)*cos(f*x + e)*sin(f*x + e) - 3*(3*a^2*c^4 - 2*a^2*c^3*d + 9*a^2*c^2*d^2 - 6*a^2*c*d^3 - 3*(3*a^2*c^2*d^2 - 2*a^2*c*d^3)*cos(f*x + e)^2 + (9*a^2*c^3*d - 6*a^2*c^2*d^2 + 3*a^2*c*d^3 - 2*a^2*d^4 - (3*a^2*c*d^3 - 2*a^2*d^4)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 12*(2*a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 - a^2*c^2*d^3 + 2*a^2*d^5)*cos(f*x + e))/(3*(c^7*d^2 + 2*c^6*d^3 - c^5*d^4 - 4*c^4*d^5 - c^3*d^6 + 2*c^2*d^7 + c*d^8)*f*cos(f*x + e)^2 - (c^9 + 2*c^8*d + 2*c^7*d^2 + 2*c^6*d^3 - 4*c^5*d^4 - 10*c^4*d^5 - 2*c^3*d^6 + 6*c^2*d^7 + 3*c*d^8)*f + ((c^6*d^3 + 2*c^5*d^4 - c^4*d^5 - 4*c^3*d^6 - c^2*d^7 + 2*c*d^8 + d^9)*f*cos(f*x + e)^2 - (3*c^8*d + 6*c^7*d^2 - 2*c^6*d^3 - 10*c^5*d^4 - 4*c^4*d^5 + 2*c^3*d^6 + 2*c^2*d^7 + 2*c*d^8 + d^9)*f)*sin(f*x + e)), -1/6*((a^2*c^4*d + 6*a^2*c^3*d^2 - 11*a^2*c^2*d^3 - 6*a^2*c*d^4 + 10*a^2*d^5)*cos(f*x + e)^3 - 3*(a^2*c^5 + 6*a^2*c^4*d - 8*a^2*c^3*d^2 - 8*a^2*c^2*d^3 + 7*a^2*c*d^4 + 2*a^2*d^5)*cos(f*x + e)*sin(f*x + e) - 3*(3*a^2*c^4 - 2*a^2*c^3*d + 9*a^2*c^2*d^2 - 6*a^2*c*d^3 - 3*(3*a^2*c^2*d^2 - 2*a^2*c*d^3)*cos(f*x + e)^2 + (9*a^2*c^3*d - 6*a^2*c^2*d^2 + 3*a^2*c*d^3 - 2*a^2*d^4 - (3*a^2*c*d^3 - 2*a^2*d^4)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - 6*(2*a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 - a^2*c^2*d^3 + 2*a^2*d^5)*cos(f*x + e))/(3*(c^7*d^2 + 2*c^6*d^3 - c^5*d^4 - 4*c^4*d^5 - c^3*d^6 + 2*c^2*d^7 + c*d^8)*f*cos(f*x + e)^2 - (c^9 + 2*c^8*d + 2*c^7*d^2 + 2*c^6*d^3 - 4*c^5*d^4 - 10*c^4*d^5 - 2*c^3*d^6 + 6*c^2*d^7 + 3*c*d^8)*f + ((c^6*d^3 + 2*c^5*d^4 - c^4*d^5 - 4*c^3*d^6 - c^2*d^7 + 2*c*d^8 + d^9)*f*cos(f*x + e)^2 - (3*c^8*d + 6*c^7*d^2 - 2*c^6*d^3 - 10*c^5*d^4 - 4*c^4*d^5 + 2*c^3*d^6 + 2*c^2*d^7 + 2*c*d^8 + d^9)*f)*sin(f*x + e))]","B",0
443,1,2151,0,1.473500," ","integrate((a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^5,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(8 \, a^{2} c^{6} d + 64 \, a^{2} c^{5} d^{2} - 208 \, a^{2} c^{4} d^{3} + 16 \, a^{2} c^{3} d^{4} + 221 \, a^{2} c^{2} d^{5} - 80 \, a^{2} c d^{6} - 21 \, a^{2} d^{7}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(12 \, a^{2} c^{6} - 16 \, a^{2} c^{5} d + 79 \, a^{2} c^{4} d^{2} - 96 \, a^{2} c^{3} d^{3} + 54 \, a^{2} c^{2} d^{4} - 16 \, a^{2} c d^{5} + 7 \, a^{2} d^{6} + {\left(12 \, a^{2} c^{2} d^{4} - 16 \, a^{2} c d^{5} + 7 \, a^{2} d^{6}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(36 \, a^{2} c^{4} d^{2} - 48 \, a^{2} c^{3} d^{3} + 33 \, a^{2} c^{2} d^{4} - 16 \, a^{2} c d^{5} + 7 \, a^{2} d^{6}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(12 \, a^{2} c^{5} d - 16 \, a^{2} c^{4} d^{2} + 19 \, a^{2} c^{3} d^{3} - 16 \, a^{2} c^{2} d^{4} + 7 \, a^{2} c d^{5} - {\left(12 \, a^{2} c^{3} d^{3} - 16 \, a^{2} c^{2} d^{4} + 7 \, a^{2} c d^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 6 \, {\left(16 \, a^{2} c^{7} - 20 \, a^{2} c^{6} d - 45 \, a^{2} c^{4} d^{3} + 16 \, a^{2} c^{3} d^{4} + 74 \, a^{2} c^{2} d^{5} - 32 \, a^{2} c d^{6} - 9 \, a^{2} d^{7}\right)} \cos\left(f x + e\right) + 2 \, {\left({\left(2 \, a^{2} c^{5} d^{2} + 16 \, a^{2} c^{4} d^{3} - 61 \, a^{2} c^{3} d^{4} + 16 \, a^{2} c^{2} d^{5} + 59 \, a^{2} c d^{6} - 32 \, a^{2} d^{7}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(4 \, a^{2} c^{7} + 32 \, a^{2} c^{6} d - 79 \, a^{2} c^{5} d^{2} - 16 \, a^{2} c^{4} d^{3} + 70 \, a^{2} c^{3} d^{4} + 5 \, a^{2} c d^{6} - 16 \, a^{2} d^{7}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, {\left({\left(c^{8} d^{4} + 2 \, c^{7} d^{5} - 2 \, c^{6} d^{6} - 6 \, c^{5} d^{7} + 6 \, c^{3} d^{9} + 2 \, c^{2} d^{10} - 2 \, c d^{11} - d^{12}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, c^{10} d^{2} + 6 \, c^{9} d^{3} - 5 \, c^{8} d^{4} - 16 \, c^{7} d^{5} - 2 \, c^{6} d^{6} + 12 \, c^{5} d^{7} + 6 \, c^{4} d^{8} - c^{2} d^{10} - 2 \, c d^{11} - d^{12}\right)} f \cos\left(f x + e\right)^{2} + {\left(c^{12} + 2 \, c^{11} d + 4 \, c^{10} d^{2} + 6 \, c^{9} d^{3} - 11 \, c^{8} d^{4} - 28 \, c^{7} d^{5} + 28 \, c^{5} d^{7} + 11 \, c^{4} d^{8} - 6 \, c^{3} d^{9} - 4 \, c^{2} d^{10} - 2 \, c d^{11} - d^{12}\right)} f - 4 \, {\left({\left(c^{9} d^{3} + 2 \, c^{8} d^{4} - 2 \, c^{7} d^{5} - 6 \, c^{6} d^{6} + 6 \, c^{4} d^{8} + 2 \, c^{3} d^{9} - 2 \, c^{2} d^{10} - c d^{11}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{11} d + 2 \, c^{10} d^{2} - c^{9} d^{3} - 4 \, c^{8} d^{4} - 2 \, c^{7} d^{5} + 2 \, c^{5} d^{7} + 4 \, c^{4} d^{8} + c^{3} d^{9} - 2 \, c^{2} d^{10} - c d^{11}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{{\left(8 \, a^{2} c^{6} d + 64 \, a^{2} c^{5} d^{2} - 208 \, a^{2} c^{4} d^{3} + 16 \, a^{2} c^{3} d^{4} + 221 \, a^{2} c^{2} d^{5} - 80 \, a^{2} c d^{6} - 21 \, a^{2} d^{7}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(12 \, a^{2} c^{6} - 16 \, a^{2} c^{5} d + 79 \, a^{2} c^{4} d^{2} - 96 \, a^{2} c^{3} d^{3} + 54 \, a^{2} c^{2} d^{4} - 16 \, a^{2} c d^{5} + 7 \, a^{2} d^{6} + {\left(12 \, a^{2} c^{2} d^{4} - 16 \, a^{2} c d^{5} + 7 \, a^{2} d^{6}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(36 \, a^{2} c^{4} d^{2} - 48 \, a^{2} c^{3} d^{3} + 33 \, a^{2} c^{2} d^{4} - 16 \, a^{2} c d^{5} + 7 \, a^{2} d^{6}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(12 \, a^{2} c^{5} d - 16 \, a^{2} c^{4} d^{2} + 19 \, a^{2} c^{3} d^{3} - 16 \, a^{2} c^{2} d^{4} + 7 \, a^{2} c d^{5} - {\left(12 \, a^{2} c^{3} d^{3} - 16 \, a^{2} c^{2} d^{4} + 7 \, a^{2} c d^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) - 3 \, {\left(16 \, a^{2} c^{7} - 20 \, a^{2} c^{6} d - 45 \, a^{2} c^{4} d^{3} + 16 \, a^{2} c^{3} d^{4} + 74 \, a^{2} c^{2} d^{5} - 32 \, a^{2} c d^{6} - 9 \, a^{2} d^{7}\right)} \cos\left(f x + e\right) + {\left({\left(2 \, a^{2} c^{5} d^{2} + 16 \, a^{2} c^{4} d^{3} - 61 \, a^{2} c^{3} d^{4} + 16 \, a^{2} c^{2} d^{5} + 59 \, a^{2} c d^{6} - 32 \, a^{2} d^{7}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(4 \, a^{2} c^{7} + 32 \, a^{2} c^{6} d - 79 \, a^{2} c^{5} d^{2} - 16 \, a^{2} c^{4} d^{3} + 70 \, a^{2} c^{3} d^{4} + 5 \, a^{2} c d^{6} - 16 \, a^{2} d^{7}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, {\left({\left(c^{8} d^{4} + 2 \, c^{7} d^{5} - 2 \, c^{6} d^{6} - 6 \, c^{5} d^{7} + 6 \, c^{3} d^{9} + 2 \, c^{2} d^{10} - 2 \, c d^{11} - d^{12}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, c^{10} d^{2} + 6 \, c^{9} d^{3} - 5 \, c^{8} d^{4} - 16 \, c^{7} d^{5} - 2 \, c^{6} d^{6} + 12 \, c^{5} d^{7} + 6 \, c^{4} d^{8} - c^{2} d^{10} - 2 \, c d^{11} - d^{12}\right)} f \cos\left(f x + e\right)^{2} + {\left(c^{12} + 2 \, c^{11} d + 4 \, c^{10} d^{2} + 6 \, c^{9} d^{3} - 11 \, c^{8} d^{4} - 28 \, c^{7} d^{5} + 28 \, c^{5} d^{7} + 11 \, c^{4} d^{8} - 6 \, c^{3} d^{9} - 4 \, c^{2} d^{10} - 2 \, c d^{11} - d^{12}\right)} f - 4 \, {\left({\left(c^{9} d^{3} + 2 \, c^{8} d^{4} - 2 \, c^{7} d^{5} - 6 \, c^{6} d^{6} + 6 \, c^{4} d^{8} + 2 \, c^{3} d^{9} - 2 \, c^{2} d^{10} - c d^{11}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{11} d + 2 \, c^{10} d^{2} - c^{9} d^{3} - 4 \, c^{8} d^{4} - 2 \, c^{7} d^{5} + 2 \, c^{5} d^{7} + 4 \, c^{4} d^{8} + c^{3} d^{9} - 2 \, c^{2} d^{10} - c d^{11}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/48*(2*(8*a^2*c^6*d + 64*a^2*c^5*d^2 - 208*a^2*c^4*d^3 + 16*a^2*c^3*d^4 + 221*a^2*c^2*d^5 - 80*a^2*c*d^6 - 21*a^2*d^7)*cos(f*x + e)^3 - 3*(12*a^2*c^6 - 16*a^2*c^5*d + 79*a^2*c^4*d^2 - 96*a^2*c^3*d^3 + 54*a^2*c^2*d^4 - 16*a^2*c*d^5 + 7*a^2*d^6 + (12*a^2*c^2*d^4 - 16*a^2*c*d^5 + 7*a^2*d^6)*cos(f*x + e)^4 - 2*(36*a^2*c^4*d^2 - 48*a^2*c^3*d^3 + 33*a^2*c^2*d^4 - 16*a^2*c*d^5 + 7*a^2*d^6)*cos(f*x + e)^2 + 4*(12*a^2*c^5*d - 16*a^2*c^4*d^2 + 19*a^2*c^3*d^3 - 16*a^2*c^2*d^4 + 7*a^2*c*d^5 - (12*a^2*c^3*d^3 - 16*a^2*c^2*d^4 + 7*a^2*c*d^5)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 6*(16*a^2*c^7 - 20*a^2*c^6*d - 45*a^2*c^4*d^3 + 16*a^2*c^3*d^4 + 74*a^2*c^2*d^5 - 32*a^2*c*d^6 - 9*a^2*d^7)*cos(f*x + e) + 2*((2*a^2*c^5*d^2 + 16*a^2*c^4*d^3 - 61*a^2*c^3*d^4 + 16*a^2*c^2*d^5 + 59*a^2*c*d^6 - 32*a^2*d^7)*cos(f*x + e)^3 - 3*(4*a^2*c^7 + 32*a^2*c^6*d - 79*a^2*c^5*d^2 - 16*a^2*c^4*d^3 + 70*a^2*c^3*d^4 + 5*a^2*c*d^6 - 16*a^2*d^7)*cos(f*x + e))*sin(f*x + e))/((c^8*d^4 + 2*c^7*d^5 - 2*c^6*d^6 - 6*c^5*d^7 + 6*c^3*d^9 + 2*c^2*d^10 - 2*c*d^11 - d^12)*f*cos(f*x + e)^4 - 2*(3*c^10*d^2 + 6*c^9*d^3 - 5*c^8*d^4 - 16*c^7*d^5 - 2*c^6*d^6 + 12*c^5*d^7 + 6*c^4*d^8 - c^2*d^10 - 2*c*d^11 - d^12)*f*cos(f*x + e)^2 + (c^12 + 2*c^11*d + 4*c^10*d^2 + 6*c^9*d^3 - 11*c^8*d^4 - 28*c^7*d^5 + 28*c^5*d^7 + 11*c^4*d^8 - 6*c^3*d^9 - 4*c^2*d^10 - 2*c*d^11 - d^12)*f - 4*((c^9*d^3 + 2*c^8*d^4 - 2*c^7*d^5 - 6*c^6*d^6 + 6*c^4*d^8 + 2*c^3*d^9 - 2*c^2*d^10 - c*d^11)*f*cos(f*x + e)^2 - (c^11*d + 2*c^10*d^2 - c^9*d^3 - 4*c^8*d^4 - 2*c^7*d^5 + 2*c^5*d^7 + 4*c^4*d^8 + c^3*d^9 - 2*c^2*d^10 - c*d^11)*f)*sin(f*x + e)), 1/24*((8*a^2*c^6*d + 64*a^2*c^5*d^2 - 208*a^2*c^4*d^3 + 16*a^2*c^3*d^4 + 221*a^2*c^2*d^5 - 80*a^2*c*d^6 - 21*a^2*d^7)*cos(f*x + e)^3 - 3*(12*a^2*c^6 - 16*a^2*c^5*d + 79*a^2*c^4*d^2 - 96*a^2*c^3*d^3 + 54*a^2*c^2*d^4 - 16*a^2*c*d^5 + 7*a^2*d^6 + (12*a^2*c^2*d^4 - 16*a^2*c*d^5 + 7*a^2*d^6)*cos(f*x + e)^4 - 2*(36*a^2*c^4*d^2 - 48*a^2*c^3*d^3 + 33*a^2*c^2*d^4 - 16*a^2*c*d^5 + 7*a^2*d^6)*cos(f*x + e)^2 + 4*(12*a^2*c^5*d - 16*a^2*c^4*d^2 + 19*a^2*c^3*d^3 - 16*a^2*c^2*d^4 + 7*a^2*c*d^5 - (12*a^2*c^3*d^3 - 16*a^2*c^2*d^4 + 7*a^2*c*d^5)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - 3*(16*a^2*c^7 - 20*a^2*c^6*d - 45*a^2*c^4*d^3 + 16*a^2*c^3*d^4 + 74*a^2*c^2*d^5 - 32*a^2*c*d^6 - 9*a^2*d^7)*cos(f*x + e) + ((2*a^2*c^5*d^2 + 16*a^2*c^4*d^3 - 61*a^2*c^3*d^4 + 16*a^2*c^2*d^5 + 59*a^2*c*d^6 - 32*a^2*d^7)*cos(f*x + e)^3 - 3*(4*a^2*c^7 + 32*a^2*c^6*d - 79*a^2*c^5*d^2 - 16*a^2*c^4*d^3 + 70*a^2*c^3*d^4 + 5*a^2*c*d^6 - 16*a^2*d^7)*cos(f*x + e))*sin(f*x + e))/((c^8*d^4 + 2*c^7*d^5 - 2*c^6*d^6 - 6*c^5*d^7 + 6*c^3*d^9 + 2*c^2*d^10 - 2*c*d^11 - d^12)*f*cos(f*x + e)^4 - 2*(3*c^10*d^2 + 6*c^9*d^3 - 5*c^8*d^4 - 16*c^7*d^5 - 2*c^6*d^6 + 12*c^5*d^7 + 6*c^4*d^8 - c^2*d^10 - 2*c*d^11 - d^12)*f*cos(f*x + e)^2 + (c^12 + 2*c^11*d + 4*c^10*d^2 + 6*c^9*d^3 - 11*c^8*d^4 - 28*c^7*d^5 + 28*c^5*d^7 + 11*c^4*d^8 - 6*c^3*d^9 - 4*c^2*d^10 - 2*c*d^11 - d^12)*f - 4*((c^9*d^3 + 2*c^8*d^4 - 2*c^7*d^5 - 6*c^6*d^6 + 6*c^4*d^8 + 2*c^3*d^9 - 2*c^2*d^10 - c*d^11)*f*cos(f*x + e)^2 - (c^11*d + 2*c^10*d^2 - c^9*d^3 - 4*c^8*d^4 - 2*c^7*d^5 + 2*c^5*d^7 + 4*c^4*d^8 + c^3*d^9 - 2*c^2*d^10 - c*d^11)*f)*sin(f*x + e))]","B",0
444,1,261,0,1.217997," ","integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{144 \, {\left(a^{3} c d^{2} + a^{3} d^{3}\right)} \cos\left(f x + e\right)^{5} - 80 \, {\left(a^{3} c^{3} + 9 \, a^{3} c^{2} d + 15 \, a^{3} c d^{2} + 7 \, a^{3} d^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right)} f x + 960 \, {\left(a^{3} c^{3} + 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} + a^{3} d^{3}\right)} \cos\left(f x + e\right) + 5 \, {\left(8 \, a^{3} d^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(18 \, a^{3} c^{2} d + 54 \, a^{3} c d^{2} + 31 \, a^{3} d^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(24 \, a^{3} c^{3} + 102 \, a^{3} c^{2} d + 114 \, a^{3} c d^{2} + 41 \, a^{3} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{240 \, f}"," ",0,"-1/240*(144*(a^3*c*d^2 + a^3*d^3)*cos(f*x + e)^5 - 80*(a^3*c^3 + 9*a^3*c^2*d + 15*a^3*c*d^2 + 7*a^3*d^3)*cos(f*x + e)^3 - 15*(40*a^3*c^3 + 90*a^3*c^2*d + 78*a^3*c*d^2 + 23*a^3*d^3)*f*x + 960*(a^3*c^3 + 3*a^3*c^2*d + 3*a^3*c*d^2 + a^3*d^3)*cos(f*x + e) + 5*(8*a^3*d^3*cos(f*x + e)^5 - 2*(18*a^3*c^2*d + 54*a^3*c*d^2 + 31*a^3*d^3)*cos(f*x + e)^3 + 3*(24*a^3*c^3 + 102*a^3*c^2*d + 114*a^3*c*d^2 + 41*a^3*d^3)*cos(f*x + e))*sin(f*x + e))/f","A",0
445,1,180,0,1.038320," ","integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{24 \, a^{3} d^{2} \cos\left(f x + e\right)^{5} - 40 \, {\left(a^{3} c^{2} + 6 \, a^{3} c d + 5 \, a^{3} d^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(20 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right)} f x + 480 \, {\left(a^{3} c^{2} + 2 \, a^{3} c d + a^{3} d^{2}\right)} \cos\left(f x + e\right) - 15 \, {\left(2 \, {\left(2 \, a^{3} c d + 3 \, a^{3} d^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(12 \, a^{3} c^{2} + 34 \, a^{3} c d + 19 \, a^{3} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{120 \, f}"," ",0,"-1/120*(24*a^3*d^2*cos(f*x + e)^5 - 40*(a^3*c^2 + 6*a^3*c*d + 5*a^3*d^2)*cos(f*x + e)^3 - 15*(20*a^3*c^2 + 30*a^3*c*d + 13*a^3*d^2)*f*x + 480*(a^3*c^2 + 2*a^3*c*d + a^3*d^2)*cos(f*x + e) - 15*(2*(2*a^3*c*d + 3*a^3*d^2)*cos(f*x + e)^3 - (12*a^3*c^2 + 34*a^3*c*d + 19*a^3*d^2)*cos(f*x + e))*sin(f*x + e))/f","A",0
446,1,108,0,1.127354," ","integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\frac{8 \, {\left(a^{3} c + 3 \, a^{3} d\right)} \cos\left(f x + e\right)^{3} + 15 \, {\left(4 \, a^{3} c + 3 \, a^{3} d\right)} f x - 96 \, {\left(a^{3} c + a^{3} d\right)} \cos\left(f x + e\right) + 3 \, {\left(2 \, a^{3} d \cos\left(f x + e\right)^{3} - {\left(12 \, a^{3} c + 17 \, a^{3} d\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, f}"," ",0,"1/24*(8*(a^3*c + 3*a^3*d)*cos(f*x + e)^3 + 15*(4*a^3*c + 3*a^3*d)*f*x - 96*(a^3*c + a^3*d)*cos(f*x + e) + 3*(2*a^3*d*cos(f*x + e)^3 - (12*a^3*c + 17*a^3*d)*cos(f*x + e))*sin(f*x + e))/f","A",0
447,1,54,0,1.087347," ","integrate((a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{2 \, a^{3} \cos\left(f x + e\right)^{3} + 15 \, a^{3} f x - 9 \, a^{3} \cos\left(f x + e\right) \sin\left(f x + e\right) - 24 \, a^{3} \cos\left(f x + e\right)}{6 \, f}"," ",0,"1/6*(2*a^3*cos(f*x + e)^3 + 15*a^3*f*x - 9*a^3*cos(f*x + e)*sin(f*x + e) - 24*a^3*cos(f*x + e))/f","A",0
448,1,404,0,1.203549," ","integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[-\frac{a^{3} d^{2} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(2 \, a^{3} c^{2} - 6 \, a^{3} c d + 7 \, a^{3} d^{2}\right)} f x - {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} \sqrt{-\frac{c - d}{c + d}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left({\left(c^{2} + c d\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{c + d}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 2 \, {\left(a^{3} c d - 3 \, a^{3} d^{2}\right)} \cos\left(f x + e\right)}{2 \, d^{3} f}, -\frac{a^{3} d^{2} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(2 \, a^{3} c^{2} - 6 \, a^{3} c d + 7 \, a^{3} d^{2}\right)} f x - 2 \, {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} \sqrt{\frac{c - d}{c + d}} \arctan\left(-\frac{{\left(c \sin\left(f x + e\right) + d\right)} \sqrt{\frac{c - d}{c + d}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - 2 \, {\left(a^{3} c d - 3 \, a^{3} d^{2}\right)} \cos\left(f x + e\right)}{2 \, d^{3} f}\right]"," ",0,"[-1/2*(a^3*d^2*cos(f*x + e)*sin(f*x + e) - (2*a^3*c^2 - 6*a^3*c*d + 7*a^3*d^2)*f*x - (a^3*c^2 - 2*a^3*c*d + a^3*d^2)*sqrt(-(c - d)/(c + d))*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*((c^2 + c*d)*cos(f*x + e)*sin(f*x + e) + (c*d + d^2)*cos(f*x + e))*sqrt(-(c - d)/(c + d)))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 2*(a^3*c*d - 3*a^3*d^2)*cos(f*x + e))/(d^3*f), -1/2*(a^3*d^2*cos(f*x + e)*sin(f*x + e) - (2*a^3*c^2 - 6*a^3*c*d + 7*a^3*d^2)*f*x - 2*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*sqrt((c - d)/(c + d))*arctan(-(c*sin(f*x + e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(f*x + e))) - 2*(a^3*c*d - 3*a^3*d^2)*cos(f*x + e))/(d^3*f)]","A",0
449,1,645,0,1.179677," ","integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(2 \, a^{3} c^{3} - a^{3} c^{2} d - 3 \, a^{3} c d^{2}\right)} f x + {\left(2 \, a^{3} c^{3} + a^{3} c^{2} d - 3 \, a^{3} c d^{2} + {\left(2 \, a^{3} c^{2} d + a^{3} c d^{2} - 3 \, a^{3} d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{c + d}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left({\left(c^{2} + c d\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{c + d}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(2 \, a^{3} c^{2} d - a^{3} c d^{2} + a^{3} d^{3}\right)} \cos\left(f x + e\right) + 2 \, {\left({\left(2 \, a^{3} c^{2} d - a^{3} c d^{2} - 3 \, a^{3} d^{3}\right)} f x + {\left(a^{3} c d^{2} + a^{3} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(c d^{4} + d^{5}\right)} f \sin\left(f x + e\right) + {\left(c^{2} d^{3} + c d^{4}\right)} f\right)}}, -\frac{{\left(2 \, a^{3} c^{3} - a^{3} c^{2} d - 3 \, a^{3} c d^{2}\right)} f x + {\left(2 \, a^{3} c^{3} + a^{3} c^{2} d - 3 \, a^{3} c d^{2} + {\left(2 \, a^{3} c^{2} d + a^{3} c d^{2} - 3 \, a^{3} d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{c + d}} \arctan\left(-\frac{{\left(c \sin\left(f x + e\right) + d\right)} \sqrt{\frac{c - d}{c + d}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) + {\left(2 \, a^{3} c^{2} d - a^{3} c d^{2} + a^{3} d^{3}\right)} \cos\left(f x + e\right) + {\left({\left(2 \, a^{3} c^{2} d - a^{3} c d^{2} - 3 \, a^{3} d^{3}\right)} f x + {\left(a^{3} c d^{2} + a^{3} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{{\left(c d^{4} + d^{5}\right)} f \sin\left(f x + e\right) + {\left(c^{2} d^{3} + c d^{4}\right)} f}\right]"," ",0,"[-1/2*(2*(2*a^3*c^3 - a^3*c^2*d - 3*a^3*c*d^2)*f*x + (2*a^3*c^3 + a^3*c^2*d - 3*a^3*c*d^2 + (2*a^3*c^2*d + a^3*c*d^2 - 3*a^3*d^3)*sin(f*x + e))*sqrt(-(c - d)/(c + d))*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*((c^2 + c*d)*cos(f*x + e)*sin(f*x + e) + (c*d + d^2)*cos(f*x + e))*sqrt(-(c - d)/(c + d)))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(2*a^3*c^2*d - a^3*c*d^2 + a^3*d^3)*cos(f*x + e) + 2*((2*a^3*c^2*d - a^3*c*d^2 - 3*a^3*d^3)*f*x + (a^3*c*d^2 + a^3*d^3)*cos(f*x + e))*sin(f*x + e))/((c*d^4 + d^5)*f*sin(f*x + e) + (c^2*d^3 + c*d^4)*f), -((2*a^3*c^3 - a^3*c^2*d - 3*a^3*c*d^2)*f*x + (2*a^3*c^3 + a^3*c^2*d - 3*a^3*c*d^2 + (2*a^3*c^2*d + a^3*c*d^2 - 3*a^3*d^3)*sin(f*x + e))*sqrt((c - d)/(c + d))*arctan(-(c*sin(f*x + e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(f*x + e))) + (2*a^3*c^2*d - a^3*c*d^2 + a^3*d^3)*cos(f*x + e) + ((2*a^3*c^2*d - a^3*c*d^2 - 3*a^3*d^3)*f*x + (a^3*c*d^2 + a^3*d^3)*cos(f*x + e))*sin(f*x + e))/((c*d^4 + d^5)*f*sin(f*x + e) + (c^2*d^3 + c*d^4)*f)]","A",0
450,1,1064,0,1.489850," ","integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4}\right)} f x \cos\left(f x + e\right)^{2} - 4 \, {\left(a^{3} c^{4} + 2 \, a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4}\right)} f x - {\left(2 \, a^{3} c^{4} + 6 \, a^{3} c^{3} d + 9 \, a^{3} c^{2} d^{2} + 6 \, a^{3} c d^{3} + 7 \, a^{3} d^{4} - {\left(2 \, a^{3} c^{2} d^{2} + 6 \, a^{3} c d^{3} + 7 \, a^{3} d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, a^{3} c^{3} d + 6 \, a^{3} c^{2} d^{2} + 7 \, a^{3} c d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{c + d}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left({\left(c^{2} + c d\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{c + d}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 2 \, {\left(2 \, a^{3} c^{3} d + 4 \, a^{3} c^{2} d^{2} - 5 \, a^{3} c d^{3} - a^{3} d^{4}\right)} \cos\left(f x + e\right) - 2 \, {\left(4 \, {\left(a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + a^{3} c d^{3}\right)} f x + 3 \, {\left(a^{3} c^{2} d^{2} + a^{3} c d^{3} - 2 \, a^{3} d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(c^{2} d^{5} + 2 \, c d^{6} + d^{7}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d^{4} + 2 \, c^{2} d^{5} + c d^{6}\right)} f \sin\left(f x + e\right) - {\left(c^{4} d^{3} + 2 \, c^{3} d^{4} + 2 \, c^{2} d^{5} + 2 \, c d^{6} + d^{7}\right)} f\right)}}, \frac{2 \, {\left(a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4}\right)} f x \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{4} + 2 \, a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} + a^{3} d^{4}\right)} f x - {\left(2 \, a^{3} c^{4} + 6 \, a^{3} c^{3} d + 9 \, a^{3} c^{2} d^{2} + 6 \, a^{3} c d^{3} + 7 \, a^{3} d^{4} - {\left(2 \, a^{3} c^{2} d^{2} + 6 \, a^{3} c d^{3} + 7 \, a^{3} d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, a^{3} c^{3} d + 6 \, a^{3} c^{2} d^{2} + 7 \, a^{3} c d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{c + d}} \arctan\left(-\frac{{\left(c \sin\left(f x + e\right) + d\right)} \sqrt{\frac{c - d}{c + d}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - {\left(2 \, a^{3} c^{3} d + 4 \, a^{3} c^{2} d^{2} - 5 \, a^{3} c d^{3} - a^{3} d^{4}\right)} \cos\left(f x + e\right) - {\left(4 \, {\left(a^{3} c^{3} d + 2 \, a^{3} c^{2} d^{2} + a^{3} c d^{3}\right)} f x + 3 \, {\left(a^{3} c^{2} d^{2} + a^{3} c d^{3} - 2 \, a^{3} d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(c^{2} d^{5} + 2 \, c d^{6} + d^{7}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d^{4} + 2 \, c^{2} d^{5} + c d^{6}\right)} f \sin\left(f x + e\right) - {\left(c^{4} d^{3} + 2 \, c^{3} d^{4} + 2 \, c^{2} d^{5} + 2 \, c d^{6} + d^{7}\right)} f\right)}}\right]"," ",0,"[1/4*(4*(a^3*c^2*d^2 + 2*a^3*c*d^3 + a^3*d^4)*f*x*cos(f*x + e)^2 - 4*(a^3*c^4 + 2*a^3*c^3*d + 2*a^3*c^2*d^2 + 2*a^3*c*d^3 + a^3*d^4)*f*x - (2*a^3*c^4 + 6*a^3*c^3*d + 9*a^3*c^2*d^2 + 6*a^3*c*d^3 + 7*a^3*d^4 - (2*a^3*c^2*d^2 + 6*a^3*c*d^3 + 7*a^3*d^4)*cos(f*x + e)^2 + 2*(2*a^3*c^3*d + 6*a^3*c^2*d^2 + 7*a^3*c*d^3)*sin(f*x + e))*sqrt(-(c - d)/(c + d))*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*((c^2 + c*d)*cos(f*x + e)*sin(f*x + e) + (c*d + d^2)*cos(f*x + e))*sqrt(-(c - d)/(c + d)))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 2*(2*a^3*c^3*d + 4*a^3*c^2*d^2 - 5*a^3*c*d^3 - a^3*d^4)*cos(f*x + e) - 2*(4*(a^3*c^3*d + 2*a^3*c^2*d^2 + a^3*c*d^3)*f*x + 3*(a^3*c^2*d^2 + a^3*c*d^3 - 2*a^3*d^4)*cos(f*x + e))*sin(f*x + e))/((c^2*d^5 + 2*c*d^6 + d^7)*f*cos(f*x + e)^2 - 2*(c^3*d^4 + 2*c^2*d^5 + c*d^6)*f*sin(f*x + e) - (c^4*d^3 + 2*c^3*d^4 + 2*c^2*d^5 + 2*c*d^6 + d^7)*f), 1/2*(2*(a^3*c^2*d^2 + 2*a^3*c*d^3 + a^3*d^4)*f*x*cos(f*x + e)^2 - 2*(a^3*c^4 + 2*a^3*c^3*d + 2*a^3*c^2*d^2 + 2*a^3*c*d^3 + a^3*d^4)*f*x - (2*a^3*c^4 + 6*a^3*c^3*d + 9*a^3*c^2*d^2 + 6*a^3*c*d^3 + 7*a^3*d^4 - (2*a^3*c^2*d^2 + 6*a^3*c*d^3 + 7*a^3*d^4)*cos(f*x + e)^2 + 2*(2*a^3*c^3*d + 6*a^3*c^2*d^2 + 7*a^3*c*d^3)*sin(f*x + e))*sqrt((c - d)/(c + d))*arctan(-(c*sin(f*x + e) + d)*sqrt((c - d)/(c + d))/((c - d)*cos(f*x + e))) - (2*a^3*c^3*d + 4*a^3*c^2*d^2 - 5*a^3*c*d^3 - a^3*d^4)*cos(f*x + e) - (4*(a^3*c^3*d + 2*a^3*c^2*d^2 + a^3*c*d^3)*f*x + 3*(a^3*c^2*d^2 + a^3*c*d^3 - 2*a^3*d^4)*cos(f*x + e))*sin(f*x + e))/((c^2*d^5 + 2*c*d^6 + d^7)*f*cos(f*x + e)^2 - 2*(c^3*d^4 + 2*c^2*d^5 + c*d^6)*f*sin(f*x + e) - (c^4*d^3 + 2*c^3*d^4 + 2*c^2*d^5 + 2*c*d^6 + d^7)*f)]","B",0
451,1,1106,0,1.156333," ","integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^4,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(2 \, a^{3} c^{4} + 9 \, a^{3} c^{3} d + 20 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} - 22 \, a^{3} d^{4}\right)} \cos\left(f x + e\right)^{3} - 6 \, {\left(3 \, a^{3} c^{4} + 16 \, a^{3} c^{3} d - 16 \, a^{3} c d^{3} - 3 \, a^{3} d^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + 15 \, {\left(3 \, a^{3} c d^{2} \cos\left(f x + e\right)^{2} - a^{3} c^{3} - 3 \, a^{3} c d^{2} + {\left(a^{3} d^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{3} c^{2} d - a^{3} d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 12 \, {\left(4 \, a^{3} c^{4} + 3 \, a^{3} c^{3} d - 3 \, a^{3} c d^{3} - 4 \, a^{3} d^{4}\right)} \cos\left(f x + e\right)}{12 \, {\left(3 \, {\left(c^{6} d^{2} + 3 \, c^{5} d^{3} + 2 \, c^{4} d^{4} - 2 \, c^{3} d^{5} - 3 \, c^{2} d^{6} - c d^{7}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{8} + 3 \, c^{7} d + 5 \, c^{6} d^{2} + 7 \, c^{5} d^{3} + 3 \, c^{4} d^{4} - 7 \, c^{3} d^{5} - 9 \, c^{2} d^{6} - 3 \, c d^{7}\right)} f + {\left({\left(c^{5} d^{3} + 3 \, c^{4} d^{4} + 2 \, c^{3} d^{5} - 2 \, c^{2} d^{6} - 3 \, c d^{7} - d^{8}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{7} d + 9 \, c^{6} d^{2} + 7 \, c^{5} d^{3} - 3 \, c^{4} d^{4} - 7 \, c^{3} d^{5} - 5 \, c^{2} d^{6} - 3 \, c d^{7} - d^{8}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left(2 \, a^{3} c^{4} + 9 \, a^{3} c^{3} d + 20 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} - 22 \, a^{3} d^{4}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(3 \, a^{3} c^{4} + 16 \, a^{3} c^{3} d - 16 \, a^{3} c d^{3} - 3 \, a^{3} d^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + 15 \, {\left(3 \, a^{3} c d^{2} \cos\left(f x + e\right)^{2} - a^{3} c^{3} - 3 \, a^{3} c d^{2} + {\left(a^{3} d^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{3} c^{2} d - a^{3} d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) - 6 \, {\left(4 \, a^{3} c^{4} + 3 \, a^{3} c^{3} d - 3 \, a^{3} c d^{3} - 4 \, a^{3} d^{4}\right)} \cos\left(f x + e\right)}{6 \, {\left(3 \, {\left(c^{6} d^{2} + 3 \, c^{5} d^{3} + 2 \, c^{4} d^{4} - 2 \, c^{3} d^{5} - 3 \, c^{2} d^{6} - c d^{7}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{8} + 3 \, c^{7} d + 5 \, c^{6} d^{2} + 7 \, c^{5} d^{3} + 3 \, c^{4} d^{4} - 7 \, c^{3} d^{5} - 9 \, c^{2} d^{6} - 3 \, c d^{7}\right)} f + {\left({\left(c^{5} d^{3} + 3 \, c^{4} d^{4} + 2 \, c^{3} d^{5} - 2 \, c^{2} d^{6} - 3 \, c d^{7} - d^{8}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{7} d + 9 \, c^{6} d^{2} + 7 \, c^{5} d^{3} - 3 \, c^{4} d^{4} - 7 \, c^{3} d^{5} - 5 \, c^{2} d^{6} - 3 \, c d^{7} - d^{8}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/12*(2*(2*a^3*c^4 + 9*a^3*c^3*d + 20*a^3*c^2*d^2 - 9*a^3*c*d^3 - 22*a^3*d^4)*cos(f*x + e)^3 - 6*(3*a^3*c^4 + 16*a^3*c^3*d - 16*a^3*c*d^3 - 3*a^3*d^4)*cos(f*x + e)*sin(f*x + e) + 15*(3*a^3*c*d^2*cos(f*x + e)^2 - a^3*c^3 - 3*a^3*c*d^2 + (a^3*d^3*cos(f*x + e)^2 - 3*a^3*c^2*d - a^3*d^3)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 12*(4*a^3*c^4 + 3*a^3*c^3*d - 3*a^3*c*d^3 - 4*a^3*d^4)*cos(f*x + e))/(3*(c^6*d^2 + 3*c^5*d^3 + 2*c^4*d^4 - 2*c^3*d^5 - 3*c^2*d^6 - c*d^7)*f*cos(f*x + e)^2 - (c^8 + 3*c^7*d + 5*c^6*d^2 + 7*c^5*d^3 + 3*c^4*d^4 - 7*c^3*d^5 - 9*c^2*d^6 - 3*c*d^7)*f + ((c^5*d^3 + 3*c^4*d^4 + 2*c^3*d^5 - 2*c^2*d^6 - 3*c*d^7 - d^8)*f*cos(f*x + e)^2 - (3*c^7*d + 9*c^6*d^2 + 7*c^5*d^3 - 3*c^4*d^4 - 7*c^3*d^5 - 5*c^2*d^6 - 3*c*d^7 - d^8)*f)*sin(f*x + e)), -1/6*((2*a^3*c^4 + 9*a^3*c^3*d + 20*a^3*c^2*d^2 - 9*a^3*c*d^3 - 22*a^3*d^4)*cos(f*x + e)^3 - 3*(3*a^3*c^4 + 16*a^3*c^3*d - 16*a^3*c*d^3 - 3*a^3*d^4)*cos(f*x + e)*sin(f*x + e) + 15*(3*a^3*c*d^2*cos(f*x + e)^2 - a^3*c^3 - 3*a^3*c*d^2 + (a^3*d^3*cos(f*x + e)^2 - 3*a^3*c^2*d - a^3*d^3)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - 6*(4*a^3*c^4 + 3*a^3*c^3*d - 3*a^3*c*d^3 - 4*a^3*d^4)*cos(f*x + e))/(3*(c^6*d^2 + 3*c^5*d^3 + 2*c^4*d^4 - 2*c^3*d^5 - 3*c^2*d^6 - c*d^7)*f*cos(f*x + e)^2 - (c^8 + 3*c^7*d + 5*c^6*d^2 + 7*c^5*d^3 + 3*c^4*d^4 - 7*c^3*d^5 - 9*c^2*d^6 - 3*c*d^7)*f + ((c^5*d^3 + 3*c^4*d^4 + 2*c^3*d^5 - 2*c^2*d^6 - 3*c*d^7 - d^8)*f*cos(f*x + e)^2 - (3*c^7*d + 9*c^6*d^2 + 7*c^5*d^3 - 3*c^4*d^4 - 7*c^3*d^5 - 5*c^2*d^6 - 3*c*d^7 - d^8)*f)*sin(f*x + e))]","B",0
452,1,2009,0,1.345876," ","integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^5,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(8 \, a^{3} c^{6} + 48 \, a^{3} c^{5} d + 164 \, a^{3} c^{4} d^{2} - 276 \, a^{3} c^{3} d^{3} - 217 \, a^{3} c^{2} d^{4} + 228 \, a^{3} c d^{5} + 45 \, a^{3} d^{6}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(4 \, a^{3} c^{5} - 3 \, a^{3} c^{4} d + 24 \, a^{3} c^{3} d^{2} - 18 \, a^{3} c^{2} d^{3} + 4 \, a^{3} c d^{4} - 3 \, a^{3} d^{5} + {\left(4 \, a^{3} c d^{4} - 3 \, a^{3} d^{5}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(12 \, a^{3} c^{3} d^{2} - 9 \, a^{3} c^{2} d^{3} + 4 \, a^{3} c d^{4} - 3 \, a^{3} d^{5}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(4 \, a^{3} c^{4} d - 3 \, a^{3} c^{3} d^{2} + 4 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} - {\left(4 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 6 \, {\left(32 \, a^{3} c^{6} + 4 \, a^{3} c^{5} d + 13 \, a^{3} c^{4} d^{2} - 88 \, a^{3} c^{3} d^{3} - 62 \, a^{3} c^{2} d^{4} + 84 \, a^{3} c d^{5} + 17 \, a^{3} d^{6}\right)} \cos\left(f x + e\right) + 2 \, {\left({\left(2 \, a^{3} c^{5} d + 12 \, a^{3} c^{4} d^{2} + 41 \, a^{3} c^{3} d^{3} - 84 \, a^{3} c^{2} d^{4} - 43 \, a^{3} c d^{5} + 72 \, a^{3} d^{6}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(12 \, a^{3} c^{6} + 79 \, a^{3} c^{5} d - 72 \, a^{3} c^{4} d^{2} - 98 \, a^{3} c^{3} d^{3} + 28 \, a^{3} c^{2} d^{4} + 19 \, a^{3} c d^{5} + 32 \, a^{3} d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, {\left({\left(c^{7} d^{4} + 3 \, c^{6} d^{5} + c^{5} d^{6} - 5 \, c^{4} d^{7} - 5 \, c^{3} d^{8} + c^{2} d^{9} + 3 \, c d^{10} + d^{11}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, c^{9} d^{2} + 9 \, c^{8} d^{3} + 4 \, c^{7} d^{4} - 12 \, c^{6} d^{5} - 14 \, c^{5} d^{6} - 2 \, c^{4} d^{7} + 4 \, c^{3} d^{8} + 4 \, c^{2} d^{9} + 3 \, c d^{10} + d^{11}\right)} f \cos\left(f x + e\right)^{2} + {\left(c^{11} + 3 \, c^{10} d + 7 \, c^{9} d^{2} + 13 \, c^{8} d^{3} + 2 \, c^{7} d^{4} - 26 \, c^{6} d^{5} - 26 \, c^{5} d^{6} + 2 \, c^{4} d^{7} + 13 \, c^{3} d^{8} + 7 \, c^{2} d^{9} + 3 \, c d^{10} + d^{11}\right)} f - 4 \, {\left({\left(c^{8} d^{3} + 3 \, c^{7} d^{4} + c^{6} d^{5} - 5 \, c^{5} d^{6} - 5 \, c^{4} d^{7} + c^{3} d^{8} + 3 \, c^{2} d^{9} + c d^{10}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{10} d + 3 \, c^{9} d^{2} + 2 \, c^{8} d^{3} - 2 \, c^{7} d^{4} - 4 \, c^{6} d^{5} - 4 \, c^{5} d^{6} - 2 \, c^{4} d^{7} + 2 \, c^{3} d^{8} + 3 \, c^{2} d^{9} + c d^{10}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{{\left(8 \, a^{3} c^{6} + 48 \, a^{3} c^{5} d + 164 \, a^{3} c^{4} d^{2} - 276 \, a^{3} c^{3} d^{3} - 217 \, a^{3} c^{2} d^{4} + 228 \, a^{3} c d^{5} + 45 \, a^{3} d^{6}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(4 \, a^{3} c^{5} - 3 \, a^{3} c^{4} d + 24 \, a^{3} c^{3} d^{2} - 18 \, a^{3} c^{2} d^{3} + 4 \, a^{3} c d^{4} - 3 \, a^{3} d^{5} + {\left(4 \, a^{3} c d^{4} - 3 \, a^{3} d^{5}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(12 \, a^{3} c^{3} d^{2} - 9 \, a^{3} c^{2} d^{3} + 4 \, a^{3} c d^{4} - 3 \, a^{3} d^{5}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(4 \, a^{3} c^{4} d - 3 \, a^{3} c^{3} d^{2} + 4 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} - {\left(4 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) - 3 \, {\left(32 \, a^{3} c^{6} + 4 \, a^{3} c^{5} d + 13 \, a^{3} c^{4} d^{2} - 88 \, a^{3} c^{3} d^{3} - 62 \, a^{3} c^{2} d^{4} + 84 \, a^{3} c d^{5} + 17 \, a^{3} d^{6}\right)} \cos\left(f x + e\right) + {\left({\left(2 \, a^{3} c^{5} d + 12 \, a^{3} c^{4} d^{2} + 41 \, a^{3} c^{3} d^{3} - 84 \, a^{3} c^{2} d^{4} - 43 \, a^{3} c d^{5} + 72 \, a^{3} d^{6}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(12 \, a^{3} c^{6} + 79 \, a^{3} c^{5} d - 72 \, a^{3} c^{4} d^{2} - 98 \, a^{3} c^{3} d^{3} + 28 \, a^{3} c^{2} d^{4} + 19 \, a^{3} c d^{5} + 32 \, a^{3} d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, {\left({\left(c^{7} d^{4} + 3 \, c^{6} d^{5} + c^{5} d^{6} - 5 \, c^{4} d^{7} - 5 \, c^{3} d^{8} + c^{2} d^{9} + 3 \, c d^{10} + d^{11}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, c^{9} d^{2} + 9 \, c^{8} d^{3} + 4 \, c^{7} d^{4} - 12 \, c^{6} d^{5} - 14 \, c^{5} d^{6} - 2 \, c^{4} d^{7} + 4 \, c^{3} d^{8} + 4 \, c^{2} d^{9} + 3 \, c d^{10} + d^{11}\right)} f \cos\left(f x + e\right)^{2} + {\left(c^{11} + 3 \, c^{10} d + 7 \, c^{9} d^{2} + 13 \, c^{8} d^{3} + 2 \, c^{7} d^{4} - 26 \, c^{6} d^{5} - 26 \, c^{5} d^{6} + 2 \, c^{4} d^{7} + 13 \, c^{3} d^{8} + 7 \, c^{2} d^{9} + 3 \, c d^{10} + d^{11}\right)} f - 4 \, {\left({\left(c^{8} d^{3} + 3 \, c^{7} d^{4} + c^{6} d^{5} - 5 \, c^{5} d^{6} - 5 \, c^{4} d^{7} + c^{3} d^{8} + 3 \, c^{2} d^{9} + c d^{10}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{10} d + 3 \, c^{9} d^{2} + 2 \, c^{8} d^{3} - 2 \, c^{7} d^{4} - 4 \, c^{6} d^{5} - 4 \, c^{5} d^{6} - 2 \, c^{4} d^{7} + 2 \, c^{3} d^{8} + 3 \, c^{2} d^{9} + c d^{10}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/48*(2*(8*a^3*c^6 + 48*a^3*c^5*d + 164*a^3*c^4*d^2 - 276*a^3*c^3*d^3 - 217*a^3*c^2*d^4 + 228*a^3*c*d^5 + 45*a^3*d^6)*cos(f*x + e)^3 - 15*(4*a^3*c^5 - 3*a^3*c^4*d + 24*a^3*c^3*d^2 - 18*a^3*c^2*d^3 + 4*a^3*c*d^4 - 3*a^3*d^5 + (4*a^3*c*d^4 - 3*a^3*d^5)*cos(f*x + e)^4 - 2*(12*a^3*c^3*d^2 - 9*a^3*c^2*d^3 + 4*a^3*c*d^4 - 3*a^3*d^5)*cos(f*x + e)^2 + 4*(4*a^3*c^4*d - 3*a^3*c^3*d^2 + 4*a^3*c^2*d^3 - 3*a^3*c*d^4 - (4*a^3*c^2*d^3 - 3*a^3*c*d^4)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 6*(32*a^3*c^6 + 4*a^3*c^5*d + 13*a^3*c^4*d^2 - 88*a^3*c^3*d^3 - 62*a^3*c^2*d^4 + 84*a^3*c*d^5 + 17*a^3*d^6)*cos(f*x + e) + 2*((2*a^3*c^5*d + 12*a^3*c^4*d^2 + 41*a^3*c^3*d^3 - 84*a^3*c^2*d^4 - 43*a^3*c*d^5 + 72*a^3*d^6)*cos(f*x + e)^3 - 3*(12*a^3*c^6 + 79*a^3*c^5*d - 72*a^3*c^4*d^2 - 98*a^3*c^3*d^3 + 28*a^3*c^2*d^4 + 19*a^3*c*d^5 + 32*a^3*d^6)*cos(f*x + e))*sin(f*x + e))/((c^7*d^4 + 3*c^6*d^5 + c^5*d^6 - 5*c^4*d^7 - 5*c^3*d^8 + c^2*d^9 + 3*c*d^10 + d^11)*f*cos(f*x + e)^4 - 2*(3*c^9*d^2 + 9*c^8*d^3 + 4*c^7*d^4 - 12*c^6*d^5 - 14*c^5*d^6 - 2*c^4*d^7 + 4*c^3*d^8 + 4*c^2*d^9 + 3*c*d^10 + d^11)*f*cos(f*x + e)^2 + (c^11 + 3*c^10*d + 7*c^9*d^2 + 13*c^8*d^3 + 2*c^7*d^4 - 26*c^6*d^5 - 26*c^5*d^6 + 2*c^4*d^7 + 13*c^3*d^8 + 7*c^2*d^9 + 3*c*d^10 + d^11)*f - 4*((c^8*d^3 + 3*c^7*d^4 + c^6*d^5 - 5*c^5*d^6 - 5*c^4*d^7 + c^3*d^8 + 3*c^2*d^9 + c*d^10)*f*cos(f*x + e)^2 - (c^10*d + 3*c^9*d^2 + 2*c^8*d^3 - 2*c^7*d^4 - 4*c^6*d^5 - 4*c^5*d^6 - 2*c^4*d^7 + 2*c^3*d^8 + 3*c^2*d^9 + c*d^10)*f)*sin(f*x + e)), 1/24*((8*a^3*c^6 + 48*a^3*c^5*d + 164*a^3*c^4*d^2 - 276*a^3*c^3*d^3 - 217*a^3*c^2*d^4 + 228*a^3*c*d^5 + 45*a^3*d^6)*cos(f*x + e)^3 - 15*(4*a^3*c^5 - 3*a^3*c^4*d + 24*a^3*c^3*d^2 - 18*a^3*c^2*d^3 + 4*a^3*c*d^4 - 3*a^3*d^5 + (4*a^3*c*d^4 - 3*a^3*d^5)*cos(f*x + e)^4 - 2*(12*a^3*c^3*d^2 - 9*a^3*c^2*d^3 + 4*a^3*c*d^4 - 3*a^3*d^5)*cos(f*x + e)^2 + 4*(4*a^3*c^4*d - 3*a^3*c^3*d^2 + 4*a^3*c^2*d^3 - 3*a^3*c*d^4 - (4*a^3*c^2*d^3 - 3*a^3*c*d^4)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - 3*(32*a^3*c^6 + 4*a^3*c^5*d + 13*a^3*c^4*d^2 - 88*a^3*c^3*d^3 - 62*a^3*c^2*d^4 + 84*a^3*c*d^5 + 17*a^3*d^6)*cos(f*x + e) + ((2*a^3*c^5*d + 12*a^3*c^4*d^2 + 41*a^3*c^3*d^3 - 84*a^3*c^2*d^4 - 43*a^3*c*d^5 + 72*a^3*d^6)*cos(f*x + e)^3 - 3*(12*a^3*c^6 + 79*a^3*c^5*d - 72*a^3*c^4*d^2 - 98*a^3*c^3*d^3 + 28*a^3*c^2*d^4 + 19*a^3*c*d^5 + 32*a^3*d^6)*cos(f*x + e))*sin(f*x + e))/((c^7*d^4 + 3*c^6*d^5 + c^5*d^6 - 5*c^4*d^7 - 5*c^3*d^8 + c^2*d^9 + 3*c*d^10 + d^11)*f*cos(f*x + e)^4 - 2*(3*c^9*d^2 + 9*c^8*d^3 + 4*c^7*d^4 - 12*c^6*d^5 - 14*c^5*d^6 - 2*c^4*d^7 + 4*c^3*d^8 + 4*c^2*d^9 + 3*c*d^10 + d^11)*f*cos(f*x + e)^2 + (c^11 + 3*c^10*d + 7*c^9*d^2 + 13*c^8*d^3 + 2*c^7*d^4 - 26*c^6*d^5 - 26*c^5*d^6 + 2*c^4*d^7 + 13*c^3*d^8 + 7*c^2*d^9 + 3*c*d^10 + d^11)*f - 4*((c^8*d^3 + 3*c^7*d^4 + c^6*d^5 - 5*c^5*d^6 - 5*c^4*d^7 + c^3*d^8 + 3*c^2*d^9 + c*d^10)*f*cos(f*x + e)^2 - (c^10*d + 3*c^9*d^2 + 2*c^8*d^3 - 2*c^7*d^4 - 4*c^6*d^5 - 4*c^5*d^6 - 2*c^4*d^7 + 2*c^3*d^8 + 3*c^2*d^9 + c*d^10)*f)*sin(f*x + e))]","B",0
453,1,349,0,0.950995," ","integrate((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","\frac{2 \, d^{4} \cos\left(f x + e\right)^{4} - 6 \, c^{4} + 24 \, c^{3} d - 36 \, c^{2} d^{2} + 24 \, c d^{3} - 6 \, d^{4} + {\left(12 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right)} f x - 12 \, {\left(3 \, c^{2} d^{2} - 2 \, c d^{3} + d^{4}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(2 \, c^{4} - 8 \, c^{3} d + 24 \, c^{2} d^{2} - 12 \, c d^{3} + 5 \, d^{4} - {\left(8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right)} f x\right)} \cos\left(f x + e\right) + {\left(2 \, d^{4} \cos\left(f x + e\right)^{3} + 6 \, c^{4} - 24 \, c^{3} d + 36 \, c^{2} d^{2} - 24 \, c d^{3} + 6 \, d^{4} + 3 \, {\left(8 \, c^{3} d - 12 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4}\right)} f x - 3 \, {\left(4 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(12 \, c^{2} d^{2} - 4 \, c d^{3} + 3 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{6 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}"," ",0,"1/6*(2*d^4*cos(f*x + e)^4 - 6*c^4 + 24*c^3*d - 36*c^2*d^2 + 24*c*d^3 - 6*d^4 + (12*c*d^3 - d^4)*cos(f*x + e)^3 + 3*(8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*f*x - 12*(3*c^2*d^2 - 2*c*d^3 + d^4)*cos(f*x + e)^2 - 3*(2*c^4 - 8*c^3*d + 24*c^2*d^2 - 12*c*d^3 + 5*d^4 - (8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*f*x)*cos(f*x + e) + (2*d^4*cos(f*x + e)^3 + 6*c^4 - 24*c^3*d + 36*c^2*d^2 - 24*c*d^3 + 6*d^4 + 3*(8*c^3*d - 12*c^2*d^2 + 12*c*d^3 - 3*d^4)*f*x - 3*(4*c*d^3 - d^4)*cos(f*x + e)^2 - 3*(12*c^2*d^2 - 4*c*d^3 + 3*d^4)*cos(f*x + e))*sin(f*x + e))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","A",0
454,1,236,0,1.053532," ","integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","\frac{d^{3} \cos\left(f x + e\right)^{3} - 2 \, c^{3} + 6 \, c^{2} d - 6 \, c d^{2} + 2 \, d^{3} + 3 \, {\left(2 \, c^{2} d - 2 \, c d^{2} + d^{3}\right)} f x - 2 \, {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(2 \, c^{3} - 6 \, c^{2} d + 12 \, c d^{2} - 3 \, d^{3} - 3 \, {\left(2 \, c^{2} d - 2 \, c d^{2} + d^{3}\right)} f x\right)} \cos\left(f x + e\right) - {\left(d^{3} \cos\left(f x + e\right)^{2} - 2 \, c^{3} + 6 \, c^{2} d - 6 \, c d^{2} + 2 \, d^{3} - 3 \, {\left(2 \, c^{2} d - 2 \, c d^{2} + d^{3}\right)} f x + {\left(6 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}"," ",0,"1/2*(d^3*cos(f*x + e)^3 - 2*c^3 + 6*c^2*d - 6*c*d^2 + 2*d^3 + 3*(2*c^2*d - 2*c*d^2 + d^3)*f*x - 2*(3*c*d^2 - d^3)*cos(f*x + e)^2 - (2*c^3 - 6*c^2*d + 12*c*d^2 - 3*d^3 - 3*(2*c^2*d - 2*c*d^2 + d^3)*f*x)*cos(f*x + e) - (d^3*cos(f*x + e)^2 - 2*c^3 + 6*c^2*d - 6*c*d^2 + 2*d^3 - 3*(2*c^2*d - 2*c*d^2 + d^3)*f*x + (6*c*d^2 - d^3)*cos(f*x + e))*sin(f*x + e))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","B",0
455,1,142,0,1.171694," ","integrate((c+d*sin(f*x+e))^2/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{d^{2} \cos\left(f x + e\right)^{2} - {\left(2 \, c d - d^{2}\right)} f x + c^{2} - 2 \, c d + d^{2} - {\left({\left(2 \, c d - d^{2}\right)} f x - c^{2} + 2 \, c d - 2 \, d^{2}\right)} \cos\left(f x + e\right) - {\left({\left(2 \, c d - d^{2}\right)} f x - d^{2} \cos\left(f x + e\right) + c^{2} - 2 \, c d + d^{2}\right)} \sin\left(f x + e\right)}{a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f}"," ",0,"-(d^2*cos(f*x + e)^2 - (2*c*d - d^2)*f*x + c^2 - 2*c*d + d^2 - ((2*c*d - d^2)*f*x - c^2 + 2*c*d - 2*d^2)*cos(f*x + e) - ((2*c*d - d^2)*f*x - d^2*cos(f*x + e) + c^2 - 2*c*d + d^2)*sin(f*x + e))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","B",0
456,1,66,0,1.075189," ","integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","\frac{d f x + {\left(d f x - c + d\right)} \cos\left(f x + e\right) + {\left(d f x + c - d\right)} \sin\left(f x + e\right) - c + d}{a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f}"," ",0,"(d*f*x + (d*f*x - c + d)*cos(f*x + e) + (d*f*x + c - d)*sin(f*x + e) - c + d)/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","A",0
457,1,42,0,0.700653," ","integrate(1/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1}{a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f}"," ",0,"-(cos(f*x + e) - sin(f*x + e) + 1)/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","A",0
458,1,489,0,0.937161," ","integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[\frac{\sqrt{-c^{2} + d^{2}} {\left(d \cos\left(f x + e\right) + d \sin\left(f x + e\right) + d\right)} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 2 \, c^{2} + 2 \, d^{2} - 2 \, {\left(c^{2} - d^{2}\right)} \cos\left(f x + e\right) + 2 \, {\left(c^{2} - d^{2}\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right)} f \cos\left(f x + e\right) + {\left(a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right)} f \sin\left(f x + e\right) + {\left(a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right)} f\right)}}, \frac{\sqrt{c^{2} - d^{2}} {\left(d \cos\left(f x + e\right) + d \sin\left(f x + e\right) + d\right)} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) - c^{2} + d^{2} - {\left(c^{2} - d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} - d^{2}\right)} \sin\left(f x + e\right)}{{\left(a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right)} f \cos\left(f x + e\right) + {\left(a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right)} f \sin\left(f x + e\right) + {\left(a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right)} f}\right]"," ",0,"[1/2*(sqrt(-c^2 + d^2)*(d*cos(f*x + e) + d*sin(f*x + e) + d)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 2*c^2 + 2*d^2 - 2*(c^2 - d^2)*cos(f*x + e) + 2*(c^2 - d^2)*sin(f*x + e))/((a*c^3 - a*c^2*d - a*c*d^2 + a*d^3)*f*cos(f*x + e) + (a*c^3 - a*c^2*d - a*c*d^2 + a*d^3)*f*sin(f*x + e) + (a*c^3 - a*c^2*d - a*c*d^2 + a*d^3)*f), (sqrt(c^2 - d^2)*(d*cos(f*x + e) + d*sin(f*x + e) + d)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - c^2 + d^2 - (c^2 - d^2)*cos(f*x + e) + (c^2 - d^2)*sin(f*x + e))/((a*c^3 - a*c^2*d - a*c*d^2 + a*d^3)*f*cos(f*x + e) + (a*c^3 - a*c^2*d - a*c*d^2 + a*d^3)*f*sin(f*x + e) + (a*c^3 - a*c^2*d - a*c*d^2 + a*d^3)*f)]","B",0
459,1,1120,0,1.268846," ","integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{2 \, c^{4} - 4 \, c^{2} d^{2} + 2 \, d^{4} + 2 \, {\left(c^{3} d + 2 \, c^{2} d^{2} - c d^{3} - 2 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(2 \, c^{2} d + 3 \, c d^{2} + d^{3} - {\left(2 \, c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(2 \, c^{2} d + c d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, c^{2} d + 3 \, c d^{2} + d^{3} + {\left(2 \, c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(-\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} - 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(c^{4} + c^{3} d - c d^{3} - d^{4}\right)} \cos\left(f x + e\right) - 2 \, {\left(c^{4} - 2 \, c^{2} d^{2} + d^{4} - {\left(c^{3} d + 2 \, c^{2} d^{2} - c d^{3} - 2 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a c^{5} d - a c^{4} d^{2} - 2 \, a c^{3} d^{3} + 2 \, a c^{2} d^{4} + a c d^{5} - a d^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{6} - a c^{5} d - 2 \, a c^{4} d^{2} + 2 \, a c^{3} d^{3} + a c^{2} d^{4} - a c d^{5}\right)} f \cos\left(f x + e\right) - {\left(a c^{6} - 3 \, a c^{4} d^{2} + 3 \, a c^{2} d^{4} - a d^{6}\right)} f - {\left({\left(a c^{5} d - a c^{4} d^{2} - 2 \, a c^{3} d^{3} + 2 \, a c^{2} d^{4} + a c d^{5} - a d^{6}\right)} f \cos\left(f x + e\right) + {\left(a c^{6} - 3 \, a c^{4} d^{2} + 3 \, a c^{2} d^{4} - a d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{c^{4} - 2 \, c^{2} d^{2} + d^{4} + {\left(c^{3} d + 2 \, c^{2} d^{2} - c d^{3} - 2 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(2 \, c^{2} d + 3 \, c d^{2} + d^{3} - {\left(2 \, c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(2 \, c^{2} d + c d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, c^{2} d + 3 \, c d^{2} + d^{3} + {\left(2 \, c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(c^{4} + c^{3} d - c d^{3} - d^{4}\right)} \cos\left(f x + e\right) - {\left(c^{4} - 2 \, c^{2} d^{2} + d^{4} - {\left(c^{3} d + 2 \, c^{2} d^{2} - c d^{3} - 2 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{{\left(a c^{5} d - a c^{4} d^{2} - 2 \, a c^{3} d^{3} + 2 \, a c^{2} d^{4} + a c d^{5} - a d^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{6} - a c^{5} d - 2 \, a c^{4} d^{2} + 2 \, a c^{3} d^{3} + a c^{2} d^{4} - a c d^{5}\right)} f \cos\left(f x + e\right) - {\left(a c^{6} - 3 \, a c^{4} d^{2} + 3 \, a c^{2} d^{4} - a d^{6}\right)} f - {\left({\left(a c^{5} d - a c^{4} d^{2} - 2 \, a c^{3} d^{3} + 2 \, a c^{2} d^{4} + a c d^{5} - a d^{6}\right)} f \cos\left(f x + e\right) + {\left(a c^{6} - 3 \, a c^{4} d^{2} + 3 \, a c^{2} d^{4} - a d^{6}\right)} f\right)} \sin\left(f x + e\right)}\right]"," ",0,"[1/2*(2*c^4 - 4*c^2*d^2 + 2*d^4 + 2*(c^3*d + 2*c^2*d^2 - c*d^3 - 2*d^4)*cos(f*x + e)^2 + (2*c^2*d + 3*c*d^2 + d^3 - (2*c*d^2 + d^3)*cos(f*x + e)^2 + (2*c^2*d + c*d^2)*cos(f*x + e) + (2*c^2*d + 3*c*d^2 + d^3 + (2*c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-c^2 + d^2)*log(-((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 - 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(c^4 + c^3*d - c*d^3 - d^4)*cos(f*x + e) - 2*(c^4 - 2*c^2*d^2 + d^4 - (c^3*d + 2*c^2*d^2 - c*d^3 - 2*d^4)*cos(f*x + e))*sin(f*x + e))/((a*c^5*d - a*c^4*d^2 - 2*a*c^3*d^3 + 2*a*c^2*d^4 + a*c*d^5 - a*d^6)*f*cos(f*x + e)^2 - (a*c^6 - a*c^5*d - 2*a*c^4*d^2 + 2*a*c^3*d^3 + a*c^2*d^4 - a*c*d^5)*f*cos(f*x + e) - (a*c^6 - 3*a*c^4*d^2 + 3*a*c^2*d^4 - a*d^6)*f - ((a*c^5*d - a*c^4*d^2 - 2*a*c^3*d^3 + 2*a*c^2*d^4 + a*c*d^5 - a*d^6)*f*cos(f*x + e) + (a*c^6 - 3*a*c^4*d^2 + 3*a*c^2*d^4 - a*d^6)*f)*sin(f*x + e)), (c^4 - 2*c^2*d^2 + d^4 + (c^3*d + 2*c^2*d^2 - c*d^3 - 2*d^4)*cos(f*x + e)^2 - (2*c^2*d + 3*c*d^2 + d^3 - (2*c*d^2 + d^3)*cos(f*x + e)^2 + (2*c^2*d + c*d^2)*cos(f*x + e) + (2*c^2*d + 3*c*d^2 + d^3 + (2*c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (c^4 + c^3*d - c*d^3 - d^4)*cos(f*x + e) - (c^4 - 2*c^2*d^2 + d^4 - (c^3*d + 2*c^2*d^2 - c*d^3 - 2*d^4)*cos(f*x + e))*sin(f*x + e))/((a*c^5*d - a*c^4*d^2 - 2*a*c^3*d^3 + 2*a*c^2*d^4 + a*c*d^5 - a*d^6)*f*cos(f*x + e)^2 - (a*c^6 - a*c^5*d - 2*a*c^4*d^2 + 2*a*c^3*d^3 + a*c^2*d^4 - a*c*d^5)*f*cos(f*x + e) - (a*c^6 - 3*a*c^4*d^2 + 3*a*c^2*d^4 - a*d^6)*f - ((a*c^5*d - a*c^4*d^2 - 2*a*c^3*d^3 + 2*a*c^2*d^4 + a*c*d^5 - a*d^6)*f*cos(f*x + e) + (a*c^6 - 3*a*c^4*d^2 + 3*a*c^2*d^4 - a*d^6)*f)*sin(f*x + e))]","B",0
460,1,2365,0,1.412703," ","integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[\frac{4 \, c^{6} - 12 \, c^{4} d^{2} + 12 \, c^{2} d^{4} - 4 \, d^{6} - 2 \, {\left(2 \, c^{4} d^{2} + 9 \, c^{3} d^{3} + 2 \, c^{2} d^{4} - 9 \, c d^{5} - 4 \, d^{6}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(4 \, c^{5} d + 12 \, c^{4} d^{2} - 2 \, c^{3} d^{3} - 15 \, c^{2} d^{4} - 2 \, c d^{5} + 3 \, d^{6}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(2 \, c^{4} d + 6 \, c^{3} d^{2} + 7 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5} - {\left(2 \, c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(4 \, c^{3} d^{2} + 6 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(2 \, c^{4} d + 2 \, c^{3} d^{2} + 3 \, c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} \cos\left(f x + e\right) + {\left(2 \, c^{4} d + 6 \, c^{3} d^{2} + 7 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5} - {\left(2 \, c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, c^{3} d^{2} + 2 \, c^{2} d^{3} + c d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(2 \, c^{6} + 4 \, c^{5} d + 8 \, c^{4} d^{2} + 7 \, c^{3} d^{3} - 7 \, c^{2} d^{4} - 11 \, c d^{5} - 3 \, d^{6}\right)} \cos\left(f x + e\right) - 2 \, {\left(2 \, c^{6} - 6 \, c^{4} d^{2} + 6 \, c^{2} d^{4} - 2 \, d^{6} - {\left(2 \, c^{4} d^{2} + 9 \, c^{3} d^{3} + 2 \, c^{2} d^{4} - 9 \, c d^{5} - 4 \, d^{6}\right)} \cos\left(f x + e\right)^{2} - {\left(4 \, c^{5} d + 14 \, c^{4} d^{2} + 7 \, c^{3} d^{3} - 13 \, c^{2} d^{4} - 11 \, c d^{5} - d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a c^{7} d^{2} - a c^{6} d^{3} - 3 \, a c^{5} d^{4} + 3 \, a c^{4} d^{5} + 3 \, a c^{3} d^{6} - 3 \, a c^{2} d^{7} - a c d^{8} + a d^{9}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, a c^{8} d - a c^{7} d^{2} - 7 \, a c^{6} d^{3} + 3 \, a c^{5} d^{4} + 9 \, a c^{4} d^{5} - 3 \, a c^{3} d^{6} - 5 \, a c^{2} d^{7} + a c d^{8} + a d^{9}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{9} - a c^{8} d - 2 \, a c^{7} d^{2} + 2 \, a c^{6} d^{3} + 2 \, a c^{3} d^{6} - 2 \, a c^{2} d^{7} - a c d^{8} + a d^{9}\right)} f \cos\left(f x + e\right) - {\left(a c^{9} + a c^{8} d - 4 \, a c^{7} d^{2} - 4 \, a c^{6} d^{3} + 6 \, a c^{5} d^{4} + 6 \, a c^{4} d^{5} - 4 \, a c^{3} d^{6} - 4 \, a c^{2} d^{7} + a c d^{8} + a d^{9}\right)} f + {\left({\left(a c^{7} d^{2} - a c^{6} d^{3} - 3 \, a c^{5} d^{4} + 3 \, a c^{4} d^{5} + 3 \, a c^{3} d^{6} - 3 \, a c^{2} d^{7} - a c d^{8} + a d^{9}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a c^{8} d - a c^{7} d^{2} - 3 \, a c^{6} d^{3} + 3 \, a c^{5} d^{4} + 3 \, a c^{4} d^{5} - 3 \, a c^{3} d^{6} - a c^{2} d^{7} + a c d^{8}\right)} f \cos\left(f x + e\right) - {\left(a c^{9} + a c^{8} d - 4 \, a c^{7} d^{2} - 4 \, a c^{6} d^{3} + 6 \, a c^{5} d^{4} + 6 \, a c^{4} d^{5} - 4 \, a c^{3} d^{6} - 4 \, a c^{2} d^{7} + a c d^{8} + a d^{9}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{2 \, c^{6} - 6 \, c^{4} d^{2} + 6 \, c^{2} d^{4} - 2 \, d^{6} - {\left(2 \, c^{4} d^{2} + 9 \, c^{3} d^{3} + 2 \, c^{2} d^{4} - 9 \, c d^{5} - 4 \, d^{6}\right)} \cos\left(f x + e\right)^{3} + {\left(4 \, c^{5} d + 12 \, c^{4} d^{2} - 2 \, c^{3} d^{3} - 15 \, c^{2} d^{4} - 2 \, c d^{5} + 3 \, d^{6}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(2 \, c^{4} d + 6 \, c^{3} d^{2} + 7 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5} - {\left(2 \, c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(4 \, c^{3} d^{2} + 6 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(2 \, c^{4} d + 2 \, c^{3} d^{2} + 3 \, c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} \cos\left(f x + e\right) + {\left(2 \, c^{4} d + 6 \, c^{3} d^{2} + 7 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5} - {\left(2 \, c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, c^{3} d^{2} + 2 \, c^{2} d^{3} + c d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(2 \, c^{6} + 4 \, c^{5} d + 8 \, c^{4} d^{2} + 7 \, c^{3} d^{3} - 7 \, c^{2} d^{4} - 11 \, c d^{5} - 3 \, d^{6}\right)} \cos\left(f x + e\right) - {\left(2 \, c^{6} - 6 \, c^{4} d^{2} + 6 \, c^{2} d^{4} - 2 \, d^{6} - {\left(2 \, c^{4} d^{2} + 9 \, c^{3} d^{3} + 2 \, c^{2} d^{4} - 9 \, c d^{5} - 4 \, d^{6}\right)} \cos\left(f x + e\right)^{2} - {\left(4 \, c^{5} d + 14 \, c^{4} d^{2} + 7 \, c^{3} d^{3} - 13 \, c^{2} d^{4} - 11 \, c d^{5} - d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a c^{7} d^{2} - a c^{6} d^{3} - 3 \, a c^{5} d^{4} + 3 \, a c^{4} d^{5} + 3 \, a c^{3} d^{6} - 3 \, a c^{2} d^{7} - a c d^{8} + a d^{9}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, a c^{8} d - a c^{7} d^{2} - 7 \, a c^{6} d^{3} + 3 \, a c^{5} d^{4} + 9 \, a c^{4} d^{5} - 3 \, a c^{3} d^{6} - 5 \, a c^{2} d^{7} + a c d^{8} + a d^{9}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{9} - a c^{8} d - 2 \, a c^{7} d^{2} + 2 \, a c^{6} d^{3} + 2 \, a c^{3} d^{6} - 2 \, a c^{2} d^{7} - a c d^{8} + a d^{9}\right)} f \cos\left(f x + e\right) - {\left(a c^{9} + a c^{8} d - 4 \, a c^{7} d^{2} - 4 \, a c^{6} d^{3} + 6 \, a c^{5} d^{4} + 6 \, a c^{4} d^{5} - 4 \, a c^{3} d^{6} - 4 \, a c^{2} d^{7} + a c d^{8} + a d^{9}\right)} f + {\left({\left(a c^{7} d^{2} - a c^{6} d^{3} - 3 \, a c^{5} d^{4} + 3 \, a c^{4} d^{5} + 3 \, a c^{3} d^{6} - 3 \, a c^{2} d^{7} - a c d^{8} + a d^{9}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a c^{8} d - a c^{7} d^{2} - 3 \, a c^{6} d^{3} + 3 \, a c^{5} d^{4} + 3 \, a c^{4} d^{5} - 3 \, a c^{3} d^{6} - a c^{2} d^{7} + a c d^{8}\right)} f \cos\left(f x + e\right) - {\left(a c^{9} + a c^{8} d - 4 \, a c^{7} d^{2} - 4 \, a c^{6} d^{3} + 6 \, a c^{5} d^{4} + 6 \, a c^{4} d^{5} - 4 \, a c^{3} d^{6} - 4 \, a c^{2} d^{7} + a c d^{8} + a d^{9}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/4*(4*c^6 - 12*c^4*d^2 + 12*c^2*d^4 - 4*d^6 - 2*(2*c^4*d^2 + 9*c^3*d^3 + 2*c^2*d^4 - 9*c*d^5 - 4*d^6)*cos(f*x + e)^3 + 2*(4*c^5*d + 12*c^4*d^2 - 2*c^3*d^3 - 15*c^2*d^4 - 2*c*d^5 + 3*d^6)*cos(f*x + e)^2 - 3*(2*c^4*d + 6*c^3*d^2 + 7*c^2*d^3 + 4*c*d^4 + d^5 - (2*c^2*d^3 + 2*c*d^4 + d^5)*cos(f*x + e)^3 - (4*c^3*d^2 + 6*c^2*d^3 + 4*c*d^4 + d^5)*cos(f*x + e)^2 + (2*c^4*d + 2*c^3*d^2 + 3*c^2*d^3 + 2*c*d^4 + d^5)*cos(f*x + e) + (2*c^4*d + 6*c^3*d^2 + 7*c^2*d^3 + 4*c*d^4 + d^5 - (2*c^2*d^3 + 2*c*d^4 + d^5)*cos(f*x + e)^2 + 2*(2*c^3*d^2 + 2*c^2*d^3 + c*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(2*c^6 + 4*c^5*d + 8*c^4*d^2 + 7*c^3*d^3 - 7*c^2*d^4 - 11*c*d^5 - 3*d^6)*cos(f*x + e) - 2*(2*c^6 - 6*c^4*d^2 + 6*c^2*d^4 - 2*d^6 - (2*c^4*d^2 + 9*c^3*d^3 + 2*c^2*d^4 - 9*c*d^5 - 4*d^6)*cos(f*x + e)^2 - (4*c^5*d + 14*c^4*d^2 + 7*c^3*d^3 - 13*c^2*d^4 - 11*c*d^5 - d^6)*cos(f*x + e))*sin(f*x + e))/((a*c^7*d^2 - a*c^6*d^3 - 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^3*d^6 - 3*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x + e)^3 + (2*a*c^8*d - a*c^7*d^2 - 7*a*c^6*d^3 + 3*a*c^5*d^4 + 9*a*c^4*d^5 - 3*a*c^3*d^6 - 5*a*c^2*d^7 + a*c*d^8 + a*d^9)*f*cos(f*x + e)^2 - (a*c^9 - a*c^8*d - 2*a*c^7*d^2 + 2*a*c^6*d^3 + 2*a*c^3*d^6 - 2*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x + e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 - 4*a*c^3*d^6 - 4*a*c^2*d^7 + a*c*d^8 + a*d^9)*f + ((a*c^7*d^2 - a*c^6*d^3 - 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^3*d^6 - 3*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x + e)^2 - 2*(a*c^8*d - a*c^7*d^2 - 3*a*c^6*d^3 + 3*a*c^5*d^4 + 3*a*c^4*d^5 - 3*a*c^3*d^6 - a*c^2*d^7 + a*c*d^8)*f*cos(f*x + e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 - 4*a*c^3*d^6 - 4*a*c^2*d^7 + a*c*d^8 + a*d^9)*f)*sin(f*x + e)), 1/2*(2*c^6 - 6*c^4*d^2 + 6*c^2*d^4 - 2*d^6 - (2*c^4*d^2 + 9*c^3*d^3 + 2*c^2*d^4 - 9*c*d^5 - 4*d^6)*cos(f*x + e)^3 + (4*c^5*d + 12*c^4*d^2 - 2*c^3*d^3 - 15*c^2*d^4 - 2*c*d^5 + 3*d^6)*cos(f*x + e)^2 - 3*(2*c^4*d + 6*c^3*d^2 + 7*c^2*d^3 + 4*c*d^4 + d^5 - (2*c^2*d^3 + 2*c*d^4 + d^5)*cos(f*x + e)^3 - (4*c^3*d^2 + 6*c^2*d^3 + 4*c*d^4 + d^5)*cos(f*x + e)^2 + (2*c^4*d + 2*c^3*d^2 + 3*c^2*d^3 + 2*c*d^4 + d^5)*cos(f*x + e) + (2*c^4*d + 6*c^3*d^2 + 7*c^2*d^3 + 4*c*d^4 + d^5 - (2*c^2*d^3 + 2*c*d^4 + d^5)*cos(f*x + e)^2 + 2*(2*c^3*d^2 + 2*c^2*d^3 + c*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (2*c^6 + 4*c^5*d + 8*c^4*d^2 + 7*c^3*d^3 - 7*c^2*d^4 - 11*c*d^5 - 3*d^6)*cos(f*x + e) - (2*c^6 - 6*c^4*d^2 + 6*c^2*d^4 - 2*d^6 - (2*c^4*d^2 + 9*c^3*d^3 + 2*c^2*d^4 - 9*c*d^5 - 4*d^6)*cos(f*x + e)^2 - (4*c^5*d + 14*c^4*d^2 + 7*c^3*d^3 - 13*c^2*d^4 - 11*c*d^5 - d^6)*cos(f*x + e))*sin(f*x + e))/((a*c^7*d^2 - a*c^6*d^3 - 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^3*d^6 - 3*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x + e)^3 + (2*a*c^8*d - a*c^7*d^2 - 7*a*c^6*d^3 + 3*a*c^5*d^4 + 9*a*c^4*d^5 - 3*a*c^3*d^6 - 5*a*c^2*d^7 + a*c*d^8 + a*d^9)*f*cos(f*x + e)^2 - (a*c^9 - a*c^8*d - 2*a*c^7*d^2 + 2*a*c^6*d^3 + 2*a*c^3*d^6 - 2*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x + e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 - 4*a*c^3*d^6 - 4*a*c^2*d^7 + a*c*d^8 + a*d^9)*f + ((a*c^7*d^2 - a*c^6*d^3 - 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^3*d^6 - 3*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x + e)^2 - 2*(a*c^8*d - a*c^7*d^2 - 3*a*c^6*d^3 + 3*a*c^5*d^4 + 3*a*c^4*d^5 - 3*a*c^3*d^6 - a*c^2*d^7 + a*c*d^8)*f*cos(f*x + e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 - 4*a*c^3*d^6 - 4*a*c^2*d^7 + a*c*d^8 + a*d^9)*f)*sin(f*x + e))]","B",0
461,1,578,0,1.037769," ","integrate((c+d*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{2 \, d^{5} \cos\left(f x + e\right)^{5} + 2 \, c^{5} - 10 \, c^{4} d + 20 \, c^{3} d^{2} - 20 \, c^{2} d^{3} + 10 \, c d^{4} - 2 \, d^{5} - {\left(15 \, c d^{4} - 4 \, d^{5}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(30 \, c^{2} d^{3} - 15 \, c d^{4} + 8 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - 30 \, {\left(4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right)} f x + {\left(2 \, c^{5} + 20 \, c^{4} d - 100 \, c^{3} d^{2} + 220 \, c^{2} d^{3} - 155 \, c d^{4} + 46 \, d^{5} + 15 \, {\left(4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right)} f x\right)} \cos\left(f x + e\right)^{2} + {\left(4 \, c^{5} + 10 \, c^{4} d - 80 \, c^{3} d^{2} + 260 \, c^{2} d^{3} - 190 \, c d^{4} + 62 \, d^{5} - 15 \, {\left(4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right)} f x\right)} \cos\left(f x + e\right) - {\left(2 \, d^{5} \cos\left(f x + e\right)^{4} + 2 \, c^{5} - 10 \, c^{4} d + 20 \, c^{3} d^{2} - 20 \, c^{2} d^{3} + 10 \, c d^{4} - 2 \, d^{5} + {\left(15 \, c d^{4} - 2 \, d^{5}\right)} \cos\left(f x + e\right)^{3} + 30 \, {\left(4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right)} f x - 3 \, {\left(20 \, c^{2} d^{3} - 15 \, c d^{4} + 6 \, d^{5}\right)} \cos\left(f x + e\right)^{2} - {\left(2 \, c^{5} + 20 \, c^{4} d - 100 \, c^{3} d^{2} + 280 \, c^{2} d^{3} - 200 \, c d^{4} + 64 \, d^{5} - 15 \, {\left(4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right)} f x\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{6 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/6*(2*d^5*cos(f*x + e)^5 + 2*c^5 - 10*c^4*d + 20*c^3*d^2 - 20*c^2*d^3 + 10*c*d^4 - 2*d^5 - (15*c*d^4 - 4*d^5)*cos(f*x + e)^4 - 2*(30*c^2*d^3 - 15*c*d^4 + 8*d^5)*cos(f*x + e)^3 - 30*(4*c^3*d^2 - 8*c^2*d^3 + 7*c*d^4 - 2*d^5)*f*x + (2*c^5 + 20*c^4*d - 100*c^3*d^2 + 220*c^2*d^3 - 155*c*d^4 + 46*d^5 + 15*(4*c^3*d^2 - 8*c^2*d^3 + 7*c*d^4 - 2*d^5)*f*x)*cos(f*x + e)^2 + (4*c^5 + 10*c^4*d - 80*c^3*d^2 + 260*c^2*d^3 - 190*c*d^4 + 62*d^5 - 15*(4*c^3*d^2 - 8*c^2*d^3 + 7*c*d^4 - 2*d^5)*f*x)*cos(f*x + e) - (2*d^5*cos(f*x + e)^4 + 2*c^5 - 10*c^4*d + 20*c^3*d^2 - 20*c^2*d^3 + 10*c*d^4 - 2*d^5 + (15*c*d^4 - 2*d^5)*cos(f*x + e)^3 + 30*(4*c^3*d^2 - 8*c^2*d^3 + 7*c*d^4 - 2*d^5)*f*x - 3*(20*c^2*d^3 - 15*c*d^4 + 6*d^5)*cos(f*x + e)^2 - (2*c^5 + 20*c^4*d - 100*c^3*d^2 + 280*c^2*d^3 - 200*c*d^4 + 64*d^5 - 15*(4*c^3*d^2 - 8*c^2*d^3 + 7*c*d^4 - 2*d^5)*f*x)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","B",0
462,1,440,0,1.112329," ","integrate((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{3 \, d^{4} \cos\left(f x + e\right)^{4} - 2 \, c^{4} + 8 \, c^{3} d - 12 \, c^{2} d^{2} + 8 \, c d^{3} - 2 \, d^{4} + 6 \, {\left(4 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} + 6 \, {\left(12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right)} f x - {\left(2 \, c^{4} + 16 \, c^{3} d - 60 \, c^{2} d^{2} + 88 \, c d^{3} - 31 \, d^{4} + 3 \, {\left(12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right)} f x\right)} \cos\left(f x + e\right)^{2} - {\left(4 \, c^{4} + 8 \, c^{3} d - 48 \, c^{2} d^{2} + 104 \, c d^{3} - 38 \, d^{4} - 3 \, {\left(12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right)} f x\right)} \cos\left(f x + e\right) + {\left(3 \, d^{4} \cos\left(f x + e\right)^{3} + 2 \, c^{4} - 8 \, c^{3} d + 12 \, c^{2} d^{2} - 8 \, c d^{3} + 2 \, d^{4} + 6 \, {\left(12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right)} f x - 3 \, {\left(8 \, c d^{3} - 3 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(2 \, c^{4} + 16 \, c^{3} d - 60 \, c^{2} d^{2} + 112 \, c d^{3} - 40 \, d^{4} - 3 \, {\left(12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right)} f x\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{6 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/6*(3*d^4*cos(f*x + e)^4 - 2*c^4 + 8*c^3*d - 12*c^2*d^2 + 8*c*d^3 - 2*d^4 + 6*(4*c*d^3 - d^4)*cos(f*x + e)^3 + 6*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*f*x - (2*c^4 + 16*c^3*d - 60*c^2*d^2 + 88*c*d^3 - 31*d^4 + 3*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*f*x)*cos(f*x + e)^2 - (4*c^4 + 8*c^3*d - 48*c^2*d^2 + 104*c*d^3 - 38*d^4 - 3*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*f*x)*cos(f*x + e) + (3*d^4*cos(f*x + e)^3 + 2*c^4 - 8*c^3*d + 12*c^2*d^2 - 8*c*d^3 + 2*d^4 + 6*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*f*x - 3*(8*c*d^3 - 3*d^4)*cos(f*x + e)^2 - (2*c^4 + 16*c^3*d - 60*c^2*d^2 + 112*c*d^3 - 40*d^4 - 3*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*f*x)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","B",0
463,1,308,0,1.433979," ","integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{3 \, d^{3} \cos\left(f x + e\right)^{3} - c^{3} + 3 \, c^{2} d - 3 \, c d^{2} + d^{3} + 6 \, {\left(3 \, c d^{2} - 2 \, d^{3}\right)} f x - {\left(c^{3} + 6 \, c^{2} d - 15 \, c d^{2} + 11 \, d^{3} + 3 \, {\left(3 \, c d^{2} - 2 \, d^{3}\right)} f x\right)} \cos\left(f x + e\right)^{2} - {\left(2 \, c^{3} + 3 \, c^{2} d - 12 \, c d^{2} + 13 \, d^{3} - 3 \, {\left(3 \, c d^{2} - 2 \, d^{3}\right)} f x\right)} \cos\left(f x + e\right) - {\left(3 \, d^{3} \cos\left(f x + e\right)^{2} - c^{3} + 3 \, c^{2} d - 3 \, c d^{2} + d^{3} - 6 \, {\left(3 \, c d^{2} - 2 \, d^{3}\right)} f x + {\left(c^{3} + 6 \, c^{2} d - 15 \, c d^{2} + 14 \, d^{3} - 3 \, {\left(3 \, c d^{2} - 2 \, d^{3}\right)} f x\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{3 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/3*(3*d^3*cos(f*x + e)^3 - c^3 + 3*c^2*d - 3*c*d^2 + d^3 + 6*(3*c*d^2 - 2*d^3)*f*x - (c^3 + 6*c^2*d - 15*c*d^2 + 11*d^3 + 3*(3*c*d^2 - 2*d^3)*f*x)*cos(f*x + e)^2 - (2*c^3 + 3*c^2*d - 12*c*d^2 + 13*d^3 - 3*(3*c*d^2 - 2*d^3)*f*x)*cos(f*x + e) - (3*d^3*cos(f*x + e)^2 - c^3 + 3*c^2*d - 3*c*d^2 + d^3 - 6*(3*c*d^2 - 2*d^3)*f*x + (c^3 + 6*c^2*d - 15*c*d^2 + 14*d^3 - 3*(3*c*d^2 - 2*d^3)*f*x)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","B",0
464,1,197,0,1.740523," ","integrate((c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{6 \, d^{2} f x - {\left(3 \, d^{2} f x + c^{2} + 4 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} + 2 \, c d - d^{2} + {\left(3 \, d^{2} f x - 2 \, c^{2} - 2 \, c d + 4 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(6 \, d^{2} f x + c^{2} - 2 \, c d + d^{2} + {\left(3 \, d^{2} f x - c^{2} - 4 \, c d + 5 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{3 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/3*(6*d^2*f*x - (3*d^2*f*x + c^2 + 4*c*d - 5*d^2)*cos(f*x + e)^2 - c^2 + 2*c*d - d^2 + (3*d^2*f*x - 2*c^2 - 2*c*d + 4*d^2)*cos(f*x + e) + (6*d^2*f*x + c^2 - 2*c*d + d^2 + (3*d^2*f*x - c^2 - 4*c*d + 5*d^2)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","B",0
465,1,117,0,1.256983," ","integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{{\left(c + 2 \, d\right)} \cos\left(f x + e\right)^{2} + {\left(2 \, c + d\right)} \cos\left(f x + e\right) + {\left({\left(c + 2 \, d\right)} \cos\left(f x + e\right) - c + d\right)} \sin\left(f x + e\right) + c - d}{3 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/3*((c + 2*d)*cos(f*x + e)^2 + (2*c + d)*cos(f*x + e) + ((c + 2*d)*cos(f*x + e) - c + d)*sin(f*x + e) + c - d)/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","A",0
466,1,95,0,0.933897," ","integrate(1/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) - 1\right)} \sin\left(f x + e\right) + 2 \, \cos\left(f x + e\right) + 1}{3 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/3*(cos(f*x + e)^2 + (cos(f*x + e) - 1)*sin(f*x + e) + 2*cos(f*x + e) + 1)/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","A",0
467,1,989,0,1.352486," ","integrate(1/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[\frac{2 \, c^{3} - 2 \, c^{2} d - 2 \, c d^{2} + 2 \, d^{3} + 2 \, {\left(c^{3} - 4 \, c^{2} d - c d^{2} + 4 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(d^{2} \cos\left(f x + e\right)^{2} - d^{2} \cos\left(f x + e\right) - 2 \, d^{2} - {\left(d^{2} \cos\left(f x + e\right) + 2 \, d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(2 \, c^{3} - 5 \, c^{2} d - 2 \, c d^{2} + 5 \, d^{3}\right)} \cos\left(f x + e\right) - 2 \, {\left(c^{3} - c^{2} d - c d^{2} + d^{3} - {\left(c^{3} - 4 \, c^{2} d - c d^{2} + 4 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{6 \, {\left({\left(a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right)} f - {\left({\left(a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{c^{3} - c^{2} d - c d^{2} + d^{3} + {\left(c^{3} - 4 \, c^{2} d - c d^{2} + 4 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(d^{2} \cos\left(f x + e\right)^{2} - d^{2} \cos\left(f x + e\right) - 2 \, d^{2} - {\left(d^{2} \cos\left(f x + e\right) + 2 \, d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(2 \, c^{3} - 5 \, c^{2} d - 2 \, c d^{2} + 5 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{3} - c^{2} d - c d^{2} + d^{3} - {\left(c^{3} - 4 \, c^{2} d - c d^{2} + 4 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{3 \, {\left({\left(a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right)} f - {\left({\left(a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/6*(2*c^3 - 2*c^2*d - 2*c*d^2 + 2*d^3 + 2*(c^3 - 4*c^2*d - c*d^2 + 4*d^3)*cos(f*x + e)^2 - 3*(d^2*cos(f*x + e)^2 - d^2*cos(f*x + e) - 2*d^2 - (d^2*cos(f*x + e) + 2*d^2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(2*c^3 - 5*c^2*d - 2*c*d^2 + 5*d^3)*cos(f*x + e) - 2*(c^3 - c^2*d - c*d^2 + d^3 - (c^3 - 4*c^2*d - c*d^2 + 4*d^3)*cos(f*x + e))*sin(f*x + e))/((a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*f*cos(f*x + e)^2 - (a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*f*cos(f*x + e) - 2*(a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*f - ((a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*f*cos(f*x + e) + 2*(a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*f)*sin(f*x + e)), 1/3*(c^3 - c^2*d - c*d^2 + d^3 + (c^3 - 4*c^2*d - c*d^2 + 4*d^3)*cos(f*x + e)^2 - 3*(d^2*cos(f*x + e)^2 - d^2*cos(f*x + e) - 2*d^2 - (d^2*cos(f*x + e) + 2*d^2)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (2*c^3 - 5*c^2*d - 2*c*d^2 + 5*d^3)*cos(f*x + e) - (c^3 - c^2*d - c*d^2 + d^3 - (c^3 - 4*c^2*d - c*d^2 + 4*d^3)*cos(f*x + e))*sin(f*x + e))/((a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*f*cos(f*x + e)^2 - (a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*f*cos(f*x + e) - 2*(a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*f - ((a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*f*cos(f*x + e) + 2*(a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*f)*sin(f*x + e))]","B",0
468,1,2297,0,1.079265," ","integrate(1/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{2 \, c^{5} - 2 \, c^{4} d - 4 \, c^{3} d^{2} + 4 \, c^{2} d^{3} + 2 \, c d^{4} - 2 \, d^{5} - 2 \, {\left(c^{4} d - 6 \, c^{3} d^{2} - 11 \, c^{2} d^{3} + 6 \, c d^{4} + 10 \, d^{5}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(c^{5} - 5 \, c^{4} d - 8 \, c^{3} d^{2} + c^{2} d^{3} + 7 \, c d^{4} + 4 \, d^{5}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(6 \, c^{2} d^{2} + 10 \, c d^{3} + 4 \, d^{4} - {\left(3 \, c d^{3} + 2 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, c^{2} d^{2} + 8 \, c d^{3} + 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, c^{2} d^{2} + 5 \, c d^{3} + 2 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(6 \, c^{2} d^{2} + 10 \, c d^{3} + 4 \, d^{4} - {\left(3 \, c d^{3} + 2 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, c^{2} d^{2} + 5 \, c d^{3} + 2 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(-\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} - 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(2 \, c^{5} - 5 \, c^{4} d - 16 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 14 \, c d^{4} + 13 \, d^{5}\right)} \cos\left(f x + e\right) - 2 \, {\left(c^{5} - c^{4} d - 2 \, c^{3} d^{2} + 2 \, c^{2} d^{3} + c d^{4} - d^{5} - {\left(c^{4} d - 6 \, c^{3} d^{2} - 11 \, c^{2} d^{3} + 6 \, c d^{4} + 10 \, d^{5}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{5} - 4 \, c^{4} d - 14 \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 13 \, c d^{4} + 14 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{6 \, {\left({\left(a^{2} c^{6} d - 2 \, a^{2} c^{5} d^{2} - a^{2} c^{4} d^{3} + 4 \, a^{2} c^{3} d^{4} - a^{2} c^{2} d^{5} - 2 \, a^{2} c d^{6} + a^{2} d^{7}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{2} c^{7} - 5 \, a^{2} c^{5} d^{2} + 2 \, a^{2} c^{4} d^{3} + 7 \, a^{2} c^{3} d^{4} - 4 \, a^{2} c^{2} d^{5} - 3 \, a^{2} c d^{6} + 2 \, a^{2} d^{7}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{7} - a^{2} c^{6} d - 3 \, a^{2} c^{5} d^{2} + 3 \, a^{2} c^{4} d^{3} + 3 \, a^{2} c^{3} d^{4} - 3 \, a^{2} c^{2} d^{5} - a^{2} c d^{6} + a^{2} d^{7}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{7} - a^{2} c^{6} d - 3 \, a^{2} c^{5} d^{2} + 3 \, a^{2} c^{4} d^{3} + 3 \, a^{2} c^{3} d^{4} - 3 \, a^{2} c^{2} d^{5} - a^{2} c d^{6} + a^{2} d^{7}\right)} f + {\left({\left(a^{2} c^{6} d - 2 \, a^{2} c^{5} d^{2} - a^{2} c^{4} d^{3} + 4 \, a^{2} c^{3} d^{4} - a^{2} c^{2} d^{5} - 2 \, a^{2} c d^{6} + a^{2} d^{7}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{7} - a^{2} c^{6} d - 3 \, a^{2} c^{5} d^{2} + 3 \, a^{2} c^{4} d^{3} + 3 \, a^{2} c^{3} d^{4} - 3 \, a^{2} c^{2} d^{5} - a^{2} c d^{6} + a^{2} d^{7}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{7} - a^{2} c^{6} d - 3 \, a^{2} c^{5} d^{2} + 3 \, a^{2} c^{4} d^{3} + 3 \, a^{2} c^{3} d^{4} - 3 \, a^{2} c^{2} d^{5} - a^{2} c d^{6} + a^{2} d^{7}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{c^{5} - c^{4} d - 2 \, c^{3} d^{2} + 2 \, c^{2} d^{3} + c d^{4} - d^{5} - {\left(c^{4} d - 6 \, c^{3} d^{2} - 11 \, c^{2} d^{3} + 6 \, c d^{4} + 10 \, d^{5}\right)} \cos\left(f x + e\right)^{3} + {\left(c^{5} - 5 \, c^{4} d - 8 \, c^{3} d^{2} + c^{2} d^{3} + 7 \, c d^{4} + 4 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + 3 \, {\left(6 \, c^{2} d^{2} + 10 \, c d^{3} + 4 \, d^{4} - {\left(3 \, c d^{3} + 2 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, c^{2} d^{2} + 8 \, c d^{3} + 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, c^{2} d^{2} + 5 \, c d^{3} + 2 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(6 \, c^{2} d^{2} + 10 \, c d^{3} + 4 \, d^{4} - {\left(3 \, c d^{3} + 2 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, c^{2} d^{2} + 5 \, c d^{3} + 2 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(2 \, c^{5} - 5 \, c^{4} d - 16 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 14 \, c d^{4} + 13 \, d^{5}\right)} \cos\left(f x + e\right) - {\left(c^{5} - c^{4} d - 2 \, c^{3} d^{2} + 2 \, c^{2} d^{3} + c d^{4} - d^{5} - {\left(c^{4} d - 6 \, c^{3} d^{2} - 11 \, c^{2} d^{3} + 6 \, c d^{4} + 10 \, d^{5}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{5} - 4 \, c^{4} d - 14 \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 13 \, c d^{4} + 14 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{3 \, {\left({\left(a^{2} c^{6} d - 2 \, a^{2} c^{5} d^{2} - a^{2} c^{4} d^{3} + 4 \, a^{2} c^{3} d^{4} - a^{2} c^{2} d^{5} - 2 \, a^{2} c d^{6} + a^{2} d^{7}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{2} c^{7} - 5 \, a^{2} c^{5} d^{2} + 2 \, a^{2} c^{4} d^{3} + 7 \, a^{2} c^{3} d^{4} - 4 \, a^{2} c^{2} d^{5} - 3 \, a^{2} c d^{6} + 2 \, a^{2} d^{7}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{7} - a^{2} c^{6} d - 3 \, a^{2} c^{5} d^{2} + 3 \, a^{2} c^{4} d^{3} + 3 \, a^{2} c^{3} d^{4} - 3 \, a^{2} c^{2} d^{5} - a^{2} c d^{6} + a^{2} d^{7}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{7} - a^{2} c^{6} d - 3 \, a^{2} c^{5} d^{2} + 3 \, a^{2} c^{4} d^{3} + 3 \, a^{2} c^{3} d^{4} - 3 \, a^{2} c^{2} d^{5} - a^{2} c d^{6} + a^{2} d^{7}\right)} f + {\left({\left(a^{2} c^{6} d - 2 \, a^{2} c^{5} d^{2} - a^{2} c^{4} d^{3} + 4 \, a^{2} c^{3} d^{4} - a^{2} c^{2} d^{5} - 2 \, a^{2} c d^{6} + a^{2} d^{7}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{7} - a^{2} c^{6} d - 3 \, a^{2} c^{5} d^{2} + 3 \, a^{2} c^{4} d^{3} + 3 \, a^{2} c^{3} d^{4} - 3 \, a^{2} c^{2} d^{5} - a^{2} c d^{6} + a^{2} d^{7}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{7} - a^{2} c^{6} d - 3 \, a^{2} c^{5} d^{2} + 3 \, a^{2} c^{4} d^{3} + 3 \, a^{2} c^{3} d^{4} - 3 \, a^{2} c^{2} d^{5} - a^{2} c d^{6} + a^{2} d^{7}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/6*(2*c^5 - 2*c^4*d - 4*c^3*d^2 + 4*c^2*d^3 + 2*c*d^4 - 2*d^5 - 2*(c^4*d - 6*c^3*d^2 - 11*c^2*d^3 + 6*c*d^4 + 10*d^5)*cos(f*x + e)^3 + 2*(c^5 - 5*c^4*d - 8*c^3*d^2 + c^2*d^3 + 7*c*d^4 + 4*d^5)*cos(f*x + e)^2 - 3*(6*c^2*d^2 + 10*c*d^3 + 4*d^4 - (3*c*d^3 + 2*d^4)*cos(f*x + e)^3 - (3*c^2*d^2 + 8*c*d^3 + 4*d^4)*cos(f*x + e)^2 + (3*c^2*d^2 + 5*c*d^3 + 2*d^4)*cos(f*x + e) + (6*c^2*d^2 + 10*c*d^3 + 4*d^4 - (3*c*d^3 + 2*d^4)*cos(f*x + e)^2 + (3*c^2*d^2 + 5*c*d^3 + 2*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(-c^2 + d^2)*log(-((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 - 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(2*c^5 - 5*c^4*d - 16*c^3*d^2 - 8*c^2*d^3 + 14*c*d^4 + 13*d^5)*cos(f*x + e) - 2*(c^5 - c^4*d - 2*c^3*d^2 + 2*c^2*d^3 + c*d^4 - d^5 - (c^4*d - 6*c^3*d^2 - 11*c^2*d^3 + 6*c*d^4 + 10*d^5)*cos(f*x + e)^2 - (c^5 - 4*c^4*d - 14*c^3*d^2 - 10*c^2*d^3 + 13*c*d^4 + 14*d^5)*cos(f*x + e))*sin(f*x + e))/((a^2*c^6*d - 2*a^2*c^5*d^2 - a^2*c^4*d^3 + 4*a^2*c^3*d^4 - a^2*c^2*d^5 - 2*a^2*c*d^6 + a^2*d^7)*f*cos(f*x + e)^3 + (a^2*c^7 - 5*a^2*c^5*d^2 + 2*a^2*c^4*d^3 + 7*a^2*c^3*d^4 - 4*a^2*c^2*d^5 - 3*a^2*c*d^6 + 2*a^2*d^7)*f*cos(f*x + e)^2 - (a^2*c^7 - a^2*c^6*d - 3*a^2*c^5*d^2 + 3*a^2*c^4*d^3 + 3*a^2*c^3*d^4 - 3*a^2*c^2*d^5 - a^2*c*d^6 + a^2*d^7)*f*cos(f*x + e) - 2*(a^2*c^7 - a^2*c^6*d - 3*a^2*c^5*d^2 + 3*a^2*c^4*d^3 + 3*a^2*c^3*d^4 - 3*a^2*c^2*d^5 - a^2*c*d^6 + a^2*d^7)*f + ((a^2*c^6*d - 2*a^2*c^5*d^2 - a^2*c^4*d^3 + 4*a^2*c^3*d^4 - a^2*c^2*d^5 - 2*a^2*c*d^6 + a^2*d^7)*f*cos(f*x + e)^2 - (a^2*c^7 - a^2*c^6*d - 3*a^2*c^5*d^2 + 3*a^2*c^4*d^3 + 3*a^2*c^3*d^4 - 3*a^2*c^2*d^5 - a^2*c*d^6 + a^2*d^7)*f*cos(f*x + e) - 2*(a^2*c^7 - a^2*c^6*d - 3*a^2*c^5*d^2 + 3*a^2*c^4*d^3 + 3*a^2*c^3*d^4 - 3*a^2*c^2*d^5 - a^2*c*d^6 + a^2*d^7)*f)*sin(f*x + e)), 1/3*(c^5 - c^4*d - 2*c^3*d^2 + 2*c^2*d^3 + c*d^4 - d^5 - (c^4*d - 6*c^3*d^2 - 11*c^2*d^3 + 6*c*d^4 + 10*d^5)*cos(f*x + e)^3 + (c^5 - 5*c^4*d - 8*c^3*d^2 + c^2*d^3 + 7*c*d^4 + 4*d^5)*cos(f*x + e)^2 + 3*(6*c^2*d^2 + 10*c*d^3 + 4*d^4 - (3*c*d^3 + 2*d^4)*cos(f*x + e)^3 - (3*c^2*d^2 + 8*c*d^3 + 4*d^4)*cos(f*x + e)^2 + (3*c^2*d^2 + 5*c*d^3 + 2*d^4)*cos(f*x + e) + (6*c^2*d^2 + 10*c*d^3 + 4*d^4 - (3*c*d^3 + 2*d^4)*cos(f*x + e)^2 + (3*c^2*d^2 + 5*c*d^3 + 2*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (2*c^5 - 5*c^4*d - 16*c^3*d^2 - 8*c^2*d^3 + 14*c*d^4 + 13*d^5)*cos(f*x + e) - (c^5 - c^4*d - 2*c^3*d^2 + 2*c^2*d^3 + c*d^4 - d^5 - (c^4*d - 6*c^3*d^2 - 11*c^2*d^3 + 6*c*d^4 + 10*d^5)*cos(f*x + e)^2 - (c^5 - 4*c^4*d - 14*c^3*d^2 - 10*c^2*d^3 + 13*c*d^4 + 14*d^5)*cos(f*x + e))*sin(f*x + e))/((a^2*c^6*d - 2*a^2*c^5*d^2 - a^2*c^4*d^3 + 4*a^2*c^3*d^4 - a^2*c^2*d^5 - 2*a^2*c*d^6 + a^2*d^7)*f*cos(f*x + e)^3 + (a^2*c^7 - 5*a^2*c^5*d^2 + 2*a^2*c^4*d^3 + 7*a^2*c^3*d^4 - 4*a^2*c^2*d^5 - 3*a^2*c*d^6 + 2*a^2*d^7)*f*cos(f*x + e)^2 - (a^2*c^7 - a^2*c^6*d - 3*a^2*c^5*d^2 + 3*a^2*c^4*d^3 + 3*a^2*c^3*d^4 - 3*a^2*c^2*d^5 - a^2*c*d^6 + a^2*d^7)*f*cos(f*x + e) - 2*(a^2*c^7 - a^2*c^6*d - 3*a^2*c^5*d^2 + 3*a^2*c^4*d^3 + 3*a^2*c^3*d^4 - 3*a^2*c^2*d^5 - a^2*c*d^6 + a^2*d^7)*f + ((a^2*c^6*d - 2*a^2*c^5*d^2 - a^2*c^4*d^3 + 4*a^2*c^3*d^4 - a^2*c^2*d^5 - 2*a^2*c*d^6 + a^2*d^7)*f*cos(f*x + e)^2 - (a^2*c^7 - a^2*c^6*d - 3*a^2*c^5*d^2 + 3*a^2*c^4*d^3 + 3*a^2*c^3*d^4 - 3*a^2*c^2*d^5 - a^2*c*d^6 + a^2*d^7)*f*cos(f*x + e) - 2*(a^2*c^7 - a^2*c^6*d - 3*a^2*c^5*d^2 + 3*a^2*c^4*d^3 + 3*a^2*c^3*d^4 - 3*a^2*c^2*d^5 - a^2*c*d^6 + a^2*d^7)*f)*sin(f*x + e))]","B",0
469,1,3540,0,1.577496," ","integrate(1/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[-\frac{4 \, c^{7} - 4 \, c^{6} d - 12 \, c^{5} d^{2} + 12 \, c^{4} d^{3} + 12 \, c^{3} d^{4} - 12 \, c^{2} d^{5} - 4 \, c d^{6} + 4 \, d^{7} - 2 \, {\left(2 \, c^{5} d^{2} - 16 \, c^{4} d^{3} - 61 \, c^{3} d^{4} - 16 \, c^{2} d^{5} + 59 \, c d^{6} + 32 \, d^{7}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(4 \, c^{6} d - 28 \, c^{5} d^{2} - 118 \, c^{4} d^{3} - 106 \, c^{3} d^{4} + 71 \, c^{2} d^{5} + 134 \, c d^{6} + 43 \, d^{7}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(2 \, c^{7} - 12 \, c^{6} d - 36 \, c^{5} d^{2} - 54 \, c^{4} d^{3} - 39 \, c^{3} d^{4} + 39 \, c^{2} d^{5} + 73 \, c d^{6} + 27 \, d^{7}\right)} \cos\left(f x + e\right)^{2} + 3 \, {\left(24 \, c^{4} d^{2} + 80 \, c^{3} d^{3} + 102 \, c^{2} d^{4} + 60 \, c d^{5} + 14 \, d^{6} + {\left(12 \, c^{2} d^{4} + 16 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(24 \, c^{3} d^{3} + 44 \, c^{2} d^{4} + 30 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(12 \, c^{4} d^{2} + 64 \, c^{3} d^{3} + 107 \, c^{2} d^{4} + 76 \, c d^{5} + 21 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + {\left(12 \, c^{4} d^{2} + 40 \, c^{3} d^{3} + 51 \, c^{2} d^{4} + 30 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right) + {\left(24 \, c^{4} d^{2} + 80 \, c^{3} d^{3} + 102 \, c^{2} d^{4} + 60 \, c d^{5} + 14 \, d^{6} - {\left(12 \, c^{2} d^{4} + 16 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(12 \, c^{3} d^{3} + 28 \, c^{2} d^{4} + 23 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + {\left(12 \, c^{4} d^{2} + 40 \, c^{3} d^{3} + 51 \, c^{2} d^{4} + 30 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 4 \, {\left(2 \, c^{7} - 5 \, c^{6} d - 36 \, c^{5} d^{2} - 75 \, c^{4} d^{3} - 39 \, c^{3} d^{4} + 60 \, c^{2} d^{5} + 73 \, c d^{6} + 20 \, d^{7}\right)} \cos\left(f x + e\right) - 2 \, {\left(2 \, c^{7} - 2 \, c^{6} d - 6 \, c^{5} d^{2} + 6 \, c^{4} d^{3} + 6 \, c^{3} d^{4} - 6 \, c^{2} d^{5} - 2 \, c d^{6} + 2 \, d^{7} + {\left(2 \, c^{5} d^{2} - 16 \, c^{4} d^{3} - 61 \, c^{3} d^{4} - 16 \, c^{2} d^{5} + 59 \, c d^{6} + 32 \, d^{7}\right)} \cos\left(f x + e\right)^{3} - {\left(4 \, c^{6} d - 30 \, c^{5} d^{2} - 102 \, c^{4} d^{3} - 45 \, c^{3} d^{4} + 87 \, c^{2} d^{5} + 75 \, c d^{6} + 11 \, d^{7}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{7} - 4 \, c^{6} d - 33 \, c^{5} d^{2} - 78 \, c^{4} d^{3} - 42 \, c^{3} d^{4} + 63 \, c^{2} d^{5} + 74 \, c d^{6} + 19 \, d^{7}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{12 \, {\left({\left(a^{2} c^{8} d^{2} - 2 \, a^{2} c^{7} d^{3} - 2 \, a^{2} c^{6} d^{4} + 6 \, a^{2} c^{5} d^{5} - 6 \, a^{2} c^{3} d^{7} + 2 \, a^{2} c^{2} d^{8} + 2 \, a^{2} c d^{9} - a^{2} d^{10}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{2} c^{9} d - 3 \, a^{2} c^{8} d^{2} - 6 \, a^{2} c^{7} d^{3} + 10 \, a^{2} c^{6} d^{4} + 6 \, a^{2} c^{5} d^{5} - 12 \, a^{2} c^{4} d^{6} - 2 \, a^{2} c^{3} d^{7} + 6 \, a^{2} c^{2} d^{8} - a^{2} d^{10}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{2} c^{10} + 2 \, a^{2} c^{9} d - 7 \, a^{2} c^{8} d^{2} - 8 \, a^{2} c^{7} d^{3} + 18 \, a^{2} c^{6} d^{4} + 12 \, a^{2} c^{5} d^{5} - 22 \, a^{2} c^{4} d^{6} - 8 \, a^{2} c^{3} d^{7} + 13 \, a^{2} c^{2} d^{8} + 2 \, a^{2} c d^{9} - 3 \, a^{2} d^{10}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} c^{10} - 5 \, a^{2} c^{8} d^{2} + 10 \, a^{2} c^{6} d^{4} - 10 \, a^{2} c^{4} d^{6} + 5 \, a^{2} c^{2} d^{8} - a^{2} d^{10}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{10} - 5 \, a^{2} c^{8} d^{2} + 10 \, a^{2} c^{6} d^{4} - 10 \, a^{2} c^{4} d^{6} + 5 \, a^{2} c^{2} d^{8} - a^{2} d^{10}\right)} f - {\left({\left(a^{2} c^{8} d^{2} - 2 \, a^{2} c^{7} d^{3} - 2 \, a^{2} c^{6} d^{4} + 6 \, a^{2} c^{5} d^{5} - 6 \, a^{2} c^{3} d^{7} + 2 \, a^{2} c^{2} d^{8} + 2 \, a^{2} c d^{9} - a^{2} d^{10}\right)} f \cos\left(f x + e\right)^{3} + 2 \, {\left(a^{2} c^{9} d - a^{2} c^{8} d^{2} - 4 \, a^{2} c^{7} d^{3} + 4 \, a^{2} c^{6} d^{4} + 6 \, a^{2} c^{5} d^{5} - 6 \, a^{2} c^{4} d^{6} - 4 \, a^{2} c^{3} d^{7} + 4 \, a^{2} c^{2} d^{8} + a^{2} c d^{9} - a^{2} d^{10}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{10} - 5 \, a^{2} c^{8} d^{2} + 10 \, a^{2} c^{6} d^{4} - 10 \, a^{2} c^{4} d^{6} + 5 \, a^{2} c^{2} d^{8} - a^{2} d^{10}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{10} - 5 \, a^{2} c^{8} d^{2} + 10 \, a^{2} c^{6} d^{4} - 10 \, a^{2} c^{4} d^{6} + 5 \, a^{2} c^{2} d^{8} - a^{2} d^{10}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{2 \, c^{7} - 2 \, c^{6} d - 6 \, c^{5} d^{2} + 6 \, c^{4} d^{3} + 6 \, c^{3} d^{4} - 6 \, c^{2} d^{5} - 2 \, c d^{6} + 2 \, d^{7} - {\left(2 \, c^{5} d^{2} - 16 \, c^{4} d^{3} - 61 \, c^{3} d^{4} - 16 \, c^{2} d^{5} + 59 \, c d^{6} + 32 \, d^{7}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, c^{6} d - 28 \, c^{5} d^{2} - 118 \, c^{4} d^{3} - 106 \, c^{3} d^{4} + 71 \, c^{2} d^{5} + 134 \, c d^{6} + 43 \, d^{7}\right)} \cos\left(f x + e\right)^{3} + {\left(2 \, c^{7} - 12 \, c^{6} d - 36 \, c^{5} d^{2} - 54 \, c^{4} d^{3} - 39 \, c^{3} d^{4} + 39 \, c^{2} d^{5} + 73 \, c d^{6} + 27 \, d^{7}\right)} \cos\left(f x + e\right)^{2} + 3 \, {\left(24 \, c^{4} d^{2} + 80 \, c^{3} d^{3} + 102 \, c^{2} d^{4} + 60 \, c d^{5} + 14 \, d^{6} + {\left(12 \, c^{2} d^{4} + 16 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(24 \, c^{3} d^{3} + 44 \, c^{2} d^{4} + 30 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(12 \, c^{4} d^{2} + 64 \, c^{3} d^{3} + 107 \, c^{2} d^{4} + 76 \, c d^{5} + 21 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + {\left(12 \, c^{4} d^{2} + 40 \, c^{3} d^{3} + 51 \, c^{2} d^{4} + 30 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right) + {\left(24 \, c^{4} d^{2} + 80 \, c^{3} d^{3} + 102 \, c^{2} d^{4} + 60 \, c d^{5} + 14 \, d^{6} - {\left(12 \, c^{2} d^{4} + 16 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(12 \, c^{3} d^{3} + 28 \, c^{2} d^{4} + 23 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + {\left(12 \, c^{4} d^{2} + 40 \, c^{3} d^{3} + 51 \, c^{2} d^{4} + 30 \, c d^{5} + 7 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + 2 \, {\left(2 \, c^{7} - 5 \, c^{6} d - 36 \, c^{5} d^{2} - 75 \, c^{4} d^{3} - 39 \, c^{3} d^{4} + 60 \, c^{2} d^{5} + 73 \, c d^{6} + 20 \, d^{7}\right)} \cos\left(f x + e\right) - {\left(2 \, c^{7} - 2 \, c^{6} d - 6 \, c^{5} d^{2} + 6 \, c^{4} d^{3} + 6 \, c^{3} d^{4} - 6 \, c^{2} d^{5} - 2 \, c d^{6} + 2 \, d^{7} + {\left(2 \, c^{5} d^{2} - 16 \, c^{4} d^{3} - 61 \, c^{3} d^{4} - 16 \, c^{2} d^{5} + 59 \, c d^{6} + 32 \, d^{7}\right)} \cos\left(f x + e\right)^{3} - {\left(4 \, c^{6} d - 30 \, c^{5} d^{2} - 102 \, c^{4} d^{3} - 45 \, c^{3} d^{4} + 87 \, c^{2} d^{5} + 75 \, c d^{6} + 11 \, d^{7}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{7} - 4 \, c^{6} d - 33 \, c^{5} d^{2} - 78 \, c^{4} d^{3} - 42 \, c^{3} d^{4} + 63 \, c^{2} d^{5} + 74 \, c d^{6} + 19 \, d^{7}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{6 \, {\left({\left(a^{2} c^{8} d^{2} - 2 \, a^{2} c^{7} d^{3} - 2 \, a^{2} c^{6} d^{4} + 6 \, a^{2} c^{5} d^{5} - 6 \, a^{2} c^{3} d^{7} + 2 \, a^{2} c^{2} d^{8} + 2 \, a^{2} c d^{9} - a^{2} d^{10}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{2} c^{9} d - 3 \, a^{2} c^{8} d^{2} - 6 \, a^{2} c^{7} d^{3} + 10 \, a^{2} c^{6} d^{4} + 6 \, a^{2} c^{5} d^{5} - 12 \, a^{2} c^{4} d^{6} - 2 \, a^{2} c^{3} d^{7} + 6 \, a^{2} c^{2} d^{8} - a^{2} d^{10}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{2} c^{10} + 2 \, a^{2} c^{9} d - 7 \, a^{2} c^{8} d^{2} - 8 \, a^{2} c^{7} d^{3} + 18 \, a^{2} c^{6} d^{4} + 12 \, a^{2} c^{5} d^{5} - 22 \, a^{2} c^{4} d^{6} - 8 \, a^{2} c^{3} d^{7} + 13 \, a^{2} c^{2} d^{8} + 2 \, a^{2} c d^{9} - 3 \, a^{2} d^{10}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} c^{10} - 5 \, a^{2} c^{8} d^{2} + 10 \, a^{2} c^{6} d^{4} - 10 \, a^{2} c^{4} d^{6} + 5 \, a^{2} c^{2} d^{8} - a^{2} d^{10}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{10} - 5 \, a^{2} c^{8} d^{2} + 10 \, a^{2} c^{6} d^{4} - 10 \, a^{2} c^{4} d^{6} + 5 \, a^{2} c^{2} d^{8} - a^{2} d^{10}\right)} f - {\left({\left(a^{2} c^{8} d^{2} - 2 \, a^{2} c^{7} d^{3} - 2 \, a^{2} c^{6} d^{4} + 6 \, a^{2} c^{5} d^{5} - 6 \, a^{2} c^{3} d^{7} + 2 \, a^{2} c^{2} d^{8} + 2 \, a^{2} c d^{9} - a^{2} d^{10}\right)} f \cos\left(f x + e\right)^{3} + 2 \, {\left(a^{2} c^{9} d - a^{2} c^{8} d^{2} - 4 \, a^{2} c^{7} d^{3} + 4 \, a^{2} c^{6} d^{4} + 6 \, a^{2} c^{5} d^{5} - 6 \, a^{2} c^{4} d^{6} - 4 \, a^{2} c^{3} d^{7} + 4 \, a^{2} c^{2} d^{8} + a^{2} c d^{9} - a^{2} d^{10}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{10} - 5 \, a^{2} c^{8} d^{2} + 10 \, a^{2} c^{6} d^{4} - 10 \, a^{2} c^{4} d^{6} + 5 \, a^{2} c^{2} d^{8} - a^{2} d^{10}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{10} - 5 \, a^{2} c^{8} d^{2} + 10 \, a^{2} c^{6} d^{4} - 10 \, a^{2} c^{4} d^{6} + 5 \, a^{2} c^{2} d^{8} - a^{2} d^{10}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/12*(4*c^7 - 4*c^6*d - 12*c^5*d^2 + 12*c^4*d^3 + 12*c^3*d^4 - 12*c^2*d^5 - 4*c*d^6 + 4*d^7 - 2*(2*c^5*d^2 - 16*c^4*d^3 - 61*c^3*d^4 - 16*c^2*d^5 + 59*c*d^6 + 32*d^7)*cos(f*x + e)^4 - 2*(4*c^6*d - 28*c^5*d^2 - 118*c^4*d^3 - 106*c^3*d^4 + 71*c^2*d^5 + 134*c*d^6 + 43*d^7)*cos(f*x + e)^3 + 2*(2*c^7 - 12*c^6*d - 36*c^5*d^2 - 54*c^4*d^3 - 39*c^3*d^4 + 39*c^2*d^5 + 73*c*d^6 + 27*d^7)*cos(f*x + e)^2 + 3*(24*c^4*d^2 + 80*c^3*d^3 + 102*c^2*d^4 + 60*c*d^5 + 14*d^6 + (12*c^2*d^4 + 16*c*d^5 + 7*d^6)*cos(f*x + e)^4 - (24*c^3*d^3 + 44*c^2*d^4 + 30*c*d^5 + 7*d^6)*cos(f*x + e)^3 - (12*c^4*d^2 + 64*c^3*d^3 + 107*c^2*d^4 + 76*c*d^5 + 21*d^6)*cos(f*x + e)^2 + (12*c^4*d^2 + 40*c^3*d^3 + 51*c^2*d^4 + 30*c*d^5 + 7*d^6)*cos(f*x + e) + (24*c^4*d^2 + 80*c^3*d^3 + 102*c^2*d^4 + 60*c*d^5 + 14*d^6 - (12*c^2*d^4 + 16*c*d^5 + 7*d^6)*cos(f*x + e)^3 - 2*(12*c^3*d^3 + 28*c^2*d^4 + 23*c*d^5 + 7*d^6)*cos(f*x + e)^2 + (12*c^4*d^2 + 40*c^3*d^3 + 51*c^2*d^4 + 30*c*d^5 + 7*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 4*(2*c^7 - 5*c^6*d - 36*c^5*d^2 - 75*c^4*d^3 - 39*c^3*d^4 + 60*c^2*d^5 + 73*c*d^6 + 20*d^7)*cos(f*x + e) - 2*(2*c^7 - 2*c^6*d - 6*c^5*d^2 + 6*c^4*d^3 + 6*c^3*d^4 - 6*c^2*d^5 - 2*c*d^6 + 2*d^7 + (2*c^5*d^2 - 16*c^4*d^3 - 61*c^3*d^4 - 16*c^2*d^5 + 59*c*d^6 + 32*d^7)*cos(f*x + e)^3 - (4*c^6*d - 30*c^5*d^2 - 102*c^4*d^3 - 45*c^3*d^4 + 87*c^2*d^5 + 75*c*d^6 + 11*d^7)*cos(f*x + e)^2 - 2*(c^7 - 4*c^6*d - 33*c^5*d^2 - 78*c^4*d^3 - 42*c^3*d^4 + 63*c^2*d^5 + 74*c*d^6 + 19*d^7)*cos(f*x + e))*sin(f*x + e))/((a^2*c^8*d^2 - 2*a^2*c^7*d^3 - 2*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^3*d^7 + 2*a^2*c^2*d^8 + 2*a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e)^4 - (2*a^2*c^9*d - 3*a^2*c^8*d^2 - 6*a^2*c^7*d^3 + 10*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 12*a^2*c^4*d^6 - 2*a^2*c^3*d^7 + 6*a^2*c^2*d^8 - a^2*d^10)*f*cos(f*x + e)^3 - (a^2*c^10 + 2*a^2*c^9*d - 7*a^2*c^8*d^2 - 8*a^2*c^7*d^3 + 18*a^2*c^6*d^4 + 12*a^2*c^5*d^5 - 22*a^2*c^4*d^6 - 8*a^2*c^3*d^7 + 13*a^2*c^2*d^8 + 2*a^2*c*d^9 - 3*a^2*d^10)*f*cos(f*x + e)^2 + (a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f*cos(f*x + e) + 2*(a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f - ((a^2*c^8*d^2 - 2*a^2*c^7*d^3 - 2*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^3*d^7 + 2*a^2*c^2*d^8 + 2*a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e)^3 + 2*(a^2*c^9*d - a^2*c^8*d^2 - 4*a^2*c^7*d^3 + 4*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^4*d^6 - 4*a^2*c^3*d^7 + 4*a^2*c^2*d^8 + a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e)^2 - (a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f*cos(f*x + e) - 2*(a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f)*sin(f*x + e)), -1/6*(2*c^7 - 2*c^6*d - 6*c^5*d^2 + 6*c^4*d^3 + 6*c^3*d^4 - 6*c^2*d^5 - 2*c*d^6 + 2*d^7 - (2*c^5*d^2 - 16*c^4*d^3 - 61*c^3*d^4 - 16*c^2*d^5 + 59*c*d^6 + 32*d^7)*cos(f*x + e)^4 - (4*c^6*d - 28*c^5*d^2 - 118*c^4*d^3 - 106*c^3*d^4 + 71*c^2*d^5 + 134*c*d^6 + 43*d^7)*cos(f*x + e)^3 + (2*c^7 - 12*c^6*d - 36*c^5*d^2 - 54*c^4*d^3 - 39*c^3*d^4 + 39*c^2*d^5 + 73*c*d^6 + 27*d^7)*cos(f*x + e)^2 + 3*(24*c^4*d^2 + 80*c^3*d^3 + 102*c^2*d^4 + 60*c*d^5 + 14*d^6 + (12*c^2*d^4 + 16*c*d^5 + 7*d^6)*cos(f*x + e)^4 - (24*c^3*d^3 + 44*c^2*d^4 + 30*c*d^5 + 7*d^6)*cos(f*x + e)^3 - (12*c^4*d^2 + 64*c^3*d^3 + 107*c^2*d^4 + 76*c*d^5 + 21*d^6)*cos(f*x + e)^2 + (12*c^4*d^2 + 40*c^3*d^3 + 51*c^2*d^4 + 30*c*d^5 + 7*d^6)*cos(f*x + e) + (24*c^4*d^2 + 80*c^3*d^3 + 102*c^2*d^4 + 60*c*d^5 + 14*d^6 - (12*c^2*d^4 + 16*c*d^5 + 7*d^6)*cos(f*x + e)^3 - 2*(12*c^3*d^3 + 28*c^2*d^4 + 23*c*d^5 + 7*d^6)*cos(f*x + e)^2 + (12*c^4*d^2 + 40*c^3*d^3 + 51*c^2*d^4 + 30*c*d^5 + 7*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + 2*(2*c^7 - 5*c^6*d - 36*c^5*d^2 - 75*c^4*d^3 - 39*c^3*d^4 + 60*c^2*d^5 + 73*c*d^6 + 20*d^7)*cos(f*x + e) - (2*c^7 - 2*c^6*d - 6*c^5*d^2 + 6*c^4*d^3 + 6*c^3*d^4 - 6*c^2*d^5 - 2*c*d^6 + 2*d^7 + (2*c^5*d^2 - 16*c^4*d^3 - 61*c^3*d^4 - 16*c^2*d^5 + 59*c*d^6 + 32*d^7)*cos(f*x + e)^3 - (4*c^6*d - 30*c^5*d^2 - 102*c^4*d^3 - 45*c^3*d^4 + 87*c^2*d^5 + 75*c*d^6 + 11*d^7)*cos(f*x + e)^2 - 2*(c^7 - 4*c^6*d - 33*c^5*d^2 - 78*c^4*d^3 - 42*c^3*d^4 + 63*c^2*d^5 + 74*c*d^6 + 19*d^7)*cos(f*x + e))*sin(f*x + e))/((a^2*c^8*d^2 - 2*a^2*c^7*d^3 - 2*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^3*d^7 + 2*a^2*c^2*d^8 + 2*a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e)^4 - (2*a^2*c^9*d - 3*a^2*c^8*d^2 - 6*a^2*c^7*d^3 + 10*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 12*a^2*c^4*d^6 - 2*a^2*c^3*d^7 + 6*a^2*c^2*d^8 - a^2*d^10)*f*cos(f*x + e)^3 - (a^2*c^10 + 2*a^2*c^9*d - 7*a^2*c^8*d^2 - 8*a^2*c^7*d^3 + 18*a^2*c^6*d^4 + 12*a^2*c^5*d^5 - 22*a^2*c^4*d^6 - 8*a^2*c^3*d^7 + 13*a^2*c^2*d^8 + 2*a^2*c*d^9 - 3*a^2*d^10)*f*cos(f*x + e)^2 + (a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f*cos(f*x + e) + 2*(a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f - ((a^2*c^8*d^2 - 2*a^2*c^7*d^3 - 2*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^3*d^7 + 2*a^2*c^2*d^8 + 2*a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e)^3 + 2*(a^2*c^9*d - a^2*c^8*d^2 - 4*a^2*c^7*d^3 + 4*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^4*d^6 - 4*a^2*c^3*d^7 + 4*a^2*c^2*d^8 + a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e)^2 - (a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f*cos(f*x + e) - 2*(a^2*c^10 - 5*a^2*c^8*d^2 + 10*a^2*c^6*d^4 - 10*a^2*c^4*d^6 + 5*a^2*c^2*d^8 - a^2*d^10)*f)*sin(f*x + e))]","B",0
470,1,823,0,1.255053," ","integrate((c+d*sin(f*x+e))^6/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{10 \, d^{6} \cos\left(f x + e\right)^{6} + 6 \, c^{6} - 36 \, c^{5} d + 90 \, c^{4} d^{2} - 120 \, c^{3} d^{3} + 90 \, c^{2} d^{4} - 36 \, c d^{5} + 6 \, d^{6} + 15 \, {\left(6 \, c d^{5} - d^{6}\right)} \cos\left(f x + e\right)^{5} - 10 \, {\left(45 \, c^{2} d^{4} - 36 \, c d^{5} + 14 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, c^{6} + 36 \, c^{5} d + 210 \, c^{4} d^{2} - 1280 \, c^{3} d^{3} + 3510 \, c^{2} d^{4} - 2694 \, c d^{5} + 839 \, d^{6} - 15 \, {\left(40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right)} f x\right)} \cos\left(f x + e\right)^{3} - 60 \, {\left(40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right)} f x + {\left(8 \, c^{6} + 72 \, c^{5} d - 30 \, c^{4} d^{2} - 760 \, c^{3} d^{3} + 2520 \, c^{2} d^{4} - 2148 \, c d^{5} + 668 \, d^{6} + 45 \, {\left(40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right)} f x\right)} \cos\left(f x + e\right)^{2} + 6 \, {\left(3 \, c^{6} + 12 \, c^{5} d + 45 \, c^{4} d^{2} - 360 \, c^{3} d^{3} + 945 \, c^{2} d^{4} - 768 \, c d^{5} + 233 \, d^{6} - 5 \, {\left(40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right)} f x\right)} \cos\left(f x + e\right) + {\left(10 \, d^{6} \cos\left(f x + e\right)^{5} - 6 \, c^{6} + 36 \, c^{5} d - 90 \, c^{4} d^{2} + 120 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 36 \, c d^{5} - 6 \, d^{6} - 5 \, {\left(18 \, c d^{5} - 5 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - 5 \, {\left(90 \, c^{2} d^{4} - 54 \, c d^{5} + 23 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - 60 \, {\left(40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right)} f x + {\left(4 \, c^{6} + 36 \, c^{5} d + 210 \, c^{4} d^{2} - 1280 \, c^{3} d^{3} + 3060 \, c^{2} d^{4} - 2424 \, c d^{5} + 724 \, d^{6} + 15 \, {\left(40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right)} f x\right)} \cos\left(f x + e\right)^{2} + 6 \, {\left(2 \, c^{6} + 18 \, c^{5} d + 30 \, c^{4} d^{2} - 340 \, c^{3} d^{3} + 930 \, c^{2} d^{4} - 762 \, c d^{5} + 232 \, d^{6} - 5 \, {\left(40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right)} f x\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{30 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/30*(10*d^6*cos(f*x + e)^6 + 6*c^6 - 36*c^5*d + 90*c^4*d^2 - 120*c^3*d^3 + 90*c^2*d^4 - 36*c*d^5 + 6*d^6 + 15*(6*c*d^5 - d^6)*cos(f*x + e)^5 - 10*(45*c^2*d^4 - 36*c*d^5 + 14*d^6)*cos(f*x + e)^4 - (4*c^6 + 36*c^5*d + 210*c^4*d^2 - 1280*c^3*d^3 + 3510*c^2*d^4 - 2694*c*d^5 + 839*d^6 - 15*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*cos(f*x + e)^3 - 60*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x + (8*c^6 + 72*c^5*d - 30*c^4*d^2 - 760*c^3*d^3 + 2520*c^2*d^4 - 2148*c*d^5 + 668*d^6 + 45*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*cos(f*x + e)^2 + 6*(3*c^6 + 12*c^5*d + 45*c^4*d^2 - 360*c^3*d^3 + 945*c^2*d^4 - 768*c*d^5 + 233*d^6 - 5*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*cos(f*x + e) + (10*d^6*cos(f*x + e)^5 - 6*c^6 + 36*c^5*d - 90*c^4*d^2 + 120*c^3*d^3 - 90*c^2*d^4 + 36*c*d^5 - 6*d^6 - 5*(18*c*d^5 - 5*d^6)*cos(f*x + e)^4 - 5*(90*c^2*d^4 - 54*c*d^5 + 23*d^6)*cos(f*x + e)^3 - 60*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x + (4*c^6 + 36*c^5*d + 210*c^4*d^2 - 1280*c^3*d^3 + 3060*c^2*d^4 - 2424*c*d^5 + 724*d^6 + 15*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*cos(f*x + e)^2 + 6*(2*c^6 + 18*c^5*d + 30*c^4*d^2 - 340*c^3*d^3 + 930*c^2*d^4 - 762*c*d^5 + 232*d^6 - 5*(40*c^3*d^3 - 90*c^2*d^4 + 78*c*d^5 - 23*d^6)*f*x)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
471,1,653,0,0.973509," ","integrate((c+d*sin(f*x+e))^5/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{15 \, d^{5} \cos\left(f x + e\right)^{5} + 6 \, c^{5} - 30 \, c^{4} d + 60 \, c^{3} d^{2} - 60 \, c^{2} d^{3} + 30 \, c d^{4} - 6 \, d^{5} - 30 \, {\left(5 \, c d^{4} - 2 \, d^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, c^{5} + 30 \, c^{4} d + 140 \, c^{3} d^{2} - 640 \, c^{2} d^{3} + 1170 \, c d^{4} - 449 \, d^{5} - 15 \, {\left(20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right)} f x\right)} \cos\left(f x + e\right)^{3} - 60 \, {\left(20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right)} f x + {\left(8 \, c^{5} + 60 \, c^{4} d - 20 \, c^{3} d^{2} - 380 \, c^{2} d^{3} + 840 \, c d^{4} - 358 \, d^{5} + 45 \, {\left(20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right)} f x\right)} \cos\left(f x + e\right)^{2} + 6 \, {\left(3 \, c^{5} + 10 \, c^{4} d + 30 \, c^{3} d^{2} - 180 \, c^{2} d^{3} + 315 \, c d^{4} - 128 \, d^{5} - 5 \, {\left(20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right)} f x\right)} \cos\left(f x + e\right) - {\left(15 \, d^{5} \cos\left(f x + e\right)^{4} + 6 \, c^{5} - 30 \, c^{4} d + 60 \, c^{3} d^{2} - 60 \, c^{2} d^{3} + 30 \, c d^{4} - 6 \, d^{5} + 15 \, {\left(10 \, c d^{4} - 3 \, d^{5}\right)} \cos\left(f x + e\right)^{3} + 60 \, {\left(20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right)} f x - {\left(4 \, c^{5} + 30 \, c^{4} d + 140 \, c^{3} d^{2} - 640 \, c^{2} d^{3} + 1020 \, c d^{4} - 404 \, d^{5} + 15 \, {\left(20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right)} f x\right)} \cos\left(f x + e\right)^{2} - 6 \, {\left(2 \, c^{5} + 15 \, c^{4} d + 20 \, c^{3} d^{2} - 170 \, c^{2} d^{3} + 310 \, c d^{4} - 127 \, d^{5} - 5 \, {\left(20 \, c^{2} d^{3} - 30 \, c d^{4} + 13 \, d^{5}\right)} f x\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{30 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/30*(15*d^5*cos(f*x + e)^5 + 6*c^5 - 30*c^4*d + 60*c^3*d^2 - 60*c^2*d^3 + 30*c*d^4 - 6*d^5 - 30*(5*c*d^4 - 2*d^5)*cos(f*x + e)^4 - (4*c^5 + 30*c^4*d + 140*c^3*d^2 - 640*c^2*d^3 + 1170*c*d^4 - 449*d^5 - 15*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x)*cos(f*x + e)^3 - 60*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x + (8*c^5 + 60*c^4*d - 20*c^3*d^2 - 380*c^2*d^3 + 840*c*d^4 - 358*d^5 + 45*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x)*cos(f*x + e)^2 + 6*(3*c^5 + 10*c^4*d + 30*c^3*d^2 - 180*c^2*d^3 + 315*c*d^4 - 128*d^5 - 5*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x)*cos(f*x + e) - (15*d^5*cos(f*x + e)^4 + 6*c^5 - 30*c^4*d + 60*c^3*d^2 - 60*c^2*d^3 + 30*c*d^4 - 6*d^5 + 15*(10*c*d^4 - 3*d^5)*cos(f*x + e)^3 + 60*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x - (4*c^5 + 30*c^4*d + 140*c^3*d^2 - 640*c^2*d^3 + 1020*c*d^4 - 404*d^5 + 15*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x)*cos(f*x + e)^2 - 6*(2*c^5 + 15*c^4*d + 20*c^3*d^2 - 170*c^2*d^3 + 310*c*d^4 - 127*d^5 - 5*(20*c^2*d^3 - 30*c*d^4 + 13*d^5)*f*x)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
472,1,494,0,1.052322," ","integrate((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{15 \, d^{4} \cos\left(f x + e\right)^{4} - 3 \, c^{4} + 12 \, c^{3} d - 18 \, c^{2} d^{2} + 12 \, c d^{3} - 3 \, d^{4} + {\left(2 \, c^{4} + 12 \, c^{3} d + 42 \, c^{2} d^{2} - 128 \, c d^{3} + 117 \, d^{4} - 15 \, {\left(4 \, c d^{3} - 3 \, d^{4}\right)} f x\right)} \cos\left(f x + e\right)^{3} + 60 \, {\left(4 \, c d^{3} - 3 \, d^{4}\right)} f x - {\left(4 \, c^{4} + 24 \, c^{3} d - 6 \, c^{2} d^{2} - 76 \, c d^{3} + 84 \, d^{4} + 45 \, {\left(4 \, c d^{3} - 3 \, d^{4}\right)} f x\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(3 \, c^{4} + 8 \, c^{3} d + 18 \, c^{2} d^{2} - 72 \, c d^{3} + 63 \, d^{4} - 10 \, {\left(4 \, c d^{3} - 3 \, d^{4}\right)} f x\right)} \cos\left(f x + e\right) + {\left(15 \, d^{4} \cos\left(f x + e\right)^{3} + 3 \, c^{4} - 12 \, c^{3} d + 18 \, c^{2} d^{2} - 12 \, c d^{3} + 3 \, d^{4} + 60 \, {\left(4 \, c d^{3} - 3 \, d^{4}\right)} f x - {\left(2 \, c^{4} + 12 \, c^{3} d + 42 \, c^{2} d^{2} - 128 \, c d^{3} + 102 \, d^{4} + 15 \, {\left(4 \, c d^{3} - 3 \, d^{4}\right)} f x\right)} \cos\left(f x + e\right)^{2} - 6 \, {\left(c^{4} + 6 \, c^{3} d + 6 \, c^{2} d^{2} - 34 \, c d^{3} + 31 \, d^{4} - 5 \, {\left(4 \, c d^{3} - 3 \, d^{4}\right)} f x\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/15*(15*d^4*cos(f*x + e)^4 - 3*c^4 + 12*c^3*d - 18*c^2*d^2 + 12*c*d^3 - 3*d^4 + (2*c^4 + 12*c^3*d + 42*c^2*d^2 - 128*c*d^3 + 117*d^4 - 15*(4*c*d^3 - 3*d^4)*f*x)*cos(f*x + e)^3 + 60*(4*c*d^3 - 3*d^4)*f*x - (4*c^4 + 24*c^3*d - 6*c^2*d^2 - 76*c*d^3 + 84*d^4 + 45*(4*c*d^3 - 3*d^4)*f*x)*cos(f*x + e)^2 - 3*(3*c^4 + 8*c^3*d + 18*c^2*d^2 - 72*c*d^3 + 63*d^4 - 10*(4*c*d^3 - 3*d^4)*f*x)*cos(f*x + e) + (15*d^4*cos(f*x + e)^3 + 3*c^4 - 12*c^3*d + 18*c^2*d^2 - 12*c*d^3 + 3*d^4 + 60*(4*c*d^3 - 3*d^4)*f*x - (2*c^4 + 12*c^3*d + 42*c^2*d^2 - 128*c*d^3 + 102*d^4 + 15*(4*c*d^3 - 3*d^4)*f*x)*cos(f*x + e)^2 - 6*(c^4 + 6*c^3*d + 6*c^2*d^2 - 34*c*d^3 + 31*d^4 - 5*(4*c*d^3 - 3*d^4)*f*x)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
473,1,352,0,1.103501," ","integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{60 \, d^{3} f x - {\left(15 \, d^{3} f x - 2 \, c^{3} - 9 \, c^{2} d - 21 \, c d^{2} + 32 \, d^{3}\right)} \cos\left(f x + e\right)^{3} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} - {\left(45 \, d^{3} f x + 4 \, c^{3} + 18 \, c^{2} d - 3 \, c d^{2} - 19 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + 3 \, {\left(10 \, d^{3} f x - 3 \, c^{3} - 6 \, c^{2} d - 9 \, c d^{2} + 18 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(60 \, d^{3} f x + 3 \, c^{3} - 9 \, c^{2} d + 9 \, c d^{2} - 3 \, d^{3} - {\left(15 \, d^{3} f x + 2 \, c^{3} + 9 \, c^{2} d + 21 \, c d^{2} - 32 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + 3 \, {\left(10 \, d^{3} f x - 2 \, c^{3} - 9 \, c^{2} d - 6 \, c d^{2} + 17 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/15*(60*d^3*f*x - (15*d^3*f*x - 2*c^3 - 9*c^2*d - 21*c*d^2 + 32*d^3)*cos(f*x + e)^3 - 3*c^3 + 9*c^2*d - 9*c*d^2 + 3*d^3 - (45*d^3*f*x + 4*c^3 + 18*c^2*d - 3*c*d^2 - 19*d^3)*cos(f*x + e)^2 + 3*(10*d^3*f*x - 3*c^3 - 6*c^2*d - 9*c*d^2 + 18*d^3)*cos(f*x + e) + (60*d^3*f*x + 3*c^3 - 9*c^2*d + 9*c*d^2 - 3*d^3 - (15*d^3*f*x + 2*c^3 + 9*c^2*d + 21*c*d^2 - 32*d^3)*cos(f*x + e)^2 + 3*(10*d^3*f*x - 2*c^3 - 9*c^2*d - 6*c*d^2 + 17*d^3)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
474,1,242,0,1.074269," ","integrate((c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{{\left(2 \, c^{2} + 6 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(4 \, c^{2} + 12 \, c d - d^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, c^{2} + 6 \, c d - 3 \, d^{2} - 3 \, {\left(3 \, c^{2} + 4 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right) - {\left({\left(2 \, c^{2} + 6 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 3 \, c^{2} + 6 \, c d - 3 \, d^{2} + 6 \, {\left(c^{2} + 3 \, c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/15*((2*c^2 + 6*c*d + 7*d^2)*cos(f*x + e)^3 - (4*c^2 + 12*c*d - d^2)*cos(f*x + e)^2 - 3*c^2 + 6*c*d - 3*d^2 - 3*(3*c^2 + 4*c*d + 3*d^2)*cos(f*x + e) - ((2*c^2 + 6*c*d + 7*d^2)*cos(f*x + e)^2 - 3*c^2 + 6*c*d - 3*d^2 + 6*(c^2 + 3*c*d + d^2)*cos(f*x + e))*sin(f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
475,1,190,0,0.885208," ","integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{{\left(2 \, c + 3 \, d\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(2 \, c + 3 \, d\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(3 \, c + 2 \, d\right)} \cos\left(f x + e\right) - {\left({\left(2 \, c + 3 \, d\right)} \cos\left(f x + e\right)^{2} + 3 \, {\left(2 \, c + 3 \, d\right)} \cos\left(f x + e\right) - 3 \, c + 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + 3 \, d}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/15*((2*c + 3*d)*cos(f*x + e)^3 - 2*(2*c + 3*d)*cos(f*x + e)^2 - 3*(3*c + 2*d)*cos(f*x + e) - ((2*c + 3*d)*cos(f*x + e)^2 + 3*(2*c + 3*d)*cos(f*x + e) - 3*c + 3*d)*sin(f*x + e) - 3*c + 3*d)/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","A",0
476,1,147,0,1.060992," ","integrate(1/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{2 \, \cos\left(f x + e\right)^{3} - 4 \, \cos\left(f x + e\right)^{2} - {\left(2 \, \cos\left(f x + e\right)^{2} + 6 \, \cos\left(f x + e\right) - 3\right)} \sin\left(f x + e\right) - 9 \, \cos\left(f x + e\right) - 3}{15 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/15*(2*cos(f*x + e)^3 - 4*cos(f*x + e)^2 - (2*cos(f*x + e)^2 + 6*cos(f*x + e) - 3)*sin(f*x + e) - 9*cos(f*x + e) - 3)/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","A",0
477,1,1744,0,1.324238," ","integrate(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[\frac{6 \, c^{4} - 12 \, c^{3} d + 12 \, c d^{3} - 6 \, d^{4} - 2 \, {\left(2 \, c^{4} - 9 \, c^{3} d + 20 \, c^{2} d^{2} + 9 \, c d^{3} - 22 \, d^{4}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(4 \, c^{4} - 18 \, c^{3} d + 25 \, c^{2} d^{2} + 18 \, c d^{3} - 29 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 15 \, {\left(d^{3} \cos\left(f x + e\right)^{3} + 3 \, d^{3} \cos\left(f x + e\right)^{2} - 2 \, d^{3} \cos\left(f x + e\right) - 4 \, d^{3} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 2 \, d^{3} \cos\left(f x + e\right) - 4 \, d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 6 \, {\left(3 \, c^{4} - 11 \, c^{3} d + 15 \, c^{2} d^{2} + 11 \, c d^{3} - 18 \, d^{4}\right)} \cos\left(f x + e\right) - 2 \, {\left(3 \, c^{4} - 6 \, c^{3} d + 6 \, c d^{3} - 3 \, d^{4} - {\left(2 \, c^{4} - 9 \, c^{3} d + 20 \, c^{2} d^{2} + 9 \, c d^{3} - 22 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(2 \, c^{4} - 9 \, c^{3} d + 15 \, c^{2} d^{2} + 9 \, c d^{3} - 17 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{30 \, {\left({\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f + {\left({\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{3 \, c^{4} - 6 \, c^{3} d + 6 \, c d^{3} - 3 \, d^{4} - {\left(2 \, c^{4} - 9 \, c^{3} d + 20 \, c^{2} d^{2} + 9 \, c d^{3} - 22 \, d^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(4 \, c^{4} - 18 \, c^{3} d + 25 \, c^{2} d^{2} + 18 \, c d^{3} - 29 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 15 \, {\left(d^{3} \cos\left(f x + e\right)^{3} + 3 \, d^{3} \cos\left(f x + e\right)^{2} - 2 \, d^{3} \cos\left(f x + e\right) - 4 \, d^{3} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 2 \, d^{3} \cos\left(f x + e\right) - 4 \, d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + 3 \, {\left(3 \, c^{4} - 11 \, c^{3} d + 15 \, c^{2} d^{2} + 11 \, c d^{3} - 18 \, d^{4}\right)} \cos\left(f x + e\right) - {\left(3 \, c^{4} - 6 \, c^{3} d + 6 \, c d^{3} - 3 \, d^{4} - {\left(2 \, c^{4} - 9 \, c^{3} d + 20 \, c^{2} d^{2} + 9 \, c d^{3} - 22 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(2 \, c^{4} - 9 \, c^{3} d + 15 \, c^{2} d^{2} + 9 \, c d^{3} - 17 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{15 \, {\left({\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f + {\left({\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{5} - 3 \, a^{3} c^{4} d + 2 \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} - 3 \, a^{3} c d^{4} + a^{3} d^{5}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/30*(6*c^4 - 12*c^3*d + 12*c*d^3 - 6*d^4 - 2*(2*c^4 - 9*c^3*d + 20*c^2*d^2 + 9*c*d^3 - 22*d^4)*cos(f*x + e)^3 + 2*(4*c^4 - 18*c^3*d + 25*c^2*d^2 + 18*c*d^3 - 29*d^4)*cos(f*x + e)^2 + 15*(d^3*cos(f*x + e)^3 + 3*d^3*cos(f*x + e)^2 - 2*d^3*cos(f*x + e) - 4*d^3 + (d^3*cos(f*x + e)^2 - 2*d^3*cos(f*x + e) - 4*d^3)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 6*(3*c^4 - 11*c^3*d + 15*c^2*d^2 + 11*c*d^3 - 18*d^4)*cos(f*x + e) - 2*(3*c^4 - 6*c^3*d + 6*c*d^3 - 3*d^4 - (2*c^4 - 9*c^3*d + 20*c^2*d^2 + 9*c*d^3 - 22*d^4)*cos(f*x + e)^2 - 3*(2*c^4 - 9*c^3*d + 15*c^2*d^2 + 9*c*d^3 - 17*d^4)*cos(f*x + e))*sin(f*x + e))/((a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f*cos(f*x + e)^3 + 3*(a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f*cos(f*x + e)^2 - 2*(a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f*cos(f*x + e) - 4*(a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f + ((a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f*cos(f*x + e)^2 - 2*(a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f*cos(f*x + e) - 4*(a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f)*sin(f*x + e)), 1/15*(3*c^4 - 6*c^3*d + 6*c*d^3 - 3*d^4 - (2*c^4 - 9*c^3*d + 20*c^2*d^2 + 9*c*d^3 - 22*d^4)*cos(f*x + e)^3 + (4*c^4 - 18*c^3*d + 25*c^2*d^2 + 18*c*d^3 - 29*d^4)*cos(f*x + e)^2 + 15*(d^3*cos(f*x + e)^3 + 3*d^3*cos(f*x + e)^2 - 2*d^3*cos(f*x + e) - 4*d^3 + (d^3*cos(f*x + e)^2 - 2*d^3*cos(f*x + e) - 4*d^3)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + 3*(3*c^4 - 11*c^3*d + 15*c^2*d^2 + 11*c*d^3 - 18*d^4)*cos(f*x + e) - (3*c^4 - 6*c^3*d + 6*c*d^3 - 3*d^4 - (2*c^4 - 9*c^3*d + 20*c^2*d^2 + 9*c*d^3 - 22*d^4)*cos(f*x + e)^2 - 3*(2*c^4 - 9*c^3*d + 15*c^2*d^2 + 9*c*d^3 - 17*d^4)*cos(f*x + e))*sin(f*x + e))/((a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f*cos(f*x + e)^3 + 3*(a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f*cos(f*x + e)^2 - 2*(a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f*cos(f*x + e) - 4*(a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f + ((a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f*cos(f*x + e)^2 - 2*(a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f*cos(f*x + e) - 4*(a^3*c^5 - 3*a^3*c^4*d + 2*a^3*c^3*d^2 + 2*a^3*c^2*d^3 - 3*a^3*c*d^4 + a^3*d^5)*f)*sin(f*x + e))]","B",0
478,1,3235,0,1.250826," ","integrate(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[-\frac{6 \, c^{6} - 12 \, c^{5} d - 6 \, c^{4} d^{2} + 24 \, c^{3} d^{3} - 6 \, c^{2} d^{4} - 12 \, c d^{5} + 6 \, d^{6} - 2 \, {\left(2 \, c^{5} d - 12 \, c^{4} d^{2} + 41 \, c^{3} d^{3} + 84 \, c^{2} d^{4} - 43 \, c d^{5} - 72 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(2 \, c^{6} - 6 \, c^{5} d + 5 \, c^{4} d^{2} + 147 \, c^{3} d^{3} + 164 \, c^{2} d^{4} - 141 \, c d^{5} - 171 \, d^{6}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(4 \, c^{6} - 19 \, c^{5} d + 22 \, c^{4} d^{2} + 128 \, c^{3} d^{3} + 64 \, c^{2} d^{4} - 109 \, c d^{5} - 90 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + 15 \, {\left(16 \, c^{2} d^{3} + 28 \, c d^{4} + 12 \, d^{5} + {\left(4 \, c d^{4} + 3 \, d^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, c^{2} d^{3} + 11 \, c d^{4} + 6 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(12 \, c^{2} d^{3} + 29 \, c d^{4} + 15 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(4 \, c^{2} d^{3} + 7 \, c d^{4} + 3 \, d^{5}\right)} \cos\left(f x + e\right) + {\left(16 \, c^{2} d^{3} + 28 \, c d^{4} + 12 \, d^{5} - {\left(4 \, c d^{4} + 3 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(4 \, c^{2} d^{3} + 15 \, c d^{4} + 9 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(4 \, c^{2} d^{3} + 7 \, c d^{4} + 3 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(-\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} - 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 6 \, {\left(3 \, c^{6} - 11 \, c^{5} d + 12 \, c^{4} d^{2} + 82 \, c^{3} d^{3} + 47 \, c^{2} d^{4} - 71 \, c d^{5} - 62 \, d^{6}\right)} \cos\left(f x + e\right) - 2 \, {\left(3 \, c^{6} - 6 \, c^{5} d - 3 \, c^{4} d^{2} + 12 \, c^{3} d^{3} - 3 \, c^{2} d^{4} - 6 \, c d^{5} + 3 \, d^{6} + {\left(2 \, c^{5} d - 12 \, c^{4} d^{2} + 41 \, c^{3} d^{3} + 84 \, c^{2} d^{4} - 43 \, c d^{5} - 72 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(2 \, c^{6} - 8 \, c^{5} d + 17 \, c^{4} d^{2} + 106 \, c^{3} d^{3} + 80 \, c^{2} d^{4} - 98 \, c d^{5} - 99 \, d^{6}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(2 \, c^{6} - 9 \, c^{5} d + 13 \, c^{4} d^{2} + 78 \, c^{3} d^{3} + 48 \, c^{2} d^{4} - 69 \, c d^{5} - 63 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{30 \, {\left({\left(a^{3} c^{7} d - 3 \, a^{3} c^{6} d^{2} + a^{3} c^{5} d^{3} + 5 \, a^{3} c^{4} d^{4} - 5 \, a^{3} c^{3} d^{5} - a^{3} c^{2} d^{6} + 3 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{8} - a^{3} c^{7} d - 5 \, a^{3} c^{6} d^{2} + 7 \, a^{3} c^{5} d^{3} + 5 \, a^{3} c^{4} d^{4} - 11 \, a^{3} c^{3} d^{5} + a^{3} c^{2} d^{6} + 5 \, a^{3} c d^{7} - 2 \, a^{3} d^{8}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{8} - 4 \, a^{3} c^{7} d - 12 \, a^{3} c^{6} d^{2} + 20 \, a^{3} c^{5} d^{3} + 10 \, a^{3} c^{4} d^{4} - 28 \, a^{3} c^{3} d^{5} + 4 \, a^{3} c^{2} d^{6} + 12 \, a^{3} c d^{7} - 5 \, a^{3} d^{8}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{8} - 2 \, a^{3} c^{7} d - 2 \, a^{3} c^{6} d^{2} + 6 \, a^{3} c^{5} d^{3} - 6 \, a^{3} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{6} + 2 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{8} - 2 \, a^{3} c^{7} d - 2 \, a^{3} c^{6} d^{2} + 6 \, a^{3} c^{5} d^{3} - 6 \, a^{3} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{6} + 2 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f - {\left({\left(a^{3} c^{7} d - 3 \, a^{3} c^{6} d^{2} + a^{3} c^{5} d^{3} + 5 \, a^{3} c^{4} d^{4} - 5 \, a^{3} c^{3} d^{5} - a^{3} c^{2} d^{6} + 3 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{3} c^{8} - 8 \, a^{3} c^{6} d^{2} + 8 \, a^{3} c^{5} d^{3} + 10 \, a^{3} c^{4} d^{4} - 16 \, a^{3} c^{3} d^{5} + 8 \, a^{3} c d^{7} - 3 \, a^{3} d^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{8} - 2 \, a^{3} c^{7} d - 2 \, a^{3} c^{6} d^{2} + 6 \, a^{3} c^{5} d^{3} - 6 \, a^{3} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{6} + 2 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{8} - 2 \, a^{3} c^{7} d - 2 \, a^{3} c^{6} d^{2} + 6 \, a^{3} c^{5} d^{3} - 6 \, a^{3} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{6} + 2 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{3 \, c^{6} - 6 \, c^{5} d - 3 \, c^{4} d^{2} + 12 \, c^{3} d^{3} - 3 \, c^{2} d^{4} - 6 \, c d^{5} + 3 \, d^{6} - {\left(2 \, c^{5} d - 12 \, c^{4} d^{2} + 41 \, c^{3} d^{3} + 84 \, c^{2} d^{4} - 43 \, c d^{5} - 72 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, c^{6} - 6 \, c^{5} d + 5 \, c^{4} d^{2} + 147 \, c^{3} d^{3} + 164 \, c^{2} d^{4} - 141 \, c d^{5} - 171 \, d^{6}\right)} \cos\left(f x + e\right)^{3} + {\left(4 \, c^{6} - 19 \, c^{5} d + 22 \, c^{4} d^{2} + 128 \, c^{3} d^{3} + 64 \, c^{2} d^{4} - 109 \, c d^{5} - 90 \, d^{6}\right)} \cos\left(f x + e\right)^{2} - 15 \, {\left(16 \, c^{2} d^{3} + 28 \, c d^{4} + 12 \, d^{5} + {\left(4 \, c d^{4} + 3 \, d^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, c^{2} d^{3} + 11 \, c d^{4} + 6 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(12 \, c^{2} d^{3} + 29 \, c d^{4} + 15 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(4 \, c^{2} d^{3} + 7 \, c d^{4} + 3 \, d^{5}\right)} \cos\left(f x + e\right) + {\left(16 \, c^{2} d^{3} + 28 \, c d^{4} + 12 \, d^{5} - {\left(4 \, c d^{4} + 3 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(4 \, c^{2} d^{3} + 15 \, c d^{4} + 9 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(4 \, c^{2} d^{3} + 7 \, c d^{4} + 3 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + 3 \, {\left(3 \, c^{6} - 11 \, c^{5} d + 12 \, c^{4} d^{2} + 82 \, c^{3} d^{3} + 47 \, c^{2} d^{4} - 71 \, c d^{5} - 62 \, d^{6}\right)} \cos\left(f x + e\right) - {\left(3 \, c^{6} - 6 \, c^{5} d - 3 \, c^{4} d^{2} + 12 \, c^{3} d^{3} - 3 \, c^{2} d^{4} - 6 \, c d^{5} + 3 \, d^{6} + {\left(2 \, c^{5} d - 12 \, c^{4} d^{2} + 41 \, c^{3} d^{3} + 84 \, c^{2} d^{4} - 43 \, c d^{5} - 72 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(2 \, c^{6} - 8 \, c^{5} d + 17 \, c^{4} d^{2} + 106 \, c^{3} d^{3} + 80 \, c^{2} d^{4} - 98 \, c d^{5} - 99 \, d^{6}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(2 \, c^{6} - 9 \, c^{5} d + 13 \, c^{4} d^{2} + 78 \, c^{3} d^{3} + 48 \, c^{2} d^{4} - 69 \, c d^{5} - 63 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{15 \, {\left({\left(a^{3} c^{7} d - 3 \, a^{3} c^{6} d^{2} + a^{3} c^{5} d^{3} + 5 \, a^{3} c^{4} d^{4} - 5 \, a^{3} c^{3} d^{5} - a^{3} c^{2} d^{6} + 3 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{8} - a^{3} c^{7} d - 5 \, a^{3} c^{6} d^{2} + 7 \, a^{3} c^{5} d^{3} + 5 \, a^{3} c^{4} d^{4} - 11 \, a^{3} c^{3} d^{5} + a^{3} c^{2} d^{6} + 5 \, a^{3} c d^{7} - 2 \, a^{3} d^{8}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{8} - 4 \, a^{3} c^{7} d - 12 \, a^{3} c^{6} d^{2} + 20 \, a^{3} c^{5} d^{3} + 10 \, a^{3} c^{4} d^{4} - 28 \, a^{3} c^{3} d^{5} + 4 \, a^{3} c^{2} d^{6} + 12 \, a^{3} c d^{7} - 5 \, a^{3} d^{8}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{8} - 2 \, a^{3} c^{7} d - 2 \, a^{3} c^{6} d^{2} + 6 \, a^{3} c^{5} d^{3} - 6 \, a^{3} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{6} + 2 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{8} - 2 \, a^{3} c^{7} d - 2 \, a^{3} c^{6} d^{2} + 6 \, a^{3} c^{5} d^{3} - 6 \, a^{3} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{6} + 2 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f - {\left({\left(a^{3} c^{7} d - 3 \, a^{3} c^{6} d^{2} + a^{3} c^{5} d^{3} + 5 \, a^{3} c^{4} d^{4} - 5 \, a^{3} c^{3} d^{5} - a^{3} c^{2} d^{6} + 3 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{3} c^{8} - 8 \, a^{3} c^{6} d^{2} + 8 \, a^{3} c^{5} d^{3} + 10 \, a^{3} c^{4} d^{4} - 16 \, a^{3} c^{3} d^{5} + 8 \, a^{3} c d^{7} - 3 \, a^{3} d^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{8} - 2 \, a^{3} c^{7} d - 2 \, a^{3} c^{6} d^{2} + 6 \, a^{3} c^{5} d^{3} - 6 \, a^{3} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{6} + 2 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{8} - 2 \, a^{3} c^{7} d - 2 \, a^{3} c^{6} d^{2} + 6 \, a^{3} c^{5} d^{3} - 6 \, a^{3} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{6} + 2 \, a^{3} c d^{7} - a^{3} d^{8}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/30*(6*c^6 - 12*c^5*d - 6*c^4*d^2 + 24*c^3*d^3 - 6*c^2*d^4 - 12*c*d^5 + 6*d^6 - 2*(2*c^5*d - 12*c^4*d^2 + 41*c^3*d^3 + 84*c^2*d^4 - 43*c*d^5 - 72*d^6)*cos(f*x + e)^4 - 2*(2*c^6 - 6*c^5*d + 5*c^4*d^2 + 147*c^3*d^3 + 164*c^2*d^4 - 141*c*d^5 - 171*d^6)*cos(f*x + e)^3 + 2*(4*c^6 - 19*c^5*d + 22*c^4*d^2 + 128*c^3*d^3 + 64*c^2*d^4 - 109*c*d^5 - 90*d^6)*cos(f*x + e)^2 + 15*(16*c^2*d^3 + 28*c*d^4 + 12*d^5 + (4*c*d^4 + 3*d^5)*cos(f*x + e)^4 - (4*c^2*d^3 + 11*c*d^4 + 6*d^5)*cos(f*x + e)^3 - (12*c^2*d^3 + 29*c*d^4 + 15*d^5)*cos(f*x + e)^2 + 2*(4*c^2*d^3 + 7*c*d^4 + 3*d^5)*cos(f*x + e) + (16*c^2*d^3 + 28*c*d^4 + 12*d^5 - (4*c*d^4 + 3*d^5)*cos(f*x + e)^3 - (4*c^2*d^3 + 15*c*d^4 + 9*d^5)*cos(f*x + e)^2 + 2*(4*c^2*d^3 + 7*c*d^4 + 3*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(-c^2 + d^2)*log(-((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 - 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 6*(3*c^6 - 11*c^5*d + 12*c^4*d^2 + 82*c^3*d^3 + 47*c^2*d^4 - 71*c*d^5 - 62*d^6)*cos(f*x + e) - 2*(3*c^6 - 6*c^5*d - 3*c^4*d^2 + 12*c^3*d^3 - 3*c^2*d^4 - 6*c*d^5 + 3*d^6 + (2*c^5*d - 12*c^4*d^2 + 41*c^3*d^3 + 84*c^2*d^4 - 43*c*d^5 - 72*d^6)*cos(f*x + e)^3 - (2*c^6 - 8*c^5*d + 17*c^4*d^2 + 106*c^3*d^3 + 80*c^2*d^4 - 98*c*d^5 - 99*d^6)*cos(f*x + e)^2 - 3*(2*c^6 - 9*c^5*d + 13*c^4*d^2 + 78*c^3*d^3 + 48*c^2*d^4 - 69*c*d^5 - 63*d^6)*cos(f*x + e))*sin(f*x + e))/((a^3*c^7*d - 3*a^3*c^6*d^2 + a^3*c^5*d^3 + 5*a^3*c^4*d^4 - 5*a^3*c^3*d^5 - a^3*c^2*d^6 + 3*a^3*c*d^7 - a^3*d^8)*f*cos(f*x + e)^4 - (a^3*c^8 - a^3*c^7*d - 5*a^3*c^6*d^2 + 7*a^3*c^5*d^3 + 5*a^3*c^4*d^4 - 11*a^3*c^3*d^5 + a^3*c^2*d^6 + 5*a^3*c*d^7 - 2*a^3*d^8)*f*cos(f*x + e)^3 - (3*a^3*c^8 - 4*a^3*c^7*d - 12*a^3*c^6*d^2 + 20*a^3*c^5*d^3 + 10*a^3*c^4*d^4 - 28*a^3*c^3*d^5 + 4*a^3*c^2*d^6 + 12*a^3*c*d^7 - 5*a^3*d^8)*f*cos(f*x + e)^2 + 2*(a^3*c^8 - 2*a^3*c^7*d - 2*a^3*c^6*d^2 + 6*a^3*c^5*d^3 - 6*a^3*c^3*d^5 + 2*a^3*c^2*d^6 + 2*a^3*c*d^7 - a^3*d^8)*f*cos(f*x + e) + 4*(a^3*c^8 - 2*a^3*c^7*d - 2*a^3*c^6*d^2 + 6*a^3*c^5*d^3 - 6*a^3*c^3*d^5 + 2*a^3*c^2*d^6 + 2*a^3*c*d^7 - a^3*d^8)*f - ((a^3*c^7*d - 3*a^3*c^6*d^2 + a^3*c^5*d^3 + 5*a^3*c^4*d^4 - 5*a^3*c^3*d^5 - a^3*c^2*d^6 + 3*a^3*c*d^7 - a^3*d^8)*f*cos(f*x + e)^3 + (a^3*c^8 - 8*a^3*c^6*d^2 + 8*a^3*c^5*d^3 + 10*a^3*c^4*d^4 - 16*a^3*c^3*d^5 + 8*a^3*c*d^7 - 3*a^3*d^8)*f*cos(f*x + e)^2 - 2*(a^3*c^8 - 2*a^3*c^7*d - 2*a^3*c^6*d^2 + 6*a^3*c^5*d^3 - 6*a^3*c^3*d^5 + 2*a^3*c^2*d^6 + 2*a^3*c*d^7 - a^3*d^8)*f*cos(f*x + e) - 4*(a^3*c^8 - 2*a^3*c^7*d - 2*a^3*c^6*d^2 + 6*a^3*c^5*d^3 - 6*a^3*c^3*d^5 + 2*a^3*c^2*d^6 + 2*a^3*c*d^7 - a^3*d^8)*f)*sin(f*x + e)), -1/15*(3*c^6 - 6*c^5*d - 3*c^4*d^2 + 12*c^3*d^3 - 3*c^2*d^4 - 6*c*d^5 + 3*d^6 - (2*c^5*d - 12*c^4*d^2 + 41*c^3*d^3 + 84*c^2*d^4 - 43*c*d^5 - 72*d^6)*cos(f*x + e)^4 - (2*c^6 - 6*c^5*d + 5*c^4*d^2 + 147*c^3*d^3 + 164*c^2*d^4 - 141*c*d^5 - 171*d^6)*cos(f*x + e)^3 + (4*c^6 - 19*c^5*d + 22*c^4*d^2 + 128*c^3*d^3 + 64*c^2*d^4 - 109*c*d^5 - 90*d^6)*cos(f*x + e)^2 - 15*(16*c^2*d^3 + 28*c*d^4 + 12*d^5 + (4*c*d^4 + 3*d^5)*cos(f*x + e)^4 - (4*c^2*d^3 + 11*c*d^4 + 6*d^5)*cos(f*x + e)^3 - (12*c^2*d^3 + 29*c*d^4 + 15*d^5)*cos(f*x + e)^2 + 2*(4*c^2*d^3 + 7*c*d^4 + 3*d^5)*cos(f*x + e) + (16*c^2*d^3 + 28*c*d^4 + 12*d^5 - (4*c*d^4 + 3*d^5)*cos(f*x + e)^3 - (4*c^2*d^3 + 15*c*d^4 + 9*d^5)*cos(f*x + e)^2 + 2*(4*c^2*d^3 + 7*c*d^4 + 3*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + 3*(3*c^6 - 11*c^5*d + 12*c^4*d^2 + 82*c^3*d^3 + 47*c^2*d^4 - 71*c*d^5 - 62*d^6)*cos(f*x + e) - (3*c^6 - 6*c^5*d - 3*c^4*d^2 + 12*c^3*d^3 - 3*c^2*d^4 - 6*c*d^5 + 3*d^6 + (2*c^5*d - 12*c^4*d^2 + 41*c^3*d^3 + 84*c^2*d^4 - 43*c*d^5 - 72*d^6)*cos(f*x + e)^3 - (2*c^6 - 8*c^5*d + 17*c^4*d^2 + 106*c^3*d^3 + 80*c^2*d^4 - 98*c*d^5 - 99*d^6)*cos(f*x + e)^2 - 3*(2*c^6 - 9*c^5*d + 13*c^4*d^2 + 78*c^3*d^3 + 48*c^2*d^4 - 69*c*d^5 - 63*d^6)*cos(f*x + e))*sin(f*x + e))/((a^3*c^7*d - 3*a^3*c^6*d^2 + a^3*c^5*d^3 + 5*a^3*c^4*d^4 - 5*a^3*c^3*d^5 - a^3*c^2*d^6 + 3*a^3*c*d^7 - a^3*d^8)*f*cos(f*x + e)^4 - (a^3*c^8 - a^3*c^7*d - 5*a^3*c^6*d^2 + 7*a^3*c^5*d^3 + 5*a^3*c^4*d^4 - 11*a^3*c^3*d^5 + a^3*c^2*d^6 + 5*a^3*c*d^7 - 2*a^3*d^8)*f*cos(f*x + e)^3 - (3*a^3*c^8 - 4*a^3*c^7*d - 12*a^3*c^6*d^2 + 20*a^3*c^5*d^3 + 10*a^3*c^4*d^4 - 28*a^3*c^3*d^5 + 4*a^3*c^2*d^6 + 12*a^3*c*d^7 - 5*a^3*d^8)*f*cos(f*x + e)^2 + 2*(a^3*c^8 - 2*a^3*c^7*d - 2*a^3*c^6*d^2 + 6*a^3*c^5*d^3 - 6*a^3*c^3*d^5 + 2*a^3*c^2*d^6 + 2*a^3*c*d^7 - a^3*d^8)*f*cos(f*x + e) + 4*(a^3*c^8 - 2*a^3*c^7*d - 2*a^3*c^6*d^2 + 6*a^3*c^5*d^3 - 6*a^3*c^3*d^5 + 2*a^3*c^2*d^6 + 2*a^3*c*d^7 - a^3*d^8)*f - ((a^3*c^7*d - 3*a^3*c^6*d^2 + a^3*c^5*d^3 + 5*a^3*c^4*d^4 - 5*a^3*c^3*d^5 - a^3*c^2*d^6 + 3*a^3*c*d^7 - a^3*d^8)*f*cos(f*x + e)^3 + (a^3*c^8 - 8*a^3*c^6*d^2 + 8*a^3*c^5*d^3 + 10*a^3*c^4*d^4 - 16*a^3*c^3*d^5 + 8*a^3*c*d^7 - 3*a^3*d^8)*f*cos(f*x + e)^2 - 2*(a^3*c^8 - 2*a^3*c^7*d - 2*a^3*c^6*d^2 + 6*a^3*c^5*d^3 - 6*a^3*c^3*d^5 + 2*a^3*c^2*d^6 + 2*a^3*c*d^7 - a^3*d^8)*f*cos(f*x + e) - 4*(a^3*c^8 - 2*a^3*c^7*d - 2*a^3*c^6*d^2 + 6*a^3*c^5*d^3 - 6*a^3*c^3*d^5 + 2*a^3*c^2*d^6 + 2*a^3*c*d^7 - a^3*d^8)*f)*sin(f*x + e))]","B",0
479,1,5226,0,1.845490," ","integrate(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[-\frac{12 \, c^{8} - 24 \, c^{7} d - 24 \, c^{6} d^{2} + 72 \, c^{5} d^{3} - 72 \, c^{3} d^{5} + 24 \, c^{2} d^{6} + 24 \, c d^{7} - 12 \, d^{8} + 2 \, {\left(4 \, c^{6} d^{2} - 30 \, c^{5} d^{3} + 138 \, c^{4} d^{4} + 555 \, c^{3} d^{5} + 162 \, c^{2} d^{6} - 525 \, c d^{7} - 304 \, d^{8}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(8 \, c^{7} d - 52 \, c^{6} d^{2} + 216 \, c^{5} d^{3} + 1086 \, c^{4} d^{4} + 984 \, c^{3} d^{5} - 621 \, c^{2} d^{6} - 1208 \, c d^{7} - 413 \, d^{8}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(4 \, c^{8} - 6 \, c^{7} d - 20 \, c^{6} d^{2} + 768 \, c^{5} d^{3} + 2676 \, c^{4} d^{4} + 2307 \, c^{3} d^{5} - 1573 \, c^{2} d^{6} - 3069 \, c d^{7} - 1087 \, d^{8}\right)} \cos\left(f x + e\right)^{3} + 4 \, {\left(4 \, c^{8} - 20 \, c^{7} d + 19 \, c^{6} d^{2} + 330 \, c^{5} d^{3} + 699 \, c^{4} d^{4} + 345 \, c^{3} d^{5} - 526 \, c^{2} d^{6} - 655 \, c d^{7} - 196 \, d^{8}\right)} \cos\left(f x + e\right)^{2} - 15 \, {\left(80 \, c^{4} d^{3} + 280 \, c^{3} d^{4} + 372 \, c^{2} d^{5} + 224 \, c d^{6} + 52 \, d^{7} + {\left(20 \, c^{2} d^{5} + 30 \, c d^{6} + 13 \, d^{7}\right)} \cos\left(f x + e\right)^{5} + {\left(40 \, c^{3} d^{4} + 120 \, c^{2} d^{5} + 116 \, c d^{6} + 39 \, d^{7}\right)} \cos\left(f x + e\right)^{4} - {\left(20 \, c^{4} d^{3} + 110 \, c^{3} d^{4} + 193 \, c^{2} d^{5} + 142 \, c d^{6} + 39 \, d^{7}\right)} \cos\left(f x + e\right)^{3} - {\left(60 \, c^{4} d^{3} + 290 \, c^{3} d^{4} + 479 \, c^{2} d^{5} + 340 \, c d^{6} + 91 \, d^{7}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(20 \, c^{4} d^{3} + 70 \, c^{3} d^{4} + 93 \, c^{2} d^{5} + 56 \, c d^{6} + 13 \, d^{7}\right)} \cos\left(f x + e\right) + {\left(80 \, c^{4} d^{3} + 280 \, c^{3} d^{4} + 372 \, c^{2} d^{5} + 224 \, c d^{6} + 52 \, d^{7} + {\left(20 \, c^{2} d^{5} + 30 \, c d^{6} + 13 \, d^{7}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(20 \, c^{3} d^{4} + 50 \, c^{2} d^{5} + 43 \, c d^{6} + 13 \, d^{7}\right)} \cos\left(f x + e\right)^{3} - {\left(20 \, c^{4} d^{3} + 150 \, c^{3} d^{4} + 293 \, c^{2} d^{5} + 228 \, c d^{6} + 65 \, d^{7}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(20 \, c^{4} d^{3} + 70 \, c^{3} d^{4} + 93 \, c^{2} d^{5} + 56 \, c d^{6} + 13 \, d^{7}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 12 \, {\left(3 \, c^{8} - 11 \, c^{7} d + 9 \, c^{6} d^{2} + 213 \, c^{5} d^{3} + 475 \, c^{4} d^{4} + 237 \, c^{3} d^{5} - 359 \, c^{2} d^{6} - 439 \, c d^{7} - 128 \, d^{8}\right)} \cos\left(f x + e\right) - 2 \, {\left(6 \, c^{8} - 12 \, c^{7} d - 12 \, c^{6} d^{2} + 36 \, c^{5} d^{3} - 36 \, c^{3} d^{5} + 12 \, c^{2} d^{6} + 12 \, c d^{7} - 6 \, d^{8} + {\left(4 \, c^{6} d^{2} - 30 \, c^{5} d^{3} + 138 \, c^{4} d^{4} + 555 \, c^{3} d^{5} + 162 \, c^{2} d^{6} - 525 \, c d^{7} - 304 \, d^{8}\right)} \cos\left(f x + e\right)^{4} + {\left(8 \, c^{7} d - 48 \, c^{6} d^{2} + 186 \, c^{5} d^{3} + 1224 \, c^{4} d^{4} + 1539 \, c^{3} d^{5} - 459 \, c^{2} d^{6} - 1733 \, c d^{7} - 717 \, d^{8}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(2 \, c^{8} - 7 \, c^{7} d + 14 \, c^{6} d^{2} + 291 \, c^{5} d^{3} + 726 \, c^{4} d^{4} + 384 \, c^{3} d^{5} - 557 \, c^{2} d^{6} - 668 \, c d^{7} - 185 \, d^{8}\right)} \cos\left(f x + e\right)^{2} - 6 \, {\left(2 \, c^{8} - 9 \, c^{7} d + 11 \, c^{6} d^{2} + 207 \, c^{5} d^{3} + 475 \, c^{4} d^{4} + 243 \, c^{3} d^{5} - 361 \, c^{2} d^{6} - 441 \, c d^{7} - 127 \, d^{8}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{60 \, {\left({\left(a^{3} c^{9} d^{2} - 3 \, a^{3} c^{8} d^{3} + 8 \, a^{3} c^{6} d^{5} - 6 \, a^{3} c^{5} d^{6} - 6 \, a^{3} c^{4} d^{7} + 8 \, a^{3} c^{3} d^{8} - 3 \, a^{3} c d^{10} + a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{5} + {\left(2 \, a^{3} c^{10} d - 3 \, a^{3} c^{9} d^{2} - 9 \, a^{3} c^{8} d^{3} + 16 \, a^{3} c^{7} d^{4} + 12 \, a^{3} c^{6} d^{5} - 30 \, a^{3} c^{5} d^{6} - 2 \, a^{3} c^{4} d^{7} + 24 \, a^{3} c^{3} d^{8} - 6 \, a^{3} c^{2} d^{9} - 7 \, a^{3} c d^{10} + 3 \, a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{11} + a^{3} c^{10} d - 9 \, a^{3} c^{9} d^{2} - a^{3} c^{8} d^{3} + 26 \, a^{3} c^{7} d^{4} - 6 \, a^{3} c^{6} d^{5} - 34 \, a^{3} c^{5} d^{6} + 14 \, a^{3} c^{4} d^{7} + 21 \, a^{3} c^{3} d^{8} - 11 \, a^{3} c^{2} d^{9} - 5 \, a^{3} c d^{10} + 3 \, a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{11} + a^{3} c^{10} d - 23 \, a^{3} c^{9} d^{2} + 3 \, a^{3} c^{8} d^{3} + 62 \, a^{3} c^{7} d^{4} - 22 \, a^{3} c^{6} d^{5} - 78 \, a^{3} c^{5} d^{6} + 38 \, a^{3} c^{4} d^{7} + 47 \, a^{3} c^{3} d^{8} - 27 \, a^{3} c^{2} d^{9} - 11 \, a^{3} c d^{10} + 7 \, a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{11} - a^{3} c^{10} d - 5 \, a^{3} c^{9} d^{2} + 5 \, a^{3} c^{8} d^{3} + 10 \, a^{3} c^{7} d^{4} - 10 \, a^{3} c^{6} d^{5} - 10 \, a^{3} c^{5} d^{6} + 10 \, a^{3} c^{4} d^{7} + 5 \, a^{3} c^{3} d^{8} - 5 \, a^{3} c^{2} d^{9} - a^{3} c d^{10} + a^{3} d^{11}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{11} - a^{3} c^{10} d - 5 \, a^{3} c^{9} d^{2} + 5 \, a^{3} c^{8} d^{3} + 10 \, a^{3} c^{7} d^{4} - 10 \, a^{3} c^{6} d^{5} - 10 \, a^{3} c^{5} d^{6} + 10 \, a^{3} c^{4} d^{7} + 5 \, a^{3} c^{3} d^{8} - 5 \, a^{3} c^{2} d^{9} - a^{3} c d^{10} + a^{3} d^{11}\right)} f + {\left({\left(a^{3} c^{9} d^{2} - 3 \, a^{3} c^{8} d^{3} + 8 \, a^{3} c^{6} d^{5} - 6 \, a^{3} c^{5} d^{6} - 6 \, a^{3} c^{4} d^{7} + 8 \, a^{3} c^{3} d^{8} - 3 \, a^{3} c d^{10} + a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} c^{10} d - 2 \, a^{3} c^{9} d^{2} - 3 \, a^{3} c^{8} d^{3} + 8 \, a^{3} c^{7} d^{4} + 2 \, a^{3} c^{6} d^{5} - 12 \, a^{3} c^{5} d^{6} + 2 \, a^{3} c^{4} d^{7} + 8 \, a^{3} c^{3} d^{8} - 3 \, a^{3} c^{2} d^{9} - 2 \, a^{3} c d^{10} + a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{3} c^{11} + 3 \, a^{3} c^{10} d - 13 \, a^{3} c^{9} d^{2} - 7 \, a^{3} c^{8} d^{3} + 42 \, a^{3} c^{7} d^{4} - 2 \, a^{3} c^{6} d^{5} - 58 \, a^{3} c^{5} d^{6} + 18 \, a^{3} c^{4} d^{7} + 37 \, a^{3} c^{3} d^{8} - 17 \, a^{3} c^{2} d^{9} - 9 \, a^{3} c d^{10} + 5 \, a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{11} - a^{3} c^{10} d - 5 \, a^{3} c^{9} d^{2} + 5 \, a^{3} c^{8} d^{3} + 10 \, a^{3} c^{7} d^{4} - 10 \, a^{3} c^{6} d^{5} - 10 \, a^{3} c^{5} d^{6} + 10 \, a^{3} c^{4} d^{7} + 5 \, a^{3} c^{3} d^{8} - 5 \, a^{3} c^{2} d^{9} - a^{3} c d^{10} + a^{3} d^{11}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{11} - a^{3} c^{10} d - 5 \, a^{3} c^{9} d^{2} + 5 \, a^{3} c^{8} d^{3} + 10 \, a^{3} c^{7} d^{4} - 10 \, a^{3} c^{6} d^{5} - 10 \, a^{3} c^{5} d^{6} + 10 \, a^{3} c^{4} d^{7} + 5 \, a^{3} c^{3} d^{8} - 5 \, a^{3} c^{2} d^{9} - a^{3} c d^{10} + a^{3} d^{11}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{6 \, c^{8} - 12 \, c^{7} d - 12 \, c^{6} d^{2} + 36 \, c^{5} d^{3} - 36 \, c^{3} d^{5} + 12 \, c^{2} d^{6} + 12 \, c d^{7} - 6 \, d^{8} + {\left(4 \, c^{6} d^{2} - 30 \, c^{5} d^{3} + 138 \, c^{4} d^{4} + 555 \, c^{3} d^{5} + 162 \, c^{2} d^{6} - 525 \, c d^{7} - 304 \, d^{8}\right)} \cos\left(f x + e\right)^{5} - {\left(8 \, c^{7} d - 52 \, c^{6} d^{2} + 216 \, c^{5} d^{3} + 1086 \, c^{4} d^{4} + 984 \, c^{3} d^{5} - 621 \, c^{2} d^{6} - 1208 \, c d^{7} - 413 \, d^{8}\right)} \cos\left(f x + e\right)^{4} - {\left(4 \, c^{8} - 6 \, c^{7} d - 20 \, c^{6} d^{2} + 768 \, c^{5} d^{3} + 2676 \, c^{4} d^{4} + 2307 \, c^{3} d^{5} - 1573 \, c^{2} d^{6} - 3069 \, c d^{7} - 1087 \, d^{8}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(4 \, c^{8} - 20 \, c^{7} d + 19 \, c^{6} d^{2} + 330 \, c^{5} d^{3} + 699 \, c^{4} d^{4} + 345 \, c^{3} d^{5} - 526 \, c^{2} d^{6} - 655 \, c d^{7} - 196 \, d^{8}\right)} \cos\left(f x + e\right)^{2} - 15 \, {\left(80 \, c^{4} d^{3} + 280 \, c^{3} d^{4} + 372 \, c^{2} d^{5} + 224 \, c d^{6} + 52 \, d^{7} + {\left(20 \, c^{2} d^{5} + 30 \, c d^{6} + 13 \, d^{7}\right)} \cos\left(f x + e\right)^{5} + {\left(40 \, c^{3} d^{4} + 120 \, c^{2} d^{5} + 116 \, c d^{6} + 39 \, d^{7}\right)} \cos\left(f x + e\right)^{4} - {\left(20 \, c^{4} d^{3} + 110 \, c^{3} d^{4} + 193 \, c^{2} d^{5} + 142 \, c d^{6} + 39 \, d^{7}\right)} \cos\left(f x + e\right)^{3} - {\left(60 \, c^{4} d^{3} + 290 \, c^{3} d^{4} + 479 \, c^{2} d^{5} + 340 \, c d^{6} + 91 \, d^{7}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(20 \, c^{4} d^{3} + 70 \, c^{3} d^{4} + 93 \, c^{2} d^{5} + 56 \, c d^{6} + 13 \, d^{7}\right)} \cos\left(f x + e\right) + {\left(80 \, c^{4} d^{3} + 280 \, c^{3} d^{4} + 372 \, c^{2} d^{5} + 224 \, c d^{6} + 52 \, d^{7} + {\left(20 \, c^{2} d^{5} + 30 \, c d^{6} + 13 \, d^{7}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(20 \, c^{3} d^{4} + 50 \, c^{2} d^{5} + 43 \, c d^{6} + 13 \, d^{7}\right)} \cos\left(f x + e\right)^{3} - {\left(20 \, c^{4} d^{3} + 150 \, c^{3} d^{4} + 293 \, c^{2} d^{5} + 228 \, c d^{6} + 65 \, d^{7}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(20 \, c^{4} d^{3} + 70 \, c^{3} d^{4} + 93 \, c^{2} d^{5} + 56 \, c d^{6} + 13 \, d^{7}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + 6 \, {\left(3 \, c^{8} - 11 \, c^{7} d + 9 \, c^{6} d^{2} + 213 \, c^{5} d^{3} + 475 \, c^{4} d^{4} + 237 \, c^{3} d^{5} - 359 \, c^{2} d^{6} - 439 \, c d^{7} - 128 \, d^{8}\right)} \cos\left(f x + e\right) - {\left(6 \, c^{8} - 12 \, c^{7} d - 12 \, c^{6} d^{2} + 36 \, c^{5} d^{3} - 36 \, c^{3} d^{5} + 12 \, c^{2} d^{6} + 12 \, c d^{7} - 6 \, d^{8} + {\left(4 \, c^{6} d^{2} - 30 \, c^{5} d^{3} + 138 \, c^{4} d^{4} + 555 \, c^{3} d^{5} + 162 \, c^{2} d^{6} - 525 \, c d^{7} - 304 \, d^{8}\right)} \cos\left(f x + e\right)^{4} + {\left(8 \, c^{7} d - 48 \, c^{6} d^{2} + 186 \, c^{5} d^{3} + 1224 \, c^{4} d^{4} + 1539 \, c^{3} d^{5} - 459 \, c^{2} d^{6} - 1733 \, c d^{7} - 717 \, d^{8}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(2 \, c^{8} - 7 \, c^{7} d + 14 \, c^{6} d^{2} + 291 \, c^{5} d^{3} + 726 \, c^{4} d^{4} + 384 \, c^{3} d^{5} - 557 \, c^{2} d^{6} - 668 \, c d^{7} - 185 \, d^{8}\right)} \cos\left(f x + e\right)^{2} - 6 \, {\left(2 \, c^{8} - 9 \, c^{7} d + 11 \, c^{6} d^{2} + 207 \, c^{5} d^{3} + 475 \, c^{4} d^{4} + 243 \, c^{3} d^{5} - 361 \, c^{2} d^{6} - 441 \, c d^{7} - 127 \, d^{8}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{30 \, {\left({\left(a^{3} c^{9} d^{2} - 3 \, a^{3} c^{8} d^{3} + 8 \, a^{3} c^{6} d^{5} - 6 \, a^{3} c^{5} d^{6} - 6 \, a^{3} c^{4} d^{7} + 8 \, a^{3} c^{3} d^{8} - 3 \, a^{3} c d^{10} + a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{5} + {\left(2 \, a^{3} c^{10} d - 3 \, a^{3} c^{9} d^{2} - 9 \, a^{3} c^{8} d^{3} + 16 \, a^{3} c^{7} d^{4} + 12 \, a^{3} c^{6} d^{5} - 30 \, a^{3} c^{5} d^{6} - 2 \, a^{3} c^{4} d^{7} + 24 \, a^{3} c^{3} d^{8} - 6 \, a^{3} c^{2} d^{9} - 7 \, a^{3} c d^{10} + 3 \, a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{11} + a^{3} c^{10} d - 9 \, a^{3} c^{9} d^{2} - a^{3} c^{8} d^{3} + 26 \, a^{3} c^{7} d^{4} - 6 \, a^{3} c^{6} d^{5} - 34 \, a^{3} c^{5} d^{6} + 14 \, a^{3} c^{4} d^{7} + 21 \, a^{3} c^{3} d^{8} - 11 \, a^{3} c^{2} d^{9} - 5 \, a^{3} c d^{10} + 3 \, a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{11} + a^{3} c^{10} d - 23 \, a^{3} c^{9} d^{2} + 3 \, a^{3} c^{8} d^{3} + 62 \, a^{3} c^{7} d^{4} - 22 \, a^{3} c^{6} d^{5} - 78 \, a^{3} c^{5} d^{6} + 38 \, a^{3} c^{4} d^{7} + 47 \, a^{3} c^{3} d^{8} - 27 \, a^{3} c^{2} d^{9} - 11 \, a^{3} c d^{10} + 7 \, a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{11} - a^{3} c^{10} d - 5 \, a^{3} c^{9} d^{2} + 5 \, a^{3} c^{8} d^{3} + 10 \, a^{3} c^{7} d^{4} - 10 \, a^{3} c^{6} d^{5} - 10 \, a^{3} c^{5} d^{6} + 10 \, a^{3} c^{4} d^{7} + 5 \, a^{3} c^{3} d^{8} - 5 \, a^{3} c^{2} d^{9} - a^{3} c d^{10} + a^{3} d^{11}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{11} - a^{3} c^{10} d - 5 \, a^{3} c^{9} d^{2} + 5 \, a^{3} c^{8} d^{3} + 10 \, a^{3} c^{7} d^{4} - 10 \, a^{3} c^{6} d^{5} - 10 \, a^{3} c^{5} d^{6} + 10 \, a^{3} c^{4} d^{7} + 5 \, a^{3} c^{3} d^{8} - 5 \, a^{3} c^{2} d^{9} - a^{3} c d^{10} + a^{3} d^{11}\right)} f + {\left({\left(a^{3} c^{9} d^{2} - 3 \, a^{3} c^{8} d^{3} + 8 \, a^{3} c^{6} d^{5} - 6 \, a^{3} c^{5} d^{6} - 6 \, a^{3} c^{4} d^{7} + 8 \, a^{3} c^{3} d^{8} - 3 \, a^{3} c d^{10} + a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} c^{10} d - 2 \, a^{3} c^{9} d^{2} - 3 \, a^{3} c^{8} d^{3} + 8 \, a^{3} c^{7} d^{4} + 2 \, a^{3} c^{6} d^{5} - 12 \, a^{3} c^{5} d^{6} + 2 \, a^{3} c^{4} d^{7} + 8 \, a^{3} c^{3} d^{8} - 3 \, a^{3} c^{2} d^{9} - 2 \, a^{3} c d^{10} + a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{3} c^{11} + 3 \, a^{3} c^{10} d - 13 \, a^{3} c^{9} d^{2} - 7 \, a^{3} c^{8} d^{3} + 42 \, a^{3} c^{7} d^{4} - 2 \, a^{3} c^{6} d^{5} - 58 \, a^{3} c^{5} d^{6} + 18 \, a^{3} c^{4} d^{7} + 37 \, a^{3} c^{3} d^{8} - 17 \, a^{3} c^{2} d^{9} - 9 \, a^{3} c d^{10} + 5 \, a^{3} d^{11}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{11} - a^{3} c^{10} d - 5 \, a^{3} c^{9} d^{2} + 5 \, a^{3} c^{8} d^{3} + 10 \, a^{3} c^{7} d^{4} - 10 \, a^{3} c^{6} d^{5} - 10 \, a^{3} c^{5} d^{6} + 10 \, a^{3} c^{4} d^{7} + 5 \, a^{3} c^{3} d^{8} - 5 \, a^{3} c^{2} d^{9} - a^{3} c d^{10} + a^{3} d^{11}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{11} - a^{3} c^{10} d - 5 \, a^{3} c^{9} d^{2} + 5 \, a^{3} c^{8} d^{3} + 10 \, a^{3} c^{7} d^{4} - 10 \, a^{3} c^{6} d^{5} - 10 \, a^{3} c^{5} d^{6} + 10 \, a^{3} c^{4} d^{7} + 5 \, a^{3} c^{3} d^{8} - 5 \, a^{3} c^{2} d^{9} - a^{3} c d^{10} + a^{3} d^{11}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/60*(12*c^8 - 24*c^7*d - 24*c^6*d^2 + 72*c^5*d^3 - 72*c^3*d^5 + 24*c^2*d^6 + 24*c*d^7 - 12*d^8 + 2*(4*c^6*d^2 - 30*c^5*d^3 + 138*c^4*d^4 + 555*c^3*d^5 + 162*c^2*d^6 - 525*c*d^7 - 304*d^8)*cos(f*x + e)^5 - 2*(8*c^7*d - 52*c^6*d^2 + 216*c^5*d^3 + 1086*c^4*d^4 + 984*c^3*d^5 - 621*c^2*d^6 - 1208*c*d^7 - 413*d^8)*cos(f*x + e)^4 - 2*(4*c^8 - 6*c^7*d - 20*c^6*d^2 + 768*c^5*d^3 + 2676*c^4*d^4 + 2307*c^3*d^5 - 1573*c^2*d^6 - 3069*c*d^7 - 1087*d^8)*cos(f*x + e)^3 + 4*(4*c^8 - 20*c^7*d + 19*c^6*d^2 + 330*c^5*d^3 + 699*c^4*d^4 + 345*c^3*d^5 - 526*c^2*d^6 - 655*c*d^7 - 196*d^8)*cos(f*x + e)^2 - 15*(80*c^4*d^3 + 280*c^3*d^4 + 372*c^2*d^5 + 224*c*d^6 + 52*d^7 + (20*c^2*d^5 + 30*c*d^6 + 13*d^7)*cos(f*x + e)^5 + (40*c^3*d^4 + 120*c^2*d^5 + 116*c*d^6 + 39*d^7)*cos(f*x + e)^4 - (20*c^4*d^3 + 110*c^3*d^4 + 193*c^2*d^5 + 142*c*d^6 + 39*d^7)*cos(f*x + e)^3 - (60*c^4*d^3 + 290*c^3*d^4 + 479*c^2*d^5 + 340*c*d^6 + 91*d^7)*cos(f*x + e)^2 + 2*(20*c^4*d^3 + 70*c^3*d^4 + 93*c^2*d^5 + 56*c*d^6 + 13*d^7)*cos(f*x + e) + (80*c^4*d^3 + 280*c^3*d^4 + 372*c^2*d^5 + 224*c*d^6 + 52*d^7 + (20*c^2*d^5 + 30*c*d^6 + 13*d^7)*cos(f*x + e)^4 - 2*(20*c^3*d^4 + 50*c^2*d^5 + 43*c*d^6 + 13*d^7)*cos(f*x + e)^3 - (20*c^4*d^3 + 150*c^3*d^4 + 293*c^2*d^5 + 228*c*d^6 + 65*d^7)*cos(f*x + e)^2 + 2*(20*c^4*d^3 + 70*c^3*d^4 + 93*c^2*d^5 + 56*c*d^6 + 13*d^7)*cos(f*x + e))*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 12*(3*c^8 - 11*c^7*d + 9*c^6*d^2 + 213*c^5*d^3 + 475*c^4*d^4 + 237*c^3*d^5 - 359*c^2*d^6 - 439*c*d^7 - 128*d^8)*cos(f*x + e) - 2*(6*c^8 - 12*c^7*d - 12*c^6*d^2 + 36*c^5*d^3 - 36*c^3*d^5 + 12*c^2*d^6 + 12*c*d^7 - 6*d^8 + (4*c^6*d^2 - 30*c^5*d^3 + 138*c^4*d^4 + 555*c^3*d^5 + 162*c^2*d^6 - 525*c*d^7 - 304*d^8)*cos(f*x + e)^4 + (8*c^7*d - 48*c^6*d^2 + 186*c^5*d^3 + 1224*c^4*d^4 + 1539*c^3*d^5 - 459*c^2*d^6 - 1733*c*d^7 - 717*d^8)*cos(f*x + e)^3 - 2*(2*c^8 - 7*c^7*d + 14*c^6*d^2 + 291*c^5*d^3 + 726*c^4*d^4 + 384*c^3*d^5 - 557*c^2*d^6 - 668*c*d^7 - 185*d^8)*cos(f*x + e)^2 - 6*(2*c^8 - 9*c^7*d + 11*c^6*d^2 + 207*c^5*d^3 + 475*c^4*d^4 + 243*c^3*d^5 - 361*c^2*d^6 - 441*c*d^7 - 127*d^8)*cos(f*x + e))*sin(f*x + e))/((a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^5 + (2*a^3*c^10*d - 3*a^3*c^9*d^2 - 9*a^3*c^8*d^3 + 16*a^3*c^7*d^4 + 12*a^3*c^6*d^5 - 30*a^3*c^5*d^6 - 2*a^3*c^4*d^7 + 24*a^3*c^3*d^8 - 6*a^3*c^2*d^9 - 7*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e)^4 - (a^3*c^11 + a^3*c^10*d - 9*a^3*c^9*d^2 - a^3*c^8*d^3 + 26*a^3*c^7*d^4 - 6*a^3*c^6*d^5 - 34*a^3*c^5*d^6 + 14*a^3*c^4*d^7 + 21*a^3*c^3*d^8 - 11*a^3*c^2*d^9 - 5*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e)^3 - (3*a^3*c^11 + a^3*c^10*d - 23*a^3*c^9*d^2 + 3*a^3*c^8*d^3 + 62*a^3*c^7*d^4 - 22*a^3*c^6*d^5 - 78*a^3*c^5*d^6 + 38*a^3*c^4*d^7 + 47*a^3*c^3*d^8 - 27*a^3*c^2*d^9 - 11*a^3*c*d^10 + 7*a^3*d^11)*f*cos(f*x + e)^2 + 2*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f + ((a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^4 - 2*(a^3*c^10*d - 2*a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^7*d^4 + 2*a^3*c^6*d^5 - 12*a^3*c^5*d^6 + 2*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c^2*d^9 - 2*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^3 - (a^3*c^11 + 3*a^3*c^10*d - 13*a^3*c^9*d^2 - 7*a^3*c^8*d^3 + 42*a^3*c^7*d^4 - 2*a^3*c^6*d^5 - 58*a^3*c^5*d^6 + 18*a^3*c^4*d^7 + 37*a^3*c^3*d^8 - 17*a^3*c^2*d^9 - 9*a^3*c*d^10 + 5*a^3*d^11)*f*cos(f*x + e)^2 + 2*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f)*sin(f*x + e)), -1/30*(6*c^8 - 12*c^7*d - 12*c^6*d^2 + 36*c^5*d^3 - 36*c^3*d^5 + 12*c^2*d^6 + 12*c*d^7 - 6*d^8 + (4*c^6*d^2 - 30*c^5*d^3 + 138*c^4*d^4 + 555*c^3*d^5 + 162*c^2*d^6 - 525*c*d^7 - 304*d^8)*cos(f*x + e)^5 - (8*c^7*d - 52*c^6*d^2 + 216*c^5*d^3 + 1086*c^4*d^4 + 984*c^3*d^5 - 621*c^2*d^6 - 1208*c*d^7 - 413*d^8)*cos(f*x + e)^4 - (4*c^8 - 6*c^7*d - 20*c^6*d^2 + 768*c^5*d^3 + 2676*c^4*d^4 + 2307*c^3*d^5 - 1573*c^2*d^6 - 3069*c*d^7 - 1087*d^8)*cos(f*x + e)^3 + 2*(4*c^8 - 20*c^7*d + 19*c^6*d^2 + 330*c^5*d^3 + 699*c^4*d^4 + 345*c^3*d^5 - 526*c^2*d^6 - 655*c*d^7 - 196*d^8)*cos(f*x + e)^2 - 15*(80*c^4*d^3 + 280*c^3*d^4 + 372*c^2*d^5 + 224*c*d^6 + 52*d^7 + (20*c^2*d^5 + 30*c*d^6 + 13*d^7)*cos(f*x + e)^5 + (40*c^3*d^4 + 120*c^2*d^5 + 116*c*d^6 + 39*d^7)*cos(f*x + e)^4 - (20*c^4*d^3 + 110*c^3*d^4 + 193*c^2*d^5 + 142*c*d^6 + 39*d^7)*cos(f*x + e)^3 - (60*c^4*d^3 + 290*c^3*d^4 + 479*c^2*d^5 + 340*c*d^6 + 91*d^7)*cos(f*x + e)^2 + 2*(20*c^4*d^3 + 70*c^3*d^4 + 93*c^2*d^5 + 56*c*d^6 + 13*d^7)*cos(f*x + e) + (80*c^4*d^3 + 280*c^3*d^4 + 372*c^2*d^5 + 224*c*d^6 + 52*d^7 + (20*c^2*d^5 + 30*c*d^6 + 13*d^7)*cos(f*x + e)^4 - 2*(20*c^3*d^4 + 50*c^2*d^5 + 43*c*d^6 + 13*d^7)*cos(f*x + e)^3 - (20*c^4*d^3 + 150*c^3*d^4 + 293*c^2*d^5 + 228*c*d^6 + 65*d^7)*cos(f*x + e)^2 + 2*(20*c^4*d^3 + 70*c^3*d^4 + 93*c^2*d^5 + 56*c*d^6 + 13*d^7)*cos(f*x + e))*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + 6*(3*c^8 - 11*c^7*d + 9*c^6*d^2 + 213*c^5*d^3 + 475*c^4*d^4 + 237*c^3*d^5 - 359*c^2*d^6 - 439*c*d^7 - 128*d^8)*cos(f*x + e) - (6*c^8 - 12*c^7*d - 12*c^6*d^2 + 36*c^5*d^3 - 36*c^3*d^5 + 12*c^2*d^6 + 12*c*d^7 - 6*d^8 + (4*c^6*d^2 - 30*c^5*d^3 + 138*c^4*d^4 + 555*c^3*d^5 + 162*c^2*d^6 - 525*c*d^7 - 304*d^8)*cos(f*x + e)^4 + (8*c^7*d - 48*c^6*d^2 + 186*c^5*d^3 + 1224*c^4*d^4 + 1539*c^3*d^5 - 459*c^2*d^6 - 1733*c*d^7 - 717*d^8)*cos(f*x + e)^3 - 2*(2*c^8 - 7*c^7*d + 14*c^6*d^2 + 291*c^5*d^3 + 726*c^4*d^4 + 384*c^3*d^5 - 557*c^2*d^6 - 668*c*d^7 - 185*d^8)*cos(f*x + e)^2 - 6*(2*c^8 - 9*c^7*d + 11*c^6*d^2 + 207*c^5*d^3 + 475*c^4*d^4 + 243*c^3*d^5 - 361*c^2*d^6 - 441*c*d^7 - 127*d^8)*cos(f*x + e))*sin(f*x + e))/((a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^5 + (2*a^3*c^10*d - 3*a^3*c^9*d^2 - 9*a^3*c^8*d^3 + 16*a^3*c^7*d^4 + 12*a^3*c^6*d^5 - 30*a^3*c^5*d^6 - 2*a^3*c^4*d^7 + 24*a^3*c^3*d^8 - 6*a^3*c^2*d^9 - 7*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e)^4 - (a^3*c^11 + a^3*c^10*d - 9*a^3*c^9*d^2 - a^3*c^8*d^3 + 26*a^3*c^7*d^4 - 6*a^3*c^6*d^5 - 34*a^3*c^5*d^6 + 14*a^3*c^4*d^7 + 21*a^3*c^3*d^8 - 11*a^3*c^2*d^9 - 5*a^3*c*d^10 + 3*a^3*d^11)*f*cos(f*x + e)^3 - (3*a^3*c^11 + a^3*c^10*d - 23*a^3*c^9*d^2 + 3*a^3*c^8*d^3 + 62*a^3*c^7*d^4 - 22*a^3*c^6*d^5 - 78*a^3*c^5*d^6 + 38*a^3*c^4*d^7 + 47*a^3*c^3*d^8 - 27*a^3*c^2*d^9 - 11*a^3*c*d^10 + 7*a^3*d^11)*f*cos(f*x + e)^2 + 2*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f + ((a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^6*d^5 - 6*a^3*c^5*d^6 - 6*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^4 - 2*(a^3*c^10*d - 2*a^3*c^9*d^2 - 3*a^3*c^8*d^3 + 8*a^3*c^7*d^4 + 2*a^3*c^6*d^5 - 12*a^3*c^5*d^6 + 2*a^3*c^4*d^7 + 8*a^3*c^3*d^8 - 3*a^3*c^2*d^9 - 2*a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e)^3 - (a^3*c^11 + 3*a^3*c^10*d - 13*a^3*c^9*d^2 - 7*a^3*c^8*d^3 + 42*a^3*c^7*d^4 - 2*a^3*c^6*d^5 - 58*a^3*c^5*d^6 + 18*a^3*c^4*d^7 + 37*a^3*c^3*d^8 - 17*a^3*c^2*d^9 - 9*a^3*c*d^10 + 5*a^3*d^11)*f*cos(f*x + e)^2 + 2*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f*cos(f*x + e) + 4*(a^3*c^11 - a^3*c^10*d - 5*a^3*c^9*d^2 + 5*a^3*c^8*d^3 + 10*a^3*c^7*d^4 - 10*a^3*c^6*d^5 - 10*a^3*c^5*d^6 + 10*a^3*c^4*d^7 + 5*a^3*c^3*d^8 - 5*a^3*c^2*d^9 - a^3*c*d^10 + a^3*d^11)*f)*sin(f*x + e))]","B",0
480,1,150,0,0.804271," ","integrate((A+B*sin(x))/(1+sin(x))^4,x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, A + 4 \, B\right)} \cos\left(x\right)^{4} + 8 \, {\left(3 \, A + 4 \, B\right)} \cos\left(x\right)^{3} - 9 \, {\left(3 \, A + 4 \, B\right)} \cos\left(x\right)^{2} - 15 \, {\left(4 \, A + 3 \, B\right)} \cos\left(x\right) + {\left(2 \, {\left(3 \, A + 4 \, B\right)} \cos\left(x\right)^{3} - 6 \, {\left(3 \, A + 4 \, B\right)} \cos\left(x\right)^{2} - 15 \, {\left(3 \, A + 4 \, B\right)} \cos\left(x\right) + 15 \, A - 15 \, B\right)} \sin\left(x\right) - 15 \, A + 15 \, B}{105 \, {\left(\cos\left(x\right)^{4} - 3 \, \cos\left(x\right)^{3} - 8 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{3} + 4 \, \cos\left(x\right)^{2} - 4 \, \cos\left(x\right) - 8\right)} \sin\left(x\right) + 4 \, \cos\left(x\right) + 8\right)}}"," ",0,"1/105*(2*(3*A + 4*B)*cos(x)^4 + 8*(3*A + 4*B)*cos(x)^3 - 9*(3*A + 4*B)*cos(x)^2 - 15*(4*A + 3*B)*cos(x) + (2*(3*A + 4*B)*cos(x)^3 - 6*(3*A + 4*B)*cos(x)^2 - 15*(3*A + 4*B)*cos(x) + 15*A - 15*B)*sin(x) - 15*A + 15*B)/(cos(x)^4 - 3*cos(x)^3 - 8*cos(x)^2 - (cos(x)^3 + 4*cos(x)^2 - 4*cos(x) - 8)*sin(x) + 4*cos(x) + 8)","B",0
481,1,150,0,0.702799," ","integrate((A+B*sin(x))/(1-sin(x))^4,x, algorithm=""fricas"")","-\frac{2 \, {\left(3 \, A - 4 \, B\right)} \cos\left(x\right)^{4} + 8 \, {\left(3 \, A - 4 \, B\right)} \cos\left(x\right)^{3} - 9 \, {\left(3 \, A - 4 \, B\right)} \cos\left(x\right)^{2} - 15 \, {\left(4 \, A - 3 \, B\right)} \cos\left(x\right) - {\left(2 \, {\left(3 \, A - 4 \, B\right)} \cos\left(x\right)^{3} - 6 \, {\left(3 \, A - 4 \, B\right)} \cos\left(x\right)^{2} - 15 \, {\left(3 \, A - 4 \, B\right)} \cos\left(x\right) + 15 \, A + 15 \, B\right)} \sin\left(x\right) - 15 \, A - 15 \, B}{105 \, {\left(\cos\left(x\right)^{4} - 3 \, \cos\left(x\right)^{3} - 8 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{3} + 4 \, \cos\left(x\right)^{2} - 4 \, \cos\left(x\right) - 8\right)} \sin\left(x\right) + 4 \, \cos\left(x\right) + 8\right)}}"," ",0,"-1/105*(2*(3*A - 4*B)*cos(x)^4 + 8*(3*A - 4*B)*cos(x)^3 - 9*(3*A - 4*B)*cos(x)^2 - 15*(4*A - 3*B)*cos(x) - (2*(3*A - 4*B)*cos(x)^3 - 6*(3*A - 4*B)*cos(x)^2 - 15*(3*A - 4*B)*cos(x) + 15*A + 15*B)*sin(x) - 15*A - 15*B)/(cos(x)^4 - 3*cos(x)^3 - 8*cos(x)^2 + (cos(x)^3 + 4*cos(x)^2 - 4*cos(x) - 8)*sin(x) + 4*cos(x) + 8)","B",0
482,0,0,0,0.829867," ","integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(a c^{2} + 2 \, a c d + a d^{2} - {\left(2 \, a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} - {\left(a d^{2} \cos\left(f x + e\right)^{2} - a c^{2} - 2 \, a c d - a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((a*c^2 + 2*a*c*d + a*d^2 - (2*a*c*d + a*d^2)*cos(f*x + e)^2 - (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
483,0,0,0,1.274883," ","integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a d \cos\left(f x + e\right)^{2} - a c - a d - {\left(a c + a d\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral(-(a*d*cos(f*x + e)^2 - a*c - a*d - (a*c + a*d)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
484,0,0,0,1.163733," ","integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c), x)","F",0
485,0,0,0,0.893472," ","integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{a \sin\left(f x + e\right) + a}{\sqrt{d \sin\left(f x + e\right) + c}}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)/sqrt(d*sin(f*x + e) + c), x)","F",0
486,0,0,0,0.981757," ","integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(f x + e\right) + a\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral(-(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
487,0,0,0,1.129945," ","integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(f x + e\right) + a\right)} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
488,0,0,0,1.192707," ","integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a \sin\left(f x + e\right) + a\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 6 \, c^{2} d^{2} + d^{4} - 2 \, {\left(3 \, c^{2} d^{2} + d^{4}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left(c d^{3} \cos\left(f x + e\right)^{2} - c^{3} d - c d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(d^4*cos(f*x + e)^4 + c^4 + 6*c^2*d^2 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)^2 - 4*(c*d^3*cos(f*x + e)^2 - c^3*d - c*d^3)*sin(f*x + e)), x)","F",0
489,0,0,0,1.267773," ","integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(a^{2} d^{2} \cos\left(f x + e\right)^{4} + 2 \, a^{2} c^{2} + 4 \, a^{2} c d + 2 \, a^{2} d^{2} - {\left(a^{2} c^{2} + 4 \, a^{2} c d + 3 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{2} c^{2} + 2 \, a^{2} c d + a^{2} d^{2} - {\left(a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((a^2*d^2*cos(f*x + e)^4 + 2*a^2*c^2 + 4*a^2*c*d + 2*a^2*d^2 - (a^2*c^2 + 4*a^2*c*d + 3*a^2*d^2)*cos(f*x + e)^2 + 2*(a^2*c^2 + 2*a^2*c*d + a^2*d^2 - (a^2*c*d + a^2*d^2)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
490,0,0,0,1.226569," ","integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(2 \, a^{2} c + 2 \, a^{2} d - {\left(a^{2} c + 2 \, a^{2} d\right)} \cos\left(f x + e\right)^{2} - {\left(a^{2} d \cos\left(f x + e\right)^{2} - 2 \, a^{2} c - 2 \, a^{2} d\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((2*a^2*c + 2*a^2*d - (a^2*c + 2*a^2*d)*cos(f*x + e)^2 - (a^2*d*cos(f*x + e)^2 - 2*a^2*c - 2*a^2*d)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
491,0,0,0,1.197066," ","integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral(-(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*sqrt(d*sin(f*x + e) + c), x)","F",0
492,0,0,0,1.145413," ","integrate((a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}{\sqrt{d \sin\left(f x + e\right) + c}}, x\right)"," ",0,"integral(-(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)/sqrt(d*sin(f*x + e) + c), x)","F",0
493,0,0,0,1.289895," ","integrate((a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral((a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*sqrt(d*sin(f*x + e) + c)/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
494,0,0,0,1.017752," ","integrate((a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*sqrt(d*sin(f*x + e) + c)/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
495,0,0,0,1.007617," ","integrate((a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 6 \, c^{2} d^{2} + d^{4} - 2 \, {\left(3 \, c^{2} d^{2} + d^{4}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left(c d^{3} \cos\left(f x + e\right)^{2} - c^{3} d - c d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*sqrt(d*sin(f*x + e) + c)/(d^4*cos(f*x + e)^4 + c^4 + 6*c^2*d^2 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)^2 - 4*(c*d^3*cos(f*x + e)^2 - c^3*d - c*d^3)*sin(f*x + e)), x)","F",0
496,0,0,0,1.219576," ","integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} + {\left(2 \, a^{3} c d + 3 \, a^{3} d^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, a^{3} c^{2} + 10 \, a^{3} c d + 7 \, a^{3} d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a^{3} d^{2} \cos\left(f x + e\right)^{4} + 4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} - {\left(a^{3} c^{2} + 6 \, a^{3} c d + 5 \, a^{3} d^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((4*a^3*c^2 + 8*a^3*c*d + 4*a^3*d^2 + (2*a^3*c*d + 3*a^3*d^2)*cos(f*x + e)^4 - (3*a^3*c^2 + 10*a^3*c*d + 7*a^3*d^2)*cos(f*x + e)^2 + (a^3*d^2*cos(f*x + e)^4 + 4*a^3*c^2 + 8*a^3*c*d + 4*a^3*d^2 - (a^3*c^2 + 6*a^3*c*d + 5*a^3*d^2)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
497,0,0,0,1.017201," ","integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(a^{3} d \cos\left(f x + e\right)^{4} + 4 \, a^{3} c + 4 \, a^{3} d - {\left(3 \, a^{3} c + 5 \, a^{3} d\right)} \cos\left(f x + e\right)^{2} + {\left(4 \, a^{3} c + 4 \, a^{3} d - {\left(a^{3} c + 3 \, a^{3} d\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((a^3*d*cos(f*x + e)^4 + 4*a^3*c + 4*a^3*d - (3*a^3*c + 5*a^3*d)*cos(f*x + e)^2 + (4*a^3*c + 4*a^3*d - (a^3*c + 3*a^3*d)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
498,0,0,0,1.218352," ","integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral(-(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
499,0,0,0,1.098433," ","integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)}{\sqrt{d \sin\left(f x + e\right) + c}}, x\right)"," ",0,"integral(-(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))/sqrt(d*sin(f*x + e) + c), x)","F",0
500,0,0,0,0.766134," ","integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral((3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c)/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
501,0,0,0,1.472339," ","integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c)/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
502,0,0,0,1.244172," ","integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 6 \, c^{2} d^{2} + d^{4} - 2 \, {\left(3 \, c^{2} d^{2} + d^{4}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left(c d^{3} \cos\left(f x + e\right)^{2} - c^{3} d - c d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c)/(d^4*cos(f*x + e)^4 + c^4 + 6*c^2*d^2 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)^2 - 4*(c*d^3*cos(f*x + e)^2 - c^3*d - c*d^3)*sin(f*x + e)), x)","F",0
503,0,0,0,1.246749," ","integrate((a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(9/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}}{5 \, c d^{4} \cos\left(f x + e\right)^{4} + c^{5} + 10 \, c^{3} d^{2} + 5 \, c d^{4} - 10 \, {\left(c^{3} d^{2} + c d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(d^{5} \cos\left(f x + e\right)^{4} + 5 \, c^{4} d + 10 \, c^{2} d^{3} + d^{5} - 2 \, {\left(5 \, c^{2} d^{3} + d^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c)/(5*c*d^4*cos(f*x + e)^4 + c^5 + 10*c^3*d^2 + 5*c*d^4 - 10*(c^3*d^2 + c*d^4)*cos(f*x + e)^2 + (d^5*cos(f*x + e)^4 + 5*c^4*d + 10*c^2*d^3 + d^5 - 2*(5*c^2*d^3 + d^5)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
504,0,0,0,1.128739," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c}}{a \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral(-(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)*sqrt(d*sin(f*x + e) + c)/(a*sin(f*x + e) + a), x)","F",0
505,0,0,0,1.115216," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(d \sin\left(f x + e\right) + c\right)}^{\frac{3}{2}}}{a \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral((d*sin(f*x + e) + c)^(3/2)/(a*sin(f*x + e) + a), x)","F",0
506,0,0,0,0.925500," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{d \sin\left(f x + e\right) + c}}{a \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral(sqrt(d*sin(f*x + e) + c)/(a*sin(f*x + e) + a), x)","F",0
507,0,0,0,0.907595," ","integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{d \sin\left(f x + e\right) + c}}{a d \cos\left(f x + e\right)^{2} - a c - a d - {\left(a c + a d\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(d*sin(f*x + e) + c)/(a*d*cos(f*x + e)^2 - a*c - a*d - (a*c + a*d)*sin(f*x + e)), x)","F",0
508,0,0,0,0.875568," ","integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{d \sin\left(f x + e\right) + c}}{a c^{2} + 2 \, a c d + a d^{2} - {\left(2 \, a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} - {\left(a d^{2} \cos\left(f x + e\right)^{2} - a c^{2} - 2 \, a c d - a d^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(d*sin(f*x + e) + c)/(a*c^2 + 2*a*c*d + a*d^2 - (2*a*c*d + a*d^2)*cos(f*x + e)^2 - (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2)*sin(f*x + e)), x)","F",0
509,0,0,0,1.149000," ","integrate(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{d \sin\left(f x + e\right) + c}}{a d^{3} \cos\left(f x + e\right)^{4} + a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3} - {\left(3 \, a c^{2} d + 3 \, a c d^{2} + 2 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3} - {\left(3 \, a c d^{2} + a d^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(d*sin(f*x + e) + c)/(a*d^3*cos(f*x + e)^4 + a*c^3 + 3*a*c^2*d + 3*a*c*d^2 + a*d^3 - (3*a*c^2*d + 3*a*c*d^2 + 2*a*d^3)*cos(f*x + e)^2 + (a*c^3 + 3*a*c^2*d + 3*a*c*d^2 + a*d^3 - (3*a*c*d^2 + a*d^3)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
510,0,0,0,1.159666," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral((d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)*sqrt(d*sin(f*x + e) + c)/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
511,0,0,0,1.038715," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(d \sin\left(f x + e\right) + c\right)}^{\frac{3}{2}}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-(d*sin(f*x + e) + c)^(3/2)/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
512,0,0,0,1.179681," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{d \sin\left(f x + e\right) + c}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-sqrt(d*sin(f*x + e) + c)/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
513,0,0,0,1.032034," ","integrate(1/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{d \sin\left(f x + e\right) + c}}{2 \, a^{2} c + 2 \, a^{2} d - {\left(a^{2} c + 2 \, a^{2} d\right)} \cos\left(f x + e\right)^{2} - {\left(a^{2} d \cos\left(f x + e\right)^{2} - 2 \, a^{2} c - 2 \, a^{2} d\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(d*sin(f*x + e) + c)/(2*a^2*c + 2*a^2*d - (a^2*c + 2*a^2*d)*cos(f*x + e)^2 - (a^2*d*cos(f*x + e)^2 - 2*a^2*c - 2*a^2*d)*sin(f*x + e)), x)","F",0
514,0,0,0,1.318361," ","integrate(1/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{d \sin\left(f x + e\right) + c}}{a^{2} d^{2} \cos\left(f x + e\right)^{4} + 2 \, a^{2} c^{2} + 4 \, a^{2} c d + 2 \, a^{2} d^{2} - {\left(a^{2} c^{2} + 4 \, a^{2} c d + 3 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{2} c^{2} + 2 \, a^{2} c d + a^{2} d^{2} - {\left(a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(d*sin(f*x + e) + c)/(a^2*d^2*cos(f*x + e)^4 + 2*a^2*c^2 + 4*a^2*c*d + 2*a^2*d^2 - (a^2*c^2 + 4*a^2*c*d + 3*a^2*d^2)*cos(f*x + e)^2 + 2*(a^2*c^2 + 2*a^2*c*d + a^2*d^2 - (a^2*c*d + a^2*d^2)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
515,0,0,0,1.417150," ","integrate(1/(a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{d \sin\left(f x + e\right) + c}}{2 \, a^{2} c^{3} + 6 \, a^{2} c^{2} d + 6 \, a^{2} c d^{2} + 2 \, a^{2} d^{3} + {\left(3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{4} - {\left(a^{2} c^{3} + 6 \, a^{2} c^{2} d + 9 \, a^{2} c d^{2} + 4 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(a^{2} d^{3} \cos\left(f x + e\right)^{4} + 2 \, a^{2} c^{3} + 6 \, a^{2} c^{2} d + 6 \, a^{2} c d^{2} + 2 \, a^{2} d^{3} - 3 \, {\left(a^{2} c^{2} d + 2 \, a^{2} c d^{2} + a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(d*sin(f*x + e) + c)/(2*a^2*c^3 + 6*a^2*c^2*d + 6*a^2*c*d^2 + 2*a^2*d^3 + (3*a^2*c*d^2 + 2*a^2*d^3)*cos(f*x + e)^4 - (a^2*c^3 + 6*a^2*c^2*d + 9*a^2*c*d^2 + 4*a^2*d^3)*cos(f*x + e)^2 + (a^2*d^3*cos(f*x + e)^4 + 2*a^2*c^3 + 6*a^2*c^2*d + 6*a^2*c*d^2 + 2*a^2*d^3 - 3*(a^2*c^2*d + 2*a^2*c*d^2 + a^2*d^3)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
516,0,0,0,1.049056," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)*sqrt(d*sin(f*x + e) + c)/(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e)), x)","F",0
517,0,0,0,1.110594," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(d \sin\left(f x + e\right) + c\right)}^{\frac{3}{2}}}{3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(d*sin(f*x + e) + c)^(3/2)/(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e)), x)","F",0
518,0,0,0,1.258474," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{d \sin\left(f x + e\right) + c}}{3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(d*sin(f*x + e) + c)/(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e)), x)","F",0
519,0,0,0,1.123124," ","integrate(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{d \sin\left(f x + e\right) + c}}{a^{3} d \cos\left(f x + e\right)^{4} + 4 \, a^{3} c + 4 \, a^{3} d - {\left(3 \, a^{3} c + 5 \, a^{3} d\right)} \cos\left(f x + e\right)^{2} + {\left(4 \, a^{3} c + 4 \, a^{3} d - {\left(a^{3} c + 3 \, a^{3} d\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(d*sin(f*x + e) + c)/(a^3*d*cos(f*x + e)^4 + 4*a^3*c + 4*a^3*d - (3*a^3*c + 5*a^3*d)*cos(f*x + e)^2 + (4*a^3*c + 4*a^3*d - (a^3*c + 3*a^3*d)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
520,0,0,0,1.023682," ","integrate(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{d \sin\left(f x + e\right) + c}}{4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} + {\left(2 \, a^{3} c d + 3 \, a^{3} d^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, a^{3} c^{2} + 10 \, a^{3} c d + 7 \, a^{3} d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a^{3} d^{2} \cos\left(f x + e\right)^{4} + 4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} - {\left(a^{3} c^{2} + 6 \, a^{3} c d + 5 \, a^{3} d^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(d*sin(f*x + e) + c)/(4*a^3*c^2 + 8*a^3*c*d + 4*a^3*d^2 + (2*a^3*c*d + 3*a^3*d^2)*cos(f*x + e)^4 - (3*a^3*c^2 + 10*a^3*c*d + 7*a^3*d^2)*cos(f*x + e)^2 + (a^3*d^2*cos(f*x + e)^4 + 4*a^3*c^2 + 8*a^3*c*d + 4*a^3*d^2 - (a^3*c^2 + 6*a^3*c*d + 5*a^3*d^2)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
521,0,0,0,1.912813," ","integrate(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{d \sin\left(f x + e\right) + c}}{a^{3} d^{3} \cos\left(f x + e\right)^{6} - 4 \, a^{3} c^{3} - 12 \, a^{3} c^{2} d - 12 \, a^{3} c d^{2} - 4 \, a^{3} d^{3} - 3 \, {\left(a^{3} c^{2} d + 3 \, a^{3} c d^{2} + 2 \, a^{3} d^{3}\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(a^{3} c^{3} + 5 \, a^{3} c^{2} d + 7 \, a^{3} c d^{2} + 3 \, a^{3} d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(4 \, a^{3} c^{3} + 12 \, a^{3} c^{2} d + 12 \, a^{3} c d^{2} + 4 \, a^{3} d^{3} + 3 \, {\left(a^{3} c d^{2} + a^{3} d^{3}\right)} \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{3} + 9 \, a^{3} c^{2} d + 15 \, a^{3} c d^{2} + 7 \, a^{3} d^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(d*sin(f*x + e) + c)/(a^3*d^3*cos(f*x + e)^6 - 4*a^3*c^3 - 12*a^3*c^2*d - 12*a^3*c*d^2 - 4*a^3*d^3 - 3*(a^3*c^2*d + 3*a^3*c*d^2 + 2*a^3*d^3)*cos(f*x + e)^4 + 3*(a^3*c^3 + 5*a^3*c^2*d + 7*a^3*c*d^2 + 3*a^3*d^3)*cos(f*x + e)^2 - (4*a^3*c^3 + 12*a^3*c^2*d + 12*a^3*c*d^2 + 4*a^3*d^3 + 3*(a^3*c*d^2 + a^3*d^3)*cos(f*x + e)^4 - (a^3*c^3 + 9*a^3*c^2*d + 15*a^3*c*d^2 + 7*a^3*d^3)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
522,1,240,0,0.923826," ","integrate((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{2 \, {\left(5 \, d^{3} \cos\left(f x + e\right)^{4} + 3 \, {\left(7 \, c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)^{3} - 35 \, c^{3} - 35 \, c^{2} d - 49 \, c d^{2} - 9 \, d^{3} - {\left(35 \, c^{2} d + 7 \, c d^{2} + 12 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(35 \, c^{3} + 70 \, c^{2} d + 77 \, c d^{2} + 22 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(5 \, d^{3} \cos\left(f x + e\right)^{3} + 35 \, c^{3} + 35 \, c^{2} d + 49 \, c d^{2} + 9 \, d^{3} - {\left(21 \, c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(35 \, c^{2} d + 28 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{35 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/35*(5*d^3*cos(f*x + e)^4 + 3*(7*c*d^2 + 2*d^3)*cos(f*x + e)^3 - 35*c^3 - 35*c^2*d - 49*c*d^2 - 9*d^3 - (35*c^2*d + 7*c*d^2 + 12*d^3)*cos(f*x + e)^2 - (35*c^3 + 70*c^2*d + 77*c*d^2 + 22*d^3)*cos(f*x + e) + (5*d^3*cos(f*x + e)^3 + 35*c^3 + 35*c^2*d + 49*c*d^2 + 9*d^3 - (21*c*d^2 + d^3)*cos(f*x + e)^2 - (35*c^2*d + 28*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
523,1,157,0,1.139511," ","integrate((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, d^{2} \cos\left(f x + e\right)^{3} - {\left(10 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - 15 \, c^{2} - 10 \, c d - 7 \, d^{2} - {\left(15 \, c^{2} + 20 \, c d + 11 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(3 \, d^{2} \cos\left(f x + e\right)^{2} - 15 \, c^{2} - 10 \, c d - 7 \, d^{2} + 2 \, {\left(5 \, c d + 2 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{15 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/15*(3*d^2*cos(f*x + e)^3 - (10*c*d + d^2)*cos(f*x + e)^2 - 15*c^2 - 10*c*d - 7*d^2 - (15*c^2 + 20*c*d + 11*d^2)*cos(f*x + e) - (3*d^2*cos(f*x + e)^2 - 15*c^2 - 10*c*d - 7*d^2 + 2*(5*c*d + 2*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
524,1,85,0,1.022778," ","integrate((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{2 \, {\left(d \cos\left(f x + e\right)^{2} + {\left(3 \, c + 2 \, d\right)} \cos\left(f x + e\right) + {\left(d \cos\left(f x + e\right) - 3 \, c - d\right)} \sin\left(f x + e\right) + 3 \, c + d\right)} \sqrt{a \sin\left(f x + e\right) + a}}{3 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"-2/3*(d*cos(f*x + e)^2 + (3*c + 2*d)*cos(f*x + e) + (d*cos(f*x + e) - 3*c - d)*sin(f*x + e) + 3*c + d)*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
525,1,50,0,1.361133," ","integrate((a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f}"," ",0,"-2*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","B",0
526,1,464,0,1.586184," ","integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[\frac{\sqrt{\frac{a}{c d + d^{2}}} \log\left(\frac{a d^{2} \cos\left(f x + e\right)^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left(6 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{a}{c d + d^{2}}} - {\left(a c^{2} + 8 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(a d^{2} \cos\left(f x + e\right)^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left(3 \, a c d + 4 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right)}{2 \, f}, -\frac{\sqrt{-\frac{a}{c d + d^{2}}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{a}{c d + d^{2}}}}{2 \, a \cos\left(f x + e\right)}\right)}{f}\right]"," ",0,"[1/2*sqrt(a/(c*d + d^2))*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e)^2 + 4*(c^2*d + 4*c*d^2 + 3*d^3 - (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d + 4*c*d^2 + 3*d^3 + (c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2)) - (a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e)))/f, -sqrt(-a/(c*d + d^2))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f*x + e)))/f]","A",0
527,1,786,0,1.374687," ","integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{{\left(d \cos\left(f x + e\right)^{2} - c \cos\left(f x + e\right) - {\left(d \cos\left(f x + e\right) + c + d\right)} \sin\left(f x + e\right) - c - d\right)} \sqrt{\frac{a}{c d + d^{2}}} \log\left(\frac{a d^{2} \cos\left(f x + e\right)^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left(6 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{a}{c d + d^{2}}} - {\left(a c^{2} + 8 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(a d^{2} \cos\left(f x + e\right)^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left(3 \, a c d + 4 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{4 \, {\left({\left(c d + d^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{2} + c d\right)} f \cos\left(f x + e\right) - {\left(c^{2} + 2 \, c d + d^{2}\right)} f - {\left({\left(c d + d^{2}\right)} f \cos\left(f x + e\right) + {\left(c^{2} + 2 \, c d + d^{2}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left(d \cos\left(f x + e\right)^{2} - c \cos\left(f x + e\right) - {\left(d \cos\left(f x + e\right) + c + d\right)} \sin\left(f x + e\right) - c - d\right)} \sqrt{-\frac{a}{c d + d^{2}}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{a}{c d + d^{2}}}}{2 \, a \cos\left(f x + e\right)}\right) - 2 \, \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{2 \, {\left({\left(c d + d^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{2} + c d\right)} f \cos\left(f x + e\right) - {\left(c^{2} + 2 \, c d + d^{2}\right)} f - {\left({\left(c d + d^{2}\right)} f \cos\left(f x + e\right) + {\left(c^{2} + 2 \, c d + d^{2}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/4*((d*cos(f*x + e)^2 - c*cos(f*x + e) - (d*cos(f*x + e) + c + d)*sin(f*x + e) - c - d)*sqrt(a/(c*d + d^2))*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e)^2 + 4*(c^2*d + 4*c*d^2 + 3*d^3 - (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d + 4*c*d^2 + 3*d^3 + (c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2)) - (a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1))/((c*d + d^2)*f*cos(f*x + e)^2 - (c^2 + c*d)*f*cos(f*x + e) - (c^2 + 2*c*d + d^2)*f - ((c*d + d^2)*f*cos(f*x + e) + (c^2 + 2*c*d + d^2)*f)*sin(f*x + e)), -1/2*((d*cos(f*x + e)^2 - c*cos(f*x + e) - (d*cos(f*x + e) + c + d)*sin(f*x + e) - c - d)*sqrt(-a/(c*d + d^2))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f*x + e))) - 2*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1))/((c*d + d^2)*f*cos(f*x + e)^2 - (c^2 + c*d)*f*cos(f*x + e) - (c^2 + 2*c*d + d^2)*f - ((c*d + d^2)*f*cos(f*x + e) + (c^2 + 2*c*d + d^2)*f)*sin(f*x + e))]","B",0
528,1,1250,0,1.917195," ","integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{c d + d^{2}}} \log\left(\frac{a d^{2} \cos\left(f x + e\right)^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left(6 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{a}{c d + d^{2}}} - {\left(a c^{2} + 8 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(a d^{2} \cos\left(f x + e\right)^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left(3 \, a c d + 4 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(3 \, d \cos\left(f x + e\right)^{2} + {\left(5 \, c + 2 \, d\right)} \cos\left(f x + e\right) + {\left(3 \, d \cos\left(f x + e\right) - 5 \, c + d\right)} \sin\left(f x + e\right) + 5 \, c - d\right)} \sqrt{a \sin\left(f x + e\right) + a}}{16 \, {\left({\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, c^{3} d + 5 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{4} + 2 \, c^{3} d + 2 \, c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right) - {\left(c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f + {\left({\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d + 2 \, c^{2} d^{2} + c d^{3}\right)} f \cos\left(f x + e\right) - {\left(c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{3 \, {\left(d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{c d + d^{2}}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{a}{c d + d^{2}}}}{2 \, a \cos\left(f x + e\right)}\right) - 2 \, {\left(3 \, d \cos\left(f x + e\right)^{2} + {\left(5 \, c + 2 \, d\right)} \cos\left(f x + e\right) + {\left(3 \, d \cos\left(f x + e\right) - 5 \, c + d\right)} \sin\left(f x + e\right) + 5 \, c - d\right)} \sqrt{a \sin\left(f x + e\right) + a}}{8 \, {\left({\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, c^{3} d + 5 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{4} + 2 \, c^{3} d + 2 \, c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right) - {\left(c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f + {\left({\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d + 2 \, c^{2} d^{2} + c d^{3}\right)} f \cos\left(f x + e\right) - {\left(c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/16*(3*(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))*sqrt(a/(c*d + d^2))*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e)^2 + 4*(c^2*d + 4*c*d^2 + 3*d^3 - (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d + 4*c*d^2 + 3*d^3 + (c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2)) - (a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(3*d*cos(f*x + e)^2 + (5*c + 2*d)*cos(f*x + e) + (3*d*cos(f*x + e) - 5*c + d)*sin(f*x + e) + 5*c - d)*sqrt(a*sin(f*x + e) + a))/((c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e)^3 + (2*c^3*d + 5*c^2*d^2 + 4*c*d^3 + d^4)*f*cos(f*x + e)^2 - (c^4 + 2*c^3*d + 2*c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e) - (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*f + ((c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e)^2 - 2*(c^3*d + 2*c^2*d^2 + c*d^3)*f*cos(f*x + e) - (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*f)*sin(f*x + e)), -1/8*(3*(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))*sqrt(-a/(c*d + d^2))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f*x + e))) - 2*(3*d*cos(f*x + e)^2 + (5*c + 2*d)*cos(f*x + e) + (3*d*cos(f*x + e) - 5*c + d)*sin(f*x + e) + 5*c - d)*sqrt(a*sin(f*x + e) + a))/((c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e)^3 + (2*c^3*d + 5*c^2*d^2 + 4*c*d^3 + d^4)*f*cos(f*x + e)^2 - (c^4 + 2*c^3*d + 2*c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e) - (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*f + ((c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e)^2 - 2*(c^3*d + 2*c^2*d^2 + c*d^3)*f*cos(f*x + e) - (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*f)*sin(f*x + e))]","B",0
529,1,339,0,1.333089," ","integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{2 \, {\left(35 \, a d^{3} \cos\left(f x + e\right)^{5} - 5 \, {\left(27 \, a c d^{2} + 10 \, a d^{3}\right)} \cos\left(f x + e\right)^{4} + 420 \, a c^{3} + 756 \, a c^{2} d + 684 \, a c d^{2} + 188 \, a d^{3} - {\left(189 \, a c^{2} d + 351 \, a c d^{2} + 172 \, a d^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(105 \, a c^{3} + 378 \, a c^{2} d + 387 \, a c d^{2} + 134 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(525 \, a c^{3} + 1323 \, a c^{2} d + 1287 \, a c d^{2} + 409 \, a d^{3}\right)} \cos\left(f x + e\right) - {\left(35 \, a d^{3} \cos\left(f x + e\right)^{4} + 420 \, a c^{3} + 756 \, a c^{2} d + 684 \, a c d^{2} + 188 \, a d^{3} + 5 \, {\left(27 \, a c d^{2} + 17 \, a d^{3}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(63 \, a c^{2} d + 72 \, a c d^{2} + 29 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(105 \, a c^{3} + 567 \, a c^{2} d + 603 \, a c d^{2} + 221 \, a d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{315 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"-2/315*(35*a*d^3*cos(f*x + e)^5 - 5*(27*a*c*d^2 + 10*a*d^3)*cos(f*x + e)^4 + 420*a*c^3 + 756*a*c^2*d + 684*a*c*d^2 + 188*a*d^3 - (189*a*c^2*d + 351*a*c*d^2 + 172*a*d^3)*cos(f*x + e)^3 + (105*a*c^3 + 378*a*c^2*d + 387*a*c*d^2 + 134*a*d^3)*cos(f*x + e)^2 + (525*a*c^3 + 1323*a*c^2*d + 1287*a*c*d^2 + 409*a*d^3)*cos(f*x + e) - (35*a*d^3*cos(f*x + e)^4 + 420*a*c^3 + 756*a*c^2*d + 684*a*c*d^2 + 188*a*d^3 + 5*(27*a*c*d^2 + 17*a*d^3)*cos(f*x + e)^3 - 3*(63*a*c^2*d + 72*a*c*d^2 + 29*a*d^3)*cos(f*x + e)^2 - (105*a*c^3 + 567*a*c^2*d + 603*a*c*d^2 + 221*a*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
530,1,229,0,1.074308," ","integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{2 \, {\left(15 \, a d^{2} \cos\left(f x + e\right)^{4} + 3 \, {\left(14 \, a c d + 13 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 140 \, a c^{2} - 168 \, a c d - 76 \, a d^{2} - {\left(35 \, a c^{2} + 84 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - {\left(175 \, a c^{2} + 294 \, a c d + 143 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(15 \, a d^{2} \cos\left(f x + e\right)^{3} + 140 \, a c^{2} + 168 \, a c d + 76 \, a d^{2} - 6 \, {\left(7 \, a c d + 4 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - {\left(35 \, a c^{2} + 126 \, a c d + 67 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{105 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/105*(15*a*d^2*cos(f*x + e)^4 + 3*(14*a*c*d + 13*a*d^2)*cos(f*x + e)^3 - 140*a*c^2 - 168*a*c*d - 76*a*d^2 - (35*a*c^2 + 84*a*c*d + 43*a*d^2)*cos(f*x + e)^2 - (175*a*c^2 + 294*a*c*d + 143*a*d^2)*cos(f*x + e) + (15*a*d^2*cos(f*x + e)^3 + 140*a*c^2 + 168*a*c*d + 76*a*d^2 - 6*(7*a*c*d + 4*a*d^2)*cos(f*x + e)^2 - (35*a*c^2 + 126*a*c*d + 67*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
531,1,136,0,0.781401," ","integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, a d \cos\left(f x + e\right)^{3} - {\left(5 \, a c + 6 \, a d\right)} \cos\left(f x + e\right)^{2} - 20 \, a c - 12 \, a d - {\left(25 \, a c + 21 \, a d\right)} \cos\left(f x + e\right) - {\left(3 \, a d \cos\left(f x + e\right)^{2} - 20 \, a c - 12 \, a d + {\left(5 \, a c + 9 \, a d\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{15 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/15*(3*a*d*cos(f*x + e)^3 - (5*a*c + 6*a*d)*cos(f*x + e)^2 - 20*a*c - 12*a*d - (25*a*c + 21*a*d)*cos(f*x + e) - (3*a*d*cos(f*x + e)^2 - 20*a*c - 12*a*d + (5*a*c + 9*a*d)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
532,1,76,0,0.945249," ","integrate((a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} + 5 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right) - 4 \, a\right)} \sin\left(f x + e\right) + 4 \, a\right)} \sqrt{a \sin\left(f x + e\right) + a}}{3 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"-2/3*(a*cos(f*x + e)^2 + 5*a*cos(f*x + e) + (a*cos(f*x + e) - 4*a)*sin(f*x + e) + 4*a)*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
533,1,651,0,1.338220," ","integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[-\frac{{\left(a c - a d + {\left(a c - a d\right)} \cos\left(f x + e\right) + {\left(a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{c d + d^{2}}} \log\left(\frac{a d^{2} \cos\left(f x + e\right)^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left(6 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{a}{c d + d^{2}}} - {\left(a c^{2} + 8 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(a d^{2} \cos\left(f x + e\right)^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left(3 \, a c d + 4 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(a \cos\left(f x + e\right) - a \sin\left(f x + e\right) + a\right)} \sqrt{a \sin\left(f x + e\right) + a}}{2 \, {\left(d f \cos\left(f x + e\right) + d f \sin\left(f x + e\right) + d f\right)}}, \frac{{\left(a c - a d + {\left(a c - a d\right)} \cos\left(f x + e\right) + {\left(a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{c d + d^{2}}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{a}{c d + d^{2}}}}{2 \, a \cos\left(f x + e\right)}\right) - 2 \, {\left(a \cos\left(f x + e\right) - a \sin\left(f x + e\right) + a\right)} \sqrt{a \sin\left(f x + e\right) + a}}{d f \cos\left(f x + e\right) + d f \sin\left(f x + e\right) + d f}\right]"," ",0,"[-1/2*((a*c - a*d + (a*c - a*d)*cos(f*x + e) + (a*c - a*d)*sin(f*x + e))*sqrt(a/(c*d + d^2))*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e)^2 + 4*(c^2*d + 4*c*d^2 + 3*d^3 - (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d + 4*c*d^2 + 3*d^3 + (c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2)) - (a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(a*cos(f*x + e) - a*sin(f*x + e) + a)*sqrt(a*sin(f*x + e) + a))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f), ((a*c - a*d + (a*c - a*d)*cos(f*x + e) + (a*c - a*d)*sin(f*x + e))*sqrt(-a/(c*d + d^2))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f*x + e))) - 2*(a*cos(f*x + e) - a*sin(f*x + e) + a)*sqrt(a*sin(f*x + e) + a))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f)]","B",0
534,1,970,0,1.108769," ","integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[-\frac{{\left(a c^{2} + 4 \, a c d + 3 \, a d^{2} - {\left(a c d + 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + 3 \, a c d\right)} \cos\left(f x + e\right) + {\left(a c^{2} + 4 \, a c d + 3 \, a d^{2} + {\left(a c d + 3 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{c d + d^{2}}} \log\left(\frac{a d^{2} \cos\left(f x + e\right)^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left(6 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{a}{c d + d^{2}}} - {\left(a c^{2} + 8 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(a d^{2} \cos\left(f x + e\right)^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left(3 \, a c d + 4 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(a c - a d + {\left(a c - a d\right)} \cos\left(f x + e\right) - {\left(a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{4 \, {\left({\left(c d^{2} + d^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{2} d + c d^{2}\right)} f \cos\left(f x + e\right) - {\left(c^{2} d + 2 \, c d^{2} + d^{3}\right)} f - {\left({\left(c d^{2} + d^{3}\right)} f \cos\left(f x + e\right) + {\left(c^{2} d + 2 \, c d^{2} + d^{3}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{{\left(a c^{2} + 4 \, a c d + 3 \, a d^{2} - {\left(a c d + 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + 3 \, a c d\right)} \cos\left(f x + e\right) + {\left(a c^{2} + 4 \, a c d + 3 \, a d^{2} + {\left(a c d + 3 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{c d + d^{2}}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{a}{c d + d^{2}}}}{2 \, a \cos\left(f x + e\right)}\right) - 2 \, {\left(a c - a d + {\left(a c - a d\right)} \cos\left(f x + e\right) - {\left(a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{2 \, {\left({\left(c d^{2} + d^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{2} d + c d^{2}\right)} f \cos\left(f x + e\right) - {\left(c^{2} d + 2 \, c d^{2} + d^{3}\right)} f - {\left({\left(c d^{2} + d^{3}\right)} f \cos\left(f x + e\right) + {\left(c^{2} d + 2 \, c d^{2} + d^{3}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/4*((a*c^2 + 4*a*c*d + 3*a*d^2 - (a*c*d + 3*a*d^2)*cos(f*x + e)^2 + (a*c^2 + 3*a*c*d)*cos(f*x + e) + (a*c^2 + 4*a*c*d + 3*a*d^2 + (a*c*d + 3*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a/(c*d + d^2))*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e)^2 + 4*(c^2*d + 4*c*d^2 + 3*d^3 - (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d + 4*c*d^2 + 3*d^3 + (c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2)) - (a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(a*c - a*d + (a*c - a*d)*cos(f*x + e) - (a*c - a*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((c*d^2 + d^3)*f*cos(f*x + e)^2 - (c^2*d + c*d^2)*f*cos(f*x + e) - (c^2*d + 2*c*d^2 + d^3)*f - ((c*d^2 + d^3)*f*cos(f*x + e) + (c^2*d + 2*c*d^2 + d^3)*f)*sin(f*x + e)), 1/2*((a*c^2 + 4*a*c*d + 3*a*d^2 - (a*c*d + 3*a*d^2)*cos(f*x + e)^2 + (a*c^2 + 3*a*c*d)*cos(f*x + e) + (a*c^2 + 4*a*c*d + 3*a*d^2 + (a*c*d + 3*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-a/(c*d + d^2))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f*x + e))) - 2*(a*c - a*d + (a*c - a*d)*cos(f*x + e) - (a*c - a*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((c*d^2 + d^3)*f*cos(f*x + e)^2 - (c^2*d + c*d^2)*f*cos(f*x + e) - (c^2*d + 2*c*d^2 + d^3)*f - ((c*d^2 + d^3)*f*cos(f*x + e) + (c^2*d + 2*c*d^2 + d^3)*f)*sin(f*x + e))]","B",0
535,1,1558,0,1.702342," ","integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[-\frac{{\left(a c^{3} + 9 \, a c^{2} d + 15 \, a c d^{2} + 7 \, a d^{3} - {\left(a c d^{2} + 7 \, a d^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(2 \, a c^{2} d + 15 \, a c d^{2} + 7 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{3} + 7 \, a c^{2} d + a c d^{2} + 7 \, a d^{3}\right)} \cos\left(f x + e\right) + {\left(a c^{3} + 9 \, a c^{2} d + 15 \, a c d^{2} + 7 \, a d^{3} - {\left(a c d^{2} + 7 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a c^{2} d + 7 \, a c d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{c d + d^{2}}} \log\left(\frac{a d^{2} \cos\left(f x + e\right)^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left(6 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{a}{c d + d^{2}}} - {\left(a c^{2} + 8 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(a d^{2} \cos\left(f x + e\right)^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left(3 \, a c d + 4 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(a c^{2} - 6 \, a c d + 5 \, a d^{2} - {\left(a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} - 7 \, a c d - 2 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(a c^{2} - 6 \, a c d + 5 \, a d^{2} + {\left(a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{16 \, {\left({\left(c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, c^{3} d^{2} + 5 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{4} d + 2 \, c^{3} d^{2} + 2 \, c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d + 4 \, c^{3} d^{2} + 6 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5}\right)} f + {\left({\left(c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d^{2} + 2 \, c^{2} d^{3} + c d^{4}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d + 4 \, c^{3} d^{2} + 6 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{{\left(a c^{3} + 9 \, a c^{2} d + 15 \, a c d^{2} + 7 \, a d^{3} - {\left(a c d^{2} + 7 \, a d^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(2 \, a c^{2} d + 15 \, a c d^{2} + 7 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{3} + 7 \, a c^{2} d + a c d^{2} + 7 \, a d^{3}\right)} \cos\left(f x + e\right) + {\left(a c^{3} + 9 \, a c^{2} d + 15 \, a c d^{2} + 7 \, a d^{3} - {\left(a c d^{2} + 7 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a c^{2} d + 7 \, a c d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{c d + d^{2}}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{a}{c d + d^{2}}}}{2 \, a \cos\left(f x + e\right)}\right) - 2 \, {\left(a c^{2} - 6 \, a c d + 5 \, a d^{2} - {\left(a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} - 7 \, a c d - 2 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(a c^{2} - 6 \, a c d + 5 \, a d^{2} + {\left(a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{8 \, {\left({\left(c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, c^{3} d^{2} + 5 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{4} d + 2 \, c^{3} d^{2} + 2 \, c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d + 4 \, c^{3} d^{2} + 6 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5}\right)} f + {\left({\left(c^{2} d^{3} + 2 \, c d^{4} + d^{5}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d^{2} + 2 \, c^{2} d^{3} + c d^{4}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d + 4 \, c^{3} d^{2} + 6 \, c^{2} d^{3} + 4 \, c d^{4} + d^{5}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/16*((a*c^3 + 9*a*c^2*d + 15*a*c*d^2 + 7*a*d^3 - (a*c*d^2 + 7*a*d^3)*cos(f*x + e)^3 - (2*a*c^2*d + 15*a*c*d^2 + 7*a*d^3)*cos(f*x + e)^2 + (a*c^3 + 7*a*c^2*d + a*c*d^2 + 7*a*d^3)*cos(f*x + e) + (a*c^3 + 9*a*c^2*d + 15*a*c*d^2 + 7*a*d^3 - (a*c*d^2 + 7*a*d^3)*cos(f*x + e)^2 + 2*(a*c^2*d + 7*a*c*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a/(c*d + d^2))*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e)^2 + 4*(c^2*d + 4*c*d^2 + 3*d^3 - (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d + 4*c*d^2 + 3*d^3 + (c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2)) - (a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(a*c^2 - 6*a*c*d + 5*a*d^2 - (a*c*d + 7*a*d^2)*cos(f*x + e)^2 + (a*c^2 - 7*a*c*d - 2*a*d^2)*cos(f*x + e) - (a*c^2 - 6*a*c*d + 5*a*d^2 + (a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((c^2*d^3 + 2*c*d^4 + d^5)*f*cos(f*x + e)^3 + (2*c^3*d^2 + 5*c^2*d^3 + 4*c*d^4 + d^5)*f*cos(f*x + e)^2 - (c^4*d + 2*c^3*d^2 + 2*c^2*d^3 + 2*c*d^4 + d^5)*f*cos(f*x + e) - (c^4*d + 4*c^3*d^2 + 6*c^2*d^3 + 4*c*d^4 + d^5)*f + ((c^2*d^3 + 2*c*d^4 + d^5)*f*cos(f*x + e)^2 - 2*(c^3*d^2 + 2*c^2*d^3 + c*d^4)*f*cos(f*x + e) - (c^4*d + 4*c^3*d^2 + 6*c^2*d^3 + 4*c*d^4 + d^5)*f)*sin(f*x + e)), 1/8*((a*c^3 + 9*a*c^2*d + 15*a*c*d^2 + 7*a*d^3 - (a*c*d^2 + 7*a*d^3)*cos(f*x + e)^3 - (2*a*c^2*d + 15*a*c*d^2 + 7*a*d^3)*cos(f*x + e)^2 + (a*c^3 + 7*a*c^2*d + a*c*d^2 + 7*a*d^3)*cos(f*x + e) + (a*c^3 + 9*a*c^2*d + 15*a*c*d^2 + 7*a*d^3 - (a*c*d^2 + 7*a*d^3)*cos(f*x + e)^2 + 2*(a*c^2*d + 7*a*c*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-a/(c*d + d^2))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f*x + e))) - 2*(a*c^2 - 6*a*c*d + 5*a*d^2 - (a*c*d + 7*a*d^2)*cos(f*x + e)^2 + (a*c^2 - 7*a*c*d - 2*a*d^2)*cos(f*x + e) - (a*c^2 - 6*a*c*d + 5*a*d^2 + (a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((c^2*d^3 + 2*c*d^4 + d^5)*f*cos(f*x + e)^3 + (2*c^3*d^2 + 5*c^2*d^3 + 4*c*d^4 + d^5)*f*cos(f*x + e)^2 - (c^4*d + 2*c^3*d^2 + 2*c^2*d^3 + 2*c*d^4 + d^5)*f*cos(f*x + e) - (c^4*d + 4*c^3*d^2 + 6*c^2*d^3 + 4*c*d^4 + d^5)*f + ((c^2*d^3 + 2*c*d^4 + d^5)*f*cos(f*x + e)^2 - 2*(c^3*d^2 + 2*c^2*d^3 + c*d^4)*f*cos(f*x + e) - (c^4*d + 4*c^3*d^2 + 6*c^2*d^3 + 4*c*d^4 + d^5)*f)*sin(f*x + e))]","B",0
536,1,493,0,0.950711," ","integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{2 \, {\left(315 \, a^{2} d^{3} \cos\left(f x + e\right)^{6} + 35 \, {\left(33 \, a^{2} c d^{2} + 32 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{5} + 7392 \, a^{2} c^{3} + 15840 \, a^{2} c^{2} d + 13728 \, a^{2} c d^{2} + 4000 \, a^{2} d^{3} - 5 \, {\left(297 \, a^{2} c^{2} d + 627 \, a^{2} c d^{2} + 320 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{4} - {\left(693 \, a^{2} c^{3} + 5940 \, a^{2} c^{2} d + 9537 \, a^{2} c d^{2} + 4370 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(2541 \, a^{2} c^{3} + 8415 \, a^{2} c^{2} d + 8679 \, a^{2} c d^{2} + 2965 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(5313 \, a^{2} c^{3} + 14355 \, a^{2} c^{2} d + 13827 \, a^{2} c d^{2} + 4465 \, a^{2} d^{3}\right)} \cos\left(f x + e\right) + {\left(315 \, a^{2} d^{3} \cos\left(f x + e\right)^{5} - 7392 \, a^{2} c^{3} - 15840 \, a^{2} c^{2} d - 13728 \, a^{2} c d^{2} - 4000 \, a^{2} d^{3} - 35 \, {\left(33 \, a^{2} c d^{2} + 23 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{4} - 5 \, {\left(297 \, a^{2} c^{2} d + 858 \, a^{2} c d^{2} + 481 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(231 \, a^{2} c^{3} + 1485 \, a^{2} c^{2} d + 1749 \, a^{2} c d^{2} + 655 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(1617 \, a^{2} c^{3} + 6435 \, a^{2} c^{2} d + 6963 \, a^{2} c d^{2} + 2465 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{3465 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"-2/3465*(315*a^2*d^3*cos(f*x + e)^6 + 35*(33*a^2*c*d^2 + 32*a^2*d^3)*cos(f*x + e)^5 + 7392*a^2*c^3 + 15840*a^2*c^2*d + 13728*a^2*c*d^2 + 4000*a^2*d^3 - 5*(297*a^2*c^2*d + 627*a^2*c*d^2 + 320*a^2*d^3)*cos(f*x + e)^4 - (693*a^2*c^3 + 5940*a^2*c^2*d + 9537*a^2*c*d^2 + 4370*a^2*d^3)*cos(f*x + e)^3 + (2541*a^2*c^3 + 8415*a^2*c^2*d + 8679*a^2*c*d^2 + 2965*a^2*d^3)*cos(f*x + e)^2 + 2*(5313*a^2*c^3 + 14355*a^2*c^2*d + 13827*a^2*c*d^2 + 4465*a^2*d^3)*cos(f*x + e) + (315*a^2*d^3*cos(f*x + e)^5 - 7392*a^2*c^3 - 15840*a^2*c^2*d - 13728*a^2*c*d^2 - 4000*a^2*d^3 - 35*(33*a^2*c*d^2 + 23*a^2*d^3)*cos(f*x + e)^4 - 5*(297*a^2*c^2*d + 858*a^2*c*d^2 + 481*a^2*d^3)*cos(f*x + e)^3 + 3*(231*a^2*c^3 + 1485*a^2*c^2*d + 1749*a^2*c*d^2 + 655*a^2*d^3)*cos(f*x + e)^2 + 2*(1617*a^2*c^3 + 6435*a^2*c^2*d + 6963*a^2*c*d^2 + 2465*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
537,1,339,0,1.233426," ","integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{2 \, {\left(35 \, a^{2} d^{2} \cos\left(f x + e\right)^{5} - 5 \, {\left(18 \, a^{2} c d + 19 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{4} + 672 \, a^{2} c^{2} + 960 \, a^{2} c d + 416 \, a^{2} d^{2} - {\left(63 \, a^{2} c^{2} + 360 \, a^{2} c d + 289 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(231 \, a^{2} c^{2} + 510 \, a^{2} c d + 263 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(483 \, a^{2} c^{2} + 870 \, a^{2} c d + 419 \, a^{2} d^{2}\right)} \cos\left(f x + e\right) - {\left(35 \, a^{2} d^{2} \cos\left(f x + e\right)^{4} + 672 \, a^{2} c^{2} + 960 \, a^{2} c d + 416 \, a^{2} d^{2} + 10 \, {\left(9 \, a^{2} c d + 13 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(21 \, a^{2} c^{2} + 90 \, a^{2} c d + 53 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(147 \, a^{2} c^{2} + 390 \, a^{2} c d + 211 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{315 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"-2/315*(35*a^2*d^2*cos(f*x + e)^5 - 5*(18*a^2*c*d + 19*a^2*d^2)*cos(f*x + e)^4 + 672*a^2*c^2 + 960*a^2*c*d + 416*a^2*d^2 - (63*a^2*c^2 + 360*a^2*c*d + 289*a^2*d^2)*cos(f*x + e)^3 + (231*a^2*c^2 + 510*a^2*c*d + 263*a^2*d^2)*cos(f*x + e)^2 + 2*(483*a^2*c^2 + 870*a^2*c*d + 419*a^2*d^2)*cos(f*x + e) - (35*a^2*d^2*cos(f*x + e)^4 + 672*a^2*c^2 + 960*a^2*c*d + 416*a^2*d^2 + 10*(9*a^2*c*d + 13*a^2*d^2)*cos(f*x + e)^3 - 3*(21*a^2*c^2 + 90*a^2*c*d + 53*a^2*d^2)*cos(f*x + e)^2 - 2*(147*a^2*c^2 + 390*a^2*c*d + 211*a^2*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
538,1,206,0,1.165623," ","integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\frac{2 \, {\left(15 \, a^{2} d \cos\left(f x + e\right)^{4} + 3 \, {\left(7 \, a^{2} c + 20 \, a^{2} d\right)} \cos\left(f x + e\right)^{3} - 224 \, a^{2} c - 160 \, a^{2} d - {\left(77 \, a^{2} c + 85 \, a^{2} d\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(161 \, a^{2} c + 145 \, a^{2} d\right)} \cos\left(f x + e\right) + {\left(15 \, a^{2} d \cos\left(f x + e\right)^{3} + 224 \, a^{2} c + 160 \, a^{2} d - 3 \, {\left(7 \, a^{2} c + 15 \, a^{2} d\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(49 \, a^{2} c + 65 \, a^{2} d\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{105 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/105*(15*a^2*d*cos(f*x + e)^4 + 3*(7*a^2*c + 20*a^2*d)*cos(f*x + e)^3 - 224*a^2*c - 160*a^2*d - (77*a^2*c + 85*a^2*d)*cos(f*x + e)^2 - 2*(161*a^2*c + 145*a^2*d)*cos(f*x + e) + (15*a^2*d*cos(f*x + e)^3 + 224*a^2*c + 160*a^2*d - 3*(7*a^2*c + 15*a^2*d)*cos(f*x + e)^2 - 2*(49*a^2*c + 65*a^2*d)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
539,1,115,0,0.967062," ","integrate((a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, a^{2} \cos\left(f x + e\right)^{3} - 11 \, a^{2} \cos\left(f x + e\right)^{2} - 46 \, a^{2} \cos\left(f x + e\right) - 32 \, a^{2} - {\left(3 \, a^{2} \cos\left(f x + e\right)^{2} + 14 \, a^{2} \cos\left(f x + e\right) - 32 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{15 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"2/15*(3*a^2*cos(f*x + e)^3 - 11*a^2*cos(f*x + e)^2 - 46*a^2*cos(f*x + e) - 32*a^2 - (3*a^2*cos(f*x + e)^2 + 14*a^2*cos(f*x + e) - 32*a^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
540,1,868,0,1.561462," ","integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2} + {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right) + {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{c d + d^{2}}} \log\left(\frac{a d^{2} \cos\left(f x + e\right)^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left(6 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{a}{c d + d^{2}}} - {\left(a c^{2} + 8 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(a d^{2} \cos\left(f x + e\right)^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left(3 \, a c d + 4 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(a^{2} d \cos\left(f x + e\right)^{2} - 3 \, a^{2} c + 7 \, a^{2} d - {\left(3 \, a^{2} c - 8 \, a^{2} d\right)} \cos\left(f x + e\right) + {\left(a^{2} d \cos\left(f x + e\right) + 3 \, a^{2} c - 7 \, a^{2} d\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{6 \, {\left(d^{2} f \cos\left(f x + e\right) + d^{2} f \sin\left(f x + e\right) + d^{2} f\right)}}, -\frac{3 \, {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2} + {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right) + {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{c d + d^{2}}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{a}{c d + d^{2}}}}{2 \, a \cos\left(f x + e\right)}\right) + 2 \, {\left(a^{2} d \cos\left(f x + e\right)^{2} - 3 \, a^{2} c + 7 \, a^{2} d - {\left(3 \, a^{2} c - 8 \, a^{2} d\right)} \cos\left(f x + e\right) + {\left(a^{2} d \cos\left(f x + e\right) + 3 \, a^{2} c - 7 \, a^{2} d\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{3 \, {\left(d^{2} f \cos\left(f x + e\right) + d^{2} f \sin\left(f x + e\right) + d^{2} f\right)}}\right]"," ",0,"[1/6*(3*(a^2*c^2 - 2*a^2*c*d + a^2*d^2 + (a^2*c^2 - 2*a^2*c*d + a^2*d^2)*cos(f*x + e) + (a^2*c^2 - 2*a^2*c*d + a^2*d^2)*sin(f*x + e))*sqrt(a/(c*d + d^2))*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e)^2 + 4*(c^2*d + 4*c*d^2 + 3*d^3 - (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d + 4*c*d^2 + 3*d^3 + (c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2)) - (a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) - 4*(a^2*d*cos(f*x + e)^2 - 3*a^2*c + 7*a^2*d - (3*a^2*c - 8*a^2*d)*cos(f*x + e) + (a^2*d*cos(f*x + e) + 3*a^2*c - 7*a^2*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f), -1/3*(3*(a^2*c^2 - 2*a^2*c*d + a^2*d^2 + (a^2*c^2 - 2*a^2*c*d + a^2*d^2)*cos(f*x + e) + (a^2*c^2 - 2*a^2*c*d + a^2*d^2)*sin(f*x + e))*sqrt(-a/(c*d + d^2))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f*x + e))) + 2*(a^2*d*cos(f*x + e)^2 - 3*a^2*c + 7*a^2*d - (3*a^2*c - 8*a^2*d)*cos(f*x + e) + (a^2*d*cos(f*x + e) + 3*a^2*c - 7*a^2*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f)]","B",0
541,1,1322,0,1.141296," ","integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{2} c^{3} + 5 \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} - {\left(3 \, a^{2} c^{2} d + 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{3} + 2 \, a^{2} c^{2} d - 5 \, a^{2} c d^{2}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{3} + 5 \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} + {\left(3 \, a^{2} c^{2} d + 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{c d + d^{2}}} \log\left(\frac{a d^{2} \cos\left(f x + e\right)^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left(6 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{a}{c d + d^{2}}} - {\left(a c^{2} + 8 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(a d^{2} \cos\left(f x + e\right)^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left(3 \, a c d + 4 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(3 \, a^{2} c^{2} - 2 \, a^{2} c d - a^{2} d^{2} + 2 \, {\left(a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{2} + a^{2} d^{2}\right)} \cos\left(f x + e\right) - {\left(3 \, a^{2} c^{2} - 2 \, a^{2} c d - a^{2} d^{2} - 2 \, {\left(a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{4 \, {\left({\left(c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{2} d^{2} + c d^{3}\right)} f \cos\left(f x + e\right) - {\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f - {\left({\left(c d^{3} + d^{4}\right)} f \cos\left(f x + e\right) + {\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left(3 \, a^{2} c^{3} + 5 \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} - {\left(3 \, a^{2} c^{2} d + 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{3} + 2 \, a^{2} c^{2} d - 5 \, a^{2} c d^{2}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{3} + 5 \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} + {\left(3 \, a^{2} c^{2} d + 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{c d + d^{2}}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{a}{c d + d^{2}}}}{2 \, a \cos\left(f x + e\right)}\right) - 2 \, {\left(3 \, a^{2} c^{2} - 2 \, a^{2} c d - a^{2} d^{2} + 2 \, {\left(a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{2} + a^{2} d^{2}\right)} \cos\left(f x + e\right) - {\left(3 \, a^{2} c^{2} - 2 \, a^{2} c d - a^{2} d^{2} - 2 \, {\left(a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{2 \, {\left({\left(c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{2} d^{2} + c d^{3}\right)} f \cos\left(f x + e\right) - {\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f - {\left({\left(c d^{3} + d^{4}\right)} f \cos\left(f x + e\right) + {\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/4*((3*a^2*c^3 + 5*a^2*c^2*d - 3*a^2*c*d^2 - 5*a^2*d^3 - (3*a^2*c^2*d + 2*a^2*c*d^2 - 5*a^2*d^3)*cos(f*x + e)^2 + (3*a^2*c^3 + 2*a^2*c^2*d - 5*a^2*c*d^2)*cos(f*x + e) + (3*a^2*c^3 + 5*a^2*c^2*d - 3*a^2*c*d^2 - 5*a^2*d^3 + (3*a^2*c^2*d + 2*a^2*c*d^2 - 5*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a/(c*d + d^2))*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e)^2 + 4*(c^2*d + 4*c*d^2 + 3*d^3 - (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d + 4*c*d^2 + 3*d^3 + (c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2)) - (a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(3*a^2*c^2 - 2*a^2*c*d - a^2*d^2 + 2*(a^2*c*d + a^2*d^2)*cos(f*x + e)^2 + (3*a^2*c^2 + a^2*d^2)*cos(f*x + e) - (3*a^2*c^2 - 2*a^2*c*d - a^2*d^2 - 2*(a^2*c*d + a^2*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((c*d^3 + d^4)*f*cos(f*x + e)^2 - (c^2*d^2 + c*d^3)*f*cos(f*x + e) - (c^2*d^2 + 2*c*d^3 + d^4)*f - ((c*d^3 + d^4)*f*cos(f*x + e) + (c^2*d^2 + 2*c*d^3 + d^4)*f)*sin(f*x + e)), -1/2*((3*a^2*c^3 + 5*a^2*c^2*d - 3*a^2*c*d^2 - 5*a^2*d^3 - (3*a^2*c^2*d + 2*a^2*c*d^2 - 5*a^2*d^3)*cos(f*x + e)^2 + (3*a^2*c^3 + 2*a^2*c^2*d - 5*a^2*c*d^2)*cos(f*x + e) + (3*a^2*c^3 + 5*a^2*c^2*d - 3*a^2*c*d^2 - 5*a^2*d^3 + (3*a^2*c^2*d + 2*a^2*c*d^2 - 5*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-a/(c*d + d^2))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f*x + e))) - 2*(3*a^2*c^2 - 2*a^2*c*d - a^2*d^2 + 2*(a^2*c*d + a^2*d^2)*cos(f*x + e)^2 + (3*a^2*c^2 + a^2*d^2)*cos(f*x + e) - (3*a^2*c^2 - 2*a^2*c*d - a^2*d^2 - 2*(a^2*c*d + a^2*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((c*d^3 + d^4)*f*cos(f*x + e)^2 - (c^2*d^2 + c*d^3)*f*cos(f*x + e) - (c^2*d^2 + 2*c*d^3 + d^4)*f - ((c*d^3 + d^4)*f*cos(f*x + e) + (c^2*d^2 + 2*c*d^3 + d^4)*f)*sin(f*x + e))]","B",0
542,1,1994,0,1.652695," ","integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[-\frac{{\left(3 \, a^{2} c^{4} + 16 \, a^{2} c^{3} d + 42 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 19 \, a^{2} d^{4} - {\left(3 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(6 \, a^{2} c^{3} d + 23 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{4} + 10 \, a^{2} c^{3} d + 22 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{4} + 16 \, a^{2} c^{3} d + 42 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 19 \, a^{2} d^{4} - {\left(3 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, a^{2} c^{3} d + 10 \, a^{2} c^{2} d^{2} + 19 \, a^{2} c d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{c d + d^{2}}} \log\left(\frac{a d^{2} \cos\left(f x + e\right)^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left(6 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left(c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{a}{c d + d^{2}}} - {\left(a c^{2} + 8 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(a d^{2} \cos\left(f x + e\right)^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left(3 \, a c d + 4 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(3 \, a^{2} c^{3} + 3 \, a^{2} c^{2} d - 15 \, a^{2} c d^{2} + 9 \, a^{2} d^{3} + {\left(5 \, a^{2} c^{2} d + 6 \, a^{2} c d^{2} - 11 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{3} + 8 \, a^{2} c^{2} d - 9 \, a^{2} c d^{2} - 2 \, a^{2} d^{3}\right)} \cos\left(f x + e\right) - {\left(3 \, a^{2} c^{3} + 3 \, a^{2} c^{2} d - 15 \, a^{2} c d^{2} + 9 \, a^{2} d^{3} - {\left(5 \, a^{2} c^{2} d + 6 \, a^{2} c d^{2} - 11 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{16 \, {\left({\left(c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, c^{3} d^{3} + 5 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{4} d^{2} + 2 \, c^{3} d^{3} + 2 \, c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d^{2} + 4 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f + {\left({\left(c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d^{3} + 2 \, c^{2} d^{4} + c d^{5}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d^{2} + 4 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{{\left(3 \, a^{2} c^{4} + 16 \, a^{2} c^{3} d + 42 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 19 \, a^{2} d^{4} - {\left(3 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(6 \, a^{2} c^{3} d + 23 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{4} + 10 \, a^{2} c^{3} d + 22 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{4} + 16 \, a^{2} c^{3} d + 42 \, a^{2} c^{2} d^{2} + 48 \, a^{2} c d^{3} + 19 \, a^{2} d^{4} - {\left(3 \, a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 19 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, a^{2} c^{3} d + 10 \, a^{2} c^{2} d^{2} + 19 \, a^{2} c d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{c d + d^{2}}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{a}{c d + d^{2}}}}{2 \, a \cos\left(f x + e\right)}\right) - 2 \, {\left(3 \, a^{2} c^{3} + 3 \, a^{2} c^{2} d - 15 \, a^{2} c d^{2} + 9 \, a^{2} d^{3} + {\left(5 \, a^{2} c^{2} d + 6 \, a^{2} c d^{2} - 11 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{3} + 8 \, a^{2} c^{2} d - 9 \, a^{2} c d^{2} - 2 \, a^{2} d^{3}\right)} \cos\left(f x + e\right) - {\left(3 \, a^{2} c^{3} + 3 \, a^{2} c^{2} d - 15 \, a^{2} c d^{2} + 9 \, a^{2} d^{3} - {\left(5 \, a^{2} c^{2} d + 6 \, a^{2} c d^{2} - 11 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{8 \, {\left({\left(c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, c^{3} d^{3} + 5 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{4} d^{2} + 2 \, c^{3} d^{3} + 2 \, c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d^{2} + 4 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f + {\left({\left(c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d^{3} + 2 \, c^{2} d^{4} + c d^{5}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d^{2} + 4 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/16*((3*a^2*c^4 + 16*a^2*c^3*d + 42*a^2*c^2*d^2 + 48*a^2*c*d^3 + 19*a^2*d^4 - (3*a^2*c^2*d^2 + 10*a^2*c*d^3 + 19*a^2*d^4)*cos(f*x + e)^3 - (6*a^2*c^3*d + 23*a^2*c^2*d^2 + 48*a^2*c*d^3 + 19*a^2*d^4)*cos(f*x + e)^2 + (3*a^2*c^4 + 10*a^2*c^3*d + 22*a^2*c^2*d^2 + 10*a^2*c*d^3 + 19*a^2*d^4)*cos(f*x + e) + (3*a^2*c^4 + 16*a^2*c^3*d + 42*a^2*c^2*d^2 + 48*a^2*c*d^3 + 19*a^2*d^4 - (3*a^2*c^2*d^2 + 10*a^2*c*d^3 + 19*a^2*d^4)*cos(f*x + e)^2 + 2*(3*a^2*c^3*d + 10*a^2*c^2*d^2 + 19*a^2*c*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a/(c*d + d^2))*log((a*d^2*cos(f*x + e)^3 - a*c^2 - 2*a*c*d - a*d^2 - (6*a*c*d + 7*a*d^2)*cos(f*x + e)^2 + 4*(c^2*d + 4*c*d^2 + 3*d^3 - (c*d^2 + d^3)*cos(f*x + e)^2 + (c^2*d + 3*c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d + 4*c*d^2 + 3*d^3 + (c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(a/(c*d + d^2)) - (a*c^2 + 8*a*c*d + 9*a*d^2)*cos(f*x + e) + (a*d^2*cos(f*x + e)^2 - a*c^2 - 2*a*c*d - a*d^2 + 2*(3*a*c*d + 4*a*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(3*a^2*c^3 + 3*a^2*c^2*d - 15*a^2*c*d^2 + 9*a^2*d^3 + (5*a^2*c^2*d + 6*a^2*c*d^2 - 11*a^2*d^3)*cos(f*x + e)^2 + (3*a^2*c^3 + 8*a^2*c^2*d - 9*a^2*c*d^2 - 2*a^2*d^3)*cos(f*x + e) - (3*a^2*c^3 + 3*a^2*c^2*d - 15*a^2*c*d^2 + 9*a^2*d^3 - (5*a^2*c^2*d + 6*a^2*c*d^2 - 11*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e)^3 + (2*c^3*d^3 + 5*c^2*d^4 + 4*c*d^5 + d^6)*f*cos(f*x + e)^2 - (c^4*d^2 + 2*c^3*d^3 + 2*c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e) - (c^4*d^2 + 4*c^3*d^3 + 6*c^2*d^4 + 4*c*d^5 + d^6)*f + ((c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e)^2 - 2*(c^3*d^3 + 2*c^2*d^4 + c*d^5)*f*cos(f*x + e) - (c^4*d^2 + 4*c^3*d^3 + 6*c^2*d^4 + 4*c*d^5 + d^6)*f)*sin(f*x + e)), 1/8*((3*a^2*c^4 + 16*a^2*c^3*d + 42*a^2*c^2*d^2 + 48*a^2*c*d^3 + 19*a^2*d^4 - (3*a^2*c^2*d^2 + 10*a^2*c*d^3 + 19*a^2*d^4)*cos(f*x + e)^3 - (6*a^2*c^3*d + 23*a^2*c^2*d^2 + 48*a^2*c*d^3 + 19*a^2*d^4)*cos(f*x + e)^2 + (3*a^2*c^4 + 10*a^2*c^3*d + 22*a^2*c^2*d^2 + 10*a^2*c*d^3 + 19*a^2*d^4)*cos(f*x + e) + (3*a^2*c^4 + 16*a^2*c^3*d + 42*a^2*c^2*d^2 + 48*a^2*c*d^3 + 19*a^2*d^4 - (3*a^2*c^2*d^2 + 10*a^2*c*d^3 + 19*a^2*d^4)*cos(f*x + e)^2 + 2*(3*a^2*c^3*d + 10*a^2*c^2*d^2 + 19*a^2*c*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-a/(c*d + d^2))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-a/(c*d + d^2))/(a*cos(f*x + e))) - 2*(3*a^2*c^3 + 3*a^2*c^2*d - 15*a^2*c*d^2 + 9*a^2*d^3 + (5*a^2*c^2*d + 6*a^2*c*d^2 - 11*a^2*d^3)*cos(f*x + e)^2 + (3*a^2*c^3 + 8*a^2*c^2*d - 9*a^2*c*d^2 - 2*a^2*d^3)*cos(f*x + e) - (3*a^2*c^3 + 3*a^2*c^2*d - 15*a^2*c*d^2 + 9*a^2*d^3 - (5*a^2*c^2*d + 6*a^2*c*d^2 - 11*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e)^3 + (2*c^3*d^3 + 5*c^2*d^4 + 4*c*d^5 + d^6)*f*cos(f*x + e)^2 - (c^4*d^2 + 2*c^3*d^3 + 2*c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e) - (c^4*d^2 + 4*c^3*d^3 + 6*c^2*d^4 + 4*c*d^5 + d^6)*f + ((c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e)^2 - 2*(c^3*d^3 + 2*c^2*d^4 + c*d^5)*f*cos(f*x + e) - (c^4*d^2 + 4*c^3*d^3 + 6*c^2*d^4 + 4*c*d^5 + d^6)*f)*sin(f*x + e))]","B",0
543,1,387,0,1.125246," ","integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","-\frac{\frac{15 \, \sqrt{2} {\left(a c^{3} - 3 \, a c^{2} d + 3 \, a c d^{2} - a d^{3} + {\left(a c^{3} - 3 \, a c^{2} d + 3 \, a c d^{2} - a d^{3}\right)} \cos\left(f x + e\right) + {\left(a c^{3} - 3 \, a c^{2} d + 3 \, a c d^{2} - a d^{3}\right)} \sin\left(f x + e\right)\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) + \frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{a}} - 4 \, {\left(3 \, d^{3} \cos\left(f x + e\right)^{3} - 45 \, c^{2} d + 30 \, c d^{2} - 17 \, d^{3} - {\left(15 \, c d^{2} - 4 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(45 \, c^{2} d - 15 \, c d^{2} + 16 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(3 \, d^{3} \cos\left(f x + e\right)^{2} - 45 \, c^{2} d + 30 \, c d^{2} - 17 \, d^{3} + {\left(15 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{30 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}"," ",0,"-1/30*(15*sqrt(2)*(a*c^3 - 3*a*c^2*d + 3*a*c*d^2 - a*d^3 + (a*c^3 - 3*a*c^2*d + 3*a*c*d^2 - a*d^3)*cos(f*x + e) + (a*c^3 - 3*a*c^2*d + 3*a*c*d^2 - a*d^3)*sin(f*x + e))*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(a) - 4*(3*d^3*cos(f*x + e)^3 - 45*c^2*d + 30*c*d^2 - 17*d^3 - (15*c*d^2 - 4*d^3)*cos(f*x + e)^2 - (45*c^2*d - 15*c*d^2 + 16*d^3)*cos(f*x + e) - (3*d^3*cos(f*x + e)^2 - 45*c^2*d + 30*c*d^2 - 17*d^3 + (15*c*d^2 - d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","B",0
544,1,286,0,0.850596," ","integrate((c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{\frac{3 \, \sqrt{2} {\left(a c^{2} - 2 \, a c d + a d^{2} + {\left(a c^{2} - 2 \, a c d + a d^{2}\right)} \cos\left(f x + e\right) + {\left(a c^{2} - 2 \, a c d + a d^{2}\right)} \sin\left(f x + e\right)\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) - \frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{a}} - 4 \, {\left(d^{2} \cos\left(f x + e\right)^{2} + 6 \, c d - 2 \, d^{2} + {\left(6 \, c d - d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right) - 6 \, c d + 2 \, d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{6 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}"," ",0,"1/6*(3*sqrt(2)*(a*c^2 - 2*a*c*d + a*d^2 + (a*c^2 - 2*a*c*d + a*d^2)*cos(f*x + e) + (a*c^2 - 2*a*c*d + a*d^2)*sin(f*x + e))*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(a) - 4*(d^2*cos(f*x + e)^2 + 6*c*d - 2*d^2 + (6*c*d - d^2)*cos(f*x + e) + (d^2*cos(f*x + e) - 6*c*d + 2*d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","B",0
545,1,214,0,1.154651," ","integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","-\frac{\frac{\sqrt{2} {\left(a c - a d + {\left(a c - a d\right)} \cos\left(f x + e\right) + {\left(a c - a d\right)} \sin\left(f x + e\right)\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) + \frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{a}} + 4 \, {\left(d \cos\left(f x + e\right) - d \sin\left(f x + e\right) + d\right)} \sqrt{a \sin\left(f x + e\right) + a}}{2 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}"," ",0,"-1/2*(sqrt(2)*(a*c - a*d + (a*c - a*d)*cos(f*x + e) + (a*c - a*d)*sin(f*x + e))*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(a) + 4*(d*cos(f*x + e) - d*sin(f*x + e) + d)*sqrt(a*sin(f*x + e) + a))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)","B",0
546,1,167,0,1.292554," ","integrate(1/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} \log\left(-\frac{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) - \frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{2 \, \sqrt{a} f}, \frac{\sqrt{2} \sqrt{-\frac{1}{a}} \arctan\left(\frac{\sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{-\frac{1}{a}}}{\cos\left(f x + e\right)}\right)}{f}\right]"," ",0,"[1/2*sqrt(2)*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/(sqrt(a)*f), sqrt(2)*sqrt(-1/a)*arctan(sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(-1/a)/cos(f*x + e))/f]","A",0
547,1,685,0,1.186164," ","integrate(1/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{\frac{d}{a c + a d}} \log\left(\frac{d^{2} \cos\left(f x + e\right)^{3} - {\left(6 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - 4 \, {\left({\left(c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 4 \, c d - 3 \, d^{2} - {\left(c^{2} + 3 \, c d + 2 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} + 4 \, c d + 3 \, d^{2} + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{d}{a c + a d}} - {\left(c^{2} + 8 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 2 \, {\left(3 \, c d + 4 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + \frac{\sqrt{2} \log\left(-\frac{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) + \frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{a}}}{2 \, {\left(c - d\right)} f}, \frac{2 \, \sqrt{-\frac{d}{a c + a d}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{d}{a c + a d}}}{2 \, d \cos\left(f x + e\right)}\right) - \frac{\sqrt{2} \log\left(-\frac{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) + \frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{a}}}{2 \, {\left(c - d\right)} f}\right]"," ",0,"[-1/2*(sqrt(d/(a*c + a*d))*log((d^2*cos(f*x + e)^3 - (6*c*d + 7*d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - 4*((c*d + d^2)*cos(f*x + e)^2 - c^2 - 4*c*d - 3*d^2 - (c^2 + 3*c*d + 2*d^2)*cos(f*x + e) + (c^2 + 4*c*d + 3*d^2 + (c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d/(a*c + a*d)) - (c^2 + 8*c*d + 9*d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 2*(3*c*d + 4*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + sqrt(2)*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(a))/((c - d)*f), 1/2*(2*sqrt(-d/(a*c + a*d))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) - sqrt(2)*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(a))/((c - d)*f)]","B",0
548,1,1494,0,1.859885," ","integrate(1/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[-\frac{{\left(3 \, a c^{2} + 4 \, a c d + a d^{2} - {\left(3 \, a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a c^{2} + a c d\right)} \cos\left(f x + e\right) + {\left(3 \, a c^{2} + 4 \, a c d + a d^{2} + {\left(3 \, a c d + a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a c + a d}} \log\left(\frac{d^{2} \cos\left(f x + e\right)^{3} - {\left(6 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 4 \, {\left({\left(c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 4 \, c d - 3 \, d^{2} - {\left(c^{2} + 3 \, c d + 2 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} + 4 \, c d + 3 \, d^{2} + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{d}{a c + a d}} - {\left(c^{2} + 8 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 2 \, {\left(3 \, c d + 4 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + \frac{2 \, \sqrt{2} {\left(a c^{2} + 2 \, a c d + a d^{2} - {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + a c d\right)} \cos\left(f x + e\right) + {\left(a c^{2} + 2 \, a c d + a d^{2} + {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) - \frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{a}} + 4 \, {\left(c d - d^{2} + {\left(c d - d^{2}\right)} \cos\left(f x + e\right) - {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{4 \, {\left({\left(a c^{3} d - a c^{2} d^{2} - a c d^{3} + a d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{4} - a c^{3} d - a c^{2} d^{2} + a c d^{3}\right)} f \cos\left(f x + e\right) - {\left(a c^{4} - 2 \, a c^{2} d^{2} + a d^{4}\right)} f - {\left({\left(a c^{3} d - a c^{2} d^{2} - a c d^{3} + a d^{4}\right)} f \cos\left(f x + e\right) + {\left(a c^{4} - 2 \, a c^{2} d^{2} + a d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left(3 \, a c^{2} + 4 \, a c d + a d^{2} - {\left(3 \, a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a c^{2} + a c d\right)} \cos\left(f x + e\right) + {\left(3 \, a c^{2} + 4 \, a c d + a d^{2} + {\left(3 \, a c d + a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a c + a d}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{d}{a c + a d}}}{2 \, d \cos\left(f x + e\right)}\right) + \frac{\sqrt{2} {\left(a c^{2} + 2 \, a c d + a d^{2} - {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + a c d\right)} \cos\left(f x + e\right) + {\left(a c^{2} + 2 \, a c d + a d^{2} + {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) - \frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{a}} + 2 \, {\left(c d - d^{2} + {\left(c d - d^{2}\right)} \cos\left(f x + e\right) - {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{2 \, {\left({\left(a c^{3} d - a c^{2} d^{2} - a c d^{3} + a d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{4} - a c^{3} d - a c^{2} d^{2} + a c d^{3}\right)} f \cos\left(f x + e\right) - {\left(a c^{4} - 2 \, a c^{2} d^{2} + a d^{4}\right)} f - {\left({\left(a c^{3} d - a c^{2} d^{2} - a c d^{3} + a d^{4}\right)} f \cos\left(f x + e\right) + {\left(a c^{4} - 2 \, a c^{2} d^{2} + a d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/4*((3*a*c^2 + 4*a*c*d + a*d^2 - (3*a*c*d + a*d^2)*cos(f*x + e)^2 + (3*a*c^2 + a*c*d)*cos(f*x + e) + (3*a*c^2 + 4*a*c*d + a*d^2 + (3*a*c*d + a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(d/(a*c + a*d))*log((d^2*cos(f*x + e)^3 - (6*c*d + 7*d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 4*((c*d + d^2)*cos(f*x + e)^2 - c^2 - 4*c*d - 3*d^2 - (c^2 + 3*c*d + 2*d^2)*cos(f*x + e) + (c^2 + 4*c*d + 3*d^2 + (c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d/(a*c + a*d)) - (c^2 + 8*c*d + 9*d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 2*(3*c*d + 4*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 2*sqrt(2)*(a*c^2 + 2*a*c*d + a*d^2 - (a*c*d + a*d^2)*cos(f*x + e)^2 + (a*c^2 + a*c*d)*cos(f*x + e) + (a*c^2 + 2*a*c*d + a*d^2 + (a*c*d + a*d^2)*cos(f*x + e))*sin(f*x + e))*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(a) + 4*(c*d - d^2 + (c*d - d^2)*cos(f*x + e) - (c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a*c^3*d - a*c^2*d^2 - a*c*d^3 + a*d^4)*f*cos(f*x + e)^2 - (a*c^4 - a*c^3*d - a*c^2*d^2 + a*c*d^3)*f*cos(f*x + e) - (a*c^4 - 2*a*c^2*d^2 + a*d^4)*f - ((a*c^3*d - a*c^2*d^2 - a*c*d^3 + a*d^4)*f*cos(f*x + e) + (a*c^4 - 2*a*c^2*d^2 + a*d^4)*f)*sin(f*x + e)), -1/2*((3*a*c^2 + 4*a*c*d + a*d^2 - (3*a*c*d + a*d^2)*cos(f*x + e)^2 + (3*a*c^2 + a*c*d)*cos(f*x + e) + (3*a*c^2 + 4*a*c*d + a*d^2 + (3*a*c*d + a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-d/(a*c + a*d))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) + sqrt(2)*(a*c^2 + 2*a*c*d + a*d^2 - (a*c*d + a*d^2)*cos(f*x + e)^2 + (a*c^2 + a*c*d)*cos(f*x + e) + (a*c^2 + 2*a*c*d + a*d^2 + (a*c*d + a*d^2)*cos(f*x + e))*sin(f*x + e))*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(a) + 2*(c*d - d^2 + (c*d - d^2)*cos(f*x + e) - (c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a*c^3*d - a*c^2*d^2 - a*c*d^3 + a*d^4)*f*cos(f*x + e)^2 - (a*c^4 - a*c^3*d - a*c^2*d^2 + a*c*d^3)*f*cos(f*x + e) - (a*c^4 - 2*a*c^2*d^2 + a*d^4)*f - ((a*c^3*d - a*c^2*d^2 - a*c*d^3 + a*d^4)*f*cos(f*x + e) + (a*c^4 - 2*a*c^2*d^2 + a*d^4)*f)*sin(f*x + e))]","B",0
549,1,2903,0,2.899149," ","integrate(1/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[\frac{{\left(15 \, a c^{4} + 40 \, a c^{3} d + 42 \, a c^{2} d^{2} + 24 \, a c d^{3} + 7 \, a d^{4} - {\left(15 \, a c^{2} d^{2} + 10 \, a c d^{3} + 7 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(30 \, a c^{3} d + 35 \, a c^{2} d^{2} + 24 \, a c d^{3} + 7 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(15 \, a c^{4} + 10 \, a c^{3} d + 22 \, a c^{2} d^{2} + 10 \, a c d^{3} + 7 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(15 \, a c^{4} + 40 \, a c^{3} d + 42 \, a c^{2} d^{2} + 24 \, a c d^{3} + 7 \, a d^{4} - {\left(15 \, a c^{2} d^{2} + 10 \, a c d^{3} + 7 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(15 \, a c^{3} d + 10 \, a c^{2} d^{2} + 7 \, a c d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a c + a d}} \log\left(\frac{d^{2} \cos\left(f x + e\right)^{3} - {\left(6 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - 4 \, {\left({\left(c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 4 \, c d - 3 \, d^{2} - {\left(c^{2} + 3 \, c d + 2 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} + 4 \, c d + 3 \, d^{2} + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{d}{a c + a d}} - {\left(c^{2} + 8 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 2 \, {\left(3 \, c d + 4 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + \frac{8 \, \sqrt{2} {\left(a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - {\left(a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(2 \, a c^{3} d + 5 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{4} + 2 \, a c^{3} d + 2 \, a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right) + {\left(a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - {\left(a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a c^{3} d + 2 \, a c^{2} d^{2} + a c d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) + \frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{a}} - 4 \, {\left(9 \, c^{3} d - 15 \, c^{2} d^{2} + 3 \, c d^{3} + 3 \, d^{4} + {\left(7 \, c^{2} d^{2} - 6 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(9 \, c^{3} d - 8 \, c^{2} d^{2} - 3 \, c d^{3} + 2 \, d^{4}\right)} \cos\left(f x + e\right) - {\left(9 \, c^{3} d - 15 \, c^{2} d^{2} + 3 \, c d^{3} + 3 \, d^{4} - {\left(7 \, c^{2} d^{2} - 6 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{16 \, {\left({\left(a c^{5} d^{2} - a c^{4} d^{3} - 2 \, a c^{3} d^{4} + 2 \, a c^{2} d^{5} + a c d^{6} - a d^{7}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, a c^{6} d - a c^{5} d^{2} - 5 \, a c^{4} d^{3} + 2 \, a c^{3} d^{4} + 4 \, a c^{2} d^{5} - a c d^{6} - a d^{7}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{7} - a c^{6} d - a c^{5} d^{2} + a c^{4} d^{3} - a c^{3} d^{4} + a c^{2} d^{5} + a c d^{6} - a d^{7}\right)} f \cos\left(f x + e\right) - {\left(a c^{7} + a c^{6} d - 3 \, a c^{5} d^{2} - 3 \, a c^{4} d^{3} + 3 \, a c^{3} d^{4} + 3 \, a c^{2} d^{5} - a c d^{6} - a d^{7}\right)} f + {\left({\left(a c^{5} d^{2} - a c^{4} d^{3} - 2 \, a c^{3} d^{4} + 2 \, a c^{2} d^{5} + a c d^{6} - a d^{7}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a c^{6} d - a c^{5} d^{2} - 2 \, a c^{4} d^{3} + 2 \, a c^{3} d^{4} + a c^{2} d^{5} - a c d^{6}\right)} f \cos\left(f x + e\right) - {\left(a c^{7} + a c^{6} d - 3 \, a c^{5} d^{2} - 3 \, a c^{4} d^{3} + 3 \, a c^{3} d^{4} + 3 \, a c^{2} d^{5} - a c d^{6} - a d^{7}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left(15 \, a c^{4} + 40 \, a c^{3} d + 42 \, a c^{2} d^{2} + 24 \, a c d^{3} + 7 \, a d^{4} - {\left(15 \, a c^{2} d^{2} + 10 \, a c d^{3} + 7 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(30 \, a c^{3} d + 35 \, a c^{2} d^{2} + 24 \, a c d^{3} + 7 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(15 \, a c^{4} + 10 \, a c^{3} d + 22 \, a c^{2} d^{2} + 10 \, a c d^{3} + 7 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(15 \, a c^{4} + 40 \, a c^{3} d + 42 \, a c^{2} d^{2} + 24 \, a c d^{3} + 7 \, a d^{4} - {\left(15 \, a c^{2} d^{2} + 10 \, a c d^{3} + 7 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(15 \, a c^{3} d + 10 \, a c^{2} d^{2} + 7 \, a c d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a c + a d}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{d}{a c + a d}}}{2 \, d \cos\left(f x + e\right)}\right) - \frac{4 \, \sqrt{2} {\left(a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - {\left(a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(2 \, a c^{3} d + 5 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{4} + 2 \, a c^{3} d + 2 \, a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right) + {\left(a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - {\left(a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a c^{3} d + 2 \, a c^{2} d^{2} + a c d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \log\left(-\frac{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) - 2\right)} \sin\left(f x + e\right) + \frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{\sqrt{a}} + 3 \, \cos\left(f x + e\right) + 2}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right)}{\sqrt{a}} + 2 \, {\left(9 \, c^{3} d - 15 \, c^{2} d^{2} + 3 \, c d^{3} + 3 \, d^{4} + {\left(7 \, c^{2} d^{2} - 6 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(9 \, c^{3} d - 8 \, c^{2} d^{2} - 3 \, c d^{3} + 2 \, d^{4}\right)} \cos\left(f x + e\right) - {\left(9 \, c^{3} d - 15 \, c^{2} d^{2} + 3 \, c d^{3} + 3 \, d^{4} - {\left(7 \, c^{2} d^{2} - 6 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{8 \, {\left({\left(a c^{5} d^{2} - a c^{4} d^{3} - 2 \, a c^{3} d^{4} + 2 \, a c^{2} d^{5} + a c d^{6} - a d^{7}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, a c^{6} d - a c^{5} d^{2} - 5 \, a c^{4} d^{3} + 2 \, a c^{3} d^{4} + 4 \, a c^{2} d^{5} - a c d^{6} - a d^{7}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{7} - a c^{6} d - a c^{5} d^{2} + a c^{4} d^{3} - a c^{3} d^{4} + a c^{2} d^{5} + a c d^{6} - a d^{7}\right)} f \cos\left(f x + e\right) - {\left(a c^{7} + a c^{6} d - 3 \, a c^{5} d^{2} - 3 \, a c^{4} d^{3} + 3 \, a c^{3} d^{4} + 3 \, a c^{2} d^{5} - a c d^{6} - a d^{7}\right)} f + {\left({\left(a c^{5} d^{2} - a c^{4} d^{3} - 2 \, a c^{3} d^{4} + 2 \, a c^{2} d^{5} + a c d^{6} - a d^{7}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a c^{6} d - a c^{5} d^{2} - 2 \, a c^{4} d^{3} + 2 \, a c^{3} d^{4} + a c^{2} d^{5} - a c d^{6}\right)} f \cos\left(f x + e\right) - {\left(a c^{7} + a c^{6} d - 3 \, a c^{5} d^{2} - 3 \, a c^{4} d^{3} + 3 \, a c^{3} d^{4} + 3 \, a c^{2} d^{5} - a c d^{6} - a d^{7}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/16*((15*a*c^4 + 40*a*c^3*d + 42*a*c^2*d^2 + 24*a*c*d^3 + 7*a*d^4 - (15*a*c^2*d^2 + 10*a*c*d^3 + 7*a*d^4)*cos(f*x + e)^3 - (30*a*c^3*d + 35*a*c^2*d^2 + 24*a*c*d^3 + 7*a*d^4)*cos(f*x + e)^2 + (15*a*c^4 + 10*a*c^3*d + 22*a*c^2*d^2 + 10*a*c*d^3 + 7*a*d^4)*cos(f*x + e) + (15*a*c^4 + 40*a*c^3*d + 42*a*c^2*d^2 + 24*a*c*d^3 + 7*a*d^4 - (15*a*c^2*d^2 + 10*a*c*d^3 + 7*a*d^4)*cos(f*x + e)^2 + 2*(15*a*c^3*d + 10*a*c^2*d^2 + 7*a*c*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(d/(a*c + a*d))*log((d^2*cos(f*x + e)^3 - (6*c*d + 7*d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - 4*((c*d + d^2)*cos(f*x + e)^2 - c^2 - 4*c*d - 3*d^2 - (c^2 + 3*c*d + 2*d^2)*cos(f*x + e) + (c^2 + 4*c*d + 3*d^2 + (c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d/(a*c + a*d)) - (c^2 + 8*c*d + 9*d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 2*(3*c*d + 4*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 8*sqrt(2)*(a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^3 - (2*a*c^3*d + 5*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*cos(f*x + e)^2 + (a*c^4 + 2*a*c^3*d + 2*a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^2 + 2*(a*c^3*d + 2*a*c^2*d^2 + a*c*d^3)*cos(f*x + e))*sin(f*x + e))*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(a) - 4*(9*c^3*d - 15*c^2*d^2 + 3*c*d^3 + 3*d^4 + (7*c^2*d^2 - 6*c*d^3 - d^4)*cos(f*x + e)^2 + (9*c^3*d - 8*c^2*d^2 - 3*c*d^3 + 2*d^4)*cos(f*x + e) - (9*c^3*d - 15*c^2*d^2 + 3*c*d^3 + 3*d^4 - (7*c^2*d^2 - 6*c*d^3 - d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a*c^5*d^2 - a*c^4*d^3 - 2*a*c^3*d^4 + 2*a*c^2*d^5 + a*c*d^6 - a*d^7)*f*cos(f*x + e)^3 + (2*a*c^6*d - a*c^5*d^2 - 5*a*c^4*d^3 + 2*a*c^3*d^4 + 4*a*c^2*d^5 - a*c*d^6 - a*d^7)*f*cos(f*x + e)^2 - (a*c^7 - a*c^6*d - a*c^5*d^2 + a*c^4*d^3 - a*c^3*d^4 + a*c^2*d^5 + a*c*d^6 - a*d^7)*f*cos(f*x + e) - (a*c^7 + a*c^6*d - 3*a*c^5*d^2 - 3*a*c^4*d^3 + 3*a*c^3*d^4 + 3*a*c^2*d^5 - a*c*d^6 - a*d^7)*f + ((a*c^5*d^2 - a*c^4*d^3 - 2*a*c^3*d^4 + 2*a*c^2*d^5 + a*c*d^6 - a*d^7)*f*cos(f*x + e)^2 - 2*(a*c^6*d - a*c^5*d^2 - 2*a*c^4*d^3 + 2*a*c^3*d^4 + a*c^2*d^5 - a*c*d^6)*f*cos(f*x + e) - (a*c^7 + a*c^6*d - 3*a*c^5*d^2 - 3*a*c^4*d^3 + 3*a*c^3*d^4 + 3*a*c^2*d^5 - a*c*d^6 - a*d^7)*f)*sin(f*x + e)), -1/8*((15*a*c^4 + 40*a*c^3*d + 42*a*c^2*d^2 + 24*a*c*d^3 + 7*a*d^4 - (15*a*c^2*d^2 + 10*a*c*d^3 + 7*a*d^4)*cos(f*x + e)^3 - (30*a*c^3*d + 35*a*c^2*d^2 + 24*a*c*d^3 + 7*a*d^4)*cos(f*x + e)^2 + (15*a*c^4 + 10*a*c^3*d + 22*a*c^2*d^2 + 10*a*c*d^3 + 7*a*d^4)*cos(f*x + e) + (15*a*c^4 + 40*a*c^3*d + 42*a*c^2*d^2 + 24*a*c*d^3 + 7*a*d^4 - (15*a*c^2*d^2 + 10*a*c*d^3 + 7*a*d^4)*cos(f*x + e)^2 + 2*(15*a*c^3*d + 10*a*c^2*d^2 + 7*a*c*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-d/(a*c + a*d))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) - 4*sqrt(2)*(a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^3 - (2*a*c^3*d + 5*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*cos(f*x + e)^2 + (a*c^4 + 2*a*c^3*d + 2*a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^2 + 2*(a*c^3*d + 2*a*c^2*d^2 + a*c*d^3)*cos(f*x + e))*sin(f*x + e))*log(-(cos(f*x + e)^2 - (cos(f*x + e) - 2)*sin(f*x + e) + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1)/sqrt(a) + 3*cos(f*x + e) + 2)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2))/sqrt(a) + 2*(9*c^3*d - 15*c^2*d^2 + 3*c*d^3 + 3*d^4 + (7*c^2*d^2 - 6*c*d^3 - d^4)*cos(f*x + e)^2 + (9*c^3*d - 8*c^2*d^2 - 3*c*d^3 + 2*d^4)*cos(f*x + e) - (9*c^3*d - 15*c^2*d^2 + 3*c*d^3 + 3*d^4 - (7*c^2*d^2 - 6*c*d^3 - d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a*c^5*d^2 - a*c^4*d^3 - 2*a*c^3*d^4 + 2*a*c^2*d^5 + a*c*d^6 - a*d^7)*f*cos(f*x + e)^3 + (2*a*c^6*d - a*c^5*d^2 - 5*a*c^4*d^3 + 2*a*c^3*d^4 + 4*a*c^2*d^5 - a*c*d^6 - a*d^7)*f*cos(f*x + e)^2 - (a*c^7 - a*c^6*d - a*c^5*d^2 + a*c^4*d^3 - a*c^3*d^4 + a*c^2*d^5 + a*c*d^6 - a*d^7)*f*cos(f*x + e) - (a*c^7 + a*c^6*d - 3*a*c^5*d^2 - 3*a*c^4*d^3 + 3*a*c^3*d^4 + 3*a*c^2*d^5 - a*c*d^6 - a*d^7)*f + ((a*c^5*d^2 - a*c^4*d^3 - 2*a*c^3*d^4 + 2*a*c^2*d^5 + a*c*d^6 - a*d^7)*f*cos(f*x + e)^2 - 2*(a*c^6*d - a*c^5*d^2 - 2*a*c^4*d^3 + 2*a*c^3*d^4 + a*c^2*d^5 - a*c*d^6)*f*cos(f*x + e) - (a*c^7 + a*c^6*d - 3*a*c^5*d^2 - 3*a*c^4*d^3 + 3*a*c^3*d^4 + 3*a*c^2*d^5 - a*c*d^6 - a*d^7)*f)*sin(f*x + e))]","B",0
550,1,494,0,1.003805," ","integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","-\frac{3 \, \sqrt{2} {\left(2 \, c^{3} + 18 \, c^{2} d - 42 \, c d^{2} + 22 \, d^{3} - {\left(c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(2 \, c^{3} + 18 \, c^{2} d - 42 \, c d^{2} + 22 \, d^{3} + {\left(c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left(4 \, d^{3} \cos\left(f x + e\right)^{3} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} - 4 \, {\left(9 \, c d^{2} - 4 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(c^{3} - 3 \, c^{2} d + 15 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(4 \, d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} + 12 \, {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{24 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/24*(3*sqrt(2)*(2*c^3 + 18*c^2*d - 42*c*d^2 + 22*d^3 - (c^3 + 9*c^2*d - 21*c*d^2 + 11*d^3)*cos(f*x + e)^2 + (c^3 + 9*c^2*d - 21*c*d^2 + 11*d^3)*cos(f*x + e) + (2*c^3 + 18*c^2*d - 42*c*d^2 + 22*d^3 + (c^3 + 9*c^2*d - 21*c*d^2 + 11*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(4*d^3*cos(f*x + e)^3 - 3*c^3 + 9*c^2*d - 9*c*d^2 + 3*d^3 - 4*(9*c*d^2 - 4*d^3)*cos(f*x + e)^2 - 3*(c^3 - 3*c^2*d + 15*c*d^2 - 5*d^3)*cos(f*x + e) - (4*d^3*cos(f*x + e)^2 - 3*c^3 + 9*c^2*d - 9*c*d^2 + 3*d^3 + 12*(3*c*d^2 - d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","B",0
551,1,380,0,0.898181," ","integrate((c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{2} {\left({\left(c^{2} + 6 \, c d - 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c^{2} - 12 \, c d + 14 \, d^{2} - {\left(c^{2} + 6 \, c d - 7 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(2 \, c^{2} + 12 \, c d - 14 \, d^{2} + {\left(c^{2} + 6 \, c d - 7 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, {\left(4 \, d^{2} \cos\left(f x + e\right)^{2} + c^{2} - 2 \, c d + d^{2} + {\left(c^{2} - 2 \, c d + 5 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(4 \, d^{2} \cos\left(f x + e\right) - c^{2} + 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{8 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/8*(sqrt(2)*((c^2 + 6*c*d - 7*d^2)*cos(f*x + e)^2 - 2*c^2 - 12*c*d + 14*d^2 - (c^2 + 6*c*d - 7*d^2)*cos(f*x + e) - (2*c^2 + 12*c*d - 14*d^2 + (c^2 + 6*c*d - 7*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(4*d^2*cos(f*x + e)^2 + c^2 - 2*c*d + d^2 + (c^2 - 2*c*d + 5*d^2)*cos(f*x + e) + (4*d^2*cos(f*x + e) - c^2 + 2*c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","B",0
552,1,293,0,1.159253," ","integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{2} {\left({\left(c + 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(c + 3 \, d\right)} \cos\left(f x + e\right) - {\left({\left(c + 3 \, d\right)} \cos\left(f x + e\right) + 2 \, c + 6 \, d\right)} \sin\left(f x + e\right) - 2 \, c - 6 \, d\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left({\left(c - d\right)} \cos\left(f x + e\right) - {\left(c - d\right)} \sin\left(f x + e\right) + c - d\right)} \sqrt{a \sin\left(f x + e\right) + a}}{8 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/8*(sqrt(2)*((c + 3*d)*cos(f*x + e)^2 - (c + 3*d)*cos(f*x + e) - ((c + 3*d)*cos(f*x + e) + 2*c + 6*d)*sin(f*x + e) - 2*c - 6*d)*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","B",0
553,1,252,0,1.152768," ","integrate(1/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{2} {\left(\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, \sqrt{a \sin\left(f x + e\right) + a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{8 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/8*(sqrt(2)*(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))","B",0
554,1,1299,0,1.957826," ","integrate(1/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{2} {\left({\left(c - 5 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(c - 5 \, d\right)} \cos\left(f x + e\right) - {\left({\left(c - 5 \, d\right)} \cos\left(f x + e\right) + 2 \, c - 10 \, d\right)} \sin\left(f x + e\right) - 2 \, c + 10 \, d\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 4 \, {\left(a d \cos\left(f x + e\right)^{2} - a d \cos\left(f x + e\right) - 2 \, a d - {\left(a d \cos\left(f x + e\right) + 2 \, a d\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a c + a d}} \log\left(\frac{d^{2} \cos\left(f x + e\right)^{3} - {\left(6 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - 4 \, {\left({\left(c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 4 \, c d - 3 \, d^{2} - {\left(c^{2} + 3 \, c d + 2 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} + 4 \, c d + 3 \, d^{2} + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{d}{a c + a d}} - {\left(c^{2} + 8 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 2 \, {\left(3 \, c d + 4 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left({\left(c - d\right)} \cos\left(f x + e\right) - {\left(c - d\right)} \sin\left(f x + e\right) + c - d\right)} \sqrt{a \sin\left(f x + e\right) + a}}{8 \, {\left({\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f - {\left({\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{\sqrt{2} {\left({\left(c - 5 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(c - 5 \, d\right)} \cos\left(f x + e\right) - {\left({\left(c - 5 \, d\right)} \cos\left(f x + e\right) + 2 \, c - 10 \, d\right)} \sin\left(f x + e\right) - 2 \, c + 10 \, d\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 8 \, {\left(a d \cos\left(f x + e\right)^{2} - a d \cos\left(f x + e\right) - 2 \, a d - {\left(a d \cos\left(f x + e\right) + 2 \, a d\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a c + a d}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{d}{a c + a d}}}{2 \, d \cos\left(f x + e\right)}\right) - 4 \, {\left({\left(c - d\right)} \cos\left(f x + e\right) - {\left(c - d\right)} \sin\left(f x + e\right) + c - d\right)} \sqrt{a \sin\left(f x + e\right) + a}}{8 \, {\left({\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f - {\left({\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/8*(sqrt(2)*((c - 5*d)*cos(f*x + e)^2 - (c - 5*d)*cos(f*x + e) - ((c - 5*d)*cos(f*x + e) + 2*c - 10*d)*sin(f*x + e) - 2*c + 10*d)*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 4*(a*d*cos(f*x + e)^2 - a*d*cos(f*x + e) - 2*a*d - (a*d*cos(f*x + e) + 2*a*d)*sin(f*x + e))*sqrt(d/(a*c + a*d))*log((d^2*cos(f*x + e)^3 - (6*c*d + 7*d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - 4*((c*d + d^2)*cos(f*x + e)^2 - c^2 - 4*c*d - 3*d^2 - (c^2 + 3*c*d + 2*d^2)*cos(f*x + e) + (c^2 + 4*c*d + 3*d^2 + (c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d/(a*c + a*d)) - (c^2 + 8*c*d + 9*d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 2*(3*c*d + 4*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) - 4*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a))/((a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e)^2 - (a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e) - 2*(a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f - ((a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e) + 2*(a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f)*sin(f*x + e)), -1/8*(sqrt(2)*((c - 5*d)*cos(f*x + e)^2 - (c - 5*d)*cos(f*x + e) - ((c - 5*d)*cos(f*x + e) + 2*c - 10*d)*sin(f*x + e) - 2*c + 10*d)*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 8*(a*d*cos(f*x + e)^2 - a*d*cos(f*x + e) - 2*a*d - (a*d*cos(f*x + e) + 2*a*d)*sin(f*x + e))*sqrt(-d/(a*c + a*d))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) - 4*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a))/((a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e)^2 - (a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e) - 2*(a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f - ((a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e) + 2*(a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f)*sin(f*x + e))]","B",0
555,1,2515,0,3.547648," ","integrate(1/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{\sqrt{2} {\left({\left(c^{2} d - 8 \, c d^{2} - 9 \, d^{3}\right)} \cos\left(f x + e\right)^{3} - 2 \, c^{3} + 14 \, c^{2} d + 34 \, c d^{2} + 18 \, d^{3} + {\left(c^{3} - 6 \, c^{2} d - 25 \, c d^{2} - 18 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d - 17 \, c d^{2} - 9 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(2 \, c^{3} - 14 \, c^{2} d - 34 \, c d^{2} - 18 \, d^{3} - {\left(c^{2} d - 8 \, c d^{2} - 9 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{3} - 7 \, c^{2} d - 17 \, c d^{2} - 9 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 2 \, {\left(10 \, a c^{2} d + 16 \, a c d^{2} + 6 \, a d^{3} - {\left(5 \, a c d^{2} + 3 \, a d^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a c^{2} d + 13 \, a c d^{2} + 6 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(5 \, a c^{2} d + 8 \, a c d^{2} + 3 \, a d^{3}\right)} \cos\left(f x + e\right) + {\left(10 \, a c^{2} d + 16 \, a c d^{2} + 6 \, a d^{3} - {\left(5 \, a c d^{2} + 3 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(5 \, a c^{2} d + 8 \, a c d^{2} + 3 \, a d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a c + a d}} \log\left(\frac{d^{2} \cos\left(f x + e\right)^{3} - {\left(6 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 4 \, {\left({\left(c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 4 \, c d - 3 \, d^{2} - {\left(c^{2} + 3 \, c d + 2 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} + 4 \, c d + 3 \, d^{2} + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{d}{a c + a d}} - {\left(c^{2} + 8 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 2 \, {\left(3 \, c d + 4 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(c^{3} - c^{2} d - c d^{2} + d^{3} + {\left(c^{2} d + 2 \, c d^{2} - 3 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{3} + c d^{2} - 2 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{3} - c^{2} d - c d^{2} + d^{3} - {\left(c^{2} d + 2 \, c d^{2} - 3 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{8 \, {\left({\left(a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{2} c^{5} - 4 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + 3 \, a^{2} c d^{4} - 2 \, a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f + {\left({\left(a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{\sqrt{2} {\left({\left(c^{2} d - 8 \, c d^{2} - 9 \, d^{3}\right)} \cos\left(f x + e\right)^{3} - 2 \, c^{3} + 14 \, c^{2} d + 34 \, c d^{2} + 18 \, d^{3} + {\left(c^{3} - 6 \, c^{2} d - 25 \, c d^{2} - 18 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d - 17 \, c d^{2} - 9 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(2 \, c^{3} - 14 \, c^{2} d - 34 \, c d^{2} - 18 \, d^{3} - {\left(c^{2} d - 8 \, c d^{2} - 9 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{3} - 7 \, c^{2} d - 17 \, c d^{2} - 9 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left(10 \, a c^{2} d + 16 \, a c d^{2} + 6 \, a d^{3} - {\left(5 \, a c d^{2} + 3 \, a d^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a c^{2} d + 13 \, a c d^{2} + 6 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(5 \, a c^{2} d + 8 \, a c d^{2} + 3 \, a d^{3}\right)} \cos\left(f x + e\right) + {\left(10 \, a c^{2} d + 16 \, a c d^{2} + 6 \, a d^{3} - {\left(5 \, a c d^{2} + 3 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(5 \, a c^{2} d + 8 \, a c d^{2} + 3 \, a d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a c + a d}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{d}{a c + a d}}}{2 \, d \cos\left(f x + e\right)}\right) + 4 \, {\left(c^{3} - c^{2} d - c d^{2} + d^{3} + {\left(c^{2} d + 2 \, c d^{2} - 3 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{3} + c d^{2} - 2 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{3} - c^{2} d - c d^{2} + d^{3} - {\left(c^{2} d + 2 \, c d^{2} - 3 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{8 \, {\left({\left(a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{2} c^{5} - 4 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + 3 \, a^{2} c d^{4} - 2 \, a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f + {\left({\left(a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/8*(sqrt(2)*((c^2*d - 8*c*d^2 - 9*d^3)*cos(f*x + e)^3 - 2*c^3 + 14*c^2*d + 34*c*d^2 + 18*d^3 + (c^3 - 6*c^2*d - 25*c*d^2 - 18*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d - 17*c*d^2 - 9*d^3)*cos(f*x + e) - (2*c^3 - 14*c^2*d - 34*c*d^2 - 18*d^3 - (c^2*d - 8*c*d^2 - 9*d^3)*cos(f*x + e)^2 + (c^3 - 7*c^2*d - 17*c*d^2 - 9*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 2*(10*a*c^2*d + 16*a*c*d^2 + 6*a*d^3 - (5*a*c*d^2 + 3*a*d^3)*cos(f*x + e)^3 - (5*a*c^2*d + 13*a*c*d^2 + 6*a*d^3)*cos(f*x + e)^2 + (5*a*c^2*d + 8*a*c*d^2 + 3*a*d^3)*cos(f*x + e) + (10*a*c^2*d + 16*a*c*d^2 + 6*a*d^3 - (5*a*c*d^2 + 3*a*d^3)*cos(f*x + e)^2 + (5*a*c^2*d + 8*a*c*d^2 + 3*a*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(d/(a*c + a*d))*log((d^2*cos(f*x + e)^3 - (6*c*d + 7*d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 4*((c*d + d^2)*cos(f*x + e)^2 - c^2 - 4*c*d - 3*d^2 - (c^2 + 3*c*d + 2*d^2)*cos(f*x + e) + (c^2 + 4*c*d + 3*d^2 + (c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d/(a*c + a*d)) - (c^2 + 8*c*d + 9*d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 2*(3*c*d + 4*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(c^3 - c^2*d - c*d^2 + d^3 + (c^2*d + 2*c*d^2 - 3*d^3)*cos(f*x + e)^2 + (c^3 + c*d^2 - 2*d^3)*cos(f*x + e) - (c^3 - c^2*d - c*d^2 + d^3 - (c^2*d + 2*c*d^2 - 3*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e)^3 + (a^2*c^5 - 4*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + 3*a^2*c*d^4 - 2*a^2*d^5)*f*cos(f*x + e)^2 - (a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e) - 2*(a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f + ((a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e)^2 - (a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e) - 2*(a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f)*sin(f*x + e)), 1/8*(sqrt(2)*((c^2*d - 8*c*d^2 - 9*d^3)*cos(f*x + e)^3 - 2*c^3 + 14*c^2*d + 34*c*d^2 + 18*d^3 + (c^3 - 6*c^2*d - 25*c*d^2 - 18*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d - 17*c*d^2 - 9*d^3)*cos(f*x + e) - (2*c^3 - 14*c^2*d - 34*c*d^2 - 18*d^3 - (c^2*d - 8*c*d^2 - 9*d^3)*cos(f*x + e)^2 + (c^3 - 7*c^2*d - 17*c*d^2 - 9*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(10*a*c^2*d + 16*a*c*d^2 + 6*a*d^3 - (5*a*c*d^2 + 3*a*d^3)*cos(f*x + e)^3 - (5*a*c^2*d + 13*a*c*d^2 + 6*a*d^3)*cos(f*x + e)^2 + (5*a*c^2*d + 8*a*c*d^2 + 3*a*d^3)*cos(f*x + e) + (10*a*c^2*d + 16*a*c*d^2 + 6*a*d^3 - (5*a*c*d^2 + 3*a*d^3)*cos(f*x + e)^2 + (5*a*c^2*d + 8*a*c*d^2 + 3*a*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-d/(a*c + a*d))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) + 4*(c^3 - c^2*d - c*d^2 + d^3 + (c^2*d + 2*c*d^2 - 3*d^3)*cos(f*x + e)^2 + (c^3 + c*d^2 - 2*d^3)*cos(f*x + e) - (c^3 - c^2*d - c*d^2 + d^3 - (c^2*d + 2*c*d^2 - 3*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e)^3 + (a^2*c^5 - 4*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + 3*a^2*c*d^4 - 2*a^2*d^5)*f*cos(f*x + e)^2 - (a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e) - 2*(a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f + ((a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e)^2 - (a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e) - 2*(a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f)*sin(f*x + e))]","B",0
556,1,4133,0,6.171340," ","integrate(1/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[-\frac{2 \, \sqrt{2} {\left(2 \, c^{5} - 18 \, c^{4} d - 92 \, c^{3} d^{2} - 148 \, c^{2} d^{3} - 102 \, c d^{4} - 26 \, d^{5} + {\left(c^{3} d^{2} - 11 \, c^{2} d^{3} - 25 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, c^{4} d - 21 \, c^{3} d^{2} - 61 \, c^{2} d^{3} - 51 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(c^{5} - 7 \, c^{4} d - 66 \, c^{3} d^{2} - 146 \, c^{2} d^{3} - 127 \, c d^{4} - 39 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{5} - 9 \, c^{4} d - 46 \, c^{3} d^{2} - 74 \, c^{2} d^{3} - 51 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right) + {\left(2 \, c^{5} - 18 \, c^{4} d - 92 \, c^{3} d^{2} - 148 \, c^{2} d^{3} - 102 \, c d^{4} - 26 \, d^{5} - {\left(c^{3} d^{2} - 11 \, c^{2} d^{3} - 25 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(c^{4} d - 10 \, c^{3} d^{2} - 36 \, c^{2} d^{3} - 38 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{5} - 9 \, c^{4} d - 46 \, c^{3} d^{2} - 74 \, c^{2} d^{3} - 51 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - {\left(70 \, a c^{4} d + 224 \, a c^{3} d^{2} + 276 \, a c^{2} d^{3} + 160 \, a c d^{4} + 38 \, a d^{5} + {\left(35 \, a c^{2} d^{3} + 42 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(70 \, a c^{3} d^{2} + 119 \, a c^{2} d^{3} + 80 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(35 \, a c^{4} d + 182 \, a c^{3} d^{2} + 292 \, a c^{2} d^{3} + 202 \, a c d^{4} + 57 \, a d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(35 \, a c^{4} d + 112 \, a c^{3} d^{2} + 138 \, a c^{2} d^{3} + 80 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right) + {\left(70 \, a c^{4} d + 224 \, a c^{3} d^{2} + 276 \, a c^{2} d^{3} + 160 \, a c d^{4} + 38 \, a d^{5} - {\left(35 \, a c^{2} d^{3} + 42 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(35 \, a c^{3} d^{2} + 77 \, a c^{2} d^{3} + 61 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(35 \, a c^{4} d + 112 \, a c^{3} d^{2} + 138 \, a c^{2} d^{3} + 80 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a c + a d}} \log\left(\frac{d^{2} \cos\left(f x + e\right)^{3} - {\left(6 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - 4 \, {\left({\left(c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 4 \, c d - 3 \, d^{2} - {\left(c^{2} + 3 \, c d + 2 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} + 4 \, c d + 3 \, d^{2} + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{d}{a c + a d}} - {\left(c^{2} + 8 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 2 \, {\left(3 \, c d + 4 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(2 \, c^{5} - 2 \, c^{4} d - 4 \, c^{3} d^{2} + 4 \, c^{2} d^{3} + 2 \, c d^{4} - 2 \, d^{5} - {\left(2 \, c^{3} d^{2} + 13 \, c^{2} d^{3} - 8 \, c d^{4} - 7 \, d^{5}\right)} \cos\left(f x + e\right)^{3} + {\left(4 \, c^{4} d + 15 \, c^{3} d^{2} - 14 \, c^{2} d^{3} - 9 \, c d^{4} + 4 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(2 \, c^{5} + 2 \, c^{4} d + 13 \, c^{3} d^{2} + 3 \, c^{2} d^{3} - 15 \, c d^{4} - 5 \, d^{5}\right)} \cos\left(f x + e\right) - {\left(2 \, c^{5} - 2 \, c^{4} d - 4 \, c^{3} d^{2} + 4 \, c^{2} d^{3} + 2 \, c d^{4} - 2 \, d^{5} - {\left(2 \, c^{3} d^{2} + 13 \, c^{2} d^{3} - 8 \, c d^{4} - 7 \, d^{5}\right)} \cos\left(f x + e\right)^{2} - {\left(4 \, c^{4} d + 17 \, c^{3} d^{2} - c^{2} d^{3} - 17 \, c d^{4} - 3 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{16 \, {\left({\left(a^{2} c^{6} d^{2} - 2 \, a^{2} c^{5} d^{3} - a^{2} c^{4} d^{4} + 4 \, a^{2} c^{3} d^{5} - a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{2} c^{7} d - 3 \, a^{2} c^{6} d^{2} - 4 \, a^{2} c^{5} d^{3} + 7 \, a^{2} c^{4} d^{4} + 2 \, a^{2} c^{3} d^{5} - 5 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{2} c^{8} + 2 \, a^{2} c^{7} d - 6 \, a^{2} c^{6} d^{2} - 6 \, a^{2} c^{5} d^{3} + 12 \, a^{2} c^{4} d^{4} + 6 \, a^{2} c^{3} d^{5} - 10 \, a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + 3 \, a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f - {\left({\left(a^{2} c^{6} d^{2} - 2 \, a^{2} c^{5} d^{3} - a^{2} c^{4} d^{4} + 4 \, a^{2} c^{3} d^{5} - a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{3} + 2 \, {\left(a^{2} c^{7} d - a^{2} c^{6} d^{2} - 3 \, a^{2} c^{5} d^{3} + 3 \, a^{2} c^{4} d^{4} + 3 \, a^{2} c^{3} d^{5} - 3 \, a^{2} c^{2} d^{6} - a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{\sqrt{2} {\left(2 \, c^{5} - 18 \, c^{4} d - 92 \, c^{3} d^{2} - 148 \, c^{2} d^{3} - 102 \, c d^{4} - 26 \, d^{5} + {\left(c^{3} d^{2} - 11 \, c^{2} d^{3} - 25 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, c^{4} d - 21 \, c^{3} d^{2} - 61 \, c^{2} d^{3} - 51 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(c^{5} - 7 \, c^{4} d - 66 \, c^{3} d^{2} - 146 \, c^{2} d^{3} - 127 \, c d^{4} - 39 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{5} - 9 \, c^{4} d - 46 \, c^{3} d^{2} - 74 \, c^{2} d^{3} - 51 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right) + {\left(2 \, c^{5} - 18 \, c^{4} d - 92 \, c^{3} d^{2} - 148 \, c^{2} d^{3} - 102 \, c d^{4} - 26 \, d^{5} - {\left(c^{3} d^{2} - 11 \, c^{2} d^{3} - 25 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(c^{4} d - 10 \, c^{3} d^{2} - 36 \, c^{2} d^{3} - 38 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{5} - 9 \, c^{4} d - 46 \, c^{3} d^{2} - 74 \, c^{2} d^{3} - 51 \, c d^{4} - 13 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + {\left(70 \, a c^{4} d + 224 \, a c^{3} d^{2} + 276 \, a c^{2} d^{3} + 160 \, a c d^{4} + 38 \, a d^{5} + {\left(35 \, a c^{2} d^{3} + 42 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(70 \, a c^{3} d^{2} + 119 \, a c^{2} d^{3} + 80 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(35 \, a c^{4} d + 182 \, a c^{3} d^{2} + 292 \, a c^{2} d^{3} + 202 \, a c d^{4} + 57 \, a d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(35 \, a c^{4} d + 112 \, a c^{3} d^{2} + 138 \, a c^{2} d^{3} + 80 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right) + {\left(70 \, a c^{4} d + 224 \, a c^{3} d^{2} + 276 \, a c^{2} d^{3} + 160 \, a c d^{4} + 38 \, a d^{5} - {\left(35 \, a c^{2} d^{3} + 42 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(35 \, a c^{3} d^{2} + 77 \, a c^{2} d^{3} + 61 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(35 \, a c^{4} d + 112 \, a c^{3} d^{2} + 138 \, a c^{2} d^{3} + 80 \, a c d^{4} + 19 \, a d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a c + a d}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{d}{a c + a d}}}{2 \, d \cos\left(f x + e\right)}\right) + 2 \, {\left(2 \, c^{5} - 2 \, c^{4} d - 4 \, c^{3} d^{2} + 4 \, c^{2} d^{3} + 2 \, c d^{4} - 2 \, d^{5} - {\left(2 \, c^{3} d^{2} + 13 \, c^{2} d^{3} - 8 \, c d^{4} - 7 \, d^{5}\right)} \cos\left(f x + e\right)^{3} + {\left(4 \, c^{4} d + 15 \, c^{3} d^{2} - 14 \, c^{2} d^{3} - 9 \, c d^{4} + 4 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(2 \, c^{5} + 2 \, c^{4} d + 13 \, c^{3} d^{2} + 3 \, c^{2} d^{3} - 15 \, c d^{4} - 5 \, d^{5}\right)} \cos\left(f x + e\right) - {\left(2 \, c^{5} - 2 \, c^{4} d - 4 \, c^{3} d^{2} + 4 \, c^{2} d^{3} + 2 \, c d^{4} - 2 \, d^{5} - {\left(2 \, c^{3} d^{2} + 13 \, c^{2} d^{3} - 8 \, c d^{4} - 7 \, d^{5}\right)} \cos\left(f x + e\right)^{2} - {\left(4 \, c^{4} d + 17 \, c^{3} d^{2} - c^{2} d^{3} - 17 \, c d^{4} - 3 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{8 \, {\left({\left(a^{2} c^{6} d^{2} - 2 \, a^{2} c^{5} d^{3} - a^{2} c^{4} d^{4} + 4 \, a^{2} c^{3} d^{5} - a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{2} c^{7} d - 3 \, a^{2} c^{6} d^{2} - 4 \, a^{2} c^{5} d^{3} + 7 \, a^{2} c^{4} d^{4} + 2 \, a^{2} c^{3} d^{5} - 5 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{2} c^{8} + 2 \, a^{2} c^{7} d - 6 \, a^{2} c^{6} d^{2} - 6 \, a^{2} c^{5} d^{3} + 12 \, a^{2} c^{4} d^{4} + 6 \, a^{2} c^{3} d^{5} - 10 \, a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + 3 \, a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f - {\left({\left(a^{2} c^{6} d^{2} - 2 \, a^{2} c^{5} d^{3} - a^{2} c^{4} d^{4} + 4 \, a^{2} c^{3} d^{5} - a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{3} + 2 \, {\left(a^{2} c^{7} d - a^{2} c^{6} d^{2} - 3 \, a^{2} c^{5} d^{3} + 3 \, a^{2} c^{4} d^{4} + 3 \, a^{2} c^{3} d^{5} - 3 \, a^{2} c^{2} d^{6} - a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/16*(2*sqrt(2)*(2*c^5 - 18*c^4*d - 92*c^3*d^2 - 148*c^2*d^3 - 102*c*d^4 - 26*d^5 + (c^3*d^2 - 11*c^2*d^3 - 25*c*d^4 - 13*d^5)*cos(f*x + e)^4 - (2*c^4*d - 21*c^3*d^2 - 61*c^2*d^3 - 51*c*d^4 - 13*d^5)*cos(f*x + e)^3 - (c^5 - 7*c^4*d - 66*c^3*d^2 - 146*c^2*d^3 - 127*c*d^4 - 39*d^5)*cos(f*x + e)^2 + (c^5 - 9*c^4*d - 46*c^3*d^2 - 74*c^2*d^3 - 51*c*d^4 - 13*d^5)*cos(f*x + e) + (2*c^5 - 18*c^4*d - 92*c^3*d^2 - 148*c^2*d^3 - 102*c*d^4 - 26*d^5 - (c^3*d^2 - 11*c^2*d^3 - 25*c*d^4 - 13*d^5)*cos(f*x + e)^3 - 2*(c^4*d - 10*c^3*d^2 - 36*c^2*d^3 - 38*c*d^4 - 13*d^5)*cos(f*x + e)^2 + (c^5 - 9*c^4*d - 46*c^3*d^2 - 74*c^2*d^3 - 51*c*d^4 - 13*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - (70*a*c^4*d + 224*a*c^3*d^2 + 276*a*c^2*d^3 + 160*a*c*d^4 + 38*a*d^5 + (35*a*c^2*d^3 + 42*a*c*d^4 + 19*a*d^5)*cos(f*x + e)^4 - (70*a*c^3*d^2 + 119*a*c^2*d^3 + 80*a*c*d^4 + 19*a*d^5)*cos(f*x + e)^3 - (35*a*c^4*d + 182*a*c^3*d^2 + 292*a*c^2*d^3 + 202*a*c*d^4 + 57*a*d^5)*cos(f*x + e)^2 + (35*a*c^4*d + 112*a*c^3*d^2 + 138*a*c^2*d^3 + 80*a*c*d^4 + 19*a*d^5)*cos(f*x + e) + (70*a*c^4*d + 224*a*c^3*d^2 + 276*a*c^2*d^3 + 160*a*c*d^4 + 38*a*d^5 - (35*a*c^2*d^3 + 42*a*c*d^4 + 19*a*d^5)*cos(f*x + e)^3 - 2*(35*a*c^3*d^2 + 77*a*c^2*d^3 + 61*a*c*d^4 + 19*a*d^5)*cos(f*x + e)^2 + (35*a*c^4*d + 112*a*c^3*d^2 + 138*a*c^2*d^3 + 80*a*c*d^4 + 19*a*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(d/(a*c + a*d))*log((d^2*cos(f*x + e)^3 - (6*c*d + 7*d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - 4*((c*d + d^2)*cos(f*x + e)^2 - c^2 - 4*c*d - 3*d^2 - (c^2 + 3*c*d + 2*d^2)*cos(f*x + e) + (c^2 + 4*c*d + 3*d^2 + (c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d/(a*c + a*d)) - (c^2 + 8*c*d + 9*d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 2*(3*c*d + 4*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(2*c^5 - 2*c^4*d - 4*c^3*d^2 + 4*c^2*d^3 + 2*c*d^4 - 2*d^5 - (2*c^3*d^2 + 13*c^2*d^3 - 8*c*d^4 - 7*d^5)*cos(f*x + e)^3 + (4*c^4*d + 15*c^3*d^2 - 14*c^2*d^3 - 9*c*d^4 + 4*d^5)*cos(f*x + e)^2 + (2*c^5 + 2*c^4*d + 13*c^3*d^2 + 3*c^2*d^3 - 15*c*d^4 - 5*d^5)*cos(f*x + e) - (2*c^5 - 2*c^4*d - 4*c^3*d^2 + 4*c^2*d^3 + 2*c*d^4 - 2*d^5 - (2*c^3*d^2 + 13*c^2*d^3 - 8*c*d^4 - 7*d^5)*cos(f*x + e)^2 - (4*c^4*d + 17*c^3*d^2 - c^2*d^3 - 17*c*d^4 - 3*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^2*c^6*d^2 - 2*a^2*c^5*d^3 - a^2*c^4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^4 - (2*a^2*c^7*d - 3*a^2*c^6*d^2 - 4*a^2*c^5*d^3 + 7*a^2*c^4*d^4 + 2*a^2*c^3*d^5 - 5*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e)^3 - (a^2*c^8 + 2*a^2*c^7*d - 6*a^2*c^6*d^2 - 6*a^2*c^5*d^3 + 12*a^2*c^4*d^4 + 6*a^2*c^3*d^5 - 10*a^2*c^2*d^6 - 2*a^2*c*d^7 + 3*a^2*d^8)*f*cos(f*x + e)^2 + (a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e) + 2*(a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f - ((a^2*c^6*d^2 - 2*a^2*c^5*d^3 - a^2*c^4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^3 + 2*(a^2*c^7*d - a^2*c^6*d^2 - 3*a^2*c^5*d^3 + 3*a^2*c^4*d^4 + 3*a^2*c^3*d^5 - 3*a^2*c^2*d^6 - a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^2 - (a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e) - 2*(a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f)*sin(f*x + e)), -1/8*(sqrt(2)*(2*c^5 - 18*c^4*d - 92*c^3*d^2 - 148*c^2*d^3 - 102*c*d^4 - 26*d^5 + (c^3*d^2 - 11*c^2*d^3 - 25*c*d^4 - 13*d^5)*cos(f*x + e)^4 - (2*c^4*d - 21*c^3*d^2 - 61*c^2*d^3 - 51*c*d^4 - 13*d^5)*cos(f*x + e)^3 - (c^5 - 7*c^4*d - 66*c^3*d^2 - 146*c^2*d^3 - 127*c*d^4 - 39*d^5)*cos(f*x + e)^2 + (c^5 - 9*c^4*d - 46*c^3*d^2 - 74*c^2*d^3 - 51*c*d^4 - 13*d^5)*cos(f*x + e) + (2*c^5 - 18*c^4*d - 92*c^3*d^2 - 148*c^2*d^3 - 102*c*d^4 - 26*d^5 - (c^3*d^2 - 11*c^2*d^3 - 25*c*d^4 - 13*d^5)*cos(f*x + e)^3 - 2*(c^4*d - 10*c^3*d^2 - 36*c^2*d^3 - 38*c*d^4 - 13*d^5)*cos(f*x + e)^2 + (c^5 - 9*c^4*d - 46*c^3*d^2 - 74*c^2*d^3 - 51*c*d^4 - 13*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + (70*a*c^4*d + 224*a*c^3*d^2 + 276*a*c^2*d^3 + 160*a*c*d^4 + 38*a*d^5 + (35*a*c^2*d^3 + 42*a*c*d^4 + 19*a*d^5)*cos(f*x + e)^4 - (70*a*c^3*d^2 + 119*a*c^2*d^3 + 80*a*c*d^4 + 19*a*d^5)*cos(f*x + e)^3 - (35*a*c^4*d + 182*a*c^3*d^2 + 292*a*c^2*d^3 + 202*a*c*d^4 + 57*a*d^5)*cos(f*x + e)^2 + (35*a*c^4*d + 112*a*c^3*d^2 + 138*a*c^2*d^3 + 80*a*c*d^4 + 19*a*d^5)*cos(f*x + e) + (70*a*c^4*d + 224*a*c^3*d^2 + 276*a*c^2*d^3 + 160*a*c*d^4 + 38*a*d^5 - (35*a*c^2*d^3 + 42*a*c*d^4 + 19*a*d^5)*cos(f*x + e)^3 - 2*(35*a*c^3*d^2 + 77*a*c^2*d^3 + 61*a*c*d^4 + 19*a*d^5)*cos(f*x + e)^2 + (35*a*c^4*d + 112*a*c^3*d^2 + 138*a*c^2*d^3 + 80*a*c*d^4 + 19*a*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(-d/(a*c + a*d))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) + 2*(2*c^5 - 2*c^4*d - 4*c^3*d^2 + 4*c^2*d^3 + 2*c*d^4 - 2*d^5 - (2*c^3*d^2 + 13*c^2*d^3 - 8*c*d^4 - 7*d^5)*cos(f*x + e)^3 + (4*c^4*d + 15*c^3*d^2 - 14*c^2*d^3 - 9*c*d^4 + 4*d^5)*cos(f*x + e)^2 + (2*c^5 + 2*c^4*d + 13*c^3*d^2 + 3*c^2*d^3 - 15*c*d^4 - 5*d^5)*cos(f*x + e) - (2*c^5 - 2*c^4*d - 4*c^3*d^2 + 4*c^2*d^3 + 2*c*d^4 - 2*d^5 - (2*c^3*d^2 + 13*c^2*d^3 - 8*c*d^4 - 7*d^5)*cos(f*x + e)^2 - (4*c^4*d + 17*c^3*d^2 - c^2*d^3 - 17*c*d^4 - 3*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^2*c^6*d^2 - 2*a^2*c^5*d^3 - a^2*c^4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^4 - (2*a^2*c^7*d - 3*a^2*c^6*d^2 - 4*a^2*c^5*d^3 + 7*a^2*c^4*d^4 + 2*a^2*c^3*d^5 - 5*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e)^3 - (a^2*c^8 + 2*a^2*c^7*d - 6*a^2*c^6*d^2 - 6*a^2*c^5*d^3 + 12*a^2*c^4*d^4 + 6*a^2*c^3*d^5 - 10*a^2*c^2*d^6 - 2*a^2*c*d^7 + 3*a^2*d^8)*f*cos(f*x + e)^2 + (a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e) + 2*(a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f - ((a^2*c^6*d^2 - 2*a^2*c^5*d^3 - a^2*c^4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^3 + 2*(a^2*c^7*d - a^2*c^6*d^2 - 3*a^2*c^5*d^3 + 3*a^2*c^4*d^4 + 3*a^2*c^3*d^5 - 3*a^2*c^2*d^6 - a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^2 - (a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e) - 2*(a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f)*sin(f*x + e))]","B",0
557,1,608,0,0.895306," ","integrate((c+d*sin(f*x+e))^3/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","-\frac{3 \, \sqrt{2} {\left({\left(c^{3} + 5 \, c^{2} d + 19 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right)^{3} - 4 \, c^{3} - 20 \, c^{2} d - 76 \, c d^{2} + 100 \, d^{3} + 3 \, {\left(c^{3} + 5 \, c^{2} d + 19 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} + 5 \, c^{2} d + 19 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{3} + 20 \, c^{2} d + 76 \, c d^{2} - 100 \, d^{3} - {\left(c^{3} + 5 \, c^{2} d + 19 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(c^{3} + 5 \, c^{2} d + 19 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left(32 \, d^{3} \cos\left(f x + e\right)^{3} - 4 \, c^{3} + 12 \, c^{2} d - 12 \, c d^{2} + 4 \, d^{3} - {\left(3 \, c^{3} + 15 \, c^{2} d - 39 \, c d^{2} + 53 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(7 \, c^{3} + 3 \, c^{2} d - 27 \, c d^{2} + 81 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(32 \, d^{3} \cos\left(f x + e\right)^{2} - 4 \, c^{3} + 12 \, c^{2} d - 12 \, c d^{2} + 4 \, d^{3} + {\left(3 \, c^{3} + 15 \, c^{2} d - 39 \, c d^{2} + 85 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{64 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-1/64*(3*sqrt(2)*((c^3 + 5*c^2*d + 19*c*d^2 - 25*d^3)*cos(f*x + e)^3 - 4*c^3 - 20*c^2*d - 76*c*d^2 + 100*d^3 + 3*(c^3 + 5*c^2*d + 19*c*d^2 - 25*d^3)*cos(f*x + e)^2 - 2*(c^3 + 5*c^2*d + 19*c*d^2 - 25*d^3)*cos(f*x + e) - (4*c^3 + 20*c^2*d + 76*c*d^2 - 100*d^3 - (c^3 + 5*c^2*d + 19*c*d^2 - 25*d^3)*cos(f*x + e)^2 + 2*(c^3 + 5*c^2*d + 19*c*d^2 - 25*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(32*d^3*cos(f*x + e)^3 - 4*c^3 + 12*c^2*d - 12*c*d^2 + 4*d^3 - (3*c^3 + 15*c^2*d - 39*c*d^2 + 53*d^3)*cos(f*x + e)^2 - (7*c^3 + 3*c^2*d - 27*c*d^2 + 81*d^3)*cos(f*x + e) - (32*d^3*cos(f*x + e)^2 - 4*c^3 + 12*c^2*d - 12*c*d^2 + 4*d^3 + (3*c^3 + 15*c^2*d - 39*c*d^2 + 85*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
558,1,492,0,0.930666," ","integrate((c+d*sin(f*x+e))^2/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{\sqrt{2} {\left({\left(3 \, c^{2} + 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, c^{2} + 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} - 40 \, c d - 76 \, d^{2} - 2 \, {\left(3 \, c^{2} + 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(3 \, c^{2} + 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} - 40 \, c d - 76 \, d^{2} - 2 \, {\left(3 \, c^{2} + 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left({\left(3 \, c^{2} + 10 \, c d - 13 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} + 2 \, c d - 9 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - {\left(3 \, c^{2} + 10 \, c d - 13 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{64 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/64*(sqrt(2)*((3*c^2 + 10*c*d + 19*d^2)*cos(f*x + e)^3 + 3*(3*c^2 + 10*c*d + 19*d^2)*cos(f*x + e)^2 - 12*c^2 - 40*c*d - 76*d^2 - 2*(3*c^2 + 10*c*d + 19*d^2)*cos(f*x + e) + ((3*c^2 + 10*c*d + 19*d^2)*cos(f*x + e)^2 - 12*c^2 - 40*c*d - 76*d^2 - 2*(3*c^2 + 10*c*d + 19*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*((3*c^2 + 10*c*d - 13*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 + 2*c*d - 9*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - (3*c^2 + 10*c*d - 13*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
559,1,392,0,1.312949," ","integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{\sqrt{2} {\left({\left(3 \, c + 5 \, d\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, c + 5 \, d\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(3 \, c + 5 \, d\right)} \cos\left(f x + e\right) + {\left({\left(3 \, c + 5 \, d\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(3 \, c + 5 \, d\right)} \cos\left(f x + e\right) - 12 \, c - 20 \, d\right)} \sin\left(f x + e\right) - 12 \, c - 20 \, d\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left({\left(3 \, c + 5 \, d\right)} \cos\left(f x + e\right)^{2} + {\left(7 \, c + d\right)} \cos\left(f x + e\right) + {\left({\left(3 \, c + 5 \, d\right)} \cos\left(f x + e\right) - 4 \, c + 4 \, d\right)} \sin\left(f x + e\right) + 4 \, c - 4 \, d\right)} \sqrt{a \sin\left(f x + e\right) + a}}{64 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/64*(sqrt(2)*((3*c + 5*d)*cos(f*x + e)^3 + 3*(3*c + 5*d)*cos(f*x + e)^2 - 2*(3*c + 5*d)*cos(f*x + e) + ((3*c + 5*d)*cos(f*x + e)^2 - 2*(3*c + 5*d)*cos(f*x + e) - 12*c - 20*d)*sin(f*x + e) - 12*c - 20*d)*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*((3*c + 5*d)*cos(f*x + e)^2 + (7*c + d)*cos(f*x + e) + ((3*c + 5*d)*cos(f*x + e) - 4*c + 4*d)*sin(f*x + e) + 4*c - 4*d)*sqrt(a*sin(f*x + e) + a))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
560,1,320,0,1.145261," ","integrate(1/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{3 \, \sqrt{2} {\left(\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 4 \, {\left(3 \, \cos\left(f x + e\right)^{2} + {\left(3 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) + 7 \, \cos\left(f x + e\right) + 4\right)} \sqrt{a \sin\left(f x + e\right) + a}}{64 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/64*(3*sqrt(2)*(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(3*cos(f*x + e)^2 + (3*cos(f*x + e) - 4)*sin(f*x + e) + 7*cos(f*x + e) + 4)*sqrt(a*sin(f*x + e) + a))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))","B",0
561,1,2015,0,2.680229," ","integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{2} {\left({\left(3 \, c^{2} - 14 \, c d + 43 \, d^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, c^{2} - 14 \, c d + 43 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 56 \, c d - 172 \, d^{2} - 2 \, {\left(3 \, c^{2} - 14 \, c d + 43 \, d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(3 \, c^{2} - 14 \, c d + 43 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 56 \, c d - 172 \, d^{2} - 2 \, {\left(3 \, c^{2} - 14 \, c d + 43 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 32 \, {\left(a d^{2} \cos\left(f x + e\right)^{3} + 3 \, a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2} + {\left(a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a c + a d}} \log\left(\frac{d^{2} \cos\left(f x + e\right)^{3} - {\left(6 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - 4 \, {\left({\left(c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 4 \, c d - 3 \, d^{2} - {\left(c^{2} + 3 \, c d + 2 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} + 4 \, c d + 3 \, d^{2} + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{d}{a c + a d}} - {\left(c^{2} + 8 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 2 \, {\left(3 \, c d + 4 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left({\left(3 \, c^{2} - 14 \, c d + 11 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} - 22 \, c d + 15 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - {\left(3 \, c^{2} - 14 \, c d + 11 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{64 \, {\left({\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f + {\left({\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{\sqrt{2} {\left({\left(3 \, c^{2} - 14 \, c d + 43 \, d^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, c^{2} - 14 \, c d + 43 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 56 \, c d - 172 \, d^{2} - 2 \, {\left(3 \, c^{2} - 14 \, c d + 43 \, d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(3 \, c^{2} - 14 \, c d + 43 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 56 \, c d - 172 \, d^{2} - 2 \, {\left(3 \, c^{2} - 14 \, c d + 43 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 64 \, {\left(a d^{2} \cos\left(f x + e\right)^{3} + 3 \, a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2} + {\left(a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a c + a d}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{d}{a c + a d}}}{2 \, d \cos\left(f x + e\right)}\right) - 4 \, {\left({\left(3 \, c^{2} - 14 \, c d + 11 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} - 22 \, c d + 15 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - {\left(3 \, c^{2} - 14 \, c d + 11 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{64 \, {\left({\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f + {\left({\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/64*(sqrt(2)*((3*c^2 - 14*c*d + 43*d^2)*cos(f*x + e)^3 + 3*(3*c^2 - 14*c*d + 43*d^2)*cos(f*x + e)^2 - 12*c^2 + 56*c*d - 172*d^2 - 2*(3*c^2 - 14*c*d + 43*d^2)*cos(f*x + e) + ((3*c^2 - 14*c*d + 43*d^2)*cos(f*x + e)^2 - 12*c^2 + 56*c*d - 172*d^2 - 2*(3*c^2 - 14*c*d + 43*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 32*(a*d^2*cos(f*x + e)^3 + 3*a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2 + (a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2)*sin(f*x + e))*sqrt(d/(a*c + a*d))*log((d^2*cos(f*x + e)^3 - (6*c*d + 7*d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - 4*((c*d + d^2)*cos(f*x + e)^2 - c^2 - 4*c*d - 3*d^2 - (c^2 + 3*c*d + 2*d^2)*cos(f*x + e) + (c^2 + 4*c*d + 3*d^2 + (c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d/(a*c + a*d)) - (c^2 + 8*c*d + 9*d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 2*(3*c*d + 4*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) - 4*((3*c^2 - 14*c*d + 11*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 - 22*c*d + 15*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - (3*c^2 - 14*c*d + 11*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^3 + 3*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^2 - 2*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e) - 4*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f + ((a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^2 - 2*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e) - 4*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f)*sin(f*x + e)), -1/64*(sqrt(2)*((3*c^2 - 14*c*d + 43*d^2)*cos(f*x + e)^3 + 3*(3*c^2 - 14*c*d + 43*d^2)*cos(f*x + e)^2 - 12*c^2 + 56*c*d - 172*d^2 - 2*(3*c^2 - 14*c*d + 43*d^2)*cos(f*x + e) + ((3*c^2 - 14*c*d + 43*d^2)*cos(f*x + e)^2 - 12*c^2 + 56*c*d - 172*d^2 - 2*(3*c^2 - 14*c*d + 43*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 64*(a*d^2*cos(f*x + e)^3 + 3*a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2 + (a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2)*sin(f*x + e))*sqrt(-d/(a*c + a*d))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) - 4*((3*c^2 - 14*c*d + 11*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 - 22*c*d + 15*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - (3*c^2 - 14*c*d + 11*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^3 + 3*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^2 - 2*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e) - 4*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f + ((a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^2 - 2*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e) - 4*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f)*sin(f*x + e))]","B",0
562,1,3719,0,4.808589," ","integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{\sqrt{2} {\left({\left(3 \, c^{3} d - 19 \, c^{2} d^{2} + 93 \, c d^{3} + 115 \, d^{4}\right)} \cos\left(f x + e\right)^{4} + 12 \, c^{4} - 64 \, c^{3} d + 296 \, c^{2} d^{2} + 832 \, c d^{3} + 460 \, d^{4} - {\left(3 \, c^{4} - 13 \, c^{3} d + 55 \, c^{2} d^{2} + 301 \, c d^{3} + 230 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(9 \, c^{4} - 42 \, c^{3} d + 184 \, c^{2} d^{2} + 810 \, c d^{3} + 575 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, c^{4} - 16 \, c^{3} d + 74 \, c^{2} d^{2} + 208 \, c d^{3} + 115 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(12 \, c^{4} - 64 \, c^{3} d + 296 \, c^{2} d^{2} + 832 \, c d^{3} + 460 \, d^{4} - {\left(3 \, c^{3} d - 19 \, c^{2} d^{2} + 93 \, c d^{3} + 115 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, c^{4} - 10 \, c^{3} d + 36 \, c^{2} d^{2} + 394 \, c d^{3} + 345 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, c^{4} - 16 \, c^{3} d + 74 \, c^{2} d^{2} + 208 \, c d^{3} + 115 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 16 \, {\left(28 \, a c^{2} d^{2} + 48 \, a c d^{3} + 20 \, a d^{4} + {\left(7 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{4} - {\left(7 \, a c^{2} d^{2} + 19 \, a c d^{3} + 10 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(21 \, a c^{2} d^{2} + 50 \, a c d^{3} + 25 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(7 \, a c^{2} d^{2} + 12 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(28 \, a c^{2} d^{2} + 48 \, a c d^{3} + 20 \, a d^{4} - {\left(7 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(7 \, a c^{2} d^{2} + 26 \, a c d^{3} + 15 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(7 \, a c^{2} d^{2} + 12 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a c + a d}} \log\left(\frac{d^{2} \cos\left(f x + e\right)^{3} - {\left(6 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 4 \, {\left({\left(c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 4 \, c d - 3 \, d^{2} - {\left(c^{2} + 3 \, c d + 2 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} + 4 \, c d + 3 \, d^{2} + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{d}{a c + a d}} - {\left(c^{2} + 8 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 2 \, {\left(3 \, c d + 4 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(4 \, c^{4} - 8 \, c^{3} d + 8 \, c d^{3} - 4 \, d^{4} - {\left(3 \, c^{3} d - 19 \, c^{2} d^{2} - 19 \, c d^{3} + 35 \, d^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, c^{4} - 15 \, c^{3} d - 7 \, c^{2} d^{2} - c d^{3} + 20 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(7 \, c^{4} - 20 \, c^{3} d - 26 \, c^{2} d^{2} - 12 \, c d^{3} + 51 \, d^{4}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{4} - 8 \, c^{3} d + 8 \, c d^{3} - 4 \, d^{4} - {\left(3 \, c^{3} d - 19 \, c^{2} d^{2} - 19 \, c d^{3} + 35 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c^{4} - 12 \, c^{3} d - 26 \, c^{2} d^{2} - 20 \, c d^{3} + 55 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{64 \, {\left({\left(a^{3} c^{5} d - 3 \, a^{3} c^{4} d^{2} + 2 \, a^{3} c^{3} d^{3} + 2 \, a^{3} c^{2} d^{4} - 3 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{6} - a^{3} c^{5} d - 4 \, a^{3} c^{4} d^{2} + 6 \, a^{3} c^{3} d^{3} + a^{3} c^{2} d^{4} - 5 \, a^{3} c d^{5} + 2 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{6} - 4 \, a^{3} c^{5} d - 9 \, a^{3} c^{4} d^{2} + 16 \, a^{3} c^{3} d^{3} + a^{3} c^{2} d^{4} - 12 \, a^{3} c d^{5} + 5 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f - {\left({\left(a^{3} c^{5} d - 3 \, a^{3} c^{4} d^{2} + 2 \, a^{3} c^{3} d^{3} + 2 \, a^{3} c^{2} d^{4} - 3 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{3} c^{6} - 7 \, a^{3} c^{4} d^{2} + 8 \, a^{3} c^{3} d^{3} + 3 \, a^{3} c^{2} d^{4} - 8 \, a^{3} c d^{5} + 3 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{\sqrt{2} {\left({\left(3 \, c^{3} d - 19 \, c^{2} d^{2} + 93 \, c d^{3} + 115 \, d^{4}\right)} \cos\left(f x + e\right)^{4} + 12 \, c^{4} - 64 \, c^{3} d + 296 \, c^{2} d^{2} + 832 \, c d^{3} + 460 \, d^{4} - {\left(3 \, c^{4} - 13 \, c^{3} d + 55 \, c^{2} d^{2} + 301 \, c d^{3} + 230 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(9 \, c^{4} - 42 \, c^{3} d + 184 \, c^{2} d^{2} + 810 \, c d^{3} + 575 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, c^{4} - 16 \, c^{3} d + 74 \, c^{2} d^{2} + 208 \, c d^{3} + 115 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(12 \, c^{4} - 64 \, c^{3} d + 296 \, c^{2} d^{2} + 832 \, c d^{3} + 460 \, d^{4} - {\left(3 \, c^{3} d - 19 \, c^{2} d^{2} + 93 \, c d^{3} + 115 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, c^{4} - 10 \, c^{3} d + 36 \, c^{2} d^{2} + 394 \, c d^{3} + 345 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, c^{4} - 16 \, c^{3} d + 74 \, c^{2} d^{2} + 208 \, c d^{3} + 115 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 32 \, {\left(28 \, a c^{2} d^{2} + 48 \, a c d^{3} + 20 \, a d^{4} + {\left(7 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{4} - {\left(7 \, a c^{2} d^{2} + 19 \, a c d^{3} + 10 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(21 \, a c^{2} d^{2} + 50 \, a c d^{3} + 25 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(7 \, a c^{2} d^{2} + 12 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(28 \, a c^{2} d^{2} + 48 \, a c d^{3} + 20 \, a d^{4} - {\left(7 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(7 \, a c^{2} d^{2} + 26 \, a c d^{3} + 15 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(7 \, a c^{2} d^{2} + 12 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a c + a d}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{d}{a c + a d}}}{2 \, d \cos\left(f x + e\right)}\right) - 4 \, {\left(4 \, c^{4} - 8 \, c^{3} d + 8 \, c d^{3} - 4 \, d^{4} - {\left(3 \, c^{3} d - 19 \, c^{2} d^{2} - 19 \, c d^{3} + 35 \, d^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, c^{4} - 15 \, c^{3} d - 7 \, c^{2} d^{2} - c d^{3} + 20 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(7 \, c^{4} - 20 \, c^{3} d - 26 \, c^{2} d^{2} - 12 \, c d^{3} + 51 \, d^{4}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{4} - 8 \, c^{3} d + 8 \, c d^{3} - 4 \, d^{4} - {\left(3 \, c^{3} d - 19 \, c^{2} d^{2} - 19 \, c d^{3} + 35 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c^{4} - 12 \, c^{3} d - 26 \, c^{2} d^{2} - 20 \, c d^{3} + 55 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{64 \, {\left({\left(a^{3} c^{5} d - 3 \, a^{3} c^{4} d^{2} + 2 \, a^{3} c^{3} d^{3} + 2 \, a^{3} c^{2} d^{4} - 3 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{6} - a^{3} c^{5} d - 4 \, a^{3} c^{4} d^{2} + 6 \, a^{3} c^{3} d^{3} + a^{3} c^{2} d^{4} - 5 \, a^{3} c d^{5} + 2 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{6} - 4 \, a^{3} c^{5} d - 9 \, a^{3} c^{4} d^{2} + 16 \, a^{3} c^{3} d^{3} + a^{3} c^{2} d^{4} - 12 \, a^{3} c d^{5} + 5 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f - {\left({\left(a^{3} c^{5} d - 3 \, a^{3} c^{4} d^{2} + 2 \, a^{3} c^{3} d^{3} + 2 \, a^{3} c^{2} d^{4} - 3 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{3} c^{6} - 7 \, a^{3} c^{4} d^{2} + 8 \, a^{3} c^{3} d^{3} + 3 \, a^{3} c^{2} d^{4} - 8 \, a^{3} c d^{5} + 3 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/64*(sqrt(2)*((3*c^3*d - 19*c^2*d^2 + 93*c*d^3 + 115*d^4)*cos(f*x + e)^4 + 12*c^4 - 64*c^3*d + 296*c^2*d^2 + 832*c*d^3 + 460*d^4 - (3*c^4 - 13*c^3*d + 55*c^2*d^2 + 301*c*d^3 + 230*d^4)*cos(f*x + e)^3 - (9*c^4 - 42*c^3*d + 184*c^2*d^2 + 810*c*d^3 + 575*d^4)*cos(f*x + e)^2 + 2*(3*c^4 - 16*c^3*d + 74*c^2*d^2 + 208*c*d^3 + 115*d^4)*cos(f*x + e) + (12*c^4 - 64*c^3*d + 296*c^2*d^2 + 832*c*d^3 + 460*d^4 - (3*c^3*d - 19*c^2*d^2 + 93*c*d^3 + 115*d^4)*cos(f*x + e)^3 - (3*c^4 - 10*c^3*d + 36*c^2*d^2 + 394*c*d^3 + 345*d^4)*cos(f*x + e)^2 + 2*(3*c^4 - 16*c^3*d + 74*c^2*d^2 + 208*c*d^3 + 115*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 16*(28*a*c^2*d^2 + 48*a*c*d^3 + 20*a*d^4 + (7*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^4 - (7*a*c^2*d^2 + 19*a*c*d^3 + 10*a*d^4)*cos(f*x + e)^3 - (21*a*c^2*d^2 + 50*a*c*d^3 + 25*a*d^4)*cos(f*x + e)^2 + 2*(7*a*c^2*d^2 + 12*a*c*d^3 + 5*a*d^4)*cos(f*x + e) + (28*a*c^2*d^2 + 48*a*c*d^3 + 20*a*d^4 - (7*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^3 - (7*a*c^2*d^2 + 26*a*c*d^3 + 15*a*d^4)*cos(f*x + e)^2 + 2*(7*a*c^2*d^2 + 12*a*c*d^3 + 5*a*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(d/(a*c + a*d))*log((d^2*cos(f*x + e)^3 - (6*c*d + 7*d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 4*((c*d + d^2)*cos(f*x + e)^2 - c^2 - 4*c*d - 3*d^2 - (c^2 + 3*c*d + 2*d^2)*cos(f*x + e) + (c^2 + 4*c*d + 3*d^2 + (c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d/(a*c + a*d)) - (c^2 + 8*c*d + 9*d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 2*(3*c*d + 4*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) - 4*(4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 19*c^2*d^2 - 19*c*d^3 + 35*d^4)*cos(f*x + e)^3 + (3*c^4 - 15*c^3*d - 7*c^2*d^2 - c*d^3 + 20*d^4)*cos(f*x + e)^2 + (7*c^4 - 20*c^3*d - 26*c^2*d^2 - 12*c*d^3 + 51*d^4)*cos(f*x + e) - (4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 19*c^2*d^2 - 19*c*d^3 + 35*d^4)*cos(f*x + e)^2 - (3*c^4 - 12*c^3*d - 26*c^2*d^2 - 20*c*d^3 + 55*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e)^4 - (a^3*c^6 - a^3*c^5*d - 4*a^3*c^4*d^2 + 6*a^3*c^3*d^3 + a^3*c^2*d^4 - 5*a^3*c*d^5 + 2*a^3*d^6)*f*cos(f*x + e)^3 - (3*a^3*c^6 - 4*a^3*c^5*d - 9*a^3*c^4*d^2 + 16*a^3*c^3*d^3 + a^3*c^2*d^4 - 12*a^3*c*d^5 + 5*a^3*d^6)*f*cos(f*x + e)^2 + 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e) + 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f - ((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e)^3 + (a^3*c^6 - 7*a^3*c^4*d^2 + 8*a^3*c^3*d^3 + 3*a^3*c^2*d^4 - 8*a^3*c*d^5 + 3*a^3*d^6)*f*cos(f*x + e)^2 - 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e) - 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f)*sin(f*x + e)), 1/64*(sqrt(2)*((3*c^3*d - 19*c^2*d^2 + 93*c*d^3 + 115*d^4)*cos(f*x + e)^4 + 12*c^4 - 64*c^3*d + 296*c^2*d^2 + 832*c*d^3 + 460*d^4 - (3*c^4 - 13*c^3*d + 55*c^2*d^2 + 301*c*d^3 + 230*d^4)*cos(f*x + e)^3 - (9*c^4 - 42*c^3*d + 184*c^2*d^2 + 810*c*d^3 + 575*d^4)*cos(f*x + e)^2 + 2*(3*c^4 - 16*c^3*d + 74*c^2*d^2 + 208*c*d^3 + 115*d^4)*cos(f*x + e) + (12*c^4 - 64*c^3*d + 296*c^2*d^2 + 832*c*d^3 + 460*d^4 - (3*c^3*d - 19*c^2*d^2 + 93*c*d^3 + 115*d^4)*cos(f*x + e)^3 - (3*c^4 - 10*c^3*d + 36*c^2*d^2 + 394*c*d^3 + 345*d^4)*cos(f*x + e)^2 + 2*(3*c^4 - 16*c^3*d + 74*c^2*d^2 + 208*c*d^3 + 115*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 32*(28*a*c^2*d^2 + 48*a*c*d^3 + 20*a*d^4 + (7*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^4 - (7*a*c^2*d^2 + 19*a*c*d^3 + 10*a*d^4)*cos(f*x + e)^3 - (21*a*c^2*d^2 + 50*a*c*d^3 + 25*a*d^4)*cos(f*x + e)^2 + 2*(7*a*c^2*d^2 + 12*a*c*d^3 + 5*a*d^4)*cos(f*x + e) + (28*a*c^2*d^2 + 48*a*c*d^3 + 20*a*d^4 - (7*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^3 - (7*a*c^2*d^2 + 26*a*c*d^3 + 15*a*d^4)*cos(f*x + e)^2 + 2*(7*a*c^2*d^2 + 12*a*c*d^3 + 5*a*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(-d/(a*c + a*d))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) - 4*(4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 19*c^2*d^2 - 19*c*d^3 + 35*d^4)*cos(f*x + e)^3 + (3*c^4 - 15*c^3*d - 7*c^2*d^2 - c*d^3 + 20*d^4)*cos(f*x + e)^2 + (7*c^4 - 20*c^3*d - 26*c^2*d^2 - 12*c*d^3 + 51*d^4)*cos(f*x + e) - (4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 19*c^2*d^2 - 19*c*d^3 + 35*d^4)*cos(f*x + e)^2 - (3*c^4 - 12*c^3*d - 26*c^2*d^2 - 20*c*d^3 + 55*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e)^4 - (a^3*c^6 - a^3*c^5*d - 4*a^3*c^4*d^2 + 6*a^3*c^3*d^3 + a^3*c^2*d^4 - 5*a^3*c*d^5 + 2*a^3*d^6)*f*cos(f*x + e)^3 - (3*a^3*c^6 - 4*a^3*c^5*d - 9*a^3*c^4*d^2 + 16*a^3*c^3*d^3 + a^3*c^2*d^4 - 12*a^3*c*d^5 + 5*a^3*d^6)*f*cos(f*x + e)^2 + 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e) + 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f - ((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e)^3 + (a^3*c^6 - 7*a^3*c^4*d^2 + 8*a^3*c^3*d^3 + 3*a^3*c^2*d^4 - 8*a^3*c*d^5 + 3*a^3*d^6)*f*cos(f*x + e)^2 - 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e) - 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f)*sin(f*x + e))]","B",0
563,1,5999,0,8.583992," ","integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[-\frac{3 \, \sqrt{2} {\left(4 \, c^{6} - 24 \, c^{5} d + 156 \, c^{4} d^{2} + 944 \, c^{3} d^{3} + 1596 \, c^{2} d^{4} + 1128 \, c d^{5} + 292 \, d^{6} + {\left(c^{4} d^{2} - 8 \, c^{3} d^{3} + 54 \, c^{2} d^{4} + 136 \, c d^{5} + 73 \, d^{6}\right)} \cos\left(f x + e\right)^{5} + {\left(2 \, c^{5} d - 13 \, c^{4} d^{2} + 84 \, c^{3} d^{3} + 434 \, c^{2} d^{4} + 554 \, c d^{5} + 219 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(c^{6} - 4 \, c^{5} d + 25 \, c^{4} d^{2} + 328 \, c^{3} d^{3} + 779 \, c^{2} d^{4} + 700 \, c d^{5} + 219 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, c^{6} - 14 \, c^{5} d + 89 \, c^{4} d^{2} + 892 \, c^{3} d^{3} + 1957 \, c^{2} d^{4} + 1682 \, c d^{5} + 511 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(c^{6} - 6 \, c^{5} d + 39 \, c^{4} d^{2} + 236 \, c^{3} d^{3} + 399 \, c^{2} d^{4} + 282 \, c d^{5} + 73 \, d^{6}\right)} \cos\left(f x + e\right) + {\left(4 \, c^{6} - 24 \, c^{5} d + 156 \, c^{4} d^{2} + 944 \, c^{3} d^{3} + 1596 \, c^{2} d^{4} + 1128 \, c d^{5} + 292 \, d^{6} + {\left(c^{4} d^{2} - 8 \, c^{3} d^{3} + 54 \, c^{2} d^{4} + 136 \, c d^{5} + 73 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(c^{5} d - 7 \, c^{4} d^{2} + 46 \, c^{3} d^{3} + 190 \, c^{2} d^{4} + 209 \, c d^{5} + 73 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(c^{6} - 2 \, c^{5} d + 11 \, c^{4} d^{2} + 420 \, c^{3} d^{3} + 1159 \, c^{2} d^{4} + 1118 \, c d^{5} + 365 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(c^{6} - 6 \, c^{5} d + 39 \, c^{4} d^{2} + 236 \, c^{3} d^{3} + 399 \, c^{2} d^{4} + 282 \, c d^{5} + 73 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 12 \, {\left(84 \, a c^{4} d^{2} + 288 \, a c^{3} d^{3} + 376 \, a c^{2} d^{4} + 224 \, a c d^{5} + 52 \, a d^{6} + {\left(21 \, a c^{2} d^{4} + 30 \, a c d^{5} + 13 \, a d^{6}\right)} \cos\left(f x + e\right)^{5} + {\left(42 \, a c^{3} d^{3} + 123 \, a c^{2} d^{4} + 116 \, a c d^{5} + 39 \, a d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(21 \, a c^{4} d^{2} + 114 \, a c^{3} d^{3} + 196 \, a c^{2} d^{4} + 142 \, a c d^{5} + 39 \, a d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(63 \, a c^{4} d^{2} + 300 \, a c^{3} d^{3} + 486 \, a c^{2} d^{4} + 340 \, a c d^{5} + 91 \, a d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(21 \, a c^{4} d^{2} + 72 \, a c^{3} d^{3} + 94 \, a c^{2} d^{4} + 56 \, a c d^{5} + 13 \, a d^{6}\right)} \cos\left(f x + e\right) + {\left(84 \, a c^{4} d^{2} + 288 \, a c^{3} d^{3} + 376 \, a c^{2} d^{4} + 224 \, a c d^{5} + 52 \, a d^{6} + {\left(21 \, a c^{2} d^{4} + 30 \, a c d^{5} + 13 \, a d^{6}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(21 \, a c^{3} d^{3} + 51 \, a c^{2} d^{4} + 43 \, a c d^{5} + 13 \, a d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(21 \, a c^{4} d^{2} + 156 \, a c^{3} d^{3} + 298 \, a c^{2} d^{4} + 228 \, a c d^{5} + 65 \, a d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(21 \, a c^{4} d^{2} + 72 \, a c^{3} d^{3} + 94 \, a c^{2} d^{4} + 56 \, a c d^{5} + 13 \, a d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a c + a d}} \log\left(\frac{d^{2} \cos\left(f x + e\right)^{3} - {\left(6 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - 4 \, {\left({\left(c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 4 \, c d - 3 \, d^{2} - {\left(c^{2} + 3 \, c d + 2 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} + 4 \, c d + 3 \, d^{2} + {\left(c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{\frac{d}{a c + a d}} - {\left(c^{2} + 8 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} + 2 \, {\left(3 \, c d + 4 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{d^{2} \cos\left(f x + e\right)^{3} + {\left(2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - c^{2} - 2 \, c d - d^{2} - {\left(c^{2} + d^{2}\right)} \cos\left(f x + e\right) + {\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \cos\left(f x + e\right) - c^{2} - 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(4 \, c^{6} - 8 \, c^{5} d - 4 \, c^{4} d^{2} + 16 \, c^{3} d^{3} - 4 \, c^{2} d^{4} - 8 \, c d^{5} + 4 \, d^{6} - 3 \, {\left(c^{4} d^{2} - 8 \, c^{3} d^{3} - 30 \, c^{2} d^{4} + 16 \, c d^{5} + 21 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(6 \, c^{5} d - 41 \, c^{4} d^{2} - 152 \, c^{3} d^{3} - 78 \, c^{2} d^{4} + 170 \, c d^{5} + 95 \, d^{6}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, c^{6} - 16 \, c^{5} d - 31 \, c^{4} d^{2} - 84 \, c^{3} d^{3} - 23 \, c^{2} d^{4} + 100 \, c d^{5} + 51 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + {\left(7 \, c^{6} - 18 \, c^{5} d - 79 \, c^{4} d^{2} - 196 \, c^{3} d^{3} - 15 \, c^{2} d^{4} + 214 \, c d^{5} + 87 \, d^{6}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{6} - 8 \, c^{5} d - 4 \, c^{4} d^{2} + 16 \, c^{3} d^{3} - 4 \, c^{2} d^{4} - 8 \, c d^{5} + 4 \, d^{6} + 3 \, {\left(c^{4} d^{2} - 8 \, c^{3} d^{3} - 30 \, c^{2} d^{4} + 16 \, c d^{5} + 21 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(3 \, c^{5} d - 22 \, c^{4} d^{2} - 64 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 61 \, c d^{5} + 16 \, d^{6}\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c^{6} - 10 \, c^{5} d - 75 \, c^{4} d^{2} - 212 \, c^{3} d^{3} - 11 \, c^{2} d^{4} + 222 \, c d^{5} + 83 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{64 \, {\left({\left(a^{3} c^{7} d^{2} - 3 \, a^{3} c^{6} d^{3} + a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 5 \, a^{3} c^{3} d^{6} - a^{3} c^{2} d^{7} + 3 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{5} + {\left(2 \, a^{3} c^{8} d - 3 \, a^{3} c^{7} d^{2} - 7 \, a^{3} c^{6} d^{3} + 13 \, a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 17 \, a^{3} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{7} + 7 \, a^{3} c d^{8} - 3 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{9} + a^{3} c^{8} d - 8 \, a^{3} c^{7} d^{2} + 18 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 16 \, a^{3} c^{3} d^{6} + 8 \, a^{3} c^{2} d^{7} + 5 \, a^{3} c d^{8} - 3 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{9} + a^{3} c^{8} d - 20 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 42 \, a^{3} c^{5} d^{4} - 18 \, a^{3} c^{4} d^{5} - 36 \, a^{3} c^{3} d^{6} + 20 \, a^{3} c^{2} d^{7} + 11 \, a^{3} c d^{8} - 7 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f + {\left({\left(a^{3} c^{7} d^{2} - 3 \, a^{3} c^{6} d^{3} + a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 5 \, a^{3} c^{3} d^{6} - a^{3} c^{2} d^{7} + 3 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} c^{8} d - 2 \, a^{3} c^{7} d^{2} - 2 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{3} d^{6} + 2 \, a^{3} c^{2} d^{7} + 2 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{3} c^{9} + 3 \, a^{3} c^{8} d - 12 \, a^{3} c^{7} d^{2} - 4 \, a^{3} c^{6} d^{3} + 30 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 28 \, a^{3} c^{3} d^{6} + 12 \, a^{3} c^{2} d^{7} + 9 \, a^{3} c d^{8} - 5 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{3 \, \sqrt{2} {\left(4 \, c^{6} - 24 \, c^{5} d + 156 \, c^{4} d^{2} + 944 \, c^{3} d^{3} + 1596 \, c^{2} d^{4} + 1128 \, c d^{5} + 292 \, d^{6} + {\left(c^{4} d^{2} - 8 \, c^{3} d^{3} + 54 \, c^{2} d^{4} + 136 \, c d^{5} + 73 \, d^{6}\right)} \cos\left(f x + e\right)^{5} + {\left(2 \, c^{5} d - 13 \, c^{4} d^{2} + 84 \, c^{3} d^{3} + 434 \, c^{2} d^{4} + 554 \, c d^{5} + 219 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(c^{6} - 4 \, c^{5} d + 25 \, c^{4} d^{2} + 328 \, c^{3} d^{3} + 779 \, c^{2} d^{4} + 700 \, c d^{5} + 219 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, c^{6} - 14 \, c^{5} d + 89 \, c^{4} d^{2} + 892 \, c^{3} d^{3} + 1957 \, c^{2} d^{4} + 1682 \, c d^{5} + 511 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(c^{6} - 6 \, c^{5} d + 39 \, c^{4} d^{2} + 236 \, c^{3} d^{3} + 399 \, c^{2} d^{4} + 282 \, c d^{5} + 73 \, d^{6}\right)} \cos\left(f x + e\right) + {\left(4 \, c^{6} - 24 \, c^{5} d + 156 \, c^{4} d^{2} + 944 \, c^{3} d^{3} + 1596 \, c^{2} d^{4} + 1128 \, c d^{5} + 292 \, d^{6} + {\left(c^{4} d^{2} - 8 \, c^{3} d^{3} + 54 \, c^{2} d^{4} + 136 \, c d^{5} + 73 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(c^{5} d - 7 \, c^{4} d^{2} + 46 \, c^{3} d^{3} + 190 \, c^{2} d^{4} + 209 \, c d^{5} + 73 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(c^{6} - 2 \, c^{5} d + 11 \, c^{4} d^{2} + 420 \, c^{3} d^{3} + 1159 \, c^{2} d^{4} + 1118 \, c d^{5} + 365 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(c^{6} - 6 \, c^{5} d + 39 \, c^{4} d^{2} + 236 \, c^{3} d^{3} + 399 \, c^{2} d^{4} + 282 \, c d^{5} + 73 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + 3 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) + 2 \, a}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - 24 \, {\left(84 \, a c^{4} d^{2} + 288 \, a c^{3} d^{3} + 376 \, a c^{2} d^{4} + 224 \, a c d^{5} + 52 \, a d^{6} + {\left(21 \, a c^{2} d^{4} + 30 \, a c d^{5} + 13 \, a d^{6}\right)} \cos\left(f x + e\right)^{5} + {\left(42 \, a c^{3} d^{3} + 123 \, a c^{2} d^{4} + 116 \, a c d^{5} + 39 \, a d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(21 \, a c^{4} d^{2} + 114 \, a c^{3} d^{3} + 196 \, a c^{2} d^{4} + 142 \, a c d^{5} + 39 \, a d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(63 \, a c^{4} d^{2} + 300 \, a c^{3} d^{3} + 486 \, a c^{2} d^{4} + 340 \, a c d^{5} + 91 \, a d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(21 \, a c^{4} d^{2} + 72 \, a c^{3} d^{3} + 94 \, a c^{2} d^{4} + 56 \, a c d^{5} + 13 \, a d^{6}\right)} \cos\left(f x + e\right) + {\left(84 \, a c^{4} d^{2} + 288 \, a c^{3} d^{3} + 376 \, a c^{2} d^{4} + 224 \, a c d^{5} + 52 \, a d^{6} + {\left(21 \, a c^{2} d^{4} + 30 \, a c d^{5} + 13 \, a d^{6}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(21 \, a c^{3} d^{3} + 51 \, a c^{2} d^{4} + 43 \, a c d^{5} + 13 \, a d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(21 \, a c^{4} d^{2} + 156 \, a c^{3} d^{3} + 298 \, a c^{2} d^{4} + 228 \, a c d^{5} + 65 \, a d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(21 \, a c^{4} d^{2} + 72 \, a c^{3} d^{3} + 94 \, a c^{2} d^{4} + 56 \, a c d^{5} + 13 \, a d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a c + a d}} \arctan\left(\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) - c - 2 \, d\right)} \sqrt{-\frac{d}{a c + a d}}}{2 \, d \cos\left(f x + e\right)}\right) + 4 \, {\left(4 \, c^{6} - 8 \, c^{5} d - 4 \, c^{4} d^{2} + 16 \, c^{3} d^{3} - 4 \, c^{2} d^{4} - 8 \, c d^{5} + 4 \, d^{6} - 3 \, {\left(c^{4} d^{2} - 8 \, c^{3} d^{3} - 30 \, c^{2} d^{4} + 16 \, c d^{5} + 21 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(6 \, c^{5} d - 41 \, c^{4} d^{2} - 152 \, c^{3} d^{3} - 78 \, c^{2} d^{4} + 170 \, c d^{5} + 95 \, d^{6}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, c^{6} - 16 \, c^{5} d - 31 \, c^{4} d^{2} - 84 \, c^{3} d^{3} - 23 \, c^{2} d^{4} + 100 \, c d^{5} + 51 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + {\left(7 \, c^{6} - 18 \, c^{5} d - 79 \, c^{4} d^{2} - 196 \, c^{3} d^{3} - 15 \, c^{2} d^{4} + 214 \, c d^{5} + 87 \, d^{6}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{6} - 8 \, c^{5} d - 4 \, c^{4} d^{2} + 16 \, c^{3} d^{3} - 4 \, c^{2} d^{4} - 8 \, c d^{5} + 4 \, d^{6} + 3 \, {\left(c^{4} d^{2} - 8 \, c^{3} d^{3} - 30 \, c^{2} d^{4} + 16 \, c d^{5} + 21 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(3 \, c^{5} d - 22 \, c^{4} d^{2} - 64 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 61 \, c d^{5} + 16 \, d^{6}\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c^{6} - 10 \, c^{5} d - 75 \, c^{4} d^{2} - 212 \, c^{3} d^{3} - 11 \, c^{2} d^{4} + 222 \, c d^{5} + 83 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{64 \, {\left({\left(a^{3} c^{7} d^{2} - 3 \, a^{3} c^{6} d^{3} + a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 5 \, a^{3} c^{3} d^{6} - a^{3} c^{2} d^{7} + 3 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{5} + {\left(2 \, a^{3} c^{8} d - 3 \, a^{3} c^{7} d^{2} - 7 \, a^{3} c^{6} d^{3} + 13 \, a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 17 \, a^{3} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{7} + 7 \, a^{3} c d^{8} - 3 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{9} + a^{3} c^{8} d - 8 \, a^{3} c^{7} d^{2} + 18 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 16 \, a^{3} c^{3} d^{6} + 8 \, a^{3} c^{2} d^{7} + 5 \, a^{3} c d^{8} - 3 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{9} + a^{3} c^{8} d - 20 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 42 \, a^{3} c^{5} d^{4} - 18 \, a^{3} c^{4} d^{5} - 36 \, a^{3} c^{3} d^{6} + 20 \, a^{3} c^{2} d^{7} + 11 \, a^{3} c d^{8} - 7 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f + {\left({\left(a^{3} c^{7} d^{2} - 3 \, a^{3} c^{6} d^{3} + a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 5 \, a^{3} c^{3} d^{6} - a^{3} c^{2} d^{7} + 3 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} c^{8} d - 2 \, a^{3} c^{7} d^{2} - 2 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{3} d^{6} + 2 \, a^{3} c^{2} d^{7} + 2 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{3} c^{9} + 3 \, a^{3} c^{8} d - 12 \, a^{3} c^{7} d^{2} - 4 \, a^{3} c^{6} d^{3} + 30 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 28 \, a^{3} c^{3} d^{6} + 12 \, a^{3} c^{2} d^{7} + 9 \, a^{3} c d^{8} - 5 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/64*(3*sqrt(2)*(4*c^6 - 24*c^5*d + 156*c^4*d^2 + 944*c^3*d^3 + 1596*c^2*d^4 + 1128*c*d^5 + 292*d^6 + (c^4*d^2 - 8*c^3*d^3 + 54*c^2*d^4 + 136*c*d^5 + 73*d^6)*cos(f*x + e)^5 + (2*c^5*d - 13*c^4*d^2 + 84*c^3*d^3 + 434*c^2*d^4 + 554*c*d^5 + 219*d^6)*cos(f*x + e)^4 - (c^6 - 4*c^5*d + 25*c^4*d^2 + 328*c^3*d^3 + 779*c^2*d^4 + 700*c*d^5 + 219*d^6)*cos(f*x + e)^3 - (3*c^6 - 14*c^5*d + 89*c^4*d^2 + 892*c^3*d^3 + 1957*c^2*d^4 + 1682*c*d^5 + 511*d^6)*cos(f*x + e)^2 + 2*(c^6 - 6*c^5*d + 39*c^4*d^2 + 236*c^3*d^3 + 399*c^2*d^4 + 282*c*d^5 + 73*d^6)*cos(f*x + e) + (4*c^6 - 24*c^5*d + 156*c^4*d^2 + 944*c^3*d^3 + 1596*c^2*d^4 + 1128*c*d^5 + 292*d^6 + (c^4*d^2 - 8*c^3*d^3 + 54*c^2*d^4 + 136*c*d^5 + 73*d^6)*cos(f*x + e)^4 - 2*(c^5*d - 7*c^4*d^2 + 46*c^3*d^3 + 190*c^2*d^4 + 209*c*d^5 + 73*d^6)*cos(f*x + e)^3 - (c^6 - 2*c^5*d + 11*c^4*d^2 + 420*c^3*d^3 + 1159*c^2*d^4 + 1118*c*d^5 + 365*d^6)*cos(f*x + e)^2 + 2*(c^6 - 6*c^5*d + 39*c^4*d^2 + 236*c^3*d^3 + 399*c^2*d^4 + 282*c*d^5 + 73*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 12*(84*a*c^4*d^2 + 288*a*c^3*d^3 + 376*a*c^2*d^4 + 224*a*c*d^5 + 52*a*d^6 + (21*a*c^2*d^4 + 30*a*c*d^5 + 13*a*d^6)*cos(f*x + e)^5 + (42*a*c^3*d^3 + 123*a*c^2*d^4 + 116*a*c*d^5 + 39*a*d^6)*cos(f*x + e)^4 - (21*a*c^4*d^2 + 114*a*c^3*d^3 + 196*a*c^2*d^4 + 142*a*c*d^5 + 39*a*d^6)*cos(f*x + e)^3 - (63*a*c^4*d^2 + 300*a*c^3*d^3 + 486*a*c^2*d^4 + 340*a*c*d^5 + 91*a*d^6)*cos(f*x + e)^2 + 2*(21*a*c^4*d^2 + 72*a*c^3*d^3 + 94*a*c^2*d^4 + 56*a*c*d^5 + 13*a*d^6)*cos(f*x + e) + (84*a*c^4*d^2 + 288*a*c^3*d^3 + 376*a*c^2*d^4 + 224*a*c*d^5 + 52*a*d^6 + (21*a*c^2*d^4 + 30*a*c*d^5 + 13*a*d^6)*cos(f*x + e)^4 - 2*(21*a*c^3*d^3 + 51*a*c^2*d^4 + 43*a*c*d^5 + 13*a*d^6)*cos(f*x + e)^3 - (21*a*c^4*d^2 + 156*a*c^3*d^3 + 298*a*c^2*d^4 + 228*a*c*d^5 + 65*a*d^6)*cos(f*x + e)^2 + 2*(21*a*c^4*d^2 + 72*a*c^3*d^3 + 94*a*c^2*d^4 + 56*a*c*d^5 + 13*a*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(d/(a*c + a*d))*log((d^2*cos(f*x + e)^3 - (6*c*d + 7*d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - 4*((c*d + d^2)*cos(f*x + e)^2 - c^2 - 4*c*d - 3*d^2 - (c^2 + 3*c*d + 2*d^2)*cos(f*x + e) + (c^2 + 4*c*d + 3*d^2 + (c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d/(a*c + a*d)) - (c^2 + 8*c*d + 9*d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 + 2*(3*c*d + 4*d^2)*cos(f*x + e))*sin(f*x + e))/(d^2*cos(f*x + e)^3 + (2*c*d + d^2)*cos(f*x + e)^2 - c^2 - 2*c*d - d^2 - (c^2 + d^2)*cos(f*x + e) + (d^2*cos(f*x + e)^2 - 2*c*d*cos(f*x + e) - c^2 - 2*c*d - d^2)*sin(f*x + e))) + 4*(4*c^6 - 8*c^5*d - 4*c^4*d^2 + 16*c^3*d^3 - 4*c^2*d^4 - 8*c*d^5 + 4*d^6 - 3*(c^4*d^2 - 8*c^3*d^3 - 30*c^2*d^4 + 16*c*d^5 + 21*d^6)*cos(f*x + e)^4 - (6*c^5*d - 41*c^4*d^2 - 152*c^3*d^3 - 78*c^2*d^4 + 170*c*d^5 + 95*d^6)*cos(f*x + e)^3 + (3*c^6 - 16*c^5*d - 31*c^4*d^2 - 84*c^3*d^3 - 23*c^2*d^4 + 100*c*d^5 + 51*d^6)*cos(f*x + e)^2 + (7*c^6 - 18*c^5*d - 79*c^4*d^2 - 196*c^3*d^3 - 15*c^2*d^4 + 214*c*d^5 + 87*d^6)*cos(f*x + e) - (4*c^6 - 8*c^5*d - 4*c^4*d^2 + 16*c^3*d^3 - 4*c^2*d^4 - 8*c*d^5 + 4*d^6 + 3*(c^4*d^2 - 8*c^3*d^3 - 30*c^2*d^4 + 16*c*d^5 + 21*d^6)*cos(f*x + e)^3 - 2*(3*c^5*d - 22*c^4*d^2 - 64*c^3*d^3 + 6*c^2*d^4 + 61*c*d^5 + 16*d^6)*cos(f*x + e)^2 - (3*c^6 - 10*c^5*d - 75*c^4*d^2 - 212*c^3*d^3 - 11*c^2*d^4 + 222*c*d^5 + 83*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^3*c^7*d^2 - 3*a^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6 - a^3*c^2*d^7 + 3*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^5 + (2*a^3*c^8*d - 3*a^3*c^7*d^2 - 7*a^3*c^6*d^3 + 13*a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 17*a^3*c^3*d^6 + 3*a^3*c^2*d^7 + 7*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^4 - (a^3*c^9 + a^3*c^8*d - 8*a^3*c^7*d^2 + 18*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 16*a^3*c^3*d^6 + 8*a^3*c^2*d^7 + 5*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^3 - (3*a^3*c^9 + a^3*c^8*d - 20*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 42*a^3*c^5*d^4 - 18*a^3*c^4*d^5 - 36*a^3*c^3*d^6 + 20*a^3*c^2*d^7 + 11*a^3*c*d^8 - 7*a^3*d^9)*f*cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f + ((a^3*c^7*d^2 - 3*a^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6 - a^3*c^2*d^7 + 3*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^4 - 2*(a^3*c^8*d - 2*a^3*c^7*d^2 - 2*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^3*d^6 + 2*a^3*c^2*d^7 + 2*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^3 - (a^3*c^9 + 3*a^3*c^8*d - 12*a^3*c^7*d^2 - 4*a^3*c^6*d^3 + 30*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 28*a^3*c^3*d^6 + 12*a^3*c^2*d^7 + 9*a^3*c*d^8 - 5*a^3*d^9)*f*cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f)*sin(f*x + e)), -1/64*(3*sqrt(2)*(4*c^6 - 24*c^5*d + 156*c^4*d^2 + 944*c^3*d^3 + 1596*c^2*d^4 + 1128*c*d^5 + 292*d^6 + (c^4*d^2 - 8*c^3*d^3 + 54*c^2*d^4 + 136*c*d^5 + 73*d^6)*cos(f*x + e)^5 + (2*c^5*d - 13*c^4*d^2 + 84*c^3*d^3 + 434*c^2*d^4 + 554*c*d^5 + 219*d^6)*cos(f*x + e)^4 - (c^6 - 4*c^5*d + 25*c^4*d^2 + 328*c^3*d^3 + 779*c^2*d^4 + 700*c*d^5 + 219*d^6)*cos(f*x + e)^3 - (3*c^6 - 14*c^5*d + 89*c^4*d^2 + 892*c^3*d^3 + 1957*c^2*d^4 + 1682*c*d^5 + 511*d^6)*cos(f*x + e)^2 + 2*(c^6 - 6*c^5*d + 39*c^4*d^2 + 236*c^3*d^3 + 399*c^2*d^4 + 282*c*d^5 + 73*d^6)*cos(f*x + e) + (4*c^6 - 24*c^5*d + 156*c^4*d^2 + 944*c^3*d^3 + 1596*c^2*d^4 + 1128*c*d^5 + 292*d^6 + (c^4*d^2 - 8*c^3*d^3 + 54*c^2*d^4 + 136*c*d^5 + 73*d^6)*cos(f*x + e)^4 - 2*(c^5*d - 7*c^4*d^2 + 46*c^3*d^3 + 190*c^2*d^4 + 209*c*d^5 + 73*d^6)*cos(f*x + e)^3 - (c^6 - 2*c^5*d + 11*c^4*d^2 + 420*c^3*d^3 + 1159*c^2*d^4 + 1118*c*d^5 + 365*d^6)*cos(f*x + e)^2 + 2*(c^6 - 6*c^5*d + 39*c^4*d^2 + 236*c^3*d^3 + 399*c^2*d^4 + 282*c*d^5 + 73*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(a)*(cos(f*x + e) - sin(f*x + e) + 1) + 3*a*cos(f*x + e) - (a*cos(f*x + e) - 2*a)*sin(f*x + e) + 2*a)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - 24*(84*a*c^4*d^2 + 288*a*c^3*d^3 + 376*a*c^2*d^4 + 224*a*c*d^5 + 52*a*d^6 + (21*a*c^2*d^4 + 30*a*c*d^5 + 13*a*d^6)*cos(f*x + e)^5 + (42*a*c^3*d^3 + 123*a*c^2*d^4 + 116*a*c*d^5 + 39*a*d^6)*cos(f*x + e)^4 - (21*a*c^4*d^2 + 114*a*c^3*d^3 + 196*a*c^2*d^4 + 142*a*c*d^5 + 39*a*d^6)*cos(f*x + e)^3 - (63*a*c^4*d^2 + 300*a*c^3*d^3 + 486*a*c^2*d^4 + 340*a*c*d^5 + 91*a*d^6)*cos(f*x + e)^2 + 2*(21*a*c^4*d^2 + 72*a*c^3*d^3 + 94*a*c^2*d^4 + 56*a*c*d^5 + 13*a*d^6)*cos(f*x + e) + (84*a*c^4*d^2 + 288*a*c^3*d^3 + 376*a*c^2*d^4 + 224*a*c*d^5 + 52*a*d^6 + (21*a*c^2*d^4 + 30*a*c*d^5 + 13*a*d^6)*cos(f*x + e)^4 - 2*(21*a*c^3*d^3 + 51*a*c^2*d^4 + 43*a*c*d^5 + 13*a*d^6)*cos(f*x + e)^3 - (21*a*c^4*d^2 + 156*a*c^3*d^3 + 298*a*c^2*d^4 + 228*a*c*d^5 + 65*a*d^6)*cos(f*x + e)^2 + 2*(21*a*c^4*d^2 + 72*a*c^3*d^3 + 94*a*c^2*d^4 + 56*a*c*d^5 + 13*a*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(-d/(a*c + a*d))*arctan(1/2*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) - c - 2*d)*sqrt(-d/(a*c + a*d))/(d*cos(f*x + e))) + 4*(4*c^6 - 8*c^5*d - 4*c^4*d^2 + 16*c^3*d^3 - 4*c^2*d^4 - 8*c*d^5 + 4*d^6 - 3*(c^4*d^2 - 8*c^3*d^3 - 30*c^2*d^4 + 16*c*d^5 + 21*d^6)*cos(f*x + e)^4 - (6*c^5*d - 41*c^4*d^2 - 152*c^3*d^3 - 78*c^2*d^4 + 170*c*d^5 + 95*d^6)*cos(f*x + e)^3 + (3*c^6 - 16*c^5*d - 31*c^4*d^2 - 84*c^3*d^3 - 23*c^2*d^4 + 100*c*d^5 + 51*d^6)*cos(f*x + e)^2 + (7*c^6 - 18*c^5*d - 79*c^4*d^2 - 196*c^3*d^3 - 15*c^2*d^4 + 214*c*d^5 + 87*d^6)*cos(f*x + e) - (4*c^6 - 8*c^5*d - 4*c^4*d^2 + 16*c^3*d^3 - 4*c^2*d^4 - 8*c*d^5 + 4*d^6 + 3*(c^4*d^2 - 8*c^3*d^3 - 30*c^2*d^4 + 16*c*d^5 + 21*d^6)*cos(f*x + e)^3 - 2*(3*c^5*d - 22*c^4*d^2 - 64*c^3*d^3 + 6*c^2*d^4 + 61*c*d^5 + 16*d^6)*cos(f*x + e)^2 - (3*c^6 - 10*c^5*d - 75*c^4*d^2 - 212*c^3*d^3 - 11*c^2*d^4 + 222*c*d^5 + 83*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/((a^3*c^7*d^2 - 3*a^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6 - a^3*c^2*d^7 + 3*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^5 + (2*a^3*c^8*d - 3*a^3*c^7*d^2 - 7*a^3*c^6*d^3 + 13*a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 17*a^3*c^3*d^6 + 3*a^3*c^2*d^7 + 7*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^4 - (a^3*c^9 + a^3*c^8*d - 8*a^3*c^7*d^2 + 18*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 16*a^3*c^3*d^6 + 8*a^3*c^2*d^7 + 5*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^3 - (3*a^3*c^9 + a^3*c^8*d - 20*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 42*a^3*c^5*d^4 - 18*a^3*c^4*d^5 - 36*a^3*c^3*d^6 + 20*a^3*c^2*d^7 + 11*a^3*c*d^8 - 7*a^3*d^9)*f*cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f + ((a^3*c^7*d^2 - 3*a^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6 - a^3*c^2*d^7 + 3*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^4 - 2*(a^3*c^8*d - 2*a^3*c^7*d^2 - 2*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^3*d^6 + 2*a^3*c^2*d^7 + 2*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^3 - (a^3*c^9 + 3*a^3*c^8*d - 12*a^3*c^7*d^2 - 4*a^3*c^6*d^3 + 30*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 28*a^3*c^3*d^6 + 12*a^3*c^2*d^7 + 9*a^3*c*d^8 - 5*a^3*d^9)*f*cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f)*sin(f*x + e))]","B",0
564,1,1257,0,2.183483," ","integrate((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3} + {\left(c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right)} \cos\left(f x + e\right) + {\left(c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 8 \, {\left(8 \, d^{2} \cos\left(f x + e\right)^{3} - 2 \, {\left(13 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - 33 \, c^{2} - 14 \, c d - 13 \, d^{2} - {\left(33 \, c^{2} + 40 \, c d + 23 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - 33 \, c^{2} - 14 \, c d - 13 \, d^{2} + 2 \, {\left(13 \, c d + 5 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{192 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}, \frac{15 \, {\left(c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3} + {\left(c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right)} \cos\left(f x + e\right) + {\left(c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(8 \, d^{2} \cos\left(f x + e\right)^{3} - 2 \, {\left(13 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - 33 \, c^{2} - 14 \, c d - 13 \, d^{2} - {\left(33 \, c^{2} + 40 \, c d + 23 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - 33 \, c^{2} - 14 \, c d - 13 \, d^{2} + 2 \, {\left(13 \, c d + 5 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{96 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}\right]"," ",0,"[1/192*(15*(c^3 + 3*c^2*d + 3*c*d^2 + d^3 + (c^3 + 3*c^2*d + 3*c*d^2 + d^3)*cos(f*x + e) + (c^3 + 3*c^2*d + 3*c*d^2 + d^3)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(8*d^2*cos(f*x + e)^3 - 2*(13*c*d + d^2)*cos(f*x + e)^2 - 33*c^2 - 14*c*d - 13*d^2 - (33*c^2 + 40*c*d + 23*d^2)*cos(f*x + e) - (8*d^2*cos(f*x + e)^2 - 33*c^2 - 14*c*d - 13*d^2 + 2*(13*c*d + 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(f*cos(f*x + e) + f*sin(f*x + e) + f), 1/96*(15*(c^3 + 3*c^2*d + 3*c*d^2 + d^3 + (c^3 + 3*c^2*d + 3*c*d^2 + d^3)*cos(f*x + e) + (c^3 + 3*c^2*d + 3*c*d^2 + d^3)*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) + 4*(8*d^2*cos(f*x + e)^3 - 2*(13*c*d + d^2)*cos(f*x + e)^2 - 33*c^2 - 14*c*d - 13*d^2 - (33*c^2 + 40*c*d + 23*d^2)*cos(f*x + e) - (8*d^2*cos(f*x + e)^2 - 33*c^2 - 14*c*d - 13*d^2 + 2*(13*c*d + 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(f*cos(f*x + e) + f*sin(f*x + e) + f)]","B",0
565,1,1069,0,2.190336," ","integrate((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(c^{2} + 2 \, c d + d^{2} + {\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} + 2 \, c d + d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) - 8 \, {\left(2 \, d \cos\left(f x + e\right)^{2} + {\left(5 \, c + 3 \, d\right)} \cos\left(f x + e\right) + {\left(2 \, d \cos\left(f x + e\right) - 5 \, c - d\right)} \sin\left(f x + e\right) + 5 \, c + d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{32 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}, \frac{3 \, {\left(c^{2} + 2 \, c d + d^{2} + {\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right) + {\left(c^{2} + 2 \, c d + d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left(2 \, d \cos\left(f x + e\right)^{2} + {\left(5 \, c + 3 \, d\right)} \cos\left(f x + e\right) + {\left(2 \, d \cos\left(f x + e\right) - 5 \, c - d\right)} \sin\left(f x + e\right) + 5 \, c + d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{16 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}\right]"," ",0,"[1/32*(3*(c^2 + 2*c*d + d^2 + (c^2 + 2*c*d + d^2)*cos(f*x + e) + (c^2 + 2*c*d + d^2)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) - 8*(2*d*cos(f*x + e)^2 + (5*c + 3*d)*cos(f*x + e) + (2*d*cos(f*x + e) - 5*c - d)*sin(f*x + e) + 5*c + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(f*cos(f*x + e) + f*sin(f*x + e) + f), 1/16*(3*(c^2 + 2*c*d + d^2 + (c^2 + 2*c*d + d^2)*cos(f*x + e) + (c^2 + 2*c*d + d^2)*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) - 4*(2*d*cos(f*x + e)^2 + (5*c + 3*d)*cos(f*x + e) + (2*d*cos(f*x + e) - 5*c - d)*sin(f*x + e) + 5*c + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(f*cos(f*x + e) + f*sin(f*x + e) + f)]","B",0
566,1,945,0,2.029693," ","integrate((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(c + d\right)} \cos\left(f x + e\right) + {\left(c + d\right)} \sin\left(f x + e\right) + c + d\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) - 8 \, \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{8 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}, \frac{{\left({\left(c + d\right)} \cos\left(f x + e\right) + {\left(c + d\right)} \sin\left(f x + e\right) + c + d\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{4 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}\right]"," ",0,"[1/8*(((c + d)*cos(f*x + e) + (c + d)*sin(f*x + e) + c + d)*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) - 8*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*(cos(f*x + e) - sin(f*x + e) + 1))/(f*cos(f*x + e) + f*sin(f*x + e) + f), 1/4*(((c + d)*cos(f*x + e) + (c + d)*sin(f*x + e) + c + d)*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) - 4*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*(cos(f*x + e) - sin(f*x + e) + 1))/(f*cos(f*x + e) + f*sin(f*x + e) + f)]","B",0
567,1,777,0,1.621672," ","integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right)}{4 \, f}, \frac{\sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right)}{2 \, f}\right]"," ",0,"[1/4*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1))/f, 1/2*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e)))/f]","B",0
568,1,129,0,0.911950," ","integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{2 \, \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{{\left(c d + d^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{2} + c d\right)} f \cos\left(f x + e\right) - {\left(c^{2} + 2 \, c d + d^{2}\right)} f - {\left({\left(c d + d^{2}\right)} f \cos\left(f x + e\right) + {\left(c^{2} + 2 \, c d + d^{2}\right)} f\right)} \sin\left(f x + e\right)}"," ",0,"2*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*(cos(f*x + e) - sin(f*x + e) + 1)/((c*d + d^2)*f*cos(f*x + e)^2 - (c^2 + c*d)*f*cos(f*x + e) - (c^2 + 2*c*d + d^2)*f - ((c*d + d^2)*f*cos(f*x + e) + (c^2 + 2*c*d + d^2)*f)*sin(f*x + e))","B",0
569,1,300,0,1.098766," ","integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{2 \, {\left(2 \, d \cos\left(f x + e\right)^{2} + {\left(3 \, c + d\right)} \cos\left(f x + e\right) + {\left(2 \, d \cos\left(f x + e\right) - 3 \, c + d\right)} \sin\left(f x + e\right) + 3 \, c - d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, {\left({\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, c^{3} d + 5 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{4} + 2 \, c^{3} d + 2 \, c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right) - {\left(c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f + {\left({\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d + 2 \, c^{2} d^{2} + c d^{3}\right)} f \cos\left(f x + e\right) - {\left(c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"2/3*(2*d*cos(f*x + e)^2 + (3*c + d)*cos(f*x + e) + (2*d*cos(f*x + e) - 3*c + d)*sin(f*x + e) + 3*c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e)^3 + (2*c^3*d + 5*c^2*d^2 + 4*c*d^3 + d^4)*f*cos(f*x + e)^2 - (c^4 + 2*c^3*d + 2*c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e) - (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*f + ((c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e)^2 - 2*(c^3*d + 2*c^2*d^2 + c*d^3)*f*cos(f*x + e) - (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*f)*sin(f*x + e))","B",0
570,1,556,0,1.184642," ","integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","\frac{2 \, {\left(8 \, d^{2} \cos\left(f x + e\right)^{3} - 4 \, {\left(5 \, c d - d^{2}\right)} \cos\left(f x + e\right)^{2} - 15 \, c^{2} + 10 \, c d - 7 \, d^{2} - {\left(15 \, c^{2} + 10 \, c d + 11 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - 15 \, c^{2} + 10 \, c d - 7 \, d^{2} + 4 \, {\left(5 \, c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{15 \, {\left({\left(c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{4} - 3 \, {\left(c^{4} d^{2} + 3 \, c^{3} d^{3} + 3 \, c^{2} d^{4} + c d^{5}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, c^{5} d + 12 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 18 \, c^{2} d^{4} + 9 \, c d^{5} + 2 \, d^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(c^{6} + 3 \, c^{5} d + 6 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 9 \, c^{2} d^{4} + 3 \, c d^{5}\right)} f \cos\left(f x + e\right) + {\left(c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right)} f - {\left({\left(c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(3 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 12 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{5} d + 9 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right) - {\left(c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"2/15*(8*d^2*cos(f*x + e)^3 - 4*(5*c*d - d^2)*cos(f*x + e)^2 - 15*c^2 + 10*c*d - 7*d^2 - (15*c^2 + 10*c*d + 11*d^2)*cos(f*x + e) - (8*d^2*cos(f*x + e)^2 - 15*c^2 + 10*c*d - 7*d^2 + 4*(5*c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^3*d^3 + 3*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e)^4 - 3*(c^4*d^2 + 3*c^3*d^3 + 3*c^2*d^4 + c*d^5)*f*cos(f*x + e)^3 - (3*c^5*d + 12*c^4*d^2 + 20*c^3*d^3 + 18*c^2*d^4 + 9*c*d^5 + 2*d^6)*f*cos(f*x + e)^2 + (c^6 + 3*c^5*d + 6*c^4*d^2 + 10*c^3*d^3 + 9*c^2*d^4 + 3*c*d^5)*f*cos(f*x + e) + (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f - ((c^3*d^3 + 3*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e)^3 + (3*c^4*d^2 + 10*c^3*d^3 + 12*c^2*d^4 + 6*c*d^5 + d^6)*f*cos(f*x + e)^2 - (3*c^5*d + 9*c^4*d^2 + 10*c^3*d^3 + 6*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e) - (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f)*sin(f*x + e))","B",0
571,1,1579,0,3.680273," ","integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(a c^{4} - 12 \, a c^{3} d - 42 \, a c^{2} d^{2} - 44 \, a c d^{3} - 15 \, a d^{4} + {\left(a c^{4} - 12 \, a c^{3} d - 42 \, a c^{2} d^{2} - 44 \, a c d^{3} - 15 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(a c^{4} - 12 \, a c^{3} d - 42 \, a c^{2} d^{2} - 44 \, a c d^{3} - 15 \, a d^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) - 8 \, {\left(48 \, a d^{3} \cos\left(f x + e\right)^{4} - 15 \, a c^{3} - 337 \, a c^{2} d - 341 \, a c d^{2} - 147 \, a d^{3} + 8 \, {\left(17 \, a c d^{2} + 15 \, a d^{3}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(59 \, a c^{2} d + 122 \, a c d^{2} + 63 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(15 \, a c^{3} + 455 \, a c^{2} d + 721 \, a c d^{2} + 345 \, a d^{3}\right)} \cos\left(f x + e\right) + {\left(48 \, a d^{3} \cos\left(f x + e\right)^{3} + 15 \, a c^{3} + 337 \, a c^{2} d + 341 \, a c d^{2} + 147 \, a d^{3} - 8 \, {\left(17 \, a c d^{2} + 9 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(59 \, a c^{2} d + 190 \, a c d^{2} + 99 \, a d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{1536 \, {\left(d f \cos\left(f x + e\right) + d f \sin\left(f x + e\right) + d f\right)}}, -\frac{15 \, {\left(a c^{4} - 12 \, a c^{3} d - 42 \, a c^{2} d^{2} - 44 \, a c d^{3} - 15 \, a d^{4} + {\left(a c^{4} - 12 \, a c^{3} d - 42 \, a c^{2} d^{2} - 44 \, a c d^{3} - 15 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(a c^{4} - 12 \, a c^{3} d - 42 \, a c^{2} d^{2} - 44 \, a c d^{3} - 15 \, a d^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left(48 \, a d^{3} \cos\left(f x + e\right)^{4} - 15 \, a c^{3} - 337 \, a c^{2} d - 341 \, a c d^{2} - 147 \, a d^{3} + 8 \, {\left(17 \, a c d^{2} + 15 \, a d^{3}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(59 \, a c^{2} d + 122 \, a c d^{2} + 63 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(15 \, a c^{3} + 455 \, a c^{2} d + 721 \, a c d^{2} + 345 \, a d^{3}\right)} \cos\left(f x + e\right) + {\left(48 \, a d^{3} \cos\left(f x + e\right)^{3} + 15 \, a c^{3} + 337 \, a c^{2} d + 341 \, a c d^{2} + 147 \, a d^{3} - 8 \, {\left(17 \, a c d^{2} + 9 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(59 \, a c^{2} d + 190 \, a c d^{2} + 99 \, a d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{768 \, {\left(d f \cos\left(f x + e\right) + d f \sin\left(f x + e\right) + d f\right)}}\right]"," ",0,"[-1/1536*(15*(a*c^4 - 12*a*c^3*d - 42*a*c^2*d^2 - 44*a*c*d^3 - 15*a*d^4 + (a*c^4 - 12*a*c^3*d - 42*a*c^2*d^2 - 44*a*c*d^3 - 15*a*d^4)*cos(f*x + e) + (a*c^4 - 12*a*c^3*d - 42*a*c^2*d^2 - 44*a*c*d^3 - 15*a*d^4)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) - 8*(48*a*d^3*cos(f*x + e)^4 - 15*a*c^3 - 337*a*c^2*d - 341*a*c*d^2 - 147*a*d^3 + 8*(17*a*c*d^2 + 15*a*d^3)*cos(f*x + e)^3 - 2*(59*a*c^2*d + 122*a*c*d^2 + 63*a*d^3)*cos(f*x + e)^2 - (15*a*c^3 + 455*a*c^2*d + 721*a*c*d^2 + 345*a*d^3)*cos(f*x + e) + (48*a*d^3*cos(f*x + e)^3 + 15*a*c^3 + 337*a*c^2*d + 341*a*c*d^2 + 147*a*d^3 - 8*(17*a*c*d^2 + 9*a*d^3)*cos(f*x + e)^2 - 2*(59*a*c^2*d + 190*a*c*d^2 + 99*a*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f), -1/768*(15*(a*c^4 - 12*a*c^3*d - 42*a*c^2*d^2 - 44*a*c*d^3 - 15*a*d^4 + (a*c^4 - 12*a*c^3*d - 42*a*c^2*d^2 - 44*a*c*d^3 - 15*a*d^4)*cos(f*x + e) + (a*c^4 - 12*a*c^3*d - 42*a*c^2*d^2 - 44*a*c*d^3 - 15*a*d^4)*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) - 4*(48*a*d^3*cos(f*x + e)^4 - 15*a*c^3 - 337*a*c^2*d - 341*a*c*d^2 - 147*a*d^3 + 8*(17*a*c*d^2 + 15*a*d^3)*cos(f*x + e)^3 - 2*(59*a*c^2*d + 122*a*c*d^2 + 63*a*d^3)*cos(f*x + e)^2 - (15*a*c^3 + 455*a*c^2*d + 721*a*c*d^2 + 345*a*d^3)*cos(f*x + e) + (48*a*d^3*cos(f*x + e)^3 + 15*a*c^3 + 337*a*c^2*d + 341*a*c*d^2 + 147*a*d^3 - 8*(17*a*c*d^2 + 9*a*d^3)*cos(f*x + e)^2 - 2*(59*a*c^2*d + 190*a*c*d^2 + 99*a*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f)]","B",0
572,1,1337,0,2.979151," ","integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3} + {\left(a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right)} \cos\left(f x + e\right) + {\left(a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) - 8 \, {\left(8 \, a d^{2} \cos\left(f x + e\right)^{3} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} - 14 \, {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, a c^{2} + 52 \, a c d + 41 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(8 \, a d^{2} \cos\left(f x + e\right)^{2} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} + 2 \, {\left(7 \, a c d + 11 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{192 \, {\left(d f \cos\left(f x + e\right) + d f \sin\left(f x + e\right) + d f\right)}}, -\frac{3 \, {\left(a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3} + {\left(a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right)} \cos\left(f x + e\right) + {\left(a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left(8 \, a d^{2} \cos\left(f x + e\right)^{3} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} - 14 \, {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, a c^{2} + 52 \, a c d + 41 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(8 \, a d^{2} \cos\left(f x + e\right)^{2} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} + 2 \, {\left(7 \, a c d + 11 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{96 \, {\left(d f \cos\left(f x + e\right) + d f \sin\left(f x + e\right) + d f\right)}}\right]"," ",0,"[-1/192*(3*(a*c^3 - 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3 + (a*c^3 - 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3)*cos(f*x + e) + (a*c^3 - 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) - 8*(8*a*d^2*cos(f*x + e)^3 - 3*a*c^2 - 38*a*c*d - 19*a*d^2 - 14*(a*c*d + a*d^2)*cos(f*x + e)^2 - (3*a*c^2 + 52*a*c*d + 41*a*d^2)*cos(f*x + e) - (8*a*d^2*cos(f*x + e)^2 - 3*a*c^2 - 38*a*c*d - 19*a*d^2 + 2*(7*a*c*d + 11*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f), -1/96*(3*(a*c^3 - 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3 + (a*c^3 - 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3)*cos(f*x + e) + (a*c^3 - 9*a*c^2*d - 21*a*c*d^2 - 11*a*d^3)*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) - 4*(8*a*d^2*cos(f*x + e)^3 - 3*a*c^2 - 38*a*c*d - 19*a*d^2 - 14*(a*c*d + a*d^2)*cos(f*x + e)^2 - (3*a*c^2 + 52*a*c*d + 41*a*d^2)*cos(f*x + e) - (8*a*d^2*cos(f*x + e)^2 - 3*a*c^2 - 38*a*c*d - 19*a*d^2 + 2*(7*a*c*d + 11*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f)]","B",0
573,1,1127,0,2.991819," ","integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a c^{2} - 6 \, a c d - 7 \, a d^{2} + {\left(a c^{2} - 6 \, a c d - 7 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(a c^{2} - 6 \, a c d - 7 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 8 \, {\left(2 \, a d \cos\left(f x + e\right)^{2} + a c + 5 \, a d + {\left(a c + 7 \, a d\right)} \cos\left(f x + e\right) + {\left(2 \, a d \cos\left(f x + e\right) - a c - 5 \, a d\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{32 \, {\left(d f \cos\left(f x + e\right) + d f \sin\left(f x + e\right) + d f\right)}}, -\frac{{\left(a c^{2} - 6 \, a c d - 7 \, a d^{2} + {\left(a c^{2} - 6 \, a c d - 7 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(a c^{2} - 6 \, a c d - 7 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(2 \, a d \cos\left(f x + e\right)^{2} + a c + 5 \, a d + {\left(a c + 7 \, a d\right)} \cos\left(f x + e\right) + {\left(2 \, a d \cos\left(f x + e\right) - a c - 5 \, a d\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{16 \, {\left(d f \cos\left(f x + e\right) + d f \sin\left(f x + e\right) + d f\right)}}\right]"," ",0,"[-1/32*((a*c^2 - 6*a*c*d - 7*a*d^2 + (a*c^2 - 6*a*c*d - 7*a*d^2)*cos(f*x + e) + (a*c^2 - 6*a*c*d - 7*a*d^2)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(2*a*d*cos(f*x + e)^2 + a*c + 5*a*d + (a*c + 7*a*d)*cos(f*x + e) + (2*a*d*cos(f*x + e) - a*c - 5*a*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f), -1/16*((a*c^2 - 6*a*c*d - 7*a*d^2 + (a*c^2 - 6*a*c*d - 7*a*d^2)*cos(f*x + e) + (a*c^2 - 6*a*c*d - 7*a*d^2)*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) + 4*(2*a*d*cos(f*x + e)^2 + a*c + 5*a*d + (a*c + 7*a*d)*cos(f*x + e) + (2*a*d*cos(f*x + e) - a*c - 5*a*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f)]","B",0
574,1,989,0,2.866925," ","integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a c - 3 \, a d + {\left(a c - 3 \, a d\right)} \cos\left(f x + e\right) + {\left(a c - 3 \, a d\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 8 \, {\left(a \cos\left(f x + e\right) - a \sin\left(f x + e\right) + a\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{8 \, {\left(d f \cos\left(f x + e\right) + d f \sin\left(f x + e\right) + d f\right)}}, -\frac{{\left(a c - 3 \, a d + {\left(a c - 3 \, a d\right)} \cos\left(f x + e\right) + {\left(a c - 3 \, a d\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(a \cos\left(f x + e\right) - a \sin\left(f x + e\right) + a\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left(d f \cos\left(f x + e\right) + d f \sin\left(f x + e\right) + d f\right)}}\right]"," ",0,"[-1/8*((a*c - 3*a*d + (a*c - 3*a*d)*cos(f*x + e) + (a*c - 3*a*d)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(a*cos(f*x + e) - a*sin(f*x + e) + a)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f), -1/4*((a*c - 3*a*d + (a*c - 3*a*d)*cos(f*x + e) + (a*c - 3*a*d)*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) + 4*(a*cos(f*x + e) - a*sin(f*x + e) + a)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d*f*cos(f*x + e) + d*f*sin(f*x + e) + d*f)]","B",0
575,1,1297,0,2.730738," ","integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a c^{2} + 2 \, a c d + a d^{2} - {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + a c d\right)} \cos\left(f x + e\right) + {\left(a c^{2} + 2 \, a c d + a d^{2} + {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 8 \, {\left(a c - a d + {\left(a c - a d\right)} \cos\left(f x + e\right) - {\left(a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left({\left(c d^{2} + d^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{2} d + c d^{2}\right)} f \cos\left(f x + e\right) - {\left(c^{2} d + 2 \, c d^{2} + d^{3}\right)} f - {\left({\left(c d^{2} + d^{3}\right)} f \cos\left(f x + e\right) + {\left(c^{2} d + 2 \, c d^{2} + d^{3}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left(a c^{2} + 2 \, a c d + a d^{2} - {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + a c d\right)} \cos\left(f x + e\right) + {\left(a c^{2} + 2 \, a c d + a d^{2} + {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(a c - a d + {\left(a c - a d\right)} \cos\left(f x + e\right) - {\left(a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{2 \, {\left({\left(c d^{2} + d^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{2} d + c d^{2}\right)} f \cos\left(f x + e\right) - {\left(c^{2} d + 2 \, c d^{2} + d^{3}\right)} f - {\left({\left(c d^{2} + d^{3}\right)} f \cos\left(f x + e\right) + {\left(c^{2} d + 2 \, c d^{2} + d^{3}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/4*((a*c^2 + 2*a*c*d + a*d^2 - (a*c*d + a*d^2)*cos(f*x + e)^2 + (a*c^2 + a*c*d)*cos(f*x + e) + (a*c^2 + 2*a*c*d + a*d^2 + (a*c*d + a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(a*c - a*d + (a*c - a*d)*cos(f*x + e) - (a*c - a*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((c*d^2 + d^3)*f*cos(f*x + e)^2 - (c^2*d + c*d^2)*f*cos(f*x + e) - (c^2*d + 2*c*d^2 + d^3)*f - ((c*d^2 + d^3)*f*cos(f*x + e) + (c^2*d + 2*c*d^2 + d^3)*f)*sin(f*x + e)), -1/2*((a*c^2 + 2*a*c*d + a*d^2 - (a*c*d + a*d^2)*cos(f*x + e)^2 + (a*c^2 + a*c*d)*cos(f*x + e) + (a*c^2 + 2*a*c*d + a*d^2 + (a*c*d + a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) + 4*(a*c - a*d + (a*c - a*d)*cos(f*x + e) - (a*c - a*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((c*d^2 + d^3)*f*cos(f*x + e)^2 - (c^2*d + c*d^2)*f*cos(f*x + e) - (c^2*d + 2*c*d^2 + d^3)*f - ((c*d^2 + d^3)*f*cos(f*x + e) + (c^2*d + 2*c*d^2 + d^3)*f)*sin(f*x + e))]","B",0
576,1,323,0,1.520638," ","integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{2 \, {\left({\left(a c + 5 \, a d\right)} \cos\left(f x + e\right)^{2} + 4 \, a c - 4 \, a d + {\left(5 \, a c + a d\right)} \cos\left(f x + e\right) - {\left(4 \, a c - 4 \, a d - {\left(a c + 5 \, a d\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, {\left({\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, c^{3} d + 5 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{4} + 2 \, c^{3} d + 2 \, c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right) - {\left(c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f + {\left({\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d + 2 \, c^{2} d^{2} + c d^{3}\right)} f \cos\left(f x + e\right) - {\left(c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"2/3*((a*c + 5*a*d)*cos(f*x + e)^2 + 4*a*c - 4*a*d + (5*a*c + a*d)*cos(f*x + e) - (4*a*c - 4*a*d - (a*c + 5*a*d)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e)^3 + (2*c^3*d + 5*c^2*d^2 + 4*c*d^3 + d^4)*f*cos(f*x + e)^2 - (c^4 + 2*c^3*d + 2*c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e) - (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*f + ((c^2*d^2 + 2*c*d^3 + d^4)*f*cos(f*x + e)^2 - 2*(c^3*d + 2*c^2*d^2 + c*d^3)*f*cos(f*x + e) - (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*f)*sin(f*x + e))","B",0
577,1,598,0,0.870219," ","integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","\frac{2 \, {\left(2 \, {\left(a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 20 \, a c^{2} + 32 \, a c d - 12 \, a d^{2} - {\left(5 \, a c^{2} + 44 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - {\left(25 \, a c^{2} + 14 \, a c d + 21 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(20 \, a c^{2} - 32 \, a c d + 12 \, a d^{2} - 2 \, {\left(a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - {\left(5 \, a c^{2} + 46 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{15 \, {\left({\left(c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{4} - 3 \, {\left(c^{4} d^{2} + 3 \, c^{3} d^{3} + 3 \, c^{2} d^{4} + c d^{5}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, c^{5} d + 12 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 18 \, c^{2} d^{4} + 9 \, c d^{5} + 2 \, d^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(c^{6} + 3 \, c^{5} d + 6 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 9 \, c^{2} d^{4} + 3 \, c d^{5}\right)} f \cos\left(f x + e\right) + {\left(c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right)} f - {\left({\left(c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(3 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 12 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{5} d + 9 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right) - {\left(c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"2/15*(2*(a*c*d + 9*a*d^2)*cos(f*x + e)^3 - 20*a*c^2 + 32*a*c*d - 12*a*d^2 - (5*a*c^2 + 44*a*c*d - 9*a*d^2)*cos(f*x + e)^2 - (25*a*c^2 + 14*a*c*d + 21*a*d^2)*cos(f*x + e) + (20*a*c^2 - 32*a*c*d + 12*a*d^2 - 2*(a*c*d + 9*a*d^2)*cos(f*x + e)^2 - (5*a*c^2 + 46*a*c*d + 9*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^3*d^3 + 3*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e)^4 - 3*(c^4*d^2 + 3*c^3*d^3 + 3*c^2*d^4 + c*d^5)*f*cos(f*x + e)^3 - (3*c^5*d + 12*c^4*d^2 + 20*c^3*d^3 + 18*c^2*d^4 + 9*c*d^5 + 2*d^6)*f*cos(f*x + e)^2 + (c^6 + 3*c^5*d + 6*c^4*d^2 + 10*c^3*d^3 + 9*c^2*d^4 + 3*c*d^5)*f*cos(f*x + e) + (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f - ((c^3*d^3 + 3*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e)^3 + (3*c^4*d^2 + 10*c^3*d^3 + 12*c^2*d^4 + 6*c*d^5 + d^6)*f*cos(f*x + e)^2 - (3*c^5*d + 9*c^4*d^2 + 10*c^3*d^3 + 6*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e) - (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f)*sin(f*x + e))","B",0
578,1,937,0,1.719804," ","integrate((a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm=""fricas"")","\frac{2 \, {\left(8 \, {\left(a c d^{2} + 13 \, a d^{3}\right)} \cos\left(f x + e\right)^{4} - 140 \, a c^{3} + 308 \, a c^{2} d - 244 \, a c d^{2} + 76 \, a d^{3} + 4 \, {\left(7 \, a c^{2} d + 92 \, a c d^{2} + 13 \, a d^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(35 \, a c^{3} + 441 \, a c^{2} d - 167 \, a c d^{2} + 195 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(175 \, a c^{3} + 161 \, a c^{2} d + 437 \, a c d^{2} + 67 \, a d^{3}\right)} \cos\left(f x + e\right) + {\left(140 \, a c^{3} - 308 \, a c^{2} d + 244 \, a c d^{2} - 76 \, a d^{3} + 8 \, {\left(a c d^{2} + 13 \, a d^{3}\right)} \cos\left(f x + e\right)^{3} - 4 \, {\left(7 \, a c^{2} d + 90 \, a c d^{2} - 13 \, a d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(35 \, a c^{3} + 469 \, a c^{2} d + 193 \, a c d^{2} + 143 \, a d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{105 \, {\left({\left(c^{4} d^{4} + 4 \, c^{3} d^{5} + 6 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{5} + {\left(4 \, c^{5} d^{3} + 17 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 22 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, c^{6} d^{2} + 12 \, c^{5} d^{3} + 19 \, c^{4} d^{4} + 16 \, c^{3} d^{5} + 9 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{3} - 2 \, {\left(2 \, c^{7} d + 11 \, c^{6} d^{2} + 28 \, c^{5} d^{3} + 43 \, c^{4} d^{4} + 42 \, c^{3} d^{5} + 25 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{2} + {\left(c^{8} + 4 \, c^{7} d + 12 \, c^{6} d^{2} + 28 \, c^{5} d^{3} + 38 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 12 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right) + {\left(c^{8} + 8 \, c^{7} d + 28 \, c^{6} d^{2} + 56 \, c^{5} d^{3} + 70 \, c^{4} d^{4} + 56 \, c^{3} d^{5} + 28 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right)} f + {\left({\left(c^{4} d^{4} + 4 \, c^{3} d^{5} + 6 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{4} - 4 \, {\left(c^{5} d^{3} + 4 \, c^{4} d^{4} + 6 \, c^{3} d^{5} + 4 \, c^{2} d^{6} + c d^{7}\right)} f \cos\left(f x + e\right)^{3} - 2 \, {\left(3 \, c^{6} d^{2} + 14 \, c^{5} d^{3} + 27 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 17 \, c^{2} d^{6} + 6 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{2} + 4 \, {\left(c^{7} d + 4 \, c^{6} d^{2} + 7 \, c^{5} d^{3} + 8 \, c^{4} d^{4} + 7 \, c^{3} d^{5} + 4 \, c^{2} d^{6} + c d^{7}\right)} f \cos\left(f x + e\right) + {\left(c^{8} + 8 \, c^{7} d + 28 \, c^{6} d^{2} + 56 \, c^{5} d^{3} + 70 \, c^{4} d^{4} + 56 \, c^{3} d^{5} + 28 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right)} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"2/105*(8*(a*c*d^2 + 13*a*d^3)*cos(f*x + e)^4 - 140*a*c^3 + 308*a*c^2*d - 244*a*c*d^2 + 76*a*d^3 + 4*(7*a*c^2*d + 92*a*c*d^2 + 13*a*d^3)*cos(f*x + e)^3 - (35*a*c^3 + 441*a*c^2*d - 167*a*c*d^2 + 195*a*d^3)*cos(f*x + e)^2 - (175*a*c^3 + 161*a*c^2*d + 437*a*c*d^2 + 67*a*d^3)*cos(f*x + e) + (140*a*c^3 - 308*a*c^2*d + 244*a*c*d^2 - 76*a*d^3 + 8*(a*c*d^2 + 13*a*d^3)*cos(f*x + e)^3 - 4*(7*a*c^2*d + 90*a*c*d^2 - 13*a*d^3)*cos(f*x + e)^2 - (35*a*c^3 + 469*a*c^2*d + 193*a*c*d^2 + 143*a*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^4*d^4 + 4*c^3*d^5 + 6*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^5 + (4*c^5*d^3 + 17*c^4*d^4 + 28*c^3*d^5 + 22*c^2*d^6 + 8*c*d^7 + d^8)*f*cos(f*x + e)^4 - 2*(3*c^6*d^2 + 12*c^5*d^3 + 19*c^4*d^4 + 16*c^3*d^5 + 9*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^3 - 2*(2*c^7*d + 11*c^6*d^2 + 28*c^5*d^3 + 43*c^4*d^4 + 42*c^3*d^5 + 25*c^2*d^6 + 8*c*d^7 + d^8)*f*cos(f*x + e)^2 + (c^8 + 4*c^7*d + 12*c^6*d^2 + 28*c^5*d^3 + 38*c^4*d^4 + 28*c^3*d^5 + 12*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e) + (c^8 + 8*c^7*d + 28*c^6*d^2 + 56*c^5*d^3 + 70*c^4*d^4 + 56*c^3*d^5 + 28*c^2*d^6 + 8*c*d^7 + d^8)*f + ((c^4*d^4 + 4*c^3*d^5 + 6*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^4 - 4*(c^5*d^3 + 4*c^4*d^4 + 6*c^3*d^5 + 4*c^2*d^6 + c*d^7)*f*cos(f*x + e)^3 - 2*(3*c^6*d^2 + 14*c^5*d^3 + 27*c^4*d^4 + 28*c^3*d^5 + 17*c^2*d^6 + 6*c*d^7 + d^8)*f*cos(f*x + e)^2 + 4*(c^7*d + 4*c^6*d^2 + 7*c^5*d^3 + 8*c^4*d^4 + 7*c^3*d^5 + 4*c^2*d^6 + c*d^7)*f*cos(f*x + e) + (c^8 + 8*c^7*d + 28*c^6*d^2 + 56*c^5*d^3 + 70*c^4*d^4 + 56*c^3*d^5 + 28*c^2*d^6 + 8*c*d^7 + d^8)*f)*sin(f*x + e))","B",0
579,1,2083,0,5.978862," ","integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[\frac{15 \, {\left(3 \, a^{2} c^{5} - 25 \, a^{2} c^{4} d + 190 \, a^{2} c^{3} d^{2} + 750 \, a^{2} c^{2} d^{3} + 815 \, a^{2} c d^{4} + 283 \, a^{2} d^{5} + {\left(3 \, a^{2} c^{5} - 25 \, a^{2} c^{4} d + 190 \, a^{2} c^{3} d^{2} + 750 \, a^{2} c^{2} d^{3} + 815 \, a^{2} c d^{4} + 283 \, a^{2} d^{5}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{5} - 25 \, a^{2} c^{4} d + 190 \, a^{2} c^{3} d^{2} + 750 \, a^{2} c^{2} d^{3} + 815 \, a^{2} c d^{4} + 283 \, a^{2} d^{5}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) - 8 \, {\left(384 \, a^{2} d^{4} \cos\left(f x + e\right)^{5} - 45 \, a^{2} c^{4} + 360 \, a^{2} c^{3} d + 5446 \, a^{2} c^{2} d^{2} + 6688 \, a^{2} c d^{3} + 2671 \, a^{2} d^{4} - 1008 \, {\left(a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(93 \, a^{2} c^{2} d^{2} + 488 \, a^{2} c d^{3} + 379 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(15 \, a^{2} c^{3} d + 1289 \, a^{2} c^{2} d^{2} + 2565 \, a^{2} c d^{3} + 1291 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(45 \, a^{2} c^{4} - 390 \, a^{2} c^{3} d - 8768 \, a^{2} c^{2} d^{2} - 14714 \, a^{2} c d^{3} - 6893 \, a^{2} d^{4}\right)} \cos\left(f x + e\right) - {\left(384 \, a^{2} d^{4} \cos\left(f x + e\right)^{4} - 45 \, a^{2} c^{4} + 360 \, a^{2} c^{3} d + 5446 \, a^{2} c^{2} d^{2} + 6688 \, a^{2} c d^{3} + 2671 \, a^{2} d^{4} + 48 \, {\left(21 \, a^{2} c d^{3} + 29 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} - 8 \, {\left(93 \, a^{2} c^{2} d^{2} + 362 \, a^{2} c d^{3} + 205 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(15 \, a^{2} c^{3} d + 1661 \, a^{2} c^{2} d^{2} + 4013 \, a^{2} c d^{3} + 2111 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{15360 \, {\left(d^{2} f \cos\left(f x + e\right) + d^{2} f \sin\left(f x + e\right) + d^{2} f\right)}}, \frac{15 \, {\left(3 \, a^{2} c^{5} - 25 \, a^{2} c^{4} d + 190 \, a^{2} c^{3} d^{2} + 750 \, a^{2} c^{2} d^{3} + 815 \, a^{2} c d^{4} + 283 \, a^{2} d^{5} + {\left(3 \, a^{2} c^{5} - 25 \, a^{2} c^{4} d + 190 \, a^{2} c^{3} d^{2} + 750 \, a^{2} c^{2} d^{3} + 815 \, a^{2} c d^{4} + 283 \, a^{2} d^{5}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{5} - 25 \, a^{2} c^{4} d + 190 \, a^{2} c^{3} d^{2} + 750 \, a^{2} c^{2} d^{3} + 815 \, a^{2} c d^{4} + 283 \, a^{2} d^{5}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left(384 \, a^{2} d^{4} \cos\left(f x + e\right)^{5} - 45 \, a^{2} c^{4} + 360 \, a^{2} c^{3} d + 5446 \, a^{2} c^{2} d^{2} + 6688 \, a^{2} c d^{3} + 2671 \, a^{2} d^{4} - 1008 \, {\left(a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(93 \, a^{2} c^{2} d^{2} + 488 \, a^{2} c d^{3} + 379 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} + 2 \, {\left(15 \, a^{2} c^{3} d + 1289 \, a^{2} c^{2} d^{2} + 2565 \, a^{2} c d^{3} + 1291 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(45 \, a^{2} c^{4} - 390 \, a^{2} c^{3} d - 8768 \, a^{2} c^{2} d^{2} - 14714 \, a^{2} c d^{3} - 6893 \, a^{2} d^{4}\right)} \cos\left(f x + e\right) - {\left(384 \, a^{2} d^{4} \cos\left(f x + e\right)^{4} - 45 \, a^{2} c^{4} + 360 \, a^{2} c^{3} d + 5446 \, a^{2} c^{2} d^{2} + 6688 \, a^{2} c d^{3} + 2671 \, a^{2} d^{4} + 48 \, {\left(21 \, a^{2} c d^{3} + 29 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} - 8 \, {\left(93 \, a^{2} c^{2} d^{2} + 362 \, a^{2} c d^{3} + 205 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(15 \, a^{2} c^{3} d + 1661 \, a^{2} c^{2} d^{2} + 4013 \, a^{2} c d^{3} + 2111 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{7680 \, {\left(d^{2} f \cos\left(f x + e\right) + d^{2} f \sin\left(f x + e\right) + d^{2} f\right)}}\right]"," ",0,"[1/15360*(15*(3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5 + (3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5)*cos(f*x + e) + (3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) - 8*(384*a^2*d^4*cos(f*x + e)^5 - 45*a^2*c^4 + 360*a^2*c^3*d + 5446*a^2*c^2*d^2 + 6688*a^2*c*d^3 + 2671*a^2*d^4 - 1008*(a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^4 - 8*(93*a^2*c^2*d^2 + 488*a^2*c*d^3 + 379*a^2*d^4)*cos(f*x + e)^3 + 2*(15*a^2*c^3*d + 1289*a^2*c^2*d^2 + 2565*a^2*c*d^3 + 1291*a^2*d^4)*cos(f*x + e)^2 - (45*a^2*c^4 - 390*a^2*c^3*d - 8768*a^2*c^2*d^2 - 14714*a^2*c*d^3 - 6893*a^2*d^4)*cos(f*x + e) - (384*a^2*d^4*cos(f*x + e)^4 - 45*a^2*c^4 + 360*a^2*c^3*d + 5446*a^2*c^2*d^2 + 6688*a^2*c*d^3 + 2671*a^2*d^4 + 48*(21*a^2*c*d^3 + 29*a^2*d^4)*cos(f*x + e)^3 - 8*(93*a^2*c^2*d^2 + 362*a^2*c*d^3 + 205*a^2*d^4)*cos(f*x + e)^2 - 2*(15*a^2*c^3*d + 1661*a^2*c^2*d^2 + 4013*a^2*c*d^3 + 2111*a^2*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f), 1/7680*(15*(3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5 + (3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5)*cos(f*x + e) + (3*a^2*c^5 - 25*a^2*c^4*d + 190*a^2*c^3*d^2 + 750*a^2*c^2*d^3 + 815*a^2*c*d^4 + 283*a^2*d^5)*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) - 4*(384*a^2*d^4*cos(f*x + e)^5 - 45*a^2*c^4 + 360*a^2*c^3*d + 5446*a^2*c^2*d^2 + 6688*a^2*c*d^3 + 2671*a^2*d^4 - 1008*(a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^4 - 8*(93*a^2*c^2*d^2 + 488*a^2*c*d^3 + 379*a^2*d^4)*cos(f*x + e)^3 + 2*(15*a^2*c^3*d + 1289*a^2*c^2*d^2 + 2565*a^2*c*d^3 + 1291*a^2*d^4)*cos(f*x + e)^2 - (45*a^2*c^4 - 390*a^2*c^3*d - 8768*a^2*c^2*d^2 - 14714*a^2*c*d^3 - 6893*a^2*d^4)*cos(f*x + e) - (384*a^2*d^4*cos(f*x + e)^4 - 45*a^2*c^4 + 360*a^2*c^3*d + 5446*a^2*c^2*d^2 + 6688*a^2*c*d^3 + 2671*a^2*d^4 + 48*(21*a^2*c*d^3 + 29*a^2*d^4)*cos(f*x + e)^3 - 8*(93*a^2*c^2*d^2 + 362*a^2*c*d^3 + 205*a^2*d^4)*cos(f*x + e)^2 - 2*(15*a^2*c^3*d + 1661*a^2*c^2*d^2 + 4013*a^2*c*d^3 + 2111*a^2*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f)]","B",0
580,1,1751,0,5.214184," ","integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(3 \, a^{2} c^{4} - 20 \, a^{2} c^{3} d + 114 \, a^{2} c^{2} d^{2} + 300 \, a^{2} c d^{3} + 163 \, a^{2} d^{4} + {\left(3 \, a^{2} c^{4} - 20 \, a^{2} c^{3} d + 114 \, a^{2} c^{2} d^{2} + 300 \, a^{2} c d^{3} + 163 \, a^{2} d^{4}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{4} - 20 \, a^{2} c^{3} d + 114 \, a^{2} c^{2} d^{2} + 300 \, a^{2} c d^{3} + 163 \, a^{2} d^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 8 \, {\left(48 \, a^{2} d^{3} \cos\left(f x + e\right)^{4} + 9 \, a^{2} c^{3} - 57 \, a^{2} c^{2} d - 493 \, a^{2} c d^{2} - 299 \, a^{2} d^{3} + 8 \, {\left(9 \, a^{2} c d^{2} + 23 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(3 \, a^{2} c^{2} d + 122 \, a^{2} c d^{2} + 119 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(9 \, a^{2} c^{3} - 63 \, a^{2} c^{2} d - 809 \, a^{2} c d^{2} - 673 \, a^{2} d^{3}\right)} \cos\left(f x + e\right) + {\left(48 \, a^{2} d^{3} \cos\left(f x + e\right)^{3} - 9 \, a^{2} c^{3} + 57 \, a^{2} c^{2} d + 493 \, a^{2} c d^{2} + 299 \, a^{2} d^{3} - 8 \, {\left(9 \, a^{2} c d^{2} + 17 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(3 \, a^{2} c^{2} d + 158 \, a^{2} c d^{2} + 187 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{1536 \, {\left(d^{2} f \cos\left(f x + e\right) + d^{2} f \sin\left(f x + e\right) + d^{2} f\right)}}, \frac{3 \, {\left(3 \, a^{2} c^{4} - 20 \, a^{2} c^{3} d + 114 \, a^{2} c^{2} d^{2} + 300 \, a^{2} c d^{3} + 163 \, a^{2} d^{4} + {\left(3 \, a^{2} c^{4} - 20 \, a^{2} c^{3} d + 114 \, a^{2} c^{2} d^{2} + 300 \, a^{2} c d^{3} + 163 \, a^{2} d^{4}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{4} - 20 \, a^{2} c^{3} d + 114 \, a^{2} c^{2} d^{2} + 300 \, a^{2} c d^{3} + 163 \, a^{2} d^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(48 \, a^{2} d^{3} \cos\left(f x + e\right)^{4} + 9 \, a^{2} c^{3} - 57 \, a^{2} c^{2} d - 493 \, a^{2} c d^{2} - 299 \, a^{2} d^{3} + 8 \, {\left(9 \, a^{2} c d^{2} + 23 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(3 \, a^{2} c^{2} d + 122 \, a^{2} c d^{2} + 119 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(9 \, a^{2} c^{3} - 63 \, a^{2} c^{2} d - 809 \, a^{2} c d^{2} - 673 \, a^{2} d^{3}\right)} \cos\left(f x + e\right) + {\left(48 \, a^{2} d^{3} \cos\left(f x + e\right)^{3} - 9 \, a^{2} c^{3} + 57 \, a^{2} c^{2} d + 493 \, a^{2} c d^{2} + 299 \, a^{2} d^{3} - 8 \, {\left(9 \, a^{2} c d^{2} + 17 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(3 \, a^{2} c^{2} d + 158 \, a^{2} c d^{2} + 187 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{768 \, {\left(d^{2} f \cos\left(f x + e\right) + d^{2} f \sin\left(f x + e\right) + d^{2} f\right)}}\right]"," ",0,"[1/1536*(3*(3*a^2*c^4 - 20*a^2*c^3*d + 114*a^2*c^2*d^2 + 300*a^2*c*d^3 + 163*a^2*d^4 + (3*a^2*c^4 - 20*a^2*c^3*d + 114*a^2*c^2*d^2 + 300*a^2*c*d^3 + 163*a^2*d^4)*cos(f*x + e) + (3*a^2*c^4 - 20*a^2*c^3*d + 114*a^2*c^2*d^2 + 300*a^2*c*d^3 + 163*a^2*d^4)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(48*a^2*d^3*cos(f*x + e)^4 + 9*a^2*c^3 - 57*a^2*c^2*d - 493*a^2*c*d^2 - 299*a^2*d^3 + 8*(9*a^2*c*d^2 + 23*a^2*d^3)*cos(f*x + e)^3 - 2*(3*a^2*c^2*d + 122*a^2*c*d^2 + 119*a^2*d^3)*cos(f*x + e)^2 + (9*a^2*c^3 - 63*a^2*c^2*d - 809*a^2*c*d^2 - 673*a^2*d^3)*cos(f*x + e) + (48*a^2*d^3*cos(f*x + e)^3 - 9*a^2*c^3 + 57*a^2*c^2*d + 493*a^2*c*d^2 + 299*a^2*d^3 - 8*(9*a^2*c*d^2 + 17*a^2*d^3)*cos(f*x + e)^2 - 2*(3*a^2*c^2*d + 158*a^2*c*d^2 + 187*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f), 1/768*(3*(3*a^2*c^4 - 20*a^2*c^3*d + 114*a^2*c^2*d^2 + 300*a^2*c*d^3 + 163*a^2*d^4 + (3*a^2*c^4 - 20*a^2*c^3*d + 114*a^2*c^2*d^2 + 300*a^2*c*d^3 + 163*a^2*d^4)*cos(f*x + e) + (3*a^2*c^4 - 20*a^2*c^3*d + 114*a^2*c^2*d^2 + 300*a^2*c*d^3 + 163*a^2*d^4)*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) + 4*(48*a^2*d^3*cos(f*x + e)^4 + 9*a^2*c^3 - 57*a^2*c^2*d - 493*a^2*c*d^2 - 299*a^2*d^3 + 8*(9*a^2*c*d^2 + 23*a^2*d^3)*cos(f*x + e)^3 - 2*(3*a^2*c^2*d + 122*a^2*c*d^2 + 119*a^2*d^3)*cos(f*x + e)^2 + (9*a^2*c^3 - 63*a^2*c^2*d - 809*a^2*c*d^2 - 673*a^2*d^3)*cos(f*x + e) + (48*a^2*d^3*cos(f*x + e)^3 - 9*a^2*c^3 + 57*a^2*c^2*d + 493*a^2*c*d^2 + 299*a^2*d^3 - 8*(9*a^2*c*d^2 + 17*a^2*d^3)*cos(f*x + e)^2 - 2*(3*a^2*c^2*d + 158*a^2*c*d^2 + 187*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f)]","B",0
581,1,1455,0,3.026948," ","integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3} + {\left(a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3}\right)} \cos\left(f x + e\right) + {\left(a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 8 \, {\left(8 \, a^{2} d^{2} \cos\left(f x + e\right)^{3} + 3 \, a^{2} c^{2} - 14 \, a^{2} c d - 49 \, a^{2} d^{2} - 2 \, {\left(a^{2} c d + 13 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{2} - 16 \, a^{2} c d - 83 \, a^{2} d^{2}\right)} \cos\left(f x + e\right) - {\left(8 \, a^{2} d^{2} \cos\left(f x + e\right)^{2} + 3 \, a^{2} c^{2} - 14 \, a^{2} c d - 49 \, a^{2} d^{2} + 2 \, {\left(a^{2} c d + 17 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{192 \, {\left(d^{2} f \cos\left(f x + e\right) + d^{2} f \sin\left(f x + e\right) + d^{2} f\right)}}, \frac{3 \, {\left(a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3} + {\left(a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3}\right)} \cos\left(f x + e\right) + {\left(a^{2} c^{3} - 5 \, a^{2} c^{2} d + 19 \, a^{2} c d^{2} + 25 \, a^{2} d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(8 \, a^{2} d^{2} \cos\left(f x + e\right)^{3} + 3 \, a^{2} c^{2} - 14 \, a^{2} c d - 49 \, a^{2} d^{2} - 2 \, {\left(a^{2} c d + 13 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{2} - 16 \, a^{2} c d - 83 \, a^{2} d^{2}\right)} \cos\left(f x + e\right) - {\left(8 \, a^{2} d^{2} \cos\left(f x + e\right)^{2} + 3 \, a^{2} c^{2} - 14 \, a^{2} c d - 49 \, a^{2} d^{2} + 2 \, {\left(a^{2} c d + 17 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{96 \, {\left(d^{2} f \cos\left(f x + e\right) + d^{2} f \sin\left(f x + e\right) + d^{2} f\right)}}\right]"," ",0,"[1/192*(3*(a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*d^3 + (a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*d^3)*cos(f*x + e) + (a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*d^3)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(8*a^2*d^2*cos(f*x + e)^3 + 3*a^2*c^2 - 14*a^2*c*d - 49*a^2*d^2 - 2*(a^2*c*d + 13*a^2*d^2)*cos(f*x + e)^2 + (3*a^2*c^2 - 16*a^2*c*d - 83*a^2*d^2)*cos(f*x + e) - (8*a^2*d^2*cos(f*x + e)^2 + 3*a^2*c^2 - 14*a^2*c*d - 49*a^2*d^2 + 2*(a^2*c*d + 17*a^2*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f), 1/96*(3*(a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*d^3 + (a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*d^3)*cos(f*x + e) + (a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*d^3)*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) + 4*(8*a^2*d^2*cos(f*x + e)^3 + 3*a^2*c^2 - 14*a^2*c*d - 49*a^2*d^2 - 2*(a^2*c*d + 13*a^2*d^2)*cos(f*x + e)^2 + (3*a^2*c^2 - 16*a^2*c*d - 83*a^2*d^2)*cos(f*x + e) - (8*a^2*d^2*cos(f*x + e)^2 + 3*a^2*c^2 - 14*a^2*c*d - 49*a^2*d^2 + 2*(a^2*c*d + 17*a^2*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f)]","B",0
582,1,1219,0,3.192562," ","integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{2} c^{2} - 10 \, a^{2} c d + 19 \, a^{2} d^{2} + {\left(3 \, a^{2} c^{2} - 10 \, a^{2} c d + 19 \, a^{2} d^{2}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{2} - 10 \, a^{2} c d + 19 \, a^{2} d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) - 8 \, {\left(2 \, a^{2} d \cos\left(f x + e\right)^{2} - 3 \, a^{2} c + 9 \, a^{2} d - {\left(3 \, a^{2} c - 11 \, a^{2} d\right)} \cos\left(f x + e\right) + {\left(2 \, a^{2} d \cos\left(f x + e\right) + 3 \, a^{2} c - 9 \, a^{2} d\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{32 \, {\left(d^{2} f \cos\left(f x + e\right) + d^{2} f \sin\left(f x + e\right) + d^{2} f\right)}}, \frac{{\left(3 \, a^{2} c^{2} - 10 \, a^{2} c d + 19 \, a^{2} d^{2} + {\left(3 \, a^{2} c^{2} - 10 \, a^{2} c d + 19 \, a^{2} d^{2}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{2} - 10 \, a^{2} c d + 19 \, a^{2} d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left(2 \, a^{2} d \cos\left(f x + e\right)^{2} - 3 \, a^{2} c + 9 \, a^{2} d - {\left(3 \, a^{2} c - 11 \, a^{2} d\right)} \cos\left(f x + e\right) + {\left(2 \, a^{2} d \cos\left(f x + e\right) + 3 \, a^{2} c - 9 \, a^{2} d\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{16 \, {\left(d^{2} f \cos\left(f x + e\right) + d^{2} f \sin\left(f x + e\right) + d^{2} f\right)}}\right]"," ",0,"[1/32*((3*a^2*c^2 - 10*a^2*c*d + 19*a^2*d^2 + (3*a^2*c^2 - 10*a^2*c*d + 19*a^2*d^2)*cos(f*x + e) + (3*a^2*c^2 - 10*a^2*c*d + 19*a^2*d^2)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) - 8*(2*a^2*d*cos(f*x + e)^2 - 3*a^2*c + 9*a^2*d - (3*a^2*c - 11*a^2*d)*cos(f*x + e) + (2*a^2*d*cos(f*x + e) + 3*a^2*c - 9*a^2*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f), 1/16*((3*a^2*c^2 - 10*a^2*c*d + 19*a^2*d^2 + (3*a^2*c^2 - 10*a^2*c*d + 19*a^2*d^2)*cos(f*x + e) + (3*a^2*c^2 - 10*a^2*c*d + 19*a^2*d^2)*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) - 4*(2*a^2*d*cos(f*x + e)^2 - 3*a^2*c + 9*a^2*d - (3*a^2*c - 11*a^2*d)*cos(f*x + e) + (2*a^2*d*cos(f*x + e) + 3*a^2*c - 9*a^2*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f)]","B",0
583,1,1665,0,2.530630," ","integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{2} c^{3} + a^{2} c^{2} d - 7 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} - {\left(3 \, a^{2} c^{2} d - 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{3} - 2 \, a^{2} c^{2} d - 5 \, a^{2} c d^{2}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{3} + a^{2} c^{2} d - 7 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} + {\left(3 \, a^{2} c^{2} d - 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 8 \, {\left(3 \, a^{2} c^{2} - 4 \, a^{2} c d + a^{2} d^{2} + {\left(a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{2} - 3 \, a^{2} c d + 2 \, a^{2} d^{2}\right)} \cos\left(f x + e\right) - {\left(3 \, a^{2} c^{2} - 4 \, a^{2} c d + a^{2} d^{2} - {\left(a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{8 \, {\left({\left(c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{2} d^{2} + c d^{3}\right)} f \cos\left(f x + e\right) - {\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f - {\left({\left(c d^{3} + d^{4}\right)} f \cos\left(f x + e\right) + {\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}, \frac{{\left(3 \, a^{2} c^{3} + a^{2} c^{2} d - 7 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} - {\left(3 \, a^{2} c^{2} d - 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{3} - 2 \, a^{2} c^{2} d - 5 \, a^{2} c d^{2}\right)} \cos\left(f x + e\right) + {\left(3 \, a^{2} c^{3} + a^{2} c^{2} d - 7 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} + {\left(3 \, a^{2} c^{2} d - 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(3 \, a^{2} c^{2} - 4 \, a^{2} c d + a^{2} d^{2} + {\left(a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{2} - 3 \, a^{2} c d + 2 \, a^{2} d^{2}\right)} \cos\left(f x + e\right) - {\left(3 \, a^{2} c^{2} - 4 \, a^{2} c d + a^{2} d^{2} - {\left(a^{2} c d + a^{2} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left({\left(c d^{3} + d^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{2} d^{2} + c d^{3}\right)} f \cos\left(f x + e\right) - {\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f - {\left({\left(c d^{3} + d^{4}\right)} f \cos\left(f x + e\right) + {\left(c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/8*((3*a^2*c^3 + a^2*c^2*d - 7*a^2*c*d^2 - 5*a^2*d^3 - (3*a^2*c^2*d - 2*a^2*c*d^2 - 5*a^2*d^3)*cos(f*x + e)^2 + (3*a^2*c^3 - 2*a^2*c^2*d - 5*a^2*c*d^2)*cos(f*x + e) + (3*a^2*c^3 + a^2*c^2*d - 7*a^2*c*d^2 - 5*a^2*d^3 + (3*a^2*c^2*d - 2*a^2*c*d^2 - 5*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(3*a^2*c^2 - 4*a^2*c*d + a^2*d^2 + (a^2*c*d + a^2*d^2)*cos(f*x + e)^2 + (3*a^2*c^2 - 3*a^2*c*d + 2*a^2*d^2)*cos(f*x + e) - (3*a^2*c^2 - 4*a^2*c*d + a^2*d^2 - (a^2*c*d + a^2*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((c*d^3 + d^4)*f*cos(f*x + e)^2 - (c^2*d^2 + c*d^3)*f*cos(f*x + e) - (c^2*d^2 + 2*c*d^3 + d^4)*f - ((c*d^3 + d^4)*f*cos(f*x + e) + (c^2*d^2 + 2*c*d^3 + d^4)*f)*sin(f*x + e)), 1/4*((3*a^2*c^3 + a^2*c^2*d - 7*a^2*c*d^2 - 5*a^2*d^3 - (3*a^2*c^2*d - 2*a^2*c*d^2 - 5*a^2*d^3)*cos(f*x + e)^2 + (3*a^2*c^3 - 2*a^2*c^2*d - 5*a^2*c*d^2)*cos(f*x + e) + (3*a^2*c^3 + a^2*c^2*d - 7*a^2*c*d^2 - 5*a^2*d^3 + (3*a^2*c^2*d - 2*a^2*c*d^2 - 5*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) + 4*(3*a^2*c^2 - 4*a^2*c*d + a^2*d^2 + (a^2*c*d + a^2*d^2)*cos(f*x + e)^2 + (3*a^2*c^2 - 3*a^2*c*d + 2*a^2*d^2)*cos(f*x + e) - (3*a^2*c^2 - 4*a^2*c*d + a^2*d^2 - (a^2*c*d + a^2*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((c*d^3 + d^4)*f*cos(f*x + e)^2 - (c^2*d^2 + c*d^3)*f*cos(f*x + e) - (c^2*d^2 + 2*c*d^3 + d^4)*f - ((c*d^3 + d^4)*f*cos(f*x + e) + (c^2*d^2 + 2*c*d^3 + d^4)*f)*sin(f*x + e))]","B",0
584,1,2297,0,2.993985," ","integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} c^{4} + 4 \, a^{2} c^{3} d + 6 \, a^{2} c^{2} d^{2} + 4 \, a^{2} c d^{3} + a^{2} d^{4} - {\left(a^{2} c^{2} d^{2} + 2 \, a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(2 \, a^{2} c^{3} d + 5 \, a^{2} c^{2} d^{2} + 4 \, a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(a^{2} c^{4} + 2 \, a^{2} c^{3} d + 2 \, a^{2} c^{2} d^{2} + 2 \, a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right) + {\left(a^{2} c^{4} + 4 \, a^{2} c^{3} d + 6 \, a^{2} c^{2} d^{2} + 4 \, a^{2} c d^{3} + a^{2} d^{4} - {\left(a^{2} c^{2} d^{2} + 2 \, a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{2} c^{3} d + 2 \, a^{2} c^{2} d^{2} + a^{2} c d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{a}{d}} \log\left(\frac{128 \, a d^{4} \cos\left(f x + e\right)^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left(2 \, a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{4} - 32 \, {\left(5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{4} \cos\left(f x + e\right)^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{4} \cos\left(f x + e\right)^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left(3 \, c d^{3} - 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{a}{d}} + {\left(a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, a d^{4} \cos\left(f x + e\right)^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left(a c d^{3} - a d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 8 \, {\left(3 \, a^{2} c^{3} + a^{2} c^{2} d - 11 \, a^{2} c d^{2} + 7 \, a^{2} d^{3} + 4 \, {\left(a^{2} c^{2} d + a^{2} c d^{2} - 2 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{3} + 5 \, a^{2} c^{2} d - 7 \, a^{2} c d^{2} - a^{2} d^{3}\right)} \cos\left(f x + e\right) - {\left(3 \, a^{2} c^{3} + a^{2} c^{2} d - 11 \, a^{2} c d^{2} + 7 \, a^{2} d^{3} - 4 \, {\left(a^{2} c^{2} d + a^{2} c d^{2} - 2 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{12 \, {\left({\left(c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, c^{3} d^{3} + 5 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{4} d^{2} + 2 \, c^{3} d^{3} + 2 \, c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d^{2} + 4 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f + {\left({\left(c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d^{3} + 2 \, c^{2} d^{4} + c d^{5}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d^{2} + 4 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{3 \, {\left(a^{2} c^{4} + 4 \, a^{2} c^{3} d + 6 \, a^{2} c^{2} d^{2} + 4 \, a^{2} c d^{3} + a^{2} d^{4} - {\left(a^{2} c^{2} d^{2} + 2 \, a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(2 \, a^{2} c^{3} d + 5 \, a^{2} c^{2} d^{2} + 4 \, a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(a^{2} c^{4} + 2 \, a^{2} c^{3} d + 2 \, a^{2} c^{2} d^{2} + 2 \, a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right) + {\left(a^{2} c^{4} + 4 \, a^{2} c^{3} d + 6 \, a^{2} c^{2} d^{2} + 4 \, a^{2} c d^{3} + a^{2} d^{4} - {\left(a^{2} c^{2} d^{2} + 2 \, a^{2} c d^{3} + a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{2} c^{3} d + 2 \, a^{2} c^{2} d^{2} + a^{2} c d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{a}{d}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{a}{d}}}{4 \, {\left(2 \, a d^{2} \cos\left(f x + e\right)^{3} - {\left(3 \, a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(a c^{2} - a c d + 2 \, a d^{2}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(3 \, a^{2} c^{3} + a^{2} c^{2} d - 11 \, a^{2} c d^{2} + 7 \, a^{2} d^{3} + 4 \, {\left(a^{2} c^{2} d + a^{2} c d^{2} - 2 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, a^{2} c^{3} + 5 \, a^{2} c^{2} d - 7 \, a^{2} c d^{2} - a^{2} d^{3}\right)} \cos\left(f x + e\right) - {\left(3 \, a^{2} c^{3} + a^{2} c^{2} d - 11 \, a^{2} c d^{2} + 7 \, a^{2} d^{3} - 4 \, {\left(a^{2} c^{2} d + a^{2} c d^{2} - 2 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{6 \, {\left({\left(c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, c^{3} d^{3} + 5 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{4} d^{2} + 2 \, c^{3} d^{3} + 2 \, c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d^{2} + 4 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f + {\left({\left(c^{2} d^{4} + 2 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{3} d^{3} + 2 \, c^{2} d^{4} + c d^{5}\right)} f \cos\left(f x + e\right) - {\left(c^{4} d^{2} + 4 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 4 \, c d^{5} + d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/12*(3*(a^2*c^4 + 4*a^2*c^3*d + 6*a^2*c^2*d^2 + 4*a^2*c*d^3 + a^2*d^4 - (a^2*c^2*d^2 + 2*a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^3 - (2*a^2*c^3*d + 5*a^2*c^2*d^2 + 4*a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^2 + (a^2*c^4 + 2*a^2*c^3*d + 2*a^2*c^2*d^2 + 2*a^2*c*d^3 + a^2*d^4)*cos(f*x + e) + (a^2*c^4 + 4*a^2*c^3*d + 6*a^2*c^2*d^2 + 4*a^2*c*d^3 + a^2*d^4 - (a^2*c^2*d^2 + 2*a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^2 + 2*(a^2*c^3*d + 2*a^2*c^2*d^2 + a^2*c*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(3*a^2*c^3 + a^2*c^2*d - 11*a^2*c*d^2 + 7*a^2*d^3 + 4*(a^2*c^2*d + a^2*c*d^2 - 2*a^2*d^3)*cos(f*x + e)^2 + (3*a^2*c^3 + 5*a^2*c^2*d - 7*a^2*c*d^2 - a^2*d^3)*cos(f*x + e) - (3*a^2*c^3 + a^2*c^2*d - 11*a^2*c*d^2 + 7*a^2*d^3 - 4*(a^2*c^2*d + a^2*c*d^2 - 2*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e)^3 + (2*c^3*d^3 + 5*c^2*d^4 + 4*c*d^5 + d^6)*f*cos(f*x + e)^2 - (c^4*d^2 + 2*c^3*d^3 + 2*c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e) - (c^4*d^2 + 4*c^3*d^3 + 6*c^2*d^4 + 4*c*d^5 + d^6)*f + ((c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e)^2 - 2*(c^3*d^3 + 2*c^2*d^4 + c*d^5)*f*cos(f*x + e) - (c^4*d^2 + 4*c^3*d^3 + 6*c^2*d^4 + 4*c*d^5 + d^6)*f)*sin(f*x + e)), -1/6*(3*(a^2*c^4 + 4*a^2*c^3*d + 6*a^2*c^2*d^2 + 4*a^2*c*d^3 + a^2*d^4 - (a^2*c^2*d^2 + 2*a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^3 - (2*a^2*c^3*d + 5*a^2*c^2*d^2 + 4*a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^2 + (a^2*c^4 + 2*a^2*c^3*d + 2*a^2*c^2*d^2 + 2*a^2*c*d^3 + a^2*d^4)*cos(f*x + e) + (a^2*c^4 + 4*a^2*c^3*d + 6*a^2*c^2*d^2 + 4*a^2*c*d^3 + a^2*d^4 - (a^2*c^2*d^2 + 2*a^2*c*d^3 + a^2*d^4)*cos(f*x + e)^2 + 2*(a^2*c^3*d + 2*a^2*c^2*d^2 + a^2*c*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) + 4*(3*a^2*c^3 + a^2*c^2*d - 11*a^2*c*d^2 + 7*a^2*d^3 + 4*(a^2*c^2*d + a^2*c*d^2 - 2*a^2*d^3)*cos(f*x + e)^2 + (3*a^2*c^3 + 5*a^2*c^2*d - 7*a^2*c*d^2 - a^2*d^3)*cos(f*x + e) - (3*a^2*c^3 + a^2*c^2*d - 11*a^2*c*d^2 + 7*a^2*d^3 - 4*(a^2*c^2*d + a^2*c*d^2 - 2*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e)^3 + (2*c^3*d^3 + 5*c^2*d^4 + 4*c*d^5 + d^6)*f*cos(f*x + e)^2 - (c^4*d^2 + 2*c^3*d^3 + 2*c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e) - (c^4*d^2 + 4*c^3*d^3 + 6*c^2*d^4 + 4*c*d^5 + d^6)*f + ((c^2*d^4 + 2*c*d^5 + d^6)*f*cos(f*x + e)^2 - 2*(c^3*d^3 + 2*c^2*d^4 + c*d^5)*f*cos(f*x + e) - (c^4*d^2 + 4*c^3*d^3 + 6*c^2*d^4 + 4*c*d^5 + d^6)*f)*sin(f*x + e))]","B",0
585,1,654,0,1.147045," ","integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(32 \, a^{2} c^{2} - 64 \, a^{2} c d + 32 \, a^{2} d^{2} - {\left(3 \, a^{2} c^{2} + 14 \, a^{2} c d + 43 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{3} + {\left(11 \, a^{2} c^{2} + 78 \, a^{2} c d - 29 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(23 \, a^{2} c^{2} + 14 \, a^{2} c d + 23 \, a^{2} d^{2}\right)} \cos\left(f x + e\right) - {\left(32 \, a^{2} c^{2} - 64 \, a^{2} c d + 32 \, a^{2} d^{2} - {\left(3 \, a^{2} c^{2} + 14 \, a^{2} c d + 43 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(7 \, a^{2} c^{2} + 46 \, a^{2} c d + 7 \, a^{2} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{15 \, {\left({\left(c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{4} - 3 \, {\left(c^{4} d^{2} + 3 \, c^{3} d^{3} + 3 \, c^{2} d^{4} + c d^{5}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, c^{5} d + 12 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 18 \, c^{2} d^{4} + 9 \, c d^{5} + 2 \, d^{6}\right)} f \cos\left(f x + e\right)^{2} + {\left(c^{6} + 3 \, c^{5} d + 6 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 9 \, c^{2} d^{4} + 3 \, c d^{5}\right)} f \cos\left(f x + e\right) + {\left(c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right)} f - {\left({\left(c^{3} d^{3} + 3 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(3 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 12 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{5} d + 9 \, c^{4} d^{2} + 10 \, c^{3} d^{3} + 6 \, c^{2} d^{4} + 3 \, c d^{5} + d^{6}\right)} f \cos\left(f x + e\right) - {\left(c^{6} + 6 \, c^{5} d + 15 \, c^{4} d^{2} + 20 \, c^{3} d^{3} + 15 \, c^{2} d^{4} + 6 \, c d^{5} + d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-2/15*(32*a^2*c^2 - 64*a^2*c*d + 32*a^2*d^2 - (3*a^2*c^2 + 14*a^2*c*d + 43*a^2*d^2)*cos(f*x + e)^3 + (11*a^2*c^2 + 78*a^2*c*d - 29*a^2*d^2)*cos(f*x + e)^2 + 2*(23*a^2*c^2 + 14*a^2*c*d + 23*a^2*d^2)*cos(f*x + e) - (32*a^2*c^2 - 64*a^2*c*d + 32*a^2*d^2 - (3*a^2*c^2 + 14*a^2*c*d + 43*a^2*d^2)*cos(f*x + e)^2 - 2*(7*a^2*c^2 + 46*a^2*c*d + 7*a^2*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^3*d^3 + 3*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e)^4 - 3*(c^4*d^2 + 3*c^3*d^3 + 3*c^2*d^4 + c*d^5)*f*cos(f*x + e)^3 - (3*c^5*d + 12*c^4*d^2 + 20*c^3*d^3 + 18*c^2*d^4 + 9*c*d^5 + 2*d^6)*f*cos(f*x + e)^2 + (c^6 + 3*c^5*d + 6*c^4*d^2 + 10*c^3*d^3 + 9*c^2*d^4 + 3*c*d^5)*f*cos(f*x + e) + (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f - ((c^3*d^3 + 3*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e)^3 + (3*c^4*d^2 + 10*c^3*d^3 + 12*c^2*d^4 + 6*c*d^5 + d^6)*f*cos(f*x + e)^2 - (3*c^5*d + 9*c^4*d^2 + 10*c^3*d^3 + 6*c^2*d^4 + 3*c*d^5 + d^6)*f*cos(f*x + e) - (c^6 + 6*c^5*d + 15*c^4*d^2 + 20*c^3*d^3 + 15*c^2*d^4 + 6*c*d^5 + d^6)*f)*sin(f*x + e))","B",0
586,1,1033,0,1.781225," ","integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(9/2),x, algorithm=""fricas"")","-\frac{2 \, {\left(224 \, a^{2} c^{3} - 608 \, a^{2} c^{2} d + 544 \, a^{2} c d^{2} - 160 \, a^{2} d^{3} - 2 \, {\left(3 \, a^{2} c^{2} d + 22 \, a^{2} c d^{2} + 115 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{4} - {\left(21 \, a^{2} c^{3} + 157 \, a^{2} c^{2} d + 827 \, a^{2} c d^{2} + 115 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(77 \, a^{2} c^{3} + 783 \, a^{2} c^{2} d - 425 \, a^{2} c d^{2} + 405 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(161 \, a^{2} c^{3} + 163 \, a^{2} c^{2} d + 451 \, a^{2} c d^{2} + 65 \, a^{2} d^{3}\right)} \cos\left(f x + e\right) - {\left(224 \, a^{2} c^{3} - 608 \, a^{2} c^{2} d + 544 \, a^{2} c d^{2} - 160 \, a^{2} d^{3} + 2 \, {\left(3 \, a^{2} c^{2} d + 22 \, a^{2} c d^{2} + 115 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(21 \, a^{2} c^{3} + 151 \, a^{2} c^{2} d + 783 \, a^{2} c d^{2} - 115 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(49 \, a^{2} c^{3} + 467 \, a^{2} c^{2} d + 179 \, a^{2} c d^{2} + 145 \, a^{2} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{105 \, {\left({\left(c^{4} d^{4} + 4 \, c^{3} d^{5} + 6 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{5} + {\left(4 \, c^{5} d^{3} + 17 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 22 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, c^{6} d^{2} + 12 \, c^{5} d^{3} + 19 \, c^{4} d^{4} + 16 \, c^{3} d^{5} + 9 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{3} - 2 \, {\left(2 \, c^{7} d + 11 \, c^{6} d^{2} + 28 \, c^{5} d^{3} + 43 \, c^{4} d^{4} + 42 \, c^{3} d^{5} + 25 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{2} + {\left(c^{8} + 4 \, c^{7} d + 12 \, c^{6} d^{2} + 28 \, c^{5} d^{3} + 38 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 12 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right) + {\left(c^{8} + 8 \, c^{7} d + 28 \, c^{6} d^{2} + 56 \, c^{5} d^{3} + 70 \, c^{4} d^{4} + 56 \, c^{3} d^{5} + 28 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right)} f + {\left({\left(c^{4} d^{4} + 4 \, c^{3} d^{5} + 6 \, c^{2} d^{6} + 4 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{4} - 4 \, {\left(c^{5} d^{3} + 4 \, c^{4} d^{4} + 6 \, c^{3} d^{5} + 4 \, c^{2} d^{6} + c d^{7}\right)} f \cos\left(f x + e\right)^{3} - 2 \, {\left(3 \, c^{6} d^{2} + 14 \, c^{5} d^{3} + 27 \, c^{4} d^{4} + 28 \, c^{3} d^{5} + 17 \, c^{2} d^{6} + 6 \, c d^{7} + d^{8}\right)} f \cos\left(f x + e\right)^{2} + 4 \, {\left(c^{7} d + 4 \, c^{6} d^{2} + 7 \, c^{5} d^{3} + 8 \, c^{4} d^{4} + 7 \, c^{3} d^{5} + 4 \, c^{2} d^{6} + c d^{7}\right)} f \cos\left(f x + e\right) + {\left(c^{8} + 8 \, c^{7} d + 28 \, c^{6} d^{2} + 56 \, c^{5} d^{3} + 70 \, c^{4} d^{4} + 56 \, c^{3} d^{5} + 28 \, c^{2} d^{6} + 8 \, c d^{7} + d^{8}\right)} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"-2/105*(224*a^2*c^3 - 608*a^2*c^2*d + 544*a^2*c*d^2 - 160*a^2*d^3 - 2*(3*a^2*c^2*d + 22*a^2*c*d^2 + 115*a^2*d^3)*cos(f*x + e)^4 - (21*a^2*c^3 + 157*a^2*c^2*d + 827*a^2*c*d^2 + 115*a^2*d^3)*cos(f*x + e)^3 + (77*a^2*c^3 + 783*a^2*c^2*d - 425*a^2*c*d^2 + 405*a^2*d^3)*cos(f*x + e)^2 + 2*(161*a^2*c^3 + 163*a^2*c^2*d + 451*a^2*c*d^2 + 65*a^2*d^3)*cos(f*x + e) - (224*a^2*c^3 - 608*a^2*c^2*d + 544*a^2*c*d^2 - 160*a^2*d^3 + 2*(3*a^2*c^2*d + 22*a^2*c*d^2 + 115*a^2*d^3)*cos(f*x + e)^3 - (21*a^2*c^3 + 151*a^2*c^2*d + 783*a^2*c*d^2 - 115*a^2*d^3)*cos(f*x + e)^2 - 2*(49*a^2*c^3 + 467*a^2*c^2*d + 179*a^2*c*d^2 + 145*a^2*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^4*d^4 + 4*c^3*d^5 + 6*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^5 + (4*c^5*d^3 + 17*c^4*d^4 + 28*c^3*d^5 + 22*c^2*d^6 + 8*c*d^7 + d^8)*f*cos(f*x + e)^4 - 2*(3*c^6*d^2 + 12*c^5*d^3 + 19*c^4*d^4 + 16*c^3*d^5 + 9*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^3 - 2*(2*c^7*d + 11*c^6*d^2 + 28*c^5*d^3 + 43*c^4*d^4 + 42*c^3*d^5 + 25*c^2*d^6 + 8*c*d^7 + d^8)*f*cos(f*x + e)^2 + (c^8 + 4*c^7*d + 12*c^6*d^2 + 28*c^5*d^3 + 38*c^4*d^4 + 28*c^3*d^5 + 12*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e) + (c^8 + 8*c^7*d + 28*c^6*d^2 + 56*c^5*d^3 + 70*c^4*d^4 + 56*c^3*d^5 + 28*c^2*d^6 + 8*c*d^7 + d^8)*f + ((c^4*d^4 + 4*c^3*d^5 + 6*c^2*d^6 + 4*c*d^7 + d^8)*f*cos(f*x + e)^4 - 4*(c^5*d^3 + 4*c^4*d^4 + 6*c^3*d^5 + 4*c^2*d^6 + c*d^7)*f*cos(f*x + e)^3 - 2*(3*c^6*d^2 + 14*c^5*d^3 + 27*c^4*d^4 + 28*c^3*d^5 + 17*c^2*d^6 + 6*c*d^7 + d^8)*f*cos(f*x + e)^2 + 4*(c^7*d + 4*c^6*d^2 + 7*c^5*d^3 + 8*c^4*d^4 + 7*c^3*d^5 + 4*c^2*d^6 + c*d^7)*f*cos(f*x + e) + (c^8 + 8*c^7*d + 28*c^6*d^2 + 56*c^5*d^3 + 70*c^4*d^4 + 56*c^3*d^5 + 28*c^2*d^6 + 8*c*d^7 + d^8)*f)*sin(f*x + e))","B",0
587,1,1492,0,1.420914," ","integrate((a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(11/2),x, algorithm=""fricas"")","\frac{2 \, {\left(672 \, a^{2} c^{4} - 2304 \, a^{2} c^{3} d + 3008 \, a^{2} c^{2} d^{2} - 1792 \, a^{2} c d^{3} + 416 \, a^{2} d^{4} + 8 \, {\left(a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 73 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{5} - 4 \, {\left(9 \, a^{2} c^{3} d + 89 \, a^{2} c^{2} d^{2} + 647 \, a^{2} c d^{3} - 73 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{4} - {\left(63 \, a^{2} c^{4} + 648 \, a^{2} c^{3} d + 4798 \, a^{2} c^{2} d^{2} + 1504 \, a^{2} c d^{3} + 1387 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(231 \, a^{2} c^{4} + 3060 \, a^{2} c^{3} d - 2158 \, a^{2} c^{2} d^{2} + 4580 \, a^{2} c d^{3} - 673 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(483 \, a^{2} c^{4} + 684 \, a^{2} c^{3} d + 2642 \, a^{2} c^{2} d^{2} + 812 \, a^{2} c d^{3} + 419 \, a^{2} d^{4}\right)} \cos\left(f x + e\right) - {\left(672 \, a^{2} c^{4} - 2304 \, a^{2} c^{3} d + 3008 \, a^{2} c^{2} d^{2} - 1792 \, a^{2} c d^{3} + 416 \, a^{2} d^{4} + 8 \, {\left(a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 73 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{4} + 4 \, {\left(9 \, a^{2} c^{3} d + 91 \, a^{2} c^{2} d^{2} + 667 \, a^{2} c d^{3} + 73 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(21 \, a^{2} c^{4} + 204 \, a^{2} c^{3} d + 1478 \, a^{2} c^{2} d^{2} - 388 \, a^{2} c d^{3} + 365 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(147 \, a^{2} c^{4} + 1836 \, a^{2} c^{3} d + 1138 \, a^{2} c^{2} d^{2} + 1708 \, a^{2} c d^{3} + 211 \, a^{2} d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{315 \, {\left({\left(c^{5} d^{5} + 5 \, c^{4} d^{6} + 10 \, c^{3} d^{7} + 10 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right)} f \cos\left(f x + e\right)^{6} - 5 \, {\left(c^{6} d^{4} + 5 \, c^{5} d^{5} + 10 \, c^{4} d^{6} + 10 \, c^{3} d^{7} + 5 \, c^{2} d^{8} + c d^{9}\right)} f \cos\left(f x + e\right)^{5} - {\left(10 \, c^{7} d^{3} + 55 \, c^{6} d^{4} + 128 \, c^{5} d^{5} + 165 \, c^{4} d^{6} + 130 \, c^{3} d^{7} + 65 \, c^{2} d^{8} + 20 \, c d^{9} + 3 \, d^{10}\right)} f \cos\left(f x + e\right)^{4} + 10 \, {\left(c^{8} d^{2} + 5 \, c^{7} d^{3} + 11 \, c^{6} d^{4} + 15 \, c^{5} d^{5} + 15 \, c^{4} d^{6} + 11 \, c^{3} d^{7} + 5 \, c^{2} d^{8} + c d^{9}\right)} f \cos\left(f x + e\right)^{3} + {\left(5 \, c^{9} d + 35 \, c^{8} d^{2} + 120 \, c^{7} d^{3} + 260 \, c^{6} d^{4} + 378 \, c^{5} d^{5} + 370 \, c^{4} d^{6} + 240 \, c^{3} d^{7} + 100 \, c^{2} d^{8} + 25 \, c d^{9} + 3 \, d^{10}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{10} + 5 \, c^{9} d + 20 \, c^{8} d^{2} + 60 \, c^{7} d^{3} + 110 \, c^{6} d^{4} + 126 \, c^{5} d^{5} + 100 \, c^{4} d^{6} + 60 \, c^{3} d^{7} + 25 \, c^{2} d^{8} + 5 \, c d^{9}\right)} f \cos\left(f x + e\right) - {\left(c^{10} + 10 \, c^{9} d + 45 \, c^{8} d^{2} + 120 \, c^{7} d^{3} + 210 \, c^{6} d^{4} + 252 \, c^{5} d^{5} + 210 \, c^{4} d^{6} + 120 \, c^{3} d^{7} + 45 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right)} f - {\left({\left(c^{5} d^{5} + 5 \, c^{4} d^{6} + 10 \, c^{3} d^{7} + 10 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right)} f \cos\left(f x + e\right)^{5} + {\left(5 \, c^{6} d^{4} + 26 \, c^{5} d^{5} + 55 \, c^{4} d^{6} + 60 \, c^{3} d^{7} + 35 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(5 \, c^{7} d^{3} + 25 \, c^{6} d^{4} + 51 \, c^{5} d^{5} + 55 \, c^{4} d^{6} + 35 \, c^{3} d^{7} + 15 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right)} f \cos\left(f x + e\right)^{3} - 2 \, {\left(5 \, c^{8} d^{2} + 30 \, c^{7} d^{3} + 80 \, c^{6} d^{4} + 126 \, c^{5} d^{5} + 130 \, c^{4} d^{6} + 90 \, c^{3} d^{7} + 40 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right)} f \cos\left(f x + e\right)^{2} + {\left(5 \, c^{9} d + 25 \, c^{8} d^{2} + 60 \, c^{7} d^{3} + 100 \, c^{6} d^{4} + 126 \, c^{5} d^{5} + 110 \, c^{4} d^{6} + 60 \, c^{3} d^{7} + 20 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right)} f \cos\left(f x + e\right) + {\left(c^{10} + 10 \, c^{9} d + 45 \, c^{8} d^{2} + 120 \, c^{7} d^{3} + 210 \, c^{6} d^{4} + 252 \, c^{5} d^{5} + 210 \, c^{4} d^{6} + 120 \, c^{3} d^{7} + 45 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right)} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"2/315*(672*a^2*c^4 - 2304*a^2*c^3*d + 3008*a^2*c^2*d^2 - 1792*a^2*c*d^3 + 416*a^2*d^4 + 8*(a^2*c^2*d^2 + 10*a^2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^5 - 4*(9*a^2*c^3*d + 89*a^2*c^2*d^2 + 647*a^2*c*d^3 - 73*a^2*d^4)*cos(f*x + e)^4 - (63*a^2*c^4 + 648*a^2*c^3*d + 4798*a^2*c^2*d^2 + 1504*a^2*c*d^3 + 1387*a^2*d^4)*cos(f*x + e)^3 + (231*a^2*c^4 + 3060*a^2*c^3*d - 2158*a^2*c^2*d^2 + 4580*a^2*c*d^3 - 673*a^2*d^4)*cos(f*x + e)^2 + 2*(483*a^2*c^4 + 684*a^2*c^3*d + 2642*a^2*c^2*d^2 + 812*a^2*c*d^3 + 419*a^2*d^4)*cos(f*x + e) - (672*a^2*c^4 - 2304*a^2*c^3*d + 3008*a^2*c^2*d^2 - 1792*a^2*c*d^3 + 416*a^2*d^4 + 8*(a^2*c^2*d^2 + 10*a^2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^4 + 4*(9*a^2*c^3*d + 91*a^2*c^2*d^2 + 667*a^2*c*d^3 + 73*a^2*d^4)*cos(f*x + e)^3 - 3*(21*a^2*c^4 + 204*a^2*c^3*d + 1478*a^2*c^2*d^2 - 388*a^2*c*d^3 + 365*a^2*d^4)*cos(f*x + e)^2 - 2*(147*a^2*c^4 + 1836*a^2*c^3*d + 1138*a^2*c^2*d^2 + 1708*a^2*c*d^3 + 211*a^2*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((c^5*d^5 + 5*c^4*d^6 + 10*c^3*d^7 + 10*c^2*d^8 + 5*c*d^9 + d^10)*f*cos(f*x + e)^6 - 5*(c^6*d^4 + 5*c^5*d^5 + 10*c^4*d^6 + 10*c^3*d^7 + 5*c^2*d^8 + c*d^9)*f*cos(f*x + e)^5 - (10*c^7*d^3 + 55*c^6*d^4 + 128*c^5*d^5 + 165*c^4*d^6 + 130*c^3*d^7 + 65*c^2*d^8 + 20*c*d^9 + 3*d^10)*f*cos(f*x + e)^4 + 10*(c^8*d^2 + 5*c^7*d^3 + 11*c^6*d^4 + 15*c^5*d^5 + 15*c^4*d^6 + 11*c^3*d^7 + 5*c^2*d^8 + c*d^9)*f*cos(f*x + e)^3 + (5*c^9*d + 35*c^8*d^2 + 120*c^7*d^3 + 260*c^6*d^4 + 378*c^5*d^5 + 370*c^4*d^6 + 240*c^3*d^7 + 100*c^2*d^8 + 25*c*d^9 + 3*d^10)*f*cos(f*x + e)^2 - (c^10 + 5*c^9*d + 20*c^8*d^2 + 60*c^7*d^3 + 110*c^6*d^4 + 126*c^5*d^5 + 100*c^4*d^6 + 60*c^3*d^7 + 25*c^2*d^8 + 5*c*d^9)*f*cos(f*x + e) - (c^10 + 10*c^9*d + 45*c^8*d^2 + 120*c^7*d^3 + 210*c^6*d^4 + 252*c^5*d^5 + 210*c^4*d^6 + 120*c^3*d^7 + 45*c^2*d^8 + 10*c*d^9 + d^10)*f - ((c^5*d^5 + 5*c^4*d^6 + 10*c^3*d^7 + 10*c^2*d^8 + 5*c*d^9 + d^10)*f*cos(f*x + e)^5 + (5*c^6*d^4 + 26*c^5*d^5 + 55*c^4*d^6 + 60*c^3*d^7 + 35*c^2*d^8 + 10*c*d^9 + d^10)*f*cos(f*x + e)^4 - 2*(5*c^7*d^3 + 25*c^6*d^4 + 51*c^5*d^5 + 55*c^4*d^6 + 35*c^3*d^7 + 15*c^2*d^8 + 5*c*d^9 + d^10)*f*cos(f*x + e)^3 - 2*(5*c^8*d^2 + 30*c^7*d^3 + 80*c^6*d^4 + 126*c^5*d^5 + 130*c^4*d^6 + 90*c^3*d^7 + 40*c^2*d^8 + 10*c*d^9 + d^10)*f*cos(f*x + e)^2 + (5*c^9*d + 25*c^8*d^2 + 60*c^7*d^3 + 100*c^6*d^4 + 126*c^5*d^5 + 110*c^4*d^6 + 60*c^3*d^7 + 20*c^2*d^8 + 5*c*d^9 + d^10)*f*cos(f*x + e) + (c^10 + 10*c^9*d + 45*c^8*d^2 + 120*c^7*d^3 + 210*c^6*d^4 + 252*c^5*d^5 + 210*c^4*d^6 + 120*c^3*d^7 + 45*c^2*d^8 + 10*c*d^9 + d^10)*f)*sin(f*x + e))","B",0
588,1,2925,0,3.661563," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{16 \, \sqrt{2} {\left(a c^{2} - 2 \, a c d + a d^{2} + {\left(a c^{2} - 2 \, a c d + a d^{2}\right)} \cos\left(f x + e\right) + {\left(a c^{2} - 2 \, a c d + a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{a}} \log\left(\frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} - {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) - 2 \, c - 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2} + {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) - 8 \, {\left(2 \, d^{2} \cos\left(f x + e\right)^{2} + 9 \, c d - 3 \, d^{2} + {\left(9 \, c d - d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, d^{2} \cos\left(f x + e\right) - 9 \, c d + 3 \, d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{32 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}, \frac{8 \, \sqrt{2} {\left(a c^{2} - 2 \, a c d + a d^{2} + {\left(a c^{2} - 2 \, a c d + a d^{2}\right)} \cos\left(f x + e\right) + {\left(a c^{2} - 2 \, a c d + a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{a}} \log\left(\frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} - {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) - 2 \, c - 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2} + {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left(2 \, d^{2} \cos\left(f x + e\right)^{2} + 9 \, c d - 3 \, d^{2} + {\left(9 \, c d - d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, d^{2} \cos\left(f x + e\right) - 9 \, c d + 3 \, d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{16 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}, -\frac{32 \, \sqrt{2} {\left(a c^{2} - 2 \, a c d + a d^{2} + {\left(a c^{2} - 2 \, a c d + a d^{2}\right)} \cos\left(f x + e\right) + {\left(a c^{2} - 2 \, a c d + a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{\sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2} + {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 8 \, {\left(2 \, d^{2} \cos\left(f x + e\right)^{2} + 9 \, c d - 3 \, d^{2} + {\left(9 \, c d - d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, d^{2} \cos\left(f x + e\right) - 9 \, c d + 3 \, d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{32 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}, -\frac{16 \, \sqrt{2} {\left(a c^{2} - 2 \, a c d + a d^{2} + {\left(a c^{2} - 2 \, a c d + a d^{2}\right)} \cos\left(f x + e\right) + {\left(a c^{2} - 2 \, a c d + a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{\sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2} + {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(15 \, a c^{2} - 10 \, a c d + 7 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(2 \, d^{2} \cos\left(f x + e\right)^{2} + 9 \, c d - 3 \, d^{2} + {\left(9 \, c d - d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, d^{2} \cos\left(f x + e\right) - 9 \, c d + 3 \, d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{16 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}\right]"," ",0,"[1/32*(16*sqrt(2)*(a*c^2 - 2*a*c*d + a*d^2 + (a*c^2 - 2*a*c*d + a*d^2)*cos(f*x + e) + (a*c^2 - 2*a*c*d + a*d^2)*sin(f*x + e))*sqrt((c - d)/a)*log((2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + (15*a*c^2 - 10*a*c*d + 7*a*d^2 + (15*a*c^2 - 10*a*c*d + 7*a*d^2)*cos(f*x + e) + (15*a*c^2 - 10*a*c*d + 7*a*d^2)*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) - 8*(2*d^2*cos(f*x + e)^2 + 9*c*d - 3*d^2 + (9*c*d - d^2)*cos(f*x + e) + (2*d^2*cos(f*x + e) - 9*c*d + 3*d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f), 1/16*(8*sqrt(2)*(a*c^2 - 2*a*c*d + a*d^2 + (a*c^2 - 2*a*c*d + a*d^2)*cos(f*x + e) + (a*c^2 - 2*a*c*d + a*d^2)*sin(f*x + e))*sqrt((c - d)/a)*log((2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + (15*a*c^2 - 10*a*c*d + 7*a*d^2 + (15*a*c^2 - 10*a*c*d + 7*a*d^2)*cos(f*x + e) + (15*a*c^2 - 10*a*c*d + 7*a*d^2)*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) - 4*(2*d^2*cos(f*x + e)^2 + 9*c*d - 3*d^2 + (9*c*d - d^2)*cos(f*x + e) + (2*d^2*cos(f*x + e) - 9*c*d + 3*d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f), -1/32*(32*sqrt(2)*(a*c^2 - 2*a*c*d + a*d^2 + (a*c^2 - 2*a*c*d + a*d^2)*cos(f*x + e) + (a*c^2 - 2*a*c*d + a*d^2)*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - (15*a*c^2 - 10*a*c*d + 7*a*d^2 + (15*a*c^2 - 10*a*c*d + 7*a*d^2)*cos(f*x + e) + (15*a*c^2 - 10*a*c*d + 7*a*d^2)*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(2*d^2*cos(f*x + e)^2 + 9*c*d - 3*d^2 + (9*c*d - d^2)*cos(f*x + e) + (2*d^2*cos(f*x + e) - 9*c*d + 3*d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f), -1/16*(16*sqrt(2)*(a*c^2 - 2*a*c*d + a*d^2 + (a*c^2 - 2*a*c*d + a*d^2)*cos(f*x + e) + (a*c^2 - 2*a*c*d + a*d^2)*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - (15*a*c^2 - 10*a*c*d + 7*a*d^2 + (15*a*c^2 - 10*a*c*d + 7*a*d^2)*cos(f*x + e) + (15*a*c^2 - 10*a*c*d + 7*a*d^2)*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) + 4*(2*d^2*cos(f*x + e)^2 + 9*c*d - 3*d^2 + (9*c*d - d^2)*cos(f*x + e) + (2*d^2*cos(f*x + e) - 9*c*d + 3*d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)]","B",0
589,1,2525,0,3.301011," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[-\frac{4 \, \sqrt{2} {\left(a c - a d + {\left(a c - a d\right)} \cos\left(f x + e\right) + {\left(a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{a}} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, c - d\right)} \cos\left(f x + e\right) - {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) + 2 \, c + 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + {\left(3 \, a c - a d + {\left(3 \, a c - a d\right)} \cos\left(f x + e\right) + {\left(3 \, a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 8 \, {\left(d \cos\left(f x + e\right) - d \sin\left(f x + e\right) + d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{8 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}, -\frac{2 \, \sqrt{2} {\left(a c - a d + {\left(a c - a d\right)} \cos\left(f x + e\right) + {\left(a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{a}} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, c - d\right)} \cos\left(f x + e\right) - {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) + 2 \, c + 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - {\left(3 \, a c - a d + {\left(3 \, a c - a d\right)} \cos\left(f x + e\right) + {\left(3 \, a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(d \cos\left(f x + e\right) - d \sin\left(f x + e\right) + d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}, -\frac{8 \, \sqrt{2} {\left(a c - a d + {\left(a c - a d\right)} \cos\left(f x + e\right) + {\left(a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{\sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) + {\left(3 \, a c - a d + {\left(3 \, a c - a d\right)} \cos\left(f x + e\right) + {\left(3 \, a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 8 \, {\left(d \cos\left(f x + e\right) - d \sin\left(f x + e\right) + d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{8 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}, -\frac{4 \, \sqrt{2} {\left(a c - a d + {\left(a c - a d\right)} \cos\left(f x + e\right) + {\left(a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{\sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - {\left(3 \, a c - a d + {\left(3 \, a c - a d\right)} \cos\left(f x + e\right) + {\left(3 \, a c - a d\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(d \cos\left(f x + e\right) - d \sin\left(f x + e\right) + d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left(a f \cos\left(f x + e\right) + a f \sin\left(f x + e\right) + a f\right)}}\right]"," ",0,"[-1/8*(4*sqrt(2)*(a*c - a*d + (a*c - a*d)*cos(f*x + e) + (a*c - a*d)*sin(f*x + e))*sqrt((c - d)/a)*log(-(2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) + (c - 3*d)*cos(f*x + e)^2 + (3*c - d)*cos(f*x + e) - ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) + 2*c + 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + (3*a*c - a*d + (3*a*c - a*d)*cos(f*x + e) + (3*a*c - a*d)*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 + 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(d*cos(f*x + e) - d*sin(f*x + e) + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f), -1/4*(2*sqrt(2)*(a*c - a*d + (a*c - a*d)*cos(f*x + e) + (a*c - a*d)*sin(f*x + e))*sqrt((c - d)/a)*log(-(2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) + (c - 3*d)*cos(f*x + e)^2 + (3*c - d)*cos(f*x + e) - ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) + 2*c + 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - (3*a*c - a*d + (3*a*c - a*d)*cos(f*x + e) + (3*a*c - a*d)*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) + 4*(d*cos(f*x + e) - d*sin(f*x + e) + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f), -1/8*(8*sqrt(2)*(a*c - a*d + (a*c - a*d)*cos(f*x + e) + (a*c - a*d)*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) + (3*a*c - a*d + (3*a*c - a*d)*cos(f*x + e) + (3*a*c - a*d)*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 + 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(d*cos(f*x + e) - d*sin(f*x + e) + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f), -1/4*(4*sqrt(2)*(a*c - a*d + (a*c - a*d)*cos(f*x + e) + (a*c - a*d)*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - (3*a*c - a*d + (3*a*c - a*d)*cos(f*x + e) + (3*a*c - a*d)*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) + 4*(d*cos(f*x + e) - d*sin(f*x + e) + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a*f*cos(f*x + e) + a*f*sin(f*x + e) + a*f)]","B",0
590,1,1944,0,2.866625," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{2} \sqrt{\frac{c - d}{a}} \log\left(\frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} - {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) - 2 \, c - 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right)}{4 \, f}, \frac{\sqrt{2} \sqrt{\frac{c - d}{a}} \log\left(\frac{2 \, \sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} - {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) - 2 \, c - 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right)}{2 \, f}, -\frac{4 \, \sqrt{2} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{\sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right)}{4 \, f}, -\frac{2 \, \sqrt{2} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{\sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right)}{2 \, f}\right]"," ",0,"[1/4*(2*sqrt(2)*sqrt((c - d)/a)*log((2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)))/f, 1/2*(sqrt(2)*sqrt((c - d)/a)*log((2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))))/f, -1/4*(4*sqrt(2)*sqrt(-(c - d)/a)*arctan(-sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)))/f, -1/2*(2*sqrt(2)*sqrt(-(c - d)/a)*arctan(-sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))))/f]","B",0
591,1,475,0,1.610639," ","integrate(1/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} \log\left(\frac{{\left(c^{2} - 14 \, c d + 17 \, d^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(13 \, c^{2} - 22 \, c d - 3 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - \frac{4 \, \sqrt{2} {\left({\left(c^{2} - 4 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, c^{2} + 8 \, c d - 4 \, d^{2} - {\left(3 \, c^{2} - 4 \, c d + d^{2}\right)} \cos\left(f x + e\right) + {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(c^{2} - 4 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{\sqrt{a c - a d}} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left(9 \, c^{2} - 14 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(c^{2} - 14 \, c d + 17 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} + 2 \, {\left(7 \, c^{2} - 18 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right)}{4 \, \sqrt{a c - a d} f}, \frac{\sqrt{2} \sqrt{-\frac{1}{a c - a d}} \arctan\left(-\frac{\sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{1}{a c - a d}}}{4 \, {\left(d \cos\left(f x + e\right) \sin\left(f x + e\right) + c \cos\left(f x + e\right)\right)}}\right)}{2 \, f}\right]"," ",0,"[1/4*sqrt(2)*log(((c^2 - 14*c*d + 17*d^2)*cos(f*x + e)^3 - (13*c^2 - 22*c*d - 3*d^2)*cos(f*x + e)^2 - 4*sqrt(2)*((c^2 - 4*c*d + 3*d^2)*cos(f*x + e)^2 - 4*c^2 + 8*c*d - 4*d^2 - (3*c^2 - 4*c*d + d^2)*cos(f*x + e) + (4*c^2 - 8*c*d + 4*d^2 + (c^2 - 4*c*d + 3*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/sqrt(a*c - a*d) - 4*c^2 - 8*c*d - 4*d^2 - 2*(9*c^2 - 14*c*d + 9*d^2)*cos(f*x + e) + ((c^2 - 14*c*d + 17*d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 + 2*(7*c^2 - 18*c*d + 7*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4))/(sqrt(a*c - a*d)*f), 1/2*sqrt(2)*sqrt(-1/(a*c - a*d))*arctan(-1/4*sqrt(2)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)*sqrt(-1/(a*c - a*d))/(d*cos(f*x + e)*sin(f*x + e) + c*cos(f*x + e)))/f]","A",0
592,1,1016,0,2.384747," ","integrate(1/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(d \cos\left(f x + e\right) - d \sin\left(f x + e\right) + d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} - \frac{\sqrt{2} {\left(a c^{2} + 2 \, a c d + a d^{2} - {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + a c d\right)} \cos\left(f x + e\right) + {\left(a c^{2} + 2 \, a c d + a d^{2} + {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \log\left(\frac{{\left(c^{2} - 14 \, c d + 17 \, d^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(13 \, c^{2} - 22 \, c d - 3 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + \frac{4 \, \sqrt{2} {\left({\left(c^{2} - 4 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, c^{2} + 8 \, c d - 4 \, d^{2} - {\left(3 \, c^{2} - 4 \, c d + d^{2}\right)} \cos\left(f x + e\right) + {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(c^{2} - 4 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{\sqrt{a c - a d}} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left(9 \, c^{2} - 14 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(c^{2} - 14 \, c d + 17 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} + 2 \, {\left(7 \, c^{2} - 18 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right)}{\sqrt{a c - a d}}}{4 \, {\left({\left(a c^{2} d - a d^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{3} - a c d^{2}\right)} f \cos\left(f x + e\right) - {\left(a c^{3} + a c^{2} d - a c d^{2} - a d^{3}\right)} f - {\left({\left(a c^{2} d - a d^{3}\right)} f \cos\left(f x + e\right) + {\left(a c^{3} + a c^{2} d - a c d^{2} - a d^{3}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{\sqrt{2} {\left(a c^{2} + 2 \, a c d + a d^{2} - {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + a c d\right)} \cos\left(f x + e\right) + {\left(a c^{2} + 2 \, a c d + a d^{2} + {\left(a c d + a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{1}{a c - a d}} \arctan\left(-\frac{\sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{1}{a c - a d}}}{4 \, {\left(d \cos\left(f x + e\right) \sin\left(f x + e\right) + c \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(d \cos\left(f x + e\right) - d \sin\left(f x + e\right) + d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{2 \, {\left({\left(a c^{2} d - a d^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{3} - a c d^{2}\right)} f \cos\left(f x + e\right) - {\left(a c^{3} + a c^{2} d - a c d^{2} - a d^{3}\right)} f - {\left({\left(a c^{2} d - a d^{3}\right)} f \cos\left(f x + e\right) + {\left(a c^{3} + a c^{2} d - a c d^{2} - a d^{3}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/4*(8*(d*cos(f*x + e) - d*sin(f*x + e) + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - sqrt(2)*(a*c^2 + 2*a*c*d + a*d^2 - (a*c*d + a*d^2)*cos(f*x + e)^2 + (a*c^2 + a*c*d)*cos(f*x + e) + (a*c^2 + 2*a*c*d + a*d^2 + (a*c*d + a*d^2)*cos(f*x + e))*sin(f*x + e))*log(((c^2 - 14*c*d + 17*d^2)*cos(f*x + e)^3 - (13*c^2 - 22*c*d - 3*d^2)*cos(f*x + e)^2 + 4*sqrt(2)*((c^2 - 4*c*d + 3*d^2)*cos(f*x + e)^2 - 4*c^2 + 8*c*d - 4*d^2 - (3*c^2 - 4*c*d + d^2)*cos(f*x + e) + (4*c^2 - 8*c*d + 4*d^2 + (c^2 - 4*c*d + 3*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/sqrt(a*c - a*d) - 4*c^2 - 8*c*d - 4*d^2 - 2*(9*c^2 - 14*c*d + 9*d^2)*cos(f*x + e) + ((c^2 - 14*c*d + 17*d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 + 2*(7*c^2 - 18*c*d + 7*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4))/sqrt(a*c - a*d))/((a*c^2*d - a*d^3)*f*cos(f*x + e)^2 - (a*c^3 - a*c*d^2)*f*cos(f*x + e) - (a*c^3 + a*c^2*d - a*c*d^2 - a*d^3)*f - ((a*c^2*d - a*d^3)*f*cos(f*x + e) + (a*c^3 + a*c^2*d - a*c*d^2 - a*d^3)*f)*sin(f*x + e)), -1/2*(sqrt(2)*(a*c^2 + 2*a*c*d + a*d^2 - (a*c*d + a*d^2)*cos(f*x + e)^2 + (a*c^2 + a*c*d)*cos(f*x + e) + (a*c^2 + 2*a*c*d + a*d^2 + (a*c*d + a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-1/(a*c - a*d))*arctan(-1/4*sqrt(2)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)*sqrt(-1/(a*c - a*d))/(d*cos(f*x + e)*sin(f*x + e) + c*cos(f*x + e))) + 4*(d*cos(f*x + e) - d*sin(f*x + e) + d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a*c^2*d - a*d^3)*f*cos(f*x + e)^2 - (a*c^3 - a*c*d^2)*f*cos(f*x + e) - (a*c^3 + a*c^2*d - a*c*d^2 - a*d^3)*f - ((a*c^2*d - a*d^3)*f*cos(f*x + e) + (a*c^3 + a*c^2*d - a*c*d^2 - a*d^3)*f)*sin(f*x + e))]","B",0
593,1,1855,0,2.386606," ","integrate(1/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(6 \, c^{2} d - 4 \, c d^{2} - 2 \, d^{3} + {\left(5 \, c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(6 \, c^{2} d + c d^{2} - d^{3}\right)} \cos\left(f x + e\right) - {\left(6 \, c^{2} d - 4 \, c d^{2} - 2 \, d^{3} - {\left(5 \, c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} + \frac{3 \, \sqrt{2} {\left(a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - {\left(a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(2 \, a c^{3} d + 5 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{4} + 2 \, a c^{3} d + 2 \, a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right) + {\left(a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - {\left(a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a c^{3} d + 2 \, a c^{2} d^{2} + a c d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \log\left(\frac{{\left(c^{2} - 14 \, c d + 17 \, d^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(13 \, c^{2} - 22 \, c d - 3 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - \frac{4 \, \sqrt{2} {\left({\left(c^{2} - 4 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, c^{2} + 8 \, c d - 4 \, d^{2} - {\left(3 \, c^{2} - 4 \, c d + d^{2}\right)} \cos\left(f x + e\right) + {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(c^{2} - 4 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{\sqrt{a c - a d}} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left(9 \, c^{2} - 14 \, c d + 9 \, d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(c^{2} - 14 \, c d + 17 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} + 2 \, {\left(7 \, c^{2} - 18 \, c d + 7 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right)}{\sqrt{a c - a d}}}{12 \, {\left({\left(a c^{4} d^{2} - 2 \, a c^{2} d^{4} + a d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, a c^{5} d + a c^{4} d^{2} - 4 \, a c^{3} d^{3} - 2 \, a c^{2} d^{4} + 2 \, a c d^{5} + a d^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{6} - a c^{4} d^{2} - a c^{2} d^{4} + a d^{6}\right)} f \cos\left(f x + e\right) - {\left(a c^{6} + 2 \, a c^{5} d - a c^{4} d^{2} - 4 \, a c^{3} d^{3} - a c^{2} d^{4} + 2 \, a c d^{5} + a d^{6}\right)} f + {\left({\left(a c^{4} d^{2} - 2 \, a c^{2} d^{4} + a d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a c^{5} d - 2 \, a c^{3} d^{3} + a c d^{5}\right)} f \cos\left(f x + e\right) - {\left(a c^{6} + 2 \, a c^{5} d - a c^{4} d^{2} - 4 \, a c^{3} d^{3} - a c^{2} d^{4} + 2 \, a c d^{5} + a d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{3 \, \sqrt{2} {\left(a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - {\left(a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(2 \, a c^{3} d + 5 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{4} + 2 \, a c^{3} d + 2 \, a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right) + {\left(a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - {\left(a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a c^{3} d + 2 \, a c^{2} d^{2} + a c d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{1}{a c - a d}} \arctan\left(-\frac{\sqrt{2} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{1}{a c - a d}}}{4 \, {\left(d \cos\left(f x + e\right) \sin\left(f x + e\right) + c \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(6 \, c^{2} d - 4 \, c d^{2} - 2 \, d^{3} + {\left(5 \, c d^{2} + d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(6 \, c^{2} d + c d^{2} - d^{3}\right)} \cos\left(f x + e\right) - {\left(6 \, c^{2} d - 4 \, c d^{2} - 2 \, d^{3} - {\left(5 \, c d^{2} + d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{6 \, {\left({\left(a c^{4} d^{2} - 2 \, a c^{2} d^{4} + a d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(2 \, a c^{5} d + a c^{4} d^{2} - 4 \, a c^{3} d^{3} - 2 \, a c^{2} d^{4} + 2 \, a c d^{5} + a d^{6}\right)} f \cos\left(f x + e\right)^{2} - {\left(a c^{6} - a c^{4} d^{2} - a c^{2} d^{4} + a d^{6}\right)} f \cos\left(f x + e\right) - {\left(a c^{6} + 2 \, a c^{5} d - a c^{4} d^{2} - 4 \, a c^{3} d^{3} - a c^{2} d^{4} + 2 \, a c d^{5} + a d^{6}\right)} f + {\left({\left(a c^{4} d^{2} - 2 \, a c^{2} d^{4} + a d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a c^{5} d - 2 \, a c^{3} d^{3} + a c d^{5}\right)} f \cos\left(f x + e\right) - {\left(a c^{6} + 2 \, a c^{5} d - a c^{4} d^{2} - 4 \, a c^{3} d^{3} - a c^{2} d^{4} + 2 \, a c d^{5} + a d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/12*(8*(6*c^2*d - 4*c*d^2 - 2*d^3 + (5*c*d^2 + d^3)*cos(f*x + e)^2 + (6*c^2*d + c*d^2 - d^3)*cos(f*x + e) - (6*c^2*d - 4*c*d^2 - 2*d^3 - (5*c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) + 3*sqrt(2)*(a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^3 - (2*a*c^3*d + 5*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*cos(f*x + e)^2 + (a*c^4 + 2*a*c^3*d + 2*a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^2 + 2*(a*c^3*d + 2*a*c^2*d^2 + a*c*d^3)*cos(f*x + e))*sin(f*x + e))*log(((c^2 - 14*c*d + 17*d^2)*cos(f*x + e)^3 - (13*c^2 - 22*c*d - 3*d^2)*cos(f*x + e)^2 - 4*sqrt(2)*((c^2 - 4*c*d + 3*d^2)*cos(f*x + e)^2 - 4*c^2 + 8*c*d - 4*d^2 - (3*c^2 - 4*c*d + d^2)*cos(f*x + e) + (4*c^2 - 8*c*d + 4*d^2 + (c^2 - 4*c*d + 3*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/sqrt(a*c - a*d) - 4*c^2 - 8*c*d - 4*d^2 - 2*(9*c^2 - 14*c*d + 9*d^2)*cos(f*x + e) + ((c^2 - 14*c*d + 17*d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 + 2*(7*c^2 - 18*c*d + 7*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4))/sqrt(a*c - a*d))/((a*c^4*d^2 - 2*a*c^2*d^4 + a*d^6)*f*cos(f*x + e)^3 + (2*a*c^5*d + a*c^4*d^2 - 4*a*c^3*d^3 - 2*a*c^2*d^4 + 2*a*c*d^5 + a*d^6)*f*cos(f*x + e)^2 - (a*c^6 - a*c^4*d^2 - a*c^2*d^4 + a*d^6)*f*cos(f*x + e) - (a*c^6 + 2*a*c^5*d - a*c^4*d^2 - 4*a*c^3*d^3 - a*c^2*d^4 + 2*a*c*d^5 + a*d^6)*f + ((a*c^4*d^2 - 2*a*c^2*d^4 + a*d^6)*f*cos(f*x + e)^2 - 2*(a*c^5*d - 2*a*c^3*d^3 + a*c*d^5)*f*cos(f*x + e) - (a*c^6 + 2*a*c^5*d - a*c^4*d^2 - 4*a*c^3*d^3 - a*c^2*d^4 + 2*a*c*d^5 + a*d^6)*f)*sin(f*x + e)), -1/6*(3*sqrt(2)*(a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^3 - (2*a*c^3*d + 5*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*cos(f*x + e)^2 + (a*c^4 + 2*a*c^3*d + 2*a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - (a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e)^2 + 2*(a*c^3*d + 2*a*c^2*d^2 + a*c*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-1/(a*c - a*d))*arctan(-1/4*sqrt(2)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)*sqrt(-1/(a*c - a*d))/(d*cos(f*x + e)*sin(f*x + e) + c*cos(f*x + e))) + 4*(6*c^2*d - 4*c*d^2 - 2*d^3 + (5*c*d^2 + d^3)*cos(f*x + e)^2 + (6*c^2*d + c*d^2 - d^3)*cos(f*x + e) - (6*c^2*d - 4*c*d^2 - 2*d^3 - (5*c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a*c^4*d^2 - 2*a*c^2*d^4 + a*d^6)*f*cos(f*x + e)^3 + (2*a*c^5*d + a*c^4*d^2 - 4*a*c^3*d^3 - 2*a*c^2*d^4 + 2*a*c*d^5 + a*d^6)*f*cos(f*x + e)^2 - (a*c^6 - a*c^4*d^2 - a*c^2*d^4 + a*d^6)*f*cos(f*x + e) - (a*c^6 + 2*a*c^5*d - a*c^4*d^2 - 4*a*c^3*d^3 - a*c^2*d^4 + 2*a*c*d^5 + a*d^6)*f + ((a*c^4*d^2 - 2*a*c^2*d^4 + a*d^6)*f*cos(f*x + e)^2 - 2*(a*c^5*d - 2*a*c^3*d^3 + a*c*d^5)*f*cos(f*x + e) - (a*c^6 + 2*a*c^5*d - a*c^4*d^2 - 4*a*c^3*d^3 - a*c^2*d^4 + 2*a*c*d^5 + a*d^6)*f)*sin(f*x + e))]","B",0
594,1,3420,0,3.506655," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{\frac{1}{2}} {\left(2 \, a c^{2} + 16 \, a c d - 18 \, a d^{2} - {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, a c^{2} + 16 \, a c d - 18 \, a d^{2} + {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{a}} \log\left(-\frac{4 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, c - d\right)} \cos\left(f x + e\right) - {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) + 2 \, c + 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + {\left(10 \, a c d - 6 \, a d^{2} - {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(10 \, a c d - 6 \, a d^{2} + {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 4 \, {\left(2 \, d^{2} \cos\left(f x + e\right)^{2} + c^{2} - 2 \, c d + d^{2} + {\left(c^{2} - 2 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, d^{2} \cos\left(f x + e\right) - c^{2} + 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{8 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}, \frac{\sqrt{\frac{1}{2}} {\left(2 \, a c^{2} + 16 \, a c d - 18 \, a d^{2} - {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, a c^{2} + 16 \, a c d - 18 \, a d^{2} + {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{a}} \log\left(-\frac{4 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} + {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, c - d\right)} \cos\left(f x + e\right) - {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) + 2 \, c + 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) - {\left(10 \, a c d - 6 \, a d^{2} - {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(10 \, a c d - 6 \, a d^{2} + {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 2 \, {\left(2 \, d^{2} \cos\left(f x + e\right)^{2} + c^{2} - 2 \, c d + d^{2} + {\left(c^{2} - 2 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, d^{2} \cos\left(f x + e\right) - c^{2} + 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}, \frac{4 \, \sqrt{\frac{1}{2}} {\left(2 \, a c^{2} + 16 \, a c d - 18 \, a d^{2} - {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, a c^{2} + 16 \, a c d - 18 \, a d^{2} + {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) + {\left(10 \, a c d - 6 \, a d^{2} - {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(10 \, a c d - 6 \, a d^{2} + {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 4 \, {\left(2 \, d^{2} \cos\left(f x + e\right)^{2} + c^{2} - 2 \, c d + d^{2} + {\left(c^{2} - 2 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, d^{2} \cos\left(f x + e\right) - c^{2} + 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{8 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}, \frac{2 \, \sqrt{\frac{1}{2}} {\left(2 \, a c^{2} + 16 \, a c d - 18 \, a d^{2} - {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, a c^{2} + 16 \, a c d - 18 \, a d^{2} + {\left(a c^{2} + 8 \, a c d - 9 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - {\left(10 \, a c d - 6 \, a d^{2} - {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right) + {\left(10 \, a c d - 6 \, a d^{2} + {\left(5 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 2 \, {\left(2 \, d^{2} \cos\left(f x + e\right)^{2} + c^{2} - 2 \, c d + d^{2} + {\left(c^{2} - 2 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right) + {\left(2 \, d^{2} \cos\left(f x + e\right) - c^{2} + 2 \, c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/8*(2*sqrt(1/2)*(2*a*c^2 + 16*a*c*d - 18*a*d^2 - (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e)^2 + (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e) + (2*a*c^2 + 16*a*c*d - 18*a*d^2 + (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt((c - d)/a)*log(-(4*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) + (c - 3*d)*cos(f*x + e)^2 + (3*c - d)*cos(f*x + e) - ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) + 2*c + 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + (10*a*c*d - 6*a*d^2 - (5*a*c*d - 3*a*d^2)*cos(f*x + e)^2 + (5*a*c*d - 3*a*d^2)*cos(f*x + e) + (10*a*c*d - 6*a*d^2 + (5*a*c*d - 3*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 + 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 4*(2*d^2*cos(f*x + e)^2 + c^2 - 2*c*d + d^2 + (c^2 - 2*c*d + 3*d^2)*cos(f*x + e) + (2*d^2*cos(f*x + e) - c^2 + 2*c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e)), 1/4*(sqrt(1/2)*(2*a*c^2 + 16*a*c*d - 18*a*d^2 - (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e)^2 + (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e) + (2*a*c^2 + 16*a*c*d - 18*a*d^2 + (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt((c - d)/a)*log(-(4*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) + (c - 3*d)*cos(f*x + e)^2 + (3*c - d)*cos(f*x + e) - ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) + 2*c + 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) - (10*a*c*d - 6*a*d^2 - (5*a*c*d - 3*a*d^2)*cos(f*x + e)^2 + (5*a*c*d - 3*a*d^2)*cos(f*x + e) + (10*a*c*d - 6*a*d^2 + (5*a*c*d - 3*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) + 2*(2*d^2*cos(f*x + e)^2 + c^2 - 2*c*d + d^2 + (c^2 - 2*c*d + 3*d^2)*cos(f*x + e) + (2*d^2*cos(f*x + e) - c^2 + 2*c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e)), 1/8*(4*sqrt(1/2)*(2*a*c^2 + 16*a*c*d - 18*a*d^2 - (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e)^2 + (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e) + (2*a*c^2 + 16*a*c*d - 18*a*d^2 + (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-2*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) + (10*a*c*d - 6*a*d^2 - (5*a*c*d - 3*a*d^2)*cos(f*x + e)^2 + (5*a*c*d - 3*a*d^2)*cos(f*x + e) + (10*a*c*d - 6*a*d^2 + (5*a*c*d - 3*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 + 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 4*(2*d^2*cos(f*x + e)^2 + c^2 - 2*c*d + d^2 + (c^2 - 2*c*d + 3*d^2)*cos(f*x + e) + (2*d^2*cos(f*x + e) - c^2 + 2*c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e)), 1/4*(2*sqrt(1/2)*(2*a*c^2 + 16*a*c*d - 18*a*d^2 - (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e)^2 + (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e) + (2*a*c^2 + 16*a*c*d - 18*a*d^2 + (a*c^2 + 8*a*c*d - 9*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-2*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - (10*a*c*d - 6*a*d^2 - (5*a*c*d - 3*a*d^2)*cos(f*x + e)^2 + (5*a*c*d - 3*a*d^2)*cos(f*x + e) + (10*a*c*d - 6*a*d^2 + (5*a*c*d - 3*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) + 2*(2*d^2*cos(f*x + e)^2 + c^2 - 2*c*d + d^2 + (c^2 - 2*c*d + 3*d^2)*cos(f*x + e) + (2*d^2*cos(f*x + e) - c^2 + 2*c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))]","B",0
595,1,2883,0,2.866137," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{\frac{1}{2}} {\left({\left(a c + 5 \, a d\right)} \cos\left(f x + e\right)^{2} - 2 \, a c - 10 \, a d - {\left(a c + 5 \, a d\right)} \cos\left(f x + e\right) - {\left(2 \, a c + 10 \, a d + {\left(a c + 5 \, a d\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{a}} \log\left(\frac{4 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} - {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) - 2 \, c - 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + {\left(a d \cos\left(f x + e\right)^{2} - a d \cos\left(f x + e\right) - 2 \, a d - {\left(a d \cos\left(f x + e\right) + 2 \, a d\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 2 \, {\left({\left(c - d\right)} \cos\left(f x + e\right) - {\left(c - d\right)} \sin\left(f x + e\right) + c - d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}, \frac{\sqrt{\frac{1}{2}} {\left({\left(a c + 5 \, a d\right)} \cos\left(f x + e\right)^{2} - 2 \, a c - 10 \, a d - {\left(a c + 5 \, a d\right)} \cos\left(f x + e\right) - {\left(2 \, a c + 10 \, a d + {\left(a c + 5 \, a d\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{a}} \log\left(\frac{4 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} - {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) - 2 \, c - 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 2 \, {\left(a d \cos\left(f x + e\right)^{2} - a d \cos\left(f x + e\right) - 2 \, a d - {\left(a d \cos\left(f x + e\right) + 2 \, a d\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 2 \, {\left({\left(c - d\right)} \cos\left(f x + e\right) - {\left(c - d\right)} \sin\left(f x + e\right) + c - d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{2 \, \sqrt{\frac{1}{2}} {\left({\left(a c + 5 \, a d\right)} \cos\left(f x + e\right)^{2} - 2 \, a c - 10 \, a d - {\left(a c + 5 \, a d\right)} \cos\left(f x + e\right) - {\left(2 \, a c + 10 \, a d + {\left(a c + 5 \, a d\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - {\left(a d \cos\left(f x + e\right)^{2} - a d \cos\left(f x + e\right) - 2 \, a d - {\left(a d \cos\left(f x + e\right) + 2 \, a d\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) - 2 \, {\left({\left(c - d\right)} \cos\left(f x + e\right) - {\left(c - d\right)} \sin\left(f x + e\right) + c - d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{\sqrt{\frac{1}{2}} {\left({\left(a c + 5 \, a d\right)} \cos\left(f x + e\right)^{2} - 2 \, a c - 10 \, a d - {\left(a c + 5 \, a d\right)} \cos\left(f x + e\right) - {\left(2 \, a c + 10 \, a d + {\left(a c + 5 \, a d\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - {\left(a d \cos\left(f x + e\right)^{2} - a d \cos\left(f x + e\right) - 2 \, a d - {\left(a d \cos\left(f x + e\right) + 2 \, a d\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right) - {\left({\left(c - d\right)} \cos\left(f x + e\right) - {\left(c - d\right)} \sin\left(f x + e\right) + c - d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{2 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - 2 \, a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + 2 \, a^{2} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/4*(sqrt(1/2)*((a*c + 5*a*d)*cos(f*x + e)^2 - 2*a*c - 10*a*d - (a*c + 5*a*d)*cos(f*x + e) - (2*a*c + 10*a*d + (a*c + 5*a*d)*cos(f*x + e))*sin(f*x + e))*sqrt((c - d)/a)*log((4*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + (a*d*cos(f*x + e)^2 - a*d*cos(f*x + e) - 2*a*d - (a*d*cos(f*x + e) + 2*a*d)*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 2*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e)), 1/4*(sqrt(1/2)*((a*c + 5*a*d)*cos(f*x + e)^2 - 2*a*c - 10*a*d - (a*c + 5*a*d)*cos(f*x + e) - (2*a*c + 10*a*d + (a*c + 5*a*d)*cos(f*x + e))*sin(f*x + e))*sqrt((c - d)/a)*log((4*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 2*(a*d*cos(f*x + e)^2 - a*d*cos(f*x + e) - 2*a*d - (a*d*cos(f*x + e) + 2*a*d)*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) + 2*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e)), -1/4*(2*sqrt(1/2)*((a*c + 5*a*d)*cos(f*x + e)^2 - 2*a*c - 10*a*d - (a*c + 5*a*d)*cos(f*x + e) - (2*a*c + 10*a*d + (a*c + 5*a*d)*cos(f*x + e))*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-2*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - (a*d*cos(f*x + e)^2 - a*d*cos(f*x + e) - 2*a*d - (a*d*cos(f*x + e) + 2*a*d)*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) - 2*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e)), -1/2*(sqrt(1/2)*((a*c + 5*a*d)*cos(f*x + e)^2 - 2*a*c - 10*a*d - (a*c + 5*a*d)*cos(f*x + e) - (2*a*c + 10*a*d + (a*c + 5*a*d)*cos(f*x + e))*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-2*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - (a*d*cos(f*x + e)^2 - a*d*cos(f*x + e) - 2*a*d - (a*d*cos(f*x + e) + 2*a*d)*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) - ((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))]","B",0
596,1,896,0,2.158526," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(c + d\right)} \cos\left(f x + e\right)^{2} - {\left(c + d\right)} \cos\left(f x + e\right) - {\left({\left(c + d\right)} \cos\left(f x + e\right) + 2 \, c + 2 \, d\right)} \sin\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sqrt{2 \, a c - 2 \, a d} \log\left(\frac{{\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 4 \, a c^{2} - 8 \, a c d - 4 \, a d^{2} - {\left(13 \, a c^{2} - 22 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) + 4 \, c - 4 \, d\right)} \sin\left(f x + e\right) - 4 \, c + 4 \, d\right)} \sqrt{2 \, a c - 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} - 2 \, {\left(9 \, a c^{2} - 14 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, a c^{2} + 8 \, a c d + 4 \, a d^{2} - {\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(7 \, a c^{2} - 18 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right) + 8 \, {\left({\left(c - d\right)} \cos\left(f x + e\right) - {\left(c - d\right)} \sin\left(f x + e\right) + c - d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{16 \, {\left({\left(a^{2} c - a^{2} d\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c - a^{2} d\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c - a^{2} d\right)} f - {\left({\left(a^{2} c - a^{2} d\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c - a^{2} d\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left({\left(c + d\right)} \cos\left(f x + e\right)^{2} - {\left(c + d\right)} \cos\left(f x + e\right) - {\left({\left(c + d\right)} \cos\left(f x + e\right) + 2 \, c + 2 \, d\right)} \sin\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sqrt{-2 \, a c + 2 \, a d} \arctan\left(\frac{\sqrt{-2 \, a c + 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left({\left(a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a c^{2} - a c d\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left({\left(c - d\right)} \cos\left(f x + e\right) - {\left(c - d\right)} \sin\left(f x + e\right) + c - d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{8 \, {\left({\left(a^{2} c - a^{2} d\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c - a^{2} d\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c - a^{2} d\right)} f - {\left({\left(a^{2} c - a^{2} d\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c - a^{2} d\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/16*(((c + d)*cos(f*x + e)^2 - (c + d)*cos(f*x + e) - ((c + d)*cos(f*x + e) + 2*c + 2*d)*sin(f*x + e) - 2*c - 2*d)*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 - 4*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^2 - 2*(7*a*c^2 - 18*a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) + 8*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^2*c - a^2*d)*f*cos(f*x + e)^2 - (a^2*c - a^2*d)*f*cos(f*x + e) - 2*(a^2*c - a^2*d)*f - ((a^2*c - a^2*d)*f*cos(f*x + e) + 2*(a^2*c - a^2*d)*f)*sin(f*x + e)), -1/8*(((c + d)*cos(f*x + e)^2 - (c + d)*cos(f*x + e) - ((c + d)*cos(f*x + e) + 2*c + 2*d)*sin(f*x + e) - 2*c - 2*d)*sqrt(-2*a*c + 2*a*d)*arctan(1/4*sqrt(-2*a*c + 2*a*d)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)/((a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) + (a*c^2 - a*c*d)*cos(f*x + e))) - 4*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^2*c - a^2*d)*f*cos(f*x + e)^2 - (a^2*c - a^2*d)*f*cos(f*x + e) - 2*(a^2*c - a^2*d)*f - ((a^2*c - a^2*d)*f*cos(f*x + e) + 2*(a^2*c - a^2*d)*f)*sin(f*x + e))]","B",0
597,1,1008,0,2.945355," ","integrate(1/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(c - 3 \, d\right)} \cos\left(f x + e\right) - {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) + 2 \, c - 6 \, d\right)} \sin\left(f x + e\right) - 2 \, c + 6 \, d\right)} \sqrt{2 \, a c - 2 \, a d} \log\left(\frac{{\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 4 \, a c^{2} - 8 \, a c d - 4 \, a d^{2} - {\left(13 \, a c^{2} - 22 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) + 4 \, c - 4 \, d\right)} \sin\left(f x + e\right) - 4 \, c + 4 \, d\right)} \sqrt{2 \, a c - 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} - 2 \, {\left(9 \, a c^{2} - 14 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, a c^{2} + 8 \, a c d + 4 \, a d^{2} - {\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(7 \, a c^{2} - 18 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right) + 8 \, {\left({\left(c - d\right)} \cos\left(f x + e\right) - {\left(c - d\right)} \sin\left(f x + e\right) + c - d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{16 \, {\left({\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f - {\left({\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(c - 3 \, d\right)} \cos\left(f x + e\right) - {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) + 2 \, c - 6 \, d\right)} \sin\left(f x + e\right) - 2 \, c + 6 \, d\right)} \sqrt{-2 \, a c + 2 \, a d} \arctan\left(\frac{\sqrt{-2 \, a c + 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left({\left(a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a c^{2} - a c d\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left({\left(c - d\right)} \cos\left(f x + e\right) - {\left(c - d\right)} \sin\left(f x + e\right) + c - d\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{8 \, {\left({\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f - {\left({\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{2} - 2 \, a^{2} c d + a^{2} d^{2}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/16*(((c - 3*d)*cos(f*x + e)^2 - (c - 3*d)*cos(f*x + e) - ((c - 3*d)*cos(f*x + e) + 2*c - 6*d)*sin(f*x + e) - 2*c + 6*d)*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 - 4*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^2 - 2*(7*a*c^2 - 18*a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) + 8*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e)^2 - (a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e) - 2*(a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f - ((a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e) + 2*(a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f)*sin(f*x + e)), -1/8*(((c - 3*d)*cos(f*x + e)^2 - (c - 3*d)*cos(f*x + e) - ((c - 3*d)*cos(f*x + e) + 2*c - 6*d)*sin(f*x + e) - 2*c + 6*d)*sqrt(-2*a*c + 2*a*d)*arctan(1/4*sqrt(-2*a*c + 2*a*d)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)/((a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) + (a*c^2 - a*c*d)*cos(f*x + e))) - 4*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e)^2 - (a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e) - 2*(a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f - ((a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f*cos(f*x + e) + 2*(a^2*c^2 - 2*a^2*c*d + a^2*d^2)*f)*sin(f*x + e))]","B",0
598,1,1954,0,3.707115," ","integrate(1/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(c^{2} d - 6 \, c d^{2} - 7 \, d^{3}\right)} \cos\left(f x + e\right)^{3} - 2 \, c^{3} + 10 \, c^{2} d + 26 \, c d^{2} + 14 \, d^{3} + {\left(c^{3} - 4 \, c^{2} d - 19 \, c d^{2} - 14 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 5 \, c^{2} d - 13 \, c d^{2} - 7 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(2 \, c^{3} - 10 \, c^{2} d - 26 \, c d^{2} - 14 \, d^{3} - {\left(c^{2} d - 6 \, c d^{2} - 7 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{3} - 5 \, c^{2} d - 13 \, c d^{2} - 7 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{2 \, a c - 2 \, a d} \log\left(\frac{{\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 4 \, a c^{2} - 8 \, a c d - 4 \, a d^{2} - {\left(13 \, a c^{2} - 22 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) + 4 \, c - 4 \, d\right)} \sin\left(f x + e\right) - 4 \, c + 4 \, d\right)} \sqrt{2 \, a c - 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} - 2 \, {\left(9 \, a c^{2} - 14 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, a c^{2} + 8 \, a c d + 4 \, a d^{2} - {\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(7 \, a c^{2} - 18 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right) - 8 \, {\left(c^{3} - c^{2} d - c d^{2} + d^{3} + {\left(c^{2} d + 4 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{3} + 3 \, c d^{2} - 4 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{3} - c^{2} d - c d^{2} + d^{3} - {\left(c^{2} d + 4 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{16 \, {\left({\left(a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{2} c^{5} - 4 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + 3 \, a^{2} c d^{4} - 2 \, a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f + {\left({\left(a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left({\left(c^{2} d - 6 \, c d^{2} - 7 \, d^{3}\right)} \cos\left(f x + e\right)^{3} - 2 \, c^{3} + 10 \, c^{2} d + 26 \, c d^{2} + 14 \, d^{3} + {\left(c^{3} - 4 \, c^{2} d - 19 \, c d^{2} - 14 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 5 \, c^{2} d - 13 \, c d^{2} - 7 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(2 \, c^{3} - 10 \, c^{2} d - 26 \, c d^{2} - 14 \, d^{3} - {\left(c^{2} d - 6 \, c d^{2} - 7 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{3} - 5 \, c^{2} d - 13 \, c d^{2} - 7 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-2 \, a c + 2 \, a d} \arctan\left(\frac{\sqrt{-2 \, a c + 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left({\left(a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a c^{2} - a c d\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left(c^{3} - c^{2} d - c d^{2} + d^{3} + {\left(c^{2} d + 4 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{3} + 3 \, c d^{2} - 4 \, d^{3}\right)} \cos\left(f x + e\right) - {\left(c^{3} - c^{2} d - c d^{2} + d^{3} - {\left(c^{2} d + 4 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{8 \, {\left({\left(a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{2} c^{5} - 4 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + 3 \, a^{2} c d^{4} - 2 \, a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f + {\left({\left(a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{5} - a^{2} c^{4} d - 2 \, a^{2} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + a^{2} c d^{4} - a^{2} d^{5}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/16*(((c^2*d - 6*c*d^2 - 7*d^3)*cos(f*x + e)^3 - 2*c^3 + 10*c^2*d + 26*c*d^2 + 14*d^3 + (c^3 - 4*c^2*d - 19*c*d^2 - 14*d^3)*cos(f*x + e)^2 - (c^3 - 5*c^2*d - 13*c*d^2 - 7*d^3)*cos(f*x + e) - (2*c^3 - 10*c^2*d - 26*c*d^2 - 14*d^3 - (c^2*d - 6*c*d^2 - 7*d^3)*cos(f*x + e)^2 + (c^3 - 5*c^2*d - 13*c*d^2 - 7*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 + 4*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^2 - 2*(7*a*c^2 - 18*a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) - 8*(c^3 - c^2*d - c*d^2 + d^3 + (c^2*d + 4*c*d^2 - 5*d^3)*cos(f*x + e)^2 + (c^3 + 3*c*d^2 - 4*d^3)*cos(f*x + e) - (c^3 - c^2*d - c*d^2 + d^3 - (c^2*d + 4*c*d^2 - 5*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e)^3 + (a^2*c^5 - 4*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + 3*a^2*c*d^4 - 2*a^2*d^5)*f*cos(f*x + e)^2 - (a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e) - 2*(a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f + ((a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e)^2 - (a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e) - 2*(a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f)*sin(f*x + e)), -1/8*(((c^2*d - 6*c*d^2 - 7*d^3)*cos(f*x + e)^3 - 2*c^3 + 10*c^2*d + 26*c*d^2 + 14*d^3 + (c^3 - 4*c^2*d - 19*c*d^2 - 14*d^3)*cos(f*x + e)^2 - (c^3 - 5*c^2*d - 13*c*d^2 - 7*d^3)*cos(f*x + e) - (2*c^3 - 10*c^2*d - 26*c*d^2 - 14*d^3 - (c^2*d - 6*c*d^2 - 7*d^3)*cos(f*x + e)^2 + (c^3 - 5*c^2*d - 13*c*d^2 - 7*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-2*a*c + 2*a*d)*arctan(1/4*sqrt(-2*a*c + 2*a*d)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)/((a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) + (a*c^2 - a*c*d)*cos(f*x + e))) - 4*(c^3 - c^2*d - c*d^2 + d^3 + (c^2*d + 4*c*d^2 - 5*d^3)*cos(f*x + e)^2 + (c^3 + 3*c*d^2 - 4*d^3)*cos(f*x + e) - (c^3 - c^2*d - c*d^2 + d^3 - (c^2*d + 4*c*d^2 - 5*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e)^3 + (a^2*c^5 - 4*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + 3*a^2*c*d^4 - 2*a^2*d^5)*f*cos(f*x + e)^2 - (a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e) - 2*(a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f + ((a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e)^2 - (a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f*cos(f*x + e) - 2*(a^2*c^5 - a^2*c^4*d - 2*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + a^2*c*d^4 - a^2*d^5)*f)*sin(f*x + e))]","B",0
599,1,3182,0,4.739012," ","integrate(1/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(2 \, c^{5} - 14 \, c^{4} d - 76 \, c^{3} d^{2} - 124 \, c^{2} d^{3} - 86 \, c d^{4} - 22 \, d^{5} + {\left(c^{3} d^{2} - 9 \, c^{2} d^{3} - 21 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, c^{4} d - 17 \, c^{3} d^{2} - 51 \, c^{2} d^{3} - 43 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(c^{5} - 5 \, c^{4} d - 54 \, c^{3} d^{2} - 122 \, c^{2} d^{3} - 107 \, c d^{4} - 33 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{5} - 7 \, c^{4} d - 38 \, c^{3} d^{2} - 62 \, c^{2} d^{3} - 43 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right) + {\left(2 \, c^{5} - 14 \, c^{4} d - 76 \, c^{3} d^{2} - 124 \, c^{2} d^{3} - 86 \, c d^{4} - 22 \, d^{5} - {\left(c^{3} d^{2} - 9 \, c^{2} d^{3} - 21 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(c^{4} d - 8 \, c^{3} d^{2} - 30 \, c^{2} d^{3} - 32 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{5} - 7 \, c^{4} d - 38 \, c^{3} d^{2} - 62 \, c^{2} d^{3} - 43 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{2 \, a c - 2 \, a d} \log\left(\frac{{\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 4 \, a c^{2} - 8 \, a c d - 4 \, a d^{2} - {\left(13 \, a c^{2} - 22 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) + 4 \, c - 4 \, d\right)} \sin\left(f x + e\right) - 4 \, c + 4 \, d\right)} \sqrt{2 \, a c - 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} - 2 \, {\left(9 \, a c^{2} - 14 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, a c^{2} + 8 \, a c d + 4 \, a d^{2} - {\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(7 \, a c^{2} - 18 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right) - 8 \, {\left(3 \, c^{5} - 3 \, c^{4} d - 6 \, c^{3} d^{2} + 6 \, c^{2} d^{3} + 3 \, c d^{4} - 3 \, d^{5} - {\left(3 \, c^{3} d^{2} + 35 \, c^{2} d^{3} - 19 \, c d^{4} - 19 \, d^{5}\right)} \cos\left(f x + e\right)^{3} + {\left(6 \, c^{4} d + 39 \, c^{3} d^{2} - 29 \, c^{2} d^{3} - 23 \, c d^{4} + 7 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + 3 \, {\left(c^{5} + c^{4} d + 12 \, c^{3} d^{2} + 4 \, c^{2} d^{3} - 13 \, c d^{4} - 5 \, d^{5}\right)} \cos\left(f x + e\right) - {\left(3 \, c^{5} - 3 \, c^{4} d - 6 \, c^{3} d^{2} + 6 \, c^{2} d^{3} + 3 \, c d^{4} - 3 \, d^{5} - {\left(3 \, c^{3} d^{2} + 35 \, c^{2} d^{3} - 19 \, c d^{4} - 19 \, d^{5}\right)} \cos\left(f x + e\right)^{2} - 6 \, {\left(c^{4} d + 7 \, c^{3} d^{2} + c^{2} d^{3} - 7 \, c d^{4} - 2 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{48 \, {\left({\left(a^{2} c^{6} d^{2} - 2 \, a^{2} c^{5} d^{3} - a^{2} c^{4} d^{4} + 4 \, a^{2} c^{3} d^{5} - a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{2} c^{7} d - 3 \, a^{2} c^{6} d^{2} - 4 \, a^{2} c^{5} d^{3} + 7 \, a^{2} c^{4} d^{4} + 2 \, a^{2} c^{3} d^{5} - 5 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{2} c^{8} + 2 \, a^{2} c^{7} d - 6 \, a^{2} c^{6} d^{2} - 6 \, a^{2} c^{5} d^{3} + 12 \, a^{2} c^{4} d^{4} + 6 \, a^{2} c^{3} d^{5} - 10 \, a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + 3 \, a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f - {\left({\left(a^{2} c^{6} d^{2} - 2 \, a^{2} c^{5} d^{3} - a^{2} c^{4} d^{4} + 4 \, a^{2} c^{3} d^{5} - a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{3} + 2 \, {\left(a^{2} c^{7} d - a^{2} c^{6} d^{2} - 3 \, a^{2} c^{5} d^{3} + 3 \, a^{2} c^{4} d^{4} + 3 \, a^{2} c^{3} d^{5} - 3 \, a^{2} c^{2} d^{6} - a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{3 \, {\left(2 \, c^{5} - 14 \, c^{4} d - 76 \, c^{3} d^{2} - 124 \, c^{2} d^{3} - 86 \, c d^{4} - 22 \, d^{5} + {\left(c^{3} d^{2} - 9 \, c^{2} d^{3} - 21 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, c^{4} d - 17 \, c^{3} d^{2} - 51 \, c^{2} d^{3} - 43 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(c^{5} - 5 \, c^{4} d - 54 \, c^{3} d^{2} - 122 \, c^{2} d^{3} - 107 \, c d^{4} - 33 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{5} - 7 \, c^{4} d - 38 \, c^{3} d^{2} - 62 \, c^{2} d^{3} - 43 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right) + {\left(2 \, c^{5} - 14 \, c^{4} d - 76 \, c^{3} d^{2} - 124 \, c^{2} d^{3} - 86 \, c d^{4} - 22 \, d^{5} - {\left(c^{3} d^{2} - 9 \, c^{2} d^{3} - 21 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(c^{4} d - 8 \, c^{3} d^{2} - 30 \, c^{2} d^{3} - 32 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(c^{5} - 7 \, c^{4} d - 38 \, c^{3} d^{2} - 62 \, c^{2} d^{3} - 43 \, c d^{4} - 11 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-2 \, a c + 2 \, a d} \arctan\left(\frac{\sqrt{-2 \, a c + 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left({\left(a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a c^{2} - a c d\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(3 \, c^{5} - 3 \, c^{4} d - 6 \, c^{3} d^{2} + 6 \, c^{2} d^{3} + 3 \, c d^{4} - 3 \, d^{5} - {\left(3 \, c^{3} d^{2} + 35 \, c^{2} d^{3} - 19 \, c d^{4} - 19 \, d^{5}\right)} \cos\left(f x + e\right)^{3} + {\left(6 \, c^{4} d + 39 \, c^{3} d^{2} - 29 \, c^{2} d^{3} - 23 \, c d^{4} + 7 \, d^{5}\right)} \cos\left(f x + e\right)^{2} + 3 \, {\left(c^{5} + c^{4} d + 12 \, c^{3} d^{2} + 4 \, c^{2} d^{3} - 13 \, c d^{4} - 5 \, d^{5}\right)} \cos\left(f x + e\right) - {\left(3 \, c^{5} - 3 \, c^{4} d - 6 \, c^{3} d^{2} + 6 \, c^{2} d^{3} + 3 \, c d^{4} - 3 \, d^{5} - {\left(3 \, c^{3} d^{2} + 35 \, c^{2} d^{3} - 19 \, c d^{4} - 19 \, d^{5}\right)} \cos\left(f x + e\right)^{2} - 6 \, {\left(c^{4} d + 7 \, c^{3} d^{2} + c^{2} d^{3} - 7 \, c d^{4} - 2 \, d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{24 \, {\left({\left(a^{2} c^{6} d^{2} - 2 \, a^{2} c^{5} d^{3} - a^{2} c^{4} d^{4} + 4 \, a^{2} c^{3} d^{5} - a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{4} - {\left(2 \, a^{2} c^{7} d - 3 \, a^{2} c^{6} d^{2} - 4 \, a^{2} c^{5} d^{3} + 7 \, a^{2} c^{4} d^{4} + 2 \, a^{2} c^{3} d^{5} - 5 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{2} c^{8} + 2 \, a^{2} c^{7} d - 6 \, a^{2} c^{6} d^{2} - 6 \, a^{2} c^{5} d^{3} + 12 \, a^{2} c^{4} d^{4} + 6 \, a^{2} c^{3} d^{5} - 10 \, a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + 3 \, a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right) + 2 \, {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f - {\left({\left(a^{2} c^{6} d^{2} - 2 \, a^{2} c^{5} d^{3} - a^{2} c^{4} d^{4} + 4 \, a^{2} c^{3} d^{5} - a^{2} c^{2} d^{6} - 2 \, a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{3} + 2 \, {\left(a^{2} c^{7} d - a^{2} c^{6} d^{2} - 3 \, a^{2} c^{5} d^{3} + 3 \, a^{2} c^{4} d^{4} + 3 \, a^{2} c^{3} d^{5} - 3 \, a^{2} c^{2} d^{6} - a^{2} c d^{7} + a^{2} d^{8}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f \cos\left(f x + e\right) - 2 \, {\left(a^{2} c^{8} - 4 \, a^{2} c^{6} d^{2} + 6 \, a^{2} c^{4} d^{4} - 4 \, a^{2} c^{2} d^{6} + a^{2} d^{8}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/48*(3*(2*c^5 - 14*c^4*d - 76*c^3*d^2 - 124*c^2*d^3 - 86*c*d^4 - 22*d^5 + (c^3*d^2 - 9*c^2*d^3 - 21*c*d^4 - 11*d^5)*cos(f*x + e)^4 - (2*c^4*d - 17*c^3*d^2 - 51*c^2*d^3 - 43*c*d^4 - 11*d^5)*cos(f*x + e)^3 - (c^5 - 5*c^4*d - 54*c^3*d^2 - 122*c^2*d^3 - 107*c*d^4 - 33*d^5)*cos(f*x + e)^2 + (c^5 - 7*c^4*d - 38*c^3*d^2 - 62*c^2*d^3 - 43*c*d^4 - 11*d^5)*cos(f*x + e) + (2*c^5 - 14*c^4*d - 76*c^3*d^2 - 124*c^2*d^3 - 86*c*d^4 - 22*d^5 - (c^3*d^2 - 9*c^2*d^3 - 21*c*d^4 - 11*d^5)*cos(f*x + e)^3 - 2*(c^4*d - 8*c^3*d^2 - 30*c^2*d^3 - 32*c*d^4 - 11*d^5)*cos(f*x + e)^2 + (c^5 - 7*c^4*d - 38*c^3*d^2 - 62*c^2*d^3 - 43*c*d^4 - 11*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 - 4*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^2 - 2*(7*a*c^2 - 18*a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) - 8*(3*c^5 - 3*c^4*d - 6*c^3*d^2 + 6*c^2*d^3 + 3*c*d^4 - 3*d^5 - (3*c^3*d^2 + 35*c^2*d^3 - 19*c*d^4 - 19*d^5)*cos(f*x + e)^3 + (6*c^4*d + 39*c^3*d^2 - 29*c^2*d^3 - 23*c*d^4 + 7*d^5)*cos(f*x + e)^2 + 3*(c^5 + c^4*d + 12*c^3*d^2 + 4*c^2*d^3 - 13*c*d^4 - 5*d^5)*cos(f*x + e) - (3*c^5 - 3*c^4*d - 6*c^3*d^2 + 6*c^2*d^3 + 3*c*d^4 - 3*d^5 - (3*c^3*d^2 + 35*c^2*d^3 - 19*c*d^4 - 19*d^5)*cos(f*x + e)^2 - 6*(c^4*d + 7*c^3*d^2 + c^2*d^3 - 7*c*d^4 - 2*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^2*c^6*d^2 - 2*a^2*c^5*d^3 - a^2*c^4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^4 - (2*a^2*c^7*d - 3*a^2*c^6*d^2 - 4*a^2*c^5*d^3 + 7*a^2*c^4*d^4 + 2*a^2*c^3*d^5 - 5*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e)^3 - (a^2*c^8 + 2*a^2*c^7*d - 6*a^2*c^6*d^2 - 6*a^2*c^5*d^3 + 12*a^2*c^4*d^4 + 6*a^2*c^3*d^5 - 10*a^2*c^2*d^6 - 2*a^2*c*d^7 + 3*a^2*d^8)*f*cos(f*x + e)^2 + (a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e) + 2*(a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f - ((a^2*c^6*d^2 - 2*a^2*c^5*d^3 - a^2*c^4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^3 + 2*(a^2*c^7*d - a^2*c^6*d^2 - 3*a^2*c^5*d^3 + 3*a^2*c^4*d^4 + 3*a^2*c^3*d^5 - 3*a^2*c^2*d^6 - a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^2 - (a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e) - 2*(a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f)*sin(f*x + e)), -1/24*(3*(2*c^5 - 14*c^4*d - 76*c^3*d^2 - 124*c^2*d^3 - 86*c*d^4 - 22*d^5 + (c^3*d^2 - 9*c^2*d^3 - 21*c*d^4 - 11*d^5)*cos(f*x + e)^4 - (2*c^4*d - 17*c^3*d^2 - 51*c^2*d^3 - 43*c*d^4 - 11*d^5)*cos(f*x + e)^3 - (c^5 - 5*c^4*d - 54*c^3*d^2 - 122*c^2*d^3 - 107*c*d^4 - 33*d^5)*cos(f*x + e)^2 + (c^5 - 7*c^4*d - 38*c^3*d^2 - 62*c^2*d^3 - 43*c*d^4 - 11*d^5)*cos(f*x + e) + (2*c^5 - 14*c^4*d - 76*c^3*d^2 - 124*c^2*d^3 - 86*c*d^4 - 22*d^5 - (c^3*d^2 - 9*c^2*d^3 - 21*c*d^4 - 11*d^5)*cos(f*x + e)^3 - 2*(c^4*d - 8*c^3*d^2 - 30*c^2*d^3 - 32*c*d^4 - 11*d^5)*cos(f*x + e)^2 + (c^5 - 7*c^4*d - 38*c^3*d^2 - 62*c^2*d^3 - 43*c*d^4 - 11*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(-2*a*c + 2*a*d)*arctan(1/4*sqrt(-2*a*c + 2*a*d)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)/((a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) + (a*c^2 - a*c*d)*cos(f*x + e))) + 4*(3*c^5 - 3*c^4*d - 6*c^3*d^2 + 6*c^2*d^3 + 3*c*d^4 - 3*d^5 - (3*c^3*d^2 + 35*c^2*d^3 - 19*c*d^4 - 19*d^5)*cos(f*x + e)^3 + (6*c^4*d + 39*c^3*d^2 - 29*c^2*d^3 - 23*c*d^4 + 7*d^5)*cos(f*x + e)^2 + 3*(c^5 + c^4*d + 12*c^3*d^2 + 4*c^2*d^3 - 13*c*d^4 - 5*d^5)*cos(f*x + e) - (3*c^5 - 3*c^4*d - 6*c^3*d^2 + 6*c^2*d^3 + 3*c*d^4 - 3*d^5 - (3*c^3*d^2 + 35*c^2*d^3 - 19*c*d^4 - 19*d^5)*cos(f*x + e)^2 - 6*(c^4*d + 7*c^3*d^2 + c^2*d^3 - 7*c*d^4 - 2*d^5)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^2*c^6*d^2 - 2*a^2*c^5*d^3 - a^2*c^4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^4 - (2*a^2*c^7*d - 3*a^2*c^6*d^2 - 4*a^2*c^5*d^3 + 7*a^2*c^4*d^4 + 2*a^2*c^3*d^5 - 5*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e)^3 - (a^2*c^8 + 2*a^2*c^7*d - 6*a^2*c^6*d^2 - 6*a^2*c^5*d^3 + 12*a^2*c^4*d^4 + 6*a^2*c^3*d^5 - 10*a^2*c^2*d^6 - 2*a^2*c*d^7 + 3*a^2*d^8)*f*cos(f*x + e)^2 + (a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e) + 2*(a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f - ((a^2*c^6*d^2 - 2*a^2*c^5*d^3 - a^2*c^4*d^4 + 4*a^2*c^3*d^5 - a^2*c^2*d^6 - 2*a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^3 + 2*(a^2*c^7*d - a^2*c^6*d^2 - 3*a^2*c^5*d^3 + 3*a^2*c^4*d^4 + 3*a^2*c^3*d^5 - 3*a^2*c^2*d^6 - a^2*c*d^7 + a^2*d^8)*f*cos(f*x + e)^2 - (a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f*cos(f*x + e) - 2*(a^2*c^8 - 4*a^2*c^6*d^2 + 6*a^2*c^4*d^4 - 4*a^2*c^2*d^6 + a^2*d^8)*f)*sin(f*x + e))]","B",0
600,1,3855,0,3.611099," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{\frac{1}{2}} {\left({\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 12 \, a c^{2} - 56 \, a c d - 172 \, a d^{2} + 3 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(12 \, a c^{2} + 56 \, a c d + 172 \, a d^{2} - {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{a}} \log\left(\frac{4 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} - {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) - 2 \, c - 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 8 \, {\left(a d^{2} \cos\left(f x + e\right)^{3} + 3 \, a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2} + {\left(a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) + 2 \, {\left(3 \, {\left(c^{2} + 4 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} + 4 \, c d - 11 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - 3 \, {\left(c^{2} + 4 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{32 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}, \frac{\sqrt{\frac{1}{2}} {\left({\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 12 \, a c^{2} - 56 \, a c d - 172 \, a d^{2} + 3 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(12 \, a c^{2} + 56 \, a c d + 172 \, a d^{2} - {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{c - d}{a}} \log\left(\frac{4 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{c - d}{a}} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)} - {\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) - 2 \, c - 2 \, d\right)} \sin\left(f x + e\right) - 2 \, c - 2 \, d}{\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 2\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 2}\right) + 16 \, {\left(a d^{2} \cos\left(f x + e\right)^{3} + 3 \, a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2} + {\left(a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 2 \, {\left(3 \, {\left(c^{2} + 4 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} + 4 \, c d - 11 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - 3 \, {\left(c^{2} + 4 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{32 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{\sqrt{\frac{1}{2}} {\left({\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 12 \, a c^{2} - 56 \, a c d - 172 \, a d^{2} + 3 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(12 \, a c^{2} + 56 \, a c d + 172 \, a d^{2} - {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - 4 \, {\left(a d^{2} \cos\left(f x + e\right)^{3} + 3 \, a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2} + {\left(a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{d}{a}} \log\left(\frac{128 \, d^{4} \cos\left(f x + e\right)^{5} + 128 \, {\left(2 \, c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 32 \, {\left(5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(c^{3} d - 2 \, c^{2} d^{2} + 9 \, c d^{3} - 4 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, d^{3} \cos\left(f x + e\right)^{4} + 24 \, {\left(c d^{2} - d^{3}\right)} \cos\left(f x + e\right)^{3} - c^{3} + 17 \, c^{2} d - 59 \, c d^{2} + 51 \, d^{3} - 2 \, {\left(5 \, c^{2} d - 26 \, c d^{2} + 33 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(c^{3} - 7 \, c^{2} d + 31 \, c d^{2} - 25 \, d^{3}\right)} \cos\left(f x + e\right) + {\left(16 \, d^{3} \cos\left(f x + e\right)^{3} + c^{3} - 17 \, c^{2} d + 59 \, c d^{2} - 51 \, d^{3} - 8 \, {\left(3 \, c d^{2} - 5 \, d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, c^{2} d - 14 \, c d^{2} + 13 \, d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{d}{a}} + {\left(c^{4} - 28 \, c^{3} d + 230 \, c^{2} d^{2} - 476 \, c d^{3} + 289 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(128 \, d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4} - 256 \, {\left(c d^{3} - d^{4}\right)} \cos\left(f x + e\right)^{3} - 32 \, {\left(5 \, c^{2} d^{2} - 6 \, c d^{3} + 5 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 32 \, {\left(c^{3} d - 7 \, c^{2} d^{2} + 15 \, c d^{3} - 9 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right) + \sin\left(f x + e\right) + 1}\right) - {\left(3 \, {\left(c^{2} + 4 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} + 4 \, c d - 11 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - 3 \, {\left(c^{2} + 4 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{16 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{\sqrt{\frac{1}{2}} {\left({\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 12 \, a c^{2} - 56 \, a c d - 172 \, a d^{2} + 3 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(12 \, a c^{2} + 56 \, a c d + 172 \, a d^{2} - {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, a c^{2} + 14 \, a c d + 43 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-\frac{c - d}{a}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{-\frac{c - d}{a}}}{{\left(c - d\right)} \cos\left(f x + e\right)}\right) - 8 \, {\left(a d^{2} \cos\left(f x + e\right)^{3} + 3 \, a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2} + {\left(a d^{2} \cos\left(f x + e\right)^{2} - 2 \, a d^{2} \cos\left(f x + e\right) - 4 \, a d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{\frac{d}{a}} \arctan\left(\frac{{\left(8 \, d^{2} \cos\left(f x + e\right)^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left(c d - d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} \sqrt{\frac{d}{a}}}{4 \, {\left(2 \, d^{3} \cos\left(f x + e\right)^{3} - {\left(3 \, c d^{2} - d^{3}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(c^{2} d - c d^{2} + 2 \, d^{3}\right)} \cos\left(f x + e\right)\right)}}\right) - {\left(3 \, {\left(c^{2} + 4 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} + 4 \, c d - 11 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - 3 \, {\left(c^{2} + 4 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{16 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 3 \, a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - 2 \, a^{3} f \cos\left(f x + e\right) - 4 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/32*(sqrt(1/2)*((3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^3 - 12*a*c^2 - 56*a*c*d - 172*a*d^2 + 3*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 - 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e) - (12*a*c^2 + 56*a*c*d + 172*a*d^2 - (3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 + 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt((c - d)/a)*log((4*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 8*(a*d^2*cos(f*x + e)^3 + 3*a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2 + (a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2)*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 2*(3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 + 4*c*d - 11*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - 3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e)), 1/32*(sqrt(1/2)*((3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^3 - 12*a*c^2 - 56*a*c*d - 172*a*d^2 + 3*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 - 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e) - (12*a*c^2 + 56*a*c*d + 172*a*d^2 - (3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 + 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt((c - d)/a)*log((4*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 16*(a*d^2*cos(f*x + e)^3 + 3*a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2 + (a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2)*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) + 2*(3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 + 4*c*d - 11*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - 3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e)), -1/16*(sqrt(1/2)*((3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^3 - 12*a*c^2 - 56*a*c*d - 172*a*d^2 + 3*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 - 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e) - (12*a*c^2 + 56*a*c*d + 172*a*d^2 - (3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 + 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-2*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - 4*(a*d^2*cos(f*x + e)^3 + 3*a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2 + (a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2)*sin(f*x + e))*sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*c*d^3 - d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) - (3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 + 4*c*d - 11*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - 3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e)), -1/16*(sqrt(1/2)*((3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^3 - 12*a*c^2 - 56*a*c*d - 172*a*d^2 + 3*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 - 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e) - (12*a*c^2 + 56*a*c*d + 172*a*d^2 - (3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e)^2 + 2*(3*a*c^2 + 14*a*c*d + 43*a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-(c - d)/a)*arctan(-2*sqrt(1/2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - 8*(a*d^2*cos(f*x + e)^3 + 3*a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2 + (a*d^2*cos(f*x + e)^2 - 2*a*d^2*cos(f*x + e) - 4*a*d^2)*sin(f*x + e))*sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))) - (3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 + 4*c*d - 11*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - 3*(c^2 + 4*c*d - 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))]","B",0
601,1,1304,0,2.408779," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{2 \, a c - 2 \, a d} \log\left(\frac{{\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 4 \, a c^{2} - 8 \, a c d - 4 \, a d^{2} - {\left(13 \, a c^{2} - 22 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) + 4 \, c - 4 \, d\right)} \sin\left(f x + e\right) - 4 \, c + 4 \, d\right)} \sqrt{2 \, a c - 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} - 2 \, {\left(9 \, a c^{2} - 14 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, a c^{2} + 8 \, a c d + 4 \, a d^{2} - {\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(7 \, a c^{2} - 18 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right) + 8 \, {\left({\left(3 \, c^{2} + 4 \, c d - 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} - 4 \, c d - 3 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - {\left(3 \, c^{2} + 4 \, c d - 7 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{128 \, {\left({\left(a^{3} c - a^{3} d\right)} f \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{3} c - a^{3} d\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c - a^{3} d\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c - a^{3} d\right)} f + {\left({\left(a^{3} c - a^{3} d\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c - a^{3} d\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c - a^{3} d\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{3 \, {\left({\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, c^{2} - 8 \, c d - 4 \, d^{2} - 2 \, {\left(c^{2} + 2 \, c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-2 \, a c + 2 \, a d} \arctan\left(\frac{\sqrt{-2 \, a c + 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left({\left(a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a c^{2} - a c d\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left({\left(3 \, c^{2} + 4 \, c d - 7 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} - 4 \, c d - 3 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - {\left(3 \, c^{2} + 4 \, c d - 7 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{64 \, {\left({\left(a^{3} c - a^{3} d\right)} f \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{3} c - a^{3} d\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c - a^{3} d\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c - a^{3} d\right)} f + {\left({\left(a^{3} c - a^{3} d\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c - a^{3} d\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c - a^{3} d\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/128*(3*((c^2 + 2*c*d + d^2)*cos(f*x + e)^3 + 3*(c^2 + 2*c*d + d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 - 2*(c^2 + 2*c*d + d^2)*cos(f*x + e) + ((c^2 + 2*c*d + d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 - 2*(c^2 + 2*c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 - 4*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^2 - 2*(7*a*c^2 - 18*a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) + 8*((3*c^2 + 4*c*d - 7*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 - 4*c*d - 3*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - (3*c^2 + 4*c*d - 7*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c - a^3*d)*f*cos(f*x + e)^3 + 3*(a^3*c - a^3*d)*f*cos(f*x + e)^2 - 2*(a^3*c - a^3*d)*f*cos(f*x + e) - 4*(a^3*c - a^3*d)*f + ((a^3*c - a^3*d)*f*cos(f*x + e)^2 - 2*(a^3*c - a^3*d)*f*cos(f*x + e) - 4*(a^3*c - a^3*d)*f)*sin(f*x + e)), -1/64*(3*((c^2 + 2*c*d + d^2)*cos(f*x + e)^3 + 3*(c^2 + 2*c*d + d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 - 2*(c^2 + 2*c*d + d^2)*cos(f*x + e) + ((c^2 + 2*c*d + d^2)*cos(f*x + e)^2 - 4*c^2 - 8*c*d - 4*d^2 - 2*(c^2 + 2*c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-2*a*c + 2*a*d)*arctan(1/4*sqrt(-2*a*c + 2*a*d)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)/((a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) + (a*c^2 - a*c*d)*cos(f*x + e))) - 4*((3*c^2 + 4*c*d - 7*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 - 4*c*d - 3*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - (3*c^2 + 4*c*d - 7*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c - a^3*d)*f*cos(f*x + e)^3 + 3*(a^3*c - a^3*d)*f*cos(f*x + e)^2 - 2*(a^3*c - a^3*d)*f*cos(f*x + e) - 4*(a^3*c - a^3*d)*f + ((a^3*c - a^3*d)*f*cos(f*x + e)^2 - 2*(a^3*c - a^3*d)*f*cos(f*x + e) - 4*(a^3*c - a^3*d)*f)*sin(f*x + e))]","B",0
602,1,1474,0,2.463933," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(3 \, c^{2} - 2 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, c^{2} - 2 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 8 \, c d + 20 \, d^{2} - 2 \, {\left(3 \, c^{2} - 2 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(3 \, c^{2} - 2 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 8 \, c d + 20 \, d^{2} - 2 \, {\left(3 \, c^{2} - 2 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{2 \, a c - 2 \, a d} \log\left(\frac{{\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 4 \, a c^{2} - 8 \, a c d - 4 \, a d^{2} - {\left(13 \, a c^{2} - 22 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) + 4 \, c - 4 \, d\right)} \sin\left(f x + e\right) - 4 \, c + 4 \, d\right)} \sqrt{2 \, a c - 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} - 2 \, {\left(9 \, a c^{2} - 14 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, a c^{2} + 8 \, a c d + 4 \, a d^{2} - {\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(7 \, a c^{2} - 18 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right) + 8 \, {\left({\left(3 \, c^{2} - 4 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} - 12 \, c d + 5 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - {\left(3 \, c^{2} - 4 \, c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{128 \, {\left({\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f + {\left({\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left({\left(3 \, c^{2} - 2 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, c^{2} - 2 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 8 \, c d + 20 \, d^{2} - 2 \, {\left(3 \, c^{2} - 2 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(3 \, c^{2} - 2 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 8 \, c d + 20 \, d^{2} - 2 \, {\left(3 \, c^{2} - 2 \, c d - 5 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-2 \, a c + 2 \, a d} \arctan\left(\frac{\sqrt{-2 \, a c + 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left({\left(a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a c^{2} - a c d\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left({\left(3 \, c^{2} - 4 \, c d + d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} - 12 \, c d + 5 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - {\left(3 \, c^{2} - 4 \, c d + d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{64 \, {\left({\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f + {\left({\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{2} - 2 \, a^{3} c d + a^{3} d^{2}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/128*(((3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e)^3 + 3*(3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e)^2 - 12*c^2 + 8*c*d + 20*d^2 - 2*(3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e) + ((3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e)^2 - 12*c^2 + 8*c*d + 20*d^2 - 2*(3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 - 4*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^2 - 2*(7*a*c^2 - 18*a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) + 8*((3*c^2 - 4*c*d + d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 - 12*c*d + 5*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - (3*c^2 - 4*c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e)^3 + 3*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e)^2 - 2*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e) - 4*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f + ((a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e)^2 - 2*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e) - 4*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f)*sin(f*x + e)), -1/64*(((3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e)^3 + 3*(3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e)^2 - 12*c^2 + 8*c*d + 20*d^2 - 2*(3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e) + ((3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e)^2 - 12*c^2 + 8*c*d + 20*d^2 - 2*(3*c^2 - 2*c*d - 5*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-2*a*c + 2*a*d)*arctan(1/4*sqrt(-2*a*c + 2*a*d)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)/((a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) + (a*c^2 - a*c*d)*cos(f*x + e))) - 4*((3*c^2 - 4*c*d + d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 - 12*c*d + 5*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - (3*c^2 - 4*c*d + d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e)^3 + 3*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e)^2 - 2*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e) - 4*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f + ((a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e)^2 - 2*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f*cos(f*x + e) - 4*(a^3*c^2 - 2*a^3*c*d + a^3*d^2)*f)*sin(f*x + e))]","B",0
603,1,1644,0,3.993620," ","integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(3 \, c^{2} - 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, c^{2} - 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 40 \, c d - 76 \, d^{2} - 2 \, {\left(3 \, c^{2} - 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(3 \, c^{2} - 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 40 \, c d - 76 \, d^{2} - 2 \, {\left(3 \, c^{2} - 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{2 \, a c - 2 \, a d} \log\left(\frac{{\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 4 \, a c^{2} - 8 \, a c d - 4 \, a d^{2} - {\left(13 \, a c^{2} - 22 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) + 4 \, c - 4 \, d\right)} \sin\left(f x + e\right) - 4 \, c + 4 \, d\right)} \sqrt{2 \, a c - 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} - 2 \, {\left(9 \, a c^{2} - 14 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, a c^{2} + 8 \, a c d + 4 \, a d^{2} - {\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(7 \, a c^{2} - 18 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right) + 8 \, {\left(3 \, {\left(c^{2} - 4 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} - 20 \, c d + 13 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - 3 \, {\left(c^{2} - 4 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{128 \, {\left({\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f + {\left({\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left({\left(3 \, c^{2} - 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, c^{2} - 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 40 \, c d - 76 \, d^{2} - 2 \, {\left(3 \, c^{2} - 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right) + {\left({\left(3 \, c^{2} - 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)^{2} - 12 \, c^{2} + 40 \, c d - 76 \, d^{2} - 2 \, {\left(3 \, c^{2} - 10 \, c d + 19 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-2 \, a c + 2 \, a d} \arctan\left(\frac{\sqrt{-2 \, a c + 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left({\left(a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a c^{2} - a c d\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left(3 \, {\left(c^{2} - 4 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, c^{2} - 8 \, c d + 4 \, d^{2} + {\left(7 \, c^{2} - 20 \, c d + 13 \, d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{2} - 8 \, c d + 4 \, d^{2} - 3 \, {\left(c^{2} - 4 \, c d + 3 \, d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{64 \, {\left({\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f + {\left({\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/128*(((3*c^2 - 10*c*d + 19*d^2)*cos(f*x + e)^3 + 3*(3*c^2 - 10*c*d + 19*d^2)*cos(f*x + e)^2 - 12*c^2 + 40*c*d - 76*d^2 - 2*(3*c^2 - 10*c*d + 19*d^2)*cos(f*x + e) + ((3*c^2 - 10*c*d + 19*d^2)*cos(f*x + e)^2 - 12*c^2 + 40*c*d - 76*d^2 - 2*(3*c^2 - 10*c*d + 19*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 - 4*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^2 - 2*(7*a*c^2 - 18*a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) + 8*(3*(c^2 - 4*c*d + 3*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 - 20*c*d + 13*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - 3*(c^2 - 4*c*d + 3*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^3 + 3*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^2 - 2*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e) - 4*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f + ((a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^2 - 2*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e) - 4*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f)*sin(f*x + e)), -1/64*(((3*c^2 - 10*c*d + 19*d^2)*cos(f*x + e)^3 + 3*(3*c^2 - 10*c*d + 19*d^2)*cos(f*x + e)^2 - 12*c^2 + 40*c*d - 76*d^2 - 2*(3*c^2 - 10*c*d + 19*d^2)*cos(f*x + e) + ((3*c^2 - 10*c*d + 19*d^2)*cos(f*x + e)^2 - 12*c^2 + 40*c*d - 76*d^2 - 2*(3*c^2 - 10*c*d + 19*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-2*a*c + 2*a*d)*arctan(1/4*sqrt(-2*a*c + 2*a*d)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)/((a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) + (a*c^2 - a*c*d)*cos(f*x + e))) - 4*(3*(c^2 - 4*c*d + 3*d^2)*cos(f*x + e)^2 + 4*c^2 - 8*c*d + 4*d^2 + (7*c^2 - 20*c*d + 13*d^2)*cos(f*x + e) - (4*c^2 - 8*c*d + 4*d^2 - 3*(c^2 - 4*c*d + 3*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^3 + 3*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^2 - 2*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e) - 4*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f + ((a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e)^2 - 2*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f*cos(f*x + e) - 4*(a^3*c^3 - 3*a^3*c^2*d + 3*a^3*c*d^2 - a^3*d^3)*f)*sin(f*x + e))]","B",0
604,1,2984,0,4.779252," ","integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(c^{3} d - 5 \, c^{2} d^{2} + 19 \, c d^{3} + 25 \, d^{4}\right)} \cos\left(f x + e\right)^{4} + 4 \, c^{4} - 16 \, c^{3} d + 56 \, c^{2} d^{2} + 176 \, c d^{3} + 100 \, d^{4} - {\left(c^{4} - 3 \, c^{3} d + 9 \, c^{2} d^{2} + 63 \, c d^{3} + 50 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, c^{4} - 10 \, c^{3} d + 32 \, c^{2} d^{2} + 170 \, c d^{3} + 125 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(c^{4} - 4 \, c^{3} d + 14 \, c^{2} d^{2} + 44 \, c d^{3} + 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(4 \, c^{4} - 16 \, c^{3} d + 56 \, c^{2} d^{2} + 176 \, c d^{3} + 100 \, d^{4} - {\left(c^{3} d - 5 \, c^{2} d^{2} + 19 \, c d^{3} + 25 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(c^{4} - 2 \, c^{3} d + 4 \, c^{2} d^{2} + 82 \, c d^{3} + 75 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(c^{4} - 4 \, c^{3} d + 14 \, c^{2} d^{2} + 44 \, c d^{3} + 25 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{2 \, a c - 2 \, a d} \log\left(\frac{{\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 4 \, a c^{2} - 8 \, a c d - 4 \, a d^{2} - {\left(13 \, a c^{2} - 22 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) + 4 \, c - 4 \, d\right)} \sin\left(f x + e\right) - 4 \, c + 4 \, d\right)} \sqrt{2 \, a c - 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} - 2 \, {\left(9 \, a c^{2} - 14 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, a c^{2} + 8 \, a c d + 4 \, a d^{2} - {\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(7 \, a c^{2} - 18 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right) + 8 \, {\left(4 \, c^{4} - 8 \, c^{3} d + 8 \, c d^{3} - 4 \, d^{4} - {\left(3 \, c^{3} d - 17 \, c^{2} d^{2} - 35 \, c d^{3} + 49 \, d^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, c^{4} - 13 \, c^{3} d - 7 \, c^{2} d^{2} - 19 \, c d^{3} + 36 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(7 \, c^{4} - 18 \, c^{3} d - 24 \, c^{2} d^{2} - 46 \, c d^{3} + 81 \, d^{4}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{4} - 8 \, c^{3} d + 8 \, c d^{3} - 4 \, d^{4} - {\left(3 \, c^{3} d - 17 \, c^{2} d^{2} - 35 \, c d^{3} + 49 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c^{4} - 10 \, c^{3} d - 24 \, c^{2} d^{2} - 54 \, c d^{3} + 85 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{128 \, {\left({\left(a^{3} c^{5} d - 3 \, a^{3} c^{4} d^{2} + 2 \, a^{3} c^{3} d^{3} + 2 \, a^{3} c^{2} d^{4} - 3 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{6} - a^{3} c^{5} d - 4 \, a^{3} c^{4} d^{2} + 6 \, a^{3} c^{3} d^{3} + a^{3} c^{2} d^{4} - 5 \, a^{3} c d^{5} + 2 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{6} - 4 \, a^{3} c^{5} d - 9 \, a^{3} c^{4} d^{2} + 16 \, a^{3} c^{3} d^{3} + a^{3} c^{2} d^{4} - 12 \, a^{3} c d^{5} + 5 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f - {\left({\left(a^{3} c^{5} d - 3 \, a^{3} c^{4} d^{2} + 2 \, a^{3} c^{3} d^{3} + 2 \, a^{3} c^{2} d^{4} - 3 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{3} c^{6} - 7 \, a^{3} c^{4} d^{2} + 8 \, a^{3} c^{3} d^{3} + 3 \, a^{3} c^{2} d^{4} - 8 \, a^{3} c d^{5} + 3 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{3 \, {\left({\left(c^{3} d - 5 \, c^{2} d^{2} + 19 \, c d^{3} + 25 \, d^{4}\right)} \cos\left(f x + e\right)^{4} + 4 \, c^{4} - 16 \, c^{3} d + 56 \, c^{2} d^{2} + 176 \, c d^{3} + 100 \, d^{4} - {\left(c^{4} - 3 \, c^{3} d + 9 \, c^{2} d^{2} + 63 \, c d^{3} + 50 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, c^{4} - 10 \, c^{3} d + 32 \, c^{2} d^{2} + 170 \, c d^{3} + 125 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(c^{4} - 4 \, c^{3} d + 14 \, c^{2} d^{2} + 44 \, c d^{3} + 25 \, d^{4}\right)} \cos\left(f x + e\right) + {\left(4 \, c^{4} - 16 \, c^{3} d + 56 \, c^{2} d^{2} + 176 \, c d^{3} + 100 \, d^{4} - {\left(c^{3} d - 5 \, c^{2} d^{2} + 19 \, c d^{3} + 25 \, d^{4}\right)} \cos\left(f x + e\right)^{3} - {\left(c^{4} - 2 \, c^{3} d + 4 \, c^{2} d^{2} + 82 \, c d^{3} + 75 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(c^{4} - 4 \, c^{3} d + 14 \, c^{2} d^{2} + 44 \, c d^{3} + 25 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-2 \, a c + 2 \, a d} \arctan\left(\frac{\sqrt{-2 \, a c + 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left({\left(a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a c^{2} - a c d\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(4 \, c^{4} - 8 \, c^{3} d + 8 \, c d^{3} - 4 \, d^{4} - {\left(3 \, c^{3} d - 17 \, c^{2} d^{2} - 35 \, c d^{3} + 49 \, d^{4}\right)} \cos\left(f x + e\right)^{3} + {\left(3 \, c^{4} - 13 \, c^{3} d - 7 \, c^{2} d^{2} - 19 \, c d^{3} + 36 \, d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(7 \, c^{4} - 18 \, c^{3} d - 24 \, c^{2} d^{2} - 46 \, c d^{3} + 81 \, d^{4}\right)} \cos\left(f x + e\right) - {\left(4 \, c^{4} - 8 \, c^{3} d + 8 \, c d^{3} - 4 \, d^{4} - {\left(3 \, c^{3} d - 17 \, c^{2} d^{2} - 35 \, c d^{3} + 49 \, d^{4}\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c^{4} - 10 \, c^{3} d - 24 \, c^{2} d^{2} - 54 \, c d^{3} + 85 \, d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{64 \, {\left({\left(a^{3} c^{5} d - 3 \, a^{3} c^{4} d^{2} + 2 \, a^{3} c^{3} d^{3} + 2 \, a^{3} c^{2} d^{4} - 3 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{6} - a^{3} c^{5} d - 4 \, a^{3} c^{4} d^{2} + 6 \, a^{3} c^{3} d^{3} + a^{3} c^{2} d^{4} - 5 \, a^{3} c d^{5} + 2 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{6} - 4 \, a^{3} c^{5} d - 9 \, a^{3} c^{4} d^{2} + 16 \, a^{3} c^{3} d^{3} + a^{3} c^{2} d^{4} - 12 \, a^{3} c d^{5} + 5 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f - {\left({\left(a^{3} c^{5} d - 3 \, a^{3} c^{4} d^{2} + 2 \, a^{3} c^{3} d^{3} + 2 \, a^{3} c^{2} d^{4} - 3 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{3} + {\left(a^{3} c^{6} - 7 \, a^{3} c^{4} d^{2} + 8 \, a^{3} c^{3} d^{3} + 3 \, a^{3} c^{2} d^{4} - 8 \, a^{3} c d^{5} + 3 \, a^{3} d^{6}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f \cos\left(f x + e\right) - 4 \, {\left(a^{3} c^{6} - 2 \, a^{3} c^{5} d - a^{3} c^{4} d^{2} + 4 \, a^{3} c^{3} d^{3} - a^{3} c^{2} d^{4} - 2 \, a^{3} c d^{5} + a^{3} d^{6}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/128*(3*((c^3*d - 5*c^2*d^2 + 19*c*d^3 + 25*d^4)*cos(f*x + e)^4 + 4*c^4 - 16*c^3*d + 56*c^2*d^2 + 176*c*d^3 + 100*d^4 - (c^4 - 3*c^3*d + 9*c^2*d^2 + 63*c*d^3 + 50*d^4)*cos(f*x + e)^3 - (3*c^4 - 10*c^3*d + 32*c^2*d^2 + 170*c*d^3 + 125*d^4)*cos(f*x + e)^2 + 2*(c^4 - 4*c^3*d + 14*c^2*d^2 + 44*c*d^3 + 25*d^4)*cos(f*x + e) + (4*c^4 - 16*c^3*d + 56*c^2*d^2 + 176*c*d^3 + 100*d^4 - (c^3*d - 5*c^2*d^2 + 19*c*d^3 + 25*d^4)*cos(f*x + e)^3 - (c^4 - 2*c^3*d + 4*c^2*d^2 + 82*c*d^3 + 75*d^4)*cos(f*x + e)^2 + 2*(c^4 - 4*c^3*d + 14*c^2*d^2 + 44*c*d^3 + 25*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 + 4*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^2 - 2*(7*a*c^2 - 18*a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) + 8*(4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 17*c^2*d^2 - 35*c*d^3 + 49*d^4)*cos(f*x + e)^3 + (3*c^4 - 13*c^3*d - 7*c^2*d^2 - 19*c*d^3 + 36*d^4)*cos(f*x + e)^2 + (7*c^4 - 18*c^3*d - 24*c^2*d^2 - 46*c*d^3 + 81*d^4)*cos(f*x + e) - (4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 17*c^2*d^2 - 35*c*d^3 + 49*d^4)*cos(f*x + e)^2 - (3*c^4 - 10*c^3*d - 24*c^2*d^2 - 54*c*d^3 + 85*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e)^4 - (a^3*c^6 - a^3*c^5*d - 4*a^3*c^4*d^2 + 6*a^3*c^3*d^3 + a^3*c^2*d^4 - 5*a^3*c*d^5 + 2*a^3*d^6)*f*cos(f*x + e)^3 - (3*a^3*c^6 - 4*a^3*c^5*d - 9*a^3*c^4*d^2 + 16*a^3*c^3*d^3 + a^3*c^2*d^4 - 12*a^3*c*d^5 + 5*a^3*d^6)*f*cos(f*x + e)^2 + 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e) + 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f - ((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e)^3 + (a^3*c^6 - 7*a^3*c^4*d^2 + 8*a^3*c^3*d^3 + 3*a^3*c^2*d^4 - 8*a^3*c*d^5 + 3*a^3*d^6)*f*cos(f*x + e)^2 - 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e) - 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f)*sin(f*x + e)), -1/64*(3*((c^3*d - 5*c^2*d^2 + 19*c*d^3 + 25*d^4)*cos(f*x + e)^4 + 4*c^4 - 16*c^3*d + 56*c^2*d^2 + 176*c*d^3 + 100*d^4 - (c^4 - 3*c^3*d + 9*c^2*d^2 + 63*c*d^3 + 50*d^4)*cos(f*x + e)^3 - (3*c^4 - 10*c^3*d + 32*c^2*d^2 + 170*c*d^3 + 125*d^4)*cos(f*x + e)^2 + 2*(c^4 - 4*c^3*d + 14*c^2*d^2 + 44*c*d^3 + 25*d^4)*cos(f*x + e) + (4*c^4 - 16*c^3*d + 56*c^2*d^2 + 176*c*d^3 + 100*d^4 - (c^3*d - 5*c^2*d^2 + 19*c*d^3 + 25*d^4)*cos(f*x + e)^3 - (c^4 - 2*c^3*d + 4*c^2*d^2 + 82*c*d^3 + 75*d^4)*cos(f*x + e)^2 + 2*(c^4 - 4*c^3*d + 14*c^2*d^2 + 44*c*d^3 + 25*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(-2*a*c + 2*a*d)*arctan(1/4*sqrt(-2*a*c + 2*a*d)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)/((a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) + (a*c^2 - a*c*d)*cos(f*x + e))) + 4*(4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 17*c^2*d^2 - 35*c*d^3 + 49*d^4)*cos(f*x + e)^3 + (3*c^4 - 13*c^3*d - 7*c^2*d^2 - 19*c*d^3 + 36*d^4)*cos(f*x + e)^2 + (7*c^4 - 18*c^3*d - 24*c^2*d^2 - 46*c*d^3 + 81*d^4)*cos(f*x + e) - (4*c^4 - 8*c^3*d + 8*c*d^3 - 4*d^4 - (3*c^3*d - 17*c^2*d^2 - 35*c*d^3 + 49*d^4)*cos(f*x + e)^2 - (3*c^4 - 10*c^3*d - 24*c^2*d^2 - 54*c*d^3 + 85*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e)^4 - (a^3*c^6 - a^3*c^5*d - 4*a^3*c^4*d^2 + 6*a^3*c^3*d^3 + a^3*c^2*d^4 - 5*a^3*c*d^5 + 2*a^3*d^6)*f*cos(f*x + e)^3 - (3*a^3*c^6 - 4*a^3*c^5*d - 9*a^3*c^4*d^2 + 16*a^3*c^3*d^3 + a^3*c^2*d^4 - 12*a^3*c*d^5 + 5*a^3*d^6)*f*cos(f*x + e)^2 + 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e) + 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f - ((a^3*c^5*d - 3*a^3*c^4*d^2 + 2*a^3*c^3*d^3 + 2*a^3*c^2*d^4 - 3*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e)^3 + (a^3*c^6 - 7*a^3*c^4*d^2 + 8*a^3*c^3*d^3 + 3*a^3*c^2*d^4 - 8*a^3*c*d^5 + 3*a^3*d^6)*f*cos(f*x + e)^2 - 2*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f*cos(f*x + e) - 4*(a^3*c^6 - 2*a^3*c^5*d - a^3*c^4*d^2 + 4*a^3*c^3*d^3 - a^3*c^2*d^4 - 2*a^3*c*d^5 + a^3*d^6)*f)*sin(f*x + e))]","B",0
605,1,4858,0,10.596517," ","integrate(1/(a+a*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(12 \, c^{6} - 56 \, c^{5} d + 308 \, c^{4} d^{2} + 2032 \, c^{3} d^{3} + 3508 \, c^{2} d^{4} + 2504 \, c d^{5} + 652 \, d^{6} + {\left(3 \, c^{4} d^{2} - 20 \, c^{3} d^{3} + 114 \, c^{2} d^{4} + 300 \, c d^{5} + 163 \, d^{6}\right)} \cos\left(f x + e\right)^{5} + {\left(6 \, c^{5} d - 31 \, c^{4} d^{2} + 168 \, c^{3} d^{3} + 942 \, c^{2} d^{4} + 1226 \, c d^{5} + 489 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, c^{6} - 8 \, c^{5} d + 43 \, c^{4} d^{2} + 696 \, c^{3} d^{3} + 1705 \, c^{2} d^{4} + 1552 \, c d^{5} + 489 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(9 \, c^{6} - 30 \, c^{5} d + 163 \, c^{4} d^{2} + 1900 \, c^{3} d^{3} + 4287 \, c^{2} d^{4} + 3730 \, c d^{5} + 1141 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, c^{6} - 14 \, c^{5} d + 77 \, c^{4} d^{2} + 508 \, c^{3} d^{3} + 877 \, c^{2} d^{4} + 626 \, c d^{5} + 163 \, d^{6}\right)} \cos\left(f x + e\right) + {\left(12 \, c^{6} - 56 \, c^{5} d + 308 \, c^{4} d^{2} + 2032 \, c^{3} d^{3} + 3508 \, c^{2} d^{4} + 2504 \, c d^{5} + 652 \, d^{6} + {\left(3 \, c^{4} d^{2} - 20 \, c^{3} d^{3} + 114 \, c^{2} d^{4} + 300 \, c d^{5} + 163 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, c^{5} d - 17 \, c^{4} d^{2} + 94 \, c^{3} d^{3} + 414 \, c^{2} d^{4} + 463 \, c d^{5} + 163 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, c^{6} - 2 \, c^{5} d + 9 \, c^{4} d^{2} + 884 \, c^{3} d^{3} + 2533 \, c^{2} d^{4} + 2478 \, c d^{5} + 815 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, c^{6} - 14 \, c^{5} d + 77 \, c^{4} d^{2} + 508 \, c^{3} d^{3} + 877 \, c^{2} d^{4} + 626 \, c d^{5} + 163 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{2 \, a c - 2 \, a d} \log\left(\frac{{\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{3} - 4 \, a c^{2} - 8 \, a c d - 4 \, a d^{2} - {\left(13 \, a c^{2} - 22 \, a c d - 3 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, c - d\right)} \cos\left(f x + e\right) + {\left({\left(c - 3 \, d\right)} \cos\left(f x + e\right) + 4 \, c - 4 \, d\right)} \sin\left(f x + e\right) - 4 \, c + 4 \, d\right)} \sqrt{2 \, a c - 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c} - 2 \, {\left(9 \, a c^{2} - 14 \, a c d + 9 \, a d^{2}\right)} \cos\left(f x + e\right) - {\left(4 \, a c^{2} + 8 \, a c d + 4 \, a d^{2} - {\left(a c^{2} - 14 \, a c d + 17 \, a d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(7 \, a c^{2} - 18 \, a c d + 7 \, a d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 4\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 4}\right) - 8 \, {\left(12 \, c^{6} - 24 \, c^{5} d - 12 \, c^{4} d^{2} + 48 \, c^{3} d^{3} - 12 \, c^{2} d^{4} - 24 \, c d^{5} + 12 \, d^{6} - {\left(9 \, c^{4} d^{2} - 66 \, c^{3} d^{3} - 436 \, c^{2} d^{4} + 194 \, c d^{5} + 299 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(18 \, c^{5} d - 111 \, c^{4} d^{2} - 618 \, c^{3} d^{3} - 520 \, c^{2} d^{4} + 728 \, c d^{5} + 503 \, d^{6}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, c^{6} - 14 \, c^{5} d - 29 \, c^{4} d^{2} - 144 \, c^{3} d^{3} - 59 \, c^{2} d^{4} + 158 \, c d^{5} + 85 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + 3 \, {\left(7 \, c^{6} - 16 \, c^{5} d - 73 \, c^{4} d^{2} - 312 \, c^{3} d^{3} - 91 \, c^{2} d^{4} + 328 \, c d^{5} + 157 \, d^{6}\right)} \cos\left(f x + e\right) - {\left(12 \, c^{6} - 24 \, c^{5} d - 12 \, c^{4} d^{2} + 48 \, c^{3} d^{3} - 12 \, c^{2} d^{4} - 24 \, c d^{5} + 12 \, d^{6} + {\left(9 \, c^{4} d^{2} - 66 \, c^{3} d^{3} - 436 \, c^{2} d^{4} + 194 \, c d^{5} + 299 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - 6 \, {\left(3 \, c^{5} d - 20 \, c^{4} d^{2} - 92 \, c^{3} d^{3} - 14 \, c^{2} d^{4} + 89 \, c d^{5} + 34 \, d^{6}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(3 \, c^{6} - 8 \, c^{5} d - 69 \, c^{4} d^{2} - 328 \, c^{3} d^{3} - 87 \, c^{2} d^{4} + 336 \, c d^{5} + 153 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{384 \, {\left({\left(a^{3} c^{7} d^{2} - 3 \, a^{3} c^{6} d^{3} + a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 5 \, a^{3} c^{3} d^{6} - a^{3} c^{2} d^{7} + 3 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{5} + {\left(2 \, a^{3} c^{8} d - 3 \, a^{3} c^{7} d^{2} - 7 \, a^{3} c^{6} d^{3} + 13 \, a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 17 \, a^{3} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{7} + 7 \, a^{3} c d^{8} - 3 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{9} + a^{3} c^{8} d - 8 \, a^{3} c^{7} d^{2} + 18 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 16 \, a^{3} c^{3} d^{6} + 8 \, a^{3} c^{2} d^{7} + 5 \, a^{3} c d^{8} - 3 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{9} + a^{3} c^{8} d - 20 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 42 \, a^{3} c^{5} d^{4} - 18 \, a^{3} c^{4} d^{5} - 36 \, a^{3} c^{3} d^{6} + 20 \, a^{3} c^{2} d^{7} + 11 \, a^{3} c d^{8} - 7 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f + {\left({\left(a^{3} c^{7} d^{2} - 3 \, a^{3} c^{6} d^{3} + a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 5 \, a^{3} c^{3} d^{6} - a^{3} c^{2} d^{7} + 3 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} c^{8} d - 2 \, a^{3} c^{7} d^{2} - 2 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{3} d^{6} + 2 \, a^{3} c^{2} d^{7} + 2 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{3} c^{9} + 3 \, a^{3} c^{8} d - 12 \, a^{3} c^{7} d^{2} - 4 \, a^{3} c^{6} d^{3} + 30 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 28 \, a^{3} c^{3} d^{6} + 12 \, a^{3} c^{2} d^{7} + 9 \, a^{3} c d^{8} - 5 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{3 \, {\left(12 \, c^{6} - 56 \, c^{5} d + 308 \, c^{4} d^{2} + 2032 \, c^{3} d^{3} + 3508 \, c^{2} d^{4} + 2504 \, c d^{5} + 652 \, d^{6} + {\left(3 \, c^{4} d^{2} - 20 \, c^{3} d^{3} + 114 \, c^{2} d^{4} + 300 \, c d^{5} + 163 \, d^{6}\right)} \cos\left(f x + e\right)^{5} + {\left(6 \, c^{5} d - 31 \, c^{4} d^{2} + 168 \, c^{3} d^{3} + 942 \, c^{2} d^{4} + 1226 \, c d^{5} + 489 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(3 \, c^{6} - 8 \, c^{5} d + 43 \, c^{4} d^{2} + 696 \, c^{3} d^{3} + 1705 \, c^{2} d^{4} + 1552 \, c d^{5} + 489 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(9 \, c^{6} - 30 \, c^{5} d + 163 \, c^{4} d^{2} + 1900 \, c^{3} d^{3} + 4287 \, c^{2} d^{4} + 3730 \, c d^{5} + 1141 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, c^{6} - 14 \, c^{5} d + 77 \, c^{4} d^{2} + 508 \, c^{3} d^{3} + 877 \, c^{2} d^{4} + 626 \, c d^{5} + 163 \, d^{6}\right)} \cos\left(f x + e\right) + {\left(12 \, c^{6} - 56 \, c^{5} d + 308 \, c^{4} d^{2} + 2032 \, c^{3} d^{3} + 3508 \, c^{2} d^{4} + 2504 \, c d^{5} + 652 \, d^{6} + {\left(3 \, c^{4} d^{2} - 20 \, c^{3} d^{3} + 114 \, c^{2} d^{4} + 300 \, c d^{5} + 163 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, c^{5} d - 17 \, c^{4} d^{2} + 94 \, c^{3} d^{3} + 414 \, c^{2} d^{4} + 463 \, c d^{5} + 163 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - {\left(3 \, c^{6} - 2 \, c^{5} d + 9 \, c^{4} d^{2} + 884 \, c^{3} d^{3} + 2533 \, c^{2} d^{4} + 2478 \, c d^{5} + 815 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(3 \, c^{6} - 14 \, c^{5} d + 77 \, c^{4} d^{2} + 508 \, c^{3} d^{3} + 877 \, c^{2} d^{4} + 626 \, c d^{5} + 163 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{-2 \, a c + 2 \, a d} \arctan\left(\frac{\sqrt{-2 \, a c + 2 \, a d} \sqrt{a \sin\left(f x + e\right) + a} {\left({\left(c - 3 \, d\right)} \sin\left(f x + e\right) - 3 \, c + d\right)} \sqrt{d \sin\left(f x + e\right) + c}}{4 \, {\left({\left(a c d - a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(a c^{2} - a c d\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(12 \, c^{6} - 24 \, c^{5} d - 12 \, c^{4} d^{2} + 48 \, c^{3} d^{3} - 12 \, c^{2} d^{4} - 24 \, c d^{5} + 12 \, d^{6} - {\left(9 \, c^{4} d^{2} - 66 \, c^{3} d^{3} - 436 \, c^{2} d^{4} + 194 \, c d^{5} + 299 \, d^{6}\right)} \cos\left(f x + e\right)^{4} - {\left(18 \, c^{5} d - 111 \, c^{4} d^{2} - 618 \, c^{3} d^{3} - 520 \, c^{2} d^{4} + 728 \, c d^{5} + 503 \, d^{6}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(3 \, c^{6} - 14 \, c^{5} d - 29 \, c^{4} d^{2} - 144 \, c^{3} d^{3} - 59 \, c^{2} d^{4} + 158 \, c d^{5} + 85 \, d^{6}\right)} \cos\left(f x + e\right)^{2} + 3 \, {\left(7 \, c^{6} - 16 \, c^{5} d - 73 \, c^{4} d^{2} - 312 \, c^{3} d^{3} - 91 \, c^{2} d^{4} + 328 \, c d^{5} + 157 \, d^{6}\right)} \cos\left(f x + e\right) - {\left(12 \, c^{6} - 24 \, c^{5} d - 12 \, c^{4} d^{2} + 48 \, c^{3} d^{3} - 12 \, c^{2} d^{4} - 24 \, c d^{5} + 12 \, d^{6} + {\left(9 \, c^{4} d^{2} - 66 \, c^{3} d^{3} - 436 \, c^{2} d^{4} + 194 \, c d^{5} + 299 \, d^{6}\right)} \cos\left(f x + e\right)^{3} - 6 \, {\left(3 \, c^{5} d - 20 \, c^{4} d^{2} - 92 \, c^{3} d^{3} - 14 \, c^{2} d^{4} + 89 \, c d^{5} + 34 \, d^{6}\right)} \cos\left(f x + e\right)^{2} - 3 \, {\left(3 \, c^{6} - 8 \, c^{5} d - 69 \, c^{4} d^{2} - 328 \, c^{3} d^{3} - 87 \, c^{2} d^{4} + 336 \, c d^{5} + 153 \, d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{192 \, {\left({\left(a^{3} c^{7} d^{2} - 3 \, a^{3} c^{6} d^{3} + a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 5 \, a^{3} c^{3} d^{6} - a^{3} c^{2} d^{7} + 3 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{5} + {\left(2 \, a^{3} c^{8} d - 3 \, a^{3} c^{7} d^{2} - 7 \, a^{3} c^{6} d^{3} + 13 \, a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 17 \, a^{3} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{7} + 7 \, a^{3} c d^{8} - 3 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{3} c^{9} + a^{3} c^{8} d - 8 \, a^{3} c^{7} d^{2} + 18 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 16 \, a^{3} c^{3} d^{6} + 8 \, a^{3} c^{2} d^{7} + 5 \, a^{3} c d^{8} - 3 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{3} - {\left(3 \, a^{3} c^{9} + a^{3} c^{8} d - 20 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 42 \, a^{3} c^{5} d^{4} - 18 \, a^{3} c^{4} d^{5} - 36 \, a^{3} c^{3} d^{6} + 20 \, a^{3} c^{2} d^{7} + 11 \, a^{3} c d^{8} - 7 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f + {\left({\left(a^{3} c^{7} d^{2} - 3 \, a^{3} c^{6} d^{3} + a^{3} c^{5} d^{4} + 5 \, a^{3} c^{4} d^{5} - 5 \, a^{3} c^{3} d^{6} - a^{3} c^{2} d^{7} + 3 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} c^{8} d - 2 \, a^{3} c^{7} d^{2} - 2 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{3} d^{6} + 2 \, a^{3} c^{2} d^{7} + 2 \, a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{3} - {\left(a^{3} c^{9} + 3 \, a^{3} c^{8} d - 12 \, a^{3} c^{7} d^{2} - 4 \, a^{3} c^{6} d^{3} + 30 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 28 \, a^{3} c^{3} d^{6} + 12 \, a^{3} c^{2} d^{7} + 9 \, a^{3} c d^{8} - 5 \, a^{3} d^{9}\right)} f \cos\left(f x + e\right)^{2} + 2 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f \cos\left(f x + e\right) + 4 \, {\left(a^{3} c^{9} - a^{3} c^{8} d - 4 \, a^{3} c^{7} d^{2} + 4 \, a^{3} c^{6} d^{3} + 6 \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 4 \, a^{3} c^{3} d^{6} + 4 \, a^{3} c^{2} d^{7} + a^{3} c d^{8} - a^{3} d^{9}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[1/384*(3*(12*c^6 - 56*c^5*d + 308*c^4*d^2 + 2032*c^3*d^3 + 3508*c^2*d^4 + 2504*c*d^5 + 652*d^6 + (3*c^4*d^2 - 20*c^3*d^3 + 114*c^2*d^4 + 300*c*d^5 + 163*d^6)*cos(f*x + e)^5 + (6*c^5*d - 31*c^4*d^2 + 168*c^3*d^3 + 942*c^2*d^4 + 1226*c*d^5 + 489*d^6)*cos(f*x + e)^4 - (3*c^6 - 8*c^5*d + 43*c^4*d^2 + 696*c^3*d^3 + 1705*c^2*d^4 + 1552*c*d^5 + 489*d^6)*cos(f*x + e)^3 - (9*c^6 - 30*c^5*d + 163*c^4*d^2 + 1900*c^3*d^3 + 4287*c^2*d^4 + 3730*c*d^5 + 1141*d^6)*cos(f*x + e)^2 + 2*(3*c^6 - 14*c^5*d + 77*c^4*d^2 + 508*c^3*d^3 + 877*c^2*d^4 + 626*c*d^5 + 163*d^6)*cos(f*x + e) + (12*c^6 - 56*c^5*d + 308*c^4*d^2 + 2032*c^3*d^3 + 3508*c^2*d^4 + 2504*c*d^5 + 652*d^6 + (3*c^4*d^2 - 20*c^3*d^3 + 114*c^2*d^4 + 300*c*d^5 + 163*d^6)*cos(f*x + e)^4 - 2*(3*c^5*d - 17*c^4*d^2 + 94*c^3*d^3 + 414*c^2*d^4 + 463*c*d^5 + 163*d^6)*cos(f*x + e)^3 - (3*c^6 - 2*c^5*d + 9*c^4*d^2 + 884*c^3*d^3 + 2533*c^2*d^4 + 2478*c*d^5 + 815*d^6)*cos(f*x + e)^2 + 2*(3*c^6 - 14*c^5*d + 77*c^4*d^2 + 508*c^3*d^3 + 877*c^2*d^4 + 626*c*d^5 + 163*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(2*a*c - 2*a*d)*log(((a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^3 - 4*a*c^2 - 8*a*c*d - 4*a*d^2 - (13*a*c^2 - 22*a*c*d - 3*a*d^2)*cos(f*x + e)^2 - 4*((c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x + e) + 4*c - 4*d)*sin(f*x + e) - 4*c + 4*d)*sqrt(2*a*c - 2*a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) - 2*(9*a*c^2 - 14*a*c*d + 9*a*d^2)*cos(f*x + e) - (4*a*c^2 + 8*a*c*d + 4*a*d^2 - (a*c^2 - 14*a*c*d + 17*a*d^2)*cos(f*x + e)^2 - 2*(7*a*c^2 - 18*a*c*d + 7*a*d^2)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) - 8*(12*c^6 - 24*c^5*d - 12*c^4*d^2 + 48*c^3*d^3 - 12*c^2*d^4 - 24*c*d^5 + 12*d^6 - (9*c^4*d^2 - 66*c^3*d^3 - 436*c^2*d^4 + 194*c*d^5 + 299*d^6)*cos(f*x + e)^4 - (18*c^5*d - 111*c^4*d^2 - 618*c^3*d^3 - 520*c^2*d^4 + 728*c*d^5 + 503*d^6)*cos(f*x + e)^3 + 3*(3*c^6 - 14*c^5*d - 29*c^4*d^2 - 144*c^3*d^3 - 59*c^2*d^4 + 158*c*d^5 + 85*d^6)*cos(f*x + e)^2 + 3*(7*c^6 - 16*c^5*d - 73*c^4*d^2 - 312*c^3*d^3 - 91*c^2*d^4 + 328*c*d^5 + 157*d^6)*cos(f*x + e) - (12*c^6 - 24*c^5*d - 12*c^4*d^2 + 48*c^3*d^3 - 12*c^2*d^4 - 24*c*d^5 + 12*d^6 + (9*c^4*d^2 - 66*c^3*d^3 - 436*c^2*d^4 + 194*c*d^5 + 299*d^6)*cos(f*x + e)^3 - 6*(3*c^5*d - 20*c^4*d^2 - 92*c^3*d^3 - 14*c^2*d^4 + 89*c*d^5 + 34*d^6)*cos(f*x + e)^2 - 3*(3*c^6 - 8*c^5*d - 69*c^4*d^2 - 328*c^3*d^3 - 87*c^2*d^4 + 336*c*d^5 + 153*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c^7*d^2 - 3*a^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6 - a^3*c^2*d^7 + 3*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^5 + (2*a^3*c^8*d - 3*a^3*c^7*d^2 - 7*a^3*c^6*d^3 + 13*a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 17*a^3*c^3*d^6 + 3*a^3*c^2*d^7 + 7*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^4 - (a^3*c^9 + a^3*c^8*d - 8*a^3*c^7*d^2 + 18*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 16*a^3*c^3*d^6 + 8*a^3*c^2*d^7 + 5*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^3 - (3*a^3*c^9 + a^3*c^8*d - 20*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 42*a^3*c^5*d^4 - 18*a^3*c^4*d^5 - 36*a^3*c^3*d^6 + 20*a^3*c^2*d^7 + 11*a^3*c*d^8 - 7*a^3*d^9)*f*cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f + ((a^3*c^7*d^2 - 3*a^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6 - a^3*c^2*d^7 + 3*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^4 - 2*(a^3*c^8*d - 2*a^3*c^7*d^2 - 2*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^3*d^6 + 2*a^3*c^2*d^7 + 2*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^3 - (a^3*c^9 + 3*a^3*c^8*d - 12*a^3*c^7*d^2 - 4*a^3*c^6*d^3 + 30*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 28*a^3*c^3*d^6 + 12*a^3*c^2*d^7 + 9*a^3*c*d^8 - 5*a^3*d^9)*f*cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f)*sin(f*x + e)), -1/192*(3*(12*c^6 - 56*c^5*d + 308*c^4*d^2 + 2032*c^3*d^3 + 3508*c^2*d^4 + 2504*c*d^5 + 652*d^6 + (3*c^4*d^2 - 20*c^3*d^3 + 114*c^2*d^4 + 300*c*d^5 + 163*d^6)*cos(f*x + e)^5 + (6*c^5*d - 31*c^4*d^2 + 168*c^3*d^3 + 942*c^2*d^4 + 1226*c*d^5 + 489*d^6)*cos(f*x + e)^4 - (3*c^6 - 8*c^5*d + 43*c^4*d^2 + 696*c^3*d^3 + 1705*c^2*d^4 + 1552*c*d^5 + 489*d^6)*cos(f*x + e)^3 - (9*c^6 - 30*c^5*d + 163*c^4*d^2 + 1900*c^3*d^3 + 4287*c^2*d^4 + 3730*c*d^5 + 1141*d^6)*cos(f*x + e)^2 + 2*(3*c^6 - 14*c^5*d + 77*c^4*d^2 + 508*c^3*d^3 + 877*c^2*d^4 + 626*c*d^5 + 163*d^6)*cos(f*x + e) + (12*c^6 - 56*c^5*d + 308*c^4*d^2 + 2032*c^3*d^3 + 3508*c^2*d^4 + 2504*c*d^5 + 652*d^6 + (3*c^4*d^2 - 20*c^3*d^3 + 114*c^2*d^4 + 300*c*d^5 + 163*d^6)*cos(f*x + e)^4 - 2*(3*c^5*d - 17*c^4*d^2 + 94*c^3*d^3 + 414*c^2*d^4 + 463*c*d^5 + 163*d^6)*cos(f*x + e)^3 - (3*c^6 - 2*c^5*d + 9*c^4*d^2 + 884*c^3*d^3 + 2533*c^2*d^4 + 2478*c*d^5 + 815*d^6)*cos(f*x + e)^2 + 2*(3*c^6 - 14*c^5*d + 77*c^4*d^2 + 508*c^3*d^3 + 877*c^2*d^4 + 626*c*d^5 + 163*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(-2*a*c + 2*a*d)*arctan(1/4*sqrt(-2*a*c + 2*a*d)*sqrt(a*sin(f*x + e) + a)*((c - 3*d)*sin(f*x + e) - 3*c + d)*sqrt(d*sin(f*x + e) + c)/((a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) + (a*c^2 - a*c*d)*cos(f*x + e))) + 4*(12*c^6 - 24*c^5*d - 12*c^4*d^2 + 48*c^3*d^3 - 12*c^2*d^4 - 24*c*d^5 + 12*d^6 - (9*c^4*d^2 - 66*c^3*d^3 - 436*c^2*d^4 + 194*c*d^5 + 299*d^6)*cos(f*x + e)^4 - (18*c^5*d - 111*c^4*d^2 - 618*c^3*d^3 - 520*c^2*d^4 + 728*c*d^5 + 503*d^6)*cos(f*x + e)^3 + 3*(3*c^6 - 14*c^5*d - 29*c^4*d^2 - 144*c^3*d^3 - 59*c^2*d^4 + 158*c*d^5 + 85*d^6)*cos(f*x + e)^2 + 3*(7*c^6 - 16*c^5*d - 73*c^4*d^2 - 312*c^3*d^3 - 91*c^2*d^4 + 328*c*d^5 + 157*d^6)*cos(f*x + e) - (12*c^6 - 24*c^5*d - 12*c^4*d^2 + 48*c^3*d^3 - 12*c^2*d^4 - 24*c*d^5 + 12*d^6 + (9*c^4*d^2 - 66*c^3*d^3 - 436*c^2*d^4 + 194*c*d^5 + 299*d^6)*cos(f*x + e)^3 - 6*(3*c^5*d - 20*c^4*d^2 - 92*c^3*d^3 - 14*c^2*d^4 + 89*c*d^5 + 34*d^6)*cos(f*x + e)^2 - 3*(3*c^6 - 8*c^5*d - 69*c^4*d^2 - 328*c^3*d^3 - 87*c^2*d^4 + 336*c*d^5 + 153*d^6)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((a^3*c^7*d^2 - 3*a^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6 - a^3*c^2*d^7 + 3*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^5 + (2*a^3*c^8*d - 3*a^3*c^7*d^2 - 7*a^3*c^6*d^3 + 13*a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 17*a^3*c^3*d^6 + 3*a^3*c^2*d^7 + 7*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^4 - (a^3*c^9 + a^3*c^8*d - 8*a^3*c^7*d^2 + 18*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 16*a^3*c^3*d^6 + 8*a^3*c^2*d^7 + 5*a^3*c*d^8 - 3*a^3*d^9)*f*cos(f*x + e)^3 - (3*a^3*c^9 + a^3*c^8*d - 20*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 42*a^3*c^5*d^4 - 18*a^3*c^4*d^5 - 36*a^3*c^3*d^6 + 20*a^3*c^2*d^7 + 11*a^3*c*d^8 - 7*a^3*d^9)*f*cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f + ((a^3*c^7*d^2 - 3*a^3*c^6*d^3 + a^3*c^5*d^4 + 5*a^3*c^4*d^5 - 5*a^3*c^3*d^6 - a^3*c^2*d^7 + 3*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^4 - 2*(a^3*c^8*d - 2*a^3*c^7*d^2 - 2*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^3*d^6 + 2*a^3*c^2*d^7 + 2*a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e)^3 - (a^3*c^9 + 3*a^3*c^8*d - 12*a^3*c^7*d^2 - 4*a^3*c^6*d^3 + 30*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 28*a^3*c^3*d^6 + 12*a^3*c^2*d^7 + 9*a^3*c*d^8 - 5*a^3*d^9)*f*cos(f*x + e)^2 + 2*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f*cos(f*x + e) + 4*(a^3*c^9 - a^3*c^8*d - 4*a^3*c^7*d^2 + 4*a^3*c^6*d^3 + 6*a^3*c^5*d^4 - 6*a^3*c^4*d^5 - 4*a^3*c^3*d^6 + 4*a^3*c^2*d^7 + a^3*c*d^8 - a^3*d^9)*f)*sin(f*x + e))]","B",0
606,0,0,0,1.830426," ","integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(d \sin\left(f x + e\right) + c\right)}^{n}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^n, x)","F",0
607,0,0,0,0.880508," ","integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)\right)} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(-(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e))*(a*sin(f*x + e) + a)^m, x)","F",0
608,0,0,0,1.417233," ","integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}\right)} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(-(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)*(a*sin(f*x + e) + a)^m, x)","F",0
609,0,0,0,1.443496," ","integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left({\left(d \sin\left(f x + e\right) + c\right)} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral((d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m, x)","F",0
610,0,0,0,1.436027," ","integrate((a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m, x)","F",0
611,0,0,0,1.337305," ","integrate((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{m}}{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m/(d*sin(f*x + e) + c), x)","F",0
612,0,0,0,1.306081," ","integrate((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{m}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral(-(a*sin(f*x + e) + a)^m/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
613,0,0,0,1.290234," ","integrate((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{m}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(a*sin(f*x + e) + a)^m/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
614,0,0,0,1.358837," ","integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(-(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)*sqrt(d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m, x)","F",0
615,0,0,0,1.264845," ","integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(d \sin\left(f x + e\right) + c\right)}^{\frac{3}{2}} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral((d*sin(f*x + e) + c)^(3/2)*(a*sin(f*x + e) + a)^m, x)","F",0
616,0,0,0,1.260169," ","integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{d \sin\left(f x + e\right) + c} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(sqrt(d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m, x)","F",0
617,0,0,0,1.370175," ","integrate((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{m}}{\sqrt{d \sin\left(f x + e\right) + c}}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m/sqrt(d*sin(f*x + e) + c), x)","F",0
618,0,0,0,1.077585," ","integrate((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{d \sin\left(f x + e\right) + c} {\left(a \sin\left(f x + e\right) + a\right)}^{m}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral(-sqrt(d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
619,0,0,0,1.637600," ","integrate((a+a*sin(f*x+e))^m/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{d \sin\left(f x + e\right) + c} {\left(a \sin\left(f x + e\right) + a\right)}^{m}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(d*sin(f*x + e) + c)*(a*sin(f*x + e) + a)^m/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
620,0,0,0,1.394972," ","integrate((1+sin(f*x+e))^m*(3+5*sin(f*x+e))^(-1-m),x, algorithm=""fricas"")","{\rm integral}\left({\left(5 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1} {\left(\sin\left(f x + e\right) + 1\right)}^{m}, x\right)"," ",0,"integral((5*sin(f*x + e) + 3)^(-m - 1)*(sin(f*x + e) + 1)^m, x)","F",0
621,0,0,0,1.457502," ","integrate((1+sin(f*x+e))^m*(3+4*sin(f*x+e))^(-1-m),x, algorithm=""fricas"")","{\rm integral}\left({\left(4 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1} {\left(\sin\left(f x + e\right) + 1\right)}^{m}, x\right)"," ",0,"integral((4*sin(f*x + e) + 3)^(-m - 1)*(sin(f*x + e) + 1)^m, x)","F",0
622,1,54,0,0.955375," ","integrate((1+sin(f*x+e))^m*(3+3*sin(f*x+e))^(-1-m),x, algorithm=""fricas"")","-\frac{3^{-m - 1} {\left(\cos\left(f x + e\right) + 1\right)} - 3^{-m - 1} \sin\left(f x + e\right)}{f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f}"," ",0,"-(3^(-m - 1)*(cos(f*x + e) + 1) - 3^(-m - 1)*sin(f*x + e))/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
623,0,0,0,1.209805," ","integrate((1+sin(f*x+e))^m*(3+2*sin(f*x+e))^(-1-m),x, algorithm=""fricas"")","{\rm integral}\left({\left(2 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1} {\left(\sin\left(f x + e\right) + 1\right)}^{m}, x\right)"," ",0,"integral((2*sin(f*x + e) + 3)^(-m - 1)*(sin(f*x + e) + 1)^m, x)","F",0
624,0,0,0,1.210258," ","integrate((1+sin(f*x+e))^m*(3+sin(f*x+e))^(-1-m),x, algorithm=""fricas"")","{\rm integral}\left({\left(\sin\left(f x + e\right) + 3\right)}^{-m - 1} {\left(\sin\left(f x + e\right) + 1\right)}^{m}, x\right)"," ",0,"integral((sin(f*x + e) + 3)^(-m - 1)*(sin(f*x + e) + 1)^m, x)","F",0
625,0,0,0,1.229597," ","integrate(3^(-1-m)*(1+sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left(3^{-m - 1} {\left(\sin\left(f x + e\right) + 1\right)}^{m}, x\right)"," ",0,"integral(3^(-m - 1)*(sin(f*x + e) + 1)^m, x)","F",0
626,0,0,0,1.172661," ","integrate((3-sin(f*x+e))^(-1-m)*(1+sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(\sin\left(f x + e\right) + 1\right)}^{m} {\left(-\sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((sin(f*x + e) + 1)^m*(-sin(f*x + e) + 3)^(-m - 1), x)","F",0
627,0,0,0,1.027264," ","integrate((3-2*sin(f*x+e))^(-1-m)*(1+sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(\sin\left(f x + e\right) + 1\right)}^{m} {\left(-2 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((sin(f*x + e) + 1)^m*(-2*sin(f*x + e) + 3)^(-m - 1), x)","F",0
628,1,41,0,0.810398," ","integrate((3-3*sin(f*x+e))^(-1-m)*(1+sin(f*x+e))^m,x, algorithm=""fricas"")","\frac{{\left(\sin\left(f x + e\right) + 1\right)}^{m} {\left(-3 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1} \cos\left(f x + e\right)}{2 \, f m + f}"," ",0,"(sin(f*x + e) + 1)^m*(-3*sin(f*x + e) + 3)^(-m - 1)*cos(f*x + e)/(2*f*m + f)","A",0
629,0,0,0,0.761921," ","integrate((3-4*sin(f*x+e))^(-1-m)*(1+sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(\sin\left(f x + e\right) + 1\right)}^{m} {\left(-4 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((sin(f*x + e) + 1)^m*(-4*sin(f*x + e) + 3)^(-m - 1), x)","F",0
630,0,0,0,1.415024," ","integrate((3-5*sin(f*x+e))^(-1-m)*(1+sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(\sin\left(f x + e\right) + 1\right)}^{m} {\left(-5 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((sin(f*x + e) + 1)^m*(-5*sin(f*x + e) + 3)^(-m - 1), x)","F",0
631,0,0,0,1.400797," ","integrate((3+5*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(5 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(5*sin(f*x + e) + 3)^(-m - 1), x)","F",0
632,0,0,0,1.382299," ","integrate((3+4*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(4 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(4*sin(f*x + e) + 3)^(-m - 1), x)","F",0
633,1,43,0,1.146952," ","integrate((3+3*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","-\frac{\left(\frac{1}{3} \, a\right)^{m} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{3 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"-1/3*(1/3*a)^m*(cos(f*x + e) - sin(f*x + e) + 1)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
634,0,0,0,1.399530," ","integrate((3+2*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(2 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(2*sin(f*x + e) + 3)^(-m - 1), x)","F",0
635,0,0,0,1.275193," ","integrate((3+sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(\sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(sin(f*x + e) + 3)^(-m - 1), x)","F",0
636,0,0,0,1.444530," ","integrate(3^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left(3^{-m - 1} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(3^(-m - 1)*(a*sin(f*x + e) + a)^m, x)","F",0
637,0,0,0,1.423583," ","integrate((3-sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-\sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(-sin(f*x + e) + 3)^(-m - 1), x)","F",0
638,0,0,0,1.471164," ","integrate((3-2*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-2 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(-2*sin(f*x + e) + 3)^(-m - 1), x)","F",0
639,1,43,0,1.402144," ","integrate((3-3*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-3 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1} \cos\left(f x + e\right)}{2 \, f m + f}"," ",0,"(a*sin(f*x + e) + a)^m*(-3*sin(f*x + e) + 3)^(-m - 1)*cos(f*x + e)/(2*f*m + f)","A",0
640,0,0,0,1.447349," ","integrate((3-4*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-4 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(-4*sin(f*x + e) + 3)^(-m - 1), x)","F",0
641,0,0,0,1.341539," ","integrate((3-5*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-5 \, \sin\left(f x + e\right) + 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(-5*sin(f*x + e) + 3)^(-m - 1), x)","F",0
642,0,0,0,1.462011," ","integrate((-3+5*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(5 \, \sin\left(f x + e\right) - 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(5*sin(f*x + e) - 3)^(-m - 1), x)","F",0
643,0,0,0,1.366710," ","integrate((-3+4*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(4 \, \sin\left(f x + e\right) - 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(4*sin(f*x + e) - 3)^(-m - 1), x)","F",0
644,1,43,0,0.847680," ","integrate((-3+3*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","\frac{{\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(3 \, \sin\left(f x + e\right) - 3\right)}^{-m - 1} \cos\left(f x + e\right)}{2 \, f m + f}"," ",0,"(a*sin(f*x + e) + a)^m*(3*sin(f*x + e) - 3)^(-m - 1)*cos(f*x + e)/(2*f*m + f)","A",0
645,0,0,0,0.836701," ","integrate((-3+2*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(2 \, \sin\left(f x + e\right) - 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(2*sin(f*x + e) - 3)^(-m - 1), x)","F",0
646,0,0,0,1.357659," ","integrate((-3+sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(\sin\left(f x + e\right) - 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(sin(f*x + e) - 3)^(-m - 1), x)","F",0
647,0,0,0,1.188529," ","integrate((-3)^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left(\left(-3\right)^{-m - 1} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral((-3)^(-m - 1)*(a*sin(f*x + e) + a)^m, x)","F",0
648,0,0,0,1.314828," ","integrate((-3-sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-\sin\left(f x + e\right) - 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(-sin(f*x + e) - 3)^(-m - 1), x)","F",0
649,0,0,0,1.276182," ","integrate((-3-2*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-2 \, \sin\left(f x + e\right) - 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(-2*sin(f*x + e) - 3)^(-m - 1), x)","F",0
650,1,43,0,1.198497," ","integrate((-3-3*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","\frac{\left(-\frac{1}{3} \, a\right)^{m} {\left(\cos\left(f x + e\right) - \sin\left(f x + e\right) + 1\right)}}{3 \, {\left(f \cos\left(f x + e\right) + f \sin\left(f x + e\right) + f\right)}}"," ",0,"1/3*(-1/3*a)^m*(cos(f*x + e) - sin(f*x + e) + 1)/(f*cos(f*x + e) + f*sin(f*x + e) + f)","A",0
651,0,0,0,1.348745," ","integrate((-3-4*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-4 \, \sin\left(f x + e\right) - 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(-4*sin(f*x + e) - 3)^(-m - 1), x)","F",0
652,0,0,0,1.005486," ","integrate((-3-5*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(-5 \, \sin\left(f x + e\right) - 3\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(-5*sin(f*x + e) - 3)^(-m - 1), x)","F",0
653,0,0,0,1.412318," ","integrate((d*sin(f*x+e))^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \left(d \sin\left(f x + e\right)\right)^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(d*sin(f*x + e))^(-m - 1), x)","F",0
654,0,0,0,1.307903," ","integrate((a+a*sin(f*x+e))^m*(c+d*sin(f*x+e))^(-1-m),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} {\left(d \sin\left(f x + e\right) + c\right)}^{-m - 1}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^(-m - 1), x)","F",0
655,0,0,0,1.107133," ","integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} {\left(d \sin\left(f x + e\right) + c\right)}^{n}, x\right)"," ",0,"integral(-(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*(d*sin(f*x + e) + c)^n, x)","F",0
656,0,0,0,1.392816," ","integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}\right)} {\left(d \sin\left(f x + e\right) + c\right)}^{n}, x\right)"," ",0,"integral(-(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*(d*sin(f*x + e) + c)^n, x)","F",0
657,0,0,0,1.070762," ","integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)} {\left(d \sin\left(f x + e\right) + c\right)}^{n}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^n, x)","F",0
658,0,0,0,1.242982," ","integrate((c+d*sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(d \sin\left(f x + e\right) + c\right)}^{n}, x\right)"," ",0,"integral((d*sin(f*x + e) + c)^n, x)","F",0
659,0,0,0,1.221461," ","integrate((c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(d \sin\left(f x + e\right) + c\right)}^{n}}{a \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral((d*sin(f*x + e) + c)^n/(a*sin(f*x + e) + a), x)","F",0
660,0,0,0,1.421232," ","integrate((c+d*sin(f*x+e))^n/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(d \sin\left(f x + e\right) + c\right)}^{n}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-(d*sin(f*x + e) + c)^n/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
661,0,0,0,1.422335," ","integrate((c+d*sin(f*x+e))^n/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(d \sin\left(f x + e\right) + c\right)}^{n}}{3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(d*sin(f*x + e) + c)^n/(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e)), x)","F",0
662,0,0,0,1.524817," ","integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}\right)} \sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) + c\right)}^{n}, x\right)"," ",0,"integral(-(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^n, x)","F",0
663,0,0,0,0.817204," ","integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{\frac{3}{2}} {\left(d \sin\left(f x + e\right) + c\right)}^{n}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^(3/2)*(d*sin(f*x + e) + c)^n, x)","F",0
664,0,0,0,1.404958," ","integrate((a+a*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) + c\right)}^{n}, x\right)"," ",0,"integral(sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^n, x)","F",0
665,0,0,0,0.859554," ","integrate((c+d*sin(f*x+e))^n/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(d \sin\left(f x + e\right) + c\right)}^{n}}{\sqrt{a \sin\left(f x + e\right) + a}}, x\right)"," ",0,"integral((d*sin(f*x + e) + c)^n/sqrt(a*sin(f*x + e) + a), x)","F",0
666,0,0,0,1.490746," ","integrate((c+d*sin(f*x+e))^n/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) + c\right)}^{n}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^n/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
667,0,0,0,1.555445," ","integrate((c+d*sin(f*x+e))^n/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{a \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) + c\right)}^{n}}{3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^n/(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e)), x)","F",0
668,0,0,0,1.420925," ","integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^(1/3),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)} {\left(d \sin\left(f x + e\right) + c\right)}^{\frac{1}{3}}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(1/3), x)","F",0
669,0,0,0,1.521649," ","integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(1/3),x, algorithm=""fricas"")","{\rm integral}\left(\frac{a \sin\left(f x + e\right) + a}{{\left(d \sin\left(f x + e\right) + c\right)}^{\frac{1}{3}}}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^(1/3), x)","F",0
670,0,0,0,1.158250," ","integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(4/3),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(a \sin\left(f x + e\right) + a\right)} {\left(d \sin\left(f x + e\right) + c\right)}^{\frac{2}{3}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral(-(a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(2/3)/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
671,1,145,0,1.427336," ","integrate((a+b*sin(f*x+e))*(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{8 \, {\left(3 \, b c d^{2} + a d^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(8 \, a c^{3} + 12 \, b c^{2} d + 12 \, a c d^{2} + 3 \, b d^{3}\right)} f x - 24 \, {\left(b c^{3} + 3 \, a c^{2} d + 3 \, b c d^{2} + a d^{3}\right)} \cos\left(f x + e\right) + 3 \, {\left(2 \, b d^{3} \cos\left(f x + e\right)^{3} - {\left(12 \, b c^{2} d + 12 \, a c d^{2} + 5 \, b d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, f}"," ",0,"1/24*(8*(3*b*c*d^2 + a*d^3)*cos(f*x + e)^3 + 3*(8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*f*x - 24*(b*c^3 + 3*a*c^2*d + 3*b*c*d^2 + a*d^3)*cos(f*x + e) + 3*(2*b*d^3*cos(f*x + e)^3 - (12*b*c^2*d + 12*a*c*d^2 + 5*b*d^3)*cos(f*x + e))*sin(f*x + e))/f","A",0
672,1,90,0,1.333467," ","integrate((a+b*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{2 \, b d^{2} \cos\left(f x + e\right)^{3} + 3 \, {\left(2 \, a c^{2} + 2 \, b c d + a d^{2}\right)} f x - 3 \, {\left(2 \, b c d + a d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 6 \, {\left(b c^{2} + 2 \, a c d + b d^{2}\right)} \cos\left(f x + e\right)}{6 \, f}"," ",0,"1/6*(2*b*d^2*cos(f*x + e)^3 + 3*(2*a*c^2 + 2*b*c*d + a*d^2)*f*x - 3*(2*b*c*d + a*d^2)*cos(f*x + e)*sin(f*x + e) - 6*(b*c^2 + 2*a*c*d + b*d^2)*cos(f*x + e))/f","A",0
673,1,48,0,1.416108," ","integrate((a+b*sin(f*x+e))*(c+d*sin(f*x+e)),x, algorithm=""fricas"")","-\frac{b d \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(2 \, a c + b d\right)} f x + 2 \, {\left(b c + a d\right)} \cos\left(f x + e\right)}{2 \, f}"," ",0,"-1/2*(b*d*cos(f*x + e)*sin(f*x + e) - (2*a*c + b*d)*f*x + 2*(b*c + a*d)*cos(f*x + e))/f","A",0
674,1,18,0,1.295214," ","integrate(a+b*sin(f*x+e),x, algorithm=""fricas"")","\frac{a f x - b \cos\left(f x + e\right)}{f}"," ",0,"(a*f*x - b*cos(f*x + e))/f","A",0
675,1,255,0,0.826458," ","integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(b c^{2} - b d^{2}\right)} f x + {\left(b c - a d\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right)}{2 \, {\left(c^{2} d - d^{3}\right)} f}, \frac{{\left(b c^{2} - b d^{2}\right)} f x + {\left(b c - a d\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right)}{{\left(c^{2} d - d^{3}\right)} f}\right]"," ",0,"[1/2*(2*(b*c^2 - b*d^2)*f*x + (b*c - a*d)*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)))/((c^2*d - d^3)*f), ((b*c^2 - b*d^2)*f*x + (b*c - a*d)*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))))/((c^2*d - d^3)*f)]","A",0
676,1,394,0,1.208735," ","integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[-\frac{{\left(a c^{2} - b c d + {\left(a c d - b d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(b c^{3} - a c^{2} d - b c d^{2} + a d^{3}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(c^{4} d - 2 \, c^{2} d^{3} + d^{5}\right)} f \sin\left(f x + e\right) + {\left(c^{5} - 2 \, c^{3} d^{2} + c d^{4}\right)} f\right)}}, -\frac{{\left(a c^{2} - b c d + {\left(a c d - b d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(b c^{3} - a c^{2} d - b c d^{2} + a d^{3}\right)} \cos\left(f x + e\right)}{{\left(c^{4} d - 2 \, c^{2} d^{3} + d^{5}\right)} f \sin\left(f x + e\right) + {\left(c^{5} - 2 \, c^{3} d^{2} + c d^{4}\right)} f}\right]"," ",0,"[-1/2*((a*c^2 - b*c*d + (a*c*d - b*d^2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(b*c^3 - a*c^2*d - b*c*d^2 + a*d^3)*cos(f*x + e))/((c^4*d - 2*c^2*d^3 + d^5)*f*sin(f*x + e) + (c^5 - 2*c^3*d^2 + c*d^4)*f), -((a*c^2 - b*c*d + (a*c*d - b*d^2)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (b*c^3 - a*c^2*d - b*c*d^2 + a*d^3)*cos(f*x + e))/((c^4*d - 2*c^2*d^3 + d^5)*f*sin(f*x + e) + (c^5 - 2*c^3*d^2 + c*d^4)*f)]","A",0
677,1,793,0,0.948096," ","integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(b c^{4} d - 3 \, a c^{3} d^{2} + b c^{2} d^{3} + 3 \, a c d^{4} - 2 \, b d^{5}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(2 \, a c^{4} - 3 \, b c^{3} d + 3 \, a c^{2} d^{2} - 3 \, b c d^{3} + a d^{4} - {\left(2 \, a c^{2} d^{2} - 3 \, b c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, a c^{3} d - 3 \, b c^{2} d^{2} + a c d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(2 \, b c^{5} - 4 \, a c^{4} d - b c^{3} d^{2} + 5 \, a c^{2} d^{3} - b c d^{4} - a d^{5}\right)} \cos\left(f x + e\right)}{4 \, {\left({\left(c^{6} d^{2} - 3 \, c^{4} d^{4} + 3 \, c^{2} d^{6} - d^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{7} d - 3 \, c^{5} d^{3} + 3 \, c^{3} d^{5} - c d^{7}\right)} f \sin\left(f x + e\right) - {\left(c^{8} - 2 \, c^{6} d^{2} + 2 \, c^{2} d^{6} - d^{8}\right)} f\right)}}, \frac{{\left(b c^{4} d - 3 \, a c^{3} d^{2} + b c^{2} d^{3} + 3 \, a c d^{4} - 2 \, b d^{5}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(2 \, a c^{4} - 3 \, b c^{3} d + 3 \, a c^{2} d^{2} - 3 \, b c d^{3} + a d^{4} - {\left(2 \, a c^{2} d^{2} - 3 \, b c d^{3} + a d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, a c^{3} d - 3 \, b c^{2} d^{2} + a c d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(2 \, b c^{5} - 4 \, a c^{4} d - b c^{3} d^{2} + 5 \, a c^{2} d^{3} - b c d^{4} - a d^{5}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(c^{6} d^{2} - 3 \, c^{4} d^{4} + 3 \, c^{2} d^{6} - d^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{7} d - 3 \, c^{5} d^{3} + 3 \, c^{3} d^{5} - c d^{7}\right)} f \sin\left(f x + e\right) - {\left(c^{8} - 2 \, c^{6} d^{2} + 2 \, c^{2} d^{6} - d^{8}\right)} f\right)}}\right]"," ",0,"[1/4*(2*(b*c^4*d - 3*a*c^3*d^2 + b*c^2*d^3 + 3*a*c*d^4 - 2*b*d^5)*cos(f*x + e)*sin(f*x + e) + (2*a*c^4 - 3*b*c^3*d + 3*a*c^2*d^2 - 3*b*c*d^3 + a*d^4 - (2*a*c^2*d^2 - 3*b*c*d^3 + a*d^4)*cos(f*x + e)^2 + 2*(2*a*c^3*d - 3*b*c^2*d^2 + a*c*d^3)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(2*b*c^5 - 4*a*c^4*d - b*c^3*d^2 + 5*a*c^2*d^3 - b*c*d^4 - a*d^5)*cos(f*x + e))/((c^6*d^2 - 3*c^4*d^4 + 3*c^2*d^6 - d^8)*f*cos(f*x + e)^2 - 2*(c^7*d - 3*c^5*d^3 + 3*c^3*d^5 - c*d^7)*f*sin(f*x + e) - (c^8 - 2*c^6*d^2 + 2*c^2*d^6 - d^8)*f), 1/2*((b*c^4*d - 3*a*c^3*d^2 + b*c^2*d^3 + 3*a*c*d^4 - 2*b*d^5)*cos(f*x + e)*sin(f*x + e) + (2*a*c^4 - 3*b*c^3*d + 3*a*c^2*d^2 - 3*b*c*d^3 + a*d^4 - (2*a*c^2*d^2 - 3*b*c*d^3 + a*d^4)*cos(f*x + e)^2 + 2*(2*a*c^3*d - 3*b*c^2*d^2 + a*c*d^3)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (2*b*c^5 - 4*a*c^4*d - b*c^3*d^2 + 5*a*c^2*d^3 - b*c*d^4 - a*d^5)*cos(f*x + e))/((c^6*d^2 - 3*c^4*d^4 + 3*c^2*d^6 - d^8)*f*cos(f*x + e)^2 - 2*(c^7*d - 3*c^5*d^3 + 3*c^3*d^5 - c*d^7)*f*sin(f*x + e) - (c^8 - 2*c^6*d^2 + 2*c^2*d^6 - d^8)*f)]","B",0
678,1,247,0,0.815717," ","integrate((a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{24 \, b^{2} d^{3} \cos\left(f x + e\right)^{5} - 40 \, {\left(3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + {\left(a^{2} + 2 \, b^{2}\right)} d^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(24 \, a b c^{2} d + 6 \, a b d^{3} + 4 \, {\left(2 \, a^{2} + b^{2}\right)} c^{3} + 3 \, {\left(4 \, a^{2} + 3 \, b^{2}\right)} c d^{2}\right)} f x + 120 \, {\left(2 \, a b c^{3} + 6 \, a b c d^{2} + 3 \, {\left(a^{2} + b^{2}\right)} c^{2} d + {\left(a^{2} + b^{2}\right)} d^{3}\right)} \cos\left(f x + e\right) - 15 \, {\left(2 \, {\left(3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(4 \, b^{2} c^{3} + 24 \, a b c^{2} d + 10 \, a b d^{3} + 3 \, {\left(4 \, a^{2} + 5 \, b^{2}\right)} c d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{120 \, f}"," ",0,"-1/120*(24*b^2*d^3*cos(f*x + e)^5 - 40*(3*b^2*c^2*d + 6*a*b*c*d^2 + (a^2 + 2*b^2)*d^3)*cos(f*x + e)^3 - 15*(24*a*b*c^2*d + 6*a*b*d^3 + 4*(2*a^2 + b^2)*c^3 + 3*(4*a^2 + 3*b^2)*c*d^2)*f*x + 120*(2*a*b*c^3 + 6*a*b*c*d^2 + 3*(a^2 + b^2)*c^2*d + (a^2 + b^2)*d^3)*cos(f*x + e) - 15*(2*(3*b^2*c*d^2 + 2*a*b*d^3)*cos(f*x + e)^3 - (4*b^2*c^3 + 24*a*b*c^2*d + 10*a*b*d^3 + 3*(4*a^2 + 5*b^2)*c*d^2)*cos(f*x + e))*sin(f*x + e))/f","A",0
679,1,163,0,1.743566," ","integrate((a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\frac{16 \, {\left(b^{2} c d + a b d^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(16 \, a b c d + 4 \, {\left(2 \, a^{2} + b^{2}\right)} c^{2} + {\left(4 \, a^{2} + 3 \, b^{2}\right)} d^{2}\right)} f x - 48 \, {\left(a b c^{2} + a b d^{2} + {\left(a^{2} + b^{2}\right)} c d\right)} \cos\left(f x + e\right) + 3 \, {\left(2 \, b^{2} d^{2} \cos\left(f x + e\right)^{3} - {\left(4 \, b^{2} c^{2} + 16 \, a b c d + {\left(4 \, a^{2} + 5 \, b^{2}\right)} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, f}"," ",0,"1/24*(16*(b^2*c*d + a*b*d^2)*cos(f*x + e)^3 + 3*(16*a*b*c*d + 4*(2*a^2 + b^2)*c^2 + (4*a^2 + 3*b^2)*d^2)*f*x - 48*(a*b*c^2 + a*b*d^2 + (a^2 + b^2)*c*d)*cos(f*x + e) + 3*(2*b^2*d^2*cos(f*x + e)^3 - (4*b^2*c^2 + 16*a*b*c*d + (4*a^2 + 5*b^2)*d^2)*cos(f*x + e))*sin(f*x + e))/f","A",0
680,1,89,0,1.424614," ","integrate((a+b*sin(f*x+e))^2*(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\frac{2 \, b^{2} d \cos\left(f x + e\right)^{3} + 3 \, {\left(2 \, a b d + {\left(2 \, a^{2} + b^{2}\right)} c\right)} f x - 3 \, {\left(b^{2} c + 2 \, a b d\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 6 \, {\left(2 \, a b c + {\left(a^{2} + b^{2}\right)} d\right)} \cos\left(f x + e\right)}{6 \, f}"," ",0,"1/6*(2*b^2*d*cos(f*x + e)^3 + 3*(2*a*b*d + (2*a^2 + b^2)*c)*f*x - 3*(b^2*c + 2*a*b*d)*cos(f*x + e)*sin(f*x + e) - 6*(2*a*b*c + (a^2 + b^2)*d)*cos(f*x + e))/f","A",0
681,1,45,0,1.317840," ","integrate((a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{b^{2} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(2 \, a^{2} + b^{2}\right)} f x + 4 \, a b \cos\left(f x + e\right)}{2 \, f}"," ",0,"-1/2*(b^2*cos(f*x + e)*sin(f*x + e) - (2*a^2 + b^2)*f*x + 4*a*b*cos(f*x + e))/f","A",0
682,1,375,0,1.500075," ","integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(b^{2} c^{3} - 2 \, a b c^{2} d - b^{2} c d^{2} + 2 \, a b d^{3}\right)} f x + {\left(b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(b^{2} c^{2} d - b^{2} d^{3}\right)} \cos\left(f x + e\right)}{2 \, {\left(c^{2} d^{2} - d^{4}\right)} f}, -\frac{{\left(b^{2} c^{3} - 2 \, a b c^{2} d - b^{2} c d^{2} + 2 \, a b d^{3}\right)} f x + {\left(b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(b^{2} c^{2} d - b^{2} d^{3}\right)} \cos\left(f x + e\right)}{{\left(c^{2} d^{2} - d^{4}\right)} f}\right]"," ",0,"[-1/2*(2*(b^2*c^3 - 2*a*b*c^2*d - b^2*c*d^2 + 2*a*b*d^3)*f*x + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(b^2*c^2*d - b^2*d^3)*cos(f*x + e))/((c^2*d^2 - d^4)*f), -((b^2*c^3 - 2*a*b*c^2*d - b^2*c*d^2 + 2*a*b*d^3)*f*x + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (b^2*c^2*d - b^2*d^3)*cos(f*x + e))/((c^2*d^2 - d^4)*f)]","A",0
683,1,677,0,1.649824," ","integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(b^{2} c^{4} d - 2 \, b^{2} c^{2} d^{3} + b^{2} d^{5}\right)} f x \sin\left(f x + e\right) + 2 \, {\left(b^{2} c^{5} - 2 \, b^{2} c^{3} d^{2} + b^{2} c d^{4}\right)} f x - {\left(b^{2} c^{4} + 2 \, a b c d^{3} - {\left(a^{2} + 2 \, b^{2}\right)} c^{2} d^{2} + {\left(b^{2} c^{3} d + 2 \, a b d^{4} - {\left(a^{2} + 2 \, b^{2}\right)} c d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(-\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} - 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + 2 \, a b c d^{4} - a^{2} d^{5} + {\left(a^{2} - b^{2}\right)} c^{2} d^{3}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(c^{4} d^{3} - 2 \, c^{2} d^{5} + d^{7}\right)} f \sin\left(f x + e\right) + {\left(c^{5} d^{2} - 2 \, c^{3} d^{4} + c d^{6}\right)} f\right)}}, \frac{{\left(b^{2} c^{4} d - 2 \, b^{2} c^{2} d^{3} + b^{2} d^{5}\right)} f x \sin\left(f x + e\right) + {\left(b^{2} c^{5} - 2 \, b^{2} c^{3} d^{2} + b^{2} c d^{4}\right)} f x + {\left(b^{2} c^{4} + 2 \, a b c d^{3} - {\left(a^{2} + 2 \, b^{2}\right)} c^{2} d^{2} + {\left(b^{2} c^{3} d + 2 \, a b d^{4} - {\left(a^{2} + 2 \, b^{2}\right)} c d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + 2 \, a b c d^{4} - a^{2} d^{5} + {\left(a^{2} - b^{2}\right)} c^{2} d^{3}\right)} \cos\left(f x + e\right)}{{\left(c^{4} d^{3} - 2 \, c^{2} d^{5} + d^{7}\right)} f \sin\left(f x + e\right) + {\left(c^{5} d^{2} - 2 \, c^{3} d^{4} + c d^{6}\right)} f}\right]"," ",0,"[1/2*(2*(b^2*c^4*d - 2*b^2*c^2*d^3 + b^2*d^5)*f*x*sin(f*x + e) + 2*(b^2*c^5 - 2*b^2*c^3*d^2 + b^2*c*d^4)*f*x - (b^2*c^4 + 2*a*b*c*d^3 - (a^2 + 2*b^2)*c^2*d^2 + (b^2*c^3*d + 2*a*b*d^4 - (a^2 + 2*b^2)*c*d^3)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(-((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 - 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(b^2*c^4*d - 2*a*b*c^3*d^2 + 2*a*b*c*d^4 - a^2*d^5 + (a^2 - b^2)*c^2*d^3)*cos(f*x + e))/((c^4*d^3 - 2*c^2*d^5 + d^7)*f*sin(f*x + e) + (c^5*d^2 - 2*c^3*d^4 + c*d^6)*f), ((b^2*c^4*d - 2*b^2*c^2*d^3 + b^2*d^5)*f*x*sin(f*x + e) + (b^2*c^5 - 2*b^2*c^3*d^2 + b^2*c*d^4)*f*x + (b^2*c^4 + 2*a*b*c*d^3 - (a^2 + 2*b^2)*c^2*d^2 + (b^2*c^3*d + 2*a*b*d^4 - (a^2 + 2*b^2)*c*d^3)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (b^2*c^4*d - 2*a*b*c^3*d^2 + 2*a*b*c*d^4 - a^2*d^5 + (a^2 - b^2)*c^2*d^3)*cos(f*x + e))/((c^4*d^3 - 2*c^2*d^5 + d^7)*f*sin(f*x + e) + (c^5*d^2 - 2*c^3*d^4 + c*d^6)*f)]","B",0
684,1,1027,0,1.051468," ","integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(b^{2} c^{5} + 2 \, a b c^{4} d + 2 \, a b c^{2} d^{3} - 4 \, a b d^{5} - {\left(3 \, a^{2} + 5 \, b^{2}\right)} c^{3} d^{2} + {\left(3 \, a^{2} + 4 \, b^{2}\right)} c d^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(6 \, a b c^{3} d + 6 \, a b c d^{3} - {\left(2 \, a^{2} + b^{2}\right)} c^{4} - 3 \, {\left(a^{2} + b^{2}\right)} c^{2} d^{2} - {\left(a^{2} + 2 \, b^{2}\right)} d^{4} - {\left(6 \, a b c d^{3} - {\left(2 \, a^{2} + b^{2}\right)} c^{2} d^{2} - {\left(a^{2} + 2 \, b^{2}\right)} d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(6 \, a b c^{2} d^{2} - {\left(2 \, a^{2} + b^{2}\right)} c^{3} d - {\left(a^{2} + 2 \, b^{2}\right)} c d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(4 \, a b c^{5} - 2 \, a b c^{3} d^{2} - 2 \, a b c d^{4} - a^{2} d^{5} - {\left(4 \, a^{2} + 3 \, b^{2}\right)} c^{4} d + {\left(5 \, a^{2} + 3 \, b^{2}\right)} c^{2} d^{3}\right)} \cos\left(f x + e\right)}{4 \, {\left({\left(c^{6} d^{2} - 3 \, c^{4} d^{4} + 3 \, c^{2} d^{6} - d^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{7} d - 3 \, c^{5} d^{3} + 3 \, c^{3} d^{5} - c d^{7}\right)} f \sin\left(f x + e\right) - {\left(c^{8} - 2 \, c^{6} d^{2} + 2 \, c^{2} d^{6} - d^{8}\right)} f\right)}}, \frac{{\left(b^{2} c^{5} + 2 \, a b c^{4} d + 2 \, a b c^{2} d^{3} - 4 \, a b d^{5} - {\left(3 \, a^{2} + 5 \, b^{2}\right)} c^{3} d^{2} + {\left(3 \, a^{2} + 4 \, b^{2}\right)} c d^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(6 \, a b c^{3} d + 6 \, a b c d^{3} - {\left(2 \, a^{2} + b^{2}\right)} c^{4} - 3 \, {\left(a^{2} + b^{2}\right)} c^{2} d^{2} - {\left(a^{2} + 2 \, b^{2}\right)} d^{4} - {\left(6 \, a b c d^{3} - {\left(2 \, a^{2} + b^{2}\right)} c^{2} d^{2} - {\left(a^{2} + 2 \, b^{2}\right)} d^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(6 \, a b c^{2} d^{2} - {\left(2 \, a^{2} + b^{2}\right)} c^{3} d - {\left(a^{2} + 2 \, b^{2}\right)} c d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(4 \, a b c^{5} - 2 \, a b c^{3} d^{2} - 2 \, a b c d^{4} - a^{2} d^{5} - {\left(4 \, a^{2} + 3 \, b^{2}\right)} c^{4} d + {\left(5 \, a^{2} + 3 \, b^{2}\right)} c^{2} d^{3}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(c^{6} d^{2} - 3 \, c^{4} d^{4} + 3 \, c^{2} d^{6} - d^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{7} d - 3 \, c^{5} d^{3} + 3 \, c^{3} d^{5} - c d^{7}\right)} f \sin\left(f x + e\right) - {\left(c^{8} - 2 \, c^{6} d^{2} + 2 \, c^{2} d^{6} - d^{8}\right)} f\right)}}\right]"," ",0,"[1/4*(2*(b^2*c^5 + 2*a*b*c^4*d + 2*a*b*c^2*d^3 - 4*a*b*d^5 - (3*a^2 + 5*b^2)*c^3*d^2 + (3*a^2 + 4*b^2)*c*d^4)*cos(f*x + e)*sin(f*x + e) - (6*a*b*c^3*d + 6*a*b*c*d^3 - (2*a^2 + b^2)*c^4 - 3*(a^2 + b^2)*c^2*d^2 - (a^2 + 2*b^2)*d^4 - (6*a*b*c*d^3 - (2*a^2 + b^2)*c^2*d^2 - (a^2 + 2*b^2)*d^4)*cos(f*x + e)^2 + 2*(6*a*b*c^2*d^2 - (2*a^2 + b^2)*c^3*d - (a^2 + 2*b^2)*c*d^3)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(4*a*b*c^5 - 2*a*b*c^3*d^2 - 2*a*b*c*d^4 - a^2*d^5 - (4*a^2 + 3*b^2)*c^4*d + (5*a^2 + 3*b^2)*c^2*d^3)*cos(f*x + e))/((c^6*d^2 - 3*c^4*d^4 + 3*c^2*d^6 - d^8)*f*cos(f*x + e)^2 - 2*(c^7*d - 3*c^5*d^3 + 3*c^3*d^5 - c*d^7)*f*sin(f*x + e) - (c^8 - 2*c^6*d^2 + 2*c^2*d^6 - d^8)*f), 1/2*((b^2*c^5 + 2*a*b*c^4*d + 2*a*b*c^2*d^3 - 4*a*b*d^5 - (3*a^2 + 5*b^2)*c^3*d^2 + (3*a^2 + 4*b^2)*c*d^4)*cos(f*x + e)*sin(f*x + e) - (6*a*b*c^3*d + 6*a*b*c*d^3 - (2*a^2 + b^2)*c^4 - 3*(a^2 + b^2)*c^2*d^2 - (a^2 + 2*b^2)*d^4 - (6*a*b*c*d^3 - (2*a^2 + b^2)*c^2*d^2 - (a^2 + 2*b^2)*d^4)*cos(f*x + e)^2 + 2*(6*a*b*c^2*d^2 - (2*a^2 + b^2)*c^3*d - (a^2 + 2*b^2)*c*d^3)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (4*a*b*c^5 - 2*a*b*c^3*d^2 - 2*a*b*c*d^4 - a^2*d^5 - (4*a^2 + 3*b^2)*c^4*d + (5*a^2 + 3*b^2)*c^2*d^3)*cos(f*x + e))/((c^6*d^2 - 3*c^4*d^4 + 3*c^2*d^6 - d^8)*f*cos(f*x + e)^2 - 2*(c^7*d - 3*c^5*d^3 + 3*c^3*d^5 - c*d^7)*f*sin(f*x + e) - (c^8 - 2*c^6*d^2 + 2*c^2*d^6 - d^8)*f)]","B",0
685,1,1724,0,1.409414," ","integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^4,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(b^{2} c^{6} d + 4 \, a b c^{5} d^{2} + 22 \, a b c^{3} d^{4} - 26 \, a b c d^{6} - 11 \, {\left(a^{2} + b^{2}\right)} c^{4} d^{3} + {\left(7 \, a^{2} + 4 \, b^{2}\right)} c^{2} d^{5} + 2 \, {\left(2 \, a^{2} + 3 \, b^{2}\right)} d^{7}\right)} \cos\left(f x + e\right)^{3} - 6 \, {\left(b^{2} c^{7} + 4 \, a b c^{6} d + 14 \, a b c^{4} d^{3} - 20 \, a b c^{2} d^{5} + 2 \, a b d^{7} - {\left(9 \, a^{2} + 10 \, b^{2}\right)} c^{5} d^{2} + {\left(8 \, a^{2} + 7 \, b^{2}\right)} c^{3} d^{4} + {\left(a^{2} + 2 \, b^{2}\right)} c d^{6}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + 3 \, {\left(8 \, a b c^{5} d + 26 \, a b c^{3} d^{3} + 6 \, a b c d^{5} - {\left(2 \, a^{2} + b^{2}\right)} c^{6} - {\left(9 \, a^{2} + 7 \, b^{2}\right)} c^{4} d^{2} - 3 \, {\left(3 \, a^{2} + 4 \, b^{2}\right)} c^{2} d^{4} - 3 \, {\left(8 \, a b c^{3} d^{3} + 2 \, a b c d^{5} - {\left(2 \, a^{2} + b^{2}\right)} c^{4} d^{2} - {\left(3 \, a^{2} + 4 \, b^{2}\right)} c^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(24 \, a b c^{4} d^{2} + 14 \, a b c^{2} d^{4} + 2 \, a b d^{6} - 3 \, {\left(2 \, a^{2} + b^{2}\right)} c^{5} d - {\left(11 \, a^{2} + 13 \, b^{2}\right)} c^{3} d^{3} - {\left(3 \, a^{2} + 4 \, b^{2}\right)} c d^{5} - {\left(8 \, a b c^{2} d^{4} + 2 \, a b d^{6} - {\left(2 \, a^{2} + b^{2}\right)} c^{3} d^{3} - {\left(3 \, a^{2} + 4 \, b^{2}\right)} c d^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 12 \, {\left(2 \, a b c^{7} + 2 \, a b c^{5} d^{2} + 2 \, a^{2} c^{4} d^{3} + b^{2} c^{2} d^{5} - 4 \, a b c d^{6} - {\left(3 \, a^{2} + 2 \, b^{2}\right)} c^{6} d + {\left(a^{2} + b^{2}\right)} d^{7}\right)} \cos\left(f x + e\right)}{12 \, {\left(3 \, {\left(c^{9} d^{2} - 4 \, c^{7} d^{4} + 6 \, c^{5} d^{6} - 4 \, c^{3} d^{8} + c d^{10}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{11} - c^{9} d^{2} - 6 \, c^{7} d^{4} + 14 \, c^{5} d^{6} - 11 \, c^{3} d^{8} + 3 \, c d^{10}\right)} f + {\left({\left(c^{8} d^{3} - 4 \, c^{6} d^{5} + 6 \, c^{4} d^{7} - 4 \, c^{2} d^{9} + d^{11}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{10} d - 11 \, c^{8} d^{3} + 14 \, c^{6} d^{5} - 6 \, c^{4} d^{7} - c^{2} d^{9} + d^{11}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left(b^{2} c^{6} d + 4 \, a b c^{5} d^{2} + 22 \, a b c^{3} d^{4} - 26 \, a b c d^{6} - 11 \, {\left(a^{2} + b^{2}\right)} c^{4} d^{3} + {\left(7 \, a^{2} + 4 \, b^{2}\right)} c^{2} d^{5} + 2 \, {\left(2 \, a^{2} + 3 \, b^{2}\right)} d^{7}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(b^{2} c^{7} + 4 \, a b c^{6} d + 14 \, a b c^{4} d^{3} - 20 \, a b c^{2} d^{5} + 2 \, a b d^{7} - {\left(9 \, a^{2} + 10 \, b^{2}\right)} c^{5} d^{2} + {\left(8 \, a^{2} + 7 \, b^{2}\right)} c^{3} d^{4} + {\left(a^{2} + 2 \, b^{2}\right)} c d^{6}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + 3 \, {\left(8 \, a b c^{5} d + 26 \, a b c^{3} d^{3} + 6 \, a b c d^{5} - {\left(2 \, a^{2} + b^{2}\right)} c^{6} - {\left(9 \, a^{2} + 7 \, b^{2}\right)} c^{4} d^{2} - 3 \, {\left(3 \, a^{2} + 4 \, b^{2}\right)} c^{2} d^{4} - 3 \, {\left(8 \, a b c^{3} d^{3} + 2 \, a b c d^{5} - {\left(2 \, a^{2} + b^{2}\right)} c^{4} d^{2} - {\left(3 \, a^{2} + 4 \, b^{2}\right)} c^{2} d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(24 \, a b c^{4} d^{2} + 14 \, a b c^{2} d^{4} + 2 \, a b d^{6} - 3 \, {\left(2 \, a^{2} + b^{2}\right)} c^{5} d - {\left(11 \, a^{2} + 13 \, b^{2}\right)} c^{3} d^{3} - {\left(3 \, a^{2} + 4 \, b^{2}\right)} c d^{5} - {\left(8 \, a b c^{2} d^{4} + 2 \, a b d^{6} - {\left(2 \, a^{2} + b^{2}\right)} c^{3} d^{3} - {\left(3 \, a^{2} + 4 \, b^{2}\right)} c d^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) - 6 \, {\left(2 \, a b c^{7} + 2 \, a b c^{5} d^{2} + 2 \, a^{2} c^{4} d^{3} + b^{2} c^{2} d^{5} - 4 \, a b c d^{6} - {\left(3 \, a^{2} + 2 \, b^{2}\right)} c^{6} d + {\left(a^{2} + b^{2}\right)} d^{7}\right)} \cos\left(f x + e\right)}{6 \, {\left(3 \, {\left(c^{9} d^{2} - 4 \, c^{7} d^{4} + 6 \, c^{5} d^{6} - 4 \, c^{3} d^{8} + c d^{10}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{11} - c^{9} d^{2} - 6 \, c^{7} d^{4} + 14 \, c^{5} d^{6} - 11 \, c^{3} d^{8} + 3 \, c d^{10}\right)} f + {\left({\left(c^{8} d^{3} - 4 \, c^{6} d^{5} + 6 \, c^{4} d^{7} - 4 \, c^{2} d^{9} + d^{11}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{10} d - 11 \, c^{8} d^{3} + 14 \, c^{6} d^{5} - 6 \, c^{4} d^{7} - c^{2} d^{9} + d^{11}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/12*(2*(b^2*c^6*d + 4*a*b*c^5*d^2 + 22*a*b*c^3*d^4 - 26*a*b*c*d^6 - 11*(a^2 + b^2)*c^4*d^3 + (7*a^2 + 4*b^2)*c^2*d^5 + 2*(2*a^2 + 3*b^2)*d^7)*cos(f*x + e)^3 - 6*(b^2*c^7 + 4*a*b*c^6*d + 14*a*b*c^4*d^3 - 20*a*b*c^2*d^5 + 2*a*b*d^7 - (9*a^2 + 10*b^2)*c^5*d^2 + (8*a^2 + 7*b^2)*c^3*d^4 + (a^2 + 2*b^2)*c*d^6)*cos(f*x + e)*sin(f*x + e) + 3*(8*a*b*c^5*d + 26*a*b*c^3*d^3 + 6*a*b*c*d^5 - (2*a^2 + b^2)*c^6 - (9*a^2 + 7*b^2)*c^4*d^2 - 3*(3*a^2 + 4*b^2)*c^2*d^4 - 3*(8*a*b*c^3*d^3 + 2*a*b*c*d^5 - (2*a^2 + b^2)*c^4*d^2 - (3*a^2 + 4*b^2)*c^2*d^4)*cos(f*x + e)^2 + (24*a*b*c^4*d^2 + 14*a*b*c^2*d^4 + 2*a*b*d^6 - 3*(2*a^2 + b^2)*c^5*d - (11*a^2 + 13*b^2)*c^3*d^3 - (3*a^2 + 4*b^2)*c*d^5 - (8*a*b*c^2*d^4 + 2*a*b*d^6 - (2*a^2 + b^2)*c^3*d^3 - (3*a^2 + 4*b^2)*c*d^5)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 12*(2*a*b*c^7 + 2*a*b*c^5*d^2 + 2*a^2*c^4*d^3 + b^2*c^2*d^5 - 4*a*b*c*d^6 - (3*a^2 + 2*b^2)*c^6*d + (a^2 + b^2)*d^7)*cos(f*x + e))/(3*(c^9*d^2 - 4*c^7*d^4 + 6*c^5*d^6 - 4*c^3*d^8 + c*d^10)*f*cos(f*x + e)^2 - (c^11 - c^9*d^2 - 6*c^7*d^4 + 14*c^5*d^6 - 11*c^3*d^8 + 3*c*d^10)*f + ((c^8*d^3 - 4*c^6*d^5 + 6*c^4*d^7 - 4*c^2*d^9 + d^11)*f*cos(f*x + e)^2 - (3*c^10*d - 11*c^8*d^3 + 14*c^6*d^5 - 6*c^4*d^7 - c^2*d^9 + d^11)*f)*sin(f*x + e)), -1/6*((b^2*c^6*d + 4*a*b*c^5*d^2 + 22*a*b*c^3*d^4 - 26*a*b*c*d^6 - 11*(a^2 + b^2)*c^4*d^3 + (7*a^2 + 4*b^2)*c^2*d^5 + 2*(2*a^2 + 3*b^2)*d^7)*cos(f*x + e)^3 - 3*(b^2*c^7 + 4*a*b*c^6*d + 14*a*b*c^4*d^3 - 20*a*b*c^2*d^5 + 2*a*b*d^7 - (9*a^2 + 10*b^2)*c^5*d^2 + (8*a^2 + 7*b^2)*c^3*d^4 + (a^2 + 2*b^2)*c*d^6)*cos(f*x + e)*sin(f*x + e) + 3*(8*a*b*c^5*d + 26*a*b*c^3*d^3 + 6*a*b*c*d^5 - (2*a^2 + b^2)*c^6 - (9*a^2 + 7*b^2)*c^4*d^2 - 3*(3*a^2 + 4*b^2)*c^2*d^4 - 3*(8*a*b*c^3*d^3 + 2*a*b*c*d^5 - (2*a^2 + b^2)*c^4*d^2 - (3*a^2 + 4*b^2)*c^2*d^4)*cos(f*x + e)^2 + (24*a*b*c^4*d^2 + 14*a*b*c^2*d^4 + 2*a*b*d^6 - 3*(2*a^2 + b^2)*c^5*d - (11*a^2 + 13*b^2)*c^3*d^3 - (3*a^2 + 4*b^2)*c*d^5 - (8*a*b*c^2*d^4 + 2*a*b*d^6 - (2*a^2 + b^2)*c^3*d^3 - (3*a^2 + 4*b^2)*c*d^5)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - 6*(2*a*b*c^7 + 2*a*b*c^5*d^2 + 2*a^2*c^4*d^3 + b^2*c^2*d^5 - 4*a*b*c*d^6 - (3*a^2 + 2*b^2)*c^6*d + (a^2 + b^2)*d^7)*cos(f*x + e))/(3*(c^9*d^2 - 4*c^7*d^4 + 6*c^5*d^6 - 4*c^3*d^8 + c*d^10)*f*cos(f*x + e)^2 - (c^11 - c^9*d^2 - 6*c^7*d^4 + 14*c^5*d^6 - 11*c^3*d^8 + 3*c*d^10)*f + ((c^8*d^3 - 4*c^6*d^5 + 6*c^4*d^7 - 4*c^2*d^9 + d^11)*f*cos(f*x + e)^2 - (3*c^10*d - 11*c^8*d^3 + 14*c^6*d^5 - 6*c^4*d^7 - c^2*d^9 + d^11)*f)*sin(f*x + e))]","B",0
686,1,375,0,1.100315," ","integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","-\frac{144 \, {\left(b^{3} c d^{2} + a b^{2} d^{3}\right)} \cos\left(f x + e\right)^{5} - 80 \, {\left(b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 3 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d^{2} + {\left(a^{3} + 6 \, a b^{2}\right)} d^{3}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(8 \, {\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{3} + 18 \, {\left(4 \, a^{2} b + b^{3}\right)} c^{2} d + 6 \, {\left(4 \, a^{3} + 9 \, a b^{2}\right)} c d^{2} + {\left(18 \, a^{2} b + 5 \, b^{3}\right)} d^{3}\right)} f x + 240 \, {\left({\left(3 \, a^{2} b + b^{3}\right)} c^{3} + 3 \, {\left(a^{3} + 3 \, a b^{2}\right)} c^{2} d + 3 \, {\left(3 \, a^{2} b + b^{3}\right)} c d^{2} + {\left(a^{3} + 3 \, a b^{2}\right)} d^{3}\right)} \cos\left(f x + e\right) + 5 \, {\left(8 \, b^{3} d^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(18 \, b^{3} c^{2} d + 54 \, a b^{2} c d^{2} + {\left(18 \, a^{2} b + 13 \, b^{3}\right)} d^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(24 \, a b^{2} c^{3} + 6 \, {\left(12 \, a^{2} b + 5 \, b^{3}\right)} c^{2} d + 6 \, {\left(4 \, a^{3} + 15 \, a b^{2}\right)} c d^{2} + {\left(30 \, a^{2} b + 11 \, b^{3}\right)} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{240 \, f}"," ",0,"-1/240*(144*(b^3*c*d^2 + a*b^2*d^3)*cos(f*x + e)^5 - 80*(b^3*c^3 + 9*a*b^2*c^2*d + 3*(3*a^2*b + 2*b^3)*c*d^2 + (a^3 + 6*a*b^2)*d^3)*cos(f*x + e)^3 - 15*(8*(2*a^3 + 3*a*b^2)*c^3 + 18*(4*a^2*b + b^3)*c^2*d + 6*(4*a^3 + 9*a*b^2)*c*d^2 + (18*a^2*b + 5*b^3)*d^3)*f*x + 240*((3*a^2*b + b^3)*c^3 + 3*(a^3 + 3*a*b^2)*c^2*d + 3*(3*a^2*b + b^3)*c*d^2 + (a^3 + 3*a*b^2)*d^3)*cos(f*x + e) + 5*(8*b^3*d^3*cos(f*x + e)^5 - 2*(18*b^3*c^2*d + 54*a*b^2*c*d^2 + (18*a^2*b + 13*b^3)*d^3)*cos(f*x + e)^3 + 3*(24*a*b^2*c^3 + 6*(12*a^2*b + 5*b^3)*c^2*d + 6*(4*a^3 + 15*a*b^2)*c*d^2 + (30*a^2*b + 11*b^3)*d^3)*cos(f*x + e))*sin(f*x + e))/f","A",0
687,1,253,0,1.712610," ","integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","-\frac{24 \, b^{3} d^{2} \cos\left(f x + e\right)^{5} - 40 \, {\left(b^{3} c^{2} + 6 \, a b^{2} c d + {\left(3 \, a^{2} b + 2 \, b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(4 \, {\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{2} + 6 \, {\left(4 \, a^{2} b + b^{3}\right)} c d + {\left(4 \, a^{3} + 9 \, a b^{2}\right)} d^{2}\right)} f x + 120 \, {\left({\left(3 \, a^{2} b + b^{3}\right)} c^{2} + 2 \, {\left(a^{3} + 3 \, a b^{2}\right)} c d + {\left(3 \, a^{2} b + b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right) - 15 \, {\left(2 \, {\left(2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(12 \, a b^{2} c^{2} + 2 \, {\left(12 \, a^{2} b + 5 \, b^{3}\right)} c d + {\left(4 \, a^{3} + 15 \, a b^{2}\right)} d^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{120 \, f}"," ",0,"-1/120*(24*b^3*d^2*cos(f*x + e)^5 - 40*(b^3*c^2 + 6*a*b^2*c*d + (3*a^2*b + 2*b^3)*d^2)*cos(f*x + e)^3 - 15*(4*(2*a^3 + 3*a*b^2)*c^2 + 6*(4*a^2*b + b^3)*c*d + (4*a^3 + 9*a*b^2)*d^2)*f*x + 120*((3*a^2*b + b^3)*c^2 + 2*(a^3 + 3*a*b^2)*c*d + (3*a^2*b + b^3)*d^2)*cos(f*x + e) - 15*(2*(2*b^3*c*d + 3*a*b^2*d^2)*cos(f*x + e)^3 - (12*a*b^2*c^2 + 2*(12*a^2*b + 5*b^3)*c*d + (4*a^3 + 15*a*b^2)*d^2)*cos(f*x + e))*sin(f*x + e))/f","A",0
688,1,148,0,1.379936," ","integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\frac{8 \, {\left(b^{3} c + 3 \, a b^{2} d\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(4 \, {\left(2 \, a^{3} + 3 \, a b^{2}\right)} c + 3 \, {\left(4 \, a^{2} b + b^{3}\right)} d\right)} f x - 24 \, {\left({\left(3 \, a^{2} b + b^{3}\right)} c + {\left(a^{3} + 3 \, a b^{2}\right)} d\right)} \cos\left(f x + e\right) + 3 \, {\left(2 \, b^{3} d \cos\left(f x + e\right)^{3} - {\left(12 \, a b^{2} c + {\left(12 \, a^{2} b + 5 \, b^{3}\right)} d\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{24 \, f}"," ",0,"1/24*(8*(b^3*c + 3*a*b^2*d)*cos(f*x + e)^3 + 3*(4*(2*a^3 + 3*a*b^2)*c + 3*(4*a^2*b + b^3)*d)*f*x - 24*((3*a^2*b + b^3)*c + (a^3 + 3*a*b^2)*d)*cos(f*x + e) + 3*(2*b^3*d*cos(f*x + e)^3 - (12*a*b^2*c + (12*a^2*b + 5*b^3)*d)*cos(f*x + e))*sin(f*x + e))/f","A",0
689,1,71,0,1.380314," ","integrate((a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\frac{2 \, b^{3} \cos\left(f x + e\right)^{3} - 9 \, a b^{2} \cos\left(f x + e\right) \sin\left(f x + e\right) + 3 \, {\left(2 \, a^{3} + 3 \, a b^{2}\right)} f x - 6 \, {\left(3 \, a^{2} b + b^{3}\right)} \cos\left(f x + e\right)}{6 \, f}"," ",0,"1/6*(2*b^3*cos(f*x + e)^3 - 9*a*b^2*cos(f*x + e)*sin(f*x + e) + 3*(2*a^3 + 3*a*b^2)*f*x - 6*(3*a^2*b + b^3)*cos(f*x + e))/f","A",0
690,1,578,0,1.514872," ","integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[\frac{{\left(2 \, b^{3} c^{4} - 6 \, a b^{2} c^{3} d + 6 \, a b^{2} c d^{3} + {\left(6 \, a^{2} b - b^{3}\right)} c^{2} d^{2} - {\left(6 \, a^{2} b + b^{3}\right)} d^{4}\right)} f x - {\left(b^{3} c^{2} d^{2} - b^{3} d^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left(b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right)} \cos\left(f x + e\right)}{2 \, {\left(c^{2} d^{3} - d^{5}\right)} f}, \frac{{\left(2 \, b^{3} c^{4} - 6 \, a b^{2} c^{3} d + 6 \, a b^{2} c d^{3} + {\left(6 \, a^{2} b - b^{3}\right)} c^{2} d^{2} - {\left(6 \, a^{2} b + b^{3}\right)} d^{4}\right)} f x - {\left(b^{3} c^{2} d^{2} - b^{3} d^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + 2 \, {\left(b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + 2 \, {\left(b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right)} \cos\left(f x + e\right)}{2 \, {\left(c^{2} d^{3} - d^{5}\right)} f}\right]"," ",0,"[1/2*((2*b^3*c^4 - 6*a*b^2*c^3*d + 6*a*b^2*c*d^3 + (6*a^2*b - b^3)*c^2*d^2 - (6*a^2*b + b^3)*d^4)*f*x - (b^3*c^2*d^2 - b^3*d^4)*cos(f*x + e)*sin(f*x + e) + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 - b^3*c*d^3 + 3*a*b^2*d^4)*cos(f*x + e))/((c^2*d^3 - d^5)*f), 1/2*((2*b^3*c^4 - 6*a*b^2*c^3*d + 6*a*b^2*c*d^3 + (6*a^2*b - b^3)*c^2*d^2 - (6*a^2*b + b^3)*d^4)*f*x - (b^3*c^2*d^2 - b^3*d^4)*cos(f*x + e)*sin(f*x + e) + 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + 2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 - b^3*c*d^3 + 3*a*b^2*d^4)*cos(f*x + e))/((c^2*d^3 - d^5)*f)]","A",0
691,1,1062,0,1.492356," ","integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(2 \, b^{3} c^{6} - 3 \, a b^{2} c^{5} d - 4 \, b^{3} c^{4} d^{2} + 6 \, a b^{2} c^{3} d^{3} + 2 \, b^{3} c^{2} d^{4} - 3 \, a b^{2} c d^{5}\right)} f x + {\left(2 \, b^{3} c^{5} - 3 \, a b^{2} c^{4} d - 3 \, b^{3} c^{3} d^{2} - 3 \, a^{2} b c d^{4} + {\left(a^{3} + 6 \, a b^{2}\right)} c^{2} d^{3} + {\left(2 \, b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} - 3 \, b^{3} c^{2} d^{3} - 3 \, a^{2} b d^{5} + {\left(a^{3} + 6 \, a b^{2}\right)} c d^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + 2 \, {\left(2 \, b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + a^{3} d^{6} + 3 \, {\left(a^{2} b - b^{3}\right)} c^{3} d^{3} - {\left(a^{3} - 3 \, a b^{2}\right)} c^{2} d^{4} - {\left(3 \, a^{2} b - b^{3}\right)} c d^{5}\right)} \cos\left(f x + e\right) + 2 \, {\left({\left(2 \, b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} - 4 \, b^{3} c^{3} d^{3} + 6 \, a b^{2} c^{2} d^{4} + 2 \, b^{3} c d^{5} - 3 \, a b^{2} d^{6}\right)} f x + {\left(b^{3} c^{4} d^{2} - 2 \, b^{3} c^{2} d^{4} + b^{3} d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(c^{4} d^{4} - 2 \, c^{2} d^{6} + d^{8}\right)} f \sin\left(f x + e\right) + {\left(c^{5} d^{3} - 2 \, c^{3} d^{5} + c d^{7}\right)} f\right)}}, -\frac{{\left(2 \, b^{3} c^{6} - 3 \, a b^{2} c^{5} d - 4 \, b^{3} c^{4} d^{2} + 6 \, a b^{2} c^{3} d^{3} + 2 \, b^{3} c^{2} d^{4} - 3 \, a b^{2} c d^{5}\right)} f x + {\left(2 \, b^{3} c^{5} - 3 \, a b^{2} c^{4} d - 3 \, b^{3} c^{3} d^{2} - 3 \, a^{2} b c d^{4} + {\left(a^{3} + 6 \, a b^{2}\right)} c^{2} d^{3} + {\left(2 \, b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} - 3 \, b^{3} c^{2} d^{3} - 3 \, a^{2} b d^{5} + {\left(a^{3} + 6 \, a b^{2}\right)} c d^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(2 \, b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + a^{3} d^{6} + 3 \, {\left(a^{2} b - b^{3}\right)} c^{3} d^{3} - {\left(a^{3} - 3 \, a b^{2}\right)} c^{2} d^{4} - {\left(3 \, a^{2} b - b^{3}\right)} c d^{5}\right)} \cos\left(f x + e\right) + {\left({\left(2 \, b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} - 4 \, b^{3} c^{3} d^{3} + 6 \, a b^{2} c^{2} d^{4} + 2 \, b^{3} c d^{5} - 3 \, a b^{2} d^{6}\right)} f x + {\left(b^{3} c^{4} d^{2} - 2 \, b^{3} c^{2} d^{4} + b^{3} d^{6}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{{\left(c^{4} d^{4} - 2 \, c^{2} d^{6} + d^{8}\right)} f \sin\left(f x + e\right) + {\left(c^{5} d^{3} - 2 \, c^{3} d^{5} + c d^{7}\right)} f}\right]"," ",0,"[-1/2*(2*(2*b^3*c^6 - 3*a*b^2*c^5*d - 4*b^3*c^4*d^2 + 6*a*b^2*c^3*d^3 + 2*b^3*c^2*d^4 - 3*a*b^2*c*d^5)*f*x + (2*b^3*c^5 - 3*a*b^2*c^4*d - 3*b^3*c^3*d^2 - 3*a^2*b*c*d^4 + (a^3 + 6*a*b^2)*c^2*d^3 + (2*b^3*c^4*d - 3*a*b^2*c^3*d^2 - 3*b^3*c^2*d^3 - 3*a^2*b*d^5 + (a^3 + 6*a*b^2)*c*d^4)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(2*b^3*c^5*d - 3*a*b^2*c^4*d^2 + a^3*d^6 + 3*(a^2*b - b^3)*c^3*d^3 - (a^3 - 3*a*b^2)*c^2*d^4 - (3*a^2*b - b^3)*c*d^5)*cos(f*x + e) + 2*((2*b^3*c^5*d - 3*a*b^2*c^4*d^2 - 4*b^3*c^3*d^3 + 6*a*b^2*c^2*d^4 + 2*b^3*c*d^5 - 3*a*b^2*d^6)*f*x + (b^3*c^4*d^2 - 2*b^3*c^2*d^4 + b^3*d^6)*cos(f*x + e))*sin(f*x + e))/((c^4*d^4 - 2*c^2*d^6 + d^8)*f*sin(f*x + e) + (c^5*d^3 - 2*c^3*d^5 + c*d^7)*f), -((2*b^3*c^6 - 3*a*b^2*c^5*d - 4*b^3*c^4*d^2 + 6*a*b^2*c^3*d^3 + 2*b^3*c^2*d^4 - 3*a*b^2*c*d^5)*f*x + (2*b^3*c^5 - 3*a*b^2*c^4*d - 3*b^3*c^3*d^2 - 3*a^2*b*c*d^4 + (a^3 + 6*a*b^2)*c^2*d^3 + (2*b^3*c^4*d - 3*a*b^2*c^3*d^2 - 3*b^3*c^2*d^3 - 3*a^2*b*d^5 + (a^3 + 6*a*b^2)*c*d^4)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (2*b^3*c^5*d - 3*a*b^2*c^4*d^2 + a^3*d^6 + 3*(a^2*b - b^3)*c^3*d^3 - (a^3 - 3*a*b^2)*c^2*d^4 - (3*a^2*b - b^3)*c*d^5)*cos(f*x + e) + ((2*b^3*c^5*d - 3*a*b^2*c^4*d^2 - 4*b^3*c^3*d^3 + 6*a*b^2*c^2*d^4 + 2*b^3*c*d^5 - 3*a*b^2*d^6)*f*x + (b^3*c^4*d^2 - 2*b^3*c^2*d^4 + b^3*d^6)*cos(f*x + e))*sin(f*x + e))/((c^4*d^4 - 2*c^2*d^6 + d^8)*f*sin(f*x + e) + (c^5*d^3 - 2*c^3*d^5 + c*d^7)*f)]","B",0
692,1,1707,0,1.779850," ","integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(b^{3} c^{6} d^{2} - 3 \, b^{3} c^{4} d^{4} + 3 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right)} f x \cos\left(f x + e\right)^{2} - 4 \, {\left(b^{3} c^{8} - 2 \, b^{3} c^{6} d^{2} + 2 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right)} f x - {\left(2 \, b^{3} c^{7} - 3 \, b^{3} c^{5} d^{2} - {\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{4} d^{3} + {\left(9 \, a^{2} b + b^{3}\right)} c^{3} d^{4} - 3 \, {\left(a^{3} + 3 \, a b^{2}\right)} c^{2} d^{5} + 3 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d^{6} - {\left(a^{3} + 6 \, a b^{2}\right)} d^{7} - {\left(2 \, b^{3} c^{5} d^{2} - 5 \, b^{3} c^{3} d^{4} - {\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{2} d^{5} + 3 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d^{6} - {\left(a^{3} + 6 \, a b^{2}\right)} d^{7}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, b^{3} c^{6} d - 5 \, b^{3} c^{4} d^{3} - {\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{3} d^{4} + 3 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c^{2} d^{5} - {\left(a^{3} + 6 \, a b^{2}\right)} c d^{6}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 2 \, {\left(2 \, b^{3} c^{7} d + 3 \, a^{2} b c d^{7} + a^{3} d^{8} - {\left(6 \, a^{2} b + 7 \, b^{3}\right)} c^{5} d^{3} + {\left(4 \, a^{3} + 9 \, a b^{2}\right)} c^{4} d^{4} + {\left(3 \, a^{2} b + 5 \, b^{3}\right)} c^{3} d^{5} - {\left(5 \, a^{3} + 9 \, a b^{2}\right)} c^{2} d^{6}\right)} \cos\left(f x + e\right) - 2 \, {\left(4 \, {\left(b^{3} c^{7} d - 3 \, b^{3} c^{5} d^{3} + 3 \, b^{3} c^{3} d^{5} - b^{3} c d^{7}\right)} f x + 3 \, {\left(b^{3} c^{6} d^{2} - a b^{2} c^{5} d^{3} + 2 \, a^{2} b d^{8} - {\left(a^{2} b + 3 \, b^{3}\right)} c^{4} d^{4} + {\left(a^{3} + 5 \, a b^{2}\right)} c^{3} d^{5} - {\left(a^{2} b - 2 \, b^{3}\right)} c^{2} d^{6} - {\left(a^{3} + 4 \, a b^{2}\right)} c d^{7}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(c^{6} d^{5} - 3 \, c^{4} d^{7} + 3 \, c^{2} d^{9} - d^{11}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{7} d^{4} - 3 \, c^{5} d^{6} + 3 \, c^{3} d^{8} - c d^{10}\right)} f \sin\left(f x + e\right) - {\left(c^{8} d^{3} - 2 \, c^{6} d^{5} + 2 \, c^{2} d^{9} - d^{11}\right)} f\right)}}, \frac{2 \, {\left(b^{3} c^{6} d^{2} - 3 \, b^{3} c^{4} d^{4} + 3 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right)} f x \cos\left(f x + e\right)^{2} - 2 \, {\left(b^{3} c^{8} - 2 \, b^{3} c^{6} d^{2} + 2 \, b^{3} c^{2} d^{6} - b^{3} d^{8}\right)} f x - {\left(2 \, b^{3} c^{7} - 3 \, b^{3} c^{5} d^{2} - {\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{4} d^{3} + {\left(9 \, a^{2} b + b^{3}\right)} c^{3} d^{4} - 3 \, {\left(a^{3} + 3 \, a b^{2}\right)} c^{2} d^{5} + 3 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d^{6} - {\left(a^{3} + 6 \, a b^{2}\right)} d^{7} - {\left(2 \, b^{3} c^{5} d^{2} - 5 \, b^{3} c^{3} d^{4} - {\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{2} d^{5} + 3 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d^{6} - {\left(a^{3} + 6 \, a b^{2}\right)} d^{7}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, b^{3} c^{6} d - 5 \, b^{3} c^{4} d^{3} - {\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{3} d^{4} + 3 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c^{2} d^{5} - {\left(a^{3} + 6 \, a b^{2}\right)} c d^{6}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) - {\left(2 \, b^{3} c^{7} d + 3 \, a^{2} b c d^{7} + a^{3} d^{8} - {\left(6 \, a^{2} b + 7 \, b^{3}\right)} c^{5} d^{3} + {\left(4 \, a^{3} + 9 \, a b^{2}\right)} c^{4} d^{4} + {\left(3 \, a^{2} b + 5 \, b^{3}\right)} c^{3} d^{5} - {\left(5 \, a^{3} + 9 \, a b^{2}\right)} c^{2} d^{6}\right)} \cos\left(f x + e\right) - {\left(4 \, {\left(b^{3} c^{7} d - 3 \, b^{3} c^{5} d^{3} + 3 \, b^{3} c^{3} d^{5} - b^{3} c d^{7}\right)} f x + 3 \, {\left(b^{3} c^{6} d^{2} - a b^{2} c^{5} d^{3} + 2 \, a^{2} b d^{8} - {\left(a^{2} b + 3 \, b^{3}\right)} c^{4} d^{4} + {\left(a^{3} + 5 \, a b^{2}\right)} c^{3} d^{5} - {\left(a^{2} b - 2 \, b^{3}\right)} c^{2} d^{6} - {\left(a^{3} + 4 \, a b^{2}\right)} c d^{7}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(c^{6} d^{5} - 3 \, c^{4} d^{7} + 3 \, c^{2} d^{9} - d^{11}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(c^{7} d^{4} - 3 \, c^{5} d^{6} + 3 \, c^{3} d^{8} - c d^{10}\right)} f \sin\left(f x + e\right) - {\left(c^{8} d^{3} - 2 \, c^{6} d^{5} + 2 \, c^{2} d^{9} - d^{11}\right)} f\right)}}\right]"," ",0,"[1/4*(4*(b^3*c^6*d^2 - 3*b^3*c^4*d^4 + 3*b^3*c^2*d^6 - b^3*d^8)*f*x*cos(f*x + e)^2 - 4*(b^3*c^8 - 2*b^3*c^6*d^2 + 2*b^3*c^2*d^6 - b^3*d^8)*f*x - (2*b^3*c^7 - 3*b^3*c^5*d^2 - (2*a^3 + 3*a*b^2)*c^4*d^3 + (9*a^2*b + b^3)*c^3*d^4 - 3*(a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)*d^7 - (2*b^3*c^5*d^2 - 5*b^3*c^3*d^4 - (2*a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)*d^7)*cos(f*x + e)^2 + 2*(2*b^3*c^6*d - 5*b^3*c^4*d^3 - (2*a^3 + 3*a*b^2)*c^3*d^4 + 3*(3*a^2*b + 2*b^3)*c^2*d^5 - (a^3 + 6*a*b^2)*c*d^6)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 2*(2*b^3*c^7*d + 3*a^2*b*c*d^7 + a^3*d^8 - (6*a^2*b + 7*b^3)*c^5*d^3 + (4*a^3 + 9*a*b^2)*c^4*d^4 + (3*a^2*b + 5*b^3)*c^3*d^5 - (5*a^3 + 9*a*b^2)*c^2*d^6)*cos(f*x + e) - 2*(4*(b^3*c^7*d - 3*b^3*c^5*d^3 + 3*b^3*c^3*d^5 - b^3*c*d^7)*f*x + 3*(b^3*c^6*d^2 - a*b^2*c^5*d^3 + 2*a^2*b*d^8 - (a^2*b + 3*b^3)*c^4*d^4 + (a^3 + 5*a*b^2)*c^3*d^5 - (a^2*b - 2*b^3)*c^2*d^6 - (a^3 + 4*a*b^2)*c*d^7)*cos(f*x + e))*sin(f*x + e))/((c^6*d^5 - 3*c^4*d^7 + 3*c^2*d^9 - d^11)*f*cos(f*x + e)^2 - 2*(c^7*d^4 - 3*c^5*d^6 + 3*c^3*d^8 - c*d^10)*f*sin(f*x + e) - (c^8*d^3 - 2*c^6*d^5 + 2*c^2*d^9 - d^11)*f), 1/2*(2*(b^3*c^6*d^2 - 3*b^3*c^4*d^4 + 3*b^3*c^2*d^6 - b^3*d^8)*f*x*cos(f*x + e)^2 - 2*(b^3*c^8 - 2*b^3*c^6*d^2 + 2*b^3*c^2*d^6 - b^3*d^8)*f*x - (2*b^3*c^7 - 3*b^3*c^5*d^2 - (2*a^3 + 3*a*b^2)*c^4*d^3 + (9*a^2*b + b^3)*c^3*d^4 - 3*(a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)*d^7 - (2*b^3*c^5*d^2 - 5*b^3*c^3*d^4 - (2*a^3 + 3*a*b^2)*c^2*d^5 + 3*(3*a^2*b + 2*b^3)*c*d^6 - (a^3 + 6*a*b^2)*d^7)*cos(f*x + e)^2 + 2*(2*b^3*c^6*d - 5*b^3*c^4*d^3 - (2*a^3 + 3*a*b^2)*c^3*d^4 + 3*(3*a^2*b + 2*b^3)*c^2*d^5 - (a^3 + 6*a*b^2)*c*d^6)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - (2*b^3*c^7*d + 3*a^2*b*c*d^7 + a^3*d^8 - (6*a^2*b + 7*b^3)*c^5*d^3 + (4*a^3 + 9*a*b^2)*c^4*d^4 + (3*a^2*b + 5*b^3)*c^3*d^5 - (5*a^3 + 9*a*b^2)*c^2*d^6)*cos(f*x + e) - (4*(b^3*c^7*d - 3*b^3*c^5*d^3 + 3*b^3*c^3*d^5 - b^3*c*d^7)*f*x + 3*(b^3*c^6*d^2 - a*b^2*c^5*d^3 + 2*a^2*b*d^8 - (a^2*b + 3*b^3)*c^4*d^4 + (a^3 + 5*a*b^2)*c^3*d^5 - (a^2*b - 2*b^3)*c^2*d^6 - (a^3 + 4*a*b^2)*c*d^7)*cos(f*x + e))*sin(f*x + e))/((c^6*d^5 - 3*c^4*d^7 + 3*c^2*d^9 - d^11)*f*cos(f*x + e)^2 - 2*(c^7*d^4 - 3*c^5*d^6 + 3*c^3*d^8 - c*d^10)*f*sin(f*x + e) - (c^8*d^3 - 2*c^6*d^5 + 2*c^2*d^9 - d^11)*f)]","B",0
693,1,2136,0,1.736296," ","integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^4,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(2 \, b^{3} c^{7} + 3 \, a b^{2} c^{6} d + {\left(6 \, a^{2} b - 7 \, b^{3}\right)} c^{5} d^{2} - 11 \, {\left(a^{3} + 3 \, a b^{2}\right)} c^{4} d^{3} + {\left(33 \, a^{2} b + 23 \, b^{3}\right)} c^{3} d^{4} + {\left(7 \, a^{3} + 12 \, a b^{2}\right)} c^{2} d^{5} - 3 \, {\left(13 \, a^{2} b + 6 \, b^{3}\right)} c d^{6} + 2 \, {\left(2 \, a^{3} + 9 \, a b^{2}\right)} d^{7}\right)} \cos\left(f x + e\right)^{3} - 6 \, {\left(3 \, a b^{2} c^{7} + 3 \, a^{2} b d^{7} + {\left(6 \, a^{2} b + b^{3}\right)} c^{6} d - 3 \, {\left(3 \, a^{3} + 10 \, a b^{2}\right)} c^{5} d^{2} + {\left(21 \, a^{2} b + 8 \, b^{3}\right)} c^{4} d^{3} + {\left(8 \, a^{3} + 21 \, a b^{2}\right)} c^{3} d^{4} - 3 \, {\left(10 \, a^{2} b + 3 \, b^{3}\right)} c^{2} d^{5} + {\left(a^{3} + 6 \, a b^{2}\right)} c d^{6}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 3 \, {\left({\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{6} - 3 \, {\left(4 \, a^{2} b + b^{3}\right)} c^{5} d + 3 \, {\left(3 \, a^{3} + 7 \, a b^{2}\right)} c^{4} d^{2} - {\left(39 \, a^{2} b + 11 \, b^{3}\right)} c^{3} d^{3} + 9 \, {\left(a^{3} + 4 \, a b^{2}\right)} c^{2} d^{4} - 3 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d^{5} - 3 \, {\left({\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{4} d^{2} - 3 \, {\left(4 \, a^{2} b + b^{3}\right)} c^{3} d^{3} + 3 \, {\left(a^{3} + 4 \, a b^{2}\right)} c^{2} d^{4} - {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, {\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{5} d - 9 \, {\left(4 \, a^{2} b + b^{3}\right)} c^{4} d^{2} + {\left(11 \, a^{3} + 39 \, a b^{2}\right)} c^{3} d^{3} - 3 \, {\left(7 \, a^{2} b + 3 \, b^{3}\right)} c^{2} d^{4} + 3 \, {\left(a^{3} + 4 \, a b^{2}\right)} c d^{5} - {\left(3 \, a^{2} b + 2 \, b^{3}\right)} d^{6} - {\left({\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{3} d^{3} - 3 \, {\left(4 \, a^{2} b + b^{3}\right)} c^{2} d^{4} + 3 \, {\left(a^{3} + 4 \, a b^{2}\right)} c d^{5} - {\left(3 \, a^{2} b + 2 \, b^{3}\right)} d^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}} \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 12 \, {\left(3 \, a^{2} b c^{5} d^{2} + 2 \, a^{3} c^{4} d^{3} + 2 \, b^{3} c^{3} d^{4} + 3 \, a b^{2} c^{2} d^{5} + {\left(3 \, a^{2} b + b^{3}\right)} c^{7} - 3 \, {\left(a^{3} + 2 \, a b^{2}\right)} c^{6} d - 3 \, {\left(2 \, a^{2} b + b^{3}\right)} c d^{6} + {\left(a^{3} + 3 \, a b^{2}\right)} d^{7}\right)} \cos\left(f x + e\right)}{12 \, {\left(3 \, {\left(c^{9} d^{2} - 4 \, c^{7} d^{4} + 6 \, c^{5} d^{6} - 4 \, c^{3} d^{8} + c d^{10}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{11} - c^{9} d^{2} - 6 \, c^{7} d^{4} + 14 \, c^{5} d^{6} - 11 \, c^{3} d^{8} + 3 \, c d^{10}\right)} f + {\left({\left(c^{8} d^{3} - 4 \, c^{6} d^{5} + 6 \, c^{4} d^{7} - 4 \, c^{2} d^{9} + d^{11}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{10} d - 11 \, c^{8} d^{3} + 14 \, c^{6} d^{5} - 6 \, c^{4} d^{7} - c^{2} d^{9} + d^{11}\right)} f\right)} \sin\left(f x + e\right)\right)}}, -\frac{{\left(2 \, b^{3} c^{7} + 3 \, a b^{2} c^{6} d + {\left(6 \, a^{2} b - 7 \, b^{3}\right)} c^{5} d^{2} - 11 \, {\left(a^{3} + 3 \, a b^{2}\right)} c^{4} d^{3} + {\left(33 \, a^{2} b + 23 \, b^{3}\right)} c^{3} d^{4} + {\left(7 \, a^{3} + 12 \, a b^{2}\right)} c^{2} d^{5} - 3 \, {\left(13 \, a^{2} b + 6 \, b^{3}\right)} c d^{6} + 2 \, {\left(2 \, a^{3} + 9 \, a b^{2}\right)} d^{7}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(3 \, a b^{2} c^{7} + 3 \, a^{2} b d^{7} + {\left(6 \, a^{2} b + b^{3}\right)} c^{6} d - 3 \, {\left(3 \, a^{3} + 10 \, a b^{2}\right)} c^{5} d^{2} + {\left(21 \, a^{2} b + 8 \, b^{3}\right)} c^{4} d^{3} + {\left(8 \, a^{3} + 21 \, a b^{2}\right)} c^{3} d^{4} - 3 \, {\left(10 \, a^{2} b + 3 \, b^{3}\right)} c^{2} d^{5} + {\left(a^{3} + 6 \, a b^{2}\right)} c d^{6}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 3 \, {\left({\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{6} - 3 \, {\left(4 \, a^{2} b + b^{3}\right)} c^{5} d + 3 \, {\left(3 \, a^{3} + 7 \, a b^{2}\right)} c^{4} d^{2} - {\left(39 \, a^{2} b + 11 \, b^{3}\right)} c^{3} d^{3} + 9 \, {\left(a^{3} + 4 \, a b^{2}\right)} c^{2} d^{4} - 3 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d^{5} - 3 \, {\left({\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{4} d^{2} - 3 \, {\left(4 \, a^{2} b + b^{3}\right)} c^{3} d^{3} + 3 \, {\left(a^{3} + 4 \, a b^{2}\right)} c^{2} d^{4} - {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d^{5}\right)} \cos\left(f x + e\right)^{2} + {\left(3 \, {\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{5} d - 9 \, {\left(4 \, a^{2} b + b^{3}\right)} c^{4} d^{2} + {\left(11 \, a^{3} + 39 \, a b^{2}\right)} c^{3} d^{3} - 3 \, {\left(7 \, a^{2} b + 3 \, b^{3}\right)} c^{2} d^{4} + 3 \, {\left(a^{3} + 4 \, a b^{2}\right)} c d^{5} - {\left(3 \, a^{2} b + 2 \, b^{3}\right)} d^{6} - {\left({\left(2 \, a^{3} + 3 \, a b^{2}\right)} c^{3} d^{3} - 3 \, {\left(4 \, a^{2} b + b^{3}\right)} c^{2} d^{4} + 3 \, {\left(a^{3} + 4 \, a b^{2}\right)} c d^{5} - {\left(3 \, a^{2} b + 2 \, b^{3}\right)} d^{6}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{c^{2} - d^{2}} \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) - 6 \, {\left(3 \, a^{2} b c^{5} d^{2} + 2 \, a^{3} c^{4} d^{3} + 2 \, b^{3} c^{3} d^{4} + 3 \, a b^{2} c^{2} d^{5} + {\left(3 \, a^{2} b + b^{3}\right)} c^{7} - 3 \, {\left(a^{3} + 2 \, a b^{2}\right)} c^{6} d - 3 \, {\left(2 \, a^{2} b + b^{3}\right)} c d^{6} + {\left(a^{3} + 3 \, a b^{2}\right)} d^{7}\right)} \cos\left(f x + e\right)}{6 \, {\left(3 \, {\left(c^{9} d^{2} - 4 \, c^{7} d^{4} + 6 \, c^{5} d^{6} - 4 \, c^{3} d^{8} + c d^{10}\right)} f \cos\left(f x + e\right)^{2} - {\left(c^{11} - c^{9} d^{2} - 6 \, c^{7} d^{4} + 14 \, c^{5} d^{6} - 11 \, c^{3} d^{8} + 3 \, c d^{10}\right)} f + {\left({\left(c^{8} d^{3} - 4 \, c^{6} d^{5} + 6 \, c^{4} d^{7} - 4 \, c^{2} d^{9} + d^{11}\right)} f \cos\left(f x + e\right)^{2} - {\left(3 \, c^{10} d - 11 \, c^{8} d^{3} + 14 \, c^{6} d^{5} - 6 \, c^{4} d^{7} - c^{2} d^{9} + d^{11}\right)} f\right)} \sin\left(f x + e\right)\right)}}\right]"," ",0,"[-1/12*(2*(2*b^3*c^7 + 3*a*b^2*c^6*d + (6*a^2*b - 7*b^3)*c^5*d^2 - 11*(a^3 + 3*a*b^2)*c^4*d^3 + (33*a^2*b + 23*b^3)*c^3*d^4 + (7*a^3 + 12*a*b^2)*c^2*d^5 - 3*(13*a^2*b + 6*b^3)*c*d^6 + 2*(2*a^3 + 9*a*b^2)*d^7)*cos(f*x + e)^3 - 6*(3*a*b^2*c^7 + 3*a^2*b*d^7 + (6*a^2*b + b^3)*c^6*d - 3*(3*a^3 + 10*a*b^2)*c^5*d^2 + (21*a^2*b + 8*b^3)*c^4*d^3 + (8*a^3 + 21*a*b^2)*c^3*d^4 - 3*(10*a^2*b + 3*b^3)*c^2*d^5 + (a^3 + 6*a*b^2)*c*d^6)*cos(f*x + e)*sin(f*x + e) - 3*((2*a^3 + 3*a*b^2)*c^6 - 3*(4*a^2*b + b^3)*c^5*d + 3*(3*a^3 + 7*a*b^2)*c^4*d^2 - (39*a^2*b + 11*b^3)*c^3*d^3 + 9*(a^3 + 4*a*b^2)*c^2*d^4 - 3*(3*a^2*b + 2*b^3)*c*d^5 - 3*((2*a^3 + 3*a*b^2)*c^4*d^2 - 3*(4*a^2*b + b^3)*c^3*d^3 + 3*(a^3 + 4*a*b^2)*c^2*d^4 - (3*a^2*b + 2*b^3)*c*d^5)*cos(f*x + e)^2 + (3*(2*a^3 + 3*a*b^2)*c^5*d - 9*(4*a^2*b + b^3)*c^4*d^2 + (11*a^3 + 39*a*b^2)*c^3*d^3 - 3*(7*a^2*b + 3*b^3)*c^2*d^4 + 3*(a^3 + 4*a*b^2)*c*d^5 - (3*a^2*b + 2*b^3)*d^6 - ((2*a^3 + 3*a*b^2)*c^3*d^3 - 3*(4*a^2*b + b^3)*c^2*d^4 + 3*(a^3 + 4*a*b^2)*c*d^5 - (3*a^2*b + 2*b^3)*d^6)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 12*(3*a^2*b*c^5*d^2 + 2*a^3*c^4*d^3 + 2*b^3*c^3*d^4 + 3*a*b^2*c^2*d^5 + (3*a^2*b + b^3)*c^7 - 3*(a^3 + 2*a*b^2)*c^6*d - 3*(2*a^2*b + b^3)*c*d^6 + (a^3 + 3*a*b^2)*d^7)*cos(f*x + e))/(3*(c^9*d^2 - 4*c^7*d^4 + 6*c^5*d^6 - 4*c^3*d^8 + c*d^10)*f*cos(f*x + e)^2 - (c^11 - c^9*d^2 - 6*c^7*d^4 + 14*c^5*d^6 - 11*c^3*d^8 + 3*c*d^10)*f + ((c^8*d^3 - 4*c^6*d^5 + 6*c^4*d^7 - 4*c^2*d^9 + d^11)*f*cos(f*x + e)^2 - (3*c^10*d - 11*c^8*d^3 + 14*c^6*d^5 - 6*c^4*d^7 - c^2*d^9 + d^11)*f)*sin(f*x + e)), -1/6*((2*b^3*c^7 + 3*a*b^2*c^6*d + (6*a^2*b - 7*b^3)*c^5*d^2 - 11*(a^3 + 3*a*b^2)*c^4*d^3 + (33*a^2*b + 23*b^3)*c^3*d^4 + (7*a^3 + 12*a*b^2)*c^2*d^5 - 3*(13*a^2*b + 6*b^3)*c*d^6 + 2*(2*a^3 + 9*a*b^2)*d^7)*cos(f*x + e)^3 - 3*(3*a*b^2*c^7 + 3*a^2*b*d^7 + (6*a^2*b + b^3)*c^6*d - 3*(3*a^3 + 10*a*b^2)*c^5*d^2 + (21*a^2*b + 8*b^3)*c^4*d^3 + (8*a^3 + 21*a*b^2)*c^3*d^4 - 3*(10*a^2*b + 3*b^3)*c^2*d^5 + (a^3 + 6*a*b^2)*c*d^6)*cos(f*x + e)*sin(f*x + e) - 3*((2*a^3 + 3*a*b^2)*c^6 - 3*(4*a^2*b + b^3)*c^5*d + 3*(3*a^3 + 7*a*b^2)*c^4*d^2 - (39*a^2*b + 11*b^3)*c^3*d^3 + 9*(a^3 + 4*a*b^2)*c^2*d^4 - 3*(3*a^2*b + 2*b^3)*c*d^5 - 3*((2*a^3 + 3*a*b^2)*c^4*d^2 - 3*(4*a^2*b + b^3)*c^3*d^3 + 3*(a^3 + 4*a*b^2)*c^2*d^4 - (3*a^2*b + 2*b^3)*c*d^5)*cos(f*x + e)^2 + (3*(2*a^3 + 3*a*b^2)*c^5*d - 9*(4*a^2*b + b^3)*c^4*d^2 + (11*a^3 + 39*a*b^2)*c^3*d^3 - 3*(7*a^2*b + 3*b^3)*c^2*d^4 + 3*(a^3 + 4*a*b^2)*c*d^5 - (3*a^2*b + 2*b^3)*d^6 - ((2*a^3 + 3*a*b^2)*c^3*d^3 - 3*(4*a^2*b + b^3)*c^2*d^4 + 3*(a^3 + 4*a*b^2)*c*d^5 - (3*a^2*b + 2*b^3)*d^6)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - 6*(3*a^2*b*c^5*d^2 + 2*a^3*c^4*d^3 + 2*b^3*c^3*d^4 + 3*a*b^2*c^2*d^5 + (3*a^2*b + b^3)*c^7 - 3*(a^3 + 2*a*b^2)*c^6*d - 3*(2*a^2*b + b^3)*c*d^6 + (a^3 + 3*a*b^2)*d^7)*cos(f*x + e))/(3*(c^9*d^2 - 4*c^7*d^4 + 6*c^5*d^6 - 4*c^3*d^8 + c*d^10)*f*cos(f*x + e)^2 - (c^11 - c^9*d^2 - 6*c^7*d^4 + 14*c^5*d^6 - 11*c^3*d^8 + 3*c*d^10)*f + ((c^8*d^3 - 4*c^6*d^5 + 6*c^4*d^7 - 4*c^2*d^9 + d^11)*f*cos(f*x + e)^2 - (3*c^10*d - 11*c^8*d^3 + 14*c^6*d^5 - 6*c^4*d^7 - c^2*d^9 + d^11)*f)*sin(f*x + e))]","B",0
694,1,163,0,1.102565," ","integrate((b*B/a+B*sin(x))/(a+b*sin(x)),x, algorithm=""fricas"")","\left[\frac{2 \, B a x + \sqrt{-a^{2} + b^{2}} 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)}{2 \, a b}, \frac{B a x + \sqrt{a^{2} - b^{2}} B \arctan\left(-\frac{a \sin\left(x\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(x\right)}\right)}{a b}\right]"," ",0,"[1/2*(2*B*a*x + sqrt(-a^2 + b^2)*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)))/(a*b), (B*a*x + sqrt(a^2 - b^2)*B*arctan(-(a*sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))))/(a*b)]","A",0
695,1,6,0,1.230258," ","integrate((a*B/b+B*sin(x))/(a+b*sin(x)),x, algorithm=""fricas"")","\frac{B x}{b}"," ",0,"B*x/b","A",0
696,1,12,0,1.236921," ","integrate((a+b*sin(x))/(b+a*sin(x))^2,x, algorithm=""fricas"")","-\frac{\cos\left(x\right)}{a \sin\left(x\right) + b}"," ",0,"-cos(x)/(a*sin(x) + b)","A",0
697,1,27,0,1.459303," ","integrate((2-sin(x))/(2+sin(x)),x, algorithm=""fricas"")","\frac{4}{3} \, \sqrt{3} \arctan\left(\frac{2 \, \sqrt{3} \sin\left(x\right) + \sqrt{3}}{3 \, \cos\left(x\right)}\right) - x"," ",0,"4/3*sqrt(3)*arctan(1/3*(2*sqrt(3)*sin(x) + sqrt(3))/cos(x)) - x","A",0
698,1,795,0,1.554164," ","integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{2} b^{3} - b^{5}\right)} d^{4} \cos\left(f x + e\right)^{3} + 3 \, {\left(8 \, {\left(a^{2} b^{3} - b^{5}\right)} c^{3} d - 12 \, {\left(a^{3} b^{2} - a b^{4}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{4} b - a^{2} b^{3} - b^{5}\right)} c d^{3} - {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} d^{4}\right)} f x - 3 \, {\left(4 \, {\left(a^{2} b^{3} - b^{5}\right)} c d^{3} - {\left(a^{3} b^{2} - a b^{4}\right)} d^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 3 \, {\left(b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) - 6 \, {\left(6 \, {\left(a^{2} b^{3} - b^{5}\right)} c^{2} d^{2} - 4 \, {\left(a^{3} b^{2} - a b^{4}\right)} c d^{3} + {\left(a^{4} b - b^{5}\right)} d^{4}\right)} \cos\left(f x + e\right)}{6 \, {\left(a^{2} b^{4} - b^{6}\right)} f}, \frac{2 \, {\left(a^{2} b^{3} - b^{5}\right)} d^{4} \cos\left(f x + e\right)^{3} + 3 \, {\left(8 \, {\left(a^{2} b^{3} - b^{5}\right)} c^{3} d - 12 \, {\left(a^{3} b^{2} - a b^{4}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{4} b - a^{2} b^{3} - b^{5}\right)} c d^{3} - {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} d^{4}\right)} f x - 3 \, {\left(4 \, {\left(a^{2} b^{3} - b^{5}\right)} c d^{3} - {\left(a^{3} b^{2} - a b^{4}\right)} d^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - 6 \, {\left(b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) - 6 \, {\left(6 \, {\left(a^{2} b^{3} - b^{5}\right)} c^{2} d^{2} - 4 \, {\left(a^{3} b^{2} - a b^{4}\right)} c d^{3} + {\left(a^{4} b - b^{5}\right)} d^{4}\right)} \cos\left(f x + e\right)}{6 \, {\left(a^{2} b^{4} - b^{6}\right)} f}\right]"," ",0,"[1/6*(2*(a^2*b^3 - b^5)*d^4*cos(f*x + e)^3 + 3*(8*(a^2*b^3 - b^5)*c^3*d - 12*(a^3*b^2 - a*b^4)*c^2*d^2 + 4*(2*a^4*b - a^2*b^3 - b^5)*c*d^3 - (2*a^5 - a^3*b^2 - a*b^4)*d^4)*f*x - 3*(4*(a^2*b^3 - b^5)*c*d^3 - (a^3*b^2 - a*b^4)*d^4)*cos(f*x + e)*sin(f*x + e) - 3*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 + 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) - 6*(6*(a^2*b^3 - b^5)*c^2*d^2 - 4*(a^3*b^2 - a*b^4)*c*d^3 + (a^4*b - b^5)*d^4)*cos(f*x + e))/((a^2*b^4 - b^6)*f), 1/6*(2*(a^2*b^3 - b^5)*d^4*cos(f*x + e)^3 + 3*(8*(a^2*b^3 - b^5)*c^3*d - 12*(a^3*b^2 - a*b^4)*c^2*d^2 + 4*(2*a^4*b - a^2*b^3 - b^5)*c*d^3 - (2*a^5 - a^3*b^2 - a*b^4)*d^4)*f*x - 3*(4*(a^2*b^3 - b^5)*c*d^3 - (a^3*b^2 - a*b^4)*d^4)*cos(f*x + e)*sin(f*x + e) - 6*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) - 6*(6*(a^2*b^3 - b^5)*c^2*d^2 - 4*(a^3*b^2 - a*b^4)*c*d^3 + (a^4*b - b^5)*d^4)*cos(f*x + e))/((a^2*b^4 - b^6)*f)]","A",0
699,1,566,0,1.451551," ","integrate((c+d*sin(f*x+e))^3/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\left[-\frac{{\left(a^{2} b^{2} - b^{4}\right)} d^{3} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(6 \, {\left(a^{2} b^{2} - b^{4}\right)} c^{2} d - 6 \, {\left(a^{3} b - a b^{3}\right)} c d^{2} + {\left(2 \, a^{4} - a^{2} b^{2} - b^{4}\right)} d^{3}\right)} f x - {\left(b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right)} \sqrt{-a^{2} + b^{2}} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) + 2 \, {\left(3 \, {\left(a^{2} b^{2} - b^{4}\right)} c d^{2} - {\left(a^{3} b - a b^{3}\right)} d^{3}\right)} \cos\left(f x + e\right)}{2 \, {\left(a^{2} b^{3} - b^{5}\right)} f}, -\frac{{\left(a^{2} b^{2} - b^{4}\right)} d^{3} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(6 \, {\left(a^{2} b^{2} - b^{4}\right)} c^{2} d - 6 \, {\left(a^{3} b - a b^{3}\right)} c d^{2} + {\left(2 \, a^{4} - a^{2} b^{2} - b^{4}\right)} d^{3}\right)} f x + 2 \, {\left(b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) + 2 \, {\left(3 \, {\left(a^{2} b^{2} - b^{4}\right)} c d^{2} - {\left(a^{3} b - a b^{3}\right)} d^{3}\right)} \cos\left(f x + e\right)}{2 \, {\left(a^{2} b^{3} - b^{5}\right)} f}\right]"," ",0,"[-1/2*((a^2*b^2 - b^4)*d^3*cos(f*x + e)*sin(f*x + e) - (6*(a^2*b^2 - b^4)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 + (2*a^4 - a^2*b^2 - b^4)*d^3)*f*x - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + 2*(3*(a^2*b^2 - b^4)*c*d^2 - (a^3*b - a*b^3)*d^3)*cos(f*x + e))/((a^2*b^3 - b^5)*f), -1/2*((a^2*b^2 - b^4)*d^3*cos(f*x + e)*sin(f*x + e) - (6*(a^2*b^2 - b^4)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 + (2*a^4 - a^2*b^2 - b^4)*d^3)*f*x + 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + 2*(3*(a^2*b^2 - b^4)*c*d^2 - (a^3*b - a*b^3)*d^3)*cos(f*x + e))/((a^2*b^3 - b^5)*f)]","A",0
700,1,368,0,1.439718," ","integrate((c+d*sin(f*x+e))^2/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a^{2} b - b^{3}\right)} d^{2} \cos\left(f x + e\right) - 2 \, {\left(2 \, {\left(a^{2} b - b^{3}\right)} c d - {\left(a^{3} - a b^{2}\right)} d^{2}\right)} f x + {\left(b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right)}{2 \, {\left(a^{2} b^{2} - b^{4}\right)} f}, -\frac{{\left(a^{2} b - b^{3}\right)} d^{2} \cos\left(f x + e\right) - {\left(2 \, {\left(a^{2} b - b^{3}\right)} c d - {\left(a^{3} - a b^{2}\right)} d^{2}\right)} f x + {\left(b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right)}{{\left(a^{2} b^{2} - b^{4}\right)} f}\right]"," ",0,"[-1/2*(2*(a^2*b - b^3)*d^2*cos(f*x + e) - 2*(2*(a^2*b - b^3)*c*d - (a^3 - a*b^2)*d^2)*f*x + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 + 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)))/((a^2*b^2 - b^4)*f), -((a^2*b - b^3)*d^2*cos(f*x + e) - (2*(a^2*b - b^3)*c*d - (a^3 - a*b^2)*d^2)*f*x + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))))/((a^2*b^2 - b^4)*f)]","A",0
701,1,253,0,0.812417," ","integrate((c+d*sin(f*x+e))/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{2} - b^{2}\right)} d f x + \sqrt{-a^{2} + b^{2}} {\left(b c - a d\right)} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right)}{2 \, {\left(a^{2} b - b^{3}\right)} f}, \frac{{\left(a^{2} - b^{2}\right)} d f x - \sqrt{a^{2} - b^{2}} {\left(b c - a d\right)} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right)}{{\left(a^{2} b - b^{3}\right)} f}\right]"," ",0,"[1/2*(2*(a^2 - b^2)*d*f*x + sqrt(-a^2 + b^2)*(b*c - a*d)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)))/((a^2*b - b^3)*f), ((a^2 - b^2)*d*f*x - sqrt(a^2 - b^2)*(b*c - a*d)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))))/((a^2*b - b^3)*f)]","A",0
702,1,190,0,1.238751," ","integrate(1/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} + b^{2}} \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right)}{2 \, {\left(a^{2} - b^{2}\right)} f}, -\frac{\arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right)}{\sqrt{a^{2} - b^{2}} f}\right]"," ",0,"[-1/2*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 + 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2))/((a^2 - b^2)*f), -arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e)))/(sqrt(a^2 - b^2)*f)]","A",0
703,1,1057,0,5.615879," ","integrate(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} - b^{2}\right)} \sqrt{-c^{2} + d^{2}} d \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) + {\left(b c^{2} - b d^{2}\right)} \sqrt{-a^{2} + b^{2}} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right)}{2 \, {\left({\left(a^{2} b - b^{3}\right)} c^{3} - {\left(a^{3} - a b^{2}\right)} c^{2} d - {\left(a^{2} b - b^{3}\right)} c d^{2} + {\left(a^{3} - a b^{2}\right)} d^{3}\right)} f}, \frac{2 \, {\left(a^{2} - b^{2}\right)} \sqrt{c^{2} - d^{2}} d \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) + {\left(b c^{2} - b d^{2}\right)} \sqrt{-a^{2} + b^{2}} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right)}{2 \, {\left({\left(a^{2} b - b^{3}\right)} c^{3} - {\left(a^{3} - a b^{2}\right)} c^{2} d - {\left(a^{2} b - b^{3}\right)} c d^{2} + {\left(a^{3} - a b^{2}\right)} d^{3}\right)} f}, \frac{{\left(a^{2} - b^{2}\right)} \sqrt{-c^{2} + d^{2}} d \log\left(\frac{{\left(2 \, c^{2} - d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2} + 2 \, {\left(c \cos\left(f x + e\right) \sin\left(f x + e\right) + d \cos\left(f x + e\right)\right)} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}\right) - 2 \, {\left(b c^{2} - b d^{2}\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right)}{2 \, {\left({\left(a^{2} b - b^{3}\right)} c^{3} - {\left(a^{3} - a b^{2}\right)} c^{2} d - {\left(a^{2} b - b^{3}\right)} c d^{2} + {\left(a^{3} - a b^{2}\right)} d^{3}\right)} f}, \frac{{\left(a^{2} - b^{2}\right)} \sqrt{c^{2} - d^{2}} d \arctan\left(-\frac{c \sin\left(f x + e\right) + d}{\sqrt{c^{2} - d^{2}} \cos\left(f x + e\right)}\right) - {\left(b c^{2} - b d^{2}\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right)}{{\left({\left(a^{2} b - b^{3}\right)} c^{3} - {\left(a^{3} - a b^{2}\right)} c^{2} d - {\left(a^{2} b - b^{3}\right)} c d^{2} + {\left(a^{3} - a b^{2}\right)} d^{3}\right)} f}\right]"," ",0,"[1/2*((a^2 - b^2)*sqrt(-c^2 + d^2)*d*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + (b*c^2 - b*d^2)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)))/(((a^2*b - b^3)*c^3 - (a^3 - a*b^2)*c^2*d - (a^2*b - b^3)*c*d^2 + (a^3 - a*b^2)*d^3)*f), 1/2*(2*(a^2 - b^2)*sqrt(c^2 - d^2)*d*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (b*c^2 - b*d^2)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)))/(((a^2*b - b^3)*c^3 - (a^3 - a*b^2)*c^2*d - (a^2*b - b^3)*c*d^2 + (a^3 - a*b^2)*d^3)*f), 1/2*((a^2 - b^2)*sqrt(-c^2 + d^2)*d*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 2*(b*c^2 - b*d^2)*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))))/(((a^2*b - b^3)*c^3 - (a^3 - a*b^2)*c^2*d - (a^2*b - b^3)*c*d^2 + (a^3 - a*b^2)*d^3)*f), ((a^2 - b^2)*sqrt(c^2 - d^2)*d*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - (b*c^2 - b*d^2)*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))))/(((a^2*b - b^3)*c^3 - (a^3 - a*b^2)*c^2*d - (a^2*b - b^3)*c*d^2 + (a^3 - a*b^2)*d^3)*f)]","A",0
704,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
705,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
706,1,1451,0,2.541605," ","integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{{\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} d^{4} \cos\left(f x + e\right)^{3} + {\left(12 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} c^{2} d^{2} - 16 \, {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} c d^{3} + {\left(6 \, a^{7} - 11 \, a^{5} b^{2} + 4 \, a^{3} b^{4} + a b^{6}\right)} d^{4}\right)} f x + {\left(a^{2} b^{4} c^{4} - 4 \, a b^{5} c^{3} d - 6 \, {\left(a^{4} b^{2} - 2 \, a^{2} b^{4}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{5} b - 3 \, a^{3} b^{3}\right)} c d^{3} - {\left(3 \, a^{6} - 4 \, a^{4} b^{2}\right)} d^{4} + {\left(a b^{5} c^{4} - 4 \, b^{6} c^{3} d - 6 \, {\left(a^{3} b^{3} - 2 \, a b^{5}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right)} c d^{3} - {\left(3 \, a^{5} b - 4 \, a^{3} b^{3}\right)} d^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) + {\left(2 \, {\left(a^{2} b^{5} - b^{7}\right)} c^{4} - 8 \, {\left(a^{3} b^{4} - a b^{6}\right)} c^{3} d + 12 \, {\left(a^{4} b^{3} - a^{2} b^{5}\right)} c^{2} d^{2} - 8 \, {\left(2 \, a^{5} b^{2} - 3 \, a^{3} b^{4} + a b^{6}\right)} c d^{3} + {\left(6 \, a^{6} b - 11 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - b^{7}\right)} d^{4}\right)} \cos\left(f x + e\right) + {\left({\left(12 \, {\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} c^{2} d^{2} - 16 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} c d^{3} + {\left(6 \, a^{6} b - 11 \, a^{4} b^{3} + 4 \, a^{2} b^{5} + b^{7}\right)} d^{4}\right)} f x - {\left(8 \, {\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} c d^{3} - 3 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right)} f \sin\left(f x + e\right) + {\left(a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right)} f\right)}}, \frac{{\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} d^{4} \cos\left(f x + e\right)^{3} + {\left(12 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} c^{2} d^{2} - 16 \, {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} c d^{3} + {\left(6 \, a^{7} - 11 \, a^{5} b^{2} + 4 \, a^{3} b^{4} + a b^{6}\right)} d^{4}\right)} f x - 2 \, {\left(a^{2} b^{4} c^{4} - 4 \, a b^{5} c^{3} d - 6 \, {\left(a^{4} b^{2} - 2 \, a^{2} b^{4}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{5} b - 3 \, a^{3} b^{3}\right)} c d^{3} - {\left(3 \, a^{6} - 4 \, a^{4} b^{2}\right)} d^{4} + {\left(a b^{5} c^{4} - 4 \, b^{6} c^{3} d - 6 \, {\left(a^{3} b^{3} - 2 \, a b^{5}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right)} c d^{3} - {\left(3 \, a^{5} b - 4 \, a^{3} b^{3}\right)} d^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) + {\left(2 \, {\left(a^{2} b^{5} - b^{7}\right)} c^{4} - 8 \, {\left(a^{3} b^{4} - a b^{6}\right)} c^{3} d + 12 \, {\left(a^{4} b^{3} - a^{2} b^{5}\right)} c^{2} d^{2} - 8 \, {\left(2 \, a^{5} b^{2} - 3 \, a^{3} b^{4} + a b^{6}\right)} c d^{3} + {\left(6 \, a^{6} b - 11 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - b^{7}\right)} d^{4}\right)} \cos\left(f x + e\right) + {\left({\left(12 \, {\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} c^{2} d^{2} - 16 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} c d^{3} + {\left(6 \, a^{6} b - 11 \, a^{4} b^{3} + 4 \, a^{2} b^{5} + b^{7}\right)} d^{4}\right)} f x - {\left(8 \, {\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} c d^{3} - 3 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right)} f \sin\left(f x + e\right) + {\left(a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right)} f\right)}}\right]"," ",0,"[1/2*((a^4*b^3 - 2*a^2*b^5 + b^7)*d^4*cos(f*x + e)^3 + (12*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^2*d^2 - 16*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 + (6*a^7 - 11*a^5*b^2 + 4*a^3*b^4 + a*b^6)*d^4)*f*x + (a^2*b^4*c^4 - 4*a*b^5*c^3*d - 6*(a^4*b^2 - 2*a^2*b^4)*c^2*d^2 + 4*(2*a^5*b - 3*a^3*b^3)*c*d^3 - (3*a^6 - 4*a^4*b^2)*d^4 + (a*b^5*c^4 - 4*b^6*c^3*d - 6*(a^3*b^3 - 2*a*b^5)*c^2*d^2 + 4*(2*a^4*b^2 - 3*a^2*b^4)*c*d^3 - (3*a^5*b - 4*a^3*b^3)*d^4)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + (2*(a^2*b^5 - b^7)*c^4 - 8*(a^3*b^4 - a*b^6)*c^3*d + 12*(a^4*b^3 - a^2*b^5)*c^2*d^2 - 8*(2*a^5*b^2 - 3*a^3*b^4 + a*b^6)*c*d^3 + (6*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 - b^7)*d^4)*cos(f*x + e) + ((12*(a^4*b^3 - 2*a^2*b^5 + b^7)*c^2*d^2 - 16*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 + (6*a^6*b - 11*a^4*b^3 + 4*a^2*b^5 + b^7)*d^4)*f*x - (8*(a^4*b^3 - 2*a^2*b^5 + b^7)*c*d^3 - 3*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^4*b^5 - 2*a^2*b^7 + b^9)*f*sin(f*x + e) + (a^5*b^4 - 2*a^3*b^6 + a*b^8)*f), 1/2*((a^4*b^3 - 2*a^2*b^5 + b^7)*d^4*cos(f*x + e)^3 + (12*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^2*d^2 - 16*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 + (6*a^7 - 11*a^5*b^2 + 4*a^3*b^4 + a*b^6)*d^4)*f*x - 2*(a^2*b^4*c^4 - 4*a*b^5*c^3*d - 6*(a^4*b^2 - 2*a^2*b^4)*c^2*d^2 + 4*(2*a^5*b - 3*a^3*b^3)*c*d^3 - (3*a^6 - 4*a^4*b^2)*d^4 + (a*b^5*c^4 - 4*b^6*c^3*d - 6*(a^3*b^3 - 2*a*b^5)*c^2*d^2 + 4*(2*a^4*b^2 - 3*a^2*b^4)*c*d^3 - (3*a^5*b - 4*a^3*b^3)*d^4)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + (2*(a^2*b^5 - b^7)*c^4 - 8*(a^3*b^4 - a*b^6)*c^3*d + 12*(a^4*b^3 - a^2*b^5)*c^2*d^2 - 8*(2*a^5*b^2 - 3*a^3*b^4 + a*b^6)*c*d^3 + (6*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 - b^7)*d^4)*cos(f*x + e) + ((12*(a^4*b^3 - 2*a^2*b^5 + b^7)*c^2*d^2 - 16*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 + (6*a^6*b - 11*a^4*b^3 + 4*a^2*b^5 + b^7)*d^4)*f*x - (8*(a^4*b^3 - 2*a^2*b^5 + b^7)*c*d^3 - 3*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^4*b^5 - 2*a^2*b^7 + b^9)*f*sin(f*x + e) + (a^5*b^4 - 2*a^3*b^6 + a*b^8)*f)]","B",0
707,1,1006,0,2.070585," ","integrate((c+d*sin(f*x+e))^3/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(3 \, {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} c d^{2} - 2 \, {\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d^{3}\right)} f x - {\left(a^{2} b^{3} c^{3} - 3 \, a b^{4} c^{2} d - 3 \, {\left(a^{4} b - 2 \, a^{2} b^{3}\right)} c d^{2} + {\left(2 \, a^{5} - 3 \, a^{3} b^{2}\right)} d^{3} + {\left(a b^{4} c^{3} - 3 \, b^{5} c^{2} d - 3 \, {\left(a^{3} b^{2} - 2 \, a b^{4}\right)} c d^{2} + {\left(2 \, a^{4} b - 3 \, a^{2} b^{3}\right)} d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) + 2 \, {\left({\left(a^{2} b^{4} - b^{6}\right)} c^{3} - 3 \, {\left(a^{3} b^{3} - a b^{5}\right)} c^{2} d + 3 \, {\left(a^{4} b^{2} - a^{2} b^{4}\right)} c d^{2} - {\left(2 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right)} d^{3}\right)} \cos\left(f x + e\right) - 2 \, {\left({\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} d^{3} \cos\left(f x + e\right) - {\left(3 \, {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} c d^{2} - 2 \, {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d^{3}\right)} f x\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right)} f \sin\left(f x + e\right) + {\left(a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right)} f\right)}}, \frac{{\left(3 \, {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} c d^{2} - 2 \, {\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d^{3}\right)} f x - {\left(a^{2} b^{3} c^{3} - 3 \, a b^{4} c^{2} d - 3 \, {\left(a^{4} b - 2 \, a^{2} b^{3}\right)} c d^{2} + {\left(2 \, a^{5} - 3 \, a^{3} b^{2}\right)} d^{3} + {\left(a b^{4} c^{3} - 3 \, b^{5} c^{2} d - 3 \, {\left(a^{3} b^{2} - 2 \, a b^{4}\right)} c d^{2} + {\left(2 \, a^{4} b - 3 \, a^{2} b^{3}\right)} d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) + {\left({\left(a^{2} b^{4} - b^{6}\right)} c^{3} - 3 \, {\left(a^{3} b^{3} - a b^{5}\right)} c^{2} d + 3 \, {\left(a^{4} b^{2} - a^{2} b^{4}\right)} c d^{2} - {\left(2 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right)} d^{3}\right)} \cos\left(f x + e\right) - {\left({\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} d^{3} \cos\left(f x + e\right) - {\left(3 \, {\left(a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right)} c d^{2} - 2 \, {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d^{3}\right)} f x\right)} \sin\left(f x + e\right)}{{\left(a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right)} f \sin\left(f x + e\right) + {\left(a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right)} f}\right]"," ",0,"[1/2*(2*(3*(a^5*b - 2*a^3*b^3 + a*b^5)*c*d^2 - 2*(a^6 - 2*a^4*b^2 + a^2*b^4)*d^3)*f*x - (a^2*b^3*c^3 - 3*a*b^4*c^2*d - 3*(a^4*b - 2*a^2*b^3)*c*d^2 + (2*a^5 - 3*a^3*b^2)*d^3 + (a*b^4*c^3 - 3*b^5*c^2*d - 3*(a^3*b^2 - 2*a*b^4)*c*d^2 + (2*a^4*b - 3*a^2*b^3)*d^3)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 + 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + 2*((a^2*b^4 - b^6)*c^3 - 3*(a^3*b^3 - a*b^5)*c^2*d + 3*(a^4*b^2 - a^2*b^4)*c*d^2 - (2*a^5*b - 3*a^3*b^3 + a*b^5)*d^3)*cos(f*x + e) - 2*((a^4*b^2 - 2*a^2*b^4 + b^6)*d^3*cos(f*x + e) - (3*(a^4*b^2 - 2*a^2*b^4 + b^6)*c*d^2 - 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d^3)*f*x)*sin(f*x + e))/((a^4*b^4 - 2*a^2*b^6 + b^8)*f*sin(f*x + e) + (a^5*b^3 - 2*a^3*b^5 + a*b^7)*f), ((3*(a^5*b - 2*a^3*b^3 + a*b^5)*c*d^2 - 2*(a^6 - 2*a^4*b^2 + a^2*b^4)*d^3)*f*x - (a^2*b^3*c^3 - 3*a*b^4*c^2*d - 3*(a^4*b - 2*a^2*b^3)*c*d^2 + (2*a^5 - 3*a^3*b^2)*d^3 + (a*b^4*c^3 - 3*b^5*c^2*d - 3*(a^3*b^2 - 2*a*b^4)*c*d^2 + (2*a^4*b - 3*a^2*b^3)*d^3)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + ((a^2*b^4 - b^6)*c^3 - 3*(a^3*b^3 - a*b^5)*c^2*d + 3*(a^4*b^2 - a^2*b^4)*c*d^2 - (2*a^5*b - 3*a^3*b^3 + a*b^5)*d^3)*cos(f*x + e) - ((a^4*b^2 - 2*a^2*b^4 + b^6)*d^3*cos(f*x + e) - (3*(a^4*b^2 - 2*a^2*b^4 + b^6)*c*d^2 - 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d^3)*f*x)*sin(f*x + e))/((a^4*b^4 - 2*a^2*b^6 + b^8)*f*sin(f*x + e) + (a^5*b^3 - 2*a^3*b^5 + a*b^7)*f)]","B",0
708,1,665,0,2.059607," ","integrate((c+d*sin(f*x+e))^2/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} d^{2} f x \sin\left(f x + e\right) + 2 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d^{2} f x + {\left(a^{2} b^{2} c^{2} - 2 \, a b^{3} c d - {\left(a^{4} - 2 \, a^{2} b^{2}\right)} d^{2} + {\left(a b^{3} c^{2} - 2 \, b^{4} c d - {\left(a^{3} b - 2 \, a b^{3}\right)} d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) + 2 \, {\left({\left(a^{2} b^{3} - b^{5}\right)} c^{2} - 2 \, {\left(a^{3} b^{2} - a b^{4}\right)} c d + {\left(a^{4} b - a^{2} b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} f \sin\left(f x + e\right) + {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} f\right)}}, \frac{{\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} d^{2} f x \sin\left(f x + e\right) + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d^{2} f x - {\left(a^{2} b^{2} c^{2} - 2 \, a b^{3} c d - {\left(a^{4} - 2 \, a^{2} b^{2}\right)} d^{2} + {\left(a b^{3} c^{2} - 2 \, b^{4} c d - {\left(a^{3} b - 2 \, a b^{3}\right)} d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) + {\left({\left(a^{2} b^{3} - b^{5}\right)} c^{2} - 2 \, {\left(a^{3} b^{2} - a b^{4}\right)} c d + {\left(a^{4} b - a^{2} b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right)}{{\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} f \sin\left(f x + e\right) + {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} f}\right]"," ",0,"[1/2*(2*(a^4*b - 2*a^2*b^3 + b^5)*d^2*f*x*sin(f*x + e) + 2*(a^5 - 2*a^3*b^2 + a*b^4)*d^2*f*x + (a^2*b^2*c^2 - 2*a*b^3*c*d - (a^4 - 2*a^2*b^2)*d^2 + (a*b^3*c^2 - 2*b^4*c*d - (a^3*b - 2*a*b^3)*d^2)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + 2*((a^2*b^3 - b^5)*c^2 - 2*(a^3*b^2 - a*b^4)*c*d + (a^4*b - a^2*b^3)*d^2)*cos(f*x + e))/((a^4*b^3 - 2*a^2*b^5 + b^7)*f*sin(f*x + e) + (a^5*b^2 - 2*a^3*b^4 + a*b^6)*f), ((a^4*b - 2*a^2*b^3 + b^5)*d^2*f*x*sin(f*x + e) + (a^5 - 2*a^3*b^2 + a*b^4)*d^2*f*x - (a^2*b^2*c^2 - 2*a*b^3*c*d - (a^4 - 2*a^2*b^2)*d^2 + (a*b^3*c^2 - 2*b^4*c*d - (a^3*b - 2*a*b^3)*d^2)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + ((a^2*b^3 - b^5)*c^2 - 2*(a^3*b^2 - a*b^4)*c*d + (a^4*b - a^2*b^3)*d^2)*cos(f*x + e))/((a^4*b^3 - 2*a^2*b^5 + b^7)*f*sin(f*x + e) + (a^5*b^2 - 2*a^3*b^4 + a*b^6)*f)]","B",0
709,1,399,0,2.102349," ","integrate((c+d*sin(f*x+e))/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[-\frac{{\left(a^{2} c - a b d + {\left(a b c - b^{2} d\right)} \sin\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) - 2 \, {\left({\left(a^{2} b - b^{3}\right)} c - {\left(a^{3} - a b^{2}\right)} d\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} f \sin\left(f x + e\right) + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f\right)}}, -\frac{{\left(a^{2} c - a b d + {\left(a b c - b^{2} d\right)} \sin\left(f x + e\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) - {\left({\left(a^{2} b - b^{3}\right)} c - {\left(a^{3} - a b^{2}\right)} d\right)} \cos\left(f x + e\right)}{{\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} f \sin\left(f x + e\right) + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f}\right]"," ",0,"[-1/2*((a^2*c - a*b*d + (a*b*c - b^2*d)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 + 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) - 2*((a^2*b - b^3)*c - (a^3 - a*b^2)*d)*cos(f*x + e))/((a^4*b - 2*a^2*b^3 + b^5)*f*sin(f*x + e) + (a^5 - 2*a^3*b^2 + a*b^4)*f), -((a^2*c - a*b*d + (a*b*c - b^2*d)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) - ((a^2*b - b^3)*c - (a^3 - a*b^2)*d)*cos(f*x + e))/((a^4*b - 2*a^2*b^3 + b^5)*f*sin(f*x + e) + (a^5 - 2*a^3*b^2 + a*b^4)*f)]","A",0
710,1,336,0,1.988288," ","integrate(1/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\left[\frac{{\left(a b \sin\left(f x + e\right) + a^{2}\right)} \sqrt{-a^{2} + b^{2}} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) + 2 \, {\left(a^{2} b - b^{3}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} f \sin\left(f x + e\right) + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f\right)}}, -\frac{{\left(a b \sin\left(f x + e\right) + a^{2}\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) - {\left(a^{2} b - b^{3}\right)} \cos\left(f x + e\right)}{{\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} f \sin\left(f x + e\right) + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} f}\right]"," ",0,"[1/2*((a*b*sin(f*x + e) + a^2)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + 2*(a^2*b - b^3)*cos(f*x + e))/((a^4*b - 2*a^2*b^3 + b^5)*f*sin(f*x + e) + (a^5 - 2*a^3*b^2 + a*b^4)*f), -((a*b*sin(f*x + e) + a^2)*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) - (a^2*b - b^3)*cos(f*x + e))/((a^4*b - 2*a^2*b^3 + b^5)*f*sin(f*x + e) + (a^5 - 2*a^3*b^2 + a*b^4)*f)]","A",0
711,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
712,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
713,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
714,1,3174,0,2.324024," ","integrate((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(20 \, {\left(a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right)} c^{2} d^{3} - 30 \, {\left(a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right)} c d^{4} + {\left(12 \, a^{8} b^{2} - 35 \, a^{6} b^{4} + 33 \, a^{4} b^{6} - 9 \, a^{2} b^{8} - b^{10}\right)} d^{5}\right)} f x \cos\left(f x + e\right)^{2} - 4 \, {\left(5 \, {\left(a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right)} c d^{4} - 2 \, {\left(a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right)} d^{5}\right)} \cos\left(f x + e\right)^{3} - 2 \, {\left(20 \, {\left(a^{8} b^{2} - 2 \, a^{6} b^{4} + 2 \, a^{2} b^{8} - b^{10}\right)} c^{2} d^{3} - 30 \, {\left(a^{9} b - 2 \, a^{7} b^{3} + 2 \, a^{3} b^{7} - a b^{9}\right)} c d^{4} + {\left(12 \, a^{10} - 23 \, a^{8} b^{2} - 2 \, a^{6} b^{4} + 24 \, a^{4} b^{6} - 10 \, a^{2} b^{8} - b^{10}\right)} d^{5}\right)} f x - {\left({\left(2 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right)} c^{5} - 15 \, {\left(a^{3} b^{6} + a b^{8}\right)} c^{4} d + 10 \, {\left(a^{4} b^{5} + 3 \, a^{2} b^{7} + 2 \, b^{9}\right)} c^{3} d^{2} - 10 \, {\left(2 \, a^{7} b^{2} - 3 \, a^{5} b^{4} + a^{3} b^{6} + 6 \, a b^{8}\right)} c^{2} d^{3} + 15 \, {\left(2 \, a^{8} b - 3 \, a^{6} b^{3} - a^{4} b^{5} + 4 \, a^{2} b^{7}\right)} c d^{4} - {\left(12 \, a^{9} - 17 \, a^{7} b^{2} - 9 \, a^{5} b^{4} + 20 \, a^{3} b^{6}\right)} d^{5} + {\left(15 \, a b^{8} c^{4} d - {\left(2 \, a^{2} b^{7} + b^{9}\right)} c^{5} - 10 \, {\left(a^{2} b^{7} + 2 \, b^{9}\right)} c^{3} d^{2} + 10 \, {\left(2 \, a^{5} b^{4} - 5 \, a^{3} b^{6} + 6 \, a b^{8}\right)} c^{2} d^{3} - 15 \, {\left(2 \, a^{6} b^{3} - 5 \, a^{4} b^{5} + 4 \, a^{2} b^{7}\right)} c d^{4} + {\left(12 \, a^{7} b^{2} - 29 \, a^{5} b^{4} + 20 \, a^{3} b^{6}\right)} d^{5}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(15 \, a^{2} b^{7} c^{4} d - {\left(2 \, a^{3} b^{6} + a b^{8}\right)} c^{5} - 10 \, {\left(a^{3} b^{6} + 2 \, a b^{8}\right)} c^{3} d^{2} + 10 \, {\left(2 \, a^{6} b^{3} - 5 \, a^{4} b^{5} + 6 \, a^{2} b^{7}\right)} c^{2} d^{3} - 15 \, {\left(2 \, a^{7} b^{2} - 5 \, a^{5} b^{4} + 4 \, a^{3} b^{6}\right)} c d^{4} + {\left(12 \, a^{8} b - 29 \, a^{6} b^{3} + 20 \, a^{4} b^{5}\right)} d^{5}\right)} \sin\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) - 2 \, {\left({\left(4 \, a^{4} b^{6} - 5 \, a^{2} b^{8} + b^{10}\right)} c^{5} - 5 \, {\left(2 \, a^{5} b^{5} - a^{3} b^{7} - a b^{9}\right)} c^{4} d + 30 \, {\left(a^{4} b^{6} - a^{2} b^{8}\right)} c^{3} d^{2} + 10 \, {\left(2 \, a^{7} b^{3} - 7 \, a^{5} b^{5} + 5 \, a^{3} b^{7}\right)} c^{2} d^{3} - 5 \, {\left(6 \, a^{8} b^{2} - 15 \, a^{6} b^{4} + 7 \, a^{4} b^{6} + 4 \, a^{2} b^{8} - 2 \, b^{10}\right)} c d^{4} + {\left(12 \, a^{9} b - 29 \, a^{7} b^{3} + 15 \, a^{5} b^{5} + 6 \, a^{3} b^{7} - 4 \, a b^{9}\right)} d^{5}\right)} \cos\left(f x + e\right) - 2 \, {\left({\left(a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right)} d^{5} \cos\left(f x + e\right)^{3} + 2 \, {\left(20 \, {\left(a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right)} c^{2} d^{3} - 30 \, {\left(a^{8} b^{2} - 3 \, a^{6} b^{4} + 3 \, a^{4} b^{6} - a^{2} b^{8}\right)} c d^{4} + {\left(12 \, a^{9} b - 35 \, a^{7} b^{3} + 33 \, a^{5} b^{5} - 9 \, a^{3} b^{7} - a b^{9}\right)} d^{5}\right)} f x + {\left(3 \, {\left(a^{3} b^{7} - a b^{9}\right)} c^{5} - 5 \, {\left(a^{4} b^{6} + a^{2} b^{8} - 2 \, b^{10}\right)} c^{4} d - 10 \, {\left(a^{5} b^{5} - 5 \, a^{3} b^{7} + 4 \, a b^{9}\right)} c^{3} d^{2} + 30 \, {\left(a^{6} b^{4} - 3 \, a^{4} b^{6} + 2 \, a^{2} b^{8}\right)} c^{2} d^{3} - 5 \, {\left(9 \, a^{7} b^{3} - 25 \, a^{5} b^{5} + 20 \, a^{3} b^{7} - 4 \, a b^{9}\right)} c d^{4} + {\left(18 \, a^{8} b^{2} - 51 \, a^{6} b^{4} + 46 \, a^{4} b^{6} - 14 \, a^{2} b^{8} + b^{10}\right)} d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{6} b^{7} - 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} - b^{13}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b^{6} - 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} - a b^{12}\right)} f \sin\left(f x + e\right) - {\left(a^{8} b^{5} - 2 \, a^{6} b^{7} + 2 \, a^{2} b^{11} - b^{13}\right)} f\right)}}, \frac{{\left(20 \, {\left(a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right)} c^{2} d^{3} - 30 \, {\left(a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right)} c d^{4} + {\left(12 \, a^{8} b^{2} - 35 \, a^{6} b^{4} + 33 \, a^{4} b^{6} - 9 \, a^{2} b^{8} - b^{10}\right)} d^{5}\right)} f x \cos\left(f x + e\right)^{2} - 2 \, {\left(5 \, {\left(a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right)} c d^{4} - 2 \, {\left(a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right)} d^{5}\right)} \cos\left(f x + e\right)^{3} - {\left(20 \, {\left(a^{8} b^{2} - 2 \, a^{6} b^{4} + 2 \, a^{2} b^{8} - b^{10}\right)} c^{2} d^{3} - 30 \, {\left(a^{9} b - 2 \, a^{7} b^{3} + 2 \, a^{3} b^{7} - a b^{9}\right)} c d^{4} + {\left(12 \, a^{10} - 23 \, a^{8} b^{2} - 2 \, a^{6} b^{4} + 24 \, a^{4} b^{6} - 10 \, a^{2} b^{8} - b^{10}\right)} d^{5}\right)} f x + {\left({\left(2 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right)} c^{5} - 15 \, {\left(a^{3} b^{6} + a b^{8}\right)} c^{4} d + 10 \, {\left(a^{4} b^{5} + 3 \, a^{2} b^{7} + 2 \, b^{9}\right)} c^{3} d^{2} - 10 \, {\left(2 \, a^{7} b^{2} - 3 \, a^{5} b^{4} + a^{3} b^{6} + 6 \, a b^{8}\right)} c^{2} d^{3} + 15 \, {\left(2 \, a^{8} b - 3 \, a^{6} b^{3} - a^{4} b^{5} + 4 \, a^{2} b^{7}\right)} c d^{4} - {\left(12 \, a^{9} - 17 \, a^{7} b^{2} - 9 \, a^{5} b^{4} + 20 \, a^{3} b^{6}\right)} d^{5} + {\left(15 \, a b^{8} c^{4} d - {\left(2 \, a^{2} b^{7} + b^{9}\right)} c^{5} - 10 \, {\left(a^{2} b^{7} + 2 \, b^{9}\right)} c^{3} d^{2} + 10 \, {\left(2 \, a^{5} b^{4} - 5 \, a^{3} b^{6} + 6 \, a b^{8}\right)} c^{2} d^{3} - 15 \, {\left(2 \, a^{6} b^{3} - 5 \, a^{4} b^{5} + 4 \, a^{2} b^{7}\right)} c d^{4} + {\left(12 \, a^{7} b^{2} - 29 \, a^{5} b^{4} + 20 \, a^{3} b^{6}\right)} d^{5}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(15 \, a^{2} b^{7} c^{4} d - {\left(2 \, a^{3} b^{6} + a b^{8}\right)} c^{5} - 10 \, {\left(a^{3} b^{6} + 2 \, a b^{8}\right)} c^{3} d^{2} + 10 \, {\left(2 \, a^{6} b^{3} - 5 \, a^{4} b^{5} + 6 \, a^{2} b^{7}\right)} c^{2} d^{3} - 15 \, {\left(2 \, a^{7} b^{2} - 5 \, a^{5} b^{4} + 4 \, a^{3} b^{6}\right)} c d^{4} + {\left(12 \, a^{8} b - 29 \, a^{6} b^{3} + 20 \, a^{4} b^{5}\right)} d^{5}\right)} \sin\left(f x + e\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) - {\left({\left(4 \, a^{4} b^{6} - 5 \, a^{2} b^{8} + b^{10}\right)} c^{5} - 5 \, {\left(2 \, a^{5} b^{5} - a^{3} b^{7} - a b^{9}\right)} c^{4} d + 30 \, {\left(a^{4} b^{6} - a^{2} b^{8}\right)} c^{3} d^{2} + 10 \, {\left(2 \, a^{7} b^{3} - 7 \, a^{5} b^{5} + 5 \, a^{3} b^{7}\right)} c^{2} d^{3} - 5 \, {\left(6 \, a^{8} b^{2} - 15 \, a^{6} b^{4} + 7 \, a^{4} b^{6} + 4 \, a^{2} b^{8} - 2 \, b^{10}\right)} c d^{4} + {\left(12 \, a^{9} b - 29 \, a^{7} b^{3} + 15 \, a^{5} b^{5} + 6 \, a^{3} b^{7} - 4 \, a b^{9}\right)} d^{5}\right)} \cos\left(f x + e\right) - {\left({\left(a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right)} d^{5} \cos\left(f x + e\right)^{3} + 2 \, {\left(20 \, {\left(a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right)} c^{2} d^{3} - 30 \, {\left(a^{8} b^{2} - 3 \, a^{6} b^{4} + 3 \, a^{4} b^{6} - a^{2} b^{8}\right)} c d^{4} + {\left(12 \, a^{9} b - 35 \, a^{7} b^{3} + 33 \, a^{5} b^{5} - 9 \, a^{3} b^{7} - a b^{9}\right)} d^{5}\right)} f x + {\left(3 \, {\left(a^{3} b^{7} - a b^{9}\right)} c^{5} - 5 \, {\left(a^{4} b^{6} + a^{2} b^{8} - 2 \, b^{10}\right)} c^{4} d - 10 \, {\left(a^{5} b^{5} - 5 \, a^{3} b^{7} + 4 \, a b^{9}\right)} c^{3} d^{2} + 30 \, {\left(a^{6} b^{4} - 3 \, a^{4} b^{6} + 2 \, a^{2} b^{8}\right)} c^{2} d^{3} - 5 \, {\left(9 \, a^{7} b^{3} - 25 \, a^{5} b^{5} + 20 \, a^{3} b^{7} - 4 \, a b^{9}\right)} c d^{4} + {\left(18 \, a^{8} b^{2} - 51 \, a^{6} b^{4} + 46 \, a^{4} b^{6} - 14 \, a^{2} b^{8} + b^{10}\right)} d^{5}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a^{6} b^{7} - 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} - b^{13}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b^{6} - 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} - a b^{12}\right)} f \sin\left(f x + e\right) - {\left(a^{8} b^{5} - 2 \, a^{6} b^{7} + 2 \, a^{2} b^{11} - b^{13}\right)} f\right)}}\right]"," ",0,"[1/4*(2*(20*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*c^2*d^3 - 30*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*c*d^4 + (12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b^8 - b^10)*d^5)*f*x*cos(f*x + e)^2 - 4*(5*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*c*d^4 - 2*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*d^5)*cos(f*x + e)^3 - 2*(20*(a^8*b^2 - 2*a^6*b^4 + 2*a^2*b^8 - b^10)*c^2*d^3 - 30*(a^9*b - 2*a^7*b^3 + 2*a^3*b^7 - a*b^9)*c*d^4 + (12*a^10 - 23*a^8*b^2 - 2*a^6*b^4 + 24*a^4*b^6 - 10*a^2*b^8 - b^10)*d^5)*f*x - ((2*a^4*b^5 + 3*a^2*b^7 + b^9)*c^5 - 15*(a^3*b^6 + a*b^8)*c^4*d + 10*(a^4*b^5 + 3*a^2*b^7 + 2*b^9)*c^3*d^2 - 10*(2*a^7*b^2 - 3*a^5*b^4 + a^3*b^6 + 6*a*b^8)*c^2*d^3 + 15*(2*a^8*b - 3*a^6*b^3 - a^4*b^5 + 4*a^2*b^7)*c*d^4 - (12*a^9 - 17*a^7*b^2 - 9*a^5*b^4 + 20*a^3*b^6)*d^5 + (15*a*b^8*c^4*d - (2*a^2*b^7 + b^9)*c^5 - 10*(a^2*b^7 + 2*b^9)*c^3*d^2 + 10*(2*a^5*b^4 - 5*a^3*b^6 + 6*a*b^8)*c^2*d^3 - 15*(2*a^6*b^3 - 5*a^4*b^5 + 4*a^2*b^7)*c*d^4 + (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*d^5)*cos(f*x + e)^2 - 2*(15*a^2*b^7*c^4*d - (2*a^3*b^6 + a*b^8)*c^5 - 10*(a^3*b^6 + 2*a*b^8)*c^3*d^2 + 10*(2*a^6*b^3 - 5*a^4*b^5 + 6*a^2*b^7)*c^2*d^3 - 15*(2*a^7*b^2 - 5*a^5*b^4 + 4*a^3*b^6)*c*d^4 + (12*a^8*b - 29*a^6*b^3 + 20*a^4*b^5)*d^5)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) - 2*((4*a^4*b^6 - 5*a^2*b^8 + b^10)*c^5 - 5*(2*a^5*b^5 - a^3*b^7 - a*b^9)*c^4*d + 30*(a^4*b^6 - a^2*b^8)*c^3*d^2 + 10*(2*a^7*b^3 - 7*a^5*b^5 + 5*a^3*b^7)*c^2*d^3 - 5*(6*a^8*b^2 - 15*a^6*b^4 + 7*a^4*b^6 + 4*a^2*b^8 - 2*b^10)*c*d^4 + (12*a^9*b - 29*a^7*b^3 + 15*a^5*b^5 + 6*a^3*b^7 - 4*a*b^9)*d^5)*cos(f*x + e) - 2*((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*d^5*cos(f*x + e)^3 + 2*(20*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*c^2*d^3 - 30*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8)*c*d^4 + (12*a^9*b - 35*a^7*b^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*d^5)*f*x + (3*(a^3*b^7 - a*b^9)*c^5 - 5*(a^4*b^6 + a^2*b^8 - 2*b^10)*c^4*d - 10*(a^5*b^5 - 5*a^3*b^7 + 4*a*b^9)*c^3*d^2 + 30*(a^6*b^4 - 3*a^4*b^6 + 2*a^2*b^8)*c^2*d^3 - 5*(9*a^7*b^3 - 25*a^5*b^5 + 20*a^3*b^7 - 4*a*b^9)*c*d^4 + (18*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8 + b^10)*d^5)*cos(f*x + e))*sin(f*x + e))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*f*cos(f*x + e)^2 - 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*f*sin(f*x + e) - (a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^13)*f), 1/2*((20*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*c^2*d^3 - 30*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*c*d^4 + (12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b^8 - b^10)*d^5)*f*x*cos(f*x + e)^2 - 2*(5*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*c*d^4 - 2*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*d^5)*cos(f*x + e)^3 - (20*(a^8*b^2 - 2*a^6*b^4 + 2*a^2*b^8 - b^10)*c^2*d^3 - 30*(a^9*b - 2*a^7*b^3 + 2*a^3*b^7 - a*b^9)*c*d^4 + (12*a^10 - 23*a^8*b^2 - 2*a^6*b^4 + 24*a^4*b^6 - 10*a^2*b^8 - b^10)*d^5)*f*x + ((2*a^4*b^5 + 3*a^2*b^7 + b^9)*c^5 - 15*(a^3*b^6 + a*b^8)*c^4*d + 10*(a^4*b^5 + 3*a^2*b^7 + 2*b^9)*c^3*d^2 - 10*(2*a^7*b^2 - 3*a^5*b^4 + a^3*b^6 + 6*a*b^8)*c^2*d^3 + 15*(2*a^8*b - 3*a^6*b^3 - a^4*b^5 + 4*a^2*b^7)*c*d^4 - (12*a^9 - 17*a^7*b^2 - 9*a^5*b^4 + 20*a^3*b^6)*d^5 + (15*a*b^8*c^4*d - (2*a^2*b^7 + b^9)*c^5 - 10*(a^2*b^7 + 2*b^9)*c^3*d^2 + 10*(2*a^5*b^4 - 5*a^3*b^6 + 6*a*b^8)*c^2*d^3 - 15*(2*a^6*b^3 - 5*a^4*b^5 + 4*a^2*b^7)*c*d^4 + (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*d^5)*cos(f*x + e)^2 - 2*(15*a^2*b^7*c^4*d - (2*a^3*b^6 + a*b^8)*c^5 - 10*(a^3*b^6 + 2*a*b^8)*c^3*d^2 + 10*(2*a^6*b^3 - 5*a^4*b^5 + 6*a^2*b^7)*c^2*d^3 - 15*(2*a^7*b^2 - 5*a^5*b^4 + 4*a^3*b^6)*c*d^4 + (12*a^8*b - 29*a^6*b^3 + 20*a^4*b^5)*d^5)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) - ((4*a^4*b^6 - 5*a^2*b^8 + b^10)*c^5 - 5*(2*a^5*b^5 - a^3*b^7 - a*b^9)*c^4*d + 30*(a^4*b^6 - a^2*b^8)*c^3*d^2 + 10*(2*a^7*b^3 - 7*a^5*b^5 + 5*a^3*b^7)*c^2*d^3 - 5*(6*a^8*b^2 - 15*a^6*b^4 + 7*a^4*b^6 + 4*a^2*b^8 - 2*b^10)*c*d^4 + (12*a^9*b - 29*a^7*b^3 + 15*a^5*b^5 + 6*a^3*b^7 - 4*a*b^9)*d^5)*cos(f*x + e) - ((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*d^5*cos(f*x + e)^3 + 2*(20*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*c^2*d^3 - 30*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8)*c*d^4 + (12*a^9*b - 35*a^7*b^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*d^5)*f*x + (3*(a^3*b^7 - a*b^9)*c^5 - 5*(a^4*b^6 + a^2*b^8 - 2*b^10)*c^4*d - 10*(a^5*b^5 - 5*a^3*b^7 + 4*a*b^9)*c^3*d^2 + 30*(a^6*b^4 - 3*a^4*b^6 + 2*a^2*b^8)*c^2*d^3 - 5*(9*a^7*b^3 - 25*a^5*b^5 + 20*a^3*b^7 - 4*a*b^9)*c*d^4 + (18*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8 + b^10)*d^5)*cos(f*x + e))*sin(f*x + e))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*f*cos(f*x + e)^2 - 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*f*sin(f*x + e) - (a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^13)*f)]","B",0
715,1,2335,0,2.094230," ","integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} d^{4} \cos\left(f x + e\right)^{3} - 4 \, {\left(4 \, {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} c d^{3} - 3 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} d^{4}\right)} f x \cos\left(f x + e\right)^{2} + 4 \, {\left(4 \, {\left(a^{8} b - 2 \, a^{6} b^{3} + 2 \, a^{2} b^{7} - b^{9}\right)} c d^{3} - 3 \, {\left(a^{9} - 2 \, a^{7} b^{2} + 2 \, a^{3} b^{6} - a b^{8}\right)} d^{4}\right)} f x - {\left({\left(2 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right)} c^{4} - 12 \, {\left(a^{3} b^{5} + a b^{7}\right)} c^{3} d + 6 \, {\left(a^{4} b^{4} + 3 \, a^{2} b^{6} + 2 \, b^{8}\right)} c^{2} d^{2} - 4 \, {\left(2 \, a^{7} b - 3 \, a^{5} b^{3} + a^{3} b^{5} + 6 \, a b^{7}\right)} c d^{3} + 3 \, {\left(2 \, a^{8} - 3 \, a^{6} b^{2} - a^{4} b^{4} + 4 \, a^{2} b^{6}\right)} d^{4} + {\left(12 \, a b^{7} c^{3} d - {\left(2 \, a^{2} b^{6} + b^{8}\right)} c^{4} - 6 \, {\left(a^{2} b^{6} + 2 \, b^{8}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{5} b^{3} - 5 \, a^{3} b^{5} + 6 \, a b^{7}\right)} c d^{3} - 3 \, {\left(2 \, a^{6} b^{2} - 5 \, a^{4} b^{4} + 4 \, a^{2} b^{6}\right)} d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(12 \, a^{2} b^{6} c^{3} d - {\left(2 \, a^{3} b^{5} + a b^{7}\right)} c^{4} - 6 \, {\left(a^{3} b^{5} + 2 \, a b^{7}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{6} b^{2} - 5 \, a^{4} b^{4} + 6 \, a^{2} b^{6}\right)} c d^{3} - 3 \, {\left(2 \, a^{7} b - 5 \, a^{5} b^{3} + 4 \, a^{3} b^{5}\right)} d^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) + 2 \, {\left({\left(4 \, a^{4} b^{5} - 5 \, a^{2} b^{7} + b^{9}\right)} c^{4} - 4 \, {\left(2 \, a^{5} b^{4} - a^{3} b^{6} - a b^{8}\right)} c^{3} d + 18 \, {\left(a^{4} b^{5} - a^{2} b^{7}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{7} b^{2} - 7 \, a^{5} b^{4} + 5 \, a^{3} b^{6}\right)} c d^{3} - {\left(6 \, a^{8} b - 15 \, a^{6} b^{3} + 7 \, a^{4} b^{5} + 4 \, a^{2} b^{7} - 2 \, b^{9}\right)} d^{4}\right)} \cos\left(f x + e\right) + 2 \, {\left(4 \, {\left(4 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} c d^{3} - 3 \, {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d^{4}\right)} f x + {\left(3 \, {\left(a^{3} b^{6} - a b^{8}\right)} c^{4} - 4 \, {\left(a^{4} b^{5} + a^{2} b^{7} - 2 \, b^{9}\right)} c^{3} d - 6 \, {\left(a^{5} b^{4} - 5 \, a^{3} b^{6} + 4 \, a b^{8}\right)} c^{2} d^{2} + 12 \, {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 2 \, a^{2} b^{7}\right)} c d^{3} - {\left(9 \, a^{7} b^{2} - 25 \, a^{5} b^{4} + 20 \, a^{3} b^{6} - 4 \, a b^{8}\right)} d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{6} b^{6} - 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} - b^{12}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b^{5} - 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} - a b^{11}\right)} f \sin\left(f x + e\right) - {\left(a^{8} b^{4} - 2 \, a^{6} b^{6} + 2 \, a^{2} b^{10} - b^{12}\right)} f\right)}}, -\frac{2 \, {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} d^{4} \cos\left(f x + e\right)^{3} - 2 \, {\left(4 \, {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right)} c d^{3} - 3 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} d^{4}\right)} f x \cos\left(f x + e\right)^{2} + 2 \, {\left(4 \, {\left(a^{8} b - 2 \, a^{6} b^{3} + 2 \, a^{2} b^{7} - b^{9}\right)} c d^{3} - 3 \, {\left(a^{9} - 2 \, a^{7} b^{2} + 2 \, a^{3} b^{6} - a b^{8}\right)} d^{4}\right)} f x - {\left({\left(2 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right)} c^{4} - 12 \, {\left(a^{3} b^{5} + a b^{7}\right)} c^{3} d + 6 \, {\left(a^{4} b^{4} + 3 \, a^{2} b^{6} + 2 \, b^{8}\right)} c^{2} d^{2} - 4 \, {\left(2 \, a^{7} b - 3 \, a^{5} b^{3} + a^{3} b^{5} + 6 \, a b^{7}\right)} c d^{3} + 3 \, {\left(2 \, a^{8} - 3 \, a^{6} b^{2} - a^{4} b^{4} + 4 \, a^{2} b^{6}\right)} d^{4} + {\left(12 \, a b^{7} c^{3} d - {\left(2 \, a^{2} b^{6} + b^{8}\right)} c^{4} - 6 \, {\left(a^{2} b^{6} + 2 \, b^{8}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{5} b^{3} - 5 \, a^{3} b^{5} + 6 \, a b^{7}\right)} c d^{3} - 3 \, {\left(2 \, a^{6} b^{2} - 5 \, a^{4} b^{4} + 4 \, a^{2} b^{6}\right)} d^{4}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(12 \, a^{2} b^{6} c^{3} d - {\left(2 \, a^{3} b^{5} + a b^{7}\right)} c^{4} - 6 \, {\left(a^{3} b^{5} + 2 \, a b^{7}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{6} b^{2} - 5 \, a^{4} b^{4} + 6 \, a^{2} b^{6}\right)} c d^{3} - 3 \, {\left(2 \, a^{7} b - 5 \, a^{5} b^{3} + 4 \, a^{3} b^{5}\right)} d^{4}\right)} \sin\left(f x + e\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) + {\left({\left(4 \, a^{4} b^{5} - 5 \, a^{2} b^{7} + b^{9}\right)} c^{4} - 4 \, {\left(2 \, a^{5} b^{4} - a^{3} b^{6} - a b^{8}\right)} c^{3} d + 18 \, {\left(a^{4} b^{5} - a^{2} b^{7}\right)} c^{2} d^{2} + 4 \, {\left(2 \, a^{7} b^{2} - 7 \, a^{5} b^{4} + 5 \, a^{3} b^{6}\right)} c d^{3} - {\left(6 \, a^{8} b - 15 \, a^{6} b^{3} + 7 \, a^{4} b^{5} + 4 \, a^{2} b^{7} - 2 \, b^{9}\right)} d^{4}\right)} \cos\left(f x + e\right) + {\left(4 \, {\left(4 \, {\left(a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right)} c d^{3} - 3 \, {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d^{4}\right)} f x + {\left(3 \, {\left(a^{3} b^{6} - a b^{8}\right)} c^{4} - 4 \, {\left(a^{4} b^{5} + a^{2} b^{7} - 2 \, b^{9}\right)} c^{3} d - 6 \, {\left(a^{5} b^{4} - 5 \, a^{3} b^{6} + 4 \, a b^{8}\right)} c^{2} d^{2} + 12 \, {\left(a^{6} b^{3} - 3 \, a^{4} b^{5} + 2 \, a^{2} b^{7}\right)} c d^{3} - {\left(9 \, a^{7} b^{2} - 25 \, a^{5} b^{4} + 20 \, a^{3} b^{6} - 4 \, a b^{8}\right)} d^{4}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a^{6} b^{6} - 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} - b^{12}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b^{5} - 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} - a b^{11}\right)} f \sin\left(f x + e\right) - {\left(a^{8} b^{4} - 2 \, a^{6} b^{6} + 2 \, a^{2} b^{10} - b^{12}\right)} f\right)}}\right]"," ",0,"[-1/4*(4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*d^4*cos(f*x + e)^3 - 4*(4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*c*d^3 - 3*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*d^4)*f*x*cos(f*x + e)^2 + 4*(4*(a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9)*c*d^3 - 3*(a^9 - 2*a^7*b^2 + 2*a^3*b^6 - a*b^8)*d^4)*f*x - ((2*a^4*b^4 + 3*a^2*b^6 + b^8)*c^4 - 12*(a^3*b^5 + a*b^7)*c^3*d + 6*(a^4*b^4 + 3*a^2*b^6 + 2*b^8)*c^2*d^2 - 4*(2*a^7*b - 3*a^5*b^3 + a^3*b^5 + 6*a*b^7)*c*d^3 + 3*(2*a^8 - 3*a^6*b^2 - a^4*b^4 + 4*a^2*b^6)*d^4 + (12*a*b^7*c^3*d - (2*a^2*b^6 + b^8)*c^4 - 6*(a^2*b^6 + 2*b^8)*c^2*d^2 + 4*(2*a^5*b^3 - 5*a^3*b^5 + 6*a*b^7)*c*d^3 - 3*(2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b^6)*d^4)*cos(f*x + e)^2 - 2*(12*a^2*b^6*c^3*d - (2*a^3*b^5 + a*b^7)*c^4 - 6*(a^3*b^5 + 2*a*b^7)*c^2*d^2 + 4*(2*a^6*b^2 - 5*a^4*b^4 + 6*a^2*b^6)*c*d^3 - 3*(2*a^7*b - 5*a^5*b^3 + 4*a^3*b^5)*d^4)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 + 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + 2*((4*a^4*b^5 - 5*a^2*b^7 + b^9)*c^4 - 4*(2*a^5*b^4 - a^3*b^6 - a*b^8)*c^3*d + 18*(a^4*b^5 - a^2*b^7)*c^2*d^2 + 4*(2*a^7*b^2 - 7*a^5*b^4 + 5*a^3*b^6)*c*d^3 - (6*a^8*b - 15*a^6*b^3 + 7*a^4*b^5 + 4*a^2*b^7 - 2*b^9)*d^4)*cos(f*x + e) + 2*(4*(4*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*c*d^3 - 3*(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d^4)*f*x + (3*(a^3*b^6 - a*b^8)*c^4 - 4*(a^4*b^5 + a^2*b^7 - 2*b^9)*c^3*d - 6*(a^5*b^4 - 5*a^3*b^6 + 4*a*b^8)*c^2*d^2 + 12*(a^6*b^3 - 3*a^4*b^5 + 2*a^2*b^7)*c*d^3 - (9*a^7*b^2 - 25*a^5*b^4 + 20*a^3*b^6 - 4*a*b^8)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*f*cos(f*x + e)^2 - 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*f*sin(f*x + e) - (a^8*b^4 - 2*a^6*b^6 + 2*a^2*b^10 - b^12)*f), -1/2*(2*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*d^4*cos(f*x + e)^3 - 2*(4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*c*d^3 - 3*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*d^4)*f*x*cos(f*x + e)^2 + 2*(4*(a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9)*c*d^3 - 3*(a^9 - 2*a^7*b^2 + 2*a^3*b^6 - a*b^8)*d^4)*f*x - ((2*a^4*b^4 + 3*a^2*b^6 + b^8)*c^4 - 12*(a^3*b^5 + a*b^7)*c^3*d + 6*(a^4*b^4 + 3*a^2*b^6 + 2*b^8)*c^2*d^2 - 4*(2*a^7*b - 3*a^5*b^3 + a^3*b^5 + 6*a*b^7)*c*d^3 + 3*(2*a^8 - 3*a^6*b^2 - a^4*b^4 + 4*a^2*b^6)*d^4 + (12*a*b^7*c^3*d - (2*a^2*b^6 + b^8)*c^4 - 6*(a^2*b^6 + 2*b^8)*c^2*d^2 + 4*(2*a^5*b^3 - 5*a^3*b^5 + 6*a*b^7)*c*d^3 - 3*(2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b^6)*d^4)*cos(f*x + e)^2 - 2*(12*a^2*b^6*c^3*d - (2*a^3*b^5 + a*b^7)*c^4 - 6*(a^3*b^5 + 2*a*b^7)*c^2*d^2 + 4*(2*a^6*b^2 - 5*a^4*b^4 + 6*a^2*b^6)*c*d^3 - 3*(2*a^7*b - 5*a^5*b^3 + 4*a^3*b^5)*d^4)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + ((4*a^4*b^5 - 5*a^2*b^7 + b^9)*c^4 - 4*(2*a^5*b^4 - a^3*b^6 - a*b^8)*c^3*d + 18*(a^4*b^5 - a^2*b^7)*c^2*d^2 + 4*(2*a^7*b^2 - 7*a^5*b^4 + 5*a^3*b^6)*c*d^3 - (6*a^8*b - 15*a^6*b^3 + 7*a^4*b^5 + 4*a^2*b^7 - 2*b^9)*d^4)*cos(f*x + e) + (4*(4*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*c*d^3 - 3*(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d^4)*f*x + (3*(a^3*b^6 - a*b^8)*c^4 - 4*(a^4*b^5 + a^2*b^7 - 2*b^9)*c^3*d - 6*(a^5*b^4 - 5*a^3*b^6 + 4*a*b^8)*c^2*d^2 + 12*(a^6*b^3 - 3*a^4*b^5 + 2*a^2*b^7)*c*d^3 - (9*a^7*b^2 - 25*a^5*b^4 + 20*a^3*b^6 - 4*a*b^8)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*f*cos(f*x + e)^2 - 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*f*sin(f*x + e) - (a^8*b^4 - 2*a^6*b^6 + 2*a^2*b^10 - b^12)*f)]","B",0
716,1,1631,0,1.472573," ","integrate((c+d*sin(f*x+e))^3/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} d^{3} f x \cos\left(f x + e\right)^{2} - 4 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} d^{3} f x - {\left({\left(2 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right)} c^{3} - 9 \, {\left(a^{3} b^{4} + a b^{6}\right)} c^{2} d + 3 \, {\left(a^{4} b^{3} + 3 \, a^{2} b^{5} + 2 \, b^{7}\right)} c d^{2} - {\left(2 \, a^{7} - 3 \, a^{5} b^{2} + a^{3} b^{4} + 6 \, a b^{6}\right)} d^{3} + {\left(9 \, a b^{6} c^{2} d - {\left(2 \, a^{2} b^{5} + b^{7}\right)} c^{3} - 3 \, {\left(a^{2} b^{5} + 2 \, b^{7}\right)} c d^{2} + {\left(2 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 6 \, a b^{6}\right)} d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(9 \, a^{2} b^{5} c^{2} d - {\left(2 \, a^{3} b^{4} + a b^{6}\right)} c^{3} - 3 \, {\left(a^{3} b^{4} + 2 \, a b^{6}\right)} c d^{2} + {\left(2 \, a^{6} b - 5 \, a^{4} b^{3} + 6 \, a^{2} b^{5}\right)} d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) - 2 \, {\left({\left(4 \, a^{4} b^{4} - 5 \, a^{2} b^{6} + b^{8}\right)} c^{3} - 3 \, {\left(2 \, a^{5} b^{3} - a^{3} b^{5} - a b^{7}\right)} c^{2} d + 9 \, {\left(a^{4} b^{4} - a^{2} b^{6}\right)} c d^{2} + {\left(2 \, a^{7} b - 7 \, a^{5} b^{3} + 5 \, a^{3} b^{5}\right)} d^{3}\right)} \cos\left(f x + e\right) - 2 \, {\left(4 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} d^{3} f x + 3 \, {\left({\left(a^{3} b^{5} - a b^{7}\right)} c^{3} - {\left(a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right)} c^{2} d - {\left(a^{5} b^{3} - 5 \, a^{3} b^{5} + 4 \, a b^{7}\right)} c d^{2} + {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 2 \, a^{2} b^{6}\right)} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{6} b^{5} - 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} - b^{11}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b^{4} - 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} - a b^{10}\right)} f \sin\left(f x + e\right) - {\left(a^{8} b^{3} - 2 \, a^{6} b^{5} + 2 \, a^{2} b^{9} - b^{11}\right)} f\right)}}, \frac{2 \, {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} d^{3} f x \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} d^{3} f x + {\left({\left(2 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right)} c^{3} - 9 \, {\left(a^{3} b^{4} + a b^{6}\right)} c^{2} d + 3 \, {\left(a^{4} b^{3} + 3 \, a^{2} b^{5} + 2 \, b^{7}\right)} c d^{2} - {\left(2 \, a^{7} - 3 \, a^{5} b^{2} + a^{3} b^{4} + 6 \, a b^{6}\right)} d^{3} + {\left(9 \, a b^{6} c^{2} d - {\left(2 \, a^{2} b^{5} + b^{7}\right)} c^{3} - 3 \, {\left(a^{2} b^{5} + 2 \, b^{7}\right)} c d^{2} + {\left(2 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 6 \, a b^{6}\right)} d^{3}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(9 \, a^{2} b^{5} c^{2} d - {\left(2 \, a^{3} b^{4} + a b^{6}\right)} c^{3} - 3 \, {\left(a^{3} b^{4} + 2 \, a b^{6}\right)} c d^{2} + {\left(2 \, a^{6} b - 5 \, a^{4} b^{3} + 6 \, a^{2} b^{5}\right)} d^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) - {\left({\left(4 \, a^{4} b^{4} - 5 \, a^{2} b^{6} + b^{8}\right)} c^{3} - 3 \, {\left(2 \, a^{5} b^{3} - a^{3} b^{5} - a b^{7}\right)} c^{2} d + 9 \, {\left(a^{4} b^{4} - a^{2} b^{6}\right)} c d^{2} + {\left(2 \, a^{7} b - 7 \, a^{5} b^{3} + 5 \, a^{3} b^{5}\right)} d^{3}\right)} \cos\left(f x + e\right) - {\left(4 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} d^{3} f x + 3 \, {\left({\left(a^{3} b^{5} - a b^{7}\right)} c^{3} - {\left(a^{4} b^{4} + a^{2} b^{6} - 2 \, b^{8}\right)} c^{2} d - {\left(a^{5} b^{3} - 5 \, a^{3} b^{5} + 4 \, a b^{7}\right)} c d^{2} + {\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 2 \, a^{2} b^{6}\right)} d^{3}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a^{6} b^{5} - 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} - b^{11}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b^{4} - 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} - a b^{10}\right)} f \sin\left(f x + e\right) - {\left(a^{8} b^{3} - 2 \, a^{6} b^{5} + 2 \, a^{2} b^{9} - b^{11}\right)} f\right)}}\right]"," ",0,"[1/4*(4*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*d^3*f*x*cos(f*x + e)^2 - 4*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*d^3*f*x - ((2*a^4*b^3 + 3*a^2*b^5 + b^7)*c^3 - 9*(a^3*b^4 + a*b^6)*c^2*d + 3*(a^4*b^3 + 3*a^2*b^5 + 2*b^7)*c*d^2 - (2*a^7 - 3*a^5*b^2 + a^3*b^4 + 6*a*b^6)*d^3 + (9*a*b^6*c^2*d - (2*a^2*b^5 + b^7)*c^3 - 3*(a^2*b^5 + 2*b^7)*c*d^2 + (2*a^5*b^2 - 5*a^3*b^4 + 6*a*b^6)*d^3)*cos(f*x + e)^2 - 2*(9*a^2*b^5*c^2*d - (2*a^3*b^4 + a*b^6)*c^3 - 3*(a^3*b^4 + 2*a*b^6)*c*d^2 + (2*a^6*b - 5*a^4*b^3 + 6*a^2*b^5)*d^3)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) - 2*((4*a^4*b^4 - 5*a^2*b^6 + b^8)*c^3 - 3*(2*a^5*b^3 - a^3*b^5 - a*b^7)*c^2*d + 9*(a^4*b^4 - a^2*b^6)*c*d^2 + (2*a^7*b - 7*a^5*b^3 + 5*a^3*b^5)*d^3)*cos(f*x + e) - 2*(4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d^3*f*x + 3*((a^3*b^5 - a*b^7)*c^3 - (a^4*b^4 + a^2*b^6 - 2*b^8)*c^2*d - (a^5*b^3 - 5*a^3*b^5 + 4*a*b^7)*c*d^2 + (a^6*b^2 - 3*a^4*b^4 + 2*a^2*b^6)*d^3)*cos(f*x + e))*sin(f*x + e))/((a^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*f*cos(f*x + e)^2 - 2*(a^7*b^4 - 3*a^5*b^6 + 3*a^3*b^8 - a*b^10)*f*sin(f*x + e) - (a^8*b^3 - 2*a^6*b^5 + 2*a^2*b^9 - b^11)*f), 1/2*(2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*d^3*f*x*cos(f*x + e)^2 - 2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*d^3*f*x + ((2*a^4*b^3 + 3*a^2*b^5 + b^7)*c^3 - 9*(a^3*b^4 + a*b^6)*c^2*d + 3*(a^4*b^3 + 3*a^2*b^5 + 2*b^7)*c*d^2 - (2*a^7 - 3*a^5*b^2 + a^3*b^4 + 6*a*b^6)*d^3 + (9*a*b^6*c^2*d - (2*a^2*b^5 + b^7)*c^3 - 3*(a^2*b^5 + 2*b^7)*c*d^2 + (2*a^5*b^2 - 5*a^3*b^4 + 6*a*b^6)*d^3)*cos(f*x + e)^2 - 2*(9*a^2*b^5*c^2*d - (2*a^3*b^4 + a*b^6)*c^3 - 3*(a^3*b^4 + 2*a*b^6)*c*d^2 + (2*a^6*b - 5*a^4*b^3 + 6*a^2*b^5)*d^3)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) - ((4*a^4*b^4 - 5*a^2*b^6 + b^8)*c^3 - 3*(2*a^5*b^3 - a^3*b^5 - a*b^7)*c^2*d + 9*(a^4*b^4 - a^2*b^6)*c*d^2 + (2*a^7*b - 7*a^5*b^3 + 5*a^3*b^5)*d^3)*cos(f*x + e) - (4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d^3*f*x + 3*((a^3*b^5 - a*b^7)*c^3 - (a^4*b^4 + a^2*b^6 - 2*b^8)*c^2*d - (a^5*b^3 - 5*a^3*b^5 + 4*a*b^7)*c*d^2 + (a^6*b^2 - 3*a^4*b^4 + 2*a^2*b^6)*d^3)*cos(f*x + e))*sin(f*x + e))/((a^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*f*cos(f*x + e)^2 - 2*(a^7*b^4 - 3*a^5*b^6 + 3*a^3*b^8 - a*b^10)*f*sin(f*x + e) - (a^8*b^3 - 2*a^6*b^5 + 2*a^2*b^9 - b^11)*f)]","B",0
717,1,1025,0,1.541085," ","integrate((c+d*sin(f*x+e))^2/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(3 \, {\left(a^{3} b^{2} - a b^{4}\right)} c^{2} - 2 \, {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} c d - {\left(a^{5} - 5 \, a^{3} b^{2} + 4 \, a b^{4}\right)} d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left({\left(2 \, a^{4} + 3 \, a^{2} b^{2} + b^{4}\right)} c^{2} - 6 \, {\left(a^{3} b + a b^{3}\right)} c d + {\left(a^{4} + 3 \, a^{2} b^{2} + 2 \, b^{4}\right)} d^{2} + {\left(6 \, a b^{3} c d - {\left(2 \, a^{2} b^{2} + b^{4}\right)} c^{2} - {\left(a^{2} b^{2} + 2 \, b^{4}\right)} d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(6 \, a^{2} b^{2} c d - {\left(2 \, a^{3} b + a b^{3}\right)} c^{2} - {\left(a^{3} b + 2 \, a b^{3}\right)} d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) + 2 \, {\left({\left(4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right)} c^{2} - 2 \, {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} c d + 3 \, {\left(a^{4} b - a^{2} b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right)}{4 \, {\left({\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} f \sin\left(f x + e\right) - {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} f\right)}}, -\frac{{\left(3 \, {\left(a^{3} b^{2} - a b^{4}\right)} c^{2} - 2 \, {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} c d - {\left(a^{5} - 5 \, a^{3} b^{2} + 4 \, a b^{4}\right)} d^{2}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left({\left(2 \, a^{4} + 3 \, a^{2} b^{2} + b^{4}\right)} c^{2} - 6 \, {\left(a^{3} b + a b^{3}\right)} c d + {\left(a^{4} + 3 \, a^{2} b^{2} + 2 \, b^{4}\right)} d^{2} + {\left(6 \, a b^{3} c d - {\left(2 \, a^{2} b^{2} + b^{4}\right)} c^{2} - {\left(a^{2} b^{2} + 2 \, b^{4}\right)} d^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, {\left(6 \, a^{2} b^{2} c d - {\left(2 \, a^{3} b + a b^{3}\right)} c^{2} - {\left(a^{3} b + 2 \, a b^{3}\right)} d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) + {\left({\left(4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right)} c^{2} - 2 \, {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} c d + 3 \, {\left(a^{4} b - a^{2} b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} f \sin\left(f x + e\right) - {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} f\right)}}\right]"," ",0,"[-1/4*(2*(3*(a^3*b^2 - a*b^4)*c^2 - 2*(a^4*b + a^2*b^3 - 2*b^5)*c*d - (a^5 - 5*a^3*b^2 + 4*a*b^4)*d^2)*cos(f*x + e)*sin(f*x + e) - ((2*a^4 + 3*a^2*b^2 + b^4)*c^2 - 6*(a^3*b + a*b^3)*c*d + (a^4 + 3*a^2*b^2 + 2*b^4)*d^2 + (6*a*b^3*c*d - (2*a^2*b^2 + b^4)*c^2 - (a^2*b^2 + 2*b^4)*d^2)*cos(f*x + e)^2 - 2*(6*a^2*b^2*c*d - (2*a^3*b + a*b^3)*c^2 - (a^3*b + 2*a*b^3)*d^2)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 + 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + 2*((4*a^4*b - 5*a^2*b^3 + b^5)*c^2 - 2*(2*a^5 - a^3*b^2 - a*b^4)*c*d + 3*(a^4*b - a^2*b^3)*d^2)*cos(f*x + e))/((a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*f*cos(f*x + e)^2 - 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*f*sin(f*x + e) - (a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*f), -1/2*((3*(a^3*b^2 - a*b^4)*c^2 - 2*(a^4*b + a^2*b^3 - 2*b^5)*c*d - (a^5 - 5*a^3*b^2 + 4*a*b^4)*d^2)*cos(f*x + e)*sin(f*x + e) - ((2*a^4 + 3*a^2*b^2 + b^4)*c^2 - 6*(a^3*b + a*b^3)*c*d + (a^4 + 3*a^2*b^2 + 2*b^4)*d^2 + (6*a*b^3*c*d - (2*a^2*b^2 + b^4)*c^2 - (a^2*b^2 + 2*b^4)*d^2)*cos(f*x + e)^2 - 2*(6*a^2*b^2*c*d - (2*a^3*b + a*b^3)*c^2 - (a^3*b + 2*a*b^3)*d^2)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + ((4*a^4*b - 5*a^2*b^3 + b^5)*c^2 - 2*(2*a^5 - a^3*b^2 - a*b^4)*c*d + 3*(a^4*b - a^2*b^3)*d^2)*cos(f*x + e))/((a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*f*cos(f*x + e)^2 - 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*f*sin(f*x + e) - (a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*f)]","B",0
718,1,799,0,1.253738," ","integrate((c+d*sin(f*x+e))/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(3 \, {\left(a^{3} b^{2} - a b^{4}\right)} c - {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} d\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) + {\left({\left(3 \, a b^{3} d - {\left(2 \, a^{2} b^{2} + b^{4}\right)} c\right)} \cos\left(f x + e\right)^{2} + {\left(2 \, a^{4} + 3 \, a^{2} b^{2} + b^{4}\right)} c - 3 \, {\left(a^{3} b + a b^{3}\right)} d - 2 \, {\left(3 \, a^{2} b^{2} d - {\left(2 \, a^{3} b + a b^{3}\right)} c\right)} \sin\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(-\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} - 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) + 2 \, {\left({\left(4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right)} c - {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} d\right)} \cos\left(f x + e\right)}{4 \, {\left({\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} f \sin\left(f x + e\right) - {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} f\right)}}, -\frac{{\left(3 \, {\left(a^{3} b^{2} - a b^{4}\right)} c - {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} d\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left({\left(3 \, a b^{3} d - {\left(2 \, a^{2} b^{2} + b^{4}\right)} c\right)} \cos\left(f x + e\right)^{2} + {\left(2 \, a^{4} + 3 \, a^{2} b^{2} + b^{4}\right)} c - 3 \, {\left(a^{3} b + a b^{3}\right)} d - 2 \, {\left(3 \, a^{2} b^{2} d - {\left(2 \, a^{3} b + a b^{3}\right)} c\right)} \sin\left(f x + e\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) + {\left({\left(4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right)} c - {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} d\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} f \sin\left(f x + e\right) - {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} f\right)}}\right]"," ",0,"[-1/4*(2*(3*(a^3*b^2 - a*b^4)*c - (a^4*b + a^2*b^3 - 2*b^5)*d)*cos(f*x + e)*sin(f*x + e) + ((3*a*b^3*d - (2*a^2*b^2 + b^4)*c)*cos(f*x + e)^2 + (2*a^4 + 3*a^2*b^2 + b^4)*c - 3*(a^3*b + a*b^3)*d - 2*(3*a^2*b^2*d - (2*a^3*b + a*b^3)*c)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + 2*((4*a^4*b - 5*a^2*b^3 + b^5)*c - (2*a^5 - a^3*b^2 - a*b^4)*d)*cos(f*x + e))/((a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*f*cos(f*x + e)^2 - 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*f*sin(f*x + e) - (a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*f), -1/2*((3*(a^3*b^2 - a*b^4)*c - (a^4*b + a^2*b^3 - 2*b^5)*d)*cos(f*x + e)*sin(f*x + e) - ((3*a*b^3*d - (2*a^2*b^2 + b^4)*c)*cos(f*x + e)^2 + (2*a^4 + 3*a^2*b^2 + b^4)*c - 3*(a^3*b + a*b^3)*d - 2*(3*a^2*b^2*d - (2*a^3*b + a*b^3)*c)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + ((4*a^4*b - 5*a^2*b^3 + b^5)*c - (2*a^5 - a^3*b^2 - a*b^4)*d)*cos(f*x + e))/((a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*f*cos(f*x + e)^2 - 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*f*sin(f*x + e) - (a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*f)]","B",0
719,1,618,0,1.085191," ","integrate(1/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\left[-\frac{6 \, {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(2 \, a^{4} + 3 \, a^{2} b^{2} + b^{4} - {\left(2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, a^{3} b + a b^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}} \log\left(\frac{{\left(2 \, a^{2} - b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2} + 2 \, {\left(a \cos\left(f x + e\right) \sin\left(f x + e\right) + b \cos\left(f x + e\right)\right)} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}\right) + 2 \, {\left(4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right)} \cos\left(f x + e\right)}{4 \, {\left({\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} f \sin\left(f x + e\right) - {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} f\right)}}, -\frac{3 \, {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(2 \, a^{4} + 3 \, a^{2} b^{2} + b^{4} - {\left(2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(2 \, a^{3} b + a b^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{a^{2} - b^{2}} \arctan\left(-\frac{a \sin\left(f x + e\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(f x + e\right)}\right) + {\left(4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right)} \cos\left(f x + e\right)}{2 \, {\left({\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} f \cos\left(f x + e\right)^{2} - 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} f \sin\left(f x + e\right) - {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} f\right)}}\right]"," ",0,"[-1/4*(6*(a^3*b^2 - a*b^4)*cos(f*x + e)*sin(f*x + e) - (2*a^4 + 3*a^2*b^2 + b^4 - (2*a^2*b^2 + b^4)*cos(f*x + e)^2 + 2*(2*a^3*b + a*b^3)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 + 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + 2*(4*a^4*b - 5*a^2*b^3 + b^5)*cos(f*x + e))/((a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*f*cos(f*x + e)^2 - 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*f*sin(f*x + e) - (a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*f), -1/2*(3*(a^3*b^2 - a*b^4)*cos(f*x + e)*sin(f*x + e) - (2*a^4 + 3*a^2*b^2 + b^4 - (2*a^2*b^2 + b^4)*cos(f*x + e)^2 + 2*(2*a^3*b + a*b^3)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + (4*a^4*b - 5*a^2*b^3 + b^5)*cos(f*x + e))/((a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*f*cos(f*x + e)^2 - 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*f*sin(f*x + e) - (a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*f)]","B",0
720,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
721,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
722,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
723,0,0,0,1.630890," ","integrate((a+b*sin(f*x+e))*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(a c^{2} + 2 \, b c d + a d^{2} - {\left(2 \, b c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} - {\left(b d^{2} \cos\left(f x + e\right)^{2} - b c^{2} - 2 \, a c d - b d^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((a*c^2 + 2*b*c*d + a*d^2 - (2*b*c*d + a*d^2)*cos(f*x + e)^2 - (b*d^2*cos(f*x + e)^2 - b*c^2 - 2*a*c*d - b*d^2)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
724,0,0,0,1.507765," ","integrate((a+b*sin(f*x+e))*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b d \cos\left(f x + e\right)^{2} - a c - b d - {\left(b c + a d\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral(-(b*d*cos(f*x + e)^2 - a*c - b*d - (b*c + a*d)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
725,0,0,0,1.345263," ","integrate((a+b*sin(f*x+e))*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right) + a\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c), x)","F",0
726,0,0,0,1.639894," ","integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{b \sin\left(f x + e\right) + a}{\sqrt{d \sin\left(f x + e\right) + c}}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)/sqrt(d*sin(f*x + e) + c), x)","F",0
727,0,0,0,1.374655," ","integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(b \sin\left(f x + e\right) + a\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral(-(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
728,0,0,0,1.356662," ","integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(b \sin\left(f x + e\right) + a\right)} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
729,0,0,0,1.549278," ","integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(b \sin\left(f x + e\right) + a\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 6 \, c^{2} d^{2} + d^{4} - 2 \, {\left(3 \, c^{2} d^{2} + d^{4}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left(c d^{3} \cos\left(f x + e\right)^{2} - c^{3} d - c d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(d^4*cos(f*x + e)^4 + c^4 + 6*c^2*d^2 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)^2 - 4*(c*d^3*cos(f*x + e)^2 - c^3*d - c*d^3)*sin(f*x + e)), x)","F",0
730,0,0,0,1.493390," ","integrate((a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b^{2} d^{2} \cos\left(f x + e\right)^{4} + 4 \, a b c d + {\left(a^{2} + b^{2}\right)} c^{2} + {\left(a^{2} + b^{2}\right)} d^{2} - {\left(b^{2} c^{2} + 4 \, a b c d + {\left(a^{2} + 2 \, b^{2}\right)} d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a b c^{2} + a b d^{2} + {\left(a^{2} + b^{2}\right)} c d - {\left(b^{2} c d + a b d^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((b^2*d^2*cos(f*x + e)^4 + 4*a*b*c*d + (a^2 + b^2)*c^2 + (a^2 + b^2)*d^2 - (b^2*c^2 + 4*a*b*c*d + (a^2 + 2*b^2)*d^2)*cos(f*x + e)^2 + 2*(a*b*c^2 + a*b*d^2 + (a^2 + b^2)*c*d - (b^2*c*d + a*b*d^2)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
731,0,0,0,1.562920," ","integrate((a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(2 \, a b d - {\left(b^{2} c + 2 \, a b d\right)} \cos\left(f x + e\right)^{2} + {\left(a^{2} + b^{2}\right)} c - {\left(b^{2} d \cos\left(f x + e\right)^{2} - 2 \, a b c - {\left(a^{2} + b^{2}\right)} d\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((2*a*b*d - (b^2*c + 2*a*b*d)*cos(f*x + e)^2 + (a^2 + b^2)*c - (b^2*d*cos(f*x + e)^2 - 2*a*b*c - (a^2 + b^2)*d)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
732,0,0,0,1.418323," ","integrate((a+b*sin(f*x+e))^2*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*sqrt(d*sin(f*x + e) + c), x)","F",0
733,0,0,0,1.395448," ","integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}{\sqrt{d \sin\left(f x + e\right) + c}}, x\right)"," ",0,"integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)/sqrt(d*sin(f*x + e) + c), x)","F",0
734,0,0,0,1.014515," ","integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral((b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*sqrt(d*sin(f*x + e) + c)/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
735,0,0,0,1.339005," ","integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*sqrt(d*sin(f*x + e) + c)/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
736,0,0,0,1.639011," ","integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 6 \, c^{2} d^{2} + d^{4} - 2 \, {\left(3 \, c^{2} d^{2} + d^{4}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left(c d^{3} \cos\left(f x + e\right)^{2} - c^{3} d - c d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*sqrt(d*sin(f*x + e) + c)/(d^4*cos(f*x + e)^4 + c^4 + 6*c^2*d^2 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)^2 - 4*(c*d^3*cos(f*x + e)^2 - c^3*d - c*d^3)*sin(f*x + e)), x)","F",0
737,0,0,0,1.669147," ","integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left({\left({\left(2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{3} + 3 \, a b^{2}\right)} c^{2} + 2 \, {\left(3 \, a^{2} b + b^{3}\right)} c d + {\left(a^{3} + 3 \, a b^{2}\right)} d^{2} - {\left(3 \, a b^{2} c^{2} + 2 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d + {\left(a^{3} + 6 \, a b^{2}\right)} d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(b^{3} d^{2} \cos\left(f x + e\right)^{4} + {\left(3 \, a^{2} b + b^{3}\right)} c^{2} + 2 \, {\left(a^{3} + 3 \, a b^{2}\right)} c d + {\left(3 \, a^{2} b + b^{3}\right)} d^{2} - {\left(b^{3} c^{2} + 6 \, a b^{2} c d + {\left(3 \, a^{2} b + 2 \, b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral(((2*b^3*c*d + 3*a*b^2*d^2)*cos(f*x + e)^4 + (a^3 + 3*a*b^2)*c^2 + 2*(3*a^2*b + b^3)*c*d + (a^3 + 3*a*b^2)*d^2 - (3*a*b^2*c^2 + 2*(3*a^2*b + 2*b^3)*c*d + (a^3 + 6*a*b^2)*d^2)*cos(f*x + e)^2 + (b^3*d^2*cos(f*x + e)^4 + (3*a^2*b + b^3)*c^2 + 2*(a^3 + 3*a*b^2)*c*d + (3*a^2*b + b^3)*d^2 - (b^3*c^2 + 6*a*b^2*c*d + (3*a^2*b + 2*b^3)*d^2)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
738,0,0,0,1.527800," ","integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b^{3} d \cos\left(f x + e\right)^{4} - {\left(3 \, a b^{2} c + {\left(3 \, a^{2} b + 2 \, b^{3}\right)} d\right)} \cos\left(f x + e\right)^{2} + {\left(a^{3} + 3 \, a b^{2}\right)} c + {\left(3 \, a^{2} b + b^{3}\right)} d - {\left({\left(b^{3} c + 3 \, a b^{2} d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, a^{2} b + b^{3}\right)} c - {\left(a^{3} + 3 \, a b^{2}\right)} d\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((b^3*d*cos(f*x + e)^4 - (3*a*b^2*c + (3*a^2*b + 2*b^3)*d)*cos(f*x + e)^2 + (a^3 + 3*a*b^2)*c + (3*a^2*b + b^3)*d - ((b^3*c + 3*a*b^2*d)*cos(f*x + e)^2 - (3*a^2*b + b^3)*c - (a^3 + 3*a*b^2)*d)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
739,0,0,0,1.354272," ","integrate((a+b*sin(f*x+e))^3*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral(-(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c), x)","F",0
740,0,0,0,0.907053," ","integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)}{\sqrt{d \sin\left(f x + e\right) + c}}, x\right)"," ",0,"integral(-(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))/sqrt(d*sin(f*x + e) + c), x)","F",0
741,0,0,0,1.583202," ","integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral((3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c)/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
742,0,0,0,1.498693," ","integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c)/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
743,0,0,0,1.686647," ","integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(7/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}}{d^{4} \cos\left(f x + e\right)^{4} + c^{4} + 6 \, c^{2} d^{2} + d^{4} - 2 \, {\left(3 \, c^{2} d^{2} + d^{4}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left(c d^{3} \cos\left(f x + e\right)^{2} - c^{3} d - c d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c)/(d^4*cos(f*x + e)^4 + c^4 + 6*c^2*d^2 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)^2 - 4*(c*d^3*cos(f*x + e)^2 - c^3*d - c*d^3)*sin(f*x + e)), x)","F",0
744,0,0,0,1.839976," ","integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(9/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)\right)} \sqrt{d \sin\left(f x + e\right) + c}}{5 \, c d^{4} \cos\left(f x + e\right)^{4} + c^{5} + 10 \, c^{3} d^{2} + 5 \, c d^{4} - 10 \, {\left(c^{3} d^{2} + c d^{4}\right)} \cos\left(f x + e\right)^{2} + {\left(d^{5} \cos\left(f x + e\right)^{4} + 5 \, c^{4} d + 10 \, c^{2} d^{3} + d^{5} - 2 \, {\left(5 \, c^{2} d^{3} + d^{5}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c)/(5*c*d^4*cos(f*x + e)^4 + c^5 + 10*c^3*d^2 + 5*c*d^4 - 10*(c^3*d^2 + c*d^4)*cos(f*x + e)^2 + (d^5*cos(f*x + e)^4 + 5*c^4*d + 10*c^2*d^3 + d^5 - 2*(5*c^2*d^3 + d^5)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
745,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
746,0,0,0,28.331915," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(d \sin\left(f x + e\right) + c\right)}^{\frac{3}{2}}}{b \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral((d*sin(f*x + e) + c)^(3/2)/(b*sin(f*x + e) + a), x)","F",0
747,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
748,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
749,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
750,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
751,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(7/2)/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
752,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
753,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
754,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
755,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
756,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
757,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
758,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(9/2)/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
759,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(7/2)/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
760,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
761,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
762,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
763,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
764,-1,0,0,0.000000," ","integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
765,-1,0,0,0.000000," ","integrate((a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
766,-1,0,0,0.000000," ","integrate((a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
767,-1,0,0,0.000000," ","integrate((a+b*sin(f*x+e))^(1/2)*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
768,0,0,0,75.379750," ","integrate((a+b*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sin\left(f x + e\right) + a}}{\sqrt{d \sin\left(f x + e\right) + c}}, x\right)"," ",0,"integral(sqrt(b*sin(f*x + e) + a)/sqrt(d*sin(f*x + e) + c), x)","F",0
769,0,0,0,1.303154," ","integrate((a+b*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
770,0,0,0,1.510380," ","integrate((a+b*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
771,-1,0,0,0.000000," ","integrate((a+b*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
772,0,0,0,53.388301," ","integrate((a+b*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b d \cos\left(f x + e\right)^{2} - a c - b d - {\left(b c + a d\right)} \sin\left(f x + e\right)\right)} \sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral(-(b*d*cos(f*x + e)^2 - a*c - b*d - (b*c + a*d)*sin(f*x + e))*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c), x)","F",0
773,0,0,0,15.145367," ","integrate((a+b*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right) + a\right)}^{\frac{3}{2}} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)^(3/2)*sqrt(d*sin(f*x + e) + c), x)","F",0
774,0,0,0,5.496654," ","integrate((a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(b \sin\left(f x + e\right) + a\right)}^{\frac{3}{2}}}{\sqrt{d \sin\left(f x + e\right) + c}}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)^(3/2)/sqrt(d*sin(f*x + e) + c), x)","F",0
775,0,0,0,3.887257," ","integrate((a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(b \sin\left(f x + e\right) + a\right)}^{\frac{3}{2}} \sqrt{d \sin\left(f x + e\right) + c}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral(-(b*sin(f*x + e) + a)^(3/2)*sqrt(d*sin(f*x + e) + c)/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
776,0,0,0,1.193380," ","integrate((a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(b \sin\left(f x + e\right) + a\right)}^{\frac{3}{2}} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(b*sin(f*x + e) + a)^(3/2)*sqrt(d*sin(f*x + e) + c)/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
777,-1,0,0,0.000000," ","integrate((a+b*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
778,-1,0,0,0.000000," ","integrate((a+b*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
779,0,0,0,71.446104," ","integrate((a+b*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}\right)} \sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c), x)","F",0
780,0,0,0,24.327338," ","integrate((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}\right)} \sqrt{b \sin\left(f x + e\right) + a}}{\sqrt{d \sin\left(f x + e\right) + c}}, x\right)"," ",0,"integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*sqrt(b*sin(f*x + e) + a)/sqrt(d*sin(f*x + e) + c), x)","F",0
781,0,0,0,10.274441," ","integrate((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}\right)} \sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral((b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
782,0,0,0,75.493126," ","integrate((a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}\right)} \sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral((b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
783,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
784,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
785,0,0,0,2.086207," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{d \sin\left(f x + e\right) + c}}{\sqrt{b \sin\left(f x + e\right) + a}}, x\right)"," ",0,"integral(sqrt(d*sin(f*x + e) + c)/sqrt(b*sin(f*x + e) + a), x)","F",0
786,0,0,0,0.949614," ","integrate(1/(a+b*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{b d \cos\left(f x + e\right)^{2} - a c - b d - {\left(b c + a d\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b*d*cos(f*x + e)^2 - a*c - b*d - (b*c + a*d)*sin(f*x + e)), x)","F",0
787,0,0,0,0.851419," ","integrate(1/(a+b*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{a c^{2} + 2 \, b c d + a d^{2} - {\left(2 \, b c d + a d^{2}\right)} \cos\left(f x + e\right)^{2} - {\left(b d^{2} \cos\left(f x + e\right)^{2} - b c^{2} - 2 \, a c d - b d^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(a*c^2 + 2*b*c*d + a*d^2 - (2*b*c*d + a*d^2)*cos(f*x + e)^2 - (b*d^2*cos(f*x + e)^2 - b*c^2 - 2*a*c*d - b*d^2)*sin(f*x + e)), x)","F",0
788,0,0,0,1.206494," ","integrate(1/(a+b*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{b d^{3} \cos\left(f x + e\right)^{4} + a c^{3} + 3 \, b c^{2} d + 3 \, a c d^{2} + b d^{3} - {\left(3 \, b c^{2} d + 3 \, a c d^{2} + 2 \, b d^{3}\right)} \cos\left(f x + e\right)^{2} + {\left(b c^{3} + 3 \, a c^{2} d + 3 \, b c d^{2} + a d^{3} - {\left(3 \, b c d^{2} + a d^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b*d^3*cos(f*x + e)^4 + a*c^3 + 3*b*c^2*d + 3*a*c*d^2 + b*d^3 - (3*b*c^2*d + 3*a*c*d^2 + 2*b*d^3)*cos(f*x + e)^2 + (b*c^3 + 3*a*c^2*d + 3*b*c*d^2 + a*d^3 - (3*b*c*d^2 + a*d^3)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
789,0,0,0,7.521782," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}\right)} \sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}, x\right)"," ",0,"integral((d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)*sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2), x)","F",0
790,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
791,0,0,0,1.409954," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}, x\right)"," ",0,"integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2), x)","F",0
792,0,0,0,1.380731," ","integrate(1/(a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{2 \, a b d - {\left(b^{2} c + 2 \, a b d\right)} \cos\left(f x + e\right)^{2} + {\left(a^{2} + b^{2}\right)} c - {\left(b^{2} d \cos\left(f x + e\right)^{2} - 2 \, a b c - {\left(a^{2} + b^{2}\right)} d\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(2*a*b*d - (b^2*c + 2*a*b*d)*cos(f*x + e)^2 + (a^2 + b^2)*c - (b^2*d*cos(f*x + e)^2 - 2*a*b*c - (a^2 + b^2)*d)*sin(f*x + e)), x)","F",0
793,0,0,0,1.671860," ","integrate(1/(a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{b^{2} d^{2} \cos\left(f x + e\right)^{4} + 4 \, a b c d + {\left(a^{2} + b^{2}\right)} c^{2} + {\left(a^{2} + b^{2}\right)} d^{2} - {\left(b^{2} c^{2} + 4 \, a b c d + {\left(a^{2} + 2 \, b^{2}\right)} d^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, {\left(a b c^{2} + a b d^{2} + {\left(a^{2} + b^{2}\right)} c d - {\left(b^{2} c d + a b d^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b^2*d^2*cos(f*x + e)^4 + 4*a*b*c*d + (a^2 + b^2)*c^2 + (a^2 + b^2)*d^2 - (b^2*c^2 + 4*a*b*c*d + (a^2 + 2*b^2)*d^2)*cos(f*x + e)^2 + 2*(a*b*c^2 + a*b*d^2 + (a^2 + b^2)*c*d - (b^2*c*d + a*b*d^2)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
794,0,0,0,2.199166," ","integrate(1/(a+b*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{6 \, a b c^{2} d + 2 \, a b d^{3} + {\left(3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{2} + b^{2}\right)} c^{3} + 3 \, {\left(a^{2} + b^{2}\right)} c d^{2} - {\left(b^{2} c^{3} + 6 \, a b c^{2} d + 4 \, a b d^{3} + 3 \, {\left(a^{2} + 2 \, b^{2}\right)} c d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(b^{2} d^{3} \cos\left(f x + e\right)^{4} + 2 \, a b c^{3} + 6 \, a b c d^{2} + 3 \, {\left(a^{2} + b^{2}\right)} c^{2} d + {\left(a^{2} + b^{2}\right)} d^{3} - {\left(3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + {\left(a^{2} + 2 \, b^{2}\right)} d^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(6*a*b*c^2*d + 2*a*b*d^3 + (3*b^2*c*d^2 + 2*a*b*d^3)*cos(f*x + e)^4 + (a^2 + b^2)*c^3 + 3*(a^2 + b^2)*c*d^2 - (b^2*c^3 + 6*a*b*c^2*d + 4*a*b*d^3 + 3*(a^2 + 2*b^2)*c*d^2)*cos(f*x + e)^2 + (b^2*d^3*cos(f*x + e)^4 + 2*a*b*c^3 + 6*a*b*c*d^2 + 3*(a^2 + b^2)*c^2*d + (a^2 + b^2)*d^3 - (3*b^2*c^2*d + 6*a*b*c*d^2 + (a^2 + 2*b^2)*d^3)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
795,-1,0,0,0.000000," ","integrate((c+d*sin(f*x+e))^(5/2)/(a+b*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
796,0,0,0,1.653227," ","integrate((c+d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{b \sin\left(f x + e\right) + a} {\left(d \sin\left(f x + e\right) + c\right)}^{\frac{3}{2}}}{3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(b*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(3/2)/(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e)), x)","F",0
797,0,0,0,1.348668," ","integrate((c+d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e)), x)","F",0
798,0,0,0,1.706931," ","integrate(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{b^{3} d \cos\left(f x + e\right)^{4} - {\left(3 \, a b^{2} c + {\left(3 \, a^{2} b + 2 \, b^{3}\right)} d\right)} \cos\left(f x + e\right)^{2} + {\left(a^{3} + 3 \, a b^{2}\right)} c + {\left(3 \, a^{2} b + b^{3}\right)} d - {\left({\left(b^{3} c + 3 \, a b^{2} d\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, a^{2} b + b^{3}\right)} c - {\left(a^{3} + 3 \, a b^{2}\right)} d\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b^3*d*cos(f*x + e)^4 - (3*a*b^2*c + (3*a^2*b + 2*b^3)*d)*cos(f*x + e)^2 + (a^3 + 3*a*b^2)*c + (3*a^2*b + b^3)*d - ((b^3*c + 3*a*b^2*d)*cos(f*x + e)^2 - (3*a^2*b + b^3)*c - (a^3 + 3*a*b^2)*d)*sin(f*x + e)), x)","F",0
799,0,0,0,2.262429," ","integrate(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{{\left(2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(a^{3} + 3 \, a b^{2}\right)} c^{2} + 2 \, {\left(3 \, a^{2} b + b^{3}\right)} c d + {\left(a^{3} + 3 \, a b^{2}\right)} d^{2} - {\left(3 \, a b^{2} c^{2} + 2 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d + {\left(a^{3} + 6 \, a b^{2}\right)} d^{2}\right)} \cos\left(f x + e\right)^{2} + {\left(b^{3} d^{2} \cos\left(f x + e\right)^{4} + {\left(3 \, a^{2} b + b^{3}\right)} c^{2} + 2 \, {\left(a^{3} + 3 \, a b^{2}\right)} c d + {\left(3 \, a^{2} b + b^{3}\right)} d^{2} - {\left(b^{3} c^{2} + 6 \, a b^{2} c d + {\left(3 \, a^{2} b + 2 \, b^{3}\right)} d^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((2*b^3*c*d + 3*a*b^2*d^2)*cos(f*x + e)^4 + (a^3 + 3*a*b^2)*c^2 + 2*(3*a^2*b + b^3)*c*d + (a^3 + 3*a*b^2)*d^2 - (3*a*b^2*c^2 + 2*(3*a^2*b + 2*b^3)*c*d + (a^3 + 6*a*b^2)*d^2)*cos(f*x + e)^2 + (b^3*d^2*cos(f*x + e)^4 + (3*a^2*b + b^3)*c^2 + 2*(a^3 + 3*a*b^2)*c*d + (3*a^2*b + b^3)*d^2 - (b^3*c^2 + 6*a*b^2*c*d + (3*a^2*b + 2*b^3)*d^2)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
800,0,0,0,3.234997," ","integrate(1/(a+b*sin(f*x+e))^(5/2)/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{b \sin\left(f x + e\right) + a} \sqrt{d \sin\left(f x + e\right) + c}}{b^{3} d^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + {\left(a^{2} b + b^{3}\right)} d^{3}\right)} \cos\left(f x + e\right)^{4} - {\left(a^{3} + 3 \, a b^{2}\right)} c^{3} - 3 \, {\left(3 \, a^{2} b + b^{3}\right)} c^{2} d - 3 \, {\left(a^{3} + 3 \, a b^{2}\right)} c d^{2} - {\left(3 \, a^{2} b + b^{3}\right)} d^{3} + 3 \, {\left(a b^{2} c^{3} + {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c^{2} d + {\left(a^{3} + 6 \, a b^{2}\right)} c d^{2} + {\left(2 \, a^{2} b + b^{3}\right)} d^{3}\right)} \cos\left(f x + e\right)^{2} - {\left(3 \, {\left(b^{3} c d^{2} + a b^{2} d^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(3 \, a^{2} b + b^{3}\right)} c^{3} + 3 \, {\left(a^{3} + 3 \, a b^{2}\right)} c^{2} d + 3 \, {\left(3 \, a^{2} b + b^{3}\right)} c d^{2} + {\left(a^{3} + 3 \, a b^{2}\right)} d^{3} - {\left(b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 3 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} c d^{2} + {\left(a^{3} + 6 \, a b^{2}\right)} d^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(b^3*d^3*cos(f*x + e)^6 - 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + (a^2*b + b^3)*d^3)*cos(f*x + e)^4 - (a^3 + 3*a*b^2)*c^3 - 3*(3*a^2*b + b^3)*c^2*d - 3*(a^3 + 3*a*b^2)*c*d^2 - (3*a^2*b + b^3)*d^3 + 3*(a*b^2*c^3 + (3*a^2*b + 2*b^3)*c^2*d + (a^3 + 6*a*b^2)*c*d^2 + (2*a^2*b + b^3)*d^3)*cos(f*x + e)^2 - (3*(b^3*c*d^2 + a*b^2*d^3)*cos(f*x + e)^4 + (3*a^2*b + b^3)*c^3 + 3*(a^3 + 3*a*b^2)*c^2*d + 3*(3*a^2*b + b^3)*c*d^2 + (a^3 + 3*a*b^2)*d^3 - (b^3*c^3 + 9*a*b^2*c^2*d + 3*(3*a^2*b + 2*b^3)*c*d^2 + (a^3 + 6*a*b^2)*d^3)*cos(f*x + e)^2)*sin(f*x + e)), x)","F",0
801,0,0,0,2.313329," ","integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right) + a\right)}^{m} {\left(d \sin\left(f x + e\right) + c\right)}^{n}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)^m*(d*sin(f*x + e) + c)^n, x)","F",0
802,0,0,0,1.423291," ","integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}\right)} {\left(b \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(-(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)*(b*sin(f*x + e) + a)^m, x)","F",0
803,0,0,0,1.294084," ","integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left({\left(d \sin\left(f x + e\right) + c\right)} {\left(b \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral((d*sin(f*x + e) + c)*(b*sin(f*x + e) + a)^m, x)","F",0
804,0,0,0,1.297798," ","integrate((a+b*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)^m, x)","F",0
805,0,0,0,1.261003," ","integrate((a+b*sin(f*x+e))^m/(c+d*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(b \sin\left(f x + e\right) + a\right)}^{m}}{d \sin\left(f x + e\right) + c}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)^m/(d*sin(f*x + e) + c), x)","F",0
806,0,0,0,1.727707," ","integrate((a+b*sin(f*x+e))^m/(c+d*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(b \sin\left(f x + e\right) + a\right)}^{m}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral(-(b*sin(f*x + e) + a)^m/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
807,0,0,0,2.516009," ","integrate((a+b*sin(f*x+e))^m/(c+d*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(b \sin\left(f x + e\right) + a\right)}^{m}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(b*sin(f*x + e) + a)^m/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
808,0,0,0,2.527539," ","integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}\right)} \sqrt{d \sin\left(f x + e\right) + c} {\left(b \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(-(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)*sqrt(d*sin(f*x + e) + c)*(b*sin(f*x + e) + a)^m, x)","F",0
809,0,0,0,1.886985," ","integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(d \sin\left(f x + e\right) + c\right)}^{\frac{3}{2}} {\left(b \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral((d*sin(f*x + e) + c)^(3/2)*(b*sin(f*x + e) + a)^m, x)","F",0
810,0,0,0,1.959387," ","integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{d \sin\left(f x + e\right) + c} {\left(b \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(sqrt(d*sin(f*x + e) + c)*(b*sin(f*x + e) + a)^m, x)","F",0
811,0,0,0,1.703645," ","integrate((a+b*sin(f*x+e))^m/(c+d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(b \sin\left(f x + e\right) + a\right)}^{m}}{\sqrt{d \sin\left(f x + e\right) + c}}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)^m/sqrt(d*sin(f*x + e) + c), x)","F",0
812,0,0,0,2.522566," ","integrate((a+b*sin(f*x+e))^m/(c+d*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{d \sin\left(f x + e\right) + c} {\left(b \sin\left(f x + e\right) + a\right)}^{m}}{d^{2} \cos\left(f x + e\right)^{2} - 2 \, c d \sin\left(f x + e\right) - c^{2} - d^{2}}, x\right)"," ",0,"integral(-sqrt(d*sin(f*x + e) + c)*(b*sin(f*x + e) + a)^m/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)","F",0
813,0,0,0,3.162837," ","integrate((a+b*sin(f*x+e))^m/(c+d*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{d \sin\left(f x + e\right) + c} {\left(b \sin\left(f x + e\right) + a\right)}^{m}}{3 \, c d^{2} \cos\left(f x + e\right)^{2} - c^{3} - 3 \, c d^{2} + {\left(d^{3} \cos\left(f x + e\right)^{2} - 3 \, c^{2} d - d^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-sqrt(d*sin(f*x + e) + c)*(b*sin(f*x + e) + a)^m/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)","F",0
814,0,0,0,1.360728," ","integrate((d*csc(f*x+e))^n*(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \left(d \csc\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral(-(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*(d*csc(f*x + e))^n, x)","F",0
815,0,0,0,1.468574," ","integrate((d*csc(f*x+e))^n*(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}\right)} \left(d \csc\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral(-(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*(d*csc(f*x + e))^n, x)","F",0
816,0,0,0,1.499351," ","integrate((d*csc(f*x+e))^n*(a+a*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)} \left(d \csc\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)*(d*csc(f*x + e))^n, x)","F",0
817,0,0,0,1.353682," ","integrate((d*csc(f*x+e))^n/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(d \csc\left(f x + e\right)\right)^{n}}{a \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral((d*csc(f*x + e))^n/(a*sin(f*x + e) + a), x)","F",0
818,0,0,0,1.050616," ","integrate((d*csc(f*x+e))^n/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(d \csc\left(f x + e\right)\right)^{n}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-(d*csc(f*x + e))^n/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
819,0,0,0,1.510016," ","integrate((c*(d*sin(f*x+e))^p)^n*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n} {\left(a \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(((d*sin(f*x + e))^p*c)^n*(a*sin(f*x + e) + a)^m, x)","F",0
820,0,0,0,1.413179," ","integrate((c*(d*sin(f*x+e))^p)^n*(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(3 \, a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3} + {\left(a^{3} \cos\left(f x + e\right)^{2} - 4 \, a^{3}\right)} \sin\left(f x + e\right)\right)} \left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n}, x\right)"," ",0,"integral(-(3*a^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e))*((d*sin(f*x + e))^p*c)^n, x)","F",0
821,0,0,0,1.506922," ","integrate((c*(d*sin(f*x+e))^p)^n*(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}\right)} \left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n}, x\right)"," ",0,"integral(-(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*((d*sin(f*x + e))^p*c)^n, x)","F",0
822,0,0,0,1.327271," ","integrate((c*(d*sin(f*x+e))^p)^n*(a+a*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)} \left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)*((d*sin(f*x + e))^p*c)^n, x)","F",0
823,0,0,0,1.271140," ","integrate((c*(d*sin(f*x+e))^p)^n/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n}}{a \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral(((d*sin(f*x + e))^p*c)^n/(a*sin(f*x + e) + a), x)","F",0
824,0,0,0,1.395830," ","integrate((c*(d*sin(f*x+e))^p)^n/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n}}{a^{2} \cos\left(f x + e\right)^{2} - 2 \, a^{2} \sin\left(f x + e\right) - 2 \, a^{2}}, x\right)"," ",0,"integral(-((d*sin(f*x + e))^p*c)^n/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
825,0,0,0,1.161618," ","integrate((d*csc(f*x+e))^n*(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)\right)} \left(d \csc\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral(-(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))*(d*csc(f*x + e))^n, x)","F",0
826,0,0,0,1.330627," ","integrate((d*csc(f*x+e))^n*(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}\right)} \left(d \csc\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*(d*csc(f*x + e))^n, x)","F",0
827,0,0,0,1.146967," ","integrate((d*csc(f*x+e))^n*(a+b*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right) + a\right)} \left(d \csc\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)*(d*csc(f*x + e))^n, x)","F",0
828,0,0,0,0.834178," ","integrate((d*csc(f*x+e))^n/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(d \csc\left(f x + e\right)\right)^{n}}{b \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral((d*csc(f*x + e))^n/(b*sin(f*x + e) + a), x)","F",0
829,0,0,0,1.366490," ","integrate((d*csc(f*x+e))^n/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(d \csc\left(f x + e\right)\right)^{n}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}, x\right)"," ",0,"integral(-(d*csc(f*x + e))^n/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2), x)","F",0
830,0,0,0,1.469497," ","integrate((d*csc(f*x+e))^n/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(d \csc\left(f x + e\right)\right)^{n}}{3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(d*csc(f*x + e))^n/(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e)), x)","F",0
831,0,0,0,1.582135," ","integrate((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n} {\left(b \sin\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(((d*sin(f*x + e))^p*c)^n*(b*sin(f*x + e) + a)^m, x)","F",0
832,0,0,0,1.226470," ","integrate((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)\right)} \left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n}, x\right)"," ",0,"integral(-(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e))*((d*sin(f*x + e))^p*c)^n, x)","F",0
833,0,0,0,1.376004," ","integrate((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}\right)} \left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n}, x\right)"," ",0,"integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*((d*sin(f*x + e))^p*c)^n, x)","F",0
834,0,0,0,1.490069," ","integrate((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right) + a\right)} \left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)*((d*sin(f*x + e))^p*c)^n, x)","F",0
835,0,0,0,1.344838," ","integrate((c*(d*sin(f*x+e))^p)^n/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n}}{b \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral(((d*sin(f*x + e))^p*c)^n/(b*sin(f*x + e) + a), x)","F",0
836,0,0,0,1.466952," ","integrate((c*(d*sin(f*x+e))^p)^n/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n}}{b^{2} \cos\left(f x + e\right)^{2} - 2 \, a b \sin\left(f x + e\right) - a^{2} - b^{2}}, x\right)"," ",0,"integral(-((d*sin(f*x + e))^p*c)^n/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2), x)","F",0
837,0,0,0,1.781107," ","integrate((c*(d*sin(f*x+e))^p)^n/(a+b*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(\left(d \sin\left(f x + e\right)\right)^{p} c\right)^{n}}{3 \, a b^{2} \cos\left(f x + e\right)^{2} - a^{3} - 3 \, a b^{2} + {\left(b^{3} \cos\left(f x + e\right)^{2} - 3 \, a^{2} b - b^{3}\right)} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-((d*sin(f*x + e))^p*c)^n/(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a*b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e)), x)","F",0
