1,1,159,0,1.111358," ","integrate((a+a*sin(d*x+c))*tan(d*x+c)^5,x, algorithm=""fricas"")","\frac{16 \, a \cos\left(d x + c\right)^{4} + 2 \, a \cos\left(d x + c\right)^{2} + 7 \, {\left(a \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) - a \cos\left(d x + c\right)^{2}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 23 \, {\left(a \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) - a \cos\left(d x + c\right)^{2}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(8 \, a \cos\left(d x + c\right)^{2} + a\right)} \sin\left(d x + c\right) - 6 \, a}{16 \, {\left(d \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) - d \cos\left(d x + c\right)^{2}\right)}}"," ",0,"1/16*(16*a*cos(d*x + c)^4 + 2*a*cos(d*x + c)^2 + 7*(a*cos(d*x + c)^2*sin(d*x + c) - a*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 23*(a*cos(d*x + c)^2*sin(d*x + c) - a*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) + 2*(8*a*cos(d*x + c)^2 + a)*sin(d*x + c) - 6*a)/(d*cos(d*x + c)^2*sin(d*x + c) - d*cos(d*x + c)^2)","A",0
2,1,87,0,1.255239," ","integrate((a+a*sin(d*x+c))*tan(d*x+c)^3,x, algorithm=""fricas"")","-\frac{4 \, a \cos\left(d x + c\right)^{2} + {\left(a \sin\left(d x + c\right) - a\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 5 \, {\left(a \sin\left(d x + c\right) - a\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 4 \, a \sin\left(d x + c\right) - 2 \, a}{4 \, {\left(d \sin\left(d x + c\right) - d\right)}}"," ",0,"-1/4*(4*a*cos(d*x + c)^2 + (a*sin(d*x + c) - a)*log(sin(d*x + c) + 1) - 5*(a*sin(d*x + c) - a)*log(-sin(d*x + c) + 1) + 4*a*sin(d*x + c) - 2*a)/(d*sin(d*x + c) - d)","A",0
3,1,27,0,1.334071," ","integrate((a+a*sin(d*x+c))*tan(d*x+c),x, algorithm=""fricas"")","-\frac{a \log\left(-\sin\left(d x + c\right) + 1\right) + a \sin\left(d x + c\right)}{d}"," ",0,"-(a*log(-sin(d*x + c) + 1) + a*sin(d*x + c))/d","A",0
4,1,24,0,1.036747," ","integrate(cot(d*x+c)*(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{a \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) + a \sin\left(d x + c\right)}{d}"," ",0,"(a*log(1/2*sin(d*x + c)) + a*sin(d*x + c))/d","A",0
5,1,69,0,1.173599," ","integrate(cot(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{2 \, {\left(a \cos\left(d x + c\right)^{2} - a\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) + 2 \, {\left(a \cos\left(d x + c\right)^{2} - 2 \, a\right)} \sin\left(d x + c\right) - a}{2 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"-1/2*(2*(a*cos(d*x + c)^2 - a)*log(1/2*sin(d*x + c)) + 2*(a*cos(d*x + c)^2 - 2*a)*sin(d*x + c) - a)/(d*cos(d*x + c)^2 - d)","A",0
6,1,110,0,1.299173," ","integrate(cot(d*x+c)^5*(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{12 \, a \cos\left(d x + c\right)^{2} - 12 \, {\left(a \cos\left(d x + c\right)^{4} - 2 \, a \cos\left(d x + c\right)^{2} + a\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 4 \, {\left(3 \, a \cos\left(d x + c\right)^{4} - 12 \, a \cos\left(d x + c\right)^{2} + 8 \, a\right)} \sin\left(d x + c\right) - 9 \, a}{12 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} + d\right)}}"," ",0,"-1/12*(12*a*cos(d*x + c)^2 - 12*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a)*log(1/2*sin(d*x + c)) - 4*(3*a*cos(d*x + c)^4 - 12*a*cos(d*x + c)^2 + 8*a)*sin(d*x + c) - 9*a)/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)","A",0
7,1,158,0,0.797486," ","integrate(cot(d*x+c)^7*(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{90 \, a \cos\left(d x + c\right)^{4} - 135 \, a \cos\left(d x + c\right)^{2} - 60 \, {\left(a \cos\left(d x + c\right)^{6} - 3 \, a \cos\left(d x + c\right)^{4} + 3 \, a \cos\left(d x + c\right)^{2} - a\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 12 \, {\left(5 \, a \cos\left(d x + c\right)^{6} - 30 \, a \cos\left(d x + c\right)^{4} + 40 \, a \cos\left(d x + c\right)^{2} - 16 \, a\right)} \sin\left(d x + c\right) + 55 \, a}{60 \, {\left(d \cos\left(d x + c\right)^{6} - 3 \, d \cos\left(d x + c\right)^{4} + 3 \, d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"1/60*(90*a*cos(d*x + c)^4 - 135*a*cos(d*x + c)^2 - 60*(a*cos(d*x + c)^6 - 3*a*cos(d*x + c)^4 + 3*a*cos(d*x + c)^2 - a)*log(1/2*sin(d*x + c)) - 12*(5*a*cos(d*x + c)^6 - 30*a*cos(d*x + c)^4 + 40*a*cos(d*x + c)^2 - 16*a)*sin(d*x + c) + 55*a)/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)","A",0
8,1,116,0,1.322594," ","integrate((a+a*sin(d*x+c))*tan(d*x+c)^6,x, algorithm=""fricas"")","\frac{15 \, a d x \cos\left(d x + c\right)^{3} - 38 \, a \cos\left(d x + c\right)^{4} - 11 \, a \cos\left(d x + c\right)^{2} - {\left(15 \, a d x \cos\left(d x + c\right)^{3} - 15 \, a \cos\left(d x + c\right)^{4} - 22 \, a \cos\left(d x + c\right)^{2} + 4 \, a\right)} \sin\left(d x + c\right) + a}{15 \, {\left(d \cos\left(d x + c\right)^{3} \sin\left(d x + c\right) - d \cos\left(d x + c\right)^{3}\right)}}"," ",0,"1/15*(15*a*d*x*cos(d*x + c)^3 - 38*a*cos(d*x + c)^4 - 11*a*cos(d*x + c)^2 - (15*a*d*x*cos(d*x + c)^3 - 15*a*cos(d*x + c)^4 - 22*a*cos(d*x + c)^2 + 4*a)*sin(d*x + c) + a)/(d*cos(d*x + c)^3*sin(d*x + c) - d*cos(d*x + c)^3)","A",0
9,1,88,0,1.136559," ","integrate((a+a*sin(d*x+c))*tan(d*x+c)^4,x, algorithm=""fricas"")","-\frac{3 \, a d x \cos\left(d x + c\right) - 7 \, a \cos\left(d x + c\right)^{2} - {\left(3 \, a d x \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right)^{2} - 2 \, a\right)} \sin\left(d x + c\right) - a}{3 \, {\left(d \cos\left(d x + c\right) \sin\left(d x + c\right) - d \cos\left(d x + c\right)\right)}}"," ",0,"-1/3*(3*a*d*x*cos(d*x + c) - 7*a*cos(d*x + c)^2 - (3*a*d*x*cos(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a)*sin(d*x + c) - a)/(d*cos(d*x + c)*sin(d*x + c) - d*cos(d*x + c))","A",0
10,1,80,0,1.454307," ","integrate((a+a*sin(d*x+c))*tan(d*x+c)^2,x, algorithm=""fricas"")","-\frac{a d x - a \cos\left(d x + c\right)^{2} + {\left(a d x - 2 \, a\right)} \cos\left(d x + c\right) - {\left(a d x - a \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) - a}{d \cos\left(d x + c\right) - d \sin\left(d x + c\right) + d}"," ",0,"-(a*d*x - a*cos(d*x + c)^2 + (a*d*x - 2*a)*cos(d*x + c) - (a*d*x - a*cos(d*x + c) + a)*sin(d*x + c) - a)/(d*cos(d*x + c) - d*sin(d*x + c) + d)","B",0
11,1,84,0,1.044298," ","integrate(cot(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{a \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - a \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 2 \, a \cos\left(d x + c\right) + 2 \, {\left(a d x - a \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{2 \, d \sin\left(d x + c\right)}"," ",0,"-1/2*(a*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - a*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 2*a*cos(d*x + c) + 2*(a*d*x - a*cos(d*x + c))*sin(d*x + c))/(d*sin(d*x + c))","B",0
12,1,160,0,1.227291," ","integrate(cot(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{16 \, a \cos\left(d x + c\right)^{3} + 9 \, {\left(a \cos\left(d x + c\right)^{2} - a\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 9 \, {\left(a \cos\left(d x + c\right)^{2} - a\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 12 \, a \cos\left(d x + c\right) + 6 \, {\left(2 \, a d x \cos\left(d x + c\right)^{2} - 2 \, a \cos\left(d x + c\right)^{3} - 2 \, a d x + 3 \, a \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{12 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}"," ",0,"1/12*(16*a*cos(d*x + c)^3 + 9*(a*cos(d*x + c)^2 - a)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 9*(a*cos(d*x + c)^2 - a)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 12*a*cos(d*x + c) + 6*(2*a*d*x*cos(d*x + c)^2 - 2*a*cos(d*x + c)^3 - 2*a*d*x + 3*a*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))","B",0
13,1,222,0,1.084686," ","integrate(cot(d*x+c)^6*(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{368 \, a \cos\left(d x + c\right)^{5} - 560 \, a \cos\left(d x + c\right)^{3} + 225 \, {\left(a \cos\left(d x + c\right)^{4} - 2 \, a \cos\left(d x + c\right)^{2} + a\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 225 \, {\left(a \cos\left(d x + c\right)^{4} - 2 \, a \cos\left(d x + c\right)^{2} + a\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 240 \, a \cos\left(d x + c\right) + 30 \, {\left(8 \, a d x \cos\left(d x + c\right)^{4} - 8 \, a \cos\left(d x + c\right)^{5} - 16 \, a d x \cos\left(d x + c\right)^{2} + 25 \, a \cos\left(d x + c\right)^{3} + 8 \, a d x - 15 \, a \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{240 \, {\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)}"," ",0,"-1/240*(368*a*cos(d*x + c)^5 - 560*a*cos(d*x + c)^3 + 225*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 225*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 240*a*cos(d*x + c) + 30*(8*a*d*x*cos(d*x + c)^4 - 8*a*cos(d*x + c)^5 - 16*a*d*x*cos(d*x + c)^2 + 25*a*cos(d*x + c)^3 + 8*a*d*x - 15*a*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))","B",0
14,1,168,0,1.181159," ","integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^5,x, algorithm=""fricas"")","\frac{4 \, a^{2} \cos\left(d x + c\right)^{4} + 22 \, a^{2} \cos\left(d x + c\right)^{2} - 12 \, a^{2} - {\left(a^{2} \cos\left(d x + c\right)^{2} + 2 \, a^{2} \sin\left(d x + c\right) - 2 \, a^{2}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 31 \, {\left(a^{2} \cos\left(d x + c\right)^{2} + 2 \, a^{2} \sin\left(d x + c\right) - 2 \, a^{2}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(4 \, a^{2} \cos\left(d x + c\right)^{2} - 5 \, a^{2}\right)} \sin\left(d x + c\right)}{8 \, {\left(d \cos\left(d x + c\right)^{2} + 2 \, d \sin\left(d x + c\right) - 2 \, d\right)}}"," ",0,"1/8*(4*a^2*cos(d*x + c)^4 + 22*a^2*cos(d*x + c)^2 - 12*a^2 - (a^2*cos(d*x + c)^2 + 2*a^2*sin(d*x + c) - 2*a^2)*log(sin(d*x + c) + 1) - 31*(a^2*cos(d*x + c)^2 + 2*a^2*sin(d*x + c) - 2*a^2)*log(-sin(d*x + c) + 1) - 2*(4*a^2*cos(d*x + c)^2 - 5*a^2)*sin(d*x + c))/(d*cos(d*x + c)^2 + 2*d*sin(d*x + c) - 2*d)","A",0
15,1,90,0,1.349536," ","integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^3,x, algorithm=""fricas"")","-\frac{6 \, a^{2} \cos\left(d x + c\right)^{2} - 3 \, a^{2} - 12 \, {\left(a^{2} \sin\left(d x + c\right) - a^{2}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + {\left(2 \, a^{2} \cos\left(d x + c\right)^{2} + 7 \, a^{2}\right)} \sin\left(d x + c\right)}{4 \, {\left(d \sin\left(d x + c\right) - d\right)}}"," ",0,"-1/4*(6*a^2*cos(d*x + c)^2 - 3*a^2 - 12*(a^2*sin(d*x + c) - a^2)*log(-sin(d*x + c) + 1) + (2*a^2*cos(d*x + c)^2 + 7*a^2)*sin(d*x + c))/(d*sin(d*x + c) - d)","A",0
16,1,45,0,1.131325," ","integrate((a+a*sin(d*x+c))^2*tan(d*x+c),x, algorithm=""fricas"")","\frac{a^{2} \cos\left(d x + c\right)^{2} - 4 \, a^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 4 \, a^{2} \sin\left(d x + c\right)}{2 \, d}"," ",0,"1/2*(a^2*cos(d*x + c)^2 - 4*a^2*log(-sin(d*x + c) + 1) - 4*a^2*sin(d*x + c))/d","A",0
17,1,76,0,0.909930," ","integrate(cot(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{2 \, a^{2} \cos\left(d x + c\right)^{4} - 3 \, a^{2} \cos\left(d x + c\right)^{2} + 3 \, a^{2} - 8 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2}\right)} \sin\left(d x + c\right)}{4 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"1/4*(2*a^2*cos(d*x + c)^4 - 3*a^2*cos(d*x + c)^2 + 3*a^2 - 8*(a^2*cos(d*x + c)^2 - 2*a^2)*sin(d*x + c))/(d*cos(d*x + c)^2 - d)","B",0
18,1,206,0,1.142819," ","integrate(cot(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{30 \, a^{2} \cos\left(d x + c\right)^{8} - 105 \, a^{2} \cos\left(d x + c\right)^{6} + 135 \, a^{2} \cos\left(d x + c\right)^{4} - 45 \, a^{2} \cos\left(d x + c\right)^{2} - 5 \, a^{2} + 120 \, {\left(a^{2} \cos\left(d x + c\right)^{6} - 3 \, a^{2} \cos\left(d x + c\right)^{4} + 3 \, a^{2} \cos\left(d x + c\right)^{2} - a^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 24 \, {\left(5 \, a^{2} \cos\left(d x + c\right)^{6} - 30 \, a^{2} \cos\left(d x + c\right)^{4} + 40 \, a^{2} \cos\left(d x + c\right)^{2} - 16 \, a^{2}\right)} \sin\left(d x + c\right)}{60 \, {\left(d \cos\left(d x + c\right)^{6} - 3 \, d \cos\left(d x + c\right)^{4} + 3 \, d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"1/60*(30*a^2*cos(d*x + c)^8 - 105*a^2*cos(d*x + c)^6 + 135*a^2*cos(d*x + c)^4 - 45*a^2*cos(d*x + c)^2 - 5*a^2 + 120*(a^2*cos(d*x + c)^6 - 3*a^2*cos(d*x + c)^4 + 3*a^2*cos(d*x + c)^2 - a^2)*log(1/2*sin(d*x + c)) - 24*(5*a^2*cos(d*x + c)^6 - 30*a^2*cos(d*x + c)^4 + 40*a^2*cos(d*x + c)^2 - 16*a^2)*sin(d*x + c))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)","A",0
19,1,152,0,0.687384," ","integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^6,x, algorithm=""fricas"")","-\frac{45 \, a^{2} d x \cos\left(d x + c\right)^{3} - 10 \, a^{2} \cos\left(d x + c\right)^{4} - 90 \, a^{2} d x \cos\left(d x + c\right) + 78 \, a^{2} \cos\left(d x + c\right)^{2} - 4 \, a^{2} - {\left(5 \, a^{2} \cos\left(d x + c\right)^{4} - 90 \, a^{2} d x \cos\left(d x + c\right) + 84 \, a^{2} \cos\left(d x + c\right)^{2} - 6 \, a^{2}\right)} \sin\left(d x + c\right)}{10 \, {\left(d \cos\left(d x + c\right)^{3} + 2 \, d \cos\left(d x + c\right) \sin\left(d x + c\right) - 2 \, d \cos\left(d x + c\right)\right)}}"," ",0,"-1/10*(45*a^2*d*x*cos(d*x + c)^3 - 10*a^2*cos(d*x + c)^4 - 90*a^2*d*x*cos(d*x + c) + 78*a^2*cos(d*x + c)^2 - 4*a^2 - (5*a^2*cos(d*x + c)^4 - 90*a^2*d*x*cos(d*x + c) + 84*a^2*cos(d*x + c)^2 - 6*a^2)*sin(d*x + c))/(d*cos(d*x + c)^3 + 2*d*cos(d*x + c)*sin(d*x + c) - 2*d*cos(d*x + c))","A",0
20,1,196,0,1.235445," ","integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^4,x, algorithm=""fricas"")","\frac{3 \, a^{2} \cos\left(d x + c\right)^{4} - 6 \, a^{2} \cos\left(d x + c\right)^{3} - 42 \, a^{2} d x + {\left(21 \, a^{2} d x + 31 \, a^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, a^{2} - {\left(21 \, a^{2} d x - 38 \, a^{2}\right)} \cos\left(d x + c\right) - {\left(3 \, a^{2} \cos\left(d x + c\right)^{3} - 42 \, a^{2} d x + 9 \, a^{2} \cos\left(d x + c\right)^{2} + 2 \, a^{2} - {\left(21 \, a^{2} d x - 40 \, a^{2}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{6 \, {\left(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) - 2 \, d\right)}}"," ",0,"1/6*(3*a^2*cos(d*x + c)^4 - 6*a^2*cos(d*x + c)^3 - 42*a^2*d*x + (21*a^2*d*x + 31*a^2)*cos(d*x + c)^2 - 2*a^2 - (21*a^2*d*x - 38*a^2)*cos(d*x + c) - (3*a^2*cos(d*x + c)^3 - 42*a^2*d*x + 9*a^2*cos(d*x + c)^2 + 2*a^2 - (21*a^2*d*x - 40*a^2)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^2 - d*cos(d*x + c) + (d*cos(d*x + c) + 2*d)*sin(d*x + c) - 2*d)","A",0
21,1,125,0,0.833836," ","integrate((a+a*sin(d*x+c))^2*tan(d*x+c)^2,x, algorithm=""fricas"")","\frac{a^{2} \cos\left(d x + c\right)^{3} - 5 \, a^{2} d x + 4 \, a^{2} \cos\left(d x + c\right)^{2} + 4 \, a^{2} - {\left(5 \, a^{2} d x - 7 \, a^{2}\right)} \cos\left(d x + c\right) + {\left(5 \, a^{2} d x + a^{2} \cos\left(d x + c\right)^{2} - 3 \, a^{2} \cos\left(d x + c\right) + 4 \, a^{2}\right)} \sin\left(d x + c\right)}{2 \, {\left(d \cos\left(d x + c\right) - d \sin\left(d x + c\right) + d\right)}}"," ",0,"1/2*(a^2*cos(d*x + c)^3 - 5*a^2*d*x + 4*a^2*cos(d*x + c)^2 + 4*a^2 - (5*a^2*d*x - 7*a^2)*cos(d*x + c) + (5*a^2*d*x + a^2*cos(d*x + c)^2 - 3*a^2*cos(d*x + c) + 4*a^2)*sin(d*x + c))/(d*cos(d*x + c) - d*sin(d*x + c) + d)","A",0
22,1,41,0,1.072859," ","integrate((a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{3 \, a^{2} d x - a^{2} \cos\left(d x + c\right) \sin\left(d x + c\right) - 4 \, a^{2} \cos\left(d x + c\right)}{2 \, d}"," ",0,"1/2*(3*a^2*d*x - a^2*cos(d*x + c)*sin(d*x + c) - 4*a^2*cos(d*x + c))/d","A",0
23,1,105,0,1.190641," ","integrate(cot(d*x+c)^2*(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","-\frac{a^{2} \cos\left(d x + c\right)^{3} + 2 \, a^{2} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 2 \, a^{2} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + a^{2} \cos\left(d x + c\right) + {\left(a^{2} d x - 4 \, a^{2} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{2 \, d \sin\left(d x + c\right)}"," ",0,"-1/2*(a^2*cos(d*x + c)^3 + 2*a^2*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 2*a^2*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + a^2*cos(d*x + c) + (a^2*d*x - 4*a^2*cos(d*x + c))*sin(d*x + c))/(d*sin(d*x + c))","A",0
24,1,192,0,1.256953," ","integrate(cot(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{3 \, a^{2} \cos\left(d x + c\right)^{5} - 4 \, a^{2} \cos\left(d x + c\right)^{3} + 3 \, a^{2} \cos\left(d x + c\right) + 9 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - a^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 9 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - a^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 3 \, {\left(a^{2} d x \cos\left(d x + c\right)^{2} + 4 \, a^{2} \cos\left(d x + c\right)^{3} - a^{2} d x - 6 \, a^{2} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{6 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}"," ",0,"1/6*(3*a^2*cos(d*x + c)^5 - 4*a^2*cos(d*x + c)^3 + 3*a^2*cos(d*x + c) + 9*(a^2*cos(d*x + c)^2 - a^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 9*(a^2*cos(d*x + c)^2 - a^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3*(a^2*d*x*cos(d*x + c)^2 + 4*a^2*cos(d*x + c)^3 - a^2*d*x - 6*a^2*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))","B",0
25,1,240,0,1.264158," ","integrate((a+a*sin(d*x+c))^3*tan(d*x+c)^7,x, algorithm=""fricas"")","-\frac{16 \, a^{3} \cos\left(d x + c\right)^{6} - 216 \, a^{3} \cos\left(d x + c\right)^{4} + 1002 \, a^{3} \cos\left(d x + c\right)^{2} - 482 \, a^{3} + 3 \, {\left(3 \, a^{3} \cos\left(d x + c\right)^{2} - 4 \, a^{3} - {\left(a^{3} \cos\left(d x + c\right)^{2} - 4 \, a^{3}\right)} \sin\left(d x + c\right)\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 627 \, {\left(3 \, a^{3} \cos\left(d x + c\right)^{2} - 4 \, a^{3} - {\left(a^{3} \cos\left(d x + c\right)^{2} - 4 \, a^{3}\right)} \sin\left(d x + c\right)\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(12 \, a^{3} \cos\left(d x + c\right)^{4} + 398 \, a^{3} \cos\left(d x + c\right)^{2} - 245 \, a^{3}\right)} \sin\left(d x + c\right)}{48 \, {\left(3 \, d \cos\left(d x + c\right)^{2} - {\left(d \cos\left(d x + c\right)^{2} - 4 \, d\right)} \sin\left(d x + c\right) - 4 \, d\right)}}"," ",0,"-1/48*(16*a^3*cos(d*x + c)^6 - 216*a^3*cos(d*x + c)^4 + 1002*a^3*cos(d*x + c)^2 - 482*a^3 + 3*(3*a^3*cos(d*x + c)^2 - 4*a^3 - (a^3*cos(d*x + c)^2 - 4*a^3)*sin(d*x + c))*log(sin(d*x + c) + 1) - 627*(3*a^3*cos(d*x + c)^2 - 4*a^3 - (a^3*cos(d*x + c)^2 - 4*a^3)*sin(d*x + c))*log(-sin(d*x + c) + 1) - 2*(12*a^3*cos(d*x + c)^4 + 398*a^3*cos(d*x + c)^2 - 245*a^3)*sin(d*x + c))/(3*d*cos(d*x + c)^2 - (d*cos(d*x + c)^2 - 4*d)*sin(d*x + c) - 4*d)","A",0
26,1,104,0,1.299764," ","integrate((a+a*sin(d*x+c))^3*tan(d*x+c)^3,x, algorithm=""fricas"")","\frac{4 \, a^{3} \cos\left(d x + c\right)^{4} - 50 \, a^{3} \cos\left(d x + c\right)^{2} + 31 \, a^{3} + 84 \, {\left(a^{3} \sin\left(d x + c\right) - a^{3}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - {\left(14 \, a^{3} \cos\left(d x + c\right)^{2} + 55 \, a^{3}\right)} \sin\left(d x + c\right)}{12 \, {\left(d \sin\left(d x + c\right) - d\right)}}"," ",0,"1/12*(4*a^3*cos(d*x + c)^4 - 50*a^3*cos(d*x + c)^2 + 31*a^3 + 84*(a^3*sin(d*x + c) - a^3)*log(-sin(d*x + c) + 1) - (14*a^3*cos(d*x + c)^2 + 55*a^3)*sin(d*x + c))/(d*sin(d*x + c) - d)","A",0
27,1,61,0,1.170544," ","integrate((a+a*sin(d*x+c))^3*tan(d*x+c),x, algorithm=""fricas"")","\frac{9 \, a^{3} \cos\left(d x + c\right)^{2} - 24 \, a^{3} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(a^{3} \cos\left(d x + c\right)^{2} - 13 \, a^{3}\right)} \sin\left(d x + c\right)}{6 \, d}"," ",0,"1/6*(9*a^3*cos(d*x + c)^2 - 24*a^3*log(-sin(d*x + c) + 1) + 2*(a^3*cos(d*x + c)^2 - 13*a^3)*sin(d*x + c))/d","A",0
28,1,118,0,1.126205," ","integrate(cot(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","\frac{18 \, a^{3} \cos\left(d x + c\right)^{4} - 27 \, a^{3} \cos\left(d x + c\right)^{2} + 15 \, a^{3} + 24 \, {\left(a^{3} \cos\left(d x + c\right)^{2} - a^{3}\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) + 4 \, {\left(a^{3} \cos\left(d x + c\right)^{4} - 8 \, a^{3} \cos\left(d x + c\right)^{2} + 16 \, a^{3}\right)} \sin\left(d x + c\right)}{12 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"1/12*(18*a^3*cos(d*x + c)^4 - 27*a^3*cos(d*x + c)^2 + 15*a^3 + 24*(a^3*cos(d*x + c)^2 - a^3)*log(1/2*sin(d*x + c)) + 4*(a^3*cos(d*x + c)^4 - 8*a^3*cos(d*x + c)^2 + 16*a^3)*sin(d*x + c))/(d*cos(d*x + c)^2 - d)","A",0
29,1,289,0,1.212628," ","integrate((a+a*sin(d*x+c))^3*tan(d*x+c)^6,x, algorithm=""fricas"")","-\frac{10 \, a^{3} \cos\left(d x + c\right)^{6} - 15 \, a^{3} \cos\left(d x + c\right)^{5} - 140 \, a^{3} \cos\left(d x + c\right)^{4} - 1380 \, a^{3} d x + {\left(345 \, a^{3} d x - 839 \, a^{3}\right)} \cos\left(d x + c\right)^{3} + 6 \, a^{3} + {\left(1035 \, a^{3} d x + 668 \, a^{3}\right)} \cos\left(d x + c\right)^{2} - 6 \, {\left(115 \, a^{3} d x - 233 \, a^{3}\right)} \cos\left(d x + c\right) - {\left(10 \, a^{3} \cos\left(d x + c\right)^{5} + 25 \, a^{3} \cos\left(d x + c\right)^{4} - 115 \, a^{3} \cos\left(d x + c\right)^{3} - 1380 \, a^{3} d x - 6 \, a^{3} + {\left(345 \, a^{3} d x + 724 \, a^{3}\right)} \cos\left(d x + c\right)^{2} - 6 \, {\left(115 \, a^{3} d x - 232 \, a^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{30 \, {\left(d \cos\left(d x + c\right)^{3} + 3 \, d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) - {\left(d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) - 4 \, d\right)} \sin\left(d x + c\right) - 4 \, d\right)}}"," ",0,"-1/30*(10*a^3*cos(d*x + c)^6 - 15*a^3*cos(d*x + c)^5 - 140*a^3*cos(d*x + c)^4 - 1380*a^3*d*x + (345*a^3*d*x - 839*a^3)*cos(d*x + c)^3 + 6*a^3 + (1035*a^3*d*x + 668*a^3)*cos(d*x + c)^2 - 6*(115*a^3*d*x - 233*a^3)*cos(d*x + c) - (10*a^3*cos(d*x + c)^5 + 25*a^3*cos(d*x + c)^4 - 115*a^3*cos(d*x + c)^3 - 1380*a^3*d*x - 6*a^3 + (345*a^3*d*x + 724*a^3)*cos(d*x + c)^2 - 6*(115*a^3*d*x - 232*a^3)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^3 + 3*d*cos(d*x + c)^2 - 2*d*cos(d*x + c) - (d*cos(d*x + c)^2 - 2*d*cos(d*x + c) - 4*d)*sin(d*x + c) - 4*d)","A",0
30,1,220,0,0.848722," ","integrate((a+a*sin(d*x+c))^3*tan(d*x+c)^4,x, algorithm=""fricas"")","\frac{2 \, a^{3} \cos\left(d x + c\right)^{5} + 7 \, a^{3} \cos\left(d x + c\right)^{4} - 22 \, a^{3} \cos\left(d x + c\right)^{3} - 102 \, a^{3} d x - 4 \, a^{3} + {\left(51 \, a^{3} d x + 77 \, a^{3}\right)} \cos\left(d x + c\right)^{2} - {\left(51 \, a^{3} d x - 100 \, a^{3}\right)} \cos\left(d x + c\right) + {\left(2 \, a^{3} \cos\left(d x + c\right)^{4} - 5 \, a^{3} \cos\left(d x + c\right)^{3} + 102 \, a^{3} d x - 27 \, a^{3} \cos\left(d x + c\right)^{2} - 4 \, a^{3} + {\left(51 \, a^{3} d x - 104 \, a^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{6 \, {\left(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) - 2 \, d\right)}}"," ",0,"1/6*(2*a^3*cos(d*x + c)^5 + 7*a^3*cos(d*x + c)^4 - 22*a^3*cos(d*x + c)^3 - 102*a^3*d*x - 4*a^3 + (51*a^3*d*x + 77*a^3)*cos(d*x + c)^2 - (51*a^3*d*x - 100*a^3)*cos(d*x + c) + (2*a^3*cos(d*x + c)^4 - 5*a^3*cos(d*x + c)^3 + 102*a^3*d*x - 27*a^3*cos(d*x + c)^2 - 4*a^3 + (51*a^3*d*x - 104*a^3)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^2 - d*cos(d*x + c) + (d*cos(d*x + c) + 2*d)*sin(d*x + c) - 2*d)","B",0
31,1,154,0,1.135082," ","integrate((a+a*sin(d*x+c))^3*tan(d*x+c)^2,x, algorithm=""fricas"")","-\frac{2 \, a^{3} \cos\left(d x + c\right)^{4} - 7 \, a^{3} \cos\left(d x + c\right)^{3} + 33 \, a^{3} d x - 30 \, a^{3} \cos\left(d x + c\right)^{2} - 24 \, a^{3} + 3 \, {\left(11 \, a^{3} d x - 15 \, a^{3}\right)} \cos\left(d x + c\right) - {\left(2 \, a^{3} \cos\left(d x + c\right)^{3} + 33 \, a^{3} d x + 9 \, a^{3} \cos\left(d x + c\right)^{2} - 21 \, a^{3} \cos\left(d x + c\right) + 24 \, a^{3}\right)} \sin\left(d x + c\right)}{6 \, {\left(d \cos\left(d x + c\right) - d \sin\left(d x + c\right) + d\right)}}"," ",0,"-1/6*(2*a^3*cos(d*x + c)^4 - 7*a^3*cos(d*x + c)^3 + 33*a^3*d*x - 30*a^3*cos(d*x + c)^2 - 24*a^3 + 3*(11*a^3*d*x - 15*a^3)*cos(d*x + c) - (2*a^3*cos(d*x + c)^3 + 33*a^3*d*x + 9*a^3*cos(d*x + c)^2 - 21*a^3*cos(d*x + c) + 24*a^3)*sin(d*x + c))/(d*cos(d*x + c) - d*sin(d*x + c) + d)","A",0
32,1,54,0,0.826077," ","integrate((a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","\frac{2 \, a^{3} \cos\left(d x + c\right)^{3} + 15 \, a^{3} d x - 9 \, a^{3} \cos\left(d x + c\right) \sin\left(d x + c\right) - 24 \, a^{3} \cos\left(d x + c\right)}{6 \, d}"," ",0,"1/6*(2*a^3*cos(d*x + c)^3 + 15*a^3*d*x - 9*a^3*cos(d*x + c)*sin(d*x + c) - 24*a^3*cos(d*x + c))/d","A",0
33,1,121,0,1.159035," ","integrate(cot(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{9 \, a^{3} \cos\left(d x + c\right)^{3} + 9 \, a^{3} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 9 \, a^{3} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 3 \, a^{3} \cos\left(d x + c\right) + {\left(2 \, a^{3} \cos\left(d x + c\right)^{3} - 3 \, a^{3} d x - 18 \, a^{3} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{6 \, d \sin\left(d x + c\right)}"," ",0,"-1/6*(9*a^3*cos(d*x + c)^3 + 9*a^3*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 9*a^3*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3*a^3*cos(d*x + c) + (2*a^3*cos(d*x + c)^3 - 3*a^3*d*x - 18*a^3*cos(d*x + c))*sin(d*x + c))/(d*sin(d*x + c))","A",0
34,1,154,0,1.181253," ","integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^5,x, algorithm=""fricas"")","-\frac{24 \, a^{4} \cos\left(d x + c\right)^{6} - 272 \, a^{4} \cos\left(d x + c\right)^{4} - 2393 \, a^{4} \cos\left(d x + c\right)^{2} + 1906 \, a^{4} + 2400 \, {\left(a^{4} \cos\left(d x + c\right)^{2} + 2 \, a^{4} \sin\left(d x + c\right) - 2 \, a^{4}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 10 \, {\left(8 \, a^{4} \cos\left(d x + c\right)^{4} - 96 \, a^{4} \cos\left(d x + c\right)^{2} + 181 \, a^{4}\right)} \sin\left(d x + c\right)}{96 \, {\left(d \cos\left(d x + c\right)^{2} + 2 \, d \sin\left(d x + c\right) - 2 \, d\right)}}"," ",0,"-1/96*(24*a^4*cos(d*x + c)^6 - 272*a^4*cos(d*x + c)^4 - 2393*a^4*cos(d*x + c)^2 + 1906*a^4 + 2400*(a^4*cos(d*x + c)^2 + 2*a^4*sin(d*x + c) - 2*a^4)*log(-sin(d*x + c) + 1) - 10*(8*a^4*cos(d*x + c)^4 - 96*a^4*cos(d*x + c)^2 + 181*a^4)*sin(d*x + c))/(d*cos(d*x + c)^2 + 2*d*sin(d*x + c) - 2*d)","A",0
35,1,116,0,1.268357," ","integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^3,x, algorithm=""fricas"")","\frac{104 \, a^{4} \cos\left(d x + c\right)^{4} - 976 \, a^{4} \cos\left(d x + c\right)^{2} + 689 \, a^{4} + 1536 \, {\left(a^{4} \sin\left(d x + c\right) - a^{4}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + {\left(24 \, a^{4} \cos\left(d x + c\right)^{4} - 304 \, a^{4} \cos\left(d x + c\right)^{2} - 1073 \, a^{4}\right)} \sin\left(d x + c\right)}{96 \, {\left(d \sin\left(d x + c\right) - d\right)}}"," ",0,"1/96*(104*a^4*cos(d*x + c)^4 - 976*a^4*cos(d*x + c)^2 + 689*a^4 + 1536*(a^4*sin(d*x + c) - a^4)*log(-sin(d*x + c) + 1) + (24*a^4*cos(d*x + c)^4 - 304*a^4*cos(d*x + c)^2 - 1073*a^4)*sin(d*x + c))/(d*sin(d*x + c) - d)","A",0
36,1,74,0,1.062797," ","integrate((a+a*sin(d*x+c))^4*tan(d*x+c),x, algorithm=""fricas"")","-\frac{3 \, a^{4} \cos\left(d x + c\right)^{4} - 48 \, a^{4} \cos\left(d x + c\right)^{2} + 96 \, a^{4} \log\left(-\sin\left(d x + c\right) + 1\right) - 16 \, {\left(a^{4} \cos\left(d x + c\right)^{2} - 7 \, a^{4}\right)} \sin\left(d x + c\right)}{12 \, d}"," ",0,"-1/12*(3*a^4*cos(d*x + c)^4 - 48*a^4*cos(d*x + c)^2 + 96*a^4*log(-sin(d*x + c) + 1) - 16*(a^4*cos(d*x + c)^2 - 7*a^4)*sin(d*x + c))/d","A",0
37,1,131,0,1.306567," ","integrate(cot(d*x+c)^3*(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","-\frac{24 \, a^{4} \cos\left(d x + c\right)^{6} - 312 \, a^{4} \cos\left(d x + c\right)^{4} + 423 \, a^{4} \cos\left(d x + c\right)^{2} - 183 \, a^{4} - 480 \, {\left(a^{4} \cos\left(d x + c\right)^{2} - a^{4}\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 128 \, {\left(a^{4} \cos\left(d x + c\right)^{4} - 2 \, a^{4} \cos\left(d x + c\right)^{2} + 4 \, a^{4}\right)} \sin\left(d x + c\right)}{96 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"-1/96*(24*a^4*cos(d*x + c)^6 - 312*a^4*cos(d*x + c)^4 + 423*a^4*cos(d*x + c)^2 - 183*a^4 - 480*(a^4*cos(d*x + c)^2 - a^4)*log(1/2*sin(d*x + c)) - 128*(a^4*cos(d*x + c)^4 - 2*a^4*cos(d*x + c)^2 + 4*a^4)*sin(d*x + c))/(d*cos(d*x + c)^2 - d)","A",0
38,1,247,0,1.095733," ","integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^4,x, algorithm=""fricas"")","-\frac{6 \, a^{4} \cos\left(d x + c\right)^{6} - 20 \, a^{4} \cos\left(d x + c\right)^{5} - 85 \, a^{4} \cos\left(d x + c\right)^{4} + 214 \, a^{4} \cos\left(d x + c\right)^{3} + 978 \, a^{4} d x + 32 \, a^{4} - {\left(489 \, a^{4} d x + 721 \, a^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(489 \, a^{4} d x - 962 \, a^{4}\right)} \cos\left(d x + c\right) - {\left(6 \, a^{4} \cos\left(d x + c\right)^{5} + 26 \, a^{4} \cos\left(d x + c\right)^{4} - 59 \, a^{4} \cos\left(d x + c\right)^{3} + 978 \, a^{4} d x - 273 \, a^{4} \cos\left(d x + c\right)^{2} - 32 \, a^{4} + {\left(489 \, a^{4} d x - 994 \, a^{4}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{24 \, {\left(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) - 2 \, d\right)}}"," ",0,"-1/24*(6*a^4*cos(d*x + c)^6 - 20*a^4*cos(d*x + c)^5 - 85*a^4*cos(d*x + c)^4 + 214*a^4*cos(d*x + c)^3 + 978*a^4*d*x + 32*a^4 - (489*a^4*d*x + 721*a^4)*cos(d*x + c)^2 + (489*a^4*d*x - 962*a^4)*cos(d*x + c) - (6*a^4*cos(d*x + c)^5 + 26*a^4*cos(d*x + c)^4 - 59*a^4*cos(d*x + c)^3 + 978*a^4*d*x - 273*a^4*cos(d*x + c)^2 - 32*a^4 + (489*a^4*d*x - 994*a^4)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^2 - d*cos(d*x + c) + (d*cos(d*x + c) + 2*d)*sin(d*x + c) - 2*d)","A",0
39,1,179,0,1.094060," ","integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^2,x, algorithm=""fricas"")","-\frac{6 \, a^{4} \cos\left(d x + c\right)^{5} + 32 \, a^{4} \cos\left(d x + c\right)^{4} - 73 \, a^{4} \cos\left(d x + c\right)^{3} + 285 \, a^{4} d x - 288 \, a^{4} \cos\left(d x + c\right)^{2} - 192 \, a^{4} + 3 \, {\left(95 \, a^{4} d x - 127 \, a^{4}\right)} \cos\left(d x + c\right) + {\left(6 \, a^{4} \cos\left(d x + c\right)^{4} - 26 \, a^{4} \cos\left(d x + c\right)^{3} - 285 \, a^{4} d x - 99 \, a^{4} \cos\left(d x + c\right)^{2} + 189 \, a^{4} \cos\left(d x + c\right) - 192 \, a^{4}\right)} \sin\left(d x + c\right)}{24 \, {\left(d \cos\left(d x + c\right) - d \sin\left(d x + c\right) + d\right)}}"," ",0,"-1/24*(6*a^4*cos(d*x + c)^5 + 32*a^4*cos(d*x + c)^4 - 73*a^4*cos(d*x + c)^3 + 285*a^4*d*x - 288*a^4*cos(d*x + c)^2 - 192*a^4 + 3*(95*a^4*d*x - 127*a^4)*cos(d*x + c) + (6*a^4*cos(d*x + c)^4 - 26*a^4*cos(d*x + c)^3 - 285*a^4*d*x - 99*a^4*cos(d*x + c)^2 + 189*a^4*cos(d*x + c) - 192*a^4)*sin(d*x + c))/(d*cos(d*x + c) - d*sin(d*x + c) + d)","A",0
40,1,70,0,0.708870," ","integrate((a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","\frac{32 \, a^{4} \cos\left(d x + c\right)^{3} + 105 \, a^{4} d x - 192 \, a^{4} \cos\left(d x + c\right) + 3 \, {\left(2 \, a^{4} \cos\left(d x + c\right)^{3} - 29 \, a^{4} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{24 \, d}"," ",0,"1/24*(32*a^4*cos(d*x + c)^3 + 105*a^4*d*x - 192*a^4*cos(d*x + c) + 3*(2*a^4*cos(d*x + c)^3 - 29*a^4*cos(d*x + c))*sin(d*x + c))/d","A",0
41,1,135,0,1.310527," ","integrate(cot(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","\frac{6 \, a^{4} \cos\left(d x + c\right)^{5} - 81 \, a^{4} \cos\left(d x + c\right)^{3} - 48 \, a^{4} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 48 \, a^{4} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 51 \, a^{4} \cos\left(d x + c\right) - {\left(32 \, a^{4} \cos\left(d x + c\right)^{3} - 51 \, a^{4} d x - 96 \, a^{4} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{24 \, d \sin\left(d x + c\right)}"," ",0,"1/24*(6*a^4*cos(d*x + c)^5 - 81*a^4*cos(d*x + c)^3 - 48*a^4*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 48*a^4*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 51*a^4*cos(d*x + c) - (32*a^4*cos(d*x + c)^3 - 51*a^4*d*x - 96*a^4*cos(d*x + c))*sin(d*x + c))/(d*sin(d*x + c))","A",0
42,1,219,0,1.079943," ","integrate(cot(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","-\frac{6 \, a^{4} \cos\left(d x + c\right)^{7} - 75 \, a^{4} \cos\left(d x + c\right)^{5} + 244 \, a^{4} \cos\left(d x + c\right)^{3} - 183 \, a^{4} \cos\left(d x + c\right) - 24 \, {\left(a^{4} \cos\left(d x + c\right)^{2} - a^{4}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 24 \, {\left(a^{4} \cos\left(d x + c\right)^{2} - a^{4}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - {\left(32 \, a^{4} \cos\left(d x + c\right)^{5} - 183 \, a^{4} d x \cos\left(d x + c\right)^{2} - 32 \, a^{4} \cos\left(d x + c\right)^{3} + 183 \, a^{4} d x + 48 \, a^{4} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{24 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}"," ",0,"-1/24*(6*a^4*cos(d*x + c)^7 - 75*a^4*cos(d*x + c)^5 + 244*a^4*cos(d*x + c)^3 - 183*a^4*cos(d*x + c) - 24*(a^4*cos(d*x + c)^2 - a^4)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 24*(a^4*cos(d*x + c)^2 - a^4)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - (32*a^4*cos(d*x + c)^5 - 183*a^4*d*x*cos(d*x + c)^2 - 32*a^4*cos(d*x + c)^3 + 183*a^4*d*x + 48*a^4*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))","A",0
43,1,291,0,1.138731," ","integrate(cot(d*x+c)^6*(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","\frac{30 \, a^{4} \cos\left(d x + c\right)^{9} - 345 \, a^{4} \cos\left(d x + c\right)^{7} + 2231 \, a^{4} \cos\left(d x + c\right)^{5} - 3395 \, a^{4} \cos\left(d x + c\right)^{3} + 1455 \, a^{4} \cos\left(d x + c\right) + 150 \, {\left(a^{4} \cos\left(d x + c\right)^{4} - 2 \, a^{4} \cos\left(d x + c\right)^{2} + a^{4}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 150 \, {\left(a^{4} \cos\left(d x + c\right)^{4} - 2 \, a^{4} \cos\left(d x + c\right)^{2} + a^{4}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 5 \, {\left(32 \, a^{4} \cos\left(d x + c\right)^{7} - 291 \, a^{4} d x \cos\left(d x + c\right)^{4} + 32 \, a^{4} \cos\left(d x + c\right)^{5} + 582 \, a^{4} d x \cos\left(d x + c\right)^{2} - 100 \, a^{4} \cos\left(d x + c\right)^{3} - 291 \, a^{4} d x + 60 \, a^{4} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{120 \, {\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)}"," ",0,"1/120*(30*a^4*cos(d*x + c)^9 - 345*a^4*cos(d*x + c)^7 + 2231*a^4*cos(d*x + c)^5 - 3395*a^4*cos(d*x + c)^3 + 1455*a^4*cos(d*x + c) + 150*(a^4*cos(d*x + c)^4 - 2*a^4*cos(d*x + c)^2 + a^4)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 150*(a^4*cos(d*x + c)^4 - 2*a^4*cos(d*x + c)^2 + a^4)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 5*(32*a^4*cos(d*x + c)^7 - 291*a^4*d*x*cos(d*x + c)^4 + 32*a^4*cos(d*x + c)^5 + 582*a^4*d*x*cos(d*x + c)^2 - 100*a^4*cos(d*x + c)^3 - 291*a^4*d*x + 60*a^4*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))","A",0
44,1,167,0,0.798276," ","integrate(tan(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{558 \, \cos\left(d x + c\right)^{6} - 826 \, \cos\left(d x + c\right)^{4} + 476 \, \cos\left(d x + c\right)^{2} + 105 \, {\left(\cos\left(d x + c\right)^{6} \sin\left(d x + c\right) + \cos\left(d x + c\right)^{6}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 105 \, {\left(\cos\left(d x + c\right)^{6} \sin\left(d x + c\right) + \cos\left(d x + c\right)^{6}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(87 \, \cos\left(d x + c\right)^{4} - 38 \, \cos\left(d x + c\right)^{2} + 8\right)} \sin\left(d x + c\right) - 112}{768 \, {\left(a d \cos\left(d x + c\right)^{6} \sin\left(d x + c\right) + a d \cos\left(d x + c\right)^{6}\right)}}"," ",0,"-1/768*(558*cos(d*x + c)^6 - 826*cos(d*x + c)^4 + 476*cos(d*x + c)^2 + 105*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(sin(d*x + c) + 1) - 105*(cos(d*x + c)^6*sin(d*x + c) + cos(d*x + c)^6)*log(-sin(d*x + c) + 1) - 2*(87*cos(d*x + c)^4 - 38*cos(d*x + c)^2 + 8)*sin(d*x + c) - 112)/(a*d*cos(d*x + c)^6*sin(d*x + c) + a*d*cos(d*x + c)^6)","A",0
45,1,147,0,0.875423," ","integrate(tan(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{66 \, \cos\left(d x + c\right)^{4} - 70 \, \cos\left(d x + c\right)^{2} + 15 \, {\left(\cos\left(d x + c\right)^{4} \sin\left(d x + c\right) + \cos\left(d x + c\right)^{4}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 15 \, {\left(\cos\left(d x + c\right)^{4} \sin\left(d x + c\right) + \cos\left(d x + c\right)^{4}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(9 \, \cos\left(d x + c\right)^{2} - 2\right)} \sin\left(d x + c\right) + 20}{96 \, {\left(a d \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) + a d \cos\left(d x + c\right)^{4}\right)}}"," ",0,"1/96*(66*cos(d*x + c)^4 - 70*cos(d*x + c)^2 + 15*(cos(d*x + c)^4*sin(d*x + c) + cos(d*x + c)^4)*log(sin(d*x + c) + 1) - 15*(cos(d*x + c)^4*sin(d*x + c) + cos(d*x + c)^4)*log(-sin(d*x + c) + 1) - 2*(9*cos(d*x + c)^2 - 2)*sin(d*x + c) + 20)/(a*d*cos(d*x + c)^4*sin(d*x + c) + a*d*cos(d*x + c)^4)","A",0
46,1,125,0,1.243948," ","integrate(tan(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{10 \, \cos\left(d x + c\right)^{2} + 3 \, {\left(\cos\left(d x + c\right)^{2} \sin\left(d x + c\right) + \cos\left(d x + c\right)^{2}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 3 \, {\left(\cos\left(d x + c\right)^{2} \sin\left(d x + c\right) + \cos\left(d x + c\right)^{2}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, \sin\left(d x + c\right) - 6}{16 \, {\left(a d \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) + a d \cos\left(d x + c\right)^{2}\right)}}"," ",0,"-1/16*(10*cos(d*x + c)^2 + 3*(cos(d*x + c)^2*sin(d*x + c) + cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 3*(cos(d*x + c)^2*sin(d*x + c) + cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*sin(d*x + c) - 6)/(a*d*cos(d*x + c)^2*sin(d*x + c) + a*d*cos(d*x + c)^2)","A",0
47,1,58,0,1.076013," ","integrate(tan(d*x+c)/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{{\left(\sin\left(d x + c\right) + 1\right)} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(\sin\left(d x + c\right) + 1\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 2}{4 \, {\left(a d \sin\left(d x + c\right) + a d\right)}}"," ",0,"1/4*((sin(d*x + c) + 1)*log(sin(d*x + c) + 1) - (sin(d*x + c) + 1)*log(-sin(d*x + c) + 1) + 2)/(a*d*sin(d*x + c) + a*d)","A",0
48,1,28,0,1.004321," ","integrate(cot(d*x+c)/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{\log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - \log\left(\sin\left(d x + c\right) + 1\right)}{a d}"," ",0,"(log(1/2*sin(d*x + c)) - log(sin(d*x + c) + 1))/(a*d)","A",0
49,1,30,0,0.931211," ","integrate(cot(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{2 \, \sin\left(d x + c\right) - 1}{2 \, {\left(a d \cos\left(d x + c\right)^{2} - a d\right)}}"," ",0,"-1/2*(2*sin(d*x + c) - 1)/(a*d*cos(d*x + c)^2 - a*d)","A",0
50,1,63,0,1.054054," ","integrate(cot(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{6 \, \cos\left(d x + c\right)^{2} - 4 \, {\left(3 \, \cos\left(d x + c\right)^{2} - 2\right)} \sin\left(d x + c\right) - 3}{12 \, {\left(a d \cos\left(d x + c\right)^{4} - 2 \, a d \cos\left(d x + c\right)^{2} + a d\right)}}"," ",0,"-1/12*(6*cos(d*x + c)^2 - 4*(3*cos(d*x + c)^2 - 2)*sin(d*x + c) - 3)/(a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^2 + a*d)","A",0
51,1,96,0,1.403876," ","integrate(cot(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{15 \, \cos\left(d x + c\right)^{4} - 15 \, \cos\left(d x + c\right)^{2} - 2 \, {\left(15 \, \cos\left(d x + c\right)^{4} - 20 \, \cos\left(d x + c\right)^{2} + 8\right)} \sin\left(d x + c\right) + 5}{30 \, {\left(a d \cos\left(d x + c\right)^{6} - 3 \, a d \cos\left(d x + c\right)^{4} + 3 \, a d \cos\left(d x + c\right)^{2} - a d\right)}}"," ",0,"1/30*(15*cos(d*x + c)^4 - 15*cos(d*x + c)^2 - 2*(15*cos(d*x + c)^4 - 20*cos(d*x + c)^2 + 8)*sin(d*x + c) + 5)/(a*d*cos(d*x + c)^6 - 3*a*d*cos(d*x + c)^4 + 3*a*d*cos(d*x + c)^2 - a*d)","A",0
52,1,127,0,1.024867," ","integrate(cot(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{140 \, \cos\left(d x + c\right)^{6} - 210 \, \cos\left(d x + c\right)^{4} + 140 \, \cos\left(d x + c\right)^{2} - 8 \, {\left(35 \, \cos\left(d x + c\right)^{6} - 70 \, \cos\left(d x + c\right)^{4} + 56 \, \cos\left(d x + c\right)^{2} - 16\right)} \sin\left(d x + c\right) - 35}{280 \, {\left(a d \cos\left(d x + c\right)^{8} - 4 \, a d \cos\left(d x + c\right)^{6} + 6 \, a d \cos\left(d x + c\right)^{4} - 4 \, a d \cos\left(d x + c\right)^{2} + a d\right)}}"," ",0,"-1/280*(140*cos(d*x + c)^6 - 210*cos(d*x + c)^4 + 140*cos(d*x + c)^2 - 8*(35*cos(d*x + c)^6 - 70*cos(d*x + c)^4 + 56*cos(d*x + c)^2 - 16)*sin(d*x + c) - 35)/(a*d*cos(d*x + c)^8 - 4*a*d*cos(d*x + c)^6 + 6*a*d*cos(d*x + c)^4 - 4*a*d*cos(d*x + c)^2 + a*d)","A",0
53,1,95,0,1.138671," ","integrate(tan(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{5 \, \cos\left(d x + c\right)^{6} + 15 \, \cos\left(d x + c\right)^{4} - 5 \, \cos\left(d x + c\right)^{2} + 2 \, {\left(15 \, \cos\left(d x + c\right)^{4} - 10 \, \cos\left(d x + c\right)^{2} + 3\right)} \sin\left(d x + c\right) + 1}{35 \, {\left(a d \cos\left(d x + c\right)^{5} \sin\left(d x + c\right) + a d \cos\left(d x + c\right)^{5}\right)}}"," ",0,"1/35*(5*cos(d*x + c)^6 + 15*cos(d*x + c)^4 - 5*cos(d*x + c)^2 + 2*(15*cos(d*x + c)^4 - 10*cos(d*x + c)^2 + 3)*sin(d*x + c) + 1)/(a*d*cos(d*x + c)^5*sin(d*x + c) + a*d*cos(d*x + c)^5)","A",0
54,1,75,0,1.013900," ","integrate(tan(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{3 \, \cos\left(d x + c\right)^{4} + 6 \, \cos\left(d x + c\right)^{2} + 4 \, {\left(3 \, \cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - 1}{15 \, {\left(a d \cos\left(d x + c\right)^{3} \sin\left(d x + c\right) + a d \cos\left(d x + c\right)^{3}\right)}}"," ",0,"-1/15*(3*cos(d*x + c)^4 + 6*cos(d*x + c)^2 + 4*(3*cos(d*x + c)^2 - 1)*sin(d*x + c) - 1)/(a*d*cos(d*x + c)^3*sin(d*x + c) + a*d*cos(d*x + c)^3)","A",0
55,1,47,0,0.930000," ","integrate(tan(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{\cos\left(d x + c\right)^{2} + 2 \, \sin\left(d x + c\right) + 1}{3 \, {\left(a d \cos\left(d x + c\right) \sin\left(d x + c\right) + a d \cos\left(d x + c\right)\right)}}"," ",0,"1/3*(cos(d*x + c)^2 + 2*sin(d*x + c) + 1)/(a*d*cos(d*x + c)*sin(d*x + c) + a*d*cos(d*x + c))","A",0
56,1,42,0,0.966434," ","integrate(1/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{\cos\left(d x + c\right) - \sin\left(d x + c\right) + 1}{a d \cos\left(d x + c\right) + a d \sin\left(d x + c\right) + a d}"," ",0,"-(cos(d*x + c) - sin(d*x + c) + 1)/(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d)","A",0
57,1,62,0,0.743812," ","integrate(cot(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{\log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right)}{2 \, a d \sin\left(d x + c\right)}"," ",0,"1/2*(log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 2*cos(d*x + c))/(a*d*sin(d*x + c))","B",0
58,1,111,0,1.105236," ","integrate(cot(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{4 \, \cos\left(d x + c\right)^{3} - 3 \, {\left(\cos\left(d x + c\right)^{2} - 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 3 \, {\left(\cos\left(d x + c\right)^{2} - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 6 \, \cos\left(d x + c\right) \sin\left(d x + c\right)}{12 \, {\left(a d \cos\left(d x + c\right)^{2} - a d\right)} \sin\left(d x + c\right)}"," ",0,"1/12*(4*cos(d*x + c)^3 - 3*(cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3*(cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 6*cos(d*x + c)*sin(d*x + c))/((a*d*cos(d*x + c)^2 - a*d)*sin(d*x + c))","B",0
59,1,155,0,1.067960," ","integrate(cot(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{16 \, \cos\left(d x + c\right)^{5} - 15 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 15 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 10 \, {\left(5 \, \cos\left(d x + c\right)^{3} - 3 \, \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{80 \, {\left(a d \cos\left(d x + c\right)^{4} - 2 \, a d \cos\left(d x + c\right)^{2} + a d\right)} \sin\left(d x + c\right)}"," ",0,"-1/80*(16*cos(d*x + c)^5 - 15*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 15*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 10*(5*cos(d*x + c)^3 - 3*cos(d*x + c))*sin(d*x + c))/((a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^2 + a*d)*sin(d*x + c))","B",0
60,1,198,0,0.613460," ","integrate(cot(d*x+c)^8/(a+a*sin(d*x+c)),x, algorithm=""fricas"")","\frac{96 \, \cos\left(d x + c\right)^{7} - 105 \, {\left(\cos\left(d x + c\right)^{6} - 3 \, \cos\left(d x + c\right)^{4} + 3 \, \cos\left(d x + c\right)^{2} - 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 105 \, {\left(\cos\left(d x + c\right)^{6} - 3 \, \cos\left(d x + c\right)^{4} + 3 \, \cos\left(d x + c\right)^{2} - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 14 \, {\left(33 \, \cos\left(d x + c\right)^{5} - 40 \, \cos\left(d x + c\right)^{3} + 15 \, \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{672 \, {\left(a d \cos\left(d x + c\right)^{6} - 3 \, a d \cos\left(d x + c\right)^{4} + 3 \, a d \cos\left(d x + c\right)^{2} - a d\right)} \sin\left(d x + c\right)}"," ",0,"1/672*(96*cos(d*x + c)^7 - 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 105*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 14*(33*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c))/((a*d*cos(d*x + c)^6 - 3*a*d*cos(d*x + c)^4 + 3*a*d*cos(d*x + c)^2 - a*d)*sin(d*x + c))","B",0
61,1,218,0,1.101520," ","integrate(tan(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{1500 \, \cos\left(d x + c\right)^{6} - 3380 \, \cos\left(d x + c\right)^{4} + 2104 \, \cos\left(d x + c\right)^{2} - 105 \, {\left(\cos\left(d x + c\right)^{8} - 2 \, \cos\left(d x + c\right)^{6} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right)^{6}\right)} \log\left(\sin\left(d x + c\right) + 1\right) + 105 \, {\left(\cos\left(d x + c\right)^{8} - 2 \, \cos\left(d x + c\right)^{6} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right)^{6}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(105 \, \cos\left(d x + c\right)^{6} + 500 \, \cos\left(d x + c\right)^{4} - 276 \, \cos\left(d x + c\right)^{2} + 64\right)} \sin\left(d x + c\right) - 512}{3840 \, {\left(a^{2} d \cos\left(d x + c\right)^{8} - 2 \, a^{2} d \cos\left(d x + c\right)^{6} \sin\left(d x + c\right) - 2 \, a^{2} d \cos\left(d x + c\right)^{6}\right)}}"," ",0,"1/3840*(1500*cos(d*x + c)^6 - 3380*cos(d*x + c)^4 + 2104*cos(d*x + c)^2 - 105*(cos(d*x + c)^8 - 2*cos(d*x + c)^6*sin(d*x + c) - 2*cos(d*x + c)^6)*log(sin(d*x + c) + 1) + 105*(cos(d*x + c)^8 - 2*cos(d*x + c)^6*sin(d*x + c) - 2*cos(d*x + c)^6)*log(-sin(d*x + c) + 1) - 2*(105*cos(d*x + c)^6 + 500*cos(d*x + c)^4 - 276*cos(d*x + c)^2 + 64)*sin(d*x + c) - 512)/(a^2*d*cos(d*x + c)^8 - 2*a^2*d*cos(d*x + c)^6*sin(d*x + c) - 2*a^2*d*cos(d*x + c)^6)","A",0
62,1,198,0,1.152174," ","integrate(tan(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","-\frac{132 \, \cos\left(d x + c\right)^{4} - 236 \, \cos\left(d x + c\right)^{2} - 15 \, {\left(\cos\left(d x + c\right)^{6} - 2 \, \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right)^{4}\right)} \log\left(\sin\left(d x + c\right) + 1\right) + 15 \, {\left(\cos\left(d x + c\right)^{6} - 2 \, \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right)^{4}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(15 \, \cos\left(d x + c\right)^{4} + 44 \, \cos\left(d x + c\right)^{2} - 12\right)} \sin\left(d x + c\right) + 72}{384 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} - 2 \, a^{2} d \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) - 2 \, a^{2} d \cos\left(d x + c\right)^{4}\right)}}"," ",0,"-1/384*(132*cos(d*x + c)^4 - 236*cos(d*x + c)^2 - 15*(cos(d*x + c)^6 - 2*cos(d*x + c)^4*sin(d*x + c) - 2*cos(d*x + c)^4)*log(sin(d*x + c) + 1) + 15*(cos(d*x + c)^6 - 2*cos(d*x + c)^4*sin(d*x + c) - 2*cos(d*x + c)^4)*log(-sin(d*x + c) + 1) - 2*(15*cos(d*x + c)^4 + 44*cos(d*x + c)^2 - 12)*sin(d*x + c) + 72)/(a^2*d*cos(d*x + c)^6 - 2*a^2*d*cos(d*x + c)^4*sin(d*x + c) - 2*a^2*d*cos(d*x + c)^4)","A",0
63,1,178,0,1.081579," ","integrate(tan(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{12 \, \cos\left(d x + c\right)^{2} - 3 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right)^{2}\right)} \log\left(\sin\left(d x + c\right) + 1\right) + 3 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right)^{2}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(3 \, \cos\left(d x + c\right)^{2} + 4\right)} \sin\left(d x + c\right) - 16}{48 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) - 2 \, a^{2} d \cos\left(d x + c\right)^{2}\right)}}"," ",0,"1/48*(12*cos(d*x + c)^2 - 3*(cos(d*x + c)^4 - 2*cos(d*x + c)^2*sin(d*x + c) - 2*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + 3*(cos(d*x + c)^4 - 2*cos(d*x + c)^2*sin(d*x + c) - 2*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(3*cos(d*x + c)^2 + 4)*sin(d*x + c) - 16)/(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2*sin(d*x + c) - 2*a^2*d*cos(d*x + c)^2)","A",0
64,1,104,0,1.127431," ","integrate(tan(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{{\left(\cos\left(d x + c\right)^{2} - 2 \, \sin\left(d x + c\right) - 2\right)} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(\cos\left(d x + c\right)^{2} - 2 \, \sin\left(d x + c\right) - 2\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, \sin\left(d x + c\right)}{8 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - 2 \, a^{2} d \sin\left(d x + c\right) - 2 \, a^{2} d\right)}}"," ",0,"1/8*((cos(d*x + c)^2 - 2*sin(d*x + c) - 2)*log(sin(d*x + c) + 1) - (cos(d*x + c)^2 - 2*sin(d*x + c) - 2)*log(-sin(d*x + c) + 1) + 2*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 - 2*a^2*d*sin(d*x + c) - 2*a^2*d)","A",0
65,1,59,0,1.181663," ","integrate(cot(d*x+c)/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{{\left(\sin\left(d x + c\right) + 1\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - {\left(\sin\left(d x + c\right) + 1\right)} \log\left(\sin\left(d x + c\right) + 1\right) + 1}{a^{2} d \sin\left(d x + c\right) + a^{2} d}"," ",0,"((sin(d*x + c) + 1)*log(1/2*sin(d*x + c)) - (sin(d*x + c) + 1)*log(sin(d*x + c) + 1) + 1)/(a^2*d*sin(d*x + c) + a^2*d)","A",0
66,1,76,0,1.268743," ","integrate(cot(d*x+c)^3/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{4 \, {\left(\cos\left(d x + c\right)^{2} - 1\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 4 \, {\left(\cos\left(d x + c\right)^{2} - 1\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 4 \, \sin\left(d x + c\right) + 1}{2 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)}}"," ",0,"1/2*(4*(cos(d*x + c)^2 - 1)*log(1/2*sin(d*x + c)) - 4*(cos(d*x + c)^2 - 1)*log(sin(d*x + c) + 1) - 4*sin(d*x + c) + 1)/(a^2*d*cos(d*x + c)^2 - a^2*d)","A",0
67,1,57,0,1.081161," ","integrate(cot(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{6 \, \cos\left(d x + c\right)^{2} + 8 \, \sin\left(d x + c\right) - 9}{12 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)}}"," ",0,"1/12*(6*cos(d*x + c)^2 + 8*sin(d*x + c) - 9)/(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)","A",0
68,1,94,0,1.094954," ","integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","-\frac{15 \, \cos\left(d x + c\right)^{4} - 30 \, \cos\left(d x + c\right)^{2} + 4 \, {\left(5 \, \cos\left(d x + c\right)^{2} - 2\right)} \sin\left(d x + c\right) + 10}{30 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} - 3 \, a^{2} d \cos\left(d x + c\right)^{4} + 3 \, a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)}}"," ",0,"-1/30*(15*cos(d*x + c)^4 - 30*cos(d*x + c)^2 + 4*(5*cos(d*x + c)^2 - 2)*sin(d*x + c) + 10)/(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)","A",0
69,1,127,0,1.114053," ","integrate(cot(d*x+c)^9/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{420 \, \cos\left(d x + c\right)^{6} - 1050 \, \cos\left(d x + c\right)^{4} + 700 \, \cos\left(d x + c\right)^{2} + 16 \, {\left(35 \, \cos\left(d x + c\right)^{4} - 28 \, \cos\left(d x + c\right)^{2} + 8\right)} \sin\left(d x + c\right) - 175}{840 \, {\left(a^{2} d \cos\left(d x + c\right)^{8} - 4 \, a^{2} d \cos\left(d x + c\right)^{6} + 6 \, a^{2} d \cos\left(d x + c\right)^{4} - 4 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)}}"," ",0,"1/840*(420*cos(d*x + c)^6 - 1050*cos(d*x + c)^4 + 700*cos(d*x + c)^2 + 16*(35*cos(d*x + c)^4 - 28*cos(d*x + c)^2 + 8)*sin(d*x + c) - 175)/(a^2*d*cos(d*x + c)^8 - 4*a^2*d*cos(d*x + c)^6 + 6*a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^2 + a^2*d)","A",0
70,1,162,0,1.166349," ","integrate(cot(d*x+c)^11/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","-\frac{630 \, \cos\left(d x + c\right)^{8} - 1890 \, \cos\left(d x + c\right)^{6} + 1890 \, \cos\left(d x + c\right)^{4} - 945 \, \cos\left(d x + c\right)^{2} + 8 \, {\left(105 \, \cos\left(d x + c\right)^{6} - 126 \, \cos\left(d x + c\right)^{4} + 72 \, \cos\left(d x + c\right)^{2} - 16\right)} \sin\left(d x + c\right) + 189}{1260 \, {\left(a^{2} d \cos\left(d x + c\right)^{10} - 5 \, a^{2} d \cos\left(d x + c\right)^{8} + 10 \, a^{2} d \cos\left(d x + c\right)^{6} - 10 \, a^{2} d \cos\left(d x + c\right)^{4} + 5 \, a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)}}"," ",0,"-1/1260*(630*cos(d*x + c)^8 - 1890*cos(d*x + c)^6 + 1890*cos(d*x + c)^4 - 945*cos(d*x + c)^2 + 8*(105*cos(d*x + c)^6 - 126*cos(d*x + c)^4 + 72*cos(d*x + c)^2 - 16)*sin(d*x + c) + 189)/(a^2*d*cos(d*x + c)^10 - 5*a^2*d*cos(d*x + c)^8 + 10*a^2*d*cos(d*x + c)^6 - 10*a^2*d*cos(d*x + c)^4 + 5*a^2*d*cos(d*x + c)^2 - a^2*d)","A",0
71,1,195,0,0.736783," ","integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{6930 \, \cos\left(d x + c\right)^{10} - 24255 \, \cos\left(d x + c\right)^{8} + 32340 \, \cos\left(d x + c\right)^{6} - 24255 \, \cos\left(d x + c\right)^{4} + 9702 \, \cos\left(d x + c\right)^{2} + 8 \, {\left(1155 \, \cos\left(d x + c\right)^{8} - 1848 \, \cos\left(d x + c\right)^{6} + 1584 \, \cos\left(d x + c\right)^{4} - 704 \, \cos\left(d x + c\right)^{2} + 128\right)} \sin\left(d x + c\right) - 1617}{13860 \, {\left(a^{2} d \cos\left(d x + c\right)^{12} - 6 \, a^{2} d \cos\left(d x + c\right)^{10} + 15 \, a^{2} d \cos\left(d x + c\right)^{8} - 20 \, a^{2} d \cos\left(d x + c\right)^{6} + 15 \, a^{2} d \cos\left(d x + c\right)^{4} - 6 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)}}"," ",0,"1/13860*(6930*cos(d*x + c)^10 - 24255*cos(d*x + c)^8 + 32340*cos(d*x + c)^6 - 24255*cos(d*x + c)^4 + 9702*cos(d*x + c)^2 + 8*(1155*cos(d*x + c)^8 - 1848*cos(d*x + c)^6 + 1584*cos(d*x + c)^4 - 704*cos(d*x + c)^2 + 128)*sin(d*x + c) - 1617)/(a^2*d*cos(d*x + c)^12 - 6*a^2*d*cos(d*x + c)^10 + 15*a^2*d*cos(d*x + c)^8 - 20*a^2*d*cos(d*x + c)^6 + 15*a^2*d*cos(d*x + c)^4 - 6*a^2*d*cos(d*x + c)^2 + a^2*d)","A",0
72,1,248,0,0.900758," ","integrate(tan(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{30 \, \cos\left(d x + c\right)^{6} + 1150 \, \cos\left(d x + c\right)^{4} - 2076 \, \cos\left(d x + c\right)^{2} - 15 \, {\left(3 \, \cos\left(d x + c\right)^{6} - 4 \, \cos\left(d x + c\right)^{4} + {\left(\cos\left(d x + c\right)^{6} - 4 \, \cos\left(d x + c\right)^{4}\right)} \sin\left(d x + c\right)\right)} \log\left(\sin\left(d x + c\right) + 1\right) + 15 \, {\left(3 \, \cos\left(d x + c\right)^{6} - 4 \, \cos\left(d x + c\right)^{4} + {\left(\cos\left(d x + c\right)^{6} - 4 \, \cos\left(d x + c\right)^{4}\right)} \sin\left(d x + c\right)\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 18 \, {\left(5 \, \cos\left(d x + c\right)^{4} + 50 \, \cos\left(d x + c\right)^{2} - 16\right)} \sin\left(d x + c\right) + 672}{3840 \, {\left(3 \, a^{3} d \cos\left(d x + c\right)^{6} - 4 \, a^{3} d \cos\left(d x + c\right)^{4} + {\left(a^{3} d \cos\left(d x + c\right)^{6} - 4 \, a^{3} d \cos\left(d x + c\right)^{4}\right)} \sin\left(d x + c\right)\right)}}"," ",0,"-1/3840*(30*cos(d*x + c)^6 + 1150*cos(d*x + c)^4 - 2076*cos(d*x + c)^2 - 15*(3*cos(d*x + c)^6 - 4*cos(d*x + c)^4 + (cos(d*x + c)^6 - 4*cos(d*x + c)^4)*sin(d*x + c))*log(sin(d*x + c) + 1) + 15*(3*cos(d*x + c)^6 - 4*cos(d*x + c)^4 + (cos(d*x + c)^6 - 4*cos(d*x + c)^4)*sin(d*x + c))*log(-sin(d*x + c) + 1) - 18*(5*cos(d*x + c)^4 + 50*cos(d*x + c)^2 - 16)*sin(d*x + c) + 672)/(3*a^3*d*cos(d*x + c)^6 - 4*a^3*d*cos(d*x + c)^4 + (a^3*d*cos(d*x + c)^6 - 4*a^3*d*cos(d*x + c)^4)*sin(d*x + c))","A",0
73,1,226,0,1.353199," ","integrate(tan(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","\frac{6 \, \cos\left(d x + c\right)^{4} + 38 \, \cos\left(d x + c\right)^{2} - 3 \, {\left(3 \, \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\sin\left(d x + c\right) + 1\right) + 3 \, {\left(3 \, \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 18 \, {\left(\cos\left(d x + c\right)^{2} + 2\right)} \sin\left(d x + c\right) - 60}{192 \, {\left(3 \, a^{3} d \cos\left(d x + c\right)^{4} - 4 \, a^{3} d \cos\left(d x + c\right)^{2} + {\left(a^{3} d \cos\left(d x + c\right)^{4} - 4 \, a^{3} d \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/192*(6*cos(d*x + c)^4 + 38*cos(d*x + c)^2 - 3*(3*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + (cos(d*x + c)^4 - 4*cos(d*x + c)^2)*sin(d*x + c))*log(sin(d*x + c) + 1) + 3*(3*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + (cos(d*x + c)^4 - 4*cos(d*x + c)^2)*sin(d*x + c))*log(-sin(d*x + c) + 1) - 18*(cos(d*x + c)^2 + 2)*sin(d*x + c) - 60)/(3*a^3*d*cos(d*x + c)^4 - 4*a^3*d*cos(d*x + c)^2 + (a^3*d*cos(d*x + c)^4 - 4*a^3*d*cos(d*x + c)^2)*sin(d*x + c))","A",0
74,1,154,0,1.170783," ","integrate(tan(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{6 \, \cos\left(d x + c\right)^{2} - 3 \, {\left(3 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 4\right)} \sin\left(d x + c\right) - 4\right)} \log\left(\sin\left(d x + c\right) + 1\right) + 3 \, {\left(3 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 4\right)} \sin\left(d x + c\right) - 4\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 18 \, \sin\left(d x + c\right) - 10}{48 \, {\left(3 \, a^{3} d \cos\left(d x + c\right)^{2} - 4 \, a^{3} d + {\left(a^{3} d \cos\left(d x + c\right)^{2} - 4 \, a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"-1/48*(6*cos(d*x + c)^2 - 3*(3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 4)*sin(d*x + c) - 4)*log(sin(d*x + c) + 1) + 3*(3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 4)*sin(d*x + c) - 4)*log(-sin(d*x + c) + 1) - 18*sin(d*x + c) - 10)/(3*a^3*d*cos(d*x + c)^2 - 4*a^3*d + (a^3*d*cos(d*x + c)^2 - 4*a^3*d)*sin(d*x + c))","B",0
75,1,104,0,1.126446," ","integrate(cot(d*x+c)/(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","\frac{2 \, {\left(\cos\left(d x + c\right)^{2} - 2 \, \sin\left(d x + c\right) - 2\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 2 \, {\left(\cos\left(d x + c\right)^{2} - 2 \, \sin\left(d x + c\right) - 2\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 2 \, \sin\left(d x + c\right) - 3}{2 \, {\left(a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \sin\left(d x + c\right) - 2 \, a^{3} d\right)}}"," ",0,"1/2*(2*(cos(d*x + c)^2 - 2*sin(d*x + c) - 2)*log(1/2*sin(d*x + c)) - 2*(cos(d*x + c)^2 - 2*sin(d*x + c) - 2)*log(sin(d*x + c) + 1) - 2*sin(d*x + c) - 3)/(a^3*d*cos(d*x + c)^2 - 2*a^3*d*sin(d*x + c) - 2*a^3*d)","A",0
76,1,147,0,1.085575," ","integrate(cot(d*x+c)^3/(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","\frac{10 \, \cos\left(d x + c\right)^{2} + 10 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - 1\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 10 \, {\left(\cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) - 1\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 5 \, \sin\left(d x + c\right) - 9}{2 \, {\left(a^{3} d \cos\left(d x + c\right)^{2} - a^{3} d + {\left(a^{3} d \cos\left(d x + c\right)^{2} - a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/2*(10*cos(d*x + c)^2 + 10*(cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - 1)*log(1/2*sin(d*x + c)) - 10*(cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - 1)*log(sin(d*x + c) + 1) - 5*sin(d*x + c) - 9)/(a^3*d*cos(d*x + c)^2 - a^3*d + (a^3*d*cos(d*x + c)^2 - a^3*d)*sin(d*x + c))","A",0
77,1,131,0,1.108781," ","integrate(cot(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","\frac{8 \, \cos\left(d x + c\right)^{2} + 16 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 16 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 4 \, {\left(4 \, \cos\left(d x + c\right)^{2} - 5\right)} \sin\left(d x + c\right) - 9}{4 \, {\left(a^{3} d \cos\left(d x + c\right)^{4} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} + a^{3} d\right)}}"," ",0,"1/4*(8*cos(d*x + c)^2 + 16*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(1/2*sin(d*x + c)) - 16*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(sin(d*x + c) + 1) - 4*(4*cos(d*x + c)^2 - 5)*sin(d*x + c) - 9)/(a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)","A",0
78,1,84,0,1.201834," ","integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{45 \, \cos\left(d x + c\right)^{2} - 4 \, {\left(5 \, \cos\left(d x + c\right)^{2} - 14\right)} \sin\left(d x + c\right) - 55}{60 \, {\left(a^{3} d \cos\left(d x + c\right)^{6} - 3 \, a^{3} d \cos\left(d x + c\right)^{4} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} - a^{3} d\right)}}"," ",0,"-1/60*(45*cos(d*x + c)^2 - 4*(5*cos(d*x + c)^2 - 14)*sin(d*x + c) - 55)/(a^3*d*cos(d*x + c)^6 - 3*a^3*d*cos(d*x + c)^4 + 3*a^3*d*cos(d*x + c)^2 - a^3*d)","A",0
79,1,117,0,0.786360," ","integrate(cot(d*x+c)^9/(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","\frac{630 \, \cos\left(d x + c\right)^{4} - 980 \, \cos\left(d x + c\right)^{2} - 8 \, {\left(35 \, \cos\left(d x + c\right)^{4} - 112 \, \cos\left(d x + c\right)^{2} + 32\right)} \sin\left(d x + c\right) + 245}{840 \, {\left(a^{3} d \cos\left(d x + c\right)^{8} - 4 \, a^{3} d \cos\left(d x + c\right)^{6} + 6 \, a^{3} d \cos\left(d x + c\right)^{4} - 4 \, a^{3} d \cos\left(d x + c\right)^{2} + a^{3} d\right)}}"," ",0,"1/840*(630*cos(d*x + c)^4 - 980*cos(d*x + c)^2 - 8*(35*cos(d*x + c)^4 - 112*cos(d*x + c)^2 + 32)*sin(d*x + c) + 245)/(a^3*d*cos(d*x + c)^8 - 4*a^3*d*cos(d*x + c)^6 + 6*a^3*d*cos(d*x + c)^4 - 4*a^3*d*cos(d*x + c)^2 + a^3*d)","A",0
80,1,152,0,0.795874," ","integrate(cot(d*x+c)^11/(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{630 \, \cos\left(d x + c\right)^{6} - 1190 \, \cos\left(d x + c\right)^{4} + 595 \, \cos\left(d x + c\right)^{2} - 8 \, {\left(35 \, \cos\left(d x + c\right)^{6} - 126 \, \cos\left(d x + c\right)^{4} + 72 \, \cos\left(d x + c\right)^{2} - 16\right)} \sin\left(d x + c\right) - 119}{840 \, {\left(a^{3} d \cos\left(d x + c\right)^{10} - 5 \, a^{3} d \cos\left(d x + c\right)^{8} + 10 \, a^{3} d \cos\left(d x + c\right)^{6} - 10 \, a^{3} d \cos\left(d x + c\right)^{4} + 5 \, a^{3} d \cos\left(d x + c\right)^{2} - a^{3} d\right)}}"," ",0,"-1/840*(630*cos(d*x + c)^6 - 1190*cos(d*x + c)^4 + 595*cos(d*x + c)^2 - 8*(35*cos(d*x + c)^6 - 126*cos(d*x + c)^4 + 72*cos(d*x + c)^2 - 16)*sin(d*x + c) - 119)/(a^3*d*cos(d*x + c)^10 - 5*a^3*d*cos(d*x + c)^8 + 10*a^3*d*cos(d*x + c)^6 - 10*a^3*d*cos(d*x + c)^4 + 5*a^3*d*cos(d*x + c)^2 - a^3*d)","A",0
81,1,185,0,1.216581," ","integrate(cot(d*x+c)^13/(a+a*sin(d*x+c))^3,x, algorithm=""fricas"")","\frac{2079 \, \cos\left(d x + c\right)^{8} - 4620 \, \cos\left(d x + c\right)^{6} + 3465 \, \cos\left(d x + c\right)^{4} - 1386 \, \cos\left(d x + c\right)^{2} - 4 \, {\left(231 \, \cos\left(d x + c\right)^{8} - 924 \, \cos\left(d x + c\right)^{6} + 792 \, \cos\left(d x + c\right)^{4} - 352 \, \cos\left(d x + c\right)^{2} + 64\right)} \sin\left(d x + c\right) + 231}{2772 \, {\left(a^{3} d \cos\left(d x + c\right)^{12} - 6 \, a^{3} d \cos\left(d x + c\right)^{10} + 15 \, a^{3} d \cos\left(d x + c\right)^{8} - 20 \, a^{3} d \cos\left(d x + c\right)^{6} + 15 \, a^{3} d \cos\left(d x + c\right)^{4} - 6 \, a^{3} d \cos\left(d x + c\right)^{2} + a^{3} d\right)}}"," ",0,"1/2772*(2079*cos(d*x + c)^8 - 4620*cos(d*x + c)^6 + 3465*cos(d*x + c)^4 - 1386*cos(d*x + c)^2 - 4*(231*cos(d*x + c)^8 - 924*cos(d*x + c)^6 + 792*cos(d*x + c)^4 - 352*cos(d*x + c)^2 + 64)*sin(d*x + c) + 231)/(a^3*d*cos(d*x + c)^12 - 6*a^3*d*cos(d*x + c)^10 + 15*a^3*d*cos(d*x + c)^8 - 20*a^3*d*cos(d*x + c)^6 + 15*a^3*d*cos(d*x + c)^4 - 6*a^3*d*cos(d*x + c)^2 + a^3*d)","A",0
82,1,290,0,1.193546," ","integrate(tan(d*x+c)^5/(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","-\frac{120 \, \cos\left(d x + c\right)^{6} - 1240 \, \cos\left(d x + c\right)^{4} + 1856 \, \cos\left(d x + c\right)^{2} + 15 \, {\left(\cos\left(d x + c\right)^{8} - 8 \, \cos\left(d x + c\right)^{6} + 8 \, \cos\left(d x + c\right)^{4} - 4 \, {\left(\cos\left(d x + c\right)^{6} - 2 \, \cos\left(d x + c\right)^{4}\right)} \sin\left(d x + c\right)\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 15 \, {\left(\cos\left(d x + c\right)^{8} - 8 \, \cos\left(d x + c\right)^{6} + 8 \, \cos\left(d x + c\right)^{4} - 4 \, {\left(\cos\left(d x + c\right)^{6} - 2 \, \cos\left(d x + c\right)^{4}\right)} \sin\left(d x + c\right)\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(15 \, \cos\left(d x + c\right)^{6} - 110 \, \cos\left(d x + c\right)^{4} + 432 \, \cos\left(d x + c\right)^{2} - 160\right)} \sin\left(d x + c\right) - 640}{3840 \, {\left(a^{4} d \cos\left(d x + c\right)^{8} - 8 \, a^{4} d \cos\left(d x + c\right)^{6} + 8 \, a^{4} d \cos\left(d x + c\right)^{4} - 4 \, {\left(a^{4} d \cos\left(d x + c\right)^{6} - 2 \, a^{4} d \cos\left(d x + c\right)^{4}\right)} \sin\left(d x + c\right)\right)}}"," ",0,"-1/3840*(120*cos(d*x + c)^6 - 1240*cos(d*x + c)^4 + 1856*cos(d*x + c)^2 + 15*(cos(d*x + c)^8 - 8*cos(d*x + c)^6 + 8*cos(d*x + c)^4 - 4*(cos(d*x + c)^6 - 2*cos(d*x + c)^4)*sin(d*x + c))*log(sin(d*x + c) + 1) - 15*(cos(d*x + c)^8 - 8*cos(d*x + c)^6 + 8*cos(d*x + c)^4 - 4*(cos(d*x + c)^6 - 2*cos(d*x + c)^4)*sin(d*x + c))*log(-sin(d*x + c) + 1) + 2*(15*cos(d*x + c)^6 - 110*cos(d*x + c)^4 + 432*cos(d*x + c)^2 - 160)*sin(d*x + c) - 640)/(a^4*d*cos(d*x + c)^8 - 8*a^4*d*cos(d*x + c)^6 + 8*a^4*d*cos(d*x + c)^4 - 4*(a^4*d*cos(d*x + c)^6 - 2*a^4*d*cos(d*x + c)^4)*sin(d*x + c))","A",0
83,1,102,0,1.145862," ","integrate(tan(d*x+c)^3/(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","-\frac{5 \, \cos\left(d x + c\right)^{2} - 4 \, \sin\left(d x + c\right) - 6}{20 \, {\left(a^{4} d \cos\left(d x + c\right)^{6} - 8 \, a^{4} d \cos\left(d x + c\right)^{4} + 8 \, a^{4} d \cos\left(d x + c\right)^{2} - 4 \, {\left(a^{4} d \cos\left(d x + c\right)^{4} - 2 \, a^{4} d \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)}}"," ",0,"-1/20*(5*cos(d*x + c)^2 - 4*sin(d*x + c) - 6)/(a^4*d*cos(d*x + c)^6 - 8*a^4*d*cos(d*x + c)^4 + 8*a^4*d*cos(d*x + c)^2 - 4*(a^4*d*cos(d*x + c)^4 - 2*a^4*d*cos(d*x + c)^2)*sin(d*x + c))","A",0
84,1,198,0,1.164980," ","integrate(tan(d*x+c)/(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","\frac{24 \, \cos\left(d x + c\right)^{2} + 3 \, {\left(\cos\left(d x + c\right)^{4} - 8 \, \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} - 2\right)} \sin\left(d x + c\right) + 8\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 3 \, {\left(\cos\left(d x + c\right)^{4} - 8 \, \cos\left(d x + c\right)^{2} - 4 \, {\left(\cos\left(d x + c\right)^{2} - 2\right)} \sin\left(d x + c\right) + 8\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(3 \, \cos\left(d x + c\right)^{2} - 22\right)} \sin\left(d x + c\right) - 32}{96 \, {\left(a^{4} d \cos\left(d x + c\right)^{4} - 8 \, a^{4} d \cos\left(d x + c\right)^{2} + 8 \, a^{4} d - 4 \, {\left(a^{4} d \cos\left(d x + c\right)^{2} - 2 \, a^{4} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/96*(24*cos(d*x + c)^2 + 3*(cos(d*x + c)^4 - 8*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)*log(sin(d*x + c) + 1) - 3*(cos(d*x + c)^4 - 8*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)*log(-sin(d*x + c) + 1) + 2*(3*cos(d*x + c)^2 - 22)*sin(d*x + c) - 32)/(a^4*d*cos(d*x + c)^4 - 8*a^4*d*cos(d*x + c)^2 + 8*a^4*d - 4*(a^4*d*cos(d*x + c)^2 - 2*a^4*d)*sin(d*x + c))","B",0
85,1,196,0,1.322672," ","integrate(cot(d*x+c)^3/(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","-\frac{27 \, \cos\left(d x + c\right)^{2} - 18 \, {\left(\cos\left(d x + c\right)^{4} - 3 \, \cos\left(d x + c\right)^{2} - 2 \, {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) + 2\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) + 18 \, {\left(\cos\left(d x + c\right)^{4} - 3 \, \cos\left(d x + c\right)^{2} - 2 \, {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right) + 2\right)} \log\left(\sin\left(d x + c\right) + 1\right) + 6 \, {\left(3 \, \cos\left(d x + c\right)^{2} - 4\right)} \sin\left(d x + c\right) - 26}{2 \, {\left(a^{4} d \cos\left(d x + c\right)^{4} - 3 \, a^{4} d \cos\left(d x + c\right)^{2} + 2 \, a^{4} d - 2 \, {\left(a^{4} d \cos\left(d x + c\right)^{2} - a^{4} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"-1/2*(27*cos(d*x + c)^2 - 18*(cos(d*x + c)^4 - 3*cos(d*x + c)^2 - 2*(cos(d*x + c)^2 - 1)*sin(d*x + c) + 2)*log(1/2*sin(d*x + c)) + 18*(cos(d*x + c)^4 - 3*cos(d*x + c)^2 - 2*(cos(d*x + c)^2 - 1)*sin(d*x + c) + 2)*log(sin(d*x + c) + 1) + 6*(3*cos(d*x + c)^2 - 4)*sin(d*x + c) - 26)/(a^4*d*cos(d*x + c)^4 - 3*a^4*d*cos(d*x + c)^2 + 2*a^4*d - 2*(a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x + c))","A",0
86,1,186,0,1.357603," ","integrate(cot(d*x+c)^7/(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","\frac{240 \, \cos\left(d x + c\right)^{4} - 585 \, \cos\left(d x + c\right)^{2} + 480 \, {\left(\cos\left(d x + c\right)^{6} - 3 \, \cos\left(d x + c\right)^{4} + 3 \, \cos\left(d x + c\right)^{2} - 1\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 480 \, {\left(\cos\left(d x + c\right)^{6} - 3 \, \cos\left(d x + c\right)^{4} + 3 \, \cos\left(d x + c\right)^{2} - 1\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 16 \, {\left(30 \, \cos\left(d x + c\right)^{4} - 70 \, \cos\left(d x + c\right)^{2} + 43\right)} \sin\left(d x + c\right) + 355}{60 \, {\left(a^{4} d \cos\left(d x + c\right)^{6} - 3 \, a^{4} d \cos\left(d x + c\right)^{4} + 3 \, a^{4} d \cos\left(d x + c\right)^{2} - a^{4} d\right)}}"," ",0,"1/60*(240*cos(d*x + c)^4 - 585*cos(d*x + c)^2 + 480*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(1/2*sin(d*x + c)) - 480*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(sin(d*x + c) + 1) - 16*(30*cos(d*x + c)^4 - 70*cos(d*x + c)^2 + 43)*sin(d*x + c) + 355)/(a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)","A",0
87,1,129,0,1.076353," ","integrate(tan(d*x+c)^2/(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","\frac{88 \, \cos\left(d x + c\right)^{4} - 220 \, \cos\left(d x + c\right)^{2} + {\left(22 \, \cos\left(d x + c\right)^{4} - 165 \, \cos\left(d x + c\right)^{2} + 175\right)} \sin\left(d x + c\right) + 140}{315 \, {\left(a^{4} d \cos\left(d x + c\right)^{5} - 8 \, a^{4} d \cos\left(d x + c\right)^{3} + 8 \, a^{4} d \cos\left(d x + c\right) - 4 \, {\left(a^{4} d \cos\left(d x + c\right)^{3} - 2 \, a^{4} d \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/315*(88*cos(d*x + c)^4 - 220*cos(d*x + c)^2 + (22*cos(d*x + c)^4 - 165*cos(d*x + c)^2 + 175)*sin(d*x + c) + 140)/(a^4*d*cos(d*x + c)^5 - 8*a^4*d*cos(d*x + c)^3 + 8*a^4*d*cos(d*x + c) - 4*(a^4*d*cos(d*x + c)^3 - 2*a^4*d*cos(d*x + c))*sin(d*x + c))","A",0
88,1,369,0,1.250840," ","integrate(cot(d*x+c)^2/(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","\frac{94 \, \cos\left(d x + c\right)^{4} + 222 \, \cos\left(d x + c\right)^{3} - 115 \, \cos\left(d x + c\right)^{2} + 30 \, {\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)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 30 \, {\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)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left(94 \, \cos\left(d x + c\right)^{3} - 128 \, \cos\left(d x + c\right)^{2} - 243 \, \cos\left(d x + c\right) - 6\right)} \sin\left(d x + c\right) - 237 \, \cos\left(d x + c\right) + 6}{15 \, {\left(a^{4} d \cos\left(d x + c\right)^{4} - 2 \, a^{4} d \cos\left(d x + c\right)^{3} - 5 \, a^{4} d \cos\left(d x + c\right)^{2} + 2 \, a^{4} d \cos\left(d x + c\right) + 4 \, a^{4} d - {\left(a^{4} d \cos\left(d x + c\right)^{3} + 3 \, a^{4} d \cos\left(d x + c\right)^{2} - 2 \, a^{4} d \cos\left(d x + c\right) - 4 \, a^{4} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/15*(94*cos(d*x + c)^4 + 222*cos(d*x + c)^3 - 115*cos(d*x + c)^2 + 30*(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)*log(1/2*cos(d*x + c) + 1/2) - 30*(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)*log(-1/2*cos(d*x + c) + 1/2) + (94*cos(d*x + c)^3 - 128*cos(d*x + c)^2 - 243*cos(d*x + c) - 6)*sin(d*x + c) - 237*cos(d*x + c) + 6)/(a^4*d*cos(d*x + c)^4 - 2*a^4*d*cos(d*x + c)^3 - 5*a^4*d*cos(d*x + c)^2 + 2*a^4*d*cos(d*x + c) + 4*a^4*d - (a^4*d*cos(d*x + c)^3 + 3*a^4*d*cos(d*x + c)^2 - 2*a^4*d*cos(d*x + c) - 4*a^4*d)*sin(d*x + c))","B",0
89,1,445,0,0.929289," ","integrate(cot(d*x+c)^4/(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","-\frac{66 \, \cos\left(d x + c\right)^{5} - 24 \, \cos\left(d x + c\right)^{4} - 147 \, \cos\left(d x + c\right)^{3} + 29 \, \cos\left(d x + c\right)^{2} - 21 \, {\left(\cos\left(d x + c\right)^{5} + 2 \, \cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{3} - 4 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{4} - \cos\left(d x + c\right)^{3} - 3 \, \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)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 21 \, {\left(\cos\left(d x + c\right)^{5} + 2 \, \cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{3} - 4 \, \cos\left(d x + c\right)^{2} + {\left(\cos\left(d x + c\right)^{4} - \cos\left(d x + c\right)^{3} - 3 \, \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)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - {\left(66 \, \cos\left(d x + c\right)^{4} + 90 \, \cos\left(d x + c\right)^{3} - 57 \, \cos\left(d x + c\right)^{2} - 86 \, \cos\left(d x + c\right) - 4\right)} \sin\left(d x + c\right) + 82 \, \cos\left(d x + c\right) - 4}{3 \, {\left(a^{4} d \cos\left(d x + c\right)^{5} + 2 \, a^{4} d \cos\left(d x + c\right)^{4} - 2 \, a^{4} d \cos\left(d x + c\right)^{3} - 4 \, a^{4} d \cos\left(d x + c\right)^{2} + a^{4} d \cos\left(d x + c\right) + 2 \, a^{4} d + {\left(a^{4} d \cos\left(d x + c\right)^{4} - a^{4} d \cos\left(d x + c\right)^{3} - 3 \, a^{4} d \cos\left(d x + c\right)^{2} + a^{4} d \cos\left(d x + c\right) + 2 \, a^{4} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"-1/3*(66*cos(d*x + c)^5 - 24*cos(d*x + c)^4 - 147*cos(d*x + c)^3 + 29*cos(d*x + c)^2 - 21*(cos(d*x + c)^5 + 2*cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 4*cos(d*x + c)^2 + (cos(d*x + c)^4 - cos(d*x + c)^3 - 3*cos(d*x + c)^2 + cos(d*x + c) + 2)*sin(d*x + c) + cos(d*x + c) + 2)*log(1/2*cos(d*x + c) + 1/2) + 21*(cos(d*x + c)^5 + 2*cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 4*cos(d*x + c)^2 + (cos(d*x + c)^4 - cos(d*x + c)^3 - 3*cos(d*x + c)^2 + cos(d*x + c) + 2)*sin(d*x + c) + cos(d*x + c) + 2)*log(-1/2*cos(d*x + c) + 1/2) - (66*cos(d*x + c)^4 + 90*cos(d*x + c)^3 - 57*cos(d*x + c)^2 - 86*cos(d*x + c) - 4)*sin(d*x + c) + 82*cos(d*x + c) - 4)/(a^4*d*cos(d*x + c)^5 + 2*a^4*d*cos(d*x + c)^4 - 2*a^4*d*cos(d*x + c)^3 - 4*a^4*d*cos(d*x + c)^2 + a^4*d*cos(d*x + c) + 2*a^4*d + (a^4*d*cos(d*x + c)^4 - a^4*d*cos(d*x + c)^3 - 3*a^4*d*cos(d*x + c)^2 + a^4*d*cos(d*x + c) + 2*a^4*d)*sin(d*x + c))","B",0
90,1,439,0,0.882669," ","integrate(cot(d*x+c)^6/(a+a*sin(d*x+c))^4,x, algorithm=""fricas"")","\frac{424 \, \cos\left(d x + c\right)^{6} + 154 \, \cos\left(d x + c\right)^{5} - 1060 \, \cos\left(d x + c\right)^{4} - 340 \, \cos\left(d x + c\right)^{3} + 800 \, \cos\left(d x + c\right)^{2} + 135 \, {\left(\cos\left(d x + c\right)^{6} - 3 \, \cos\left(d x + c\right)^{4} + 3 \, \cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right)^{5} + \cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right) + 1\right)} \sin\left(d x + c\right) - 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 135 \, {\left(\cos\left(d x + c\right)^{6} - 3 \, \cos\left(d x + c\right)^{4} + 3 \, \cos\left(d x + c\right)^{2} - {\left(\cos\left(d x + c\right)^{5} + \cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right) + 1\right)} \sin\left(d x + c\right) - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 2 \, {\left(212 \, \cos\left(d x + c\right)^{5} + 135 \, \cos\left(d x + c\right)^{4} - 395 \, \cos\left(d x + c\right)^{3} - 225 \, \cos\left(d x + c\right)^{2} + 175 \, \cos\left(d x + c\right) + 80\right)} \sin\left(d x + c\right) + 190 \, \cos\left(d x + c\right) - 160}{20 \, {\left(a^{4} d \cos\left(d x + c\right)^{6} - 3 \, a^{4} d \cos\left(d x + c\right)^{4} + 3 \, a^{4} d \cos\left(d x + c\right)^{2} - a^{4} d - {\left(a^{4} d \cos\left(d x + c\right)^{5} + a^{4} d \cos\left(d x + c\right)^{4} - 2 \, a^{4} d \cos\left(d x + c\right)^{3} - 2 \, a^{4} d \cos\left(d x + c\right)^{2} + a^{4} d \cos\left(d x + c\right) + a^{4} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/20*(424*cos(d*x + c)^6 + 154*cos(d*x + c)^5 - 1060*cos(d*x + c)^4 - 340*cos(d*x + c)^3 + 800*cos(d*x + c)^2 + 135*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - (cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 + cos(d*x + c) + 1)*sin(d*x + c) - 1)*log(1/2*cos(d*x + c) + 1/2) - 135*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - (cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 + cos(d*x + c) + 1)*sin(d*x + c) - 1)*log(-1/2*cos(d*x + c) + 1/2) + 2*(212*cos(d*x + c)^5 + 135*cos(d*x + c)^4 - 395*cos(d*x + c)^3 - 225*cos(d*x + c)^2 + 175*cos(d*x + c) + 80)*sin(d*x + c) + 190*cos(d*x + c) - 160)/(a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d - (a^4*d*cos(d*x + c)^5 + a^4*d*cos(d*x + c)^4 - 2*a^4*d*cos(d*x + c)^3 - 2*a^4*d*cos(d*x + c)^2 + a^4*d*cos(d*x + c) + a^4*d)*sin(d*x + c))","B",0
91,1,200,0,0.818012," ","integrate((a+a*sin(f*x+e))^(1/2)*tan(f*x+e)^4,x, algorithm=""fricas"")","\frac{33 \, \sqrt{2} \sqrt{a} \cos\left(f x + e\right)^{3} \log\left(-\frac{a \cos\left(f x + e\right)^{2} + 2 \, \sqrt{a \sin\left(f x + e\right) + a} {\left(\sqrt{2} \cos\left(f x + e\right) - \sqrt{2} \sin\left(f x + e\right) + \sqrt{2}\right)} \sqrt{a} + 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(81 \, \cos\left(f x + e\right)^{2} - 2 \, {\left(24 \, \cos\left(f x + e\right)^{2} + 5\right)} \sin\left(f x + e\right) + 2\right)} \sqrt{a \sin\left(f x + e\right) + a}}{96 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/96*(33*sqrt(2)*sqrt(a)*cos(f*x + e)^3*log(-(a*cos(f*x + e)^2 + 2*sqrt(a*sin(f*x + e) + a)*(sqrt(2)*cos(f*x + e) - sqrt(2)*sin(f*x + e) + sqrt(2))*sqrt(a) + 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*(81*cos(f*x + e)^2 - 2*(24*cos(f*x + e)^2 + 5)*sin(f*x + e) + 2)*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)^3)","A",0
92,1,169,0,0.880622," ","integrate((a+a*sin(f*x+e))^(1/2)*tan(f*x+e)^2,x, algorithm=""fricas"")","\frac{\sqrt{2} \sqrt{a} \cos\left(f x + e\right) \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(2 \, \sin\left(f x + e\right) - 3\right)}}{4 \, f \cos\left(f x + e\right)}"," ",0,"1/4*(sqrt(2)*sqrt(a)*cos(f*x + e)*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)*(2*sin(f*x + e) - 3))/(f*cos(f*x + e))","A",0
93,1,279,0,1.091936," ","integrate(cot(f*x+e)^2*(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{{\left(\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 1\right)} \sin\left(f x + e\right) - 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} - 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right) - 4 \, {\left(2 \, \cos\left(f x + e\right)^{2} + {\left(2 \, \cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a}}{4 \, {\left(f \cos\left(f x + e\right)^{2} - {\left(f \cos\left(f x + e\right) + f\right)} \sin\left(f x + e\right) - f\right)}}"," ",0,"1/4*((cos(f*x + e)^2 - (cos(f*x + e) + 1)*sin(f*x + e) - 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) - 4*(2*cos(f*x + e)^2 + (2*cos(f*x + e) + 3)*sin(f*x + e) - cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)^2 - (f*cos(f*x + e) + f)*sin(f*x + e) - f)","B",0
94,1,380,0,0.911945," ","integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{33 \, {\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} - \cos\left(f x + e\right) - 1\right)} \sin\left(f x + e\right) + 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} + 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right) + 4 \, {\left(48 \, \cos\left(f x + e\right)^{4} - 33 \, \cos\left(f x + e\right)^{3} - 139 \, \cos\left(f x + e\right)^{2} + {\left(48 \, \cos\left(f x + e\right)^{3} + 81 \, \cos\left(f x + e\right)^{2} - 58 \, \cos\left(f x + e\right) - 83\right)} \sin\left(f x + e\right) + 25 \, \cos\left(f x + e\right) + 83\right)} \sqrt{a \sin\left(f x + e\right) + a}}{96 \, {\left(f \cos\left(f x + e\right)^{4} - 2 \, f \cos\left(f x + e\right)^{2} - {\left(f \cos\left(f x + e\right)^{3} + f \cos\left(f x + e\right)^{2} - f \cos\left(f x + e\right) - f\right)} \sin\left(f x + e\right) + f\right)}}"," ",0,"1/96*(33*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 - (cos(f*x + e)^3 + cos(f*x + e)^2 - cos(f*x + e) - 1)*sin(f*x + e) + 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) + 4*(48*cos(f*x + e)^4 - 33*cos(f*x + e)^3 - 139*cos(f*x + e)^2 + (48*cos(f*x + e)^3 + 81*cos(f*x + e)^2 - 58*cos(f*x + e) - 83)*sin(f*x + e) + 25*cos(f*x + e) + 83)*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 - (f*cos(f*x + e)^3 + f*cos(f*x + e)^2 - f*cos(f*x + e) - f)*sin(f*x + e) + f)","B",0
95,1,239,0,1.116819," ","integrate((a+a*sin(f*x+e))^(3/2)*tan(f*x+e)^4,x, algorithm=""fricas"")","\frac{3 \, {\left(\sqrt{2} a \cos\left(f x + e\right) \sin\left(f x + e\right) - \sqrt{2} a \cos\left(f x + e\right)\right)} \sqrt{a} \log\left(-\frac{a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{a \sin\left(f x + e\right) + a} {\left(\sqrt{2} \cos\left(f x + e\right) - \sqrt{2} \sin\left(f x + e\right) + \sqrt{2}\right)} \sqrt{a} + 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(12 \, a \cos\left(f x + e\right)^{2} + {\left(4 \, a \cos\left(f x + e\right)^{2} + 53 \, a\right)} \sin\left(f x + e\right) - 51 \, a\right)} \sqrt{a \sin\left(f x + e\right) + a}}{24 \, {\left(f \cos\left(f x + e\right) \sin\left(f x + e\right) - f \cos\left(f x + e\right)\right)}}"," ",0,"1/24*(3*(sqrt(2)*a*cos(f*x + e)*sin(f*x + e) - sqrt(2)*a*cos(f*x + e))*sqrt(a)*log(-(a*cos(f*x + e)^2 - 2*sqrt(a*sin(f*x + e) + a)*(sqrt(2)*cos(f*x + e) - sqrt(2)*sin(f*x + e) + sqrt(2))*sqrt(a) + 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*(12*a*cos(f*x + e)^2 + (4*a*cos(f*x + e)^2 + 53*a)*sin(f*x + e) - 51*a)*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)*sin(f*x + e) - f*cos(f*x + e))","A",0
96,1,48,0,0.986511," ","integrate((a+a*sin(f*x+e))^(3/2)*tan(f*x+e)^2,x, algorithm=""fricas"")","\frac{2 \, {\left(a \cos\left(f x + e\right)^{2} - 4 \, a \sin\left(f x + e\right) + 7 \, a\right)} \sqrt{a \sin\left(f x + e\right) + a}}{3 \, f \cos\left(f x + e\right)}"," ",0,"2/3*(a*cos(f*x + e)^2 - 4*a*sin(f*x + e) + 7*a)*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e))","A",0
97,1,315,0,1.224200," ","integrate(cot(f*x+e)^2*(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{9 \, {\left(a \cos\left(f x + e\right)^{2} - {\left(a \cos\left(f x + e\right) + a\right)} \sin\left(f x + e\right) - a\right)} \sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} - 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right) + 4 \, {\left(2 \, a \cos\left(f x + e\right)^{3} - 8 \, a \cos\left(f x + e\right)^{2} + a \cos\left(f x + e\right) - {\left(2 \, a \cos\left(f x + e\right)^{2} + 10 \, a \cos\left(f x + e\right) + 11 \, a\right)} \sin\left(f x + e\right) + 11 \, a\right)} \sqrt{a \sin\left(f x + e\right) + a}}{12 \, {\left(f \cos\left(f x + e\right)^{2} - {\left(f \cos\left(f x + e\right) + f\right)} \sin\left(f x + e\right) - f\right)}}"," ",0,"1/12*(9*(a*cos(f*x + e)^2 - (a*cos(f*x + e) + a)*sin(f*x + e) - a)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) + 4*(2*a*cos(f*x + e)^3 - 8*a*cos(f*x + e)^2 + a*cos(f*x + e) - (2*a*cos(f*x + e)^2 + 10*a*cos(f*x + e) + 11*a)*sin(f*x + e) + 11*a)*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)^2 - (f*cos(f*x + e) + f)*sin(f*x + e) - f)","B",0
98,1,424,0,1.153635," ","integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{111 \, {\left(a \cos\left(f x + e\right)^{4} - 2 \, a \cos\left(f x + e\right)^{2} - {\left(a \cos\left(f x + e\right)^{3} + a \cos\left(f x + e\right)^{2} - a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) + a\right)} \sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} + 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right) - 4 \, {\left(16 \, a \cos\left(f x + e\right)^{5} - 64 \, a \cos\left(f x + e\right)^{4} - 17 \, a \cos\left(f x + e\right)^{3} + 165 \, a \cos\left(f x + e\right)^{2} + 9 \, a \cos\left(f x + e\right) - {\left(16 \, a \cos\left(f x + e\right)^{4} + 80 \, a \cos\left(f x + e\right)^{3} + 63 \, a \cos\left(f x + e\right)^{2} - 102 \, a \cos\left(f x + e\right) - 93 \, a\right)} \sin\left(f x + e\right) - 93 \, a\right)} \sqrt{a \sin\left(f x + e\right) + a}}{96 \, {\left(f \cos\left(f x + e\right)^{4} - 2 \, f \cos\left(f x + e\right)^{2} - {\left(f \cos\left(f x + e\right)^{3} + f \cos\left(f x + e\right)^{2} - f \cos\left(f x + e\right) - f\right)} \sin\left(f x + e\right) + f\right)}}"," ",0,"1/96*(111*(a*cos(f*x + e)^4 - 2*a*cos(f*x + e)^2 - (a*cos(f*x + e)^3 + a*cos(f*x + e)^2 - a*cos(f*x + e) - a)*sin(f*x + e) + a)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) - 4*(16*a*cos(f*x + e)^5 - 64*a*cos(f*x + e)^4 - 17*a*cos(f*x + e)^3 + 165*a*cos(f*x + e)^2 + 9*a*cos(f*x + e) - (16*a*cos(f*x + e)^4 + 80*a*cos(f*x + e)^3 + 63*a*cos(f*x + e)^2 - 102*a*cos(f*x + e) - 93*a)*sin(f*x + e) - 93*a)*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 - (f*cos(f*x + e)^3 + f*cos(f*x + e)^2 - f*cos(f*x + e) - f)*sin(f*x + e) + f)","B",0
99,1,98,0,0.768294," ","integrate((a+a*sin(f*x+e))^(5/2)*tan(f*x+e)^4,x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, a^{2} \cos\left(f x + e\right)^{4} - 54 \, a^{2} \cos\left(f x + e\right)^{2} + 179 \, a^{2} - 8 \, {\left(a^{2} \cos\left(f x + e\right)^{2} + 23 \, 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) \sin\left(f x + e\right) - f \cos\left(f x + e\right)\right)}}"," ",0,"2/15*(3*a^2*cos(f*x + e)^4 - 54*a^2*cos(f*x + e)^2 + 179*a^2 - 8*(a^2*cos(f*x + e)^2 + 23*a^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e)*sin(f*x + e) - f*cos(f*x + e))","A",0
100,1,70,0,1.191596," ","integrate((a+a*sin(f*x+e))^(5/2)*tan(f*x+e)^2,x, algorithm=""fricas"")","\frac{2 \, {\left(11 \, a^{2} \cos\left(f x + e\right)^{2} + 77 \, a^{2} + {\left(3 \, a^{2} \cos\left(f x + e\right)^{2} - 47 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{15 \, f \cos\left(f x + e\right)}"," ",0,"2/15*(11*a^2*cos(f*x + e)^2 + 77*a^2 + (3*a^2*cos(f*x + e)^2 - 47*a^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e))","A",0
101,1,363,0,1.121797," ","integrate(cot(f*x+e)^2*(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{75 \, {\left(a^{2} \cos\left(f x + e\right)^{2} - a^{2} - {\left(a^{2} \cos\left(f x + e\right) + a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} - 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right) + 4 \, {\left(6 \, a^{2} \cos\left(f x + e\right)^{4} + 28 \, a^{2} \cos\left(f x + e\right)^{3} - 40 \, a^{2} \cos\left(f x + e\right)^{2} - 13 \, a^{2} \cos\left(f x + e\right) + 49 \, a^{2} + {\left(6 \, a^{2} \cos\left(f x + e\right)^{3} - 22 \, a^{2} \cos\left(f x + e\right)^{2} - 62 \, a^{2} \cos\left(f x + e\right) - 49 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{60 \, {\left(f \cos\left(f x + e\right)^{2} - {\left(f \cos\left(f x + e\right) + f\right)} \sin\left(f x + e\right) - f\right)}}"," ",0,"1/60*(75*(a^2*cos(f*x + e)^2 - a^2 - (a^2*cos(f*x + e) + a^2)*sin(f*x + e))*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) + 4*(6*a^2*cos(f*x + e)^4 + 28*a^2*cos(f*x + e)^3 - 40*a^2*cos(f*x + e)^2 - 13*a^2*cos(f*x + e) + 49*a^2 + (6*a^2*cos(f*x + e)^3 - 22*a^2*cos(f*x + e)^2 - 62*a^2*cos(f*x + e) - 49*a^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)^2 - (f*cos(f*x + e) + f)*sin(f*x + e) - f)","B",0
102,1,485,0,1.573529," ","integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{825 \, {\left(a^{2} \cos\left(f x + e\right)^{4} - 2 \, a^{2} \cos\left(f x + e\right)^{2} + a^{2} - {\left(a^{2} \cos\left(f x + e\right)^{3} + a^{2} \cos\left(f x + e\right)^{2} - a^{2} \cos\left(f x + e\right) - a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} + 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right) - 4 \, {\left(48 \, a^{2} \cos\left(f x + e\right)^{6} + 224 \, a^{2} \cos\left(f x + e\right)^{5} - 128 \, a^{2} \cos\left(f x + e\right)^{4} - 583 \, a^{2} \cos\left(f x + e\right)^{3} + 147 \, a^{2} \cos\left(f x + e\right)^{2} + 399 \, a^{2} \cos\left(f x + e\right) - 27 \, a^{2} + {\left(48 \, a^{2} \cos\left(f x + e\right)^{5} - 176 \, a^{2} \cos\left(f x + e\right)^{4} - 304 \, a^{2} \cos\left(f x + e\right)^{3} + 279 \, a^{2} \cos\left(f x + e\right)^{2} + 426 \, a^{2} \cos\left(f x + e\right) + 27 \, a^{2}\right)} \sin\left(f x + e\right)\right)} \sqrt{a \sin\left(f x + e\right) + a}}{480 \, {\left(f \cos\left(f x + e\right)^{4} - 2 \, f \cos\left(f x + e\right)^{2} - {\left(f \cos\left(f x + e\right)^{3} + f \cos\left(f x + e\right)^{2} - f \cos\left(f x + e\right) - f\right)} \sin\left(f x + e\right) + f\right)}}"," ",0,"1/480*(825*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2 - (a^2*cos(f*x + e)^3 + a^2*cos(f*x + e)^2 - a^2*cos(f*x + e) - a^2)*sin(f*x + e))*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) - 4*(48*a^2*cos(f*x + e)^6 + 224*a^2*cos(f*x + e)^5 - 128*a^2*cos(f*x + e)^4 - 583*a^2*cos(f*x + e)^3 + 147*a^2*cos(f*x + e)^2 + 399*a^2*cos(f*x + e) - 27*a^2 + (48*a^2*cos(f*x + e)^5 - 176*a^2*cos(f*x + e)^4 - 304*a^2*cos(f*x + e)^3 + 279*a^2*cos(f*x + e)^2 + 426*a^2*cos(f*x + e) + 27*a^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 - (f*cos(f*x + e)^3 + f*cos(f*x + e)^2 - f*cos(f*x + e) - f)*sin(f*x + e) + f)","B",0
103,1,229,0,1.216794," ","integrate(tan(f*x+e)^4/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{201 \, \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{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(61 \, \cos\left(f x + e\right)^{2} + {\left(183 \, \cos\left(f x + e\right)^{2} - 56\right)} \sin\left(f x + e\right) - 8\right)} \sqrt{a \sin\left(f x + e\right) + a}}{768 \, {\left(a f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) + a f \cos\left(f x + e\right)^{3}\right)}}"," ",0,"1/768*(201*sqrt(2)*(cos(f*x + e)^3*sin(f*x + e) + cos(f*x + e)^3)*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*(61*cos(f*x + e)^2 + (183*cos(f*x + e)^2 - 56)*sin(f*x + e) - 8)*sqrt(a*sin(f*x + e) + a))/(a*f*cos(f*x + e)^3*sin(f*x + e) + a*f*cos(f*x + e)^3)","A",0
104,1,200,0,1.171064," ","integrate(tan(f*x+e)^2/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{5 \, \sqrt{2} {\left(\cos\left(f x + e\right) \sin\left(f x + e\right) + \cos\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 \, \sqrt{a \sin\left(f x + e\right) + a} {\left(3 \, \sin\left(f x + e\right) + 1\right)}}{16 \, {\left(a f \cos\left(f x + e\right) \sin\left(f x + e\right) + a f \cos\left(f x + e\right)\right)}}"," ",0,"1/16*(5*sqrt(2)*(cos(f*x + e)*sin(f*x + e) + cos(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*sqrt(a*sin(f*x + e) + a)*(3*sin(f*x + e) + 1))/(a*f*cos(f*x + e)*sin(f*x + e) + a*f*cos(f*x + e))","B",0
105,1,263,0,1.254422," ","integrate(cot(f*x+e)^2/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{{\left(\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 1\right)} \sin\left(f x + e\right) - 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} + 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\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(a f \cos\left(f x + e\right)^{2} - a f - {\left(a f \cos\left(f x + e\right) + a f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/4*((cos(f*x + e)^2 - (cos(f*x + e) + 1)*sin(f*x + e) - 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) + 4*sqrt(a*sin(f*x + e) + a)*(cos(f*x + e) - sin(f*x + e) + 1))/(a*f*cos(f*x + e)^2 - a*f - (a*f*cos(f*x + e) + a*f)*sin(f*x + e))","B",0
106,1,369,0,1.109030," ","integrate(cot(f*x+e)^4/(a+a*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","\frac{21 \, {\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} - \cos\left(f x + e\right) - 1\right)} \sin\left(f x + e\right) + 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} - 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right) - 4 \, {\left(27 \, \cos\left(f x + e\right)^{3} + 25 \, \cos\left(f x + e\right)^{2} - {\left(27 \, \cos\left(f x + e\right)^{2} + 2 \, \cos\left(f x + e\right) - 17\right)} \sin\left(f x + e\right) - 19 \, \cos\left(f x + e\right) - 17\right)} \sqrt{a \sin\left(f x + e\right) + a}}{96 \, {\left(a f \cos\left(f x + e\right)^{4} - 2 \, a f \cos\left(f x + e\right)^{2} + a f - {\left(a f \cos\left(f x + e\right)^{3} + a f \cos\left(f x + e\right)^{2} - a f \cos\left(f x + e\right) - a f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/96*(21*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 - (cos(f*x + e)^3 + cos(f*x + e)^2 - cos(f*x + e) - 1)*sin(f*x + e) + 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) - 4*(27*cos(f*x + e)^3 + 25*cos(f*x + e)^2 - (27*cos(f*x + e)^2 + 2*cos(f*x + e) - 17)*sin(f*x + e) - 19*cos(f*x + e) - 17)*sqrt(a*sin(f*x + e) + a))/(a*f*cos(f*x + e)^4 - 2*a*f*cos(f*x + e)^2 + a*f - (a*f*cos(f*x + e)^3 + a*f*cos(f*x + e)^2 - a*f*cos(f*x + e) - a*f)*sin(f*x + e))","B",0
107,1,270,0,1.500208," ","integrate(tan(f*x+e)^4/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{21 \, \sqrt{2} {\left(\cos\left(f x + e\right)^{5} - 2 \, \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right)^{3}\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(21 \, \cos\left(f x + e\right)^{4} - 324 \, \cos\left(f x + e\right)^{2} - 12 \, {\left(45 \, \cos\left(f x + e\right)^{2} - 16\right)} \sin\left(f x + e\right) + 64\right)} \sqrt{a \sin\left(f x + e\right) + a}}{3072 \, {\left(a^{2} f \cos\left(f x + e\right)^{5} - 2 \, a^{2} f \cos\left(f x + e\right)^{3} \sin\left(f x + e\right) - 2 \, a^{2} f \cos\left(f x + e\right)^{3}\right)}}"," ",0,"1/3072*(21*sqrt(2)*(cos(f*x + e)^5 - 2*cos(f*x + e)^3*sin(f*x + e) - 2*cos(f*x + e)^3)*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*(21*cos(f*x + e)^4 - 324*cos(f*x + e)^2 - 12*(45*cos(f*x + e)^2 - 16)*sin(f*x + e) + 64)*sqrt(a*sin(f*x + e) + a))/(a^2*f*cos(f*x + e)^5 - 2*a^2*f*cos(f*x + e)^3*sin(f*x + e) - 2*a^2*f*cos(f*x + e)^3)","A",0
108,1,237,0,0.960819," ","integrate(tan(f*x+e)^2/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{\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{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(\cos\left(f x + e\right)^{2} + 20 \, \sin\left(f x + e\right) + 12\right)} \sqrt{a \sin\left(f x + e\right) + a}}{128 \, {\left(a^{2} f \cos\left(f x + e\right)^{3} - 2 \, a^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right) - 2 \, a^{2} f \cos\left(f x + e\right)\right)}}"," ",0,"1/128*(sqrt(2)*(cos(f*x + e)^3 - 2*cos(f*x + e)*sin(f*x + e) - 2*cos(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*(cos(f*x + e)^2 + 20*sin(f*x + e) + 12)*sqrt(a*sin(f*x + e) + a))/(a^2*f*cos(f*x + e)^3 - 2*a^2*f*cos(f*x + e)*sin(f*x + e) - 2*a^2*f*cos(f*x + e))","B",0
109,1,421,0,1.010850," ","integrate(cot(f*x+e)^2/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{3 \, {\left(\cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right) + 1\right)} \sin\left(f x + e\right) - 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} + 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right) + \frac{4 \, \sqrt{2} {\left(a \cos\left(f x + e\right)^{2} - {\left(a \cos\left(f x + e\right) + a\right)} \sin\left(f x + e\right) - a\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 \, \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(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f - {\left(a^{2} f \cos\left(f x + e\right) + a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/4*(3*(cos(f*x + e)^2 - (cos(f*x + e) + 1)*sin(f*x + e) - 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) + 4*sqrt(2)*(a*cos(f*x + e)^2 - (a*cos(f*x + e) + a)*sin(f*x + e) - a)*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*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 - (a^2*f*cos(f*x + e) + a^2*f)*sin(f*x + e))","B",0
110,1,383,0,0.927660," ","integrate(cot(f*x+e)^4/(a+a*sin(f*x+e))^(3/2),x, algorithm=""fricas"")","\frac{3 \, {\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} - \cos\left(f x + e\right) - 1\right)} \sin\left(f x + e\right) + 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} - 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right) + 4 \, {\left(3 \, \cos\left(f x + e\right)^{3} + 17 \, \cos\left(f x + e\right)^{2} - {\left(3 \, \cos\left(f x + e\right)^{2} - 14 \, \cos\left(f x + e\right) - 25\right)} \sin\left(f x + e\right) - 11 \, \cos\left(f x + e\right) - 25\right)} \sqrt{a \sin\left(f x + e\right) + a}}{96 \, {\left(a^{2} f \cos\left(f x + e\right)^{4} - 2 \, a^{2} f \cos\left(f x + e\right)^{2} + a^{2} f - {\left(a^{2} f \cos\left(f x + e\right)^{3} + a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f \cos\left(f x + e\right) - a^{2} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/96*(3*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 - (cos(f*x + e)^3 + cos(f*x + e)^2 - cos(f*x + e) - 1)*sin(f*x + e) + 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) + 4*(3*cos(f*x + e)^3 + 17*cos(f*x + e)^2 - (3*cos(f*x + e)^2 - 14*cos(f*x + e) - 25)*sin(f*x + e) - 11*cos(f*x + e) - 25)*sqrt(a*sin(f*x + e) + a))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f - (a^2*f*cos(f*x + e)^3 + a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - a^2*f)*sin(f*x + e))","B",0
111,1,307,0,1.069902," ","integrate(tan(f*x+e)^4/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{951 \, \sqrt{2} {\left(3 \, \cos\left(f x + e\right)^{5} - 4 \, \cos\left(f x + e\right)^{3} + {\left(\cos\left(f x + e\right)^{5} - 4 \, \cos\left(f x + e\right)^{3}\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(2219 \, \cos\left(f x + e\right)^{4} - 4960 \, \cos\left(f x + e\right)^{2} + {\left(951 \, \cos\left(f x + e\right)^{4} - 6944 \, \cos\left(f x + e\right)^{2} + 2816\right)} \sin\left(f x + e\right) + 1280\right)} \sqrt{a \sin\left(f x + e\right) + a}}{49152 \, {\left(3 \, a^{3} f \cos\left(f x + e\right)^{5} - 4 \, a^{3} f \cos\left(f x + e\right)^{3} + {\left(a^{3} f \cos\left(f x + e\right)^{5} - 4 \, a^{3} f \cos\left(f x + e\right)^{3}\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/49152*(951*sqrt(2)*(3*cos(f*x + e)^5 - 4*cos(f*x + e)^3 + (cos(f*x + e)^5 - 4*cos(f*x + e)^3)*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*(2219*cos(f*x + e)^4 - 4960*cos(f*x + e)^2 + (951*cos(f*x + e)^4 - 6944*cos(f*x + e)^2 + 2816)*sin(f*x + e) + 1280)*sqrt(a*sin(f*x + e) + a))/(3*a^3*f*cos(f*x + e)^5 - 4*a^3*f*cos(f*x + e)^3 + (a^3*f*cos(f*x + e)^5 - 4*a^3*f*cos(f*x + e)^3)*sin(f*x + e))","A",0
112,1,279,0,1.311610," ","integrate(tan(f*x+e)^2/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{33 \, \sqrt{2} {\left(3 \, \cos\left(f x + e\right)^{3} + {\left(\cos\left(f x + e\right)^{3} - 4 \, \cos\left(f x + e\right)\right)} \sin\left(f x + e\right) - 4 \, \cos\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(77 \, \cos\left(f x + e\right)^{2} + {\left(33 \, \cos\left(f x + e\right)^{2} - 224\right)} \sin\left(f x + e\right) - 160\right)} \sqrt{a \sin\left(f x + e\right) + a}}{1536 \, {\left(3 \, a^{3} f \cos\left(f x + e\right)^{3} - 4 \, a^{3} f \cos\left(f x + e\right) + {\left(a^{3} f \cos\left(f x + e\right)^{3} - 4 \, a^{3} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/1536*(33*sqrt(2)*(3*cos(f*x + e)^3 + (cos(f*x + e)^3 - 4*cos(f*x + e))*sin(f*x + e) - 4*cos(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*(77*cos(f*x + e)^2 + (33*cos(f*x + e)^2 - 224)*sin(f*x + e) - 160)*sqrt(a*sin(f*x + e) + a))/(3*a^3*f*cos(f*x + e)^3 - 4*a^3*f*cos(f*x + e) + (a^3*f*cos(f*x + e)^3 - 4*a^3*f*cos(f*x + e))*sin(f*x + e))","A",0
113,1,539,0,1.025177," ","integrate(cot(f*x+e)^2/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{5 \, {\left(\cos\left(f x + e\right)^{3} + 2 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - \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)^{3} - 7 \, a \cos\left(f x + e\right)^{2} + 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right) + \frac{7 \, \sqrt{2} {\left(a \cos\left(f x + e\right)^{3} + 2 \, 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 \cos\left(f x + e\right) - 2 \, a\right)} \sin\left(f x + e\right) - 2 \, a\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(2 \, \cos\left(f x + e\right)^{2} + {\left(2 \, \cos\left(f x + e\right) + 1\right)} \sin\left(f x + e\right) + \cos\left(f x + e\right) - 1\right)} \sqrt{a \sin\left(f x + e\right) + a}}{4 \, {\left(a^{3} f \cos\left(f x + e\right)^{3} + 2 \, a^{3} f \cos\left(f x + e\right)^{2} - a^{3} f \cos\left(f x + e\right) - 2 \, a^{3} f + {\left(a^{3} f \cos\left(f x + e\right)^{2} - a^{3} f \cos\left(f x + e\right) - 2 \, a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/4*(5*(cos(f*x + e)^3 + 2*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)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) + 7*sqrt(2)*(a*cos(f*x + e)^3 + 2*a*cos(f*x + e)^2 - a*cos(f*x + e) + (a*cos(f*x + e)^2 - a*cos(f*x + e) - 2*a)*sin(f*x + e) - 2*a)*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*(2*cos(f*x + e)^2 + (2*cos(f*x + e) + 1)*sin(f*x + e) + cos(f*x + e) - 1)*sqrt(a*sin(f*x + e) + a))/(a^3*f*cos(f*x + e)^3 + 2*a^3*f*cos(f*x + e)^2 - a^3*f*cos(f*x + e) - 2*a^3*f + (a^3*f*cos(f*x + e)^2 - a^3*f*cos(f*x + e) - 2*a^3*f)*sin(f*x + e))","B",0
114,1,564,0,1.217989," ","integrate(cot(f*x+e)^4/(a+a*sin(f*x+e))^(5/2),x, algorithm=""fricas"")","\frac{135 \, {\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} - \cos\left(f x + e\right) - 1\right)} \sin\left(f x + e\right) + 1\right)} \sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} + 4 \, {\left(\cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right) + 3\right)} \sin\left(f x + e\right) - 2 \, \cos\left(f x + e\right) - 3\right)} \sqrt{a \sin\left(f x + e\right) + a} \sqrt{a} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right) + \frac{192 \, \sqrt{2} {\left(a \cos\left(f x + e\right)^{4} - 2 \, a \cos\left(f x + e\right)^{2} - {\left(a \cos\left(f x + e\right)^{3} + a \cos\left(f x + e\right)^{2} - a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) + a\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(57 \, \cos\left(f x + e\right)^{3} + 83 \, \cos\left(f x + e\right)^{2} - {\left(57 \, \cos\left(f x + e\right)^{2} - 26 \, \cos\left(f x + e\right) - 91\right)} \sin\left(f x + e\right) - 65 \, \cos\left(f x + e\right) - 91\right)} \sqrt{a \sin\left(f x + e\right) + a}}{96 \, {\left(a^{3} f \cos\left(f x + e\right)^{4} - 2 \, a^{3} f \cos\left(f x + e\right)^{2} + a^{3} f - {\left(a^{3} f \cos\left(f x + e\right)^{3} + a^{3} f \cos\left(f x + e\right)^{2} - a^{3} f \cos\left(f x + e\right) - a^{3} f\right)} \sin\left(f x + e\right)\right)}}"," ",0,"1/96*(135*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 - (cos(f*x + e)^3 + cos(f*x + e)^2 - cos(f*x + e) - 1)*sin(f*x + e) + 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) + 192*sqrt(2)*(a*cos(f*x + e)^4 - 2*a*cos(f*x + e)^2 - (a*cos(f*x + e)^3 + a*cos(f*x + e)^2 - a*cos(f*x + e) - a)*sin(f*x + e) + a)*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*(57*cos(f*x + e)^3 + 83*cos(f*x + e)^2 - (57*cos(f*x + e)^2 - 26*cos(f*x + e) - 91)*sin(f*x + e) - 65*cos(f*x + e) - 91)*sqrt(a*sin(f*x + e) + a))/(a^3*f*cos(f*x + e)^4 - 2*a^3*f*cos(f*x + e)^2 + a^3*f - (a^3*f*cos(f*x + e)^3 + a^3*f*cos(f*x + e)^2 - a^3*f*cos(f*x + e) - a^3*f)*sin(f*x + e))","B",0
115,0,0,0,0.867629," ","integrate((a+a*sin(f*x+e))^(1/3)*tan(f*x+e)^4,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{\frac{1}{3}} \tan\left(f x + e\right)^{4}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^(1/3)*tan(f*x + e)^4, x)","F",0
116,0,0,0,1.461598," ","integrate((a+a*sin(f*x+e))^(1/3)*tan(f*x+e)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{\frac{1}{3}} \tan\left(f x + e\right)^{2}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^(1/3)*tan(f*x + e)^2, x)","F",0
117,-1,0,0,0.000000," ","integrate(cot(f*x+e)^2*(a+a*sin(f*x+e))^(1/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
118,-1,0,0,0.000000," ","integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(1/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
119,0,0,0,1.929872," ","integrate(tan(f*x+e)^4/(a+a*sin(f*x+e))^(1/3),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\tan\left(f x + e\right)^{4}}{{\left(a \sin\left(f x + e\right) + a\right)}^{\frac{1}{3}}}, x\right)"," ",0,"integral(tan(f*x + e)^4/(a*sin(f*x + e) + a)^(1/3), x)","F",0
120,0,0,0,1.726138," ","integrate(tan(f*x+e)^2/(a+a*sin(f*x+e))^(1/3),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\tan\left(f x + e\right)^{2}}{{\left(a \sin\left(f x + e\right) + a\right)}^{\frac{1}{3}}}, x\right)"," ",0,"integral(tan(f*x + e)^2/(a*sin(f*x + e) + a)^(1/3), x)","F",0
121,-1,0,0,0.000000," ","integrate(cot(f*x+e)^2/(a+a*sin(f*x+e))^(1/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
122,-1,0,0,0.000000," ","integrate(cot(f*x+e)^4/(a+a*sin(f*x+e))^(1/3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
123,0,0,0,1.738881," ","integrate((a+a*sin(f*x+e))^3*(g*tan(f*x+e))^p,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(g \tan\left(f x + e\right)\right)^{p}, 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))*(g*tan(f*x + e))^p, x)","F",0
124,0,0,0,1.560894," ","integrate((a+a*sin(f*x+e))^2*(g*tan(f*x+e))^p,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(g \tan\left(f x + e\right)\right)^{p}, x\right)"," ",0,"integral(-(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2)*(g*tan(f*x + e))^p, x)","F",0
125,0,0,0,0.933839," ","integrate((a+a*sin(f*x+e))*(g*tan(f*x+e))^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)} \left(g \tan\left(f x + e\right)\right)^{p}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)*(g*tan(f*x + e))^p, x)","F",0
126,0,0,0,1.428559," ","integrate((g*tan(f*x+e))^p/(a+a*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(g \tan\left(f x + e\right)\right)^{p}}{a \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral((g*tan(f*x + e))^p/(a*sin(f*x + e) + a), x)","F",0
127,0,0,0,1.492353," ","integrate((g*tan(f*x+e))^p/(a+a*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(g \tan\left(f x + e\right)\right)^{p}}{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(-(g*tan(f*x + e))^p/(a^2*cos(f*x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)","F",0
128,0,0,0,1.361887," ","integrate((g*tan(f*x+e))^p/(a+a*sin(f*x+e))^3,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(g \tan\left(f x + e\right)\right)^{p}}{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(-(g*tan(f*x + e))^p/(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
129,0,0,0,1.112256," ","integrate((a+a*sin(f*x+e))^m*(g*tan(f*x+e))^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \left(g \tan\left(f x + e\right)\right)^{p}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*(g*tan(f*x + e))^p, x)","F",0
130,0,0,0,1.349844," ","integrate((a+a*sin(f*x+e))^m*tan(f*x+e)^3,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \tan\left(f x + e\right)^{3}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*tan(f*x + e)^3, x)","F",0
131,0,0,0,1.364827," ","integrate((a+a*sin(f*x+e))^m*tan(f*x+e),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \tan\left(f x + e\right), x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*tan(f*x + e), x)","F",0
132,0,0,0,1.329981," ","integrate(cot(f*x+e)*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \cot\left(f x + e\right), x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*cot(f*x + e), x)","F",0
133,0,0,0,1.189036," ","integrate(cot(f*x+e)^3*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \cot\left(f x + e\right)^{3}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*cot(f*x + e)^3, x)","F",0
134,0,0,0,1.490150," ","integrate(cot(f*x+e)^5*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \cot\left(f x + e\right)^{5}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*cot(f*x + e)^5, x)","F",0
135,0,0,0,1.257886," ","integrate((a+a*sin(f*x+e))^m*tan(f*x+e)^4,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \tan\left(f x + e\right)^{4}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*tan(f*x + e)^4, x)","F",0
136,0,0,0,1.333003," ","integrate((a+a*sin(f*x+e))^m*tan(f*x+e)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \tan\left(f x + e\right)^{2}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*tan(f*x + e)^2, x)","F",0
137,0,0,0,1.229056," ","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
138,0,0,0,0.815748," ","integrate(cot(f*x+e)^2*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \cot\left(f x + e\right)^{2}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*cot(f*x + e)^2, x)","F",0
139,0,0,0,1.361017," ","integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sin\left(f x + e\right) + a\right)}^{m} \cot\left(f x + e\right)^{4}, x\right)"," ",0,"integral((a*sin(f*x + e) + a)^m*cot(f*x + e)^4, x)","F",0
140,1,90,0,1.180138," ","integrate((a+b*sin(d*x+c))*tan(d*x+c)^3,x, algorithm=""fricas"")","\frac{{\left(2 \, a - 3 \, b\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) + {\left(2 \, a + 3 \, b\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(2 \, b \cos\left(d x + c\right)^{2} + b\right)} \sin\left(d x + c\right) + 2 \, a}{4 \, d \cos\left(d x + c\right)^{2}}"," ",0,"1/4*((2*a - 3*b)*cos(d*x + c)^2*log(sin(d*x + c) + 1) + (2*a + 3*b)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + 2*(2*b*cos(d*x + c)^2 + b)*sin(d*x + c) + 2*a)/(d*cos(d*x + c)^2)","A",0
141,1,45,0,1.207396," ","integrate((a+b*sin(d*x+c))*tan(d*x+c),x, algorithm=""fricas"")","-\frac{{\left(a - b\right)} \log\left(\sin\left(d x + c\right) + 1\right) + {\left(a + b\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, b \sin\left(d x + c\right)}{2 \, d}"," ",0,"-1/2*((a - b)*log(sin(d*x + c) + 1) + (a + b)*log(-sin(d*x + c) + 1) + 2*b*sin(d*x + c))/d","A",0
142,1,24,0,1.382271," ","integrate(cot(d*x+c)*(a+b*sin(d*x+c)),x, algorithm=""fricas"")","\frac{a \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) + b \sin\left(d x + c\right)}{d}"," ",0,"(a*log(1/2*sin(d*x + c)) + b*sin(d*x + c))/d","A",0
143,1,69,0,1.246694," ","integrate(cot(d*x+c)^3*(a+b*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{2 \, {\left(a \cos\left(d x + c\right)^{2} - a\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) + 2 \, {\left(b \cos\left(d x + c\right)^{2} - 2 \, b\right)} \sin\left(d x + c\right) - a}{2 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"-1/2*(2*(a*cos(d*x + c)^2 - a)*log(1/2*sin(d*x + c)) + 2*(b*cos(d*x + c)^2 - 2*b)*sin(d*x + c) - a)/(d*cos(d*x + c)^2 - d)","A",0
144,1,110,0,1.332350," ","integrate(cot(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{12 \, a \cos\left(d x + c\right)^{2} - 12 \, {\left(a \cos\left(d x + c\right)^{4} - 2 \, a \cos\left(d x + c\right)^{2} + a\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 4 \, {\left(3 \, b \cos\left(d x + c\right)^{4} - 12 \, b \cos\left(d x + c\right)^{2} + 8 \, b\right)} \sin\left(d x + c\right) - 9 \, a}{12 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} + d\right)}}"," ",0,"-1/12*(12*a*cos(d*x + c)^2 - 12*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a)*log(1/2*sin(d*x + c)) - 4*(3*b*cos(d*x + c)^4 - 12*b*cos(d*x + c)^2 + 8*b)*sin(d*x + c) - 9*a)/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)","A",0
145,1,73,0,1.045875," ","integrate((a+b*sin(d*x+c))*tan(d*x+c)^4,x, algorithm=""fricas"")","\frac{3 \, a d x \cos\left(d x + c\right)^{3} - 3 \, b \cos\left(d x + c\right)^{4} - 6 \, b \cos\left(d x + c\right)^{2} - {\left(4 \, a \cos\left(d x + c\right)^{2} - a\right)} \sin\left(d x + c\right) + b}{3 \, d \cos\left(d x + c\right)^{3}}"," ",0,"1/3*(3*a*d*x*cos(d*x + c)^3 - 3*b*cos(d*x + c)^4 - 6*b*cos(d*x + c)^2 - (4*a*cos(d*x + c)^2 - a)*sin(d*x + c) + b)/(d*cos(d*x + c)^3)","A",0
146,1,47,0,1.403224," ","integrate((a+b*sin(d*x+c))*tan(d*x+c)^2,x, algorithm=""fricas"")","-\frac{a d x \cos\left(d x + c\right) - b \cos\left(d x + c\right)^{2} - a \sin\left(d x + c\right) - b}{d \cos\left(d x + c\right)}"," ",0,"-(a*d*x*cos(d*x + c) - b*cos(d*x + c)^2 - a*sin(d*x + c) - b)/(d*cos(d*x + c))","A",0
147,1,84,0,1.352624," ","integrate(cot(d*x+c)^2*(a+b*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{b \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - b \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 2 \, a \cos\left(d x + c\right) + 2 \, {\left(a d x - b \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{2 \, d \sin\left(d x + c\right)}"," ",0,"-1/2*(b*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - b*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 2*a*cos(d*x + c) + 2*(a*d*x - b*cos(d*x + c))*sin(d*x + c))/(d*sin(d*x + c))","B",0
148,1,160,0,1.278830," ","integrate(cot(d*x+c)^4*(a+b*sin(d*x+c)),x, algorithm=""fricas"")","\frac{16 \, a \cos\left(d x + c\right)^{3} + 9 \, {\left(b \cos\left(d x + c\right)^{2} - b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 9 \, {\left(b \cos\left(d x + c\right)^{2} - b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 12 \, a \cos\left(d x + c\right) + 6 \, {\left(2 \, a d x \cos\left(d x + c\right)^{2} - 2 \, b \cos\left(d x + c\right)^{3} - 2 \, a d x + 3 \, b \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{12 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}"," ",0,"1/12*(16*a*cos(d*x + c)^3 + 9*(b*cos(d*x + c)^2 - b)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 9*(b*cos(d*x + c)^2 - b)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 12*a*cos(d*x + c) + 6*(2*a*d*x*cos(d*x + c)^2 - 2*b*cos(d*x + c)^3 - 2*a*d*x + 3*b*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))","B",0
149,1,222,0,0.968257," ","integrate(cot(d*x+c)^6*(a+b*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{368 \, a \cos\left(d x + c\right)^{5} - 560 \, a \cos\left(d x + c\right)^{3} + 225 \, {\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 225 \, {\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 240 \, a \cos\left(d x + c\right) + 30 \, {\left(8 \, a d x \cos\left(d x + c\right)^{4} - 8 \, b \cos\left(d x + c\right)^{5} - 16 \, a d x \cos\left(d x + c\right)^{2} + 25 \, b \cos\left(d x + c\right)^{3} + 8 \, a d x - 15 \, b \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{240 \, {\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)}"," ",0,"-1/240*(368*a*cos(d*x + c)^5 - 560*a*cos(d*x + c)^3 + 225*(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + b)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 225*(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + b)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 240*a*cos(d*x + c) + 30*(8*a*d*x*cos(d*x + c)^4 - 8*b*cos(d*x + c)^5 - 16*a*d*x*cos(d*x + c)^2 + 25*b*cos(d*x + c)^3 + 8*a*d*x - 15*b*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))","B",0
150,1,140,0,1.329508," ","integrate((a+b*sin(d*x+c))^2*tan(d*x+c)^3,x, algorithm=""fricas"")","-\frac{2 \, b^{2} \cos\left(d x + c\right)^{4} - b^{2} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{2} - 3 \, a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - 2 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, a^{2} - 2 \, b^{2} - 4 \, {\left(2 \, a b \cos\left(d x + c\right)^{2} + a b\right)} \sin\left(d x + c\right)}{4 \, d \cos\left(d x + c\right)^{2}}"," ",0,"-1/4*(2*b^2*cos(d*x + c)^4 - b^2*cos(d*x + c)^2 - 2*(a^2 - 3*a*b + 2*b^2)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - 2*(a^2 + 3*a*b + 2*b^2)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 2*a^2 - 2*b^2 - 4*(2*a*b*cos(d*x + c)^2 + a*b)*sin(d*x + c))/(d*cos(d*x + c)^2)","A",0
151,1,74,0,1.423627," ","integrate((a+b*sin(d*x+c))^2*tan(d*x+c),x, algorithm=""fricas"")","\frac{b^{2} \cos\left(d x + c\right)^{2} - 4 \, a b \sin\left(d x + c\right) - {\left(a^{2} - 2 \, a b + b^{2}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(a^{2} + 2 \, a b + b^{2}\right)} \log\left(-\sin\left(d x + c\right) + 1\right)}{2 \, d}"," ",0,"1/2*(b^2*cos(d*x + c)^2 - 4*a*b*sin(d*x + c) - (a^2 - 2*a*b + b^2)*log(sin(d*x + c) + 1) - (a^2 + 2*a*b + b^2)*log(-sin(d*x + c) + 1))/d","A",0
152,1,42,0,1.103979," ","integrate(cot(d*x+c)*(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","-\frac{b^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 4 \, a b \sin\left(d x + c\right)}{2 \, d}"," ",0,"-1/2*(b^2*cos(d*x + c)^2 - 2*a^2*log(1/2*sin(d*x + c)) - 4*a*b*sin(d*x + c))/d","A",0
153,1,115,0,1.213625," ","integrate(cot(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{2 \, b^{2} \cos\left(d x + c\right)^{4} - 3 \, b^{2} \cos\left(d x + c\right)^{2} + 2 \, a^{2} + b^{2} - 4 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 8 \, {\left(a b \cos\left(d x + c\right)^{2} - 2 \, a b\right)} \sin\left(d x + c\right)}{4 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"1/4*(2*b^2*cos(d*x + c)^4 - 3*b^2*cos(d*x + c)^2 + 2*a^2 + b^2 - 4*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*log(1/2*sin(d*x + c)) - 8*(a*b*cos(d*x + c)^2 - 2*a*b)*sin(d*x + c))/(d*cos(d*x + c)^2 - d)","A",0
154,1,177,0,1.308566," ","integrate(cot(d*x+c)^5*(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","-\frac{6 \, b^{2} \cos\left(d x + c\right)^{6} - 15 \, b^{2} \cos\left(d x + c\right)^{4} + 6 \, {\left(2 \, a^{2} + b^{2}\right)} \cos\left(d x + c\right)^{2} - 9 \, a^{2} + 3 \, b^{2} - 12 \, {\left({\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} - 2 \, b^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) - 8 \, {\left(3 \, a b \cos\left(d x + c\right)^{4} - 12 \, a b \cos\left(d x + c\right)^{2} + 8 \, a b\right)} \sin\left(d x + c\right)}{12 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} + d\right)}}"," ",0,"-1/12*(6*b^2*cos(d*x + c)^6 - 15*b^2*cos(d*x + c)^4 + 6*(2*a^2 + b^2)*cos(d*x + c)^2 - 9*a^2 + 3*b^2 - 12*((a^2 - 2*b^2)*cos(d*x + c)^4 - 2*(a^2 - 2*b^2)*cos(d*x + c)^2 + a^2 - 2*b^2)*log(1/2*sin(d*x + c)) - 8*(3*a*b*cos(d*x + c)^4 - 12*a*b*cos(d*x + c)^2 + 8*a*b)*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)","A",0
155,1,118,0,1.212795," ","integrate((a+b*sin(d*x+c))^2*tan(d*x+c)^4,x, algorithm=""fricas"")","\frac{3 \, {\left(2 \, a^{2} + 5 \, b^{2}\right)} d x \cos\left(d x + c\right)^{3} - 12 \, a b \cos\left(d x + c\right)^{4} - 24 \, a b \cos\left(d x + c\right)^{2} + 4 \, a b - {\left(3 \, b^{2} \cos\left(d x + c\right)^{4} + 2 \, {\left(4 \, a^{2} + 7 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, a^{2} - 2 \, b^{2}\right)} \sin\left(d x + c\right)}{6 \, d \cos\left(d x + c\right)^{3}}"," ",0,"1/6*(3*(2*a^2 + 5*b^2)*d*x*cos(d*x + c)^3 - 12*a*b*cos(d*x + c)^4 - 24*a*b*cos(d*x + c)^2 + 4*a*b - (3*b^2*cos(d*x + c)^4 + 2*(4*a^2 + 7*b^2)*cos(d*x + c)^2 - 2*a^2 - 2*b^2)*sin(d*x + c))/(d*cos(d*x + c)^3)","A",0
156,1,81,0,1.144169," ","integrate((a+b*sin(d*x+c))^2*tan(d*x+c)^2,x, algorithm=""fricas"")","-\frac{{\left(2 \, a^{2} + 3 \, b^{2}\right)} d x \cos\left(d x + c\right) - 4 \, a b \cos\left(d x + c\right)^{2} - 4 \, a b - {\left(b^{2} \cos\left(d x + c\right)^{2} + 2 \, a^{2} + 2 \, b^{2}\right)} \sin\left(d x + c\right)}{2 \, d \cos\left(d x + c\right)}"," ",0,"-1/2*((2*a^2 + 3*b^2)*d*x*cos(d*x + c) - 4*a*b*cos(d*x + c)^2 - 4*a*b - (b^2*cos(d*x + c)^2 + 2*a^2 + 2*b^2)*sin(d*x + c))/(d*cos(d*x + c))","A",0
157,1,118,0,1.674103," ","integrate(cot(d*x+c)^2*(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","-\frac{b^{2} \cos\left(d x + c\right)^{3} + 2 \, a b \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 2 \, a b \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + {\left(2 \, a^{2} - b^{2}\right)} \cos\left(d x + c\right) + {\left({\left(2 \, a^{2} - b^{2}\right)} d x - 4 \, a b \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{2 \, d \sin\left(d x + c\right)}"," ",0,"-1/2*(b^2*cos(d*x + c)^3 + 2*a*b*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 2*a*b*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + (2*a^2 - b^2)*cos(d*x + c) + ((2*a^2 - b^2)*d*x - 4*a*b*cos(d*x + c))*sin(d*x + c))/(d*sin(d*x + c))","A",0
158,1,218,0,1.453091," ","integrate(cot(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{3 \, b^{2} \cos\left(d x + c\right)^{5} + 4 \, {\left(2 \, a^{2} - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{3} + 9 \, {\left(a b \cos\left(d x + c\right)^{2} - a b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 9 \, {\left(a b \cos\left(d x + c\right)^{2} - a b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 3 \, {\left(2 \, a^{2} - 3 \, b^{2}\right)} \cos\left(d x + c\right) + 3 \, {\left({\left(2 \, a^{2} - 3 \, b^{2}\right)} d x \cos\left(d x + c\right)^{2} - 4 \, a b \cos\left(d x + c\right)^{3} - {\left(2 \, a^{2} - 3 \, b^{2}\right)} d x + 6 \, a b \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{6 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}"," ",0,"1/6*(3*b^2*cos(d*x + c)^5 + 4*(2*a^2 - 3*b^2)*cos(d*x + c)^3 + 9*(a*b*cos(d*x + c)^2 - a*b)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 9*(a*b*cos(d*x + c)^2 - a*b)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3*(2*a^2 - 3*b^2)*cos(d*x + c) + 3*((2*a^2 - 3*b^2)*d*x*cos(d*x + c)^2 - 4*a*b*cos(d*x + c)^3 - (2*a^2 - 3*b^2)*d*x + 6*a*b*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))","A",0
159,1,306,0,1.558477," ","integrate(cot(d*x+c)^6*(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","-\frac{60 \, b^{2} \cos\left(d x + c\right)^{7} + 92 \, {\left(2 \, a^{2} - 5 \, b^{2}\right)} \cos\left(d x + c\right)^{5} - 140 \, {\left(2 \, a^{2} - 5 \, b^{2}\right)} \cos\left(d x + c\right)^{3} + 225 \, {\left(a b \cos\left(d x + c\right)^{4} - 2 \, a b \cos\left(d x + c\right)^{2} + a b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 225 \, {\left(a b \cos\left(d x + c\right)^{4} - 2 \, a b \cos\left(d x + c\right)^{2} + a b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 60 \, {\left(2 \, a^{2} - 5 \, b^{2}\right)} \cos\left(d x + c\right) + 30 \, {\left(2 \, {\left(2 \, a^{2} - 5 \, b^{2}\right)} d x \cos\left(d x + c\right)^{4} - 8 \, a b \cos\left(d x + c\right)^{5} - 4 \, {\left(2 \, a^{2} - 5 \, b^{2}\right)} d x \cos\left(d x + c\right)^{2} + 25 \, a b \cos\left(d x + c\right)^{3} + 2 \, {\left(2 \, a^{2} - 5 \, b^{2}\right)} d x - 15 \, a b \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{120 \, {\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)}"," ",0,"-1/120*(60*b^2*cos(d*x + c)^7 + 92*(2*a^2 - 5*b^2)*cos(d*x + c)^5 - 140*(2*a^2 - 5*b^2)*cos(d*x + c)^3 + 225*(a*b*cos(d*x + c)^4 - 2*a*b*cos(d*x + c)^2 + a*b)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 225*(a*b*cos(d*x + c)^4 - 2*a*b*cos(d*x + c)^2 + a*b)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 60*(2*a^2 - 5*b^2)*cos(d*x + c) + 30*(2*(2*a^2 - 5*b^2)*d*x*cos(d*x + c)^4 - 8*a*b*cos(d*x + c)^5 - 4*(2*a^2 - 5*b^2)*d*x*cos(d*x + c)^2 + 25*a*b*cos(d*x + c)^3 + 2*(2*a^2 - 5*b^2)*d*x - 15*a*b*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))","A",0
160,1,194,0,1.577128," ","integrate((a+b*sin(d*x+c))^3*tan(d*x+c)^3,x, algorithm=""fricas"")","-\frac{18 \, a b^{2} \cos\left(d x + c\right)^{4} - 9 \, a b^{2} \cos\left(d x + c\right)^{2} - 3 \, {\left(2 \, a^{3} - 9 \, a^{2} b + 12 \, a b^{2} - 5 \, b^{3}\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - 3 \, {\left(2 \, a^{3} + 9 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 6 \, a^{3} - 18 \, a b^{2} + 2 \, {\left(2 \, b^{3} \cos\left(d x + c\right)^{4} - 9 \, a^{2} b - 3 \, b^{3} - 2 \, {\left(9 \, a^{2} b + 7 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{12 \, d \cos\left(d x + c\right)^{2}}"," ",0,"-1/12*(18*a*b^2*cos(d*x + c)^4 - 9*a*b^2*cos(d*x + c)^2 - 3*(2*a^3 - 9*a^2*b + 12*a*b^2 - 5*b^3)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - 3*(2*a^3 + 9*a^2*b + 12*a*b^2 + 5*b^3)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 6*a^3 - 18*a*b^2 + 2*(2*b^3*cos(d*x + c)^4 - 9*a^2*b - 3*b^3 - 2*(9*a^2*b + 7*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^2)","A",0
161,1,116,0,1.629751," ","integrate((a+b*sin(d*x+c))^3*tan(d*x+c),x, algorithm=""fricas"")","\frac{9 \, a b^{2} \cos\left(d x + c\right)^{2} - 3 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 3 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(b^{3} \cos\left(d x + c\right)^{2} - 9 \, a^{2} b - 4 \, b^{3}\right)} \sin\left(d x + c\right)}{6 \, d}"," ",0,"1/6*(9*a*b^2*cos(d*x + c)^2 - 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*log(sin(d*x + c) + 1) - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*log(-sin(d*x + c) + 1) + 2*(b^3*cos(d*x + c)^2 - 9*a^2*b - 4*b^3)*sin(d*x + c))/d","A",0
162,1,66,0,1.631490," ","integrate(cot(d*x+c)*(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{9 \, a b^{2} \cos\left(d x + c\right)^{2} - 6 \, a^{3} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) + 2 \, {\left(b^{3} \cos\left(d x + c\right)^{2} - 9 \, a^{2} b - b^{3}\right)} \sin\left(d x + c\right)}{6 \, d}"," ",0,"-1/6*(9*a*b^2*cos(d*x + c)^2 - 6*a^3*log(1/2*sin(d*x + c)) + 2*(b^3*cos(d*x + c)^2 - 9*a^2*b - b^3)*sin(d*x + c))/d","A",0
163,1,153,0,1.739221," ","integrate(cot(d*x+c)^3*(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","\frac{18 \, a b^{2} \cos\left(d x + c\right)^{4} - 27 \, a b^{2} \cos\left(d x + c\right)^{2} + 6 \, a^{3} + 9 \, a b^{2} + 12 \, {\left(a^{3} - 3 \, a b^{2} - {\left(a^{3} - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) + 4 \, {\left(b^{3} \cos\left(d x + c\right)^{4} + 18 \, a^{2} b - 2 \, b^{3} - {\left(9 \, a^{2} b - b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{12 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"1/12*(18*a*b^2*cos(d*x + c)^4 - 27*a*b^2*cos(d*x + c)^2 + 6*a^3 + 9*a*b^2 + 12*(a^3 - 3*a*b^2 - (a^3 - 3*a*b^2)*cos(d*x + c)^2)*log(1/2*sin(d*x + c)) + 4*(b^3*cos(d*x + c)^4 + 18*a^2*b - 2*b^3 - (9*a^2*b - b^3)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^2 - d)","A",0
164,1,225,0,1.464460," ","integrate(cot(d*x+c)^5*(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{18 \, a b^{2} \cos\left(d x + c\right)^{6} - 45 \, a b^{2} \cos\left(d x + c\right)^{4} - 9 \, a^{3} + 9 \, a b^{2} + 6 \, {\left(2 \, a^{3} + 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{2} - 12 \, {\left({\left(a^{3} - 6 \, a b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{3} - 6 \, a b^{2} - 2 \, {\left(a^{3} - 6 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) + 4 \, {\left(b^{3} \cos\left(d x + c\right)^{6} - 3 \, {\left(3 \, a^{2} b - b^{3}\right)} \cos\left(d x + c\right)^{4} - 24 \, a^{2} b + 8 \, b^{3} + 12 \, {\left(3 \, a^{2} b - b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{12 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} + d\right)}}"," ",0,"-1/12*(18*a*b^2*cos(d*x + c)^6 - 45*a*b^2*cos(d*x + c)^4 - 9*a^3 + 9*a*b^2 + 6*(2*a^3 + 3*a*b^2)*cos(d*x + c)^2 - 12*((a^3 - 6*a*b^2)*cos(d*x + c)^4 + a^3 - 6*a*b^2 - 2*(a^3 - 6*a*b^2)*cos(d*x + c)^2)*log(1/2*sin(d*x + c)) + 4*(b^3*cos(d*x + c)^6 - 3*(3*a^2*b - b^3)*cos(d*x + c)^4 - 24*a^2*b + 8*b^3 + 12*(3*a^2*b - b^3)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)","A",0
165,1,157,0,1.492220," ","integrate((a+b*sin(d*x+c))^3*tan(d*x+c)^4,x, algorithm=""fricas"")","\frac{2 \, b^{3} \cos\left(d x + c\right)^{6} + 3 \, {\left(2 \, a^{3} + 15 \, a b^{2}\right)} d x \cos\left(d x + c\right)^{3} - 18 \, {\left(a^{2} b + b^{3}\right)} \cos\left(d x + c\right)^{4} + 6 \, a^{2} b + 2 \, b^{3} - 18 \, {\left(2 \, a^{2} b + b^{3}\right)} \cos\left(d x + c\right)^{2} - {\left(9 \, a b^{2} \cos\left(d x + c\right)^{4} - 2 \, a^{3} - 6 \, a b^{2} + 2 \, {\left(4 \, a^{3} + 21 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{6 \, d \cos\left(d x + c\right)^{3}}"," ",0,"1/6*(2*b^3*cos(d*x + c)^6 + 3*(2*a^3 + 15*a*b^2)*d*x*cos(d*x + c)^3 - 18*(a^2*b + b^3)*cos(d*x + c)^4 + 6*a^2*b + 2*b^3 - 18*(2*a^2*b + b^3)*cos(d*x + c)^2 - (9*a*b^2*cos(d*x + c)^4 - 2*a^3 - 6*a*b^2 + 2*(4*a^3 + 21*a*b^2)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^3)","A",0
166,1,116,0,1.315703," ","integrate((a+b*sin(d*x+c))^3*tan(d*x+c)^2,x, algorithm=""fricas"")","-\frac{2 \, b^{3} \cos\left(d x + c\right)^{4} + 3 \, {\left(2 \, a^{3} + 9 \, a b^{2}\right)} d x \cos\left(d x + c\right) - 18 \, a^{2} b - 6 \, b^{3} - 6 \, {\left(3 \, a^{2} b + 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2} - 3 \, {\left(3 \, a b^{2} \cos\left(d x + c\right)^{2} + 2 \, a^{3} + 6 \, a b^{2}\right)} \sin\left(d x + c\right)}{6 \, d \cos\left(d x + c\right)}"," ",0,"-1/6*(2*b^3*cos(d*x + c)^4 + 3*(2*a^3 + 9*a*b^2)*d*x*cos(d*x + c) - 18*a^2*b - 6*b^3 - 6*(3*a^2*b + 2*b^3)*cos(d*x + c)^2 - 3*(3*a*b^2*cos(d*x + c)^2 + 2*a^3 + 6*a*b^2)*sin(d*x + c))/(d*cos(d*x + c))","A",0
167,1,143,0,1.261778," ","integrate(cot(d*x+c)^2*(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{9 \, a b^{2} \cos\left(d x + c\right)^{3} + 9 \, a^{2} b \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 9 \, a^{2} b \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 3 \, {\left(2 \, a^{3} - 3 \, a b^{2}\right)} \cos\left(d x + c\right) + {\left(2 \, b^{3} \cos\left(d x + c\right)^{3} - 18 \, a^{2} b \cos\left(d x + c\right) + 3 \, {\left(2 \, a^{3} - 3 \, a b^{2}\right)} d x\right)} \sin\left(d x + c\right)}{6 \, d \sin\left(d x + c\right)}"," ",0,"-1/6*(9*a*b^2*cos(d*x + c)^3 + 9*a^2*b*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 9*a^2*b*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3*(2*a^3 - 3*a*b^2)*cos(d*x + c) + (2*b^3*cos(d*x + c)^3 - 18*a^2*b*cos(d*x + c) + 3*(2*a^3 - 3*a*b^2)*d*x)*sin(d*x + c))/(d*sin(d*x + c))","A",0
168,1,293,0,1.698638," ","integrate(cot(d*x+c)^4*(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","\frac{18 \, a b^{2} \cos\left(d x + c\right)^{5} + 8 \, {\left(2 \, a^{3} - 9 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(9 \, a^{2} b - 2 \, b^{3} - {\left(9 \, a^{2} b - 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 3 \, {\left(9 \, a^{2} b - 2 \, b^{3} - {\left(9 \, a^{2} b - 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 6 \, {\left(2 \, a^{3} - 9 \, a b^{2}\right)} \cos\left(d x + c\right) + 2 \, {\left(2 \, b^{3} \cos\left(d x + c\right)^{5} + 3 \, {\left(2 \, a^{3} - 9 \, a b^{2}\right)} d x \cos\left(d x + c\right)^{2} - 2 \, {\left(9 \, a^{2} b - 2 \, b^{3}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(2 \, a^{3} - 9 \, a b^{2}\right)} d x + 3 \, {\left(9 \, a^{2} b - 2 \, b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{12 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}"," ",0,"1/12*(18*a*b^2*cos(d*x + c)^5 + 8*(2*a^3 - 9*a*b^2)*cos(d*x + c)^3 - 3*(9*a^2*b - 2*b^3 - (9*a^2*b - 2*b^3)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3*(9*a^2*b - 2*b^3 - (9*a^2*b - 2*b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 6*(2*a^3 - 9*a*b^2)*cos(d*x + c) + 2*(2*b^3*cos(d*x + c)^5 + 3*(2*a^3 - 9*a*b^2)*d*x*cos(d*x + c)^2 - 2*(9*a^2*b - 2*b^3)*cos(d*x + c)^3 - 3*(2*a^3 - 9*a*b^2)*d*x + 3*(9*a^2*b - 2*b^3)*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))","A",0
169,1,412,0,1.356496," ","integrate(cot(d*x+c)^6*(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{360 \, a b^{2} \cos\left(d x + c\right)^{7} + 184 \, {\left(2 \, a^{3} - 15 \, a b^{2}\right)} \cos\left(d x + c\right)^{5} - 280 \, {\left(2 \, a^{3} - 15 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} + 75 \, {\left({\left(9 \, a^{2} b - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{4} + 9 \, a^{2} b - 4 \, b^{3} - 2 \, {\left(9 \, a^{2} b - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 75 \, {\left({\left(9 \, a^{2} b - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{4} + 9 \, a^{2} b - 4 \, b^{3} - 2 \, {\left(9 \, a^{2} b - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 120 \, {\left(2 \, a^{3} - 15 \, a b^{2}\right)} \cos\left(d x + c\right) + 10 \, {\left(8 \, b^{3} \cos\left(d x + c\right)^{7} + 12 \, {\left(2 \, a^{3} - 15 \, a b^{2}\right)} d x \cos\left(d x + c\right)^{4} - 8 \, {\left(9 \, a^{2} b - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{5} - 24 \, {\left(2 \, a^{3} - 15 \, a b^{2}\right)} d x \cos\left(d x + c\right)^{2} + 25 \, {\left(9 \, a^{2} b - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{3} + 12 \, {\left(2 \, a^{3} - 15 \, a b^{2}\right)} d x - 15 \, {\left(9 \, a^{2} b - 4 \, b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{240 \, {\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)}"," ",0,"-1/240*(360*a*b^2*cos(d*x + c)^7 + 184*(2*a^3 - 15*a*b^2)*cos(d*x + c)^5 - 280*(2*a^3 - 15*a*b^2)*cos(d*x + c)^3 + 75*((9*a^2*b - 4*b^3)*cos(d*x + c)^4 + 9*a^2*b - 4*b^3 - 2*(9*a^2*b - 4*b^3)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 75*((9*a^2*b - 4*b^3)*cos(d*x + c)^4 + 9*a^2*b - 4*b^3 - 2*(9*a^2*b - 4*b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 120*(2*a^3 - 15*a*b^2)*cos(d*x + c) + 10*(8*b^3*cos(d*x + c)^7 + 12*(2*a^3 - 15*a*b^2)*d*x*cos(d*x + c)^4 - 8*(9*a^2*b - 4*b^3)*cos(d*x + c)^5 - 24*(2*a^3 - 15*a*b^2)*d*x*cos(d*x + c)^2 + 25*(9*a^2*b - 4*b^3)*cos(d*x + c)^3 + 12*(2*a^3 - 15*a*b^2)*d*x - 15*(9*a^2*b - 4*b^3)*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))","A",0
170,1,261,0,2.261092," ","integrate(tan(d*x+c)^5/(a+b*sin(d*x+c)),x, algorithm=""fricas"")","\frac{16 \, a^{5} \cos\left(d x + c\right)^{4} \log\left(b \sin\left(d x + c\right) + a\right) - {\left(8 \, a^{5} + 15 \, a^{4} b - 10 \, a^{2} b^{3} + 3 \, b^{5}\right)} \cos\left(d x + c\right)^{4} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(8 \, a^{5} - 15 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5}\right)} \cos\left(d x + c\right)^{4} \log\left(-\sin\left(d x + c\right) + 1\right) + 4 \, a^{5} - 8 \, a^{3} b^{2} + 4 \, a b^{4} - 8 \, {\left(2 \, a^{5} - 3 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(2 \, a^{4} b - 4 \, a^{2} b^{3} + 2 \, b^{5} - {\left(9 \, a^{4} b - 14 \, a^{2} b^{3} + 5 \, b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{16 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} d \cos\left(d x + c\right)^{4}}"," ",0,"1/16*(16*a^5*cos(d*x + c)^4*log(b*sin(d*x + c) + a) - (8*a^5 + 15*a^4*b - 10*a^2*b^3 + 3*b^5)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - (8*a^5 - 15*a^4*b + 10*a^2*b^3 - 3*b^5)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 4*a^5 - 8*a^3*b^2 + 4*a*b^4 - 8*(2*a^5 - 3*a^3*b^2 + a*b^4)*cos(d*x + c)^2 - 2*(2*a^4*b - 4*a^2*b^3 + 2*b^5 - (9*a^4*b - 14*a^2*b^3 + 5*b^5)*cos(d*x + c)^2)*sin(d*x + c))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d*cos(d*x + c)^4)","A",0
171,1,157,0,1.741762," ","integrate(tan(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{4 \, a^{3} \cos\left(d x + c\right)^{2} \log\left(b \sin\left(d x + c\right) + a\right) - {\left(2 \, a^{3} + 3 \, a^{2} b - b^{3}\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(2 \, a^{3} - 3 \, a^{2} b + b^{3}\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, a^{3} + 2 \, a b^{2} + 2 \, {\left(a^{2} b - b^{3}\right)} \sin\left(d x + c\right)}{4 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d \cos\left(d x + c\right)^{2}}"," ",0,"-1/4*(4*a^3*cos(d*x + c)^2*log(b*sin(d*x + c) + a) - (2*a^3 + 3*a^2*b - b^3)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (2*a^3 - 3*a^2*b + b^3)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 2*a^3 + 2*a*b^2 + 2*(a^2*b - b^3)*sin(d*x + c))/((a^4 - 2*a^2*b^2 + b^4)*d*cos(d*x + c)^2)","A",0
172,1,63,0,1.328883," ","integrate(tan(d*x+c)/(a+b*sin(d*x+c)),x, algorithm=""fricas"")","\frac{2 \, a \log\left(b \sin\left(d x + c\right) + a\right) - {\left(a + b\right)} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(a - b\right)} \log\left(-\sin\left(d x + c\right) + 1\right)}{2 \, {\left(a^{2} - b^{2}\right)} d}"," ",0,"1/2*(2*a*log(b*sin(d*x + c) + a) - (a + b)*log(sin(d*x + c) + 1) - (a - b)*log(-sin(d*x + c) + 1))/((a^2 - b^2)*d)","A",0
173,1,31,0,1.528866," ","integrate(cot(d*x+c)/(a+b*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{\log\left(b \sin\left(d x + c\right) + a\right) - \log\left(-\frac{1}{2} \, \sin\left(d x + c\right)\right)}{a d}"," ",0,"-(log(b*sin(d*x + c) + a) - log(-1/2*sin(d*x + c)))/(a*d)","A",0
174,1,118,0,1.077243," ","integrate(cot(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm=""fricas"")","-\frac{2 \, a b \sin\left(d x + c\right) - a^{2} - 2 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\right)} \log\left(b \sin\left(d x + c\right) + a\right) + 2 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\right)} \log\left(-\frac{1}{2} \, \sin\left(d x + c\right)\right)}{2 \, {\left(a^{3} d \cos\left(d x + c\right)^{2} - a^{3} d\right)}}"," ",0,"-1/2*(2*a*b*sin(d*x + c) - a^2 - 2*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*log(b*sin(d*x + c) + a) + 2*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*log(-1/2*sin(d*x + c)))/(a^3*d*cos(d*x + c)^2 - a^3*d)","A",0
175,1,271,0,1.371580," ","integrate(cot(d*x+c)^5/(a+b*sin(d*x+c)),x, algorithm=""fricas"")","\frac{9 \, a^{4} - 6 \, a^{2} b^{2} - 6 \, {\left(2 \, a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2} - 12 \, {\left({\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(b \sin\left(d x + c\right) + a\right) + 12 \, {\left({\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\frac{1}{2} \, \sin\left(d x + c\right)\right) - 4 \, {\left(5 \, a^{3} b - 3 \, a b^{3} - 3 \, {\left(2 \, a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{12 \, {\left(a^{5} d \cos\left(d x + c\right)^{4} - 2 \, a^{5} d \cos\left(d x + c\right)^{2} + a^{5} d\right)}}"," ",0,"1/12*(9*a^4 - 6*a^2*b^2 - 6*(2*a^4 - a^2*b^2)*cos(d*x + c)^2 - 12*((a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*log(b*sin(d*x + c) + a) + 12*((a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*log(-1/2*sin(d*x + c)) - 4*(5*a^3*b - 3*a*b^3 - 3*(2*a^3*b - a*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a^5*d*cos(d*x + c)^4 - 2*a^5*d*cos(d*x + c)^2 + a^5*d)","A",0
176,1,476,0,1.712114," ","integrate(tan(d*x+c)^4/(a+b*sin(d*x+c)),x, algorithm=""fricas"")","\left[-\frac{3 \, \sqrt{-a^{2} + b^{2}} a^{4} \cos\left(d x + c\right)^{3} \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) + 2 \, a^{4} b - 4 \, a^{2} b^{3} + 2 \, b^{5} - 6 \, {\left(2 \, a^{4} b - 3 \, a^{2} b^{3} + b^{5}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4} - {\left(4 \, a^{5} - 5 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{6 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} d \cos\left(d x + c\right)^{3}}, -\frac{3 \, \sqrt{a^{2} - b^{2}} a^{4} \arctan\left(-\frac{a \sin\left(d x + c\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(d x + c\right)}\right) \cos\left(d x + c\right)^{3} + a^{4} b - 2 \, a^{2} b^{3} + b^{5} - 3 \, {\left(2 \, a^{4} b - 3 \, a^{2} b^{3} + b^{5}\right)} \cos\left(d x + c\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4} - {\left(4 \, a^{5} - 5 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{3 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} d \cos\left(d x + c\right)^{3}}\right]"," ",0,"[-1/6*(3*sqrt(-a^2 + b^2)*a^4*cos(d*x + c)^3*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)) + 2*a^4*b - 4*a^2*b^3 + 2*b^5 - 6*(2*a^4*b - 3*a^2*b^3 + b^5)*cos(d*x + c)^2 - 2*(a^5 - 2*a^3*b^2 + a*b^4 - (4*a^5 - 5*a^3*b^2 + a*b^4)*cos(d*x + c)^2)*sin(d*x + c))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d*cos(d*x + c)^3), -1/3*(3*sqrt(a^2 - b^2)*a^4*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))*cos(d*x + c)^3 + a^4*b - 2*a^2*b^3 + b^5 - 3*(2*a^4*b - 3*a^2*b^3 + b^5)*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4 - (4*a^5 - 5*a^3*b^2 + a*b^4)*cos(d*x + c)^2)*sin(d*x + c))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d*cos(d*x + c)^3)]","A",0
177,1,305,0,1.879369," ","integrate(tan(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a^{2} + b^{2}} a^{2} \cos\left(d x + c\right) \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) - 2 \, a^{2} b + 2 \, b^{3} + 2 \, {\left(a^{3} - a b^{2}\right)} \sin\left(d x + c\right)}{2 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d \cos\left(d x + c\right)}, \frac{\sqrt{a^{2} - b^{2}} a^{2} \arctan\left(-\frac{a \sin\left(d x + c\right) + b}{\sqrt{a^{2} - b^{2}} \cos\left(d x + c\right)}\right) \cos\left(d x + c\right) - a^{2} b + b^{3} + {\left(a^{3} - a b^{2}\right)} \sin\left(d x + c\right)}{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d \cos\left(d x + c\right)}\right]"," ",0,"[1/2*(sqrt(-a^2 + b^2)*a^2*cos(d*x + c)*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)) - 2*a^2*b + 2*b^3 + 2*(a^3 - a*b^2)*sin(d*x + c))/((a^4 - 2*a^2*b^2 + b^4)*d*cos(d*x + c)), (sqrt(a^2 - b^2)*a^2*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))*cos(d*x + c) - a^2*b + b^3 + (a^3 - a*b^2)*sin(d*x + c))/((a^4 - 2*a^2*b^2 + b^4)*d*cos(d*x + c))]","A",0
178,1,314,0,1.476411," ","integrate(cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm=""fricas"")","\left[\frac{b \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - b \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\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) \sin\left(d x + c\right) - 2 \, a \cos\left(d x + c\right)}{2 \, a^{2} d \sin\left(d x + c\right)}, \frac{b \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - b \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 2 \, \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) \sin\left(d x + c\right) - 2 \, a \cos\left(d x + c\right)}{2 \, a^{2} d \sin\left(d x + c\right)}\right]"," ",0,"[1/2*(b*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - b*log(-1/2*cos(d*x + c) + 1/2)*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))*sin(d*x + c) - 2*a*cos(d*x + c))/(a^2*d*sin(d*x + c)), 1/2*(b*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - b*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 2*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))*sin(d*x + c) - 2*a*cos(d*x + c))/(a^2*d*sin(d*x + c))]","A",0
179,1,633,0,2.146119," ","integrate(cot(d*x+c)^4/(a+b*sin(d*x+c)),x, algorithm=""fricas"")","\left[-\frac{6 \, a^{2} b \cos\left(d x + c\right) \sin\left(d x + c\right) - 4 \, {\left(4 \, a^{3} - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} + 6 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\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) \sin\left(d x + c\right) - 3 \, {\left(3 \, a^{2} b - 2 \, b^{3} - {\left(3 \, a^{2} b - 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 3 \, {\left(3 \, a^{2} b - 2 \, b^{3} - {\left(3 \, a^{2} b - 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 12 \, {\left(a^{3} - a b^{2}\right)} \cos\left(d x + c\right)}{12 \, {\left(a^{4} d \cos\left(d x + c\right)^{2} - a^{4} d\right)} \sin\left(d x + c\right)}, -\frac{6 \, a^{2} b \cos\left(d x + c\right) \sin\left(d x + c\right) - 4 \, {\left(4 \, a^{3} - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} + 12 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\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) \sin\left(d x + c\right) - 3 \, {\left(3 \, a^{2} b - 2 \, b^{3} - {\left(3 \, a^{2} b - 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 3 \, {\left(3 \, a^{2} b - 2 \, b^{3} - {\left(3 \, a^{2} b - 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 12 \, {\left(a^{3} - a b^{2}\right)} \cos\left(d x + c\right)}{12 \, {\left(a^{4} d \cos\left(d x + c\right)^{2} - a^{4} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/12*(6*a^2*b*cos(d*x + c)*sin(d*x + c) - 4*(4*a^3 - 3*a*b^2)*cos(d*x + c)^3 + 6*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*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))*sin(d*x + c) - 3*(3*a^2*b - 2*b^3 - (3*a^2*b - 2*b^3)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3*(3*a^2*b - 2*b^3 - (3*a^2*b - 2*b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 12*(a^3 - a*b^2)*cos(d*x + c))/((a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x + c)), -1/12*(6*a^2*b*cos(d*x + c)*sin(d*x + c) - 4*(4*a^3 - 3*a*b^2)*cos(d*x + c)^3 + 12*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))*sin(d*x + c) - 3*(3*a^2*b - 2*b^3 - (3*a^2*b - 2*b^3)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3*(3*a^2*b - 2*b^3 - (3*a^2*b - 2*b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 12*(a^3 - a*b^2)*cos(d*x + c))/((a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x + c))]","A",0
180,1,1079,0,3.209015," ","integrate(cot(d*x+c)^6/(a+b*sin(d*x+c)),x, algorithm=""fricas"")","\left[-\frac{16 \, {\left(23 \, a^{5} - 35 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(d x + c\right)^{5} - 80 \, {\left(7 \, a^{5} - 13 \, a^{3} b^{2} + 6 \, a b^{4}\right)} \cos\left(d x + c\right)^{3} - 120 \, {\left({\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\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) \sin\left(d x + c\right) - 15 \, {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5} + {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 15 \, {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5} + {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 240 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right) - 30 \, {\left({\left(9 \, a^{4} b - 4 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{3} - {\left(7 \, a^{4} b - 4 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{240 \, {\left(a^{6} d \cos\left(d x + c\right)^{4} - 2 \, a^{6} d \cos\left(d x + c\right)^{2} + a^{6} d\right)} \sin\left(d x + c\right)}, -\frac{16 \, {\left(23 \, a^{5} - 35 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(d x + c\right)^{5} - 80 \, {\left(7 \, a^{5} - 13 \, a^{3} b^{2} + 6 \, a b^{4}\right)} \cos\left(d x + c\right)^{3} - 240 \, {\left({\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\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) \sin\left(d x + c\right) - 15 \, {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5} + {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 15 \, {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5} + {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(15 \, a^{4} b - 20 \, a^{2} b^{3} + 8 \, b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 240 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right) - 30 \, {\left({\left(9 \, a^{4} b - 4 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{3} - {\left(7 \, a^{4} b - 4 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{240 \, {\left(a^{6} d \cos\left(d x + c\right)^{4} - 2 \, a^{6} d \cos\left(d x + c\right)^{2} + a^{6} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/240*(16*(23*a^5 - 35*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 - 80*(7*a^5 - 13*a^3*b^2 + 6*a*b^4)*cos(d*x + c)^3 - 120*((a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*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))*sin(d*x + c) - 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 240*(a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c) - 30*((9*a^4*b - 4*a^2*b^3)*cos(d*x + c)^3 - (7*a^4*b - 4*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/((a^6*d*cos(d*x + c)^4 - 2*a^6*d*cos(d*x + c)^2 + a^6*d)*sin(d*x + c)), -1/240*(16*(23*a^5 - 35*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 - 80*(7*a^5 - 13*a^3*b^2 + 6*a*b^4)*cos(d*x + c)^3 - 240*((a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))*sin(d*x + c) - 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 15*(15*a^4*b - 20*a^2*b^3 + 8*b^5 + (15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^4 - 2*(15*a^4*b - 20*a^2*b^3 + 8*b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 240*(a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c) - 30*((9*a^4*b - 4*a^2*b^3)*cos(d*x + c)^3 - (7*a^4*b - 4*a^2*b^3)*cos(d*x + c))*sin(d*x + c))/((a^6*d*cos(d*x + c)^4 - 2*a^6*d*cos(d*x + c)^2 + a^6*d)*sin(d*x + c))]","A",0
181,1,555,0,2.872354," ","integrate(tan(d*x+c)^5/(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{2 \, a^{7} - 6 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 2 \, a b^{6} - 2 \, {\left(4 \, a^{7} + 5 \, a^{5} b^{2} - 10 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{7} - 9 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)^{2} + 8 \, {\left({\left(a^{6} b + 5 \, a^{4} b^{3}\right)} \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) + {\left(a^{7} + 5 \, a^{5} b^{2}\right)} \cos\left(d x + c\right)^{4}\right)} \log\left(b \sin\left(d x + c\right) + a\right) - {\left({\left(4 \, a^{6} b + 15 \, a^{5} b^{2} + 20 \, a^{4} b^{3} + 10 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) + {\left(4 \, a^{7} + 15 \, a^{6} b + 20 \, a^{5} b^{2} + 10 \, a^{4} b^{3} - a^{2} b^{5}\right)} \cos\left(d x + c\right)^{4}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - {\left({\left(4 \, a^{6} b - 15 \, a^{5} b^{2} + 20 \, a^{4} b^{3} - 10 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) + {\left(4 \, a^{7} - 15 \, a^{6} b + 20 \, a^{5} b^{2} - 10 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(d x + c\right)^{4}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left(5 \, a^{6} b - 12 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - 2 \, b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{8 \, {\left({\left(a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right)} d \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) + {\left(a^{9} - 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} + a b^{8}\right)} d \cos\left(d x + c\right)^{4}\right)}}"," ",0,"1/8*(2*a^7 - 6*a^5*b^2 + 6*a^3*b^4 - 2*a*b^6 - 2*(4*a^7 + 5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*cos(d*x + c)^4 - 2*(4*a^7 - 9*a^5*b^2 + 6*a^3*b^4 - a*b^6)*cos(d*x + c)^2 + 8*((a^6*b + 5*a^4*b^3)*cos(d*x + c)^4*sin(d*x + c) + (a^7 + 5*a^5*b^2)*cos(d*x + c)^4)*log(b*sin(d*x + c) + a) - ((4*a^6*b + 15*a^5*b^2 + 20*a^4*b^3 + 10*a^3*b^4 - a*b^6)*cos(d*x + c)^4*sin(d*x + c) + (4*a^7 + 15*a^6*b + 20*a^5*b^2 + 10*a^4*b^3 - a^2*b^5)*cos(d*x + c)^4)*log(sin(d*x + c) + 1) - ((4*a^6*b - 15*a^5*b^2 + 20*a^4*b^3 - 10*a^3*b^4 + a*b^6)*cos(d*x + c)^4*sin(d*x + c) + (4*a^7 - 15*a^6*b + 20*a^5*b^2 - 10*a^4*b^3 + a^2*b^5)*cos(d*x + c)^4)*log(-sin(d*x + c) + 1) - 2*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (5*a^6*b - 12*a^4*b^3 + 9*a^2*b^5 - 2*b^7)*cos(d*x + c)^2)*sin(d*x + c))/((a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*cos(d*x + c)^4*sin(d*x + c) + (a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c)^4)","B",0
182,1,388,0,1.622745," ","integrate(tan(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{a^{5} - 2 \, a^{3} b^{2} + a b^{4} + 2 \, {\left(a^{5} - a b^{4}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left({\left(a^{4} b + 3 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) + {\left(a^{5} + 3 \, a^{3} b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(b \sin\left(d x + c\right) + a\right) + {\left({\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) + {\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\sin\left(d x + c\right) + 1\right) + {\left({\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} \sin\left(d x + c\right)}{2 \, {\left({\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} d \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} d \cos\left(d x + c\right)^{2}\right)}}"," ",0,"1/2*(a^5 - 2*a^3*b^2 + a*b^4 + 2*(a^5 - a*b^4)*cos(d*x + c)^2 - 2*((a^4*b + 3*a^2*b^3)*cos(d*x + c)^2*sin(d*x + c) + (a^5 + 3*a^3*b^2)*cos(d*x + c)^2)*log(b*sin(d*x + c) + a) + ((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(d*x + c)^2*sin(d*x + c) + (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + ((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*cos(d*x + c)^2*sin(d*x + c) + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - (a^4*b - 2*a^2*b^3 + b^5)*sin(d*x + c))/((a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*d*cos(d*x + c)^2*sin(d*x + c) + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d*cos(d*x + c)^2)","B",0
183,1,195,0,1.790814," ","integrate(tan(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","-\frac{2 \, a^{3} - 2 \, a b^{2} - 2 \, {\left(a^{3} + a b^{2} + {\left(a^{2} b + b^{3}\right)} \sin\left(d x + c\right)\right)} \log\left(b \sin\left(d x + c\right) + a\right) + {\left(a^{3} + 2 \, a^{2} b + a b^{2} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \sin\left(d x + c\right)\right)} \log\left(\sin\left(d x + c\right) + 1\right) + {\left(a^{3} - 2 \, a^{2} b + a b^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \sin\left(d x + c\right)\right)} \log\left(-\sin\left(d x + c\right) + 1\right)}{2 \, {\left({\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} d \sin\left(d x + c\right) + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d\right)}}"," ",0,"-1/2*(2*a^3 - 2*a*b^2 - 2*(a^3 + a*b^2 + (a^2*b + b^3)*sin(d*x + c))*log(b*sin(d*x + c) + a) + (a^3 + 2*a^2*b + a*b^2 + (a^2*b + 2*a*b^2 + b^3)*sin(d*x + c))*log(sin(d*x + c) + 1) + (a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*sin(d*x + c))*log(-sin(d*x + c) + 1))/((a^4*b - 2*a^2*b^3 + b^5)*d*sin(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d)","A",0
184,1,69,0,1.467966," ","integrate(cot(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","-\frac{{\left(b \sin\left(d x + c\right) + a\right)} \log\left(b \sin\left(d x + c\right) + a\right) - {\left(b \sin\left(d x + c\right) + a\right)} \log\left(-\frac{1}{2} \, \sin\left(d x + c\right)\right) - a}{a^{2} b d \sin\left(d x + c\right) + a^{3} d}"," ",0,"-((b*sin(d*x + c) + a)*log(b*sin(d*x + c) + a) - (b*sin(d*x + c) + a)*log(-1/2*sin(d*x + c)) - a)/(a^2*b*d*sin(d*x + c) + a^3*d)","A",0
185,1,259,0,1.190408," ","integrate(cot(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","-\frac{3 \, a^{2} b \sin\left(d x + c\right) - 3 \, a^{3} + 6 \, a b^{2} + 2 \, {\left(a^{3} - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, {\left(a^{3} - 3 \, a b^{2} - {\left(a^{3} - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{2} b - 3 \, b^{3} - {\left(a^{2} b - 3 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(b \sin\left(d x + c\right) + a\right) - 2 \, {\left(a^{3} - 3 \, a b^{2} - {\left(a^{3} - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{2} b - 3 \, b^{3} - {\left(a^{2} b - 3 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \sin\left(d x + c\right)\right)}{2 \, {\left(a^{5} d \cos\left(d x + c\right)^{2} - a^{5} d + {\left(a^{4} b d \cos\left(d x + c\right)^{2} - a^{4} b d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"-1/2*(3*a^2*b*sin(d*x + c) - 3*a^3 + 6*a*b^2 + 2*(a^3 - 3*a*b^2)*cos(d*x + c)^2 + 2*(a^3 - 3*a*b^2 - (a^3 - 3*a*b^2)*cos(d*x + c)^2 + (a^2*b - 3*b^3 - (a^2*b - 3*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(b*sin(d*x + c) + a) - 2*(a^3 - 3*a*b^2 - (a^3 - 3*a*b^2)*cos(d*x + c)^2 + (a^2*b - 3*b^3 - (a^2*b - 3*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*sin(d*x + c)))/(a^5*d*cos(d*x + c)^2 - a^5*d + (a^4*b*d*cos(d*x + c)^2 - a^4*b*d)*sin(d*x + c))","B",0
186,1,542,0,1.380626," ","integrate(cot(d*x+c)^5/(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","\frac{21 \, a^{5} - 82 \, a^{3} b^{2} + 60 \, a b^{4} + 12 \, {\left(a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(18 \, a^{5} - 77 \, a^{3} b^{2} + 60 \, a b^{4}\right)} \cos\left(d x + c\right)^{2} - 12 \, {\left(a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4} + {\left(a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5} + {\left(a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(b \sin\left(d x + c\right) + a\right) + 12 \, {\left(a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4} + {\left(a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{5} - 6 \, a^{3} b^{2} + 5 \, a b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5} + {\left(a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{4} b - 6 \, a^{2} b^{3} + 5 \, b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \sin\left(d x + c\right)\right) - {\left(31 \, a^{4} b - 30 \, a^{2} b^{3} - 6 \, {\left(6 \, a^{4} b - 5 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{12 \, {\left(a^{7} d \cos\left(d x + c\right)^{4} - 2 \, a^{7} d \cos\left(d x + c\right)^{2} + a^{7} d + {\left(a^{6} b d \cos\left(d x + c\right)^{4} - 2 \, a^{6} b d \cos\left(d x + c\right)^{2} + a^{6} b d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/12*(21*a^5 - 82*a^3*b^2 + 60*a*b^4 + 12*(a^5 - 6*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^4 - 2*(18*a^5 - 77*a^3*b^2 + 60*a*b^4)*cos(d*x + c)^2 - 12*(a^5 - 6*a^3*b^2 + 5*a*b^4 + (a^5 - 6*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^4 - 2*(a^5 - 6*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^2 + (a^4*b - 6*a^2*b^3 + 5*b^5 + (a^4*b - 6*a^2*b^3 + 5*b^5)*cos(d*x + c)^4 - 2*(a^4*b - 6*a^2*b^3 + 5*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(b*sin(d*x + c) + a) + 12*(a^5 - 6*a^3*b^2 + 5*a*b^4 + (a^5 - 6*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^4 - 2*(a^5 - 6*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^2 + (a^4*b - 6*a^2*b^3 + 5*b^5 + (a^4*b - 6*a^2*b^3 + 5*b^5)*cos(d*x + c)^4 - 2*(a^4*b - 6*a^2*b^3 + 5*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*sin(d*x + c)) - (31*a^4*b - 30*a^2*b^3 - 6*(6*a^4*b - 5*a^2*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a^7*d*cos(d*x + c)^4 - 2*a^7*d*cos(d*x + c)^2 + a^7*d + (a^6*b*d*cos(d*x + c)^4 - 2*a^6*b*d*cos(d*x + c)^2 + a^6*b*d)*sin(d*x + c))","B",0
187,1,815,0,1.919536," ","integrate(tan(d*x+c)^4/(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","\left[-\frac{2 \, a^{6} b - 6 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - 2 \, b^{7} - 2 \, {\left(7 \, a^{6} b + 2 \, a^{4} b^{3} - 10 \, a^{2} b^{5} + b^{7}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(7 \, a^{6} b - 16 \, a^{4} b^{3} + 11 \, a^{2} b^{5} - 2 \, b^{7}\right)} \cos\left(d x + c\right)^{2} - 3 \, {\left({\left(a^{5} b + 4 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{3} \sin\left(d x + c\right) + {\left(a^{6} + 4 \, a^{4} b^{2}\right)} \cos\left(d x + c\right)^{3}\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) - 2 \, {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6} - {\left(4 \, a^{7} - 7 \, a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{6 \, {\left({\left(a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right)} d \cos\left(d x + c\right)^{3} \sin\left(d x + c\right) + {\left(a^{9} - 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} + a b^{8}\right)} d \cos\left(d x + c\right)^{3}\right)}}, -\frac{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left(7 \, a^{6} b + 2 \, a^{4} b^{3} - 10 \, a^{2} b^{5} + b^{7}\right)} \cos\left(d x + c\right)^{4} - {\left(7 \, a^{6} b - 16 \, a^{4} b^{3} + 11 \, a^{2} b^{5} - 2 \, b^{7}\right)} \cos\left(d x + c\right)^{2} + 3 \, {\left({\left(a^{5} b + 4 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{3} \sin\left(d x + c\right) + {\left(a^{6} + 4 \, a^{4} b^{2}\right)} \cos\left(d x + c\right)^{3}\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) - {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6} - {\left(4 \, a^{7} - 7 \, a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{3 \, {\left({\left(a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right)} d \cos\left(d x + c\right)^{3} \sin\left(d x + c\right) + {\left(a^{9} - 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} + a b^{8}\right)} d \cos\left(d x + c\right)^{3}\right)}}\right]"," ",0,"[-1/6*(2*a^6*b - 6*a^4*b^3 + 6*a^2*b^5 - 2*b^7 - 2*(7*a^6*b + 2*a^4*b^3 - 10*a^2*b^5 + b^7)*cos(d*x + c)^4 - 2*(7*a^6*b - 16*a^4*b^3 + 11*a^2*b^5 - 2*b^7)*cos(d*x + c)^2 - 3*((a^5*b + 4*a^3*b^3)*cos(d*x + c)^3*sin(d*x + c) + (a^6 + 4*a^4*b^2)*cos(d*x + c)^3)*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)) - 2*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6 - (4*a^7 - 7*a^5*b^2 + 2*a^3*b^4 + a*b^6)*cos(d*x + c)^2)*sin(d*x + c))/((a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*cos(d*x + c)^3*sin(d*x + c) + (a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c)^3), -1/3*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (7*a^6*b + 2*a^4*b^3 - 10*a^2*b^5 + b^7)*cos(d*x + c)^4 - (7*a^6*b - 16*a^4*b^3 + 11*a^2*b^5 - 2*b^7)*cos(d*x + c)^2 + 3*((a^5*b + 4*a^3*b^3)*cos(d*x + c)^3*sin(d*x + c) + (a^6 + 4*a^4*b^2)*cos(d*x + c)^3)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6 - (4*a^7 - 7*a^5*b^2 + 2*a^3*b^4 + a*b^6)*cos(d*x + c)^2)*sin(d*x + c))/((a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*cos(d*x + c)^3*sin(d*x + c) + (a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c)^3)]","A",0
188,1,569,0,1.709977," ","integrate(tan(d*x+c)^2/(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","\left[-\frac{2 \, a^{4} b - 4 \, a^{2} b^{3} + 2 \, b^{5} + 2 \, {\left(2 \, a^{4} b - a^{2} b^{3} - b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left({\left(a^{3} b + 2 \, a b^{3}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a^{4} + 2 \, a^{2} b^{2}\right)} \cos\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) - 2 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \sin\left(d x + c\right)}{2 \, {\left({\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} d \cos\left(d x + c\right)\right)}}, -\frac{a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left(2 \, a^{4} b - a^{2} b^{3} - b^{5}\right)} \cos\left(d x + c\right)^{2} - {\left({\left(a^{3} b + 2 \, a b^{3}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a^{4} + 2 \, a^{2} b^{2}\right)} \cos\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) - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \sin\left(d x + c\right)}{{\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} d \cos\left(d x + c\right)}\right]"," ",0,"[-1/2*(2*a^4*b - 4*a^2*b^3 + 2*b^5 + 2*(2*a^4*b - a^2*b^3 - b^5)*cos(d*x + c)^2 + ((a^3*b + 2*a*b^3)*cos(d*x + c)*sin(d*x + c) + (a^4 + 2*a^2*b^2)*cos(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)) - 2*(a^5 - 2*a^3*b^2 + a*b^4)*sin(d*x + c))/((a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*d*cos(d*x + c)*sin(d*x + c) + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d*cos(d*x + c)), -(a^4*b - 2*a^2*b^3 + b^5 + (2*a^4*b - a^2*b^3 - b^5)*cos(d*x + c)^2 - ((a^3*b + 2*a*b^3)*cos(d*x + c)*sin(d*x + c) + (a^4 + 2*a^2*b^2)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - (a^5 - 2*a^3*b^2 + a*b^4)*sin(d*x + c))/((a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*d*cos(d*x + c)*sin(d*x + c) + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d*cos(d*x + c))]","A",0
189,1,768,0,2.053357," ","integrate(cot(d*x+c)^2/(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a^{2} b - 2 \, b^{3} - {\left(a^{2} b - 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{3} - 2 \, a b^{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) + 2 \, {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right) - 2 \, {\left(a^{2} b^{2} - b^{4} - {\left(a^{2} b^{2} - b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{3} b - a b^{3}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 2 \, {\left(a^{2} b^{2} - b^{4} - {\left(a^{2} b^{2} - b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{3} b - a b^{3}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{2 \, {\left({\left(a^{5} b - a^{3} b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{6} - a^{4} b^{2}\right)} d \sin\left(d x + c\right) - {\left(a^{5} b - a^{3} b^{3}\right)} d\right)}}, \frac{2 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a^{2} b - 2 \, b^{3} - {\left(a^{2} b - 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{3} - 2 \, a b^{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) + {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right) - {\left(a^{2} b^{2} - b^{4} - {\left(a^{2} b^{2} - b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{3} b - a b^{3}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left(a^{2} b^{2} - b^{4} - {\left(a^{2} b^{2} - b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{3} b - a b^{3}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{{\left(a^{5} b - a^{3} b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{6} - a^{4} b^{2}\right)} d \sin\left(d x + c\right) - {\left(a^{5} b - a^{3} b^{3}\right)} d}\right]"," ",0,"[1/2*(4*(a^3*b - a*b^3)*cos(d*x + c)*sin(d*x + c) - (a^2*b - 2*b^3 - (a^2*b - 2*b^3)*cos(d*x + c)^2 + (a^3 - 2*a*b^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)) + 2*(a^4 - a^2*b^2)*cos(d*x + c) - 2*(a^2*b^2 - b^4 - (a^2*b^2 - b^4)*cos(d*x + c)^2 + (a^3*b - a*b^3)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 2*(a^2*b^2 - b^4 - (a^2*b^2 - b^4)*cos(d*x + c)^2 + (a^3*b - a*b^3)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^5*b - a^3*b^3)*d*cos(d*x + c)^2 - (a^6 - a^4*b^2)*d*sin(d*x + c) - (a^5*b - a^3*b^3)*d), (2*(a^3*b - a*b^3)*cos(d*x + c)*sin(d*x + c) - (a^2*b - 2*b^3 - (a^2*b - 2*b^3)*cos(d*x + c)^2 + (a^3 - 2*a*b^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))) + (a^4 - a^2*b^2)*cos(d*x + c) - (a^2*b^2 - b^4 - (a^2*b^2 - b^4)*cos(d*x + c)^2 + (a^3*b - a*b^3)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (a^2*b^2 - b^4 - (a^2*b^2 - b^4)*cos(d*x + c)^2 + (a^3*b - a*b^3)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^5*b - a^3*b^3)*d*cos(d*x + c)^2 - (a^6 - a^4*b^2)*d*sin(d*x + c) - (a^5*b - a^3*b^3)*d)]","B",0
190,1,1149,0,2.306200," ","integrate(cot(d*x+c)^4/(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(2 \, a^{4} - 3 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left({\left(a^{2} b - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{4} + a^{2} b - 4 \, b^{3} - 2 \, {\left(a^{2} b - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{3} - 4 \, a b^{2} - {\left(a^{3} - 4 \, a b^{2}\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) - 6 \, {\left(a^{4} - 2 \, a^{2} b^{2}\right)} \cos\left(d x + c\right) + 3 \, {\left({\left(3 \, a^{2} b^{2} - 4 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, a^{2} b^{2} - 4 \, b^{4} - 2 \, {\left(3 \, a^{2} b^{2} - 4 \, b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(3 \, a^{3} b - 4 \, a b^{3} - {\left(3 \, a^{3} b - 4 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 3 \, {\left({\left(3 \, a^{2} b^{2} - 4 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, a^{2} b^{2} - 4 \, b^{4} - 2 \, {\left(3 \, a^{2} b^{2} - 4 \, b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(3 \, a^{3} b - 4 \, a b^{3} - {\left(3 \, a^{3} b - 4 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 2 \, {\left({\left(7 \, a^{3} b - 12 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(3 \, a^{3} b - 4 \, a b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{6 \, {\left(a^{5} b d \cos\left(d x + c\right)^{4} - 2 \, a^{5} b d \cos\left(d x + c\right)^{2} + a^{5} b d - {\left(a^{6} d \cos\left(d x + c\right)^{2} - a^{6} d\right)} \sin\left(d x + c\right)\right)}}, -\frac{4 \, {\left(2 \, a^{4} - 3 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{3} + 6 \, {\left({\left(a^{2} b - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{4} + a^{2} b - 4 \, b^{3} - 2 \, {\left(a^{2} b - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{3} - 4 \, a b^{2} - {\left(a^{3} - 4 \, a b^{2}\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(a^{4} - 2 \, a^{2} b^{2}\right)} \cos\left(d x + c\right) + 3 \, {\left({\left(3 \, a^{2} b^{2} - 4 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, a^{2} b^{2} - 4 \, b^{4} - 2 \, {\left(3 \, a^{2} b^{2} - 4 \, b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(3 \, a^{3} b - 4 \, a b^{3} - {\left(3 \, a^{3} b - 4 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 3 \, {\left({\left(3 \, a^{2} b^{2} - 4 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, a^{2} b^{2} - 4 \, b^{4} - 2 \, {\left(3 \, a^{2} b^{2} - 4 \, b^{4}\right)} \cos\left(d x + c\right)^{2} + {\left(3 \, a^{3} b - 4 \, a b^{3} - {\left(3 \, a^{3} b - 4 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 2 \, {\left({\left(7 \, a^{3} b - 12 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(3 \, a^{3} b - 4 \, a b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{6 \, {\left(a^{5} b d \cos\left(d x + c\right)^{4} - 2 \, a^{5} b d \cos\left(d x + c\right)^{2} + a^{5} b d - {\left(a^{6} d \cos\left(d x + c\right)^{2} - a^{6} d\right)} \sin\left(d x + c\right)\right)}}\right]"," ",0,"[-1/6*(4*(2*a^4 - 3*a^2*b^2)*cos(d*x + c)^3 + 3*((a^2*b - 4*b^3)*cos(d*x + c)^4 + a^2*b - 4*b^3 - 2*(a^2*b - 4*b^3)*cos(d*x + c)^2 + (a^3 - 4*a*b^2 - (a^3 - 4*a*b^2)*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)) - 6*(a^4 - 2*a^2*b^2)*cos(d*x + c) + 3*((3*a^2*b^2 - 4*b^4)*cos(d*x + c)^4 + 3*a^2*b^2 - 4*b^4 - 2*(3*a^2*b^2 - 4*b^4)*cos(d*x + c)^2 + (3*a^3*b - 4*a*b^3 - (3*a^3*b - 4*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*((3*a^2*b^2 - 4*b^4)*cos(d*x + c)^4 + 3*a^2*b^2 - 4*b^4 - 2*(3*a^2*b^2 - 4*b^4)*cos(d*x + c)^2 + (3*a^3*b - 4*a*b^3 - (3*a^3*b - 4*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*((7*a^3*b - 12*a*b^3)*cos(d*x + c)^3 - 3*(3*a^3*b - 4*a*b^3)*cos(d*x + c))*sin(d*x + c))/(a^5*b*d*cos(d*x + c)^4 - 2*a^5*b*d*cos(d*x + c)^2 + a^5*b*d - (a^6*d*cos(d*x + c)^2 - a^6*d)*sin(d*x + c)), -1/6*(4*(2*a^4 - 3*a^2*b^2)*cos(d*x + c)^3 + 6*((a^2*b - 4*b^3)*cos(d*x + c)^4 + a^2*b - 4*b^3 - 2*(a^2*b - 4*b^3)*cos(d*x + c)^2 + (a^3 - 4*a*b^2 - (a^3 - 4*a*b^2)*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*(a^4 - 2*a^2*b^2)*cos(d*x + c) + 3*((3*a^2*b^2 - 4*b^4)*cos(d*x + c)^4 + 3*a^2*b^2 - 4*b^4 - 2*(3*a^2*b^2 - 4*b^4)*cos(d*x + c)^2 + (3*a^3*b - 4*a*b^3 - (3*a^3*b - 4*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*((3*a^2*b^2 - 4*b^4)*cos(d*x + c)^4 + 3*a^2*b^2 - 4*b^4 - 2*(3*a^2*b^2 - 4*b^4)*cos(d*x + c)^2 + (3*a^3*b - 4*a*b^3 - (3*a^3*b - 4*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*((7*a^3*b - 12*a*b^3)*cos(d*x + c)^3 - 3*(3*a^3*b - 4*a*b^3)*cos(d*x + c))*sin(d*x + c))/(a^5*b*d*cos(d*x + c)^4 - 2*a^5*b*d*cos(d*x + c)^2 + a^5*b*d - (a^6*d*cos(d*x + c)^2 - a^6*d)*sin(d*x + c))]","B",0
191,1,2011,0,3.347361," ","integrate(cot(d*x+c)^6/(a+b*sin(d*x+c))^2,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(92 \, a^{6} - 285 \, a^{4} b^{2} + 180 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{5} - 40 \, {\left(7 \, a^{6} - 27 \, a^{4} b^{2} + 18 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} + 60 \, {\left({\left(a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right)} \cos\left(d x + c\right)^{6} - a^{4} b + 7 \, a^{2} b^{3} - 6 \, b^{5} - 3 \, {\left(a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right)} \cos\left(d x + c\right)^{2} - {\left(a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4} + {\left(a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4}\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) + 30 \, {\left(4 \, a^{6} - 17 \, a^{4} b^{2} + 12 \, a^{2} b^{4}\right)} \cos\left(d x + c\right) + 15 \, {\left({\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{6} - 15 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 24 \, b^{6} - 3 \, {\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{2} - {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5} + {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 15 \, {\left({\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{6} - 15 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 24 \, b^{6} - 3 \, {\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{2} - {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5} + {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 2 \, {\left(4 \, {\left(38 \, a^{5} b - 135 \, a^{3} b^{3} + 90 \, a b^{5}\right)} \cos\left(d x + c\right)^{5} - 5 \, {\left(79 \, a^{5} b - 228 \, a^{3} b^{3} + 144 \, a b^{5}\right)} \cos\left(d x + c\right)^{3} + 15 \, {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{120 \, {\left(a^{7} b d \cos\left(d x + c\right)^{6} - 3 \, a^{7} b d \cos\left(d x + c\right)^{4} + 3 \, a^{7} b d \cos\left(d x + c\right)^{2} - a^{7} b d - {\left(a^{8} d \cos\left(d x + c\right)^{4} - 2 \, a^{8} d \cos\left(d x + c\right)^{2} + a^{8} d\right)} \sin\left(d x + c\right)\right)}}, \frac{2 \, {\left(92 \, a^{6} - 285 \, a^{4} b^{2} + 180 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{5} - 40 \, {\left(7 \, a^{6} - 27 \, a^{4} b^{2} + 18 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} + 120 \, {\left({\left(a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right)} \cos\left(d x + c\right)^{6} - a^{4} b + 7 \, a^{2} b^{3} - 6 \, b^{5} - 3 \, {\left(a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right)} \cos\left(d x + c\right)^{2} - {\left(a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4} + {\left(a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4}\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) + 30 \, {\left(4 \, a^{6} - 17 \, a^{4} b^{2} + 12 \, a^{2} b^{4}\right)} \cos\left(d x + c\right) + 15 \, {\left({\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{6} - 15 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 24 \, b^{6} - 3 \, {\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{2} - {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5} + {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 15 \, {\left({\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{6} - 15 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 24 \, b^{6} - 3 \, {\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(15 \, a^{4} b^{2} - 40 \, a^{2} b^{4} + 24 \, b^{6}\right)} \cos\left(d x + c\right)^{2} - {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5} + {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 2 \, {\left(4 \, {\left(38 \, a^{5} b - 135 \, a^{3} b^{3} + 90 \, a b^{5}\right)} \cos\left(d x + c\right)^{5} - 5 \, {\left(79 \, a^{5} b - 228 \, a^{3} b^{3} + 144 \, a b^{5}\right)} \cos\left(d x + c\right)^{3} + 15 \, {\left(15 \, a^{5} b - 40 \, a^{3} b^{3} + 24 \, a b^{5}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{120 \, {\left(a^{7} b d \cos\left(d x + c\right)^{6} - 3 \, a^{7} b d \cos\left(d x + c\right)^{4} + 3 \, a^{7} b d \cos\left(d x + c\right)^{2} - a^{7} b d - {\left(a^{8} d \cos\left(d x + c\right)^{4} - 2 \, a^{8} d \cos\left(d x + c\right)^{2} + a^{8} d\right)} \sin\left(d x + c\right)\right)}}\right]"," ",0,"[1/120*(2*(92*a^6 - 285*a^4*b^2 + 180*a^2*b^4)*cos(d*x + c)^5 - 40*(7*a^6 - 27*a^4*b^2 + 18*a^2*b^4)*cos(d*x + c)^3 + 60*((a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^6 - a^4*b + 7*a^2*b^3 - 6*b^5 - 3*(a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^4 + 3*(a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^2 - (a^5 - 7*a^3*b^2 + 6*a*b^4 + (a^5 - 7*a^3*b^2 + 6*a*b^4)*cos(d*x + c)^4 - 2*(a^5 - 7*a^3*b^2 + 6*a*b^4)*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)) + 30*(4*a^6 - 17*a^4*b^2 + 12*a^2*b^4)*cos(d*x + c) + 15*((15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^6 - 15*a^4*b^2 + 40*a^2*b^4 - 24*b^6 - 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^4 + 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^2 - (15*a^5*b - 40*a^3*b^3 + 24*a*b^5 + (15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^4 - 2*(15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 15*((15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^6 - 15*a^4*b^2 + 40*a^2*b^4 - 24*b^6 - 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^4 + 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^2 - (15*a^5*b - 40*a^3*b^3 + 24*a*b^5 + (15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^4 - 2*(15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*(4*(38*a^5*b - 135*a^3*b^3 + 90*a*b^5)*cos(d*x + c)^5 - 5*(79*a^5*b - 228*a^3*b^3 + 144*a*b^5)*cos(d*x + c)^3 + 15*(15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c))*sin(d*x + c))/(a^7*b*d*cos(d*x + c)^6 - 3*a^7*b*d*cos(d*x + c)^4 + 3*a^7*b*d*cos(d*x + c)^2 - a^7*b*d - (a^8*d*cos(d*x + c)^4 - 2*a^8*d*cos(d*x + c)^2 + a^8*d)*sin(d*x + c)), 1/120*(2*(92*a^6 - 285*a^4*b^2 + 180*a^2*b^4)*cos(d*x + c)^5 - 40*(7*a^6 - 27*a^4*b^2 + 18*a^2*b^4)*cos(d*x + c)^3 + 120*((a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^6 - a^4*b + 7*a^2*b^3 - 6*b^5 - 3*(a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^4 + 3*(a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^2 - (a^5 - 7*a^3*b^2 + 6*a*b^4 + (a^5 - 7*a^3*b^2 + 6*a*b^4)*cos(d*x + c)^4 - 2*(a^5 - 7*a^3*b^2 + 6*a*b^4)*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))) + 30*(4*a^6 - 17*a^4*b^2 + 12*a^2*b^4)*cos(d*x + c) + 15*((15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^6 - 15*a^4*b^2 + 40*a^2*b^4 - 24*b^6 - 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^4 + 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^2 - (15*a^5*b - 40*a^3*b^3 + 24*a*b^5 + (15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^4 - 2*(15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 15*((15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^6 - 15*a^4*b^2 + 40*a^2*b^4 - 24*b^6 - 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^4 + 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^2 - (15*a^5*b - 40*a^3*b^3 + 24*a*b^5 + (15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^4 - 2*(15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*(4*(38*a^5*b - 135*a^3*b^3 + 90*a*b^5)*cos(d*x + c)^5 - 5*(79*a^5*b - 228*a^3*b^3 + 144*a*b^5)*cos(d*x + c)^3 + 15*(15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c))*sin(d*x + c))/(a^7*b*d*cos(d*x + c)^6 - 3*a^7*b*d*cos(d*x + c)^4 + 3*a^7*b*d*cos(d*x + c)^2 - a^7*b*d - (a^8*d*cos(d*x + c)^4 - 2*a^8*d*cos(d*x + c)^2 + a^8*d)*sin(d*x + c))]","B",0
192,1,981,0,3.806986," ","integrate(tan(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{4 \, a^{9} - 16 \, a^{7} b^{2} + 24 \, a^{5} b^{4} - 16 \, a^{3} b^{6} + 4 \, a b^{8} - 4 \, {\left(6 \, a^{9} + 35 \, a^{7} b^{2} - 39 \, a^{5} b^{4} - 3 \, a^{3} b^{6} + a b^{8}\right)} \cos\left(d x + c\right)^{4} - 16 \, {\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} \cos\left(d x + c\right)^{2} - 16 \, {\left({\left(a^{7} b^{2} + 13 \, a^{5} b^{4} + 10 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{8} b + 13 \, a^{6} b^{3} + 10 \, a^{4} b^{5}\right)} \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) - {\left(a^{9} + 14 \, a^{7} b^{2} + 23 \, a^{5} b^{4} + 10 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)^{4}\right)} \log\left(b \sin\left(d x + c\right) + a\right) + {\left({\left(8 \, a^{7} b^{2} + 45 \, a^{6} b^{3} + 104 \, a^{5} b^{4} + 125 \, a^{4} b^{5} + 80 \, a^{3} b^{6} + 23 \, a^{2} b^{7} - b^{9}\right)} \cos\left(d x + c\right)^{6} - 2 \, {\left(8 \, a^{8} b + 45 \, a^{7} b^{2} + 104 \, a^{6} b^{3} + 125 \, a^{5} b^{4} + 80 \, a^{4} b^{5} + 23 \, a^{3} b^{6} - a b^{8}\right)} \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) - {\left(8 \, a^{9} + 45 \, a^{8} b + 112 \, a^{7} b^{2} + 170 \, a^{6} b^{3} + 184 \, a^{5} b^{4} + 148 \, a^{4} b^{5} + 80 \, a^{3} b^{6} + 22 \, a^{2} b^{7} - b^{9}\right)} \cos\left(d x + c\right)^{4}\right)} \log\left(\sin\left(d x + c\right) + 1\right) + {\left({\left(8 \, a^{7} b^{2} - 45 \, a^{6} b^{3} + 104 \, a^{5} b^{4} - 125 \, a^{4} b^{5} + 80 \, a^{3} b^{6} - 23 \, a^{2} b^{7} + b^{9}\right)} \cos\left(d x + c\right)^{6} - 2 \, {\left(8 \, a^{8} b - 45 \, a^{7} b^{2} + 104 \, a^{6} b^{3} - 125 \, a^{5} b^{4} + 80 \, a^{4} b^{5} - 23 \, a^{3} b^{6} + a b^{8}\right)} \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) - {\left(8 \, a^{9} - 45 \, a^{8} b + 112 \, a^{7} b^{2} - 170 \, a^{6} b^{3} + 184 \, a^{5} b^{4} - 148 \, a^{4} b^{5} + 80 \, a^{3} b^{6} - 22 \, a^{2} b^{7} + b^{9}\right)} \cos\left(d x + c\right)^{4}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(2 \, a^{8} b - 8 \, a^{6} b^{3} + 12 \, a^{4} b^{5} - 8 \, a^{2} b^{7} + 2 \, b^{9} + {\left(8 \, a^{8} b + 59 \, a^{6} b^{3} - 45 \, a^{4} b^{5} - 23 \, a^{2} b^{7} + b^{9}\right)} \cos\left(d x + c\right)^{4} - {\left(11 \, a^{8} b - 36 \, a^{6} b^{3} + 42 \, a^{4} b^{5} - 20 \, a^{2} b^{7} + 3 \, b^{9}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{16 \, {\left({\left(a^{10} b^{2} - 5 \, a^{8} b^{4} + 10 \, a^{6} b^{6} - 10 \, a^{4} b^{8} + 5 \, a^{2} b^{10} - b^{12}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{11} b - 5 \, a^{9} b^{3} + 10 \, a^{7} b^{5} - 10 \, a^{5} b^{7} + 5 \, a^{3} b^{9} - a b^{11}\right)} d \cos\left(d x + c\right)^{4} \sin\left(d x + c\right) - {\left(a^{12} - 4 \, a^{10} b^{2} + 5 \, a^{8} b^{4} - 5 \, a^{4} b^{8} + 4 \, a^{2} b^{10} - b^{12}\right)} d \cos\left(d x + c\right)^{4}\right)}}"," ",0,"-1/16*(4*a^9 - 16*a^7*b^2 + 24*a^5*b^4 - 16*a^3*b^6 + 4*a*b^8 - 4*(6*a^9 + 35*a^7*b^2 - 39*a^5*b^4 - 3*a^3*b^6 + a*b^8)*cos(d*x + c)^4 - 16*(a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*cos(d*x + c)^2 - 16*((a^7*b^2 + 13*a^5*b^4 + 10*a^3*b^6)*cos(d*x + c)^6 - 2*(a^8*b + 13*a^6*b^3 + 10*a^4*b^5)*cos(d*x + c)^4*sin(d*x + c) - (a^9 + 14*a^7*b^2 + 23*a^5*b^4 + 10*a^3*b^6)*cos(d*x + c)^4)*log(b*sin(d*x + c) + a) + ((8*a^7*b^2 + 45*a^6*b^3 + 104*a^5*b^4 + 125*a^4*b^5 + 80*a^3*b^6 + 23*a^2*b^7 - b^9)*cos(d*x + c)^6 - 2*(8*a^8*b + 45*a^7*b^2 + 104*a^6*b^3 + 125*a^5*b^4 + 80*a^4*b^5 + 23*a^3*b^6 - a*b^8)*cos(d*x + c)^4*sin(d*x + c) - (8*a^9 + 45*a^8*b + 112*a^7*b^2 + 170*a^6*b^3 + 184*a^5*b^4 + 148*a^4*b^5 + 80*a^3*b^6 + 22*a^2*b^7 - b^9)*cos(d*x + c)^4)*log(sin(d*x + c) + 1) + ((8*a^7*b^2 - 45*a^6*b^3 + 104*a^5*b^4 - 125*a^4*b^5 + 80*a^3*b^6 - 23*a^2*b^7 + b^9)*cos(d*x + c)^6 - 2*(8*a^8*b - 45*a^7*b^2 + 104*a^6*b^3 - 125*a^5*b^4 + 80*a^4*b^5 - 23*a^3*b^6 + a*b^8)*cos(d*x + c)^4*sin(d*x + c) - (8*a^9 - 45*a^8*b + 112*a^7*b^2 - 170*a^6*b^3 + 184*a^5*b^4 - 148*a^4*b^5 + 80*a^3*b^6 - 22*a^2*b^7 + b^9)*cos(d*x + c)^4)*log(-sin(d*x + c) + 1) - 2*(2*a^8*b - 8*a^6*b^3 + 12*a^4*b^5 - 8*a^2*b^7 + 2*b^9 + (8*a^8*b + 59*a^6*b^3 - 45*a^4*b^5 - 23*a^2*b^7 + b^9)*cos(d*x + c)^4 - (11*a^8*b - 36*a^6*b^3 + 42*a^4*b^5 - 20*a^2*b^7 + 3*b^9)*cos(d*x + c)^2)*sin(d*x + c))/((a^10*b^2 - 5*a^8*b^4 + 10*a^6*b^6 - 10*a^4*b^8 + 5*a^2*b^10 - b^12)*d*cos(d*x + c)^6 - 2*(a^11*b - 5*a^9*b^3 + 10*a^7*b^5 - 10*a^5*b^7 + 5*a^3*b^9 - a*b^11)*d*cos(d*x + c)^4*sin(d*x + c) - (a^12 - 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 + 4*a^2*b^10 - b^12)*d*cos(d*x + c)^4)","B",0
193,1,788,0,3.011370," ","integrate(tan(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{2 \, a^{7} - 6 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 2 \, a b^{6} + 2 \, {\left(3 \, a^{7} + 7 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a^{5} b^{2} + 8 \, a^{3} b^{4} + 3 \, a b^{6}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{6} b + 8 \, a^{4} b^{3} + 3 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) - {\left(a^{7} + 9 \, a^{5} b^{2} + 11 \, a^{3} b^{4} + 3 \, a b^{6}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(b \sin\left(d x + c\right) + a\right) - {\left({\left(2 \, a^{5} b^{2} + 9 \, a^{4} b^{3} + 16 \, a^{3} b^{4} + 14 \, a^{2} b^{5} + 6 \, a b^{6} + b^{7}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(2 \, a^{6} b + 9 \, a^{5} b^{2} + 16 \, a^{4} b^{3} + 14 \, a^{3} b^{4} + 6 \, a^{2} b^{5} + a b^{6}\right)} \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) - {\left(2 \, a^{7} + 9 \, a^{6} b + 18 \, a^{5} b^{2} + 23 \, a^{4} b^{3} + 22 \, a^{3} b^{4} + 15 \, a^{2} b^{5} + 6 \, a b^{6} + b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - {\left({\left(2 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 16 \, a^{3} b^{4} - 14 \, a^{2} b^{5} + 6 \, a b^{6} - b^{7}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(2 \, a^{6} b - 9 \, a^{5} b^{2} + 16 \, a^{4} b^{3} - 14 \, a^{3} b^{4} + 6 \, a^{2} b^{5} - a b^{6}\right)} \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) - {\left(2 \, a^{7} - 9 \, a^{6} b + 18 \, a^{5} b^{2} - 23 \, a^{4} b^{3} + 22 \, a^{3} b^{4} - 15 \, a^{2} b^{5} + 6 \, a b^{6} - b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left(2 \, a^{6} b + 7 \, a^{4} b^{3} - 8 \, a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{4 \, {\left({\left(a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right)} d \cos\left(d x + c\right)^{2} \sin\left(d x + c\right) - {\left(a^{10} - 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} + 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} + b^{10}\right)} d \cos\left(d x + c\right)^{2}\right)}}"," ",0,"-1/4*(2*a^7 - 6*a^5*b^2 + 6*a^3*b^4 - 2*a*b^6 + 2*(3*a^7 + 7*a^5*b^2 - 11*a^3*b^4 + a*b^6)*cos(d*x + c)^2 + 4*((a^5*b^2 + 8*a^3*b^4 + 3*a*b^6)*cos(d*x + c)^4 - 2*(a^6*b + 8*a^4*b^3 + 3*a^2*b^5)*cos(d*x + c)^2*sin(d*x + c) - (a^7 + 9*a^5*b^2 + 11*a^3*b^4 + 3*a*b^6)*cos(d*x + c)^2)*log(b*sin(d*x + c) + a) - ((2*a^5*b^2 + 9*a^4*b^3 + 16*a^3*b^4 + 14*a^2*b^5 + 6*a*b^6 + b^7)*cos(d*x + c)^4 - 2*(2*a^6*b + 9*a^5*b^2 + 16*a^4*b^3 + 14*a^3*b^4 + 6*a^2*b^5 + a*b^6)*cos(d*x + c)^2*sin(d*x + c) - (2*a^7 + 9*a^6*b + 18*a^5*b^2 + 23*a^4*b^3 + 22*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - ((2*a^5*b^2 - 9*a^4*b^3 + 16*a^3*b^4 - 14*a^2*b^5 + 6*a*b^6 - b^7)*cos(d*x + c)^4 - 2*(2*a^6*b - 9*a^5*b^2 + 16*a^4*b^3 - 14*a^3*b^4 + 6*a^2*b^5 - a*b^6)*cos(d*x + c)^2*sin(d*x + c) - (2*a^7 - 9*a^6*b + 18*a^5*b^2 - 23*a^4*b^3 + 22*a^3*b^4 - 15*a^2*b^5 + 6*a*b^6 - b^7)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (2*a^6*b + 7*a^4*b^3 - 8*a^2*b^5 - b^7)*cos(d*x + c)^2)*sin(d*x + c))/((a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b^10)*d*cos(d*x + c)^4 - 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*d*cos(d*x + c)^2*sin(d*x + c) - (a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10)*d*cos(d*x + c)^2)","B",0
194,1,462,0,1.867326," ","integrate(tan(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","\frac{3 \, a^{5} - 2 \, a^{3} b^{2} - a b^{4} - 2 \, {\left(a^{5} + 4 \, a^{3} b^{2} + 3 \, a b^{4} - {\left(a^{3} b^{2} + 3 \, a b^{4}\right)} \cos\left(d x + c\right)^{2} + 2 \, {\left(a^{4} b + 3 \, a^{2} b^{3}\right)} \sin\left(d x + c\right)\right)} \log\left(b \sin\left(d x + c\right) + a\right) + {\left(a^{5} + 3 \, a^{4} b + 4 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5} - {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} \cos\left(d x + c\right)^{2} + 2 \, {\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \sin\left(d x + c\right)\right)} \log\left(\sin\left(d x + c\right) + 1\right) + {\left(a^{5} - 3 \, a^{4} b + 4 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5} - {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} \cos\left(d x + c\right)^{2} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \sin\left(d x + c\right)\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(a^{4} b - b^{5}\right)} \sin\left(d x + c\right)}{2 \, {\left({\left(a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right)} d \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} d \sin\left(d x + c\right) - {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} d\right)}}"," ",0,"1/2*(3*a^5 - 2*a^3*b^2 - a*b^4 - 2*(a^5 + 4*a^3*b^2 + 3*a*b^4 - (a^3*b^2 + 3*a*b^4)*cos(d*x + c)^2 + 2*(a^4*b + 3*a^2*b^3)*sin(d*x + c))*log(b*sin(d*x + c) + a) + (a^5 + 3*a^4*b + 4*a^3*b^2 + 4*a^2*b^3 + 3*a*b^4 + b^5 - (a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*cos(d*x + c)^2 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*sin(d*x + c))*log(sin(d*x + c) + 1) + (a^5 - 3*a^4*b + 4*a^3*b^2 - 4*a^2*b^3 + 3*a*b^4 - b^5 - (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*cos(d*x + c)^2 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*sin(d*x + c))*log(-sin(d*x + c) + 1) + 2*(a^4*b - b^5)*sin(d*x + c))/((a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*d*cos(d*x + c)^2 - 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*sin(d*x + c) - (a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*d)","B",0
195,1,154,0,1.438456," ","integrate(cot(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{2 \, a b \sin\left(d x + c\right) + 3 \, a^{2} + 2 \, {\left(b^{2} \cos\left(d x + c\right)^{2} - 2 \, a b \sin\left(d x + c\right) - a^{2} - b^{2}\right)} \log\left(b \sin\left(d x + c\right) + a\right) - 2 \, {\left(b^{2} \cos\left(d x + c\right)^{2} - 2 \, a b \sin\left(d x + c\right) - a^{2} - b^{2}\right)} \log\left(-\frac{1}{2} \, \sin\left(d x + c\right)\right)}{2 \, {\left(a^{3} b^{2} d \cos\left(d x + c\right)^{2} - 2 \, a^{4} b d \sin\left(d x + c\right) - {\left(a^{5} + a^{3} b^{2}\right)} d\right)}}"," ",0,"-1/2*(2*a*b*sin(d*x + c) + 3*a^2 + 2*(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)*log(b*sin(d*x + c) + a) - 2*(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)*log(-1/2*sin(d*x + c)))/(a^3*b^2*d*cos(d*x + c)^2 - 2*a^4*b*d*sin(d*x + c) - (a^5 + a^3*b^2)*d)","B",0
196,1,404,0,1.683662," ","integrate(cot(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{4 \, a^{4} - 18 \, a^{2} b^{2} - 3 \, {\left(a^{4} - 6 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left({\left(a^{2} b^{2} - 6 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + a^{4} - 5 \, a^{2} b^{2} - 6 \, b^{4} - {\left(a^{4} - 4 \, a^{2} b^{2} - 12 \, b^{4}\right)} \cos\left(d x + c\right)^{2} + 2 \, {\left(a^{3} b - 6 \, a b^{3} - {\left(a^{3} b - 6 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(b \sin\left(d x + c\right) + a\right) + 2 \, {\left({\left(a^{2} b^{2} - 6 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + a^{4} - 5 \, a^{2} b^{2} - 6 \, b^{4} - {\left(a^{4} - 4 \, a^{2} b^{2} - 12 \, b^{4}\right)} \cos\left(d x + c\right)^{2} + 2 \, {\left(a^{3} b - 6 \, a b^{3} - {\left(a^{3} b - 6 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \sin\left(d x + c\right)\right) - 2 \, {\left(a^{3} b + 6 \, a b^{3} + {\left(a^{3} b - 6 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{2 \, {\left(a^{5} b^{2} d \cos\left(d x + c\right)^{4} - {\left(a^{7} + 2 \, a^{5} b^{2}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} + a^{5} b^{2}\right)} d - 2 \, {\left(a^{6} b d \cos\left(d x + c\right)^{2} - a^{6} b d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"-1/2*(4*a^4 - 18*a^2*b^2 - 3*(a^4 - 6*a^2*b^2)*cos(d*x + c)^2 - 2*((a^2*b^2 - 6*b^4)*cos(d*x + c)^4 + a^4 - 5*a^2*b^2 - 6*b^4 - (a^4 - 4*a^2*b^2 - 12*b^4)*cos(d*x + c)^2 + 2*(a^3*b - 6*a*b^3 - (a^3*b - 6*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(b*sin(d*x + c) + a) + 2*((a^2*b^2 - 6*b^4)*cos(d*x + c)^4 + a^4 - 5*a^2*b^2 - 6*b^4 - (a^4 - 4*a^2*b^2 - 12*b^4)*cos(d*x + c)^2 + 2*(a^3*b - 6*a*b^3 - (a^3*b - 6*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*sin(d*x + c)) - 2*(a^3*b + 6*a*b^3 + (a^3*b - 6*a*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(a^5*b^2*d*cos(d*x + c)^4 - (a^7 + 2*a^5*b^2)*d*cos(d*x + c)^2 + (a^7 + a^5*b^2)*d - 2*(a^6*b*d*cos(d*x + c)^2 - a^6*b*d)*sin(d*x + c))","B",0
197,1,754,0,1.019028," ","integrate(cot(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","-\frac{9 \, a^{6} - 77 \, a^{4} b^{2} + 90 \, a^{2} b^{4} + 6 \, {\left(a^{6} - 12 \, a^{4} b^{2} + 15 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{4} - {\left(16 \, a^{6} - 149 \, a^{4} b^{2} + 180 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a^{4} b^{2} - 12 \, a^{2} b^{4} + 15 \, b^{6}\right)} \cos\left(d x + c\right)^{6} - a^{6} + 11 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - 15 \, b^{6} - {\left(a^{6} - 9 \, a^{4} b^{2} - 21 \, a^{2} b^{4} + 45 \, b^{6}\right)} \cos\left(d x + c\right)^{4} + {\left(2 \, a^{6} - 21 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + 45 \, b^{6}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5} + {\left(a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(b \sin\left(d x + c\right) + a\right) - 4 \, {\left({\left(a^{4} b^{2} - 12 \, a^{2} b^{4} + 15 \, b^{6}\right)} \cos\left(d x + c\right)^{6} - a^{6} + 11 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - 15 \, b^{6} - {\left(a^{6} - 9 \, a^{4} b^{2} - 21 \, a^{2} b^{4} + 45 \, b^{6}\right)} \cos\left(d x + c\right)^{4} + {\left(2 \, a^{6} - 21 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + 45 \, b^{6}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5} + {\left(a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \sin\left(d x + c\right)\right) - 2 \, {\left(5 \, a^{5} b + 14 \, a^{3} b^{3} - 30 \, a b^{5} - 2 \, {\left(a^{5} b - 12 \, a^{3} b^{3} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(2 \, a^{5} b + 19 \, a^{3} b^{3} - 30 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{4 \, {\left(a^{7} b^{2} d \cos\left(d x + c\right)^{6} - {\left(a^{9} + 3 \, a^{7} b^{2}\right)} d \cos\left(d x + c\right)^{4} + {\left(2 \, a^{9} + 3 \, a^{7} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{9} + a^{7} b^{2}\right)} d - 2 \, {\left(a^{8} b d \cos\left(d x + c\right)^{4} - 2 \, a^{8} b d \cos\left(d x + c\right)^{2} + a^{8} b d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"-1/4*(9*a^6 - 77*a^4*b^2 + 90*a^2*b^4 + 6*(a^6 - 12*a^4*b^2 + 15*a^2*b^4)*cos(d*x + c)^4 - (16*a^6 - 149*a^4*b^2 + 180*a^2*b^4)*cos(d*x + c)^2 + 4*((a^4*b^2 - 12*a^2*b^4 + 15*b^6)*cos(d*x + c)^6 - a^6 + 11*a^4*b^2 - 3*a^2*b^4 - 15*b^6 - (a^6 - 9*a^4*b^2 - 21*a^2*b^4 + 45*b^6)*cos(d*x + c)^4 + (2*a^6 - 21*a^4*b^2 - 6*a^2*b^4 + 45*b^6)*cos(d*x + c)^2 - 2*(a^5*b - 12*a^3*b^3 + 15*a*b^5 + (a^5*b - 12*a^3*b^3 + 15*a*b^5)*cos(d*x + c)^4 - 2*(a^5*b - 12*a^3*b^3 + 15*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(b*sin(d*x + c) + a) - 4*((a^4*b^2 - 12*a^2*b^4 + 15*b^6)*cos(d*x + c)^6 - a^6 + 11*a^4*b^2 - 3*a^2*b^4 - 15*b^6 - (a^6 - 9*a^4*b^2 - 21*a^2*b^4 + 45*b^6)*cos(d*x + c)^4 + (2*a^6 - 21*a^4*b^2 - 6*a^2*b^4 + 45*b^6)*cos(d*x + c)^2 - 2*(a^5*b - 12*a^3*b^3 + 15*a*b^5 + (a^5*b - 12*a^3*b^3 + 15*a*b^5)*cos(d*x + c)^4 - 2*(a^5*b - 12*a^3*b^3 + 15*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*sin(d*x + c)) - 2*(5*a^5*b + 14*a^3*b^3 - 30*a*b^5 - 2*(a^5*b - 12*a^3*b^3 + 15*a*b^5)*cos(d*x + c)^4 - 2*(2*a^5*b + 19*a^3*b^3 - 30*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))/(a^7*b^2*d*cos(d*x + c)^6 - (a^9 + 3*a^7*b^2)*d*cos(d*x + c)^4 + (2*a^9 + 3*a^7*b^2)*d*cos(d*x + c)^2 - (a^9 + a^7*b^2)*d - 2*(a^8*b*d*cos(d*x + c)^4 - 2*a^8*b*d*cos(d*x + c)^2 + a^8*b*d)*sin(d*x + c))","B",0
198,1,1249,0,1.587654," ","integrate(tan(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","\left[\frac{4 \, a^{8} b - 16 \, a^{6} b^{3} + 24 \, a^{4} b^{5} - 16 \, a^{2} b^{7} + 4 \, b^{9} - 2 \, {\left(28 \, a^{8} b + 61 \, a^{6} b^{3} - 101 \, a^{4} b^{5} + 12 \, a^{2} b^{7}\right)} \cos\left(d x + c\right)^{4} - 4 \, {\left(8 \, a^{8} b - 25 \, a^{6} b^{3} + 27 \, a^{4} b^{5} - 11 \, a^{2} b^{7} + b^{9}\right)} \cos\left(d x + c\right)^{2} - 3 \, {\left({\left(2 \, a^{6} b^{2} + 21 \, a^{4} b^{4} + 12 \, a^{2} b^{6}\right)} \cos\left(d x + c\right)^{5} - 2 \, {\left(2 \, a^{7} b + 21 \, a^{5} b^{3} + 12 \, a^{3} b^{5}\right)} \cos\left(d x + c\right)^{3} \sin\left(d x + c\right) - {\left(2 \, a^{8} + 23 \, a^{6} b^{2} + 33 \, a^{4} b^{4} + 12 \, a^{2} b^{6}\right)} \cos\left(d x + c\right)^{3}\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) - 2 \, {\left(2 \, a^{9} - 8 \, a^{7} b^{2} + 12 \, a^{5} b^{4} - 8 \, a^{3} b^{6} + 2 \, a b^{8} + {\left(17 \, a^{7} b^{2} + 59 \, a^{5} b^{4} - 64 \, a^{3} b^{6} - 12 \, a b^{8}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{9} - 9 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + 5 \, a^{3} b^{6} - 3 \, a b^{8}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{12 \, {\left({\left(a^{10} b^{2} - 5 \, a^{8} b^{4} + 10 \, a^{6} b^{6} - 10 \, a^{4} b^{8} + 5 \, a^{2} b^{10} - b^{12}\right)} d \cos\left(d x + c\right)^{5} - 2 \, {\left(a^{11} b - 5 \, a^{9} b^{3} + 10 \, a^{7} b^{5} - 10 \, a^{5} b^{7} + 5 \, a^{3} b^{9} - a b^{11}\right)} d \cos\left(d x + c\right)^{3} \sin\left(d x + c\right) - {\left(a^{12} - 4 \, a^{10} b^{2} + 5 \, a^{8} b^{4} - 5 \, a^{4} b^{8} + 4 \, a^{2} b^{10} - b^{12}\right)} d \cos\left(d x + c\right)^{3}\right)}}, \frac{2 \, a^{8} b - 8 \, a^{6} b^{3} + 12 \, a^{4} b^{5} - 8 \, a^{2} b^{7} + 2 \, b^{9} - {\left(28 \, a^{8} b + 61 \, a^{6} b^{3} - 101 \, a^{4} b^{5} + 12 \, a^{2} b^{7}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(8 \, a^{8} b - 25 \, a^{6} b^{3} + 27 \, a^{4} b^{5} - 11 \, a^{2} b^{7} + b^{9}\right)} \cos\left(d x + c\right)^{2} - 3 \, {\left({\left(2 \, a^{6} b^{2} + 21 \, a^{4} b^{4} + 12 \, a^{2} b^{6}\right)} \cos\left(d x + c\right)^{5} - 2 \, {\left(2 \, a^{7} b + 21 \, a^{5} b^{3} + 12 \, a^{3} b^{5}\right)} \cos\left(d x + c\right)^{3} \sin\left(d x + c\right) - {\left(2 \, a^{8} + 23 \, a^{6} b^{2} + 33 \, a^{4} b^{4} + 12 \, a^{2} b^{6}\right)} \cos\left(d x + c\right)^{3}\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) - {\left(2 \, a^{9} - 8 \, a^{7} b^{2} + 12 \, a^{5} b^{4} - 8 \, a^{3} b^{6} + 2 \, a b^{8} + {\left(17 \, a^{7} b^{2} + 59 \, a^{5} b^{4} - 64 \, a^{3} b^{6} - 12 \, a b^{8}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{9} - 9 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + 5 \, a^{3} b^{6} - 3 \, a b^{8}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{6 \, {\left({\left(a^{10} b^{2} - 5 \, a^{8} b^{4} + 10 \, a^{6} b^{6} - 10 \, a^{4} b^{8} + 5 \, a^{2} b^{10} - b^{12}\right)} d \cos\left(d x + c\right)^{5} - 2 \, {\left(a^{11} b - 5 \, a^{9} b^{3} + 10 \, a^{7} b^{5} - 10 \, a^{5} b^{7} + 5 \, a^{3} b^{9} - a b^{11}\right)} d \cos\left(d x + c\right)^{3} \sin\left(d x + c\right) - {\left(a^{12} - 4 \, a^{10} b^{2} + 5 \, a^{8} b^{4} - 5 \, a^{4} b^{8} + 4 \, a^{2} b^{10} - b^{12}\right)} d \cos\left(d x + c\right)^{3}\right)}}\right]"," ",0,"[1/12*(4*a^8*b - 16*a^6*b^3 + 24*a^4*b^5 - 16*a^2*b^7 + 4*b^9 - 2*(28*a^8*b + 61*a^6*b^3 - 101*a^4*b^5 + 12*a^2*b^7)*cos(d*x + c)^4 - 4*(8*a^8*b - 25*a^6*b^3 + 27*a^4*b^5 - 11*a^2*b^7 + b^9)*cos(d*x + c)^2 - 3*((2*a^6*b^2 + 21*a^4*b^4 + 12*a^2*b^6)*cos(d*x + c)^5 - 2*(2*a^7*b + 21*a^5*b^3 + 12*a^3*b^5)*cos(d*x + c)^3*sin(d*x + c) - (2*a^8 + 23*a^6*b^2 + 33*a^4*b^4 + 12*a^2*b^6)*cos(d*x + c)^3)*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)) - 2*(2*a^9 - 8*a^7*b^2 + 12*a^5*b^4 - 8*a^3*b^6 + 2*a*b^8 + (17*a^7*b^2 + 59*a^5*b^4 - 64*a^3*b^6 - 12*a*b^8)*cos(d*x + c)^4 - 2*(4*a^9 - 9*a^7*b^2 + 3*a^5*b^4 + 5*a^3*b^6 - 3*a*b^8)*cos(d*x + c)^2)*sin(d*x + c))/((a^10*b^2 - 5*a^8*b^4 + 10*a^6*b^6 - 10*a^4*b^8 + 5*a^2*b^10 - b^12)*d*cos(d*x + c)^5 - 2*(a^11*b - 5*a^9*b^3 + 10*a^7*b^5 - 10*a^5*b^7 + 5*a^3*b^9 - a*b^11)*d*cos(d*x + c)^3*sin(d*x + c) - (a^12 - 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 + 4*a^2*b^10 - b^12)*d*cos(d*x + c)^3), 1/6*(2*a^8*b - 8*a^6*b^3 + 12*a^4*b^5 - 8*a^2*b^7 + 2*b^9 - (28*a^8*b + 61*a^6*b^3 - 101*a^4*b^5 + 12*a^2*b^7)*cos(d*x + c)^4 - 2*(8*a^8*b - 25*a^6*b^3 + 27*a^4*b^5 - 11*a^2*b^7 + b^9)*cos(d*x + c)^2 - 3*((2*a^6*b^2 + 21*a^4*b^4 + 12*a^2*b^6)*cos(d*x + c)^5 - 2*(2*a^7*b + 21*a^5*b^3 + 12*a^3*b^5)*cos(d*x + c)^3*sin(d*x + c) - (2*a^8 + 23*a^6*b^2 + 33*a^4*b^4 + 12*a^2*b^6)*cos(d*x + c)^3)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - (2*a^9 - 8*a^7*b^2 + 12*a^5*b^4 - 8*a^3*b^6 + 2*a*b^8 + (17*a^7*b^2 + 59*a^5*b^4 - 64*a^3*b^6 - 12*a*b^8)*cos(d*x + c)^4 - 2*(4*a^9 - 9*a^7*b^2 + 3*a^5*b^4 + 5*a^3*b^6 - 3*a*b^8)*cos(d*x + c)^2)*sin(d*x + c))/((a^10*b^2 - 5*a^8*b^4 + 10*a^6*b^6 - 10*a^4*b^8 + 5*a^2*b^10 - b^12)*d*cos(d*x + c)^5 - 2*(a^11*b - 5*a^9*b^3 + 10*a^7*b^5 - 10*a^5*b^7 + 5*a^3*b^9 - a*b^11)*d*cos(d*x + c)^3*sin(d*x + c) - (a^12 - 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 + 4*a^2*b^10 - b^12)*d*cos(d*x + c)^3)]","A",0
199,1,934,0,1.771450," ","integrate(tan(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","\left[\frac{4 \, a^{6} b - 12 \, a^{4} b^{3} + 12 \, a^{2} b^{5} - 4 \, b^{7} + 2 \, {\left(8 \, a^{6} b + a^{4} b^{3} - 11 \, a^{2} b^{5} + 2 \, b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left({\left(2 \, a^{4} b^{2} + 11 \, a^{2} b^{4} + 2 \, b^{6}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(2 \, a^{5} b + 11 \, a^{3} b^{3} + 2 \, a b^{5}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(2 \, a^{6} + 13 \, a^{4} b^{2} + 13 \, a^{2} b^{4} + 2 \, b^{6}\right)} \cos\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) - 2 \, {\left(2 \, a^{7} - 6 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 2 \, a b^{6} - 5 \, {\left(a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{4 \, {\left({\left(a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right)} d \cos\left(d x + c\right)^{3} - 2 \, {\left(a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a^{10} - 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} + 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} + b^{10}\right)} d \cos\left(d x + c\right)\right)}}, \frac{2 \, a^{6} b - 6 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - 2 \, b^{7} + {\left(8 \, a^{6} b + a^{4} b^{3} - 11 \, a^{2} b^{5} + 2 \, b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left({\left(2 \, a^{4} b^{2} + 11 \, a^{2} b^{4} + 2 \, b^{6}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(2 \, a^{5} b + 11 \, a^{3} b^{3} + 2 \, a b^{5}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(2 \, a^{6} + 13 \, a^{4} b^{2} + 13 \, a^{2} b^{4} + 2 \, b^{6}\right)} \cos\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) - {\left(2 \, a^{7} - 6 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 2 \, a b^{6} - 5 \, {\left(a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{2 \, {\left({\left(a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right)} d \cos\left(d x + c\right)^{3} - 2 \, {\left(a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a^{10} - 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} + 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} + b^{10}\right)} d \cos\left(d x + c\right)\right)}}\right]"," ",0,"[1/4*(4*a^6*b - 12*a^4*b^3 + 12*a^2*b^5 - 4*b^7 + 2*(8*a^6*b + a^4*b^3 - 11*a^2*b^5 + 2*b^7)*cos(d*x + c)^2 + ((2*a^4*b^2 + 11*a^2*b^4 + 2*b^6)*cos(d*x + c)^3 - 2*(2*a^5*b + 11*a^3*b^3 + 2*a*b^5)*cos(d*x + c)*sin(d*x + c) - (2*a^6 + 13*a^4*b^2 + 13*a^2*b^4 + 2*b^6)*cos(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)) - 2*(2*a^7 - 6*a^5*b^2 + 6*a^3*b^4 - 2*a*b^6 - 5*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))/((a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b^10)*d*cos(d*x + c)^3 - 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*d*cos(d*x + c)*sin(d*x + c) - (a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10)*d*cos(d*x + c)), 1/2*(2*a^6*b - 6*a^4*b^3 + 6*a^2*b^5 - 2*b^7 + (8*a^6*b + a^4*b^3 - 11*a^2*b^5 + 2*b^7)*cos(d*x + c)^2 + ((2*a^4*b^2 + 11*a^2*b^4 + 2*b^6)*cos(d*x + c)^3 - 2*(2*a^5*b + 11*a^3*b^3 + 2*a*b^5)*cos(d*x + c)*sin(d*x + c) - (2*a^6 + 13*a^4*b^2 + 13*a^2*b^4 + 2*b^6)*cos(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - (2*a^7 - 6*a^5*b^2 + 6*a^3*b^4 - 2*a*b^6 - 5*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))/((a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b^10)*d*cos(d*x + c)^3 - 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*d*cos(d*x + c)*sin(d*x + c) - (a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10)*d*cos(d*x + c))]","A",0
200,1,1394,0,2.747251," ","integrate(cot(d*x+c)^2/(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(5 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(8 \, a^{6} b - 17 \, a^{4} b^{3} + 9 \, a^{2} b^{5}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(4 \, a^{5} b - 18 \, a^{3} b^{3} + 12 \, a b^{5} - 2 \, {\left(2 \, a^{5} b - 9 \, a^{3} b^{3} + 6 \, a b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(2 \, a^{6} - 7 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + 6 \, b^{6} - {\left(2 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 6 \, b^{6}\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) - 2 \, {\left(2 \, a^{7} + a^{5} b^{2} - 9 \, a^{3} b^{4} + 6 \, a b^{6}\right)} \cos\left(d x + c\right) + 6 \, {\left(2 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + 2 \, a b^{6} - 2 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{6} b - a^{4} b^{3} - a^{2} b^{5} + b^{7} - {\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 6 \, {\left(2 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + 2 \, a b^{6} - 2 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{6} b - a^{4} b^{3} - a^{2} b^{5} + b^{7} - {\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{4 \, {\left(2 \, {\left(a^{9} b - 2 \, a^{7} b^{3} + a^{5} b^{5}\right)} d \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{9} b - 2 \, a^{7} b^{3} + a^{5} b^{5}\right)} d + {\left({\left(a^{8} b^{2} - 2 \, a^{6} b^{4} + a^{4} b^{6}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{10} - a^{8} b^{2} - a^{6} b^{4} + a^{4} b^{6}\right)} d\right)} \sin\left(d x + c\right)\right)}}, -\frac{{\left(5 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(8 \, a^{6} b - 17 \, a^{4} b^{3} + 9 \, a^{2} b^{5}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(4 \, a^{5} b - 18 \, a^{3} b^{3} + 12 \, a b^{5} - 2 \, {\left(2 \, a^{5} b - 9 \, a^{3} b^{3} + 6 \, a b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(2 \, a^{6} - 7 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + 6 \, b^{6} - {\left(2 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 6 \, b^{6}\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) - {\left(2 \, a^{7} + a^{5} b^{2} - 9 \, a^{3} b^{4} + 6 \, a b^{6}\right)} \cos\left(d x + c\right) + 3 \, {\left(2 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + 2 \, a b^{6} - 2 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{6} b - a^{4} b^{3} - a^{2} b^{5} + b^{7} - {\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 3 \, {\left(2 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + 2 \, a b^{6} - 2 \, {\left(a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{6} b - a^{4} b^{3} - a^{2} b^{5} + b^{7} - {\left(a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{2 \, {\left(2 \, {\left(a^{9} b - 2 \, a^{7} b^{3} + a^{5} b^{5}\right)} d \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{9} b - 2 \, a^{7} b^{3} + a^{5} b^{5}\right)} d + {\left({\left(a^{8} b^{2} - 2 \, a^{6} b^{4} + a^{4} b^{6}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{10} - a^{8} b^{2} - a^{6} b^{4} + a^{4} b^{6}\right)} d\right)} \sin\left(d x + c\right)\right)}}\right]"," ",0,"[-1/4*(2*(5*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^3 - 2*(8*a^6*b - 17*a^4*b^3 + 9*a^2*b^5)*cos(d*x + c)*sin(d*x + c) + (4*a^5*b - 18*a^3*b^3 + 12*a*b^5 - 2*(2*a^5*b - 9*a^3*b^3 + 6*a*b^5)*cos(d*x + c)^2 + (2*a^6 - 7*a^4*b^2 - 3*a^2*b^4 + 6*b^6 - (2*a^4*b^2 - 9*a^2*b^4 + 6*b^6)*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)) - 2*(2*a^7 + a^5*b^2 - 9*a^3*b^4 + 6*a*b^6)*cos(d*x + c) + 6*(2*a^5*b^2 - 4*a^3*b^4 + 2*a*b^6 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(d*x + c)^2 + (a^6*b - a^4*b^3 - a^2*b^5 + b^7 - (a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 6*(2*a^5*b^2 - 4*a^3*b^4 + 2*a*b^6 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(d*x + c)^2 + (a^6*b - a^4*b^3 - a^2*b^5 + b^7 - (a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/(2*(a^9*b - 2*a^7*b^3 + a^5*b^5)*d*cos(d*x + c)^2 - 2*(a^9*b - 2*a^7*b^3 + a^5*b^5)*d + ((a^8*b^2 - 2*a^6*b^4 + a^4*b^6)*d*cos(d*x + c)^2 - (a^10 - a^8*b^2 - a^6*b^4 + a^4*b^6)*d)*sin(d*x + c)), -1/2*((5*a^5*b^2 - 11*a^3*b^4 + 6*a*b^6)*cos(d*x + c)^3 - (8*a^6*b - 17*a^4*b^3 + 9*a^2*b^5)*cos(d*x + c)*sin(d*x + c) + (4*a^5*b - 18*a^3*b^3 + 12*a*b^5 - 2*(2*a^5*b - 9*a^3*b^3 + 6*a*b^5)*cos(d*x + c)^2 + (2*a^6 - 7*a^4*b^2 - 3*a^2*b^4 + 6*b^6 - (2*a^4*b^2 - 9*a^2*b^4 + 6*b^6)*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))) - (2*a^7 + a^5*b^2 - 9*a^3*b^4 + 6*a*b^6)*cos(d*x + c) + 3*(2*a^5*b^2 - 4*a^3*b^4 + 2*a*b^6 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(d*x + c)^2 + (a^6*b - a^4*b^3 - a^2*b^5 + b^7 - (a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*(2*a^5*b^2 - 4*a^3*b^4 + 2*a*b^6 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(d*x + c)^2 + (a^6*b - a^4*b^3 - a^2*b^5 + b^7 - (a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/(2*(a^9*b - 2*a^7*b^3 + a^5*b^5)*d*cos(d*x + c)^2 - 2*(a^9*b - 2*a^7*b^3 + a^5*b^5)*d + ((a^8*b^2 - 2*a^6*b^4 + a^4*b^6)*d*cos(d*x + c)^2 - (a^10 - a^8*b^2 - a^6*b^4 + a^4*b^6)*d)*sin(d*x + c))]","B",0
201,1,2027,0,2.822995," ","integrate(cot(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(17 \, a^{5} b^{2} - 77 \, a^{3} b^{4} + 60 \, a b^{6}\right)} \cos\left(d x + c\right)^{5} - 4 \, {\left(4 \, a^{7} + 3 \, a^{5} b^{2} - 67 \, a^{3} b^{4} + 60 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(4 \, a^{5} b - 38 \, a^{3} b^{3} + 40 \, a b^{5} + 2 \, {\left(2 \, a^{5} b - 19 \, a^{3} b^{3} + 20 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} - 4 \, {\left(2 \, a^{5} b - 19 \, a^{3} b^{3} + 20 \, a b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(2 \, a^{6} - 17 \, a^{4} b^{2} + a^{2} b^{4} + 20 \, b^{6} + {\left(2 \, a^{4} b^{2} - 19 \, a^{2} b^{4} + 20 \, b^{6}\right)} \cos\left(d x + c\right)^{4} - {\left(2 \, a^{6} - 15 \, a^{4} b^{2} - 18 \, a^{2} b^{4} + 40 \, b^{6}\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) + 6 \, {\left(2 \, a^{7} - 3 \, a^{5} b^{2} - 19 \, a^{3} b^{4} + 20 \, a b^{6}\right)} \cos\left(d x + c\right) - 3 \, {\left(18 \, a^{5} b^{2} - 58 \, a^{3} b^{4} + 40 \, a b^{6} + 2 \, {\left(9 \, a^{5} b^{2} - 29 \, a^{3} b^{4} + 20 \, a b^{6}\right)} \cos\left(d x + c\right)^{4} - 4 \, {\left(9 \, a^{5} b^{2} - 29 \, a^{3} b^{4} + 20 \, a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(9 \, a^{6} b - 20 \, a^{4} b^{3} - 9 \, a^{2} b^{5} + 20 \, b^{7} + {\left(9 \, a^{4} b^{3} - 29 \, a^{2} b^{5} + 20 \, b^{7}\right)} \cos\left(d x + c\right)^{4} - {\left(9 \, a^{6} b - 11 \, a^{4} b^{3} - 38 \, a^{2} b^{5} + 40 \, b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 3 \, {\left(18 \, a^{5} b^{2} - 58 \, a^{3} b^{4} + 40 \, a b^{6} + 2 \, {\left(9 \, a^{5} b^{2} - 29 \, a^{3} b^{4} + 20 \, a b^{6}\right)} \cos\left(d x + c\right)^{4} - 4 \, {\left(9 \, a^{5} b^{2} - 29 \, a^{3} b^{4} + 20 \, a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(9 \, a^{6} b - 20 \, a^{4} b^{3} - 9 \, a^{2} b^{5} + 20 \, b^{7} + {\left(9 \, a^{4} b^{3} - 29 \, a^{2} b^{5} + 20 \, b^{7}\right)} \cos\left(d x + c\right)^{4} - {\left(9 \, a^{6} b - 11 \, a^{4} b^{3} - 38 \, a^{2} b^{5} + 40 \, b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 2 \, {\left(2 \, {\left(14 \, a^{6} b - 59 \, a^{4} b^{3} + 45 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(11 \, a^{6} b - 41 \, a^{4} b^{3} + 30 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{12 \, {\left(2 \, {\left(a^{9} b - a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{4} - 4 \, {\left(a^{9} b - a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{2} + 2 \, {\left(a^{9} b - a^{7} b^{3}\right)} d + {\left({\left(a^{8} b^{2} - a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{4} - {\left(a^{10} + a^{8} b^{2} - 2 \, a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{10} - a^{6} b^{4}\right)} d\right)} \sin\left(d x + c\right)\right)}}, \frac{2 \, {\left(17 \, a^{5} b^{2} - 77 \, a^{3} b^{4} + 60 \, a b^{6}\right)} \cos\left(d x + c\right)^{5} - 4 \, {\left(4 \, a^{7} + 3 \, a^{5} b^{2} - 67 \, a^{3} b^{4} + 60 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - 6 \, {\left(4 \, a^{5} b - 38 \, a^{3} b^{3} + 40 \, a b^{5} + 2 \, {\left(2 \, a^{5} b - 19 \, a^{3} b^{3} + 20 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} - 4 \, {\left(2 \, a^{5} b - 19 \, a^{3} b^{3} + 20 \, a b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(2 \, a^{6} - 17 \, a^{4} b^{2} + a^{2} b^{4} + 20 \, b^{6} + {\left(2 \, a^{4} b^{2} - 19 \, a^{2} b^{4} + 20 \, b^{6}\right)} \cos\left(d x + c\right)^{4} - {\left(2 \, a^{6} - 15 \, a^{4} b^{2} - 18 \, a^{2} b^{4} + 40 \, b^{6}\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(2 \, a^{7} - 3 \, a^{5} b^{2} - 19 \, a^{3} b^{4} + 20 \, a b^{6}\right)} \cos\left(d x + c\right) - 3 \, {\left(18 \, a^{5} b^{2} - 58 \, a^{3} b^{4} + 40 \, a b^{6} + 2 \, {\left(9 \, a^{5} b^{2} - 29 \, a^{3} b^{4} + 20 \, a b^{6}\right)} \cos\left(d x + c\right)^{4} - 4 \, {\left(9 \, a^{5} b^{2} - 29 \, a^{3} b^{4} + 20 \, a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(9 \, a^{6} b - 20 \, a^{4} b^{3} - 9 \, a^{2} b^{5} + 20 \, b^{7} + {\left(9 \, a^{4} b^{3} - 29 \, a^{2} b^{5} + 20 \, b^{7}\right)} \cos\left(d x + c\right)^{4} - {\left(9 \, a^{6} b - 11 \, a^{4} b^{3} - 38 \, a^{2} b^{5} + 40 \, b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 3 \, {\left(18 \, a^{5} b^{2} - 58 \, a^{3} b^{4} + 40 \, a b^{6} + 2 \, {\left(9 \, a^{5} b^{2} - 29 \, a^{3} b^{4} + 20 \, a b^{6}\right)} \cos\left(d x + c\right)^{4} - 4 \, {\left(9 \, a^{5} b^{2} - 29 \, a^{3} b^{4} + 20 \, a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(9 \, a^{6} b - 20 \, a^{4} b^{3} - 9 \, a^{2} b^{5} + 20 \, b^{7} + {\left(9 \, a^{4} b^{3} - 29 \, a^{2} b^{5} + 20 \, b^{7}\right)} \cos\left(d x + c\right)^{4} - {\left(9 \, a^{6} b - 11 \, a^{4} b^{3} - 38 \, a^{2} b^{5} + 40 \, b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 2 \, {\left(2 \, {\left(14 \, a^{6} b - 59 \, a^{4} b^{3} + 45 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(11 \, a^{6} b - 41 \, a^{4} b^{3} + 30 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{12 \, {\left(2 \, {\left(a^{9} b - a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{4} - 4 \, {\left(a^{9} b - a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{2} + 2 \, {\left(a^{9} b - a^{7} b^{3}\right)} d + {\left({\left(a^{8} b^{2} - a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{4} - {\left(a^{10} + a^{8} b^{2} - 2 \, a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{10} - a^{6} b^{4}\right)} d\right)} \sin\left(d x + c\right)\right)}}\right]"," ",0,"[1/12*(2*(17*a^5*b^2 - 77*a^3*b^4 + 60*a*b^6)*cos(d*x + c)^5 - 4*(4*a^7 + 3*a^5*b^2 - 67*a^3*b^4 + 60*a*b^6)*cos(d*x + c)^3 - 3*(4*a^5*b - 38*a^3*b^3 + 40*a*b^5 + 2*(2*a^5*b - 19*a^3*b^3 + 20*a*b^5)*cos(d*x + c)^4 - 4*(2*a^5*b - 19*a^3*b^3 + 20*a*b^5)*cos(d*x + c)^2 + (2*a^6 - 17*a^4*b^2 + a^2*b^4 + 20*b^6 + (2*a^4*b^2 - 19*a^2*b^4 + 20*b^6)*cos(d*x + c)^4 - (2*a^6 - 15*a^4*b^2 - 18*a^2*b^4 + 40*b^6)*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)) + 6*(2*a^7 - 3*a^5*b^2 - 19*a^3*b^4 + 20*a*b^6)*cos(d*x + c) - 3*(18*a^5*b^2 - 58*a^3*b^4 + 40*a*b^6 + 2*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^4 - 4*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^2 + (9*a^6*b - 20*a^4*b^3 - 9*a^2*b^5 + 20*b^7 + (9*a^4*b^3 - 29*a^2*b^5 + 20*b^7)*cos(d*x + c)^4 - (9*a^6*b - 11*a^4*b^3 - 38*a^2*b^5 + 40*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 3*(18*a^5*b^2 - 58*a^3*b^4 + 40*a*b^6 + 2*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^4 - 4*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^2 + (9*a^6*b - 20*a^4*b^3 - 9*a^2*b^5 + 20*b^7 + (9*a^4*b^3 - 29*a^2*b^5 + 20*b^7)*cos(d*x + c)^4 - (9*a^6*b - 11*a^4*b^3 - 38*a^2*b^5 + 40*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*(2*(14*a^6*b - 59*a^4*b^3 + 45*a^2*b^5)*cos(d*x + c)^3 - 3*(11*a^6*b - 41*a^4*b^3 + 30*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/(2*(a^9*b - a^7*b^3)*d*cos(d*x + c)^4 - 4*(a^9*b - a^7*b^3)*d*cos(d*x + c)^2 + 2*(a^9*b - a^7*b^3)*d + ((a^8*b^2 - a^6*b^4)*d*cos(d*x + c)^4 - (a^10 + a^8*b^2 - 2*a^6*b^4)*d*cos(d*x + c)^2 + (a^10 - a^6*b^4)*d)*sin(d*x + c)), 1/12*(2*(17*a^5*b^2 - 77*a^3*b^4 + 60*a*b^6)*cos(d*x + c)^5 - 4*(4*a^7 + 3*a^5*b^2 - 67*a^3*b^4 + 60*a*b^6)*cos(d*x + c)^3 - 6*(4*a^5*b - 38*a^3*b^3 + 40*a*b^5 + 2*(2*a^5*b - 19*a^3*b^3 + 20*a*b^5)*cos(d*x + c)^4 - 4*(2*a^5*b - 19*a^3*b^3 + 20*a*b^5)*cos(d*x + c)^2 + (2*a^6 - 17*a^4*b^2 + a^2*b^4 + 20*b^6 + (2*a^4*b^2 - 19*a^2*b^4 + 20*b^6)*cos(d*x + c)^4 - (2*a^6 - 15*a^4*b^2 - 18*a^2*b^4 + 40*b^6)*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*(2*a^7 - 3*a^5*b^2 - 19*a^3*b^4 + 20*a*b^6)*cos(d*x + c) - 3*(18*a^5*b^2 - 58*a^3*b^4 + 40*a*b^6 + 2*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^4 - 4*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^2 + (9*a^6*b - 20*a^4*b^3 - 9*a^2*b^5 + 20*b^7 + (9*a^4*b^3 - 29*a^2*b^5 + 20*b^7)*cos(d*x + c)^4 - (9*a^6*b - 11*a^4*b^3 - 38*a^2*b^5 + 40*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 3*(18*a^5*b^2 - 58*a^3*b^4 + 40*a*b^6 + 2*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^4 - 4*(9*a^5*b^2 - 29*a^3*b^4 + 20*a*b^6)*cos(d*x + c)^2 + (9*a^6*b - 20*a^4*b^3 - 9*a^2*b^5 + 20*b^7 + (9*a^4*b^3 - 29*a^2*b^5 + 20*b^7)*cos(d*x + c)^4 - (9*a^6*b - 11*a^4*b^3 - 38*a^2*b^5 + 40*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*(2*(14*a^6*b - 59*a^4*b^3 + 45*a^2*b^5)*cos(d*x + c)^3 - 3*(11*a^6*b - 41*a^4*b^3 + 30*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/(2*(a^9*b - a^7*b^3)*d*cos(d*x + c)^4 - 4*(a^9*b - a^7*b^3)*d*cos(d*x + c)^2 + 2*(a^9*b - a^7*b^3)*d + ((a^8*b^2 - a^6*b^4)*d*cos(d*x + c)^4 - (a^10 + a^8*b^2 - 2*a^6*b^4)*d*cos(d*x + c)^2 + (a^10 - a^6*b^4)*d)*sin(d*x + c))]","B",0
202,1,2571,0,3.188009," ","integrate(cot(d*x+c)^6/(a+b*sin(d*x+c))^3,x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(91 \, a^{5} b^{2} - 645 \, a^{3} b^{4} + 630 \, a b^{6}\right)} \cos\left(d x + c\right)^{7} - 4 \, {\left(92 \, a^{7} + 67 \, a^{5} b^{2} - 3450 \, a^{3} b^{4} + 3780 \, a b^{6}\right)} \cos\left(d x + c\right)^{5} + 40 \, {\left(14 \, a^{7} - 37 \, a^{5} b^{2} - 303 \, a^{3} b^{4} + 378 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - 60 \, {\left(2 \, {\left(2 \, a^{5} b - 29 \, a^{3} b^{3} + 42 \, a b^{5}\right)} \cos\left(d x + c\right)^{6} - 4 \, a^{5} b + 58 \, a^{3} b^{3} - 84 \, a b^{5} - 6 \, {\left(2 \, a^{5} b - 29 \, a^{3} b^{3} + 42 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} + 6 \, {\left(2 \, a^{5} b - 29 \, a^{3} b^{3} + 42 \, a b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left({\left(2 \, a^{4} b^{2} - 29 \, a^{2} b^{4} + 42 \, b^{6}\right)} \cos\left(d x + c\right)^{6} - 2 \, a^{6} + 27 \, a^{4} b^{2} - 13 \, a^{2} b^{4} - 42 \, b^{6} - {\left(2 \, a^{6} - 23 \, a^{4} b^{2} - 45 \, a^{2} b^{4} + 126 \, b^{6}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{6} - 52 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + 126 \, b^{6}\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) - 60 \, {\left(4 \, a^{7} - 17 \, a^{5} b^{2} - 58 \, a^{3} b^{4} + 84 \, a b^{6}\right)} \cos\left(d x + c\right) + 15 \, {\left(90 \, a^{5} b^{2} - 400 \, a^{3} b^{4} + 336 \, a b^{6} - 2 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{6} + 6 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{4} - 6 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(45 \, a^{6} b - 155 \, a^{4} b^{3} - 32 \, a^{2} b^{5} + 168 \, b^{7} - {\left(45 \, a^{4} b^{3} - 200 \, a^{2} b^{5} + 168 \, b^{7}\right)} \cos\left(d x + c\right)^{6} + {\left(45 \, a^{6} b - 65 \, a^{4} b^{3} - 432 \, a^{2} b^{5} + 504 \, b^{7}\right)} \cos\left(d x + c\right)^{4} - {\left(90 \, a^{6} b - 265 \, a^{4} b^{3} - 264 \, a^{2} b^{5} + 504 \, b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 15 \, {\left(90 \, a^{5} b^{2} - 400 \, a^{3} b^{4} + 336 \, a b^{6} - 2 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{6} + 6 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{4} - 6 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(45 \, a^{6} b - 155 \, a^{4} b^{3} - 32 \, a^{2} b^{5} + 168 \, b^{7} - {\left(45 \, a^{4} b^{3} - 200 \, a^{2} b^{5} + 168 \, b^{7}\right)} \cos\left(d x + c\right)^{6} + {\left(45 \, a^{6} b - 65 \, a^{4} b^{3} - 432 \, a^{2} b^{5} + 504 \, b^{7}\right)} \cos\left(d x + c\right)^{4} - {\left(90 \, a^{6} b - 265 \, a^{4} b^{3} - 264 \, a^{2} b^{5} + 504 \, b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 2 \, {\left({\left(608 \, a^{6} b - 3975 \, a^{4} b^{3} + 3780 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{5} - 5 \, {\left(289 \, a^{6} b - 1632 \, a^{4} b^{3} + 1512 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{3} + 15 \, {\left(53 \, a^{6} b - 279 \, a^{4} b^{3} + 252 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{240 \, {\left(2 \, a^{9} b d \cos\left(d x + c\right)^{6} - 6 \, a^{9} b d \cos\left(d x + c\right)^{4} + 6 \, a^{9} b d \cos\left(d x + c\right)^{2} - 2 \, a^{9} b d + {\left(a^{8} b^{2} d \cos\left(d x + c\right)^{6} - {\left(a^{10} + 3 \, a^{8} b^{2}\right)} d \cos\left(d x + c\right)^{4} + {\left(2 \, a^{10} + 3 \, a^{8} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{10} + a^{8} b^{2}\right)} d\right)} \sin\left(d x + c\right)\right)}}, -\frac{8 \, {\left(91 \, a^{5} b^{2} - 645 \, a^{3} b^{4} + 630 \, a b^{6}\right)} \cos\left(d x + c\right)^{7} - 4 \, {\left(92 \, a^{7} + 67 \, a^{5} b^{2} - 3450 \, a^{3} b^{4} + 3780 \, a b^{6}\right)} \cos\left(d x + c\right)^{5} + 40 \, {\left(14 \, a^{7} - 37 \, a^{5} b^{2} - 303 \, a^{3} b^{4} + 378 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - 120 \, {\left(2 \, {\left(2 \, a^{5} b - 29 \, a^{3} b^{3} + 42 \, a b^{5}\right)} \cos\left(d x + c\right)^{6} - 4 \, a^{5} b + 58 \, a^{3} b^{3} - 84 \, a b^{5} - 6 \, {\left(2 \, a^{5} b - 29 \, a^{3} b^{3} + 42 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} + 6 \, {\left(2 \, a^{5} b - 29 \, a^{3} b^{3} + 42 \, a b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left({\left(2 \, a^{4} b^{2} - 29 \, a^{2} b^{4} + 42 \, b^{6}\right)} \cos\left(d x + c\right)^{6} - 2 \, a^{6} + 27 \, a^{4} b^{2} - 13 \, a^{2} b^{4} - 42 \, b^{6} - {\left(2 \, a^{6} - 23 \, a^{4} b^{2} - 45 \, a^{2} b^{4} + 126 \, b^{6}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{6} - 52 \, a^{4} b^{2} - 3 \, a^{2} b^{4} + 126 \, b^{6}\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) - 60 \, {\left(4 \, a^{7} - 17 \, a^{5} b^{2} - 58 \, a^{3} b^{4} + 84 \, a b^{6}\right)} \cos\left(d x + c\right) + 15 \, {\left(90 \, a^{5} b^{2} - 400 \, a^{3} b^{4} + 336 \, a b^{6} - 2 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{6} + 6 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{4} - 6 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(45 \, a^{6} b - 155 \, a^{4} b^{3} - 32 \, a^{2} b^{5} + 168 \, b^{7} - {\left(45 \, a^{4} b^{3} - 200 \, a^{2} b^{5} + 168 \, b^{7}\right)} \cos\left(d x + c\right)^{6} + {\left(45 \, a^{6} b - 65 \, a^{4} b^{3} - 432 \, a^{2} b^{5} + 504 \, b^{7}\right)} \cos\left(d x + c\right)^{4} - {\left(90 \, a^{6} b - 265 \, a^{4} b^{3} - 264 \, a^{2} b^{5} + 504 \, b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 15 \, {\left(90 \, a^{5} b^{2} - 400 \, a^{3} b^{4} + 336 \, a b^{6} - 2 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{6} + 6 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{4} - 6 \, {\left(45 \, a^{5} b^{2} - 200 \, a^{3} b^{4} + 168 \, a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(45 \, a^{6} b - 155 \, a^{4} b^{3} - 32 \, a^{2} b^{5} + 168 \, b^{7} - {\left(45 \, a^{4} b^{3} - 200 \, a^{2} b^{5} + 168 \, b^{7}\right)} \cos\left(d x + c\right)^{6} + {\left(45 \, a^{6} b - 65 \, a^{4} b^{3} - 432 \, a^{2} b^{5} + 504 \, b^{7}\right)} \cos\left(d x + c\right)^{4} - {\left(90 \, a^{6} b - 265 \, a^{4} b^{3} - 264 \, a^{2} b^{5} + 504 \, b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 2 \, {\left({\left(608 \, a^{6} b - 3975 \, a^{4} b^{3} + 3780 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{5} - 5 \, {\left(289 \, a^{6} b - 1632 \, a^{4} b^{3} + 1512 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{3} + 15 \, {\left(53 \, a^{6} b - 279 \, a^{4} b^{3} + 252 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{240 \, {\left(2 \, a^{9} b d \cos\left(d x + c\right)^{6} - 6 \, a^{9} b d \cos\left(d x + c\right)^{4} + 6 \, a^{9} b d \cos\left(d x + c\right)^{2} - 2 \, a^{9} b d + {\left(a^{8} b^{2} d \cos\left(d x + c\right)^{6} - {\left(a^{10} + 3 \, a^{8} b^{2}\right)} d \cos\left(d x + c\right)^{4} + {\left(2 \, a^{10} + 3 \, a^{8} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{10} + a^{8} b^{2}\right)} d\right)} \sin\left(d x + c\right)\right)}}\right]"," ",0,"[-1/240*(8*(91*a^5*b^2 - 645*a^3*b^4 + 630*a*b^6)*cos(d*x + c)^7 - 4*(92*a^7 + 67*a^5*b^2 - 3450*a^3*b^4 + 3780*a*b^6)*cos(d*x + c)^5 + 40*(14*a^7 - 37*a^5*b^2 - 303*a^3*b^4 + 378*a*b^6)*cos(d*x + c)^3 - 60*(2*(2*a^5*b - 29*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^6 - 4*a^5*b + 58*a^3*b^3 - 84*a*b^5 - 6*(2*a^5*b - 29*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^4 + 6*(2*a^5*b - 29*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^2 + ((2*a^4*b^2 - 29*a^2*b^4 + 42*b^6)*cos(d*x + c)^6 - 2*a^6 + 27*a^4*b^2 - 13*a^2*b^4 - 42*b^6 - (2*a^6 - 23*a^4*b^2 - 45*a^2*b^4 + 126*b^6)*cos(d*x + c)^4 + (4*a^6 - 52*a^4*b^2 - 3*a^2*b^4 + 126*b^6)*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)) - 60*(4*a^7 - 17*a^5*b^2 - 58*a^3*b^4 + 84*a*b^6)*cos(d*x + c) + 15*(90*a^5*b^2 - 400*a^3*b^4 + 336*a*b^6 - 2*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^6 + 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^4 - 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^2 + (45*a^6*b - 155*a^4*b^3 - 32*a^2*b^5 + 168*b^7 - (45*a^4*b^3 - 200*a^2*b^5 + 168*b^7)*cos(d*x + c)^6 + (45*a^6*b - 65*a^4*b^3 - 432*a^2*b^5 + 504*b^7)*cos(d*x + c)^4 - (90*a^6*b - 265*a^4*b^3 - 264*a^2*b^5 + 504*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 15*(90*a^5*b^2 - 400*a^3*b^4 + 336*a*b^6 - 2*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^6 + 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^4 - 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^2 + (45*a^6*b - 155*a^4*b^3 - 32*a^2*b^5 + 168*b^7 - (45*a^4*b^3 - 200*a^2*b^5 + 168*b^7)*cos(d*x + c)^6 + (45*a^6*b - 65*a^4*b^3 - 432*a^2*b^5 + 504*b^7)*cos(d*x + c)^4 - (90*a^6*b - 265*a^4*b^3 - 264*a^2*b^5 + 504*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*((608*a^6*b - 3975*a^4*b^3 + 3780*a^2*b^5)*cos(d*x + c)^5 - 5*(289*a^6*b - 1632*a^4*b^3 + 1512*a^2*b^5)*cos(d*x + c)^3 + 15*(53*a^6*b - 279*a^4*b^3 + 252*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/(2*a^9*b*d*cos(d*x + c)^6 - 6*a^9*b*d*cos(d*x + c)^4 + 6*a^9*b*d*cos(d*x + c)^2 - 2*a^9*b*d + (a^8*b^2*d*cos(d*x + c)^6 - (a^10 + 3*a^8*b^2)*d*cos(d*x + c)^4 + (2*a^10 + 3*a^8*b^2)*d*cos(d*x + c)^2 - (a^10 + a^8*b^2)*d)*sin(d*x + c)), -1/240*(8*(91*a^5*b^2 - 645*a^3*b^4 + 630*a*b^6)*cos(d*x + c)^7 - 4*(92*a^7 + 67*a^5*b^2 - 3450*a^3*b^4 + 3780*a*b^6)*cos(d*x + c)^5 + 40*(14*a^7 - 37*a^5*b^2 - 303*a^3*b^4 + 378*a*b^6)*cos(d*x + c)^3 - 120*(2*(2*a^5*b - 29*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^6 - 4*a^5*b + 58*a^3*b^3 - 84*a*b^5 - 6*(2*a^5*b - 29*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^4 + 6*(2*a^5*b - 29*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^2 + ((2*a^4*b^2 - 29*a^2*b^4 + 42*b^6)*cos(d*x + c)^6 - 2*a^6 + 27*a^4*b^2 - 13*a^2*b^4 - 42*b^6 - (2*a^6 - 23*a^4*b^2 - 45*a^2*b^4 + 126*b^6)*cos(d*x + c)^4 + (4*a^6 - 52*a^4*b^2 - 3*a^2*b^4 + 126*b^6)*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))) - 60*(4*a^7 - 17*a^5*b^2 - 58*a^3*b^4 + 84*a*b^6)*cos(d*x + c) + 15*(90*a^5*b^2 - 400*a^3*b^4 + 336*a*b^6 - 2*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^6 + 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^4 - 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^2 + (45*a^6*b - 155*a^4*b^3 - 32*a^2*b^5 + 168*b^7 - (45*a^4*b^3 - 200*a^2*b^5 + 168*b^7)*cos(d*x + c)^6 + (45*a^6*b - 65*a^4*b^3 - 432*a^2*b^5 + 504*b^7)*cos(d*x + c)^4 - (90*a^6*b - 265*a^4*b^3 - 264*a^2*b^5 + 504*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 15*(90*a^5*b^2 - 400*a^3*b^4 + 336*a*b^6 - 2*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^6 + 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^4 - 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^2 + (45*a^6*b - 155*a^4*b^3 - 32*a^2*b^5 + 168*b^7 - (45*a^4*b^3 - 200*a^2*b^5 + 168*b^7)*cos(d*x + c)^6 + (45*a^6*b - 65*a^4*b^3 - 432*a^2*b^5 + 504*b^7)*cos(d*x + c)^4 - (90*a^6*b - 265*a^4*b^3 - 264*a^2*b^5 + 504*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*((608*a^6*b - 3975*a^4*b^3 + 3780*a^2*b^5)*cos(d*x + c)^5 - 5*(289*a^6*b - 1632*a^4*b^3 + 1512*a^2*b^5)*cos(d*x + c)^3 + 15*(53*a^6*b - 279*a^4*b^3 + 252*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/(2*a^9*b*d*cos(d*x + c)^6 - 6*a^9*b*d*cos(d*x + c)^4 + 6*a^9*b*d*cos(d*x + c)^2 - 2*a^9*b*d + (a^8*b^2*d*cos(d*x + c)^6 - (a^10 + 3*a^8*b^2)*d*cos(d*x + c)^4 + (2*a^10 + 3*a^8*b^2)*d*cos(d*x + c)^2 - (a^10 + a^8*b^2)*d)*sin(d*x + c))]","B",0
203,0,0,0,1.408124," ","integrate((a+b*sin(f*x+e))^3*(g*tan(f*x+e))^p,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(g \tan\left(f x + e\right)\right)^{p}, 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))*(g*tan(f*x + e))^p, x)","F",0
204,0,0,0,1.348333," ","integrate((a+b*sin(f*x+e))^2*(g*tan(f*x+e))^p,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(g \tan\left(f x + e\right)\right)^{p}, x\right)"," ",0,"integral(-(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)*(g*tan(f*x + e))^p, x)","F",0
205,0,0,0,1.291649," ","integrate((a+b*sin(f*x+e))*(g*tan(f*x+e))^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right) + a\right)} \left(g \tan\left(f x + e\right)\right)^{p}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)*(g*tan(f*x + e))^p, x)","F",0
206,0,0,0,1.318991," ","integrate((g*tan(f*x+e))^p/(a+b*sin(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(g \tan\left(f x + e\right)\right)^{p}}{b \sin\left(f x + e\right) + a}, x\right)"," ",0,"integral((g*tan(f*x + e))^p/(b*sin(f*x + e) + a), x)","F",0
207,0,0,0,1.594958," ","integrate((g*tan(f*x+e))^p/(a+b*sin(f*x+e))^2,x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\left(g \tan\left(f x + e\right)\right)^{p}}{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(-(g*tan(f*x + e))^p/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2), x)","F",0
208,0,0,0,2.172121," ","integrate((a+b*sin(f*x+e))^m*(g*tan(f*x+e))^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right) + a\right)}^{m} \left(g \tan\left(f x + e\right)\right)^{p}, x\right)"," ",0,"integral((b*sin(f*x + e) + a)^m*(g*tan(f*x + e))^p, x)","F",0
