1,1,123,0,0.708948," ","integrate((a+a*sec(d*x+c))*tan(d*x+c)^9,x, algorithm=""fricas"")","-\frac{2520 \, a \cos\left(d x + c\right)^{9} \log\left(-\cos\left(d x + c\right)\right) - 2520 \, a \cos\left(d x + c\right)^{8} + 5040 \, a \cos\left(d x + c\right)^{7} + 3360 \, a \cos\left(d x + c\right)^{6} - 3780 \, a \cos\left(d x + c\right)^{5} - 3024 \, a \cos\left(d x + c\right)^{4} + 1680 \, a \cos\left(d x + c\right)^{3} + 1440 \, a \cos\left(d x + c\right)^{2} - 315 \, a \cos\left(d x + c\right) - 280 \, a}{2520 \, d \cos\left(d x + c\right)^{9}}"," ",0,"-1/2520*(2520*a*cos(d*x + c)^9*log(-cos(d*x + c)) - 2520*a*cos(d*x + c)^8 + 5040*a*cos(d*x + c)^7 + 3360*a*cos(d*x + c)^6 - 3780*a*cos(d*x + c)^5 - 3024*a*cos(d*x + c)^4 + 1680*a*cos(d*x + c)^3 + 1440*a*cos(d*x + c)^2 - 315*a*cos(d*x + c) - 280*a)/(d*cos(d*x + c)^9)","A",0
2,1,101,0,0.797130," ","integrate((a+a*sec(d*x+c))*tan(d*x+c)^7,x, algorithm=""fricas"")","\frac{420 \, a \cos\left(d x + c\right)^{7} \log\left(-\cos\left(d x + c\right)\right) - 420 \, a \cos\left(d x + c\right)^{6} + 630 \, a \cos\left(d x + c\right)^{5} + 420 \, a \cos\left(d x + c\right)^{4} - 315 \, a \cos\left(d x + c\right)^{3} - 252 \, a \cos\left(d x + c\right)^{2} + 70 \, a \cos\left(d x + c\right) + 60 \, a}{420 \, d \cos\left(d x + c\right)^{7}}"," ",0,"1/420*(420*a*cos(d*x + c)^7*log(-cos(d*x + c)) - 420*a*cos(d*x + c)^6 + 630*a*cos(d*x + c)^5 + 420*a*cos(d*x + c)^4 - 315*a*cos(d*x + c)^3 - 252*a*cos(d*x + c)^2 + 70*a*cos(d*x + c) + 60*a)/(d*cos(d*x + c)^7)","A",0
3,1,79,0,0.563891," ","integrate((a+a*sec(d*x+c))*tan(d*x+c)^5,x, algorithm=""fricas"")","-\frac{60 \, a \cos\left(d x + c\right)^{5} \log\left(-\cos\left(d x + c\right)\right) - 60 \, a \cos\left(d x + c\right)^{4} + 60 \, a \cos\left(d x + c\right)^{3} + 40 \, a \cos\left(d x + c\right)^{2} - 15 \, a \cos\left(d x + c\right) - 12 \, a}{60 \, d \cos\left(d x + c\right)^{5}}"," ",0,"-1/60*(60*a*cos(d*x + c)^5*log(-cos(d*x + c)) - 60*a*cos(d*x + c)^4 + 60*a*cos(d*x + c)^3 + 40*a*cos(d*x + c)^2 - 15*a*cos(d*x + c) - 12*a)/(d*cos(d*x + c)^5)","A",0
4,1,57,0,0.710042," ","integrate((a+a*sec(d*x+c))*tan(d*x+c)^3,x, algorithm=""fricas"")","\frac{6 \, a \cos\left(d x + c\right)^{3} \log\left(-\cos\left(d x + c\right)\right) - 6 \, a \cos\left(d x + c\right)^{2} + 3 \, a \cos\left(d x + c\right) + 2 \, a}{6 \, d \cos\left(d x + c\right)^{3}}"," ",0,"1/6*(6*a*cos(d*x + c)^3*log(-cos(d*x + c)) - 6*a*cos(d*x + c)^2 + 3*a*cos(d*x + c) + 2*a)/(d*cos(d*x + c)^3)","A",0
5,1,34,0,0.719805," ","integrate((a+a*sec(d*x+c))*tan(d*x+c),x, algorithm=""fricas"")","-\frac{a \cos\left(d x + c\right) \log\left(-\cos\left(d x + c\right)\right) - a}{d \cos\left(d x + c\right)}"," ",0,"-(a*cos(d*x + c)*log(-cos(d*x + c)) - a)/(d*cos(d*x + c))","A",0
6,1,16,0,0.670980," ","integrate(cot(d*x+c)*(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\frac{a \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{d}"," ",0,"a*log(-1/2*cos(d*x + c) + 1/2)/d","A",0
7,1,69,0,0.612186," ","integrate(cot(d*x+c)^3*(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{{\left(a \cos\left(d x + c\right) - a\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 3 \, {\left(a \cos\left(d x + c\right) - a\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 2 \, a}{4 \, {\left(d \cos\left(d x + c\right) - d\right)}}"," ",0,"-1/4*((a*cos(d*x + c) - a)*log(1/2*cos(d*x + c) + 1/2) + 3*(a*cos(d*x + c) - a)*log(-1/2*cos(d*x + c) + 1/2) - 2*a)/(d*cos(d*x + c) - d)","A",0
8,1,150,0,0.614841," ","integrate(cot(d*x+c)^5*(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{10 \, a \cos\left(d x + c\right)^{2} + 6 \, a \cos\left(d x + c\right) - 5 \, {\left(a \cos\left(d x + c\right)^{3} - a \cos\left(d x + c\right)^{2} - a \cos\left(d x + c\right) + a\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 11 \, {\left(a \cos\left(d x + c\right)^{3} - a \cos\left(d x + c\right)^{2} - a \cos\left(d x + c\right) + a\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 12 \, a}{16 \, {\left(d \cos\left(d x + c\right)^{3} - d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) + d\right)}}"," ",0,"-1/16*(10*a*cos(d*x + c)^2 + 6*a*cos(d*x + c) - 5*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*log(1/2*cos(d*x + c) + 1/2) - 11*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*log(-1/2*cos(d*x + c) + 1/2) - 12*a)/(d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*cos(d*x + c) + d)","A",0
9,1,241,0,0.616306," ","integrate(cot(d*x+c)^7*(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\frac{66 \, a \cos\left(d x + c\right)^{4} + 78 \, a \cos\left(d x + c\right)^{3} - 158 \, a \cos\left(d x + c\right)^{2} - 58 \, a \cos\left(d x + c\right) - 33 \, {\left(a \cos\left(d x + c\right)^{5} - a \cos\left(d x + c\right)^{4} - 2 \, a \cos\left(d x + c\right)^{3} + 2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 63 \, {\left(a \cos\left(d x + c\right)^{5} - a \cos\left(d x + c\right)^{4} - 2 \, a \cos\left(d x + c\right)^{3} + 2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 88 \, a}{96 \, {\left(d \cos\left(d x + c\right)^{5} - d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{3} + 2 \, d \cos\left(d x + c\right)^{2} + d \cos\left(d x + c\right) - d\right)}}"," ",0,"1/96*(66*a*cos(d*x + c)^4 + 78*a*cos(d*x + c)^3 - 158*a*cos(d*x + c)^2 - 58*a*cos(d*x + c) - 33*(a*cos(d*x + c)^5 - a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 + 2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a)*log(1/2*cos(d*x + c) + 1/2) - 63*(a*cos(d*x + c)^5 - a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^3 + 2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a)*log(-1/2*cos(d*x + c) + 1/2) + 88*a)/(d*cos(d*x + c)^5 - d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^2 + d*cos(d*x + c) - d)","B",0
10,1,156,0,0.757581," ","integrate((a+a*sec(d*x+c))*tan(d*x+c)^8,x, algorithm=""fricas"")","\frac{26880 \, a d x \cos\left(d x + c\right)^{8} + 3675 \, a \cos\left(d x + c\right)^{8} \log\left(\sin\left(d x + c\right) + 1\right) - 3675 \, a \cos\left(d x + c\right)^{8} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(22528 \, a \cos\left(d x + c\right)^{7} + 9765 \, a \cos\left(d x + c\right)^{6} - 15616 \, a \cos\left(d x + c\right)^{5} - 11410 \, a \cos\left(d x + c\right)^{4} + 8448 \, a \cos\left(d x + c\right)^{3} + 7000 \, a \cos\left(d x + c\right)^{2} - 1920 \, a \cos\left(d x + c\right) - 1680 \, a\right)} \sin\left(d x + c\right)}{26880 \, d \cos\left(d x + c\right)^{8}}"," ",0,"1/26880*(26880*a*d*x*cos(d*x + c)^8 + 3675*a*cos(d*x + c)^8*log(sin(d*x + c) + 1) - 3675*a*cos(d*x + c)^8*log(-sin(d*x + c) + 1) - 2*(22528*a*cos(d*x + c)^7 + 9765*a*cos(d*x + c)^6 - 15616*a*cos(d*x + c)^5 - 11410*a*cos(d*x + c)^4 + 8448*a*cos(d*x + c)^3 + 7000*a*cos(d*x + c)^2 - 1920*a*cos(d*x + c) - 1680*a)*sin(d*x + c))/(d*cos(d*x + c)^8)","A",0
11,1,134,0,0.694958," ","integrate((a+a*sec(d*x+c))*tan(d*x+c)^6,x, algorithm=""fricas"")","-\frac{480 \, a d x \cos\left(d x + c\right)^{6} + 75 \, a \cos\left(d x + c\right)^{6} \log\left(\sin\left(d x + c\right) + 1\right) - 75 \, a \cos\left(d x + c\right)^{6} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(368 \, a \cos\left(d x + c\right)^{5} + 165 \, a \cos\left(d x + c\right)^{4} - 176 \, a \cos\left(d x + c\right)^{3} - 130 \, a \cos\left(d x + c\right)^{2} + 48 \, a \cos\left(d x + c\right) + 40 \, a\right)} \sin\left(d x + c\right)}{480 \, d \cos\left(d x + c\right)^{6}}"," ",0,"-1/480*(480*a*d*x*cos(d*x + c)^6 + 75*a*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 75*a*cos(d*x + c)^6*log(-sin(d*x + c) + 1) - 2*(368*a*cos(d*x + c)^5 + 165*a*cos(d*x + c)^4 - 176*a*cos(d*x + c)^3 - 130*a*cos(d*x + c)^2 + 48*a*cos(d*x + c) + 40*a)*sin(d*x + c))/(d*cos(d*x + c)^6)","A",0
12,1,112,0,0.804554," ","integrate((a+a*sec(d*x+c))*tan(d*x+c)^4,x, algorithm=""fricas"")","\frac{48 \, a d x \cos\left(d x + c\right)^{4} + 9 \, a \cos\left(d x + c\right)^{4} \log\left(\sin\left(d x + c\right) + 1\right) - 9 \, a \cos\left(d x + c\right)^{4} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(32 \, a \cos\left(d x + c\right)^{3} + 15 \, a \cos\left(d x + c\right)^{2} - 8 \, a \cos\left(d x + c\right) - 6 \, a\right)} \sin\left(d x + c\right)}{48 \, d \cos\left(d x + c\right)^{4}}"," ",0,"1/48*(48*a*d*x*cos(d*x + c)^4 + 9*a*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 9*a*cos(d*x + c)^4*log(-sin(d*x + c) + 1) - 2*(32*a*cos(d*x + c)^3 + 15*a*cos(d*x + c)^2 - 8*a*cos(d*x + c) - 6*a)*sin(d*x + c))/(d*cos(d*x + c)^4)","A",0
13,1,87,0,0.828384," ","integrate((a+a*sec(d*x+c))*tan(d*x+c)^2,x, algorithm=""fricas"")","-\frac{4 \, a d x \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - a \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(2 \, a \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right)}{4 \, d \cos\left(d x + c\right)^{2}}"," ",0,"-1/4*(4*a*d*x*cos(d*x + c)^2 + a*cos(d*x + c)^2*log(sin(d*x + c) + 1) - a*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 2*(2*a*cos(d*x + c) + a)*sin(d*x + c))/(d*cos(d*x + c)^2)","B",0
14,1,33,0,0.941493," ","integrate(cot(d*x+c)^2*(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{a d x \sin\left(d x + c\right) + a \cos\left(d x + c\right) + a}{d \sin\left(d x + c\right)}"," ",0,"-(a*d*x*sin(d*x + c) + a*cos(d*x + c) + a)/(d*sin(d*x + c))","A",0
15,1,72,0,0.634780," ","integrate(cot(d*x+c)^4*(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\frac{4 \, a \cos\left(d x + c\right)^{2} - a \cos\left(d x + c\right) + 3 \, {\left(a d x \cos\left(d x + c\right) - a d x\right)} \sin\left(d x + c\right) - 2 \, a}{3 \, {\left(d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right)}"," ",0,"1/3*(4*a*cos(d*x + c)^2 - a*cos(d*x + c) + 3*(a*d*x*cos(d*x + c) - a*d*x)*sin(d*x + c) - 2*a)/((d*cos(d*x + c) - d)*sin(d*x + c))","A",0
16,1,139,0,0.508656," ","integrate(cot(d*x+c)^6*(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{23 \, a \cos\left(d x + c\right)^{4} - 8 \, a \cos\left(d x + c\right)^{3} - 27 \, a \cos\left(d x + c\right)^{2} + 7 \, a \cos\left(d x + c\right) + 15 \, {\left(a d x \cos\left(d x + c\right)^{3} - a d x \cos\left(d x + c\right)^{2} - a d x \cos\left(d x + c\right) + a d x\right)} \sin\left(d x + c\right) + 8 \, a}{15 \, {\left(d \cos\left(d x + c\right)^{3} - d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right)}"," ",0,"-1/15*(23*a*cos(d*x + c)^4 - 8*a*cos(d*x + c)^3 - 27*a*cos(d*x + c)^2 + 7*a*cos(d*x + c) + 15*(a*d*x*cos(d*x + c)^3 - a*d*x*cos(d*x + c)^2 - a*d*x*cos(d*x + c) + a*d*x)*sin(d*x + c) + 8*a)/((d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*cos(d*x + c) + d)*sin(d*x + c))","A",0
17,1,210,0,0.738164," ","integrate(cot(d*x+c)^8*(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\frac{176 \, a \cos\left(d x + c\right)^{6} - 71 \, a \cos\left(d x + c\right)^{5} - 335 \, a \cos\left(d x + c\right)^{4} + 125 \, a \cos\left(d x + c\right)^{3} + 225 \, a \cos\left(d x + c\right)^{2} - 57 \, a \cos\left(d x + c\right) + 105 \, {\left(a d x \cos\left(d x + c\right)^{5} - a d x \cos\left(d x + c\right)^{4} - 2 \, a d x \cos\left(d x + c\right)^{3} + 2 \, a d x \cos\left(d x + c\right)^{2} + a d x \cos\left(d x + c\right) - a d x\right)} \sin\left(d x + c\right) - 48 \, a}{105 \, {\left(d \cos\left(d x + c\right)^{5} - d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{3} + 2 \, d \cos\left(d x + c\right)^{2} + d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right)}"," ",0,"1/105*(176*a*cos(d*x + c)^6 - 71*a*cos(d*x + c)^5 - 335*a*cos(d*x + c)^4 + 125*a*cos(d*x + c)^3 + 225*a*cos(d*x + c)^2 - 57*a*cos(d*x + c) + 105*(a*d*x*cos(d*x + c)^5 - a*d*x*cos(d*x + c)^4 - 2*a*d*x*cos(d*x + c)^3 + 2*a*d*x*cos(d*x + c)^2 + a*d*x*cos(d*x + c) - a*d*x)*sin(d*x + c) - 48*a)/((d*cos(d*x + c)^5 - d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^2 + d*cos(d*x + c) - d)*sin(d*x + c))","B",0
18,1,279,0,0.893156," ","integrate(cot(d*x+c)^10*(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{563 \, a \cos\left(d x + c\right)^{8} - 248 \, a \cos\left(d x + c\right)^{7} - 1498 \, a \cos\left(d x + c\right)^{6} + 658 \, a \cos\left(d x + c\right)^{5} + 1610 \, a \cos\left(d x + c\right)^{4} - 602 \, a \cos\left(d x + c\right)^{3} - 763 \, a \cos\left(d x + c\right)^{2} + 187 \, a \cos\left(d x + c\right) + 315 \, {\left(a d x \cos\left(d x + c\right)^{7} - a d x \cos\left(d x + c\right)^{6} - 3 \, a d x \cos\left(d x + c\right)^{5} + 3 \, a d x \cos\left(d x + c\right)^{4} + 3 \, a d x \cos\left(d x + c\right)^{3} - 3 \, a d x \cos\left(d x + c\right)^{2} - a d x \cos\left(d x + c\right) + a d x\right)} \sin\left(d x + c\right) + 128 \, a}{315 \, {\left(d \cos\left(d x + c\right)^{7} - d \cos\left(d x + c\right)^{6} - 3 \, d \cos\left(d x + c\right)^{5} + 3 \, d \cos\left(d x + c\right)^{4} + 3 \, d \cos\left(d x + c\right)^{3} - 3 \, d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right)}"," ",0,"-1/315*(563*a*cos(d*x + c)^8 - 248*a*cos(d*x + c)^7 - 1498*a*cos(d*x + c)^6 + 658*a*cos(d*x + c)^5 + 1610*a*cos(d*x + c)^4 - 602*a*cos(d*x + c)^3 - 763*a*cos(d*x + c)^2 + 187*a*cos(d*x + c) + 315*(a*d*x*cos(d*x + c)^7 - a*d*x*cos(d*x + c)^6 - 3*a*d*x*cos(d*x + c)^5 + 3*a*d*x*cos(d*x + c)^4 + 3*a*d*x*cos(d*x + c)^3 - 3*a*d*x*cos(d*x + c)^2 - a*d*x*cos(d*x + c) + a*d*x)*sin(d*x + c) + 128*a)/((d*cos(d*x + c)^7 - d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^5 + 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^3 - 3*d*cos(d*x + c)^2 - d*cos(d*x + c) + d)*sin(d*x + c))","B",0
19,1,156,0,0.873042," ","integrate((a+a*sec(d*x+c))^2*tan(d*x+c)^9,x, algorithm=""fricas"")","-\frac{2520 \, a^{2} \cos\left(d x + c\right)^{10} \log\left(-\cos\left(d x + c\right)\right) - 5040 \, a^{2} \cos\left(d x + c\right)^{9} + 3780 \, a^{2} \cos\left(d x + c\right)^{8} + 6720 \, a^{2} \cos\left(d x + c\right)^{7} - 1260 \, a^{2} \cos\left(d x + c\right)^{6} - 6048 \, a^{2} \cos\left(d x + c\right)^{5} - 840 \, a^{2} \cos\left(d x + c\right)^{4} + 2880 \, a^{2} \cos\left(d x + c\right)^{3} + 945 \, a^{2} \cos\left(d x + c\right)^{2} - 560 \, a^{2} \cos\left(d x + c\right) - 252 \, a^{2}}{2520 \, d \cos\left(d x + c\right)^{10}}"," ",0,"-1/2520*(2520*a^2*cos(d*x + c)^10*log(-cos(d*x + c)) - 5040*a^2*cos(d*x + c)^9 + 3780*a^2*cos(d*x + c)^8 + 6720*a^2*cos(d*x + c)^7 - 1260*a^2*cos(d*x + c)^6 - 6048*a^2*cos(d*x + c)^5 - 840*a^2*cos(d*x + c)^4 + 2880*a^2*cos(d*x + c)^3 + 945*a^2*cos(d*x + c)^2 - 560*a^2*cos(d*x + c) - 252*a^2)/(d*cos(d*x + c)^10)","A",0
20,1,117,0,0.726553," ","integrate((a+a*sec(d*x+c))^2*tan(d*x+c)^7,x, algorithm=""fricas"")","\frac{840 \, a^{2} \cos\left(d x + c\right)^{8} \log\left(-\cos\left(d x + c\right)\right) - 1680 \, a^{2} \cos\left(d x + c\right)^{7} + 840 \, a^{2} \cos\left(d x + c\right)^{6} + 1680 \, a^{2} \cos\left(d x + c\right)^{5} - 1008 \, a^{2} \cos\left(d x + c\right)^{3} - 280 \, a^{2} \cos\left(d x + c\right)^{2} + 240 \, a^{2} \cos\left(d x + c\right) + 105 \, a^{2}}{840 \, d \cos\left(d x + c\right)^{8}}"," ",0,"1/840*(840*a^2*cos(d*x + c)^8*log(-cos(d*x + c)) - 1680*a^2*cos(d*x + c)^7 + 840*a^2*cos(d*x + c)^6 + 1680*a^2*cos(d*x + c)^5 - 1008*a^2*cos(d*x + c)^3 - 280*a^2*cos(d*x + c)^2 + 240*a^2*cos(d*x + c) + 105*a^2)/(d*cos(d*x + c)^8)","A",0
21,1,104,0,0.747889," ","integrate((a+a*sec(d*x+c))^2*tan(d*x+c)^5,x, algorithm=""fricas"")","-\frac{60 \, a^{2} \cos\left(d x + c\right)^{6} \log\left(-\cos\left(d x + c\right)\right) - 120 \, a^{2} \cos\left(d x + c\right)^{5} + 30 \, a^{2} \cos\left(d x + c\right)^{4} + 80 \, a^{2} \cos\left(d x + c\right)^{3} + 15 \, a^{2} \cos\left(d x + c\right)^{2} - 24 \, a^{2} \cos\left(d x + c\right) - 10 \, a^{2}}{60 \, d \cos\left(d x + c\right)^{6}}"," ",0,"-1/60*(60*a^2*cos(d*x + c)^6*log(-cos(d*x + c)) - 120*a^2*cos(d*x + c)^5 + 30*a^2*cos(d*x + c)^4 + 80*a^2*cos(d*x + c)^3 + 15*a^2*cos(d*x + c)^2 - 24*a^2*cos(d*x + c) - 10*a^2)/(d*cos(d*x + c)^6)","A",0
22,1,65,0,0.631622," ","integrate((a+a*sec(d*x+c))^2*tan(d*x+c)^3,x, algorithm=""fricas"")","\frac{12 \, a^{2} \cos\left(d x + c\right)^{4} \log\left(-\cos\left(d x + c\right)\right) - 24 \, a^{2} \cos\left(d x + c\right)^{3} + 8 \, a^{2} \cos\left(d x + c\right) + 3 \, a^{2}}{12 \, d \cos\left(d x + c\right)^{4}}"," ",0,"1/12*(12*a^2*cos(d*x + c)^4*log(-cos(d*x + c)) - 24*a^2*cos(d*x + c)^3 + 8*a^2*cos(d*x + c) + 3*a^2)/(d*cos(d*x + c)^4)","A",0
23,1,52,0,0.622406," ","integrate((a+a*sec(d*x+c))^2*tan(d*x+c),x, algorithm=""fricas"")","-\frac{2 \, a^{2} \cos\left(d x + c\right)^{2} \log\left(-\cos\left(d x + c\right)\right) - 4 \, a^{2} \cos\left(d x + c\right) - a^{2}}{2 \, d \cos\left(d x + c\right)^{2}}"," ",0,"-1/2*(2*a^2*cos(d*x + c)^2*log(-cos(d*x + c)) - 4*a^2*cos(d*x + c) - a^2)/(d*cos(d*x + c)^2)","A",0
24,1,35,0,0.773654," ","integrate(cot(d*x+c)*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{a^{2} \log\left(-\cos\left(d x + c\right)\right) - 2 \, a^{2} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{d}"," ",0,"-(a^2*log(-cos(d*x + c)) - 2*a^2*log(-1/2*cos(d*x + c) + 1/2))/d","A",0
25,1,48,0,0.669437," ","integrate(cot(d*x+c)^3*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{a^{2} - {\left(a^{2} \cos\left(d x + c\right) - a^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{d \cos\left(d x + c\right) - d}"," ",0,"(a^2 - (a^2*cos(d*x + c) - a^2)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c) - d)","A",0
26,1,122,0,0.583089," ","integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{10 \, a^{2} \cos\left(d x + c\right) - 8 \, a^{2} - {\left(a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \cos\left(d x + c\right) + a^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 7 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \cos\left(d x + c\right) + a^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{8 \, {\left(d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) + d\right)}}"," ",0,"-1/8*(10*a^2*cos(d*x + c) - 8*a^2 - (a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*log(1/2*cos(d*x + c) + 1/2) - 7*(a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)","A",0
27,1,191,0,0.702150," ","integrate(cot(d*x+c)^7*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{66 \, a^{2} \cos\left(d x + c\right)^{3} - 36 \, a^{2} \cos\left(d x + c\right)^{2} - 74 \, a^{2} \cos\left(d x + c\right) + 52 \, a^{2} - 9 \, {\left(a^{2} \cos\left(d x + c\right)^{4} - 2 \, a^{2} \cos\left(d x + c\right)^{3} + 2 \, a^{2} \cos\left(d x + c\right) - a^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 39 \, {\left(a^{2} \cos\left(d x + c\right)^{4} - 2 \, a^{2} \cos\left(d x + c\right)^{3} + 2 \, a^{2} \cos\left(d x + c\right) - a^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{48 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{3} + 2 \, d \cos\left(d x + c\right) - d\right)}}"," ",0,"1/48*(66*a^2*cos(d*x + c)^3 - 36*a^2*cos(d*x + c)^2 - 74*a^2*cos(d*x + c) + 52*a^2 - 9*(a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^3 + 2*a^2*cos(d*x + c) - a^2)*log(1/2*cos(d*x + c) + 1/2) - 39*(a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^3 + 2*a^2*cos(d*x + c) - a^2)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c) - d)","A",0
28,1,322,0,0.604015," ","integrate(cot(d*x+c)^9*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{558 \, a^{2} \cos\left(d x + c\right)^{5} - 156 \, a^{2} \cos\left(d x + c\right)^{4} - 1268 \, a^{2} \cos\left(d x + c\right)^{3} + 676 \, a^{2} \cos\left(d x + c\right)^{2} + 686 \, a^{2} \cos\left(d x + c\right) - 448 \, a^{2} - 87 \, {\left(a^{2} \cos\left(d x + c\right)^{6} - 2 \, a^{2} \cos\left(d x + c\right)^{5} - a^{2} \cos\left(d x + c\right)^{4} + 4 \, a^{2} \cos\left(d x + c\right)^{3} - a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \cos\left(d x + c\right) + a^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 297 \, {\left(a^{2} \cos\left(d x + c\right)^{6} - 2 \, a^{2} \cos\left(d x + c\right)^{5} - a^{2} \cos\left(d x + c\right)^{4} + 4 \, a^{2} \cos\left(d x + c\right)^{3} - a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \cos\left(d x + c\right) + a^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{384 \, {\left(d \cos\left(d x + c\right)^{6} - 2 \, d \cos\left(d x + c\right)^{5} - d \cos\left(d x + c\right)^{4} + 4 \, d \cos\left(d x + c\right)^{3} - d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) + d\right)}}"," ",0,"-1/384*(558*a^2*cos(d*x + c)^5 - 156*a^2*cos(d*x + c)^4 - 1268*a^2*cos(d*x + c)^3 + 676*a^2*cos(d*x + c)^2 + 686*a^2*cos(d*x + c) - 448*a^2 - 87*(a^2*cos(d*x + c)^6 - 2*a^2*cos(d*x + c)^5 - a^2*cos(d*x + c)^4 + 4*a^2*cos(d*x + c)^3 - a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*log(1/2*cos(d*x + c) + 1/2) - 297*(a^2*cos(d*x + c)^6 - 2*a^2*cos(d*x + c)^5 - a^2*cos(d*x + c)^4 + 4*a^2*cos(d*x + c)^3 - a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^6 - 2*d*cos(d*x + c)^5 - d*cos(d*x + c)^4 + 4*d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)","B",0
29,1,165,0,0.669194," ","integrate((a+a*sec(d*x+c))^2*tan(d*x+c)^6,x, algorithm=""fricas"")","-\frac{1680 \, a^{2} d x \cos\left(d x + c\right)^{7} + 525 \, a^{2} \cos\left(d x + c\right)^{7} \log\left(\sin\left(d x + c\right) + 1\right) - 525 \, a^{2} \cos\left(d x + c\right)^{7} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(1168 \, a^{2} \cos\left(d x + c\right)^{6} + 1155 \, a^{2} \cos\left(d x + c\right)^{5} - 256 \, a^{2} \cos\left(d x + c\right)^{4} - 910 \, a^{2} \cos\left(d x + c\right)^{3} - 192 \, a^{2} \cos\left(d x + c\right)^{2} + 280 \, a^{2} \cos\left(d x + c\right) + 120 \, a^{2}\right)} \sin\left(d x + c\right)}{1680 \, d \cos\left(d x + c\right)^{7}}"," ",0,"-1/1680*(1680*a^2*d*x*cos(d*x + c)^7 + 525*a^2*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 525*a^2*cos(d*x + c)^7*log(-sin(d*x + c) + 1) - 2*(1168*a^2*cos(d*x + c)^6 + 1155*a^2*cos(d*x + c)^5 - 256*a^2*cos(d*x + c)^4 - 910*a^2*cos(d*x + c)^3 - 192*a^2*cos(d*x + c)^2 + 280*a^2*cos(d*x + c) + 120*a^2)*sin(d*x + c))/(d*cos(d*x + c)^7)","A",0
30,1,139,0,0.609171," ","integrate((a+a*sec(d*x+c))^2*tan(d*x+c)^4,x, algorithm=""fricas"")","\frac{120 \, a^{2} d x \cos\left(d x + c\right)^{5} + 45 \, a^{2} \cos\left(d x + c\right)^{5} \log\left(\sin\left(d x + c\right) + 1\right) - 45 \, a^{2} \cos\left(d x + c\right)^{5} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(68 \, a^{2} \cos\left(d x + c\right)^{4} + 75 \, a^{2} \cos\left(d x + c\right)^{3} + 4 \, a^{2} \cos\left(d x + c\right)^{2} - 30 \, a^{2} \cos\left(d x + c\right) - 12 \, a^{2}\right)} \sin\left(d x + c\right)}{120 \, d \cos\left(d x + c\right)^{5}}"," ",0,"1/120*(120*a^2*d*x*cos(d*x + c)^5 + 45*a^2*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 45*a^2*cos(d*x + c)^5*log(-sin(d*x + c) + 1) - 2*(68*a^2*cos(d*x + c)^4 + 75*a^2*cos(d*x + c)^3 + 4*a^2*cos(d*x + c)^2 - 30*a^2*cos(d*x + c) - 12*a^2)*sin(d*x + c))/(d*cos(d*x + c)^5)","A",0
31,1,111,0,0.725610," ","integrate((a+a*sec(d*x+c))^2*tan(d*x+c)^2,x, algorithm=""fricas"")","-\frac{6 \, a^{2} d x \cos\left(d x + c\right)^{3} + 3 \, a^{2} \cos\left(d x + c\right)^{3} \log\left(\sin\left(d x + c\right) + 1\right) - 3 \, a^{2} \cos\left(d x + c\right)^{3} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(2 \, a^{2} \cos\left(d x + c\right)^{2} + 3 \, a^{2} \cos\left(d x + c\right) + a^{2}\right)} \sin\left(d x + c\right)}{6 \, d \cos\left(d x + c\right)^{3}}"," ",0,"-1/6*(6*a^2*d*x*cos(d*x + c)^3 + 3*a^2*cos(d*x + c)^3*log(sin(d*x + c) + 1) - 3*a^2*cos(d*x + c)^3*log(-sin(d*x + c) + 1) - 2*(2*a^2*cos(d*x + c)^2 + 3*a^2*cos(d*x + c) + a^2)*sin(d*x + c))/(d*cos(d*x + c)^3)","A",0
32,1,42,0,0.681574," ","integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{a^{2} d x \sin\left(d x + c\right) + 2 \, a^{2} \cos\left(d x + c\right) + 2 \, a^{2}}{d \sin\left(d x + c\right)}"," ",0,"-(a^2*d*x*sin(d*x + c) + 2*a^2*cos(d*x + c) + 2*a^2)/(d*sin(d*x + c))","A",0
33,1,81,0,0.511287," ","integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{5 \, a^{2} \cos\left(d x + c\right)^{2} + a^{2} \cos\left(d x + c\right) - 4 \, a^{2} + 3 \, {\left(a^{2} d x \cos\left(d x + c\right) - a^{2} d x\right)} \sin\left(d x + c\right)}{3 \, {\left(d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right)}"," ",0,"1/3*(5*a^2*cos(d*x + c)^2 + a^2*cos(d*x + c) - 4*a^2 + 3*(a^2*d*x*cos(d*x + c) - a^2*d*x)*sin(d*x + c))/((d*cos(d*x + c) - d)*sin(d*x + c))","A",0
34,1,118,0,0.715646," ","integrate(cot(d*x+c)^6*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{26 \, a^{2} \cos\left(d x + c\right)^{3} - 22 \, a^{2} \cos\left(d x + c\right)^{2} - 17 \, a^{2} \cos\left(d x + c\right) + 16 \, a^{2} + 15 \, {\left(a^{2} d x \cos\left(d x + c\right)^{2} - 2 \, a^{2} d x \cos\left(d x + c\right) + a^{2} d x\right)} \sin\left(d x + c\right)}{15 \, {\left(d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right)}"," ",0,"-1/15*(26*a^2*cos(d*x + c)^3 - 22*a^2*cos(d*x + c)^2 - 17*a^2*cos(d*x + c) + 16*a^2 + 15*(a^2*d*x*cos(d*x + c)^2 - 2*a^2*d*x*cos(d*x + c) + a^2*d*x)*sin(d*x + c))/((d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)*sin(d*x + c))","A",0
35,1,173,0,0.590474," ","integrate(cot(d*x+c)^8*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{191 \, a^{2} \cos\left(d x + c\right)^{5} - 172 \, a^{2} \cos\left(d x + c\right)^{4} - 253 \, a^{2} \cos\left(d x + c\right)^{3} + 258 \, a^{2} \cos\left(d x + c\right)^{2} + 87 \, a^{2} \cos\left(d x + c\right) - 96 \, a^{2} + 105 \, {\left(a^{2} d x \cos\left(d x + c\right)^{4} - 2 \, a^{2} d x \cos\left(d x + c\right)^{3} + 2 \, a^{2} d x \cos\left(d x + c\right) - a^{2} d x\right)} \sin\left(d x + c\right)}{105 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{3} + 2 \, d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right)}"," ",0,"1/105*(191*a^2*cos(d*x + c)^5 - 172*a^2*cos(d*x + c)^4 - 253*a^2*cos(d*x + c)^3 + 258*a^2*cos(d*x + c)^2 + 87*a^2*cos(d*x + c) - 96*a^2 + 105*(a^2*d*x*cos(d*x + c)^4 - 2*a^2*d*x*cos(d*x + c)^3 + 2*a^2*d*x*cos(d*x + c) - a^2*d*x)*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c) - d)*sin(d*x + c))","A",0
36,1,274,0,0.613266," ","integrate(cot(d*x+c)^10*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{598 \, a^{2} \cos\left(d x + c\right)^{7} - 566 \, a^{2} \cos\left(d x + c\right)^{6} - 1212 \, a^{2} \cos\left(d x + c\right)^{5} + 1310 \, a^{2} \cos\left(d x + c\right)^{4} + 860 \, a^{2} \cos\left(d x + c\right)^{3} - 1014 \, a^{2} \cos\left(d x + c\right)^{2} - 197 \, a^{2} \cos\left(d x + c\right) + 256 \, a^{2} + 315 \, {\left(a^{2} d x \cos\left(d x + c\right)^{6} - 2 \, a^{2} d x \cos\left(d x + c\right)^{5} - a^{2} d x \cos\left(d x + c\right)^{4} + 4 \, a^{2} d x \cos\left(d x + c\right)^{3} - a^{2} d x \cos\left(d x + c\right)^{2} - 2 \, a^{2} d x \cos\left(d x + c\right) + a^{2} d x\right)} \sin\left(d x + c\right)}{315 \, {\left(d \cos\left(d x + c\right)^{6} - 2 \, d \cos\left(d x + c\right)^{5} - d \cos\left(d x + c\right)^{4} + 4 \, d \cos\left(d x + c\right)^{3} - d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right)}"," ",0,"-1/315*(598*a^2*cos(d*x + c)^7 - 566*a^2*cos(d*x + c)^6 - 1212*a^2*cos(d*x + c)^5 + 1310*a^2*cos(d*x + c)^4 + 860*a^2*cos(d*x + c)^3 - 1014*a^2*cos(d*x + c)^2 - 197*a^2*cos(d*x + c) + 256*a^2 + 315*(a^2*d*x*cos(d*x + c)^6 - 2*a^2*d*x*cos(d*x + c)^5 - a^2*d*x*cos(d*x + c)^4 + 4*a^2*d*x*cos(d*x + c)^3 - a^2*d*x*cos(d*x + c)^2 - 2*a^2*d*x*cos(d*x + c) + a^2*d*x)*sin(d*x + c))/((d*cos(d*x + c)^6 - 2*d*cos(d*x + c)^5 - d*cos(d*x + c)^4 + 4*d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)*sin(d*x + c))","A",0
37,1,169,0,0.873456," ","integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^9,x, algorithm=""fricas"")","-\frac{27720 \, a^{3} \cos\left(d x + c\right)^{11} \log\left(-\cos\left(d x + c\right)\right) - 83160 \, a^{3} \cos\left(d x + c\right)^{10} + 13860 \, a^{3} \cos\left(d x + c\right)^{9} + 101640 \, a^{3} \cos\left(d x + c\right)^{8} + 41580 \, a^{3} \cos\left(d x + c\right)^{7} - 77616 \, a^{3} \cos\left(d x + c\right)^{6} - 64680 \, a^{3} \cos\left(d x + c\right)^{5} + 23760 \, a^{3} \cos\left(d x + c\right)^{4} + 38115 \, a^{3} \cos\left(d x + c\right)^{3} + 3080 \, a^{3} \cos\left(d x + c\right)^{2} - 8316 \, a^{3} \cos\left(d x + c\right) - 2520 \, a^{3}}{27720 \, d \cos\left(d x + c\right)^{11}}"," ",0,"-1/27720*(27720*a^3*cos(d*x + c)^11*log(-cos(d*x + c)) - 83160*a^3*cos(d*x + c)^10 + 13860*a^3*cos(d*x + c)^9 + 101640*a^3*cos(d*x + c)^8 + 41580*a^3*cos(d*x + c)^7 - 77616*a^3*cos(d*x + c)^6 - 64680*a^3*cos(d*x + c)^5 + 23760*a^3*cos(d*x + c)^4 + 38115*a^3*cos(d*x + c)^3 + 3080*a^3*cos(d*x + c)^2 - 8316*a^3*cos(d*x + c) - 2520*a^3)/(d*cos(d*x + c)^11)","A",0
38,1,117,0,0.831963," ","integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^7,x, algorithm=""fricas"")","\frac{360 \, a^{3} \cos\left(d x + c\right)^{9} \log\left(-\cos\left(d x + c\right)\right) - 1080 \, a^{3} \cos\left(d x + c\right)^{8} + 960 \, a^{3} \cos\left(d x + c\right)^{6} + 540 \, a^{3} \cos\left(d x + c\right)^{5} - 432 \, a^{3} \cos\left(d x + c\right)^{4} - 480 \, a^{3} \cos\left(d x + c\right)^{3} + 135 \, a^{3} \cos\left(d x + c\right) + 40 \, a^{3}}{360 \, d \cos\left(d x + c\right)^{9}}"," ",0,"1/360*(360*a^3*cos(d*x + c)^9*log(-cos(d*x + c)) - 1080*a^3*cos(d*x + c)^8 + 960*a^3*cos(d*x + c)^6 + 540*a^3*cos(d*x + c)^5 - 432*a^3*cos(d*x + c)^4 - 480*a^3*cos(d*x + c)^3 + 135*a^3*cos(d*x + c) + 40*a^3)/(d*cos(d*x + c)^9)","A",0
39,1,117,0,0.485213," ","integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^5,x, algorithm=""fricas"")","-\frac{420 \, a^{3} \cos\left(d x + c\right)^{7} \log\left(-\cos\left(d x + c\right)\right) - 1260 \, a^{3} \cos\left(d x + c\right)^{6} - 210 \, a^{3} \cos\left(d x + c\right)^{5} + 700 \, a^{3} \cos\left(d x + c\right)^{4} + 525 \, a^{3} \cos\left(d x + c\right)^{3} - 84 \, a^{3} \cos\left(d x + c\right)^{2} - 210 \, a^{3} \cos\left(d x + c\right) - 60 \, a^{3}}{420 \, d \cos\left(d x + c\right)^{7}}"," ",0,"-1/420*(420*a^3*cos(d*x + c)^7*log(-cos(d*x + c)) - 1260*a^3*cos(d*x + c)^6 - 210*a^3*cos(d*x + c)^5 + 700*a^3*cos(d*x + c)^4 + 525*a^3*cos(d*x + c)^3 - 84*a^3*cos(d*x + c)^2 - 210*a^3*cos(d*x + c) - 60*a^3)/(d*cos(d*x + c)^7)","A",0
40,1,91,0,0.486794," ","integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^3,x, algorithm=""fricas"")","\frac{60 \, a^{3} \cos\left(d x + c\right)^{5} \log\left(-\cos\left(d x + c\right)\right) - 180 \, a^{3} \cos\left(d x + c\right)^{4} - 60 \, a^{3} \cos\left(d x + c\right)^{3} + 40 \, a^{3} \cos\left(d x + c\right)^{2} + 45 \, a^{3} \cos\left(d x + c\right) + 12 \, a^{3}}{60 \, d \cos\left(d x + c\right)^{5}}"," ",0,"1/60*(60*a^3*cos(d*x + c)^5*log(-cos(d*x + c)) - 180*a^3*cos(d*x + c)^4 - 60*a^3*cos(d*x + c)^3 + 40*a^3*cos(d*x + c)^2 + 45*a^3*cos(d*x + c) + 12*a^3)/(d*cos(d*x + c)^5)","A",0
41,1,65,0,0.658825," ","integrate((a+a*sec(d*x+c))^3*tan(d*x+c),x, algorithm=""fricas"")","-\frac{6 \, a^{3} \cos\left(d x + c\right)^{3} \log\left(-\cos\left(d x + c\right)\right) - 18 \, a^{3} \cos\left(d x + c\right)^{2} - 9 \, a^{3} \cos\left(d x + c\right) - 2 \, a^{3}}{6 \, d \cos\left(d x + c\right)^{3}}"," ",0,"-1/6*(6*a^3*cos(d*x + c)^3*log(-cos(d*x + c)) - 18*a^3*cos(d*x + c)^2 - 9*a^3*cos(d*x + c) - 2*a^3)/(d*cos(d*x + c)^3)","A",0
42,1,61,0,0.785511," ","integrate(cot(d*x+c)*(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{3 \, a^{3} \cos\left(d x + c\right) \log\left(-\cos\left(d x + c\right)\right) - 4 \, a^{3} \cos\left(d x + c\right) \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - a^{3}}{d \cos\left(d x + c\right)}"," ",0,"-(3*a^3*cos(d*x + c)*log(-cos(d*x + c)) - 4*a^3*cos(d*x + c)*log(-1/2*cos(d*x + c) + 1/2) - a^3)/(d*cos(d*x + c))","A",0
43,1,50,0,0.730777," ","integrate(cot(d*x+c)^3*(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{2 \, a^{3} - {\left(a^{3} \cos\left(d x + c\right) - a^{3}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{d \cos\left(d x + c\right) - d}"," ",0,"(2*a^3 - (a^3*cos(d*x + c) - a^3)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c) - d)","A",0
44,1,82,0,0.856907," ","integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{4 \, a^{3} \cos\left(d x + c\right) - 3 \, a^{3} - 2 \, {\left(a^{3} \cos\left(d x + c\right)^{2} - 2 \, a^{3} \cos\left(d x + c\right) + a^{3}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{2 \, {\left(d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) + d\right)}}"," ",0,"-1/2*(4*a^3*cos(d*x + c) - 3*a^3 - 2*(a^3*cos(d*x + c)^2 - 2*a^3*cos(d*x + c) + a^3)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)","A",0
45,1,178,0,1.172327," ","integrate(cot(d*x+c)^7*(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{102 \, a^{3} \cos\left(d x + c\right)^{2} - 162 \, a^{3} \cos\left(d x + c\right) + 68 \, a^{3} - 3 \, {\left(a^{3} \cos\left(d x + c\right)^{3} - 3 \, a^{3} \cos\left(d x + c\right)^{2} + 3 \, a^{3} \cos\left(d x + c\right) - a^{3}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 45 \, {\left(a^{3} \cos\left(d x + c\right)^{3} - 3 \, a^{3} \cos\left(d x + c\right)^{2} + 3 \, a^{3} \cos\left(d x + c\right) - a^{3}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{48 \, {\left(d \cos\left(d x + c\right)^{3} - 3 \, d \cos\left(d x + c\right)^{2} + 3 \, d \cos\left(d x + c\right) - d\right)}}"," ",0,"1/48*(102*a^3*cos(d*x + c)^2 - 162*a^3*cos(d*x + c) + 68*a^3 - 3*(a^3*cos(d*x + c)^3 - 3*a^3*cos(d*x + c)^2 + 3*a^3*cos(d*x + c) - a^3)*log(1/2*cos(d*x + c) + 1/2) - 45*(a^3*cos(d*x + c)^3 - 3*a^3*cos(d*x + c)^2 + 3*a^3*cos(d*x + c) - a^3)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^3 - 3*d*cos(d*x + c)^2 + 3*d*cos(d*x + c) - d)","A",0
46,1,272,0,0.759223," ","integrate(cot(d*x+c)^9*(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{426 \, a^{3} \cos\left(d x + c\right)^{4} - 606 \, a^{3} \cos\left(d x + c\right)^{3} - 190 \, a^{3} \cos\left(d x + c\right)^{2} + 666 \, a^{3} \cos\left(d x + c\right) - 272 \, a^{3} - 21 \, {\left(a^{3} \cos\left(d x + c\right)^{5} - 3 \, a^{3} \cos\left(d x + c\right)^{4} + 2 \, a^{3} \cos\left(d x + c\right)^{3} + 2 \, a^{3} \cos\left(d x + c\right)^{2} - 3 \, a^{3} \cos\left(d x + c\right) + a^{3}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 171 \, {\left(a^{3} \cos\left(d x + c\right)^{5} - 3 \, a^{3} \cos\left(d x + c\right)^{4} + 2 \, a^{3} \cos\left(d x + c\right)^{3} + 2 \, a^{3} \cos\left(d x + c\right)^{2} - 3 \, a^{3} \cos\left(d x + c\right) + a^{3}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{192 \, {\left(d \cos\left(d x + c\right)^{5} - 3 \, d \cos\left(d x + c\right)^{4} + 2 \, d \cos\left(d x + c\right)^{3} + 2 \, d \cos\left(d x + c\right)^{2} - 3 \, d \cos\left(d x + c\right) + d\right)}}"," ",0,"-1/192*(426*a^3*cos(d*x + c)^4 - 606*a^3*cos(d*x + c)^3 - 190*a^3*cos(d*x + c)^2 + 666*a^3*cos(d*x + c) - 272*a^3 - 21*(a^3*cos(d*x + c)^5 - 3*a^3*cos(d*x + c)^4 + 2*a^3*cos(d*x + c)^3 + 2*a^3*cos(d*x + c)^2 - 3*a^3*cos(d*x + c) + a^3)*log(1/2*cos(d*x + c) + 1/2) - 171*(a^3*cos(d*x + c)^5 - 3*a^3*cos(d*x + c)^4 + 2*a^3*cos(d*x + c)^3 + 2*a^3*cos(d*x + c)^2 - 3*a^3*cos(d*x + c) + a^3)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^5 - 3*d*cos(d*x + c)^4 + 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^2 - 3*d*cos(d*x + c) + d)","B",0
47,1,178,0,0.823959," ","integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^6,x, algorithm=""fricas"")","-\frac{26880 \, a^{3} d x \cos\left(d x + c\right)^{8} + 13125 \, a^{3} \cos\left(d x + c\right)^{8} \log\left(\sin\left(d x + c\right) + 1\right) - 13125 \, a^{3} \cos\left(d x + c\right)^{8} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(14848 \, a^{3} \cos\left(d x + c\right)^{7} + 27195 \, a^{3} \cos\left(d x + c\right)^{6} + 7424 \, a^{3} \cos\left(d x + c\right)^{5} - 17710 \, a^{3} \cos\left(d x + c\right)^{4} - 14592 \, a^{3} \cos\left(d x + c\right)^{3} + 1960 \, a^{3} \cos\left(d x + c\right)^{2} + 5760 \, a^{3} \cos\left(d x + c\right) + 1680 \, a^{3}\right)} \sin\left(d x + c\right)}{26880 \, d \cos\left(d x + c\right)^{8}}"," ",0,"-1/26880*(26880*a^3*d*x*cos(d*x + c)^8 + 13125*a^3*cos(d*x + c)^8*log(sin(d*x + c) + 1) - 13125*a^3*cos(d*x + c)^8*log(-sin(d*x + c) + 1) - 2*(14848*a^3*cos(d*x + c)^7 + 27195*a^3*cos(d*x + c)^6 + 7424*a^3*cos(d*x + c)^5 - 17710*a^3*cos(d*x + c)^4 - 14592*a^3*cos(d*x + c)^3 + 1960*a^3*cos(d*x + c)^2 + 5760*a^3*cos(d*x + c) + 1680*a^3)*sin(d*x + c))/(d*cos(d*x + c)^8)","A",0
48,1,152,0,0.822320," ","integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^4,x, algorithm=""fricas"")","\frac{480 \, a^{3} d x \cos\left(d x + c\right)^{6} + 285 \, a^{3} \cos\left(d x + c\right)^{6} \log\left(\sin\left(d x + c\right) + 1\right) - 285 \, a^{3} \cos\left(d x + c\right)^{6} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(176 \, a^{3} \cos\left(d x + c\right)^{5} + 435 \, a^{3} \cos\left(d x + c\right)^{4} + 208 \, a^{3} \cos\left(d x + c\right)^{3} - 110 \, a^{3} \cos\left(d x + c\right)^{2} - 144 \, a^{3} \cos\left(d x + c\right) - 40 \, a^{3}\right)} \sin\left(d x + c\right)}{480 \, d \cos\left(d x + c\right)^{6}}"," ",0,"1/480*(480*a^3*d*x*cos(d*x + c)^6 + 285*a^3*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 285*a^3*cos(d*x + c)^6*log(-sin(d*x + c) + 1) - 2*(176*a^3*cos(d*x + c)^5 + 435*a^3*cos(d*x + c)^4 + 208*a^3*cos(d*x + c)^3 - 110*a^3*cos(d*x + c)^2 - 144*a^3*cos(d*x + c) - 40*a^3)*sin(d*x + c))/(d*cos(d*x + c)^6)","A",0
49,1,113,0,0.558748," ","integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^2,x, algorithm=""fricas"")","-\frac{16 \, a^{3} d x \cos\left(d x + c\right)^{4} + 13 \, a^{3} \cos\left(d x + c\right)^{4} \log\left(\sin\left(d x + c\right) + 1\right) - 13 \, a^{3} \cos\left(d x + c\right)^{4} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(11 \, a^{3} \cos\left(d x + c\right)^{2} + 8 \, a^{3} \cos\left(d x + c\right) + 2 \, a^{3}\right)} \sin\left(d x + c\right)}{16 \, d \cos\left(d x + c\right)^{4}}"," ",0,"-1/16*(16*a^3*d*x*cos(d*x + c)^4 + 13*a^3*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 13*a^3*cos(d*x + c)^4*log(-sin(d*x + c) + 1) - 2*(11*a^3*cos(d*x + c)^2 + 8*a^3*cos(d*x + c) + 2*a^3)*sin(d*x + c))/(d*cos(d*x + c)^4)","A",0
50,1,84,0,0.731812," ","integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{2 \, a^{3} d x \sin\left(d x + c\right) - a^{3} \log\left(\sin\left(d x + c\right) + 1\right) \sin\left(d x + c\right) + a^{3} \log\left(-\sin\left(d x + c\right) + 1\right) \sin\left(d x + c\right) + 8 \, a^{3} \cos\left(d x + c\right) + 8 \, a^{3}}{2 \, d \sin\left(d x + c\right)}"," ",0,"-1/2*(2*a^3*d*x*sin(d*x + c) - a^3*log(sin(d*x + c) + 1)*sin(d*x + c) + a^3*log(-sin(d*x + c) + 1)*sin(d*x + c) + 8*a^3*cos(d*x + c) + 8*a^3)/(d*sin(d*x + c))","A",0
51,1,82,0,0.749613," ","integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{7 \, a^{3} \cos\left(d x + c\right)^{2} + 2 \, a^{3} \cos\left(d x + c\right) - 5 \, a^{3} + 3 \, {\left(a^{3} d x \cos\left(d x + c\right) - a^{3} d x\right)} \sin\left(d x + c\right)}{3 \, {\left(d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right)}"," ",0,"1/3*(7*a^3*cos(d*x + c)^2 + 2*a^3*cos(d*x + c) - 5*a^3 + 3*(a^3*d*x*cos(d*x + c) - a^3*d*x)*sin(d*x + c))/((d*cos(d*x + c) - d)*sin(d*x + c))","A",0
52,1,118,0,0.749247," ","integrate(cot(d*x+c)^6*(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{32 \, a^{3} \cos\left(d x + c\right)^{3} - 19 \, a^{3} \cos\left(d x + c\right)^{2} - 29 \, a^{3} \cos\left(d x + c\right) + 22 \, a^{3} + 15 \, {\left(a^{3} d x \cos\left(d x + c\right)^{2} - 2 \, a^{3} d x \cos\left(d x + c\right) + a^{3} d x\right)} \sin\left(d x + c\right)}{15 \, {\left(d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right)}"," ",0,"-1/15*(32*a^3*cos(d*x + c)^3 - 19*a^3*cos(d*x + c)^2 - 29*a^3*cos(d*x + c) + 22*a^3 + 15*(a^3*d*x*cos(d*x + c)^2 - 2*a^3*d*x*cos(d*x + c) + a^3*d*x)*sin(d*x + c))/((d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)*sin(d*x + c))","A",0
53,1,160,0,0.485726," ","integrate(cot(d*x+c)^8*(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{221 \, a^{3} \cos\left(d x + c\right)^{4} - 348 \, a^{3} \cos\left(d x + c\right)^{3} - 25 \, a^{3} \cos\left(d x + c\right)^{2} + 303 \, a^{3} \cos\left(d x + c\right) - 136 \, a^{3} + 105 \, {\left(a^{3} d x \cos\left(d x + c\right)^{3} - 3 \, a^{3} d x \cos\left(d x + c\right)^{2} + 3 \, a^{3} d x \cos\left(d x + c\right) - a^{3} d x\right)} \sin\left(d x + c\right)}{105 \, {\left(d \cos\left(d x + c\right)^{3} - 3 \, d \cos\left(d x + c\right)^{2} + 3 \, d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right)}"," ",0,"1/105*(221*a^3*cos(d*x + c)^4 - 348*a^3*cos(d*x + c)^3 - 25*a^3*cos(d*x + c)^2 + 303*a^3*cos(d*x + c) - 136*a^3 + 105*(a^3*d*x*cos(d*x + c)^3 - 3*a^3*d*x*cos(d*x + c)^2 + 3*a^3*d*x*cos(d*x + c) - a^3*d*x)*sin(d*x + c))/((d*cos(d*x + c)^3 - 3*d*cos(d*x + c)^2 + 3*d*cos(d*x + c) - d)*sin(d*x + c))","A",0
54,1,235,0,0.548996," ","integrate(cot(d*x+c)^10*(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{668 \, a^{3} \cos\left(d x + c\right)^{6} - 1059 \, a^{3} \cos\left(d x + c\right)^{5} - 573 \, a^{3} \cos\left(d x + c\right)^{4} + 1813 \, a^{3} \cos\left(d x + c\right)^{3} - 393 \, a^{3} \cos\left(d x + c\right)^{2} - 789 \, a^{3} \cos\left(d x + c\right) + 368 \, a^{3} + 315 \, {\left(a^{3} d x \cos\left(d x + c\right)^{5} - 3 \, a^{3} d x \cos\left(d x + c\right)^{4} + 2 \, a^{3} d x \cos\left(d x + c\right)^{3} + 2 \, a^{3} d x \cos\left(d x + c\right)^{2} - 3 \, a^{3} d x \cos\left(d x + c\right) + a^{3} d x\right)} \sin\left(d x + c\right)}{315 \, {\left(d \cos\left(d x + c\right)^{5} - 3 \, d \cos\left(d x + c\right)^{4} + 2 \, d \cos\left(d x + c\right)^{3} + 2 \, d \cos\left(d x + c\right)^{2} - 3 \, d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right)}"," ",0,"-1/315*(668*a^3*cos(d*x + c)^6 - 1059*a^3*cos(d*x + c)^5 - 573*a^3*cos(d*x + c)^4 + 1813*a^3*cos(d*x + c)^3 - 393*a^3*cos(d*x + c)^2 - 789*a^3*cos(d*x + c) + 368*a^3 + 315*(a^3*d*x*cos(d*x + c)^5 - 3*a^3*d*x*cos(d*x + c)^4 + 2*a^3*d*x*cos(d*x + c)^3 + 2*a^3*d*x*cos(d*x + c)^2 - 3*a^3*d*x*cos(d*x + c) + a^3*d*x)*sin(d*x + c))/((d*cos(d*x + c)^5 - 3*d*cos(d*x + c)^4 + 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^2 - 3*d*cos(d*x + c) + d)*sin(d*x + c))","A",0
55,1,314,0,0.522728," ","integrate(cot(d*x+c)^12*(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{7453 \, a^{3} \cos\left(d x + c\right)^{8} - 11964 \, a^{3} \cos\left(d x + c\right)^{7} - 11866 \, a^{3} \cos\left(d x + c\right)^{6} + 30542 \, a^{3} \cos\left(d x + c\right)^{5} + 90 \, a^{3} \cos\left(d x + c\right)^{4} - 26438 \, a^{3} \cos\left(d x + c\right)^{3} + 8539 \, a^{3} \cos\left(d x + c\right)^{2} + 7671 \, a^{3} \cos\left(d x + c\right) - 3712 \, a^{3} + 3465 \, {\left(a^{3} d x \cos\left(d x + c\right)^{7} - 3 \, a^{3} d x \cos\left(d x + c\right)^{6} + a^{3} d x \cos\left(d x + c\right)^{5} + 5 \, a^{3} d x \cos\left(d x + c\right)^{4} - 5 \, a^{3} d x \cos\left(d x + c\right)^{3} - a^{3} d x \cos\left(d x + c\right)^{2} + 3 \, a^{3} d x \cos\left(d x + c\right) - a^{3} d x\right)} \sin\left(d x + c\right)}{3465 \, {\left(d \cos\left(d x + c\right)^{7} - 3 \, d \cos\left(d x + c\right)^{6} + d \cos\left(d x + c\right)^{5} + 5 \, d \cos\left(d x + c\right)^{4} - 5 \, d \cos\left(d x + c\right)^{3} - d \cos\left(d x + c\right)^{2} + 3 \, d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right)}"," ",0,"1/3465*(7453*a^3*cos(d*x + c)^8 - 11964*a^3*cos(d*x + c)^7 - 11866*a^3*cos(d*x + c)^6 + 30542*a^3*cos(d*x + c)^5 + 90*a^3*cos(d*x + c)^4 - 26438*a^3*cos(d*x + c)^3 + 8539*a^3*cos(d*x + c)^2 + 7671*a^3*cos(d*x + c) - 3712*a^3 + 3465*(a^3*d*x*cos(d*x + c)^7 - 3*a^3*d*x*cos(d*x + c)^6 + a^3*d*x*cos(d*x + c)^5 + 5*a^3*d*x*cos(d*x + c)^4 - 5*a^3*d*x*cos(d*x + c)^3 - a^3*d*x*cos(d*x + c)^2 + 3*a^3*d*x*cos(d*x + c) - a^3*d*x)*sin(d*x + c))/((d*cos(d*x + c)^7 - 3*d*cos(d*x + c)^6 + d*cos(d*x + c)^5 + 5*d*cos(d*x + c)^4 - 5*d*cos(d*x + c)^3 - d*cos(d*x + c)^2 + 3*d*cos(d*x + c) - d)*sin(d*x + c))","A",0
56,1,95,0,0.502532," ","integrate(tan(d*x+c)^9/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{420 \, \cos\left(d x + c\right)^{7} \log\left(-\cos\left(d x + c\right)\right) + 420 \, \cos\left(d x + c\right)^{6} + 630 \, \cos\left(d x + c\right)^{5} - 420 \, \cos\left(d x + c\right)^{4} - 315 \, \cos\left(d x + c\right)^{3} + 252 \, \cos\left(d x + c\right)^{2} + 70 \, \cos\left(d x + c\right) - 60}{420 \, a d \cos\left(d x + c\right)^{7}}"," ",0,"-1/420*(420*cos(d*x + c)^7*log(-cos(d*x + c)) + 420*cos(d*x + c)^6 + 630*cos(d*x + c)^5 - 420*cos(d*x + c)^4 - 315*cos(d*x + c)^3 + 252*cos(d*x + c)^2 + 70*cos(d*x + c) - 60)/(a*d*cos(d*x + c)^7)","A",0
57,1,75,0,0.499445," ","integrate(tan(d*x+c)^7/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\frac{60 \, \cos\left(d x + c\right)^{5} \log\left(-\cos\left(d x + c\right)\right) + 60 \, \cos\left(d x + c\right)^{4} + 60 \, \cos\left(d x + c\right)^{3} - 40 \, \cos\left(d x + c\right)^{2} - 15 \, \cos\left(d x + c\right) + 12}{60 \, a d \cos\left(d x + c\right)^{5}}"," ",0,"1/60*(60*cos(d*x + c)^5*log(-cos(d*x + c)) + 60*cos(d*x + c)^4 + 60*cos(d*x + c)^3 - 40*cos(d*x + c)^2 - 15*cos(d*x + c) + 12)/(a*d*cos(d*x + c)^5)","A",0
58,1,55,0,0.490768," ","integrate(tan(d*x+c)^5/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{6 \, \cos\left(d x + c\right)^{3} \log\left(-\cos\left(d x + c\right)\right) + 6 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) - 2}{6 \, a d \cos\left(d x + c\right)^{3}}"," ",0,"-1/6*(6*cos(d*x + c)^3*log(-cos(d*x + c)) + 6*cos(d*x + c)^2 + 3*cos(d*x + c) - 2)/(a*d*cos(d*x + c)^3)","A",0
59,1,33,0,0.496222," ","integrate(tan(d*x+c)^3/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\frac{\cos\left(d x + c\right) \log\left(-\cos\left(d x + c\right)\right) + 1}{a d \cos\left(d x + c\right)}"," ",0,"(cos(d*x + c)*log(-cos(d*x + c)) + 1)/(a*d*cos(d*x + c))","A",0
60,1,19,0,0.764246," ","integrate(tan(d*x+c)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{\log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{a d}"," ",0,"-log(1/2*cos(d*x + c) + 1/2)/(a*d)","A",0
61,1,60,0,0.735150," ","integrate(cot(d*x+c)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\frac{3 \, {\left(\cos\left(d x + c\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left(\cos\left(d x + c\right) + 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 2}{4 \, {\left(a d \cos\left(d x + c\right) + a d\right)}}"," ",0,"1/4*(3*(cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + (cos(d*x + c) + 1)*log(-1/2*cos(d*x + c) + 1/2) + 2)/(a*d*cos(d*x + c) + a*d)","A",0
62,1,139,0,0.707029," ","integrate(cot(d*x+c)^3/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{10 \, \cos\left(d x + c\right)^{2} + 11 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 5 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 6 \, \cos\left(d x + c\right) - 12}{16 \, {\left(a d \cos\left(d x + c\right)^{3} + a d \cos\left(d x + c\right)^{2} - a d \cos\left(d x + c\right) - a d\right)}}"," ",0,"-1/16*(10*cos(d*x + c)^2 + 11*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*log(1/2*cos(d*x + c) + 1/2) + 5*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*log(-1/2*cos(d*x + c) + 1/2) - 6*cos(d*x + c) - 12)/(a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)^2 - a*d*cos(d*x + c) - a*d)","A",0
63,1,217,0,0.894247," ","integrate(cot(d*x+c)^5/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\frac{66 \, \cos\left(d x + c\right)^{4} - 78 \, \cos\left(d x + c\right)^{3} - 158 \, \cos\left(d x + c\right)^{2} + 63 \, {\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)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 33 \, {\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)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 58 \, \cos\left(d x + c\right) + 88}{96 \, {\left(a d \cos\left(d x + c\right)^{5} + a d \cos\left(d x + c\right)^{4} - 2 \, a d \cos\left(d x + c\right)^{3} - 2 \, a d \cos\left(d x + c\right)^{2} + a d \cos\left(d x + c\right) + a d\right)}}"," ",0,"1/96*(66*cos(d*x + c)^4 - 78*cos(d*x + c)^3 - 158*cos(d*x + c)^2 + 63*(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)*log(1/2*cos(d*x + c) + 1/2) + 33*(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)*log(-1/2*cos(d*x + c) + 1/2) + 58*cos(d*x + c) + 88)/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d)","A",0
64,1,127,0,0.574966," ","integrate(tan(d*x+c)^8/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\frac{480 \, d x \cos\left(d x + c\right)^{6} - 75 \, \cos\left(d x + c\right)^{6} \log\left(\sin\left(d x + c\right) + 1\right) + 75 \, \cos\left(d x + c\right)^{6} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(368 \, \cos\left(d x + c\right)^{5} - 165 \, \cos\left(d x + c\right)^{4} - 176 \, \cos\left(d x + c\right)^{3} + 130 \, \cos\left(d x + c\right)^{2} + 48 \, \cos\left(d x + c\right) - 40\right)} \sin\left(d x + c\right)}{480 \, a d \cos\left(d x + c\right)^{6}}"," ",0,"1/480*(480*d*x*cos(d*x + c)^6 - 75*cos(d*x + c)^6*log(sin(d*x + c) + 1) + 75*cos(d*x + c)^6*log(-sin(d*x + c) + 1) - 2*(368*cos(d*x + c)^5 - 165*cos(d*x + c)^4 - 176*cos(d*x + c)^3 + 130*cos(d*x + c)^2 + 48*cos(d*x + c) - 40)*sin(d*x + c))/(a*d*cos(d*x + c)^6)","A",0
65,1,107,0,0.807164," ","integrate(tan(d*x+c)^6/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{48 \, d x \cos\left(d x + c\right)^{4} - 9 \, \cos\left(d x + c\right)^{4} \log\left(\sin\left(d x + c\right) + 1\right) + 9 \, \cos\left(d x + c\right)^{4} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(32 \, \cos\left(d x + c\right)^{3} - 15 \, \cos\left(d x + c\right)^{2} - 8 \, \cos\left(d x + c\right) + 6\right)} \sin\left(d x + c\right)}{48 \, a d \cos\left(d x + c\right)^{4}}"," ",0,"-1/48*(48*d*x*cos(d*x + c)^4 - 9*cos(d*x + c)^4*log(sin(d*x + c) + 1) + 9*cos(d*x + c)^4*log(-sin(d*x + c) + 1) - 2*(32*cos(d*x + c)^3 - 15*cos(d*x + c)^2 - 8*cos(d*x + c) + 6)*sin(d*x + c))/(a*d*cos(d*x + c)^4)","A",0
66,1,86,0,0.614531," ","integrate(tan(d*x+c)^4/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\frac{4 \, d x \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) + \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(2 \, \cos\left(d x + c\right) - 1\right)} \sin\left(d x + c\right)}{4 \, a d \cos\left(d x + c\right)^{2}}"," ",0,"1/4*(4*d*x*cos(d*x + c)^2 - cos(d*x + c)^2*log(sin(d*x + c) + 1) + cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 2*(2*cos(d*x + c) - 1)*sin(d*x + c))/(a*d*cos(d*x + c)^2)","A",0
67,1,35,0,0.828371," ","integrate(tan(d*x+c)^2/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{2 \, d x - \log\left(\sin\left(d x + c\right) + 1\right) + \log\left(-\sin\left(d x + c\right) + 1\right)}{2 \, a d}"," ",0,"-1/2*(2*d*x - log(sin(d*x + c) + 1) + log(-sin(d*x + c) + 1))/(a*d)","A",0
68,1,64,0,0.489805," ","integrate(cot(d*x+c)^2/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{4 \, \cos\left(d x + c\right)^{2} + 3 \, {\left(d x \cos\left(d x + c\right) + d x\right)} \sin\left(d x + c\right) + \cos\left(d x + c\right) - 2}{3 \, {\left(a d \cos\left(d x + c\right) + a d\right)} \sin\left(d x + c\right)}"," ",0,"-1/3*(4*cos(d*x + c)^2 + 3*(d*x*cos(d*x + c) + d*x)*sin(d*x + c) + cos(d*x + c) - 2)/((a*d*cos(d*x + c) + a*d)*sin(d*x + c))","A",0
69,1,134,0,0.458260," ","integrate(cot(d*x+c)^4/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\frac{23 \, \cos\left(d x + c\right)^{4} + 8 \, \cos\left(d x + c\right)^{3} - 27 \, \cos\left(d x + c\right)^{2} + 15 \, {\left(d x \cos\left(d x + c\right)^{3} + d x \cos\left(d x + c\right)^{2} - d x \cos\left(d x + c\right) - d x\right)} \sin\left(d x + c\right) - 7 \, \cos\left(d x + c\right) + 8}{15 \, {\left(a d \cos\left(d x + c\right)^{3} + a d \cos\left(d x + c\right)^{2} - a d \cos\left(d x + c\right) - a d\right)} \sin\left(d x + c\right)}"," ",0,"1/15*(23*cos(d*x + c)^4 + 8*cos(d*x + c)^3 - 27*cos(d*x + c)^2 + 15*(d*x*cos(d*x + c)^3 + d*x*cos(d*x + c)^2 - d*x*cos(d*x + c) - d*x)*sin(d*x + c) - 7*cos(d*x + c) + 8)/((a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)^2 - a*d*cos(d*x + c) - a*d)*sin(d*x + c))","A",0
70,1,198,0,0.465848," ","integrate(cot(d*x+c)^6/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{176 \, \cos\left(d x + c\right)^{6} + 71 \, \cos\left(d x + c\right)^{5} - 335 \, \cos\left(d x + c\right)^{4} - 125 \, \cos\left(d x + c\right)^{3} + 225 \, \cos\left(d x + c\right)^{2} + 105 \, {\left(d x \cos\left(d x + c\right)^{5} + d x \cos\left(d x + c\right)^{4} - 2 \, d x \cos\left(d x + c\right)^{3} - 2 \, d x \cos\left(d x + c\right)^{2} + d x \cos\left(d x + c\right) + d x\right)} \sin\left(d x + c\right) + 57 \, \cos\left(d x + c\right) - 48}{105 \, {\left(a d \cos\left(d x + c\right)^{5} + a d \cos\left(d x + c\right)^{4} - 2 \, a d \cos\left(d x + c\right)^{3} - 2 \, a d \cos\left(d x + c\right)^{2} + a d \cos\left(d x + c\right) + a d\right)} \sin\left(d x + c\right)}"," ",0,"-1/105*(176*cos(d*x + c)^6 + 71*cos(d*x + c)^5 - 335*cos(d*x + c)^4 - 125*cos(d*x + c)^3 + 225*cos(d*x + c)^2 + 105*(d*x*cos(d*x + c)^5 + d*x*cos(d*x + c)^4 - 2*d*x*cos(d*x + c)^3 - 2*d*x*cos(d*x + c)^2 + d*x*cos(d*x + c) + d*x)*sin(d*x + c) + 57*cos(d*x + c) - 48)/((a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d)*sin(d*x + c))","A",0
71,1,85,0,0.500443," ","integrate(tan(d*x+c)^9/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{60 \, \cos\left(d x + c\right)^{6} \log\left(-\cos\left(d x + c\right)\right) + 120 \, \cos\left(d x + c\right)^{5} + 30 \, \cos\left(d x + c\right)^{4} - 80 \, \cos\left(d x + c\right)^{3} + 15 \, \cos\left(d x + c\right)^{2} + 24 \, \cos\left(d x + c\right) - 10}{60 \, a^{2} d \cos\left(d x + c\right)^{6}}"," ",0,"-1/60*(60*cos(d*x + c)^6*log(-cos(d*x + c)) + 120*cos(d*x + c)^5 + 30*cos(d*x + c)^4 - 80*cos(d*x + c)^3 + 15*cos(d*x + c)^2 + 24*cos(d*x + c) - 10)/(a^2*d*cos(d*x + c)^6)","A",0
72,1,55,0,0.485998," ","integrate(tan(d*x+c)^7/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{12 \, \cos\left(d x + c\right)^{4} \log\left(-\cos\left(d x + c\right)\right) + 24 \, \cos\left(d x + c\right)^{3} - 8 \, \cos\left(d x + c\right) + 3}{12 \, a^{2} d \cos\left(d x + c\right)^{4}}"," ",0,"1/12*(12*cos(d*x + c)^4*log(-cos(d*x + c)) + 24*cos(d*x + c)^3 - 8*cos(d*x + c) + 3)/(a^2*d*cos(d*x + c)^4)","A",0
73,1,45,0,0.475077," ","integrate(tan(d*x+c)^5/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{2 \, \cos\left(d x + c\right)^{2} \log\left(-\cos\left(d x + c\right)\right) + 4 \, \cos\left(d x + c\right) - 1}{2 \, a^{2} d \cos\left(d x + c\right)^{2}}"," ",0,"-1/2*(2*cos(d*x + c)^2*log(-cos(d*x + c)) + 4*cos(d*x + c) - 1)/(a^2*d*cos(d*x + c)^2)","A",0
74,1,31,0,0.464740," ","integrate(tan(d*x+c)^3/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{\log\left(-\cos\left(d x + c\right)\right) - 2 \, \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{a^{2} d}"," ",0,"-(log(-cos(d*x + c)) - 2*log(1/2*cos(d*x + c) + 1/2))/(a^2*d)","A",0
75,1,43,0,0.483208," ","integrate(tan(d*x+c)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{{\left(\cos\left(d x + c\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 1}{a^{2} d \cos\left(d x + c\right) + a^{2} d}"," ",0,"-((cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + 1)/(a^2*d*cos(d*x + c) + a^2*d)","A",0
76,1,106,0,0.491656," ","integrate(cot(d*x+c)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{7 \, {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 10 \, \cos\left(d x + c\right) + 8}{8 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)}}"," ",0,"1/8*(7*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + (cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*log(-1/2*cos(d*x + c) + 1/2) + 10*cos(d*x + c) + 8)/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)","A",0
77,1,162,0,0.538034," ","integrate(cot(d*x+c)^3/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{66 \, \cos\left(d x + c\right)^{3} + 36 \, \cos\left(d x + c\right)^{2} + 39 \, {\left(\cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right) - 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 9 \, {\left(\cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right) - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 74 \, \cos\left(d x + c\right) - 52}{48 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} + 2 \, a^{2} d \cos\left(d x + c\right)^{3} - 2 \, a^{2} d \cos\left(d x + c\right) - a^{2} d\right)}}"," ",0,"-1/48*(66*cos(d*x + c)^3 + 36*cos(d*x + c)^2 + 39*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*log(1/2*cos(d*x + c) + 1/2) + 9*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*log(-1/2*cos(d*x + c) + 1/2) - 74*cos(d*x + c) - 52)/(a^2*d*cos(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x + c) - a^2*d)","A",0
78,1,283,0,0.604952," ","integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{558 \, \cos\left(d x + c\right)^{5} + 156 \, \cos\left(d x + c\right)^{4} - 1268 \, \cos\left(d x + c\right)^{3} - 676 \, \cos\left(d x + c\right)^{2} + 297 \, {\left(\cos\left(d x + c\right)^{6} + 2 \, \cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 87 \, {\left(\cos\left(d x + c\right)^{6} + 2 \, \cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 686 \, \cos\left(d x + c\right) + 448}{384 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} + 2 \, a^{2} d \cos\left(d x + c\right)^{5} - a^{2} d \cos\left(d x + c\right)^{4} - 4 \, a^{2} d \cos\left(d x + c\right)^{3} - a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)}}"," ",0,"1/384*(558*cos(d*x + c)^5 + 156*cos(d*x + c)^4 - 1268*cos(d*x + c)^3 - 676*cos(d*x + c)^2 + 297*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + 87*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*log(-1/2*cos(d*x + c) + 1/2) + 686*cos(d*x + c) + 448)/(a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)","A",0
79,1,117,0,0.760413," ","integrate(tan(d*x+c)^8/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{120 \, d x \cos\left(d x + c\right)^{5} - 45 \, \cos\left(d x + c\right)^{5} \log\left(\sin\left(d x + c\right) + 1\right) + 45 \, \cos\left(d x + c\right)^{5} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(68 \, \cos\left(d x + c\right)^{4} - 75 \, \cos\left(d x + c\right)^{3} + 4 \, \cos\left(d x + c\right)^{2} + 30 \, \cos\left(d x + c\right) - 12\right)} \sin\left(d x + c\right)}{120 \, a^{2} d \cos\left(d x + c\right)^{5}}"," ",0,"1/120*(120*d*x*cos(d*x + c)^5 - 45*cos(d*x + c)^5*log(sin(d*x + c) + 1) + 45*cos(d*x + c)^5*log(-sin(d*x + c) + 1) - 2*(68*cos(d*x + c)^4 - 75*cos(d*x + c)^3 + 4*cos(d*x + c)^2 + 30*cos(d*x + c) - 12)*sin(d*x + c))/(a^2*d*cos(d*x + c)^5)","A",0
80,1,97,0,0.763160," ","integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{6 \, d x \cos\left(d x + c\right)^{3} - 3 \, \cos\left(d x + c\right)^{3} \log\left(\sin\left(d x + c\right) + 1\right) + 3 \, \cos\left(d x + c\right)^{3} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(2 \, \cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) + 1\right)} \sin\left(d x + c\right)}{6 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"-1/6*(6*d*x*cos(d*x + c)^3 - 3*cos(d*x + c)^3*log(sin(d*x + c) + 1) + 3*cos(d*x + c)^3*log(-sin(d*x + c) + 1) - 2*(2*cos(d*x + c)^2 - 3*cos(d*x + c) + 1)*sin(d*x + c))/(a^2*d*cos(d*x + c)^3)","A",0
81,1,66,0,0.571232," ","integrate(tan(d*x+c)^4/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{d x \cos\left(d x + c\right) - \cos\left(d x + c\right) \log\left(\sin\left(d x + c\right) + 1\right) + \cos\left(d x + c\right) \log\left(-\sin\left(d x + c\right) + 1\right) + \sin\left(d x + c\right)}{a^{2} d \cos\left(d x + c\right)}"," ",0,"(d*x*cos(d*x + c) - cos(d*x + c)*log(sin(d*x + c) + 1) + cos(d*x + c)*log(-sin(d*x + c) + 1) + sin(d*x + c))/(a^2*d*cos(d*x + c))","A",0
82,1,42,0,0.773622," ","integrate(tan(d*x+c)^2/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{d x \cos\left(d x + c\right) + d x - 2 \, \sin\left(d x + c\right)}{a^{2} d \cos\left(d x + c\right) + a^{2} d}"," ",0,"-(d*x*cos(d*x + c) + d*x - 2*sin(d*x + c))/(a^2*d*cos(d*x + c) + a^2*d)","A",0
83,1,106,0,0.665561," ","integrate(cot(d*x+c)^2/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{26 \, \cos\left(d x + c\right)^{3} + 22 \, \cos\left(d x + c\right)^{2} + 15 \, {\left(d x \cos\left(d x + c\right)^{2} + 2 \, d x \cos\left(d x + c\right) + d x\right)} \sin\left(d x + c\right) - 17 \, \cos\left(d x + c\right) - 16}{15 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)} \sin\left(d x + c\right)}"," ",0,"-1/15*(26*cos(d*x + c)^3 + 22*cos(d*x + c)^2 + 15*(d*x*cos(d*x + c)^2 + 2*d*x*cos(d*x + c) + d*x)*sin(d*x + c) - 17*cos(d*x + c) - 16)/((a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)*sin(d*x + c))","A",0
84,1,154,0,0.770997," ","integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{191 \, \cos\left(d x + c\right)^{5} + 172 \, \cos\left(d x + c\right)^{4} - 253 \, \cos\left(d x + c\right)^{3} - 258 \, \cos\left(d x + c\right)^{2} + 105 \, {\left(d x \cos\left(d x + c\right)^{4} + 2 \, d x \cos\left(d x + c\right)^{3} - 2 \, d x \cos\left(d x + c\right) - d x\right)} \sin\left(d x + c\right) + 87 \, \cos\left(d x + c\right) + 96}{105 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} + 2 \, a^{2} d \cos\left(d x + c\right)^{3} - 2 \, a^{2} d \cos\left(d x + c\right) - a^{2} d\right)} \sin\left(d x + c\right)}"," ",0,"1/105*(191*cos(d*x + c)^5 + 172*cos(d*x + c)^4 - 253*cos(d*x + c)^3 - 258*cos(d*x + c)^2 + 105*(d*x*cos(d*x + c)^4 + 2*d*x*cos(d*x + c)^3 - 2*d*x*cos(d*x + c) - d*x)*sin(d*x + c) + 87*cos(d*x + c) + 96)/((a^2*d*cos(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c))","A",0
85,1,250,0,0.697616," ","integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{598 \, \cos\left(d x + c\right)^{7} + 566 \, \cos\left(d x + c\right)^{6} - 1212 \, \cos\left(d x + c\right)^{5} - 1310 \, \cos\left(d x + c\right)^{4} + 860 \, \cos\left(d x + c\right)^{3} + 1014 \, \cos\left(d x + c\right)^{2} + 315 \, {\left(d x \cos\left(d x + c\right)^{6} + 2 \, d x \cos\left(d x + c\right)^{5} - d x \cos\left(d x + c\right)^{4} - 4 \, d x \cos\left(d x + c\right)^{3} - d x \cos\left(d x + c\right)^{2} + 2 \, d x \cos\left(d x + c\right) + d x\right)} \sin\left(d x + c\right) - 197 \, \cos\left(d x + c\right) - 256}{315 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} + 2 \, a^{2} d \cos\left(d x + c\right)^{5} - a^{2} d \cos\left(d x + c\right)^{4} - 4 \, a^{2} d \cos\left(d x + c\right)^{3} - a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)} \sin\left(d x + c\right)}"," ",0,"-1/315*(598*cos(d*x + c)^7 + 566*cos(d*x + c)^6 - 1212*cos(d*x + c)^5 - 1310*cos(d*x + c)^4 + 860*cos(d*x + c)^3 + 1014*cos(d*x + c)^2 + 315*(d*x*cos(d*x + c)^6 + 2*d*x*cos(d*x + c)^5 - d*x*cos(d*x + c)^4 - 4*d*x*cos(d*x + c)^3 - d*x*cos(d*x + c)^2 + 2*d*x*cos(d*x + c) + d*x)*sin(d*x + c) - 197*cos(d*x + c) - 256)/((a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)*sin(d*x + c))","A",0
86,1,95,0,0.657794," ","integrate(tan(d*x+c)^11/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{420 \, \cos\left(d x + c\right)^{7} \log\left(-\cos\left(d x + c\right)\right) + 1260 \, \cos\left(d x + c\right)^{6} - 210 \, \cos\left(d x + c\right)^{5} - 700 \, \cos\left(d x + c\right)^{4} + 525 \, \cos\left(d x + c\right)^{3} + 84 \, \cos\left(d x + c\right)^{2} - 210 \, \cos\left(d x + c\right) + 60}{420 \, a^{3} d \cos\left(d x + c\right)^{7}}"," ",0,"1/420*(420*cos(d*x + c)^7*log(-cos(d*x + c)) + 1260*cos(d*x + c)^6 - 210*cos(d*x + c)^5 - 700*cos(d*x + c)^4 + 525*cos(d*x + c)^3 + 84*cos(d*x + c)^2 - 210*cos(d*x + c) + 60)/(a^3*d*cos(d*x + c)^7)","A",0
87,1,75,0,0.710849," ","integrate(tan(d*x+c)^9/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{60 \, \cos\left(d x + c\right)^{5} \log\left(-\cos\left(d x + c\right)\right) + 180 \, \cos\left(d x + c\right)^{4} - 60 \, \cos\left(d x + c\right)^{3} - 40 \, \cos\left(d x + c\right)^{2} + 45 \, \cos\left(d x + c\right) - 12}{60 \, a^{3} d \cos\left(d x + c\right)^{5}}"," ",0,"-1/60*(60*cos(d*x + c)^5*log(-cos(d*x + c)) + 180*cos(d*x + c)^4 - 60*cos(d*x + c)^3 - 40*cos(d*x + c)^2 + 45*cos(d*x + c) - 12)/(a^3*d*cos(d*x + c)^5)","A",0
88,1,55,0,0.998697," ","integrate(tan(d*x+c)^7/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{6 \, \cos\left(d x + c\right)^{3} \log\left(-\cos\left(d x + c\right)\right) + 18 \, \cos\left(d x + c\right)^{2} - 9 \, \cos\left(d x + c\right) + 2}{6 \, a^{3} d \cos\left(d x + c\right)^{3}}"," ",0,"1/6*(6*cos(d*x + c)^3*log(-cos(d*x + c)) + 18*cos(d*x + c)^2 - 9*cos(d*x + c) + 2)/(a^3*d*cos(d*x + c)^3)","A",0
89,1,53,0,0.657558," ","integrate(tan(d*x+c)^5/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{3 \, \cos\left(d x + c\right) \log\left(-\cos\left(d x + c\right)\right) - 4 \, \cos\left(d x + c\right) \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 1}{a^{3} d \cos\left(d x + c\right)}"," ",0,"(3*cos(d*x + c)*log(-cos(d*x + c)) - 4*cos(d*x + c)*log(1/2*cos(d*x + c) + 1/2) + 1)/(a^3*d*cos(d*x + c))","A",0
90,1,42,0,0.821215," ","integrate(tan(d*x+c)^3/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{{\left(\cos\left(d x + c\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 2}{a^{3} d \cos\left(d x + c\right) + a^{3} d}"," ",0,"((cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + 2)/(a^3*d*cos(d*x + c) + a^3*d)","A",0
91,1,76,0,0.696757," ","integrate(tan(d*x+c)/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{2 \, {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 4 \, \cos\left(d x + c\right) + 3}{2 \, {\left(a^{3} d \cos\left(d x + c\right)^{2} + 2 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}"," ",0,"-1/2*(2*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + 4*cos(d*x + c) + 3)/(a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + a^3*d)","A",0
92,1,151,0,0.661086," ","integrate(cot(d*x+c)/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{102 \, \cos\left(d x + c\right)^{2} + 45 \, {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 3 \, {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 162 \, \cos\left(d x + c\right) + 68}{48 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}"," ",0,"1/48*(102*cos(d*x + c)^2 + 45*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + 3*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*log(-1/2*cos(d*x + c) + 1/2) + 162*cos(d*x + c) + 68)/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)","A",0
93,1,240,0,0.669364," ","integrate(cot(d*x+c)^3/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{426 \, \cos\left(d x + c\right)^{4} + 606 \, \cos\left(d x + c\right)^{3} - 190 \, \cos\left(d x + c\right)^{2} + 171 \, {\left(\cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) - 1\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} + 3 \, \cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 666 \, \cos\left(d x + c\right) - 272}{192 \, {\left(a^{3} d \cos\left(d x + c\right)^{5} + 3 \, a^{3} d \cos\left(d x + c\right)^{4} + 2 \, a^{3} d \cos\left(d x + c\right)^{3} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} - 3 \, a^{3} d \cos\left(d x + c\right) - a^{3} d\right)}}"," ",0,"-1/192*(426*cos(d*x + c)^4 + 606*cos(d*x + c)^3 - 190*cos(d*x + c)^2 + 171*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*log(1/2*cos(d*x + c) + 1/2) + 21*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*log(-1/2*cos(d*x + c) + 1/2) - 666*cos(d*x + c) - 272)/(a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + 2*a^3*d*cos(d*x + c)^3 - 2*a^3*d*cos(d*x + c)^2 - 3*a^3*d*cos(d*x + c) - a^3*d)","A",0
94,1,317,0,0.813991," ","integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{8790 \, \cos\left(d x + c\right)^{6} + 11010 \, \cos\left(d x + c\right)^{5} - 13880 \, \cos\left(d x + c\right)^{4} - 25560 \, \cos\left(d x + c\right)^{3} - 734 \, \cos\left(d x + c\right)^{2} + 3285 \, {\left(\cos\left(d x + c\right)^{7} + 3 \, \cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5} - 5 \, \cos\left(d x + c\right)^{4} - 5 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 555 \, {\left(\cos\left(d x + c\right)^{7} + 3 \, \cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5} - 5 \, \cos\left(d x + c\right)^{4} - 5 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 13878 \, \cos\left(d x + c\right) + 5536}{3840 \, {\left(a^{3} d \cos\left(d x + c\right)^{7} + 3 \, a^{3} d \cos\left(d x + c\right)^{6} + a^{3} d \cos\left(d x + c\right)^{5} - 5 \, a^{3} d \cos\left(d x + c\right)^{4} - 5 \, a^{3} d \cos\left(d x + c\right)^{3} + a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}"," ",0,"1/3840*(8790*cos(d*x + c)^6 + 11010*cos(d*x + c)^5 - 13880*cos(d*x + c)^4 - 25560*cos(d*x + c)^3 - 734*cos(d*x + c)^2 + 3285*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*log(1/2*cos(d*x + c) + 1/2) + 555*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*log(-1/2*cos(d*x + c) + 1/2) + 13878*cos(d*x + c) + 5536)/(a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c)^6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)","A",0
95,1,147,0,0.600928," ","integrate(tan(d*x+c)^12/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{26880 \, d x \cos\left(d x + c\right)^{8} - 13125 \, \cos\left(d x + c\right)^{8} \log\left(\sin\left(d x + c\right) + 1\right) + 13125 \, \cos\left(d x + c\right)^{8} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(14848 \, \cos\left(d x + c\right)^{7} - 27195 \, \cos\left(d x + c\right)^{6} + 7424 \, \cos\left(d x + c\right)^{5} + 17710 \, \cos\left(d x + c\right)^{4} - 14592 \, \cos\left(d x + c\right)^{3} - 1960 \, \cos\left(d x + c\right)^{2} + 5760 \, \cos\left(d x + c\right) - 1680\right)} \sin\left(d x + c\right)}{26880 \, a^{3} d \cos\left(d x + c\right)^{8}}"," ",0,"1/26880*(26880*d*x*cos(d*x + c)^8 - 13125*cos(d*x + c)^8*log(sin(d*x + c) + 1) + 13125*cos(d*x + c)^8*log(-sin(d*x + c) + 1) - 2*(14848*cos(d*x + c)^7 - 27195*cos(d*x + c)^6 + 7424*cos(d*x + c)^5 + 17710*cos(d*x + c)^4 - 14592*cos(d*x + c)^3 - 1960*cos(d*x + c)^2 + 5760*cos(d*x + c) - 1680)*sin(d*x + c))/(a^3*d*cos(d*x + c)^8)","A",0
96,1,127,0,0.537749," ","integrate(tan(d*x+c)^10/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{480 \, d x \cos\left(d x + c\right)^{6} - 285 \, \cos\left(d x + c\right)^{6} \log\left(\sin\left(d x + c\right) + 1\right) + 285 \, \cos\left(d x + c\right)^{6} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(176 \, \cos\left(d x + c\right)^{5} - 435 \, \cos\left(d x + c\right)^{4} + 208 \, \cos\left(d x + c\right)^{3} + 110 \, \cos\left(d x + c\right)^{2} - 144 \, \cos\left(d x + c\right) + 40\right)} \sin\left(d x + c\right)}{480 \, a^{3} d \cos\left(d x + c\right)^{6}}"," ",0,"-1/480*(480*d*x*cos(d*x + c)^6 - 285*cos(d*x + c)^6*log(sin(d*x + c) + 1) + 285*cos(d*x + c)^6*log(-sin(d*x + c) + 1) - 2*(176*cos(d*x + c)^5 - 435*cos(d*x + c)^4 + 208*cos(d*x + c)^3 + 110*cos(d*x + c)^2 - 144*cos(d*x + c) + 40)*sin(d*x + c))/(a^3*d*cos(d*x + c)^6)","A",0
97,1,97,0,0.783440," ","integrate(tan(d*x+c)^8/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{16 \, d x \cos\left(d x + c\right)^{4} - 13 \, \cos\left(d x + c\right)^{4} \log\left(\sin\left(d x + c\right) + 1\right) + 13 \, \cos\left(d x + c\right)^{4} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(11 \, \cos\left(d x + c\right)^{2} - 8 \, \cos\left(d x + c\right) + 2\right)} \sin\left(d x + c\right)}{16 \, a^{3} d \cos\left(d x + c\right)^{4}}"," ",0,"1/16*(16*d*x*cos(d*x + c)^4 - 13*cos(d*x + c)^4*log(sin(d*x + c) + 1) + 13*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(11*cos(d*x + c)^2 - 8*cos(d*x + c) + 2)*sin(d*x + c))/(a^3*d*cos(d*x + c)^4)","A",0
98,1,87,0,0.618505," ","integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{4 \, d x \cos\left(d x + c\right)^{2} - 7 \, \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) + 7 \, \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(6 \, \cos\left(d x + c\right) - 1\right)} \sin\left(d x + c\right)}{4 \, a^{3} d \cos\left(d x + c\right)^{2}}"," ",0,"-1/4*(4*d*x*cos(d*x + c)^2 - 7*cos(d*x + c)^2*log(sin(d*x + c) + 1) + 7*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + 2*(6*cos(d*x + c) - 1)*sin(d*x + c))/(a^3*d*cos(d*x + c)^2)","A",0
99,1,83,0,0.784719," ","integrate(tan(d*x+c)^4/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{2 \, d x \cos\left(d x + c\right) + 2 \, d x + {\left(\cos\left(d x + c\right) + 1\right)} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(\cos\left(d x + c\right) + 1\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 8 \, \sin\left(d x + c\right)}{2 \, {\left(a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}"," ",0,"1/2*(2*d*x*cos(d*x + c) + 2*d*x + (cos(d*x + c) + 1)*log(sin(d*x + c) + 1) - (cos(d*x + c) + 1)*log(-sin(d*x + c) + 1) - 8*sin(d*x + c))/(a^3*d*cos(d*x + c) + a^3*d)","A",0
100,1,80,0,0.597919," ","integrate(tan(d*x+c)^2/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{3 \, d x \cos\left(d x + c\right)^{2} + 6 \, d x \cos\left(d x + c\right) + 3 \, d x - {\left(7 \, \cos\left(d x + c\right) + 5\right)} \sin\left(d x + c\right)}{3 \, {\left(a^{3} d \cos\left(d x + c\right)^{2} + 2 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}"," ",0,"-1/3*(3*d*x*cos(d*x + c)^2 + 6*d*x*cos(d*x + c) + 3*d*x - (7*cos(d*x + c) + 5)*sin(d*x + c))/(a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + a^3*d)","A",0
101,1,142,0,0.752431," ","integrate(cot(d*x+c)^2/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{221 \, \cos\left(d x + c\right)^{4} + 348 \, \cos\left(d x + c\right)^{3} - 25 \, \cos\left(d x + c\right)^{2} + 105 \, {\left(d x \cos\left(d x + c\right)^{3} + 3 \, d x \cos\left(d x + c\right)^{2} + 3 \, d x \cos\left(d x + c\right) + d x\right)} \sin\left(d x + c\right) - 303 \, \cos\left(d x + c\right) - 136}{105 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)} \sin\left(d x + c\right)}"," ",0,"-1/105*(221*cos(d*x + c)^4 + 348*cos(d*x + c)^3 - 25*cos(d*x + c)^2 + 105*(d*x*cos(d*x + c)^3 + 3*d*x*cos(d*x + c)^2 + 3*d*x*cos(d*x + c) + d*x)*sin(d*x + c) - 303*cos(d*x + c) - 136)/((a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c))","A",0
102,1,216,0,1.208149," ","integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","\frac{668 \, \cos\left(d x + c\right)^{6} + 1059 \, \cos\left(d x + c\right)^{5} - 573 \, \cos\left(d x + c\right)^{4} - 1813 \, \cos\left(d x + c\right)^{3} - 393 \, \cos\left(d x + c\right)^{2} + 315 \, {\left(d x \cos\left(d x + c\right)^{5} + 3 \, d x \cos\left(d x + c\right)^{4} + 2 \, d x \cos\left(d x + c\right)^{3} - 2 \, d x \cos\left(d x + c\right)^{2} - 3 \, d x \cos\left(d x + c\right) - d x\right)} \sin\left(d x + c\right) + 789 \, \cos\left(d x + c\right) + 368}{315 \, {\left(a^{3} d \cos\left(d x + c\right)^{5} + 3 \, a^{3} d \cos\left(d x + c\right)^{4} + 2 \, a^{3} d \cos\left(d x + c\right)^{3} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} - 3 \, a^{3} d \cos\left(d x + c\right) - a^{3} d\right)} \sin\left(d x + c\right)}"," ",0,"1/315*(668*cos(d*x + c)^6 + 1059*cos(d*x + c)^5 - 573*cos(d*x + c)^4 - 1813*cos(d*x + c)^3 - 393*cos(d*x + c)^2 + 315*(d*x*cos(d*x + c)^5 + 3*d*x*cos(d*x + c)^4 + 2*d*x*cos(d*x + c)^3 - 2*d*x*cos(d*x + c)^2 - 3*d*x*cos(d*x + c) - d*x)*sin(d*x + c) + 789*cos(d*x + c) + 368)/((a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + 2*a^3*d*cos(d*x + c)^3 - 2*a^3*d*cos(d*x + c)^2 - 3*a^3*d*cos(d*x + c) - a^3*d)*sin(d*x + c))","A",0
103,1,282,0,1.158014," ","integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","-\frac{7453 \, \cos\left(d x + c\right)^{8} + 11964 \, \cos\left(d x + c\right)^{7} - 11866 \, \cos\left(d x + c\right)^{6} - 30542 \, \cos\left(d x + c\right)^{5} + 90 \, \cos\left(d x + c\right)^{4} + 26438 \, \cos\left(d x + c\right)^{3} + 8539 \, \cos\left(d x + c\right)^{2} + 3465 \, {\left(d x \cos\left(d x + c\right)^{7} + 3 \, d x \cos\left(d x + c\right)^{6} + d x \cos\left(d x + c\right)^{5} - 5 \, d x \cos\left(d x + c\right)^{4} - 5 \, d x \cos\left(d x + c\right)^{3} + d x \cos\left(d x + c\right)^{2} + 3 \, d x \cos\left(d x + c\right) + d x\right)} \sin\left(d x + c\right) - 7671 \, \cos\left(d x + c\right) - 3712}{3465 \, {\left(a^{3} d \cos\left(d x + c\right)^{7} + 3 \, a^{3} d \cos\left(d x + c\right)^{6} + a^{3} d \cos\left(d x + c\right)^{5} - 5 \, a^{3} d \cos\left(d x + c\right)^{4} - 5 \, a^{3} d \cos\left(d x + c\right)^{3} + a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)} \sin\left(d x + c\right)}"," ",0,"-1/3465*(7453*cos(d*x + c)^8 + 11964*cos(d*x + c)^7 - 11866*cos(d*x + c)^6 - 30542*cos(d*x + c)^5 + 90*cos(d*x + c)^4 + 26438*cos(d*x + c)^3 + 8539*cos(d*x + c)^2 + 3465*(d*x*cos(d*x + c)^7 + 3*d*x*cos(d*x + c)^6 + d*x*cos(d*x + c)^5 - 5*d*x*cos(d*x + c)^4 - 5*d*x*cos(d*x + c)^3 + d*x*cos(d*x + c)^2 + 3*d*x*cos(d*x + c) + d*x)*sin(d*x + c) - 7671*cos(d*x + c) - 3712)/((a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c)^6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c))","A",0
104,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))*(e*tan(d*x+c))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
105,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))*(e*tan(d*x+c))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
106,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))*(e*tan(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
107,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))/(e*tan(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
108,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))/(e*tan(d*x+c))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
109,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))/(e*tan(d*x+c))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
110,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))/(e*tan(d*x+c))^(7/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
111,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
112,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
113,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
114,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))^2/(e*tan(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
115,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))^2/(e*tan(d*x+c))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
116,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))^2/(e*tan(d*x+c))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
117,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))^2/(e*tan(d*x+c))^(7/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
118,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(11/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
119,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(9/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
120,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(7/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
121,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(5/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
122,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
123,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(1/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
124,-1,0,0,0.000000," ","integrate(1/(a+a*sec(d*x+c))/(e*tan(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
125,-1,0,0,0.000000," ","integrate(1/(a+a*sec(d*x+c))/(e*tan(d*x+c))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
126,-1,0,0,0.000000," ","integrate(1/(a+a*sec(d*x+c))/(e*tan(d*x+c))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
127,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(13/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
128,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(11/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
129,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(9/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
130,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(7/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
131,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(5/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
132,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(3/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
133,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(1/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
134,-1,0,0,0.000000," ","integrate(1/(a+a*sec(d*x+c))^2/(e*tan(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
135,1,299,0,1.021694," ","integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^5,x, algorithm=""fricas"")","\left[\frac{315 \, \sqrt{a} \cos\left(d x + c\right)^{4} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, {\left(383 \, \cos\left(d x + c\right)^{4} - 34 \, \cos\left(d x + c\right)^{3} - 132 \, \cos\left(d x + c\right)^{2} + 5 \, \cos\left(d x + c\right) + 35\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{630 \, d \cos\left(d x + c\right)^{4}}, \frac{315 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right)^{4} + 2 \, {\left(383 \, \cos\left(d x + c\right)^{4} - 34 \, \cos\left(d x + c\right)^{3} - 132 \, \cos\left(d x + c\right)^{2} + 5 \, \cos\left(d x + c\right) + 35\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{315 \, d \cos\left(d x + c\right)^{4}}\right]"," ",0,"[1/630*(315*sqrt(a)*cos(d*x + c)^4*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(383*cos(d*x + c)^4 - 34*cos(d*x + c)^3 - 132*cos(d*x + c)^2 + 5*cos(d*x + c) + 35)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^4), 1/315*(315*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c)^4 + 2*(383*cos(d*x + c)^4 - 34*cos(d*x + c)^3 - 132*cos(d*x + c)^2 + 5*cos(d*x + c) + 35)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^4)]","A",0
136,1,259,0,0.980473," ","integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^3,x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{a} \cos\left(d x + c\right)^{2} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) - 4 \, {\left(17 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 3\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{30 \, d \cos\left(d x + c\right)^{2}}, -\frac{15 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right)^{2} + 2 \, {\left(17 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 3\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{15 \, d \cos\left(d x + c\right)^{2}}\right]"," ",0,"[1/30*(15*sqrt(a)*cos(d*x + c)^2*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(17*cos(d*x + c)^2 - cos(d*x + c) - 3)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^2), -1/15*(15*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c)^2 + 2*(17*cos(d*x + c)^2 - cos(d*x + c) - 3)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^2)]","A",0
137,1,184,0,1.084023," ","integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{2 \, d}, \frac{\sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) + 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{d}\right]"," ",0,"[1/2*(sqrt(a)*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/d, (sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) + 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/d]","A",0
138,1,242,0,0.833453," ","integrate(cot(d*x+c)*(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} \sqrt{a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) - 1}\right) + 2 \, \sqrt{a} \log\left(-2 \, a \cos\left(d x + c\right) - 2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - a\right)}{2 \, d}, \frac{\sqrt{2} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 2 \, \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right)}{d}\right]"," ",0,"[1/2*(sqrt(2)*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + 2*sqrt(a)*log(-2*a*cos(d*x + c) - 2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - a))/d, (sqrt(2)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 2*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)))/d]","A",0
139,1,426,0,1.126367," ","integrate(cot(d*x+c)^3*(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{8 \, {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 7 \, {\left(\sqrt{2} \cos\left(d x + c\right)^{2} - \sqrt{2}\right)} \sqrt{a} \log\left(\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + 3 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) - 1}\right) + 4 \, {\left(3 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{16 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}, -\frac{7 \, {\left(\sqrt{2} \cos\left(d x + c\right)^{2} - \sqrt{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 8 \, {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(3 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{8 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}\right]"," ",0,"[1/16*(8*(cos(d*x + c)^2 - 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 7*(sqrt(2)*cos(d*x + c)^2 - sqrt(2))*sqrt(a)*log((2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) + 3*a*cos(d*x + c) + a)/(cos(d*x + c) - 1)) + 4*(3*cos(d*x + c)^2 - cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^2 - d), -1/8*(7*(sqrt(2)*cos(d*x + c)^2 - sqrt(2))*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 8*(cos(d*x + c)^2 - 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(3*cos(d*x + c)^2 - cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^2 - d)]","A",0
140,1,529,0,0.967862," ","integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{384 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1\right)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 321 \, {\left(\sqrt{2} \cos\left(d x + c\right)^{4} - 2 \, \sqrt{2} \cos\left(d x + c\right)^{2} + \sqrt{2}\right)} \sqrt{a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) - 1}\right) - 4 \, {\left(205 \, \cos\left(d x + c\right)^{4} - 71 \, \cos\left(d x + c\right)^{3} - 149 \, \cos\left(d x + c\right)^{2} + 63 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{768 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} + d\right)}}, \frac{321 \, {\left(\sqrt{2} \cos\left(d x + c\right)^{4} - 2 \, \sqrt{2} \cos\left(d x + c\right)^{2} + \sqrt{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 384 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(205 \, \cos\left(d x + c\right)^{4} - 71 \, \cos\left(d x + c\right)^{3} - 149 \, \cos\left(d x + c\right)^{2} + 63 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{384 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} + d\right)}}\right]"," ",0,"[1/768*(384*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 321*(sqrt(2)*cos(d*x + c)^4 - 2*sqrt(2)*cos(d*x + c)^2 + sqrt(2))*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) - 4*(205*cos(d*x + c)^4 - 71*cos(d*x + c)^3 - 149*cos(d*x + c)^2 + 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d), 1/384*(321*(sqrt(2)*cos(d*x + c)^4 - 2*sqrt(2)*cos(d*x + c)^2 + sqrt(2))*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 384*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(205*cos(d*x + c)^4 - 71*cos(d*x + c)^3 - 149*cos(d*x + c)^2 + 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)]","A",0
141,1,371,0,0.776443," ","integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^6,x, algorithm=""fricas"")","\left[\frac{495 \, {\left(\cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) + 2 \, {\left(494 \, \cos\left(d x + c\right)^{5} + 247 \, \cos\left(d x + c\right)^{4} - 186 \, \cos\left(d x + c\right)^{3} - 155 \, \cos\left(d x + c\right)^{2} + 50 \, \cos\left(d x + c\right) + 45\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{495 \, {\left(d \cos\left(d x + c\right)^{6} + d \cos\left(d x + c\right)^{5}\right)}}, \frac{2 \, {\left(495 \, {\left(\cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + {\left(494 \, \cos\left(d x + c\right)^{5} + 247 \, \cos\left(d x + c\right)^{4} - 186 \, \cos\left(d x + c\right)^{3} - 155 \, \cos\left(d x + c\right)^{2} + 50 \, \cos\left(d x + c\right) + 45\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{495 \, {\left(d \cos\left(d x + c\right)^{6} + d \cos\left(d x + c\right)^{5}\right)}}\right]"," ",0,"[1/495*(495*(cos(d*x + c)^6 + cos(d*x + c)^5)*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*(494*cos(d*x + c)^5 + 247*cos(d*x + c)^4 - 186*cos(d*x + c)^3 - 155*cos(d*x + c)^2 + 50*cos(d*x + c) + 45)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5), 2/495*(495*(cos(d*x + c)^6 + cos(d*x + c)^5)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (494*cos(d*x + c)^5 + 247*cos(d*x + c)^4 - 186*cos(d*x + c)^3 - 155*cos(d*x + c)^2 + 50*cos(d*x + c) + 45)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5)]","A",0
142,1,331,0,0.692403," ","integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^4,x, algorithm=""fricas"")","\left[\frac{105 \, {\left(\cos\left(d x + c\right)^{4} + \cos\left(d x + c\right)^{3}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) - 2 \, {\left(92 \, \cos\left(d x + c\right)^{3} + 46 \, \cos\left(d x + c\right)^{2} - 18 \, \cos\left(d x + c\right) - 15\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{105 \, {\left(d \cos\left(d x + c\right)^{4} + d \cos\left(d x + c\right)^{3}\right)}}, -\frac{2 \, {\left(105 \, {\left(\cos\left(d x + c\right)^{4} + \cos\left(d x + c\right)^{3}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + {\left(92 \, \cos\left(d x + c\right)^{3} + 46 \, \cos\left(d x + c\right)^{2} - 18 \, \cos\left(d x + c\right) - 15\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{105 \, {\left(d \cos\left(d x + c\right)^{4} + d \cos\left(d x + c\right)^{3}\right)}}\right]"," ",0,"[1/105*(105*(cos(d*x + c)^4 + cos(d*x + c)^3)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) - 2*(92*cos(d*x + c)^3 + 46*cos(d*x + c)^2 - 18*cos(d*x + c) - 15)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 + d*cos(d*x + c)^3), -2/105*(105*(cos(d*x + c)^4 + cos(d*x + c)^3)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (92*cos(d*x + c)^3 + 46*cos(d*x + c)^2 - 18*cos(d*x + c) - 15)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 + d*cos(d*x + c)^3)]","A",0
143,1,283,0,0.797769," ","integrate((a+a*sec(d*x+c))^(1/2)*tan(d*x+c)^2,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(\cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) + 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} {\left(2 \, \cos\left(d x + c\right) + 1\right)} \sin\left(d x + c\right)}{3 \, {\left(d \cos\left(d x + c\right)^{2} + d \cos\left(d x + c\right)\right)}}, \frac{2 \, {\left(3 \, {\left(\cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} {\left(2 \, \cos\left(d x + c\right) + 1\right)} \sin\left(d x + c\right)\right)}}{3 \, {\left(d \cos\left(d x + c\right)^{2} + d \cos\left(d x + c\right)\right)}}\right]"," ",0,"[1/3*(3*(cos(d*x + c)^2 + cos(d*x + c))*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(2*cos(d*x + c) + 1)*sin(d*x + c))/(d*cos(d*x + c)^2 + d*cos(d*x + c)), 2/3*(3*(cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(2*cos(d*x + c) + 1)*sin(d*x + c))/(d*cos(d*x + c)^2 + d*cos(d*x + c))]","A",0
144,1,422,0,0.971247," ","integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} \sqrt{-a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 3 \, a \cos\left(d x + c\right)^{2} - 2 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 2 \, \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) - 4 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{4 \, d \sin\left(d x + c\right)}, -\frac{\sqrt{2} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 2 \, \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, d \sin\left(d x + c\right)}\right]"," ",0,"[1/4*(sqrt(2)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c) + a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 2*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(d*sin(d*x + c)), -1/2*(sqrt(2)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 2*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(d*sin(d*x + c))]","A",0
145,1,547,0,1.339998," ","integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{27 \, {\left(\sqrt{2} \cos\left(d x + c\right)^{2} - \sqrt{2}\right)} \sqrt{-a} \log\left(\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 3 \, a \cos\left(d x + c\right)^{2} + 2 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 48 \, {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 4 \, {\left(31 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 21 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{96 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}, \frac{48 \, {\left(\cos\left(d x + c\right)^{2} - 1\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 27 \, {\left(\sqrt{2} \cos\left(d x + c\right)^{2} - \sqrt{2}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 2 \, {\left(31 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 21 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{48 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[1/96*(27*(sqrt(2)*cos(d*x + c)^2 - sqrt(2))*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + 3*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 48*(cos(d*x + c)^2 - 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 - 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) + 4*(31*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 21*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^2 - d)*sin(d*x + c)), 1/48*(48*(cos(d*x + c)^2 - 1)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 27*(sqrt(2)*cos(d*x + c)^2 - sqrt(2))*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 2*(31*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 21*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))]","A",0
146,1,646,0,2.963723," ","integrate(cot(d*x+c)^6*(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{2265 \, {\left(\sqrt{2} \cos\left(d x + c\right)^{4} - 2 \, \sqrt{2} \cos\left(d x + c\right)^{2} + \sqrt{2}\right)} \sqrt{-a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 3 \, a \cos\left(d x + c\right)^{2} - 2 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 3840 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) - 4 \, {\left(2821 \, \cos\left(d x + c\right)^{5} - 278 \, \cos\left(d x + c\right)^{4} - 3964 \, \cos\left(d x + c\right)^{3} + 230 \, \cos\left(d x + c\right)^{2} + 1575 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{7680 \, {\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)}, -\frac{3840 \, {\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2265 \, {\left(\sqrt{2} \cos\left(d x + c\right)^{4} - 2 \, \sqrt{2} \cos\left(d x + c\right)^{2} + \sqrt{2}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 2 \, {\left(2821 \, \cos\left(d x + c\right)^{5} - 278 \, \cos\left(d x + c\right)^{4} - 3964 \, \cos\left(d x + c\right)^{3} + 230 \, \cos\left(d x + c\right)^{2} + 1575 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{3840 \, {\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)}\right]"," ",0,"[1/7680*(2265*(sqrt(2)*cos(d*x + c)^4 - 2*sqrt(2)*cos(d*x + c)^2 + sqrt(2))*sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c) + a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 3840*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*(2821*cos(d*x + c)^5 - 278*cos(d*x + c)^4 - 3964*cos(d*x + c)^3 + 230*cos(d*x + c)^2 + 1575*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c)), -1/3840*(3840*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2265*(sqrt(2)*cos(d*x + c)^4 - 2*sqrt(2)*cos(d*x + c)^2 + sqrt(2))*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 2*(2821*cos(d*x + c)^5 - 278*cos(d*x + c)^4 - 3964*cos(d*x + c)^3 + 230*cos(d*x + c)^2 + 1575*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))]","A",0
147,1,334,0,1.693351," ","integrate((a+a*sec(d*x+c))^(3/2)*tan(d*x+c)^5,x, algorithm=""fricas"")","\left[\frac{1155 \, a^{\frac{3}{2}} \cos\left(d x + c\right)^{5} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, {\left(1656 \, a \cos\left(d x + c\right)^{5} + 327 \, a \cos\left(d x + c\right)^{4} - 534 \, a \cos\left(d x + c\right)^{3} - 325 \, a \cos\left(d x + c\right)^{2} + 140 \, a \cos\left(d x + c\right) + 105 \, a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{2310 \, d \cos\left(d x + c\right)^{5}}, \frac{1155 \, \sqrt{-a} a \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right)^{5} + 2 \, {\left(1656 \, a \cos\left(d x + c\right)^{5} + 327 \, a \cos\left(d x + c\right)^{4} - 534 \, a \cos\left(d x + c\right)^{3} - 325 \, a \cos\left(d x + c\right)^{2} + 140 \, a \cos\left(d x + c\right) + 105 \, a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{1155 \, d \cos\left(d x + c\right)^{5}}\right]"," ",0,"[1/2310*(1155*a^(3/2)*cos(d*x + c)^5*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(1656*a*cos(d*x + c)^5 + 327*a*cos(d*x + c)^4 - 534*a*cos(d*x + c)^3 - 325*a*cos(d*x + c)^2 + 140*a*cos(d*x + c) + 105*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^5), 1/1155*(1155*sqrt(-a)*a*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c)^5 + 2*(1656*a*cos(d*x + c)^5 + 327*a*cos(d*x + c)^4 - 534*a*cos(d*x + c)^3 - 325*a*cos(d*x + c)^2 + 140*a*cos(d*x + c) + 105*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^5)]","A",0
148,1,290,0,1.333249," ","integrate((a+a*sec(d*x+c))^(3/2)*tan(d*x+c)^3,x, algorithm=""fricas"")","\left[\frac{105 \, a^{\frac{3}{2}} \cos\left(d x + c\right)^{3} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) - 4 \, {\left(146 \, a \cos\left(d x + c\right)^{3} + 32 \, a \cos\left(d x + c\right)^{2} - 24 \, a \cos\left(d x + c\right) - 15 \, a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{210 \, d \cos\left(d x + c\right)^{3}}, -\frac{105 \, \sqrt{-a} a \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right)^{3} + 2 \, {\left(146 \, a \cos\left(d x + c\right)^{3} + 32 \, a \cos\left(d x + c\right)^{2} - 24 \, a \cos\left(d x + c\right) - 15 \, a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{105 \, d \cos\left(d x + c\right)^{3}}\right]"," ",0,"[1/210*(105*a^(3/2)*cos(d*x + c)^3*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(146*a*cos(d*x + c)^3 + 32*a*cos(d*x + c)^2 - 24*a*cos(d*x + c) - 15*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^3), -1/105*(105*sqrt(-a)*a*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c)^3 + 2*(146*a*cos(d*x + c)^3 + 32*a*cos(d*x + c)^2 - 24*a*cos(d*x + c) - 15*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^3)]","A",0
149,1,238,0,0.695270," ","integrate((a+a*sec(d*x+c))^(3/2)*tan(d*x+c),x, algorithm=""fricas"")","\left[\frac{3 \, a^{\frac{3}{2}} \cos\left(d x + c\right) \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, {\left(4 \, a \cos\left(d x + c\right) + a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{6 \, d \cos\left(d x + c\right)}, \frac{3 \, \sqrt{-a} a \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right) + 2 \, {\left(4 \, a \cos\left(d x + c\right) + a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{3 \, d \cos\left(d x + c\right)}\right]"," ",0,"[1/6*(3*a^(3/2)*cos(d*x + c)*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(4*a*cos(d*x + c) + a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)), 1/3*(3*sqrt(-a)*a*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c) + 2*(4*a*cos(d*x + c) + a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c))]","A",0
150,1,243,0,0.748359," ","integrate(cot(d*x+c)*(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} a^{\frac{3}{2}} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) - 1}\right) + a^{\frac{3}{2}} \log\left(-2 \, a \cos\left(d x + c\right) - 2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - a\right)}{d}, \frac{2 \, {\left(\sqrt{2} \sqrt{-a} a \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - \sqrt{-a} a \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right)\right)}}{d}\right]"," ",0,"[(sqrt(2)*a^(3/2)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + a^(3/2)*log(-2*a*cos(d*x + c) - 2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - a))/d, 2*(sqrt(2)*sqrt(-a)*a*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - sqrt(-a)*a*arctan(sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)))/d]","B",0
151,1,378,0,0.680162," ","integrate(cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{4 \, a \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + 8 \, {\left(a \cos\left(d x + c\right) - a\right)} \sqrt{a} \log\left(-2 \, a \cos\left(d x + c\right) + 2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - a\right) + 5 \, {\left(\sqrt{2} a \cos\left(d x + c\right) - \sqrt{2} a\right)} \sqrt{a} \log\left(\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + 3 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) - 1}\right)}{8 \, {\left(d \cos\left(d x + c\right) - d\right)}}, -\frac{5 \, {\left(\sqrt{2} a \cos\left(d x + c\right) - \sqrt{2} a\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 8 \, {\left(a \cos\left(d x + c\right) - a\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 2 \, a \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{4 \, {\left(d \cos\left(d x + c\right) - d\right)}}\right]"," ",0,"[1/8*(4*a*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) + 8*(a*cos(d*x + c) - a)*sqrt(a)*log(-2*a*cos(d*x + c) + 2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - a) + 5*(sqrt(2)*a*cos(d*x + c) - sqrt(2)*a)*sqrt(a)*log((2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) + 3*a*cos(d*x + c) + a)/(cos(d*x + c) - 1)))/(d*cos(d*x + c) - d), -1/4*(5*(sqrt(2)*a*cos(d*x + c) - sqrt(2)*a)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 8*(a*cos(d*x + c) - a)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 2*a*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(d*cos(d*x + c) - d)]","B",0
152,1,589,0,0.846978," ","integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{64 \, {\left(a \cos\left(d x + c\right)^{3} - a \cos\left(d x + c\right)^{2} - a \cos\left(d x + c\right) + a\right)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 71 \, {\left(\sqrt{2} a \cos\left(d x + c\right)^{3} - \sqrt{2} a \cos\left(d x + c\right)^{2} - \sqrt{2} a \cos\left(d x + c\right) + \sqrt{2} a\right)} \sqrt{a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) - 1}\right) - 4 \, {\left(27 \, a \cos\left(d x + c\right)^{3} - 12 \, a \cos\left(d x + c\right)^{2} - 7 \, a \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{128 \, {\left(d \cos\left(d x + c\right)^{3} - d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) + d\right)}}, \frac{71 \, {\left(\sqrt{2} a \cos\left(d x + c\right)^{3} - \sqrt{2} a \cos\left(d x + c\right)^{2} - \sqrt{2} a \cos\left(d x + c\right) + \sqrt{2} a\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 64 \, {\left(a \cos\left(d x + c\right)^{3} - a \cos\left(d x + c\right)^{2} - a \cos\left(d x + c\right) + a\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(27 \, a \cos\left(d x + c\right)^{3} - 12 \, a \cos\left(d x + c\right)^{2} - 7 \, a \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{64 \, {\left(d \cos\left(d x + c\right)^{3} - d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) + d\right)}}\right]"," ",0,"[1/128*(64*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 71*(sqrt(2)*a*cos(d*x + c)^3 - sqrt(2)*a*cos(d*x + c)^2 - sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) - 4*(27*a*cos(d*x + c)^3 - 12*a*cos(d*x + c)^2 - 7*a*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*cos(d*x + c) + d), 1/64*(71*(sqrt(2)*a*cos(d*x + c)^3 - sqrt(2)*a*cos(d*x + c)^2 - sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 64*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(27*a*cos(d*x + c)^3 - 12*a*cos(d*x + c)^2 - 7*a*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*cos(d*x + c) + d)]","B",0
153,1,415,0,0.720004," ","integrate((a+a*sec(d*x+c))^(3/2)*tan(d*x+c)^6,x, algorithm=""fricas"")","\left[\frac{45045 \, {\left(a \cos\left(d x + c\right)^{7} + a \cos\left(d x + c\right)^{6}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) + 2 \, {\left(32429 \, a \cos\left(d x + c\right)^{6} + 38737 \, a \cos\left(d x + c\right)^{5} - 4731 \, a \cos\left(d x + c\right)^{4} - 26465 \, a \cos\left(d x + c\right)^{3} - 6265 \, a \cos\left(d x + c\right)^{2} + 7875 \, a \cos\left(d x + c\right) + 3465 \, a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{45045 \, {\left(d \cos\left(d x + c\right)^{7} + d \cos\left(d x + c\right)^{6}\right)}}, \frac{2 \, {\left(45045 \, {\left(a \cos\left(d x + c\right)^{7} + a \cos\left(d x + c\right)^{6}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + {\left(32429 \, a \cos\left(d x + c\right)^{6} + 38737 \, a \cos\left(d x + c\right)^{5} - 4731 \, a \cos\left(d x + c\right)^{4} - 26465 \, a \cos\left(d x + c\right)^{3} - 6265 \, a \cos\left(d x + c\right)^{2} + 7875 \, a \cos\left(d x + c\right) + 3465 \, a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{45045 \, {\left(d \cos\left(d x + c\right)^{7} + d \cos\left(d x + c\right)^{6}\right)}}\right]"," ",0,"[1/45045*(45045*(a*cos(d*x + c)^7 + a*cos(d*x + c)^6)*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*(32429*a*cos(d*x + c)^6 + 38737*a*cos(d*x + c)^5 - 4731*a*cos(d*x + c)^4 - 26465*a*cos(d*x + c)^3 - 6265*a*cos(d*x + c)^2 + 7875*a*cos(d*x + c) + 3465*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^7 + d*cos(d*x + c)^6), 2/45045*(45045*(a*cos(d*x + c)^7 + a*cos(d*x + c)^6)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (32429*a*cos(d*x + c)^6 + 38737*a*cos(d*x + c)^5 - 4731*a*cos(d*x + c)^4 - 26465*a*cos(d*x + c)^3 - 6265*a*cos(d*x + c)^2 + 7875*a*cos(d*x + c) + 3465*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^7 + d*cos(d*x + c)^6)]","A",0
154,1,371,0,0.765529," ","integrate((a+a*sec(d*x+c))^(3/2)*tan(d*x+c)^4,x, algorithm=""fricas"")","\left[\frac{315 \, {\left(a \cos\left(d x + c\right)^{5} + a \cos\left(d x + c\right)^{4}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) - 2 \, {\left(169 \, a \cos\left(d x + c\right)^{4} + 242 \, a \cos\left(d x + c\right)^{3} + 24 \, a \cos\left(d x + c\right)^{2} - 85 \, a \cos\left(d x + c\right) - 35 \, a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{315 \, {\left(d \cos\left(d x + c\right)^{5} + d \cos\left(d x + c\right)^{4}\right)}}, -\frac{2 \, {\left(315 \, {\left(a \cos\left(d x + c\right)^{5} + a \cos\left(d x + c\right)^{4}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + {\left(169 \, a \cos\left(d x + c\right)^{4} + 242 \, a \cos\left(d x + c\right)^{3} + 24 \, a \cos\left(d x + c\right)^{2} - 85 \, a \cos\left(d x + c\right) - 35 \, a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{315 \, {\left(d \cos\left(d x + c\right)^{5} + d \cos\left(d x + c\right)^{4}\right)}}\right]"," ",0,"[1/315*(315*(a*cos(d*x + c)^5 + a*cos(d*x + c)^4)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) - 2*(169*a*cos(d*x + c)^4 + 242*a*cos(d*x + c)^3 + 24*a*cos(d*x + c)^2 - 85*a*cos(d*x + c) - 35*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^5 + d*cos(d*x + c)^4), -2/315*(315*(a*cos(d*x + c)^5 + a*cos(d*x + c)^4)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (169*a*cos(d*x + c)^4 + 242*a*cos(d*x + c)^3 + 24*a*cos(d*x + c)^2 - 85*a*cos(d*x + c) - 35*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^5 + d*cos(d*x + c)^4)]","A",0
155,1,321,0,0.577400," ","integrate((a+a*sec(d*x+c))^(3/2)*tan(d*x+c)^2,x, algorithm=""fricas"")","\left[\frac{5 \, {\left(a \cos\left(d x + c\right)^{3} + a \cos\left(d x + c\right)^{2}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) + 2 \, {\left(a \cos\left(d x + c\right)^{2} + 3 \, a \cos\left(d x + c\right) + a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{5 \, {\left(d \cos\left(d x + c\right)^{3} + d \cos\left(d x + c\right)^{2}\right)}}, \frac{2 \, {\left(5 \, {\left(a \cos\left(d x + c\right)^{3} + a \cos\left(d x + c\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + {\left(a \cos\left(d x + c\right)^{2} + 3 \, a \cos\left(d x + c\right) + a\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{5 \, {\left(d \cos\left(d x + c\right)^{3} + d \cos\left(d x + c\right)^{2}\right)}}\right]"," ",0,"[1/5*(5*(a*cos(d*x + c)^3 + a*cos(d*x + c)^2)*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*(a*cos(d*x + c)^2 + 3*a*cos(d*x + c) + a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^3 + d*cos(d*x + c)^2), 2/5*(5*(a*cos(d*x + c)^3 + a*cos(d*x + c)^2)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (a*cos(d*x + c)^2 + 3*a*cos(d*x + c) + a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^3 + d*cos(d*x + c)^2)]","A",0
156,1,264,0,0.666202," ","integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} a \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) - 4 \, a \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, d \sin\left(d x + c\right)}, -\frac{a^{\frac{3}{2}} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, a \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{d \sin\left(d x + c\right)}\right]"," ",0,"[1/2*(sqrt(-a)*a*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*a*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(d*sin(d*x + c)), -(a^(3/2)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*a*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(d*sin(d*x + c))]","B",0
157,1,531,0,0.777961," ","integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(\sqrt{2} a \cos\left(d x + c\right) - \sqrt{2} a\right)} \sqrt{-a} \log\left(\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 3 \, a \cos\left(d x + c\right)^{2} + 2 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 12 \, {\left(a \cos\left(d x + c\right) - a\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 4 \, {\left(11 \, a \cos\left(d x + c\right)^{2} - 9 \, a \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{24 \, {\left(d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right)}, \frac{12 \, {\left(a \cos\left(d x + c\right) - a\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 3 \, {\left(\sqrt{2} a \cos\left(d x + c\right) - \sqrt{2} a\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 2 \, {\left(11 \, a \cos\left(d x + c\right)^{2} - 9 \, a \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{12 \, {\left(d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[1/24*(3*(sqrt(2)*a*cos(d*x + c) - sqrt(2)*a)*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + 3*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 12*(a*cos(d*x + c) - a)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 - 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) + 4*(11*a*cos(d*x + c)^2 - 9*a*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c) - d)*sin(d*x + c)), 1/12*(12*(a*cos(d*x + c) - a)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 3*(sqrt(2)*a*cos(d*x + c) - sqrt(2)*a)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 2*(11*a*cos(d*x + c)^2 - 9*a*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c) - d)*sin(d*x + c))]","A",0
158,1,708,0,0.724126," ","integrate(cot(d*x+c)^6*(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{165 \, {\left(\sqrt{2} a \cos\left(d x + c\right)^{3} - \sqrt{2} a \cos\left(d x + c\right)^{2} - \sqrt{2} a \cos\left(d x + c\right) + \sqrt{2} a\right)} \sqrt{-a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 3 \, a \cos\left(d x + c\right)^{2} - 2 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 480 \, {\left(a \cos\left(d x + c\right)^{3} - a \cos\left(d x + c\right)^{2} - a \cos\left(d x + c\right) + a\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) - 4 \, {\left(449 \, a \cos\left(d x + c\right)^{4} - 351 \, a \cos\left(d x + c\right)^{3} - 365 \, a \cos\left(d x + c\right)^{2} + 315 \, a \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{960 \, {\left(d \cos\left(d x + c\right)^{3} - d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right)}, -\frac{480 \, {\left(a \cos\left(d x + c\right)^{3} - a \cos\left(d x + c\right)^{2} - a \cos\left(d x + c\right) + a\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 165 \, {\left(\sqrt{2} a \cos\left(d x + c\right)^{3} - \sqrt{2} a \cos\left(d x + c\right)^{2} - \sqrt{2} a \cos\left(d x + c\right) + \sqrt{2} a\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 2 \, {\left(449 \, a \cos\left(d x + c\right)^{4} - 351 \, a \cos\left(d x + c\right)^{3} - 365 \, a \cos\left(d x + c\right)^{2} + 315 \, a \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{480 \, {\left(d \cos\left(d x + c\right)^{3} - d \cos\left(d x + c\right)^{2} - d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[1/960*(165*(sqrt(2)*a*cos(d*x + c)^3 - sqrt(2)*a*cos(d*x + c)^2 - sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c) + a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 480*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*(449*a*cos(d*x + c)^4 - 351*a*cos(d*x + c)^3 - 365*a*cos(d*x + c)^2 + 315*a*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*cos(d*x + c) + d)*sin(d*x + c)), -1/480*(480*(a*cos(d*x + c)^3 - a*cos(d*x + c)^2 - a*cos(d*x + c) + a)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 165*(sqrt(2)*a*cos(d*x + c)^3 - sqrt(2)*a*cos(d*x + c)^2 - sqrt(2)*a*cos(d*x + c) + sqrt(2)*a)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 2*(449*a*cos(d*x + c)^4 - 351*a*cos(d*x + c)^3 - 365*a*cos(d*x + c)^2 + 315*a*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^3 - d*cos(d*x + c)^2 - d*cos(d*x + c) + d)*sin(d*x + c))]","A",0
159,1,386,0,0.729748," ","integrate((a+a*sec(d*x+c))^(5/2)*tan(d*x+c)^5,x, algorithm=""fricas"")","\left[\frac{45045 \, a^{\frac{5}{2}} \cos\left(d x + c\right)^{6} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, {\left(71689 \, a^{2} \cos\left(d x + c\right)^{6} + 31723 \, a^{2} \cos\left(d x + c\right)^{5} - 12531 \, a^{2} \cos\left(d x + c\right)^{4} - 27095 \, a^{2} \cos\left(d x + c\right)^{3} - 4445 \, a^{2} \cos\left(d x + c\right)^{2} + 8505 \, a^{2} \cos\left(d x + c\right) + 3465 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{90090 \, d \cos\left(d x + c\right)^{6}}, \frac{45045 \, \sqrt{-a} a^{2} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right)^{6} + 2 \, {\left(71689 \, a^{2} \cos\left(d x + c\right)^{6} + 31723 \, a^{2} \cos\left(d x + c\right)^{5} - 12531 \, a^{2} \cos\left(d x + c\right)^{4} - 27095 \, a^{2} \cos\left(d x + c\right)^{3} - 4445 \, a^{2} \cos\left(d x + c\right)^{2} + 8505 \, a^{2} \cos\left(d x + c\right) + 3465 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{45045 \, d \cos\left(d x + c\right)^{6}}\right]"," ",0,"[1/90090*(45045*a^(5/2)*cos(d*x + c)^6*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(71689*a^2*cos(d*x + c)^6 + 31723*a^2*cos(d*x + c)^5 - 12531*a^2*cos(d*x + c)^4 - 27095*a^2*cos(d*x + c)^3 - 4445*a^2*cos(d*x + c)^2 + 8505*a^2*cos(d*x + c) + 3465*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^6), 1/45045*(45045*sqrt(-a)*a^2*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c)^6 + 2*(71689*a^2*cos(d*x + c)^6 + 31723*a^2*cos(d*x + c)^5 - 12531*a^2*cos(d*x + c)^4 - 27095*a^2*cos(d*x + c)^3 - 4445*a^2*cos(d*x + c)^2 + 8505*a^2*cos(d*x + c) + 3465*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^6)]","A",0
160,1,334,0,0.786683," ","integrate((a+a*sec(d*x+c))^(5/2)*tan(d*x+c)^3,x, algorithm=""fricas"")","\left[\frac{315 \, a^{\frac{5}{2}} \cos\left(d x + c\right)^{4} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) - 4 \, {\left(493 \, a^{2} \cos\left(d x + c\right)^{4} + 226 \, a^{2} \cos\left(d x + c\right)^{3} - 12 \, a^{2} \cos\left(d x + c\right)^{2} - 95 \, a^{2} \cos\left(d x + c\right) - 35 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{630 \, d \cos\left(d x + c\right)^{4}}, -\frac{315 \, \sqrt{-a} a^{2} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right)^{4} + 2 \, {\left(493 \, a^{2} \cos\left(d x + c\right)^{4} + 226 \, a^{2} \cos\left(d x + c\right)^{3} - 12 \, a^{2} \cos\left(d x + c\right)^{2} - 95 \, a^{2} \cos\left(d x + c\right) - 35 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{315 \, d \cos\left(d x + c\right)^{4}}\right]"," ",0,"[1/630*(315*a^(5/2)*cos(d*x + c)^4*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(493*a^2*cos(d*x + c)^4 + 226*a^2*cos(d*x + c)^3 - 12*a^2*cos(d*x + c)^2 - 95*a^2*cos(d*x + c) - 35*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^4), -1/315*(315*sqrt(-a)*a^2*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c)^4 + 2*(493*a^2*cos(d*x + c)^4 + 226*a^2*cos(d*x + c)^3 - 12*a^2*cos(d*x + c)^2 - 95*a^2*cos(d*x + c) - 35*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^4)]","A",0
161,1,282,0,1.256691," ","integrate((a+a*sec(d*x+c))^(5/2)*tan(d*x+c),x, algorithm=""fricas"")","\left[\frac{15 \, a^{\frac{5}{2}} \cos\left(d x + c\right)^{2} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, {\left(23 \, a^{2} \cos\left(d x + c\right)^{2} + 11 \, a^{2} \cos\left(d x + c\right) + 3 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{30 \, d \cos\left(d x + c\right)^{2}}, \frac{15 \, \sqrt{-a} a^{2} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right)^{2} + 2 \, {\left(23 \, a^{2} \cos\left(d x + c\right)^{2} + 11 \, a^{2} \cos\left(d x + c\right) + 3 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{15 \, d \cos\left(d x + c\right)^{2}}\right]"," ",0,"[1/30*(15*a^(5/2)*cos(d*x + c)^2*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(23*a^2*cos(d*x + c)^2 + 11*a^2*cos(d*x + c) + 3*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^2), 1/15*(15*sqrt(-a)*a^2*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c)^2 + 2*(23*a^2*cos(d*x + c)^2 + 11*a^2*cos(d*x + c) + 3*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^2)]","A",0
162,1,300,0,0.600485," ","integrate(cot(d*x+c)*(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{2} a^{\frac{5}{2}} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) - 1}\right) + a^{\frac{5}{2}} \log\left(-2 \, a \cos\left(d x + c\right) - 2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - a\right) + 2 \, a^{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{d}, \frac{2 \, {\left(2 \, \sqrt{2} \sqrt{-a} a^{2} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - \sqrt{-a} a^{2} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) + a^{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}\right)}}{d}\right]"," ",0,"[(2*sqrt(2)*a^(5/2)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + a^(5/2)*log(-2*a*cos(d*x + c) - 2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - a) + 2*a^2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/d, 2*(2*sqrt(2)*sqrt(-a)*a^2*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - sqrt(-a)*a^2*arctan(sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) + a^2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/d]","A",0
163,1,398,0,0.763370," ","integrate(cot(d*x+c)^3*(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{4 \, a^{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + 4 \, {\left(a^{2} \cos\left(d x + c\right) - a^{2}\right)} \sqrt{a} \log\left(-2 \, a \cos\left(d x + c\right) + 2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - a\right) + 3 \, {\left(\sqrt{2} a^{2} \cos\left(d x + c\right) - \sqrt{2} a^{2}\right)} \sqrt{a} \log\left(\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + 3 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) - 1}\right)}{4 \, {\left(d \cos\left(d x + c\right) - d\right)}}, \frac{2 \, a^{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, {\left(\sqrt{2} a^{2} \cos\left(d x + c\right) - \sqrt{2} a^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) + 4 \, {\left(a^{2} \cos\left(d x + c\right) - a^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right)}{2 \, {\left(d \cos\left(d x + c\right) - d\right)}}\right]"," ",0,"[1/4*(4*a^2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) + 4*(a^2*cos(d*x + c) - a^2)*sqrt(a)*log(-2*a*cos(d*x + c) + 2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - a) + 3*(sqrt(2)*a^2*cos(d*x + c) - sqrt(2)*a^2)*sqrt(a)*log((2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) + 3*a*cos(d*x + c) + a)/(cos(d*x + c) - 1)))/(d*cos(d*x + c) - d), 1/2*(2*a^2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*(sqrt(2)*a^2*cos(d*x + c) - sqrt(2)*a^2)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) + 4*(a^2*cos(d*x + c) - a^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)))/(d*cos(d*x + c) - d)]","B",0
164,1,503,0,0.586885," ","integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{64 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \cos\left(d x + c\right) + a^{2}\right)} \sqrt{a} \log\left(-2 \, a \cos\left(d x + c\right) - 2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - a\right) + 43 \, {\left(\sqrt{2} a^{2} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} a^{2} \cos\left(d x + c\right) + \sqrt{2} a^{2}\right)} \sqrt{a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) - 1}\right) - 4 \, {\left(15 \, a^{2} \cos\left(d x + c\right)^{2} - 11 \, a^{2} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{64 \, {\left(d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) + d\right)}}, \frac{43 \, {\left(\sqrt{2} a^{2} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} a^{2} \cos\left(d x + c\right) + \sqrt{2} a^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 64 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \cos\left(d x + c\right) + a^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(15 \, a^{2} \cos\left(d x + c\right)^{2} - 11 \, a^{2} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{32 \, {\left(d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) + d\right)}}\right]"," ",0,"[1/64*(64*(a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*sqrt(a)*log(-2*a*cos(d*x + c) - 2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - a) + 43*(sqrt(2)*a^2*cos(d*x + c)^2 - 2*sqrt(2)*a^2*cos(d*x + c) + sqrt(2)*a^2)*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) - 4*(15*a^2*cos(d*x + c)^2 - 11*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d), 1/32*(43*(sqrt(2)*a^2*cos(d*x + c)^2 - 2*sqrt(2)*a^2*cos(d*x + c) + sqrt(2)*a^2)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 64*(a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 2*(15*a^2*cos(d*x + c)^2 - 11*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)]","B",0
165,1,477,0,0.799954," ","integrate((a+a*sec(d*x+c))^(5/2)*tan(d*x+c)^6,x, algorithm=""fricas"")","\left[\frac{45045 \, {\left(a^{2} \cos\left(d x + c\right)^{8} + a^{2} \cos\left(d x + c\right)^{7}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) + 2 \, {\left(16066 \, a^{2} \cos\left(d x + c\right)^{7} + 53078 \, a^{2} \cos\left(d x + c\right)^{6} + 17286 \, a^{2} \cos\left(d x + c\right)^{5} - 30640 \, a^{2} \cos\left(d x + c\right)^{4} - 26810 \, a^{2} \cos\left(d x + c\right)^{3} + 2898 \, a^{2} \cos\left(d x + c\right)^{2} + 10164 \, a^{2} \cos\left(d x + c\right) + 3003 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{45045 \, {\left(d \cos\left(d x + c\right)^{8} + d \cos\left(d x + c\right)^{7}\right)}}, \frac{2 \, {\left(45045 \, {\left(a^{2} \cos\left(d x + c\right)^{8} + a^{2} \cos\left(d x + c\right)^{7}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + {\left(16066 \, a^{2} \cos\left(d x + c\right)^{7} + 53078 \, a^{2} \cos\left(d x + c\right)^{6} + 17286 \, a^{2} \cos\left(d x + c\right)^{5} - 30640 \, a^{2} \cos\left(d x + c\right)^{4} - 26810 \, a^{2} \cos\left(d x + c\right)^{3} + 2898 \, a^{2} \cos\left(d x + c\right)^{2} + 10164 \, a^{2} \cos\left(d x + c\right) + 3003 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{45045 \, {\left(d \cos\left(d x + c\right)^{8} + d \cos\left(d x + c\right)^{7}\right)}}\right]"," ",0,"[1/45045*(45045*(a^2*cos(d*x + c)^8 + a^2*cos(d*x + c)^7)*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*(16066*a^2*cos(d*x + c)^7 + 53078*a^2*cos(d*x + c)^6 + 17286*a^2*cos(d*x + c)^5 - 30640*a^2*cos(d*x + c)^4 - 26810*a^2*cos(d*x + c)^3 + 2898*a^2*cos(d*x + c)^2 + 10164*a^2*cos(d*x + c) + 3003*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^8 + d*cos(d*x + c)^7), 2/45045*(45045*(a^2*cos(d*x + c)^8 + a^2*cos(d*x + c)^7)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (16066*a^2*cos(d*x + c)^7 + 53078*a^2*cos(d*x + c)^6 + 17286*a^2*cos(d*x + c)^5 - 30640*a^2*cos(d*x + c)^4 - 26810*a^2*cos(d*x + c)^3 + 2898*a^2*cos(d*x + c)^2 + 10164*a^2*cos(d*x + c) + 3003*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^8 + d*cos(d*x + c)^7)]","A",0
166,1,425,0,0.496128," ","integrate((a+a*sec(d*x+c))^(5/2)*tan(d*x+c)^4,x, algorithm=""fricas"")","\left[\frac{693 \, {\left(a^{2} \cos\left(d x + c\right)^{6} + a^{2} \cos\left(d x + c\right)^{5}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) - 2 \, {\left(52 \, a^{2} \cos\left(d x + c\right)^{5} + 719 \, a^{2} \cos\left(d x + c\right)^{4} + 366 \, a^{2} \cos\left(d x + c\right)^{3} - 157 \, a^{2} \cos\left(d x + c\right)^{2} - 224 \, a^{2} \cos\left(d x + c\right) - 63 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{693 \, {\left(d \cos\left(d x + c\right)^{6} + d \cos\left(d x + c\right)^{5}\right)}}, -\frac{2 \, {\left(693 \, {\left(a^{2} \cos\left(d x + c\right)^{6} + a^{2} \cos\left(d x + c\right)^{5}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + {\left(52 \, a^{2} \cos\left(d x + c\right)^{5} + 719 \, a^{2} \cos\left(d x + c\right)^{4} + 366 \, a^{2} \cos\left(d x + c\right)^{3} - 157 \, a^{2} \cos\left(d x + c\right)^{2} - 224 \, a^{2} \cos\left(d x + c\right) - 63 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{693 \, {\left(d \cos\left(d x + c\right)^{6} + d \cos\left(d x + c\right)^{5}\right)}}\right]"," ",0,"[1/693*(693*(a^2*cos(d*x + c)^6 + a^2*cos(d*x + c)^5)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) - 2*(52*a^2*cos(d*x + c)^5 + 719*a^2*cos(d*x + c)^4 + 366*a^2*cos(d*x + c)^3 - 157*a^2*cos(d*x + c)^2 - 224*a^2*cos(d*x + c) - 63*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5), -2/693*(693*(a^2*cos(d*x + c)^6 + a^2*cos(d*x + c)^5)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (52*a^2*cos(d*x + c)^5 + 719*a^2*cos(d*x + c)^4 + 366*a^2*cos(d*x + c)^3 - 157*a^2*cos(d*x + c)^2 - 224*a^2*cos(d*x + c) - 63*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5)]","A",0
167,1,374,0,0.650683," ","integrate((a+a*sec(d*x+c))^(5/2)*tan(d*x+c)^2,x, algorithm=""fricas"")","\left[\frac{21 \, {\left(a^{2} \cos\left(d x + c\right)^{4} + a^{2} \cos\left(d x + c\right)^{3}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) - 2 \, {\left(10 \, a^{2} \cos\left(d x + c\right)^{3} - 16 \, a^{2} \cos\left(d x + c\right)^{2} - 12 \, a^{2} \cos\left(d x + c\right) - 3 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{21 \, {\left(d \cos\left(d x + c\right)^{4} + d \cos\left(d x + c\right)^{3}\right)}}, \frac{2 \, {\left(21 \, {\left(a^{2} \cos\left(d x + c\right)^{4} + a^{2} \cos\left(d x + c\right)^{3}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) - {\left(10 \, a^{2} \cos\left(d x + c\right)^{3} - 16 \, a^{2} \cos\left(d x + c\right)^{2} - 12 \, a^{2} \cos\left(d x + c\right) - 3 \, a^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{21 \, {\left(d \cos\left(d x + c\right)^{4} + d \cos\left(d x + c\right)^{3}\right)}}\right]"," ",0,"[1/21*(21*(a^2*cos(d*x + c)^4 + a^2*cos(d*x + c)^3)*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) - 2*(10*a^2*cos(d*x + c)^3 - 16*a^2*cos(d*x + c)^2 - 12*a^2*cos(d*x + c) - 3*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 + d*cos(d*x + c)^3), 2/21*(21*(a^2*cos(d*x + c)^4 + a^2*cos(d*x + c)^3)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) - (10*a^2*cos(d*x + c)^3 - 16*a^2*cos(d*x + c)^2 - 12*a^2*cos(d*x + c) - 3*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 + d*cos(d*x + c)^3)]","A",0
168,1,270,0,0.707257," ","integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} a^{2} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) - 8 \, a^{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, d \sin\left(d x + c\right)}, -\frac{a^{\frac{5}{2}} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 4 \, a^{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{d \sin\left(d x + c\right)}\right]"," ",0,"[1/2*(sqrt(-a)*a^2*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 8*a^2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(d*sin(d*x + c)), -(a^(5/2)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 4*a^2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(d*sin(d*x + c))]","B",0
169,1,355,0,0.635589," ","integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} \cos\left(d x + c\right) - a^{2}\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 4 \, {\left(4 \, a^{2} \cos\left(d x + c\right)^{2} - 3 \, a^{2} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{6 \, {\left(d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right)}, \frac{3 \, {\left(a^{2} \cos\left(d x + c\right) - a^{2}\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, {\left(4 \, a^{2} \cos\left(d x + c\right)^{2} - 3 \, a^{2} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{3 \, {\left(d \cos\left(d x + c\right) - d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[1/6*(3*(a^2*cos(d*x + c) - a^2)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 - 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) + 4*(4*a^2*cos(d*x + c)^2 - 3*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c) - d)*sin(d*x + c)), 1/3*(3*(a^2*cos(d*x + c) - a^2)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(4*a^2*cos(d*x + c)^2 - 3*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c) - d)*sin(d*x + c))]","A",0
170,1,650,0,0.864789," ","integrate(cot(d*x+c)^6*(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{5 \, {\left(\sqrt{2} a^{2} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} a^{2} \cos\left(d x + c\right) + \sqrt{2} a^{2}\right)} \sqrt{-a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 3 \, a \cos\left(d x + c\right)^{2} - 2 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 40 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \cos\left(d x + c\right) + a^{2}\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) - 4 \, {\left(49 \, a^{2} \cos\left(d x + c\right)^{3} - 80 \, a^{2} \cos\left(d x + c\right)^{2} + 35 \, a^{2} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{80 \, {\left(d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right)}, -\frac{40 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - 2 \, a^{2} \cos\left(d x + c\right) + a^{2}\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 5 \, {\left(\sqrt{2} a^{2} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{2} a^{2} \cos\left(d x + c\right) + \sqrt{2} a^{2}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 2 \, {\left(49 \, a^{2} \cos\left(d x + c\right)^{3} - 80 \, a^{2} \cos\left(d x + c\right)^{2} + 35 \, a^{2} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{40 \, {\left(d \cos\left(d x + c\right)^{2} - 2 \, d \cos\left(d x + c\right) + d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[1/80*(5*(sqrt(2)*a^2*cos(d*x + c)^2 - 2*sqrt(2)*a^2*cos(d*x + c) + sqrt(2)*a^2)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c) + a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 40*(a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*(49*a^2*cos(d*x + c)^3 - 80*a^2*cos(d*x + c)^2 + 35*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)*sin(d*x + c)), -1/40*(40*(a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 5*(sqrt(2)*a^2*cos(d*x + c)^2 - 2*sqrt(2)*a^2*cos(d*x + c) + sqrt(2)*a^2)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 2*(49*a^2*cos(d*x + c)^3 - 80*a^2*cos(d*x + c)^2 + 35*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)*sin(d*x + c))]","A",0
171,1,285,0,0.791640," ","integrate(tan(d*x+c)^5/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{105 \, \sqrt{a} \cos\left(d x + c\right)^{3} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, {\left(92 \, \cos\left(d x + c\right)^{3} - 46 \, \cos\left(d x + c\right)^{2} - 18 \, \cos\left(d x + c\right) + 15\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{210 \, a d \cos\left(d x + c\right)^{3}}, \frac{105 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right)^{3} + 2 \, {\left(92 \, \cos\left(d x + c\right)^{3} - 46 \, \cos\left(d x + c\right)^{2} - 18 \, \cos\left(d x + c\right) + 15\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{105 \, a d \cos\left(d x + c\right)^{3}}\right]"," ",0,"[1/210*(105*sqrt(a)*cos(d*x + c)^3*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(92*cos(d*x + c)^3 - 46*cos(d*x + c)^2 - 18*cos(d*x + c) + 15)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a*d*cos(d*x + c)^3), 1/105*(105*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c)^3 + 2*(92*cos(d*x + c)^3 - 46*cos(d*x + c)^2 - 18*cos(d*x + c) + 15)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a*d*cos(d*x + c)^3)]","A",0
172,1,241,0,0.527701," ","integrate(tan(d*x+c)^3/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a} \cos\left(d x + c\right) \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) - 4 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} {\left(2 \, \cos\left(d x + c\right) - 1\right)}}{6 \, a d \cos\left(d x + c\right)}, -\frac{3 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right) + 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} {\left(2 \, \cos\left(d x + c\right) - 1\right)}}{3 \, a d \cos\left(d x + c\right)}\right]"," ",0,"[1/6*(3*sqrt(a)*cos(d*x + c)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(2*cos(d*x + c) - 1))/(a*d*cos(d*x + c)), -1/3*(3*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c) + 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(2*cos(d*x + c) - 1))/(a*d*cos(d*x + c))]","A",0
173,1,137,0,0.467930," ","integrate(tan(d*x+c)/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right)}{2 \, \sqrt{a} d}, \frac{\sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right)}{a d}\right]"," ",0,"[1/2*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a)/(sqrt(a)*d), sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))/(a*d)]","B",0
174,1,384,0,0.729970," ","integrate(cot(d*x+c)/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{a} {\left(\cos\left(d x + c\right) + 1\right)} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + \frac{\sqrt{2} {\left(a \cos\left(d x + c\right) + a\right)} \log\left(-\frac{\frac{2 \, \sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a}} - 3 \, \cos\left(d x + c\right) - 1}{\cos\left(d x + c\right) - 1}\right)}{\sqrt{a}} - 4 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{4 \, {\left(a d \cos\left(d x + c\right) + a d\right)}}, \frac{\sqrt{2} {\left(a \cos\left(d x + c\right) + a\right)} \sqrt{-\frac{1}{a}} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sqrt{-\frac{1}{a}} \cos\left(d x + c\right)}{\cos\left(d x + c\right) + 1}\right) - 2 \, \sqrt{-a} {\left(\cos\left(d x + c\right) + 1\right)} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, {\left(a d \cos\left(d x + c\right) + a d\right)}}\right]"," ",0,"[1/4*(2*sqrt(a)*(cos(d*x + c) + 1)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + sqrt(2)*(a*cos(d*x + c) + a)*log(-(2*sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/sqrt(a) - 3*cos(d*x + c) - 1)/(cos(d*x + c) - 1))/sqrt(a) - 4*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(a*d*cos(d*x + c) + a*d), 1/2*(sqrt(2)*(a*cos(d*x + c) + a)*sqrt(-1/a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(-1/a)*cos(d*x + c)/(cos(d*x + c) + 1)) - 2*sqrt(-a)*(cos(d*x + c) + 1)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(a*d*cos(d*x + c) + a*d)]","B",0
175,1,546,0,0.592989," ","integrate(cot(d*x+c)^3/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{27 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 1\right)} \sqrt{a} \log\left(\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + 3 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) - 1}\right) + 48 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 1\right)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, {\left(31 \, \cos\left(d x + c\right)^{3} + 2 \, \cos\left(d x + c\right)^{2} - 21 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{96 \, {\left(a d \cos\left(d x + c\right)^{3} + a d \cos\left(d x + c\right)^{2} - a d \cos\left(d x + c\right) - a d\right)}}, -\frac{27 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 48 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(31 \, \cos\left(d x + c\right)^{3} + 2 \, \cos\left(d x + c\right)^{2} - 21 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{48 \, {\left(a d \cos\left(d x + c\right)^{3} + a d \cos\left(d x + c\right)^{2} - a d \cos\left(d x + c\right) - a d\right)}}\right]"," ",0,"[1/96*(27*sqrt(2)*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sqrt(a)*log((2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) + 3*a*cos(d*x + c) + a)/(cos(d*x + c) - 1)) + 48*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(31*cos(d*x + c)^3 + 2*cos(d*x + c)^2 - 21*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)^2 - a*d*cos(d*x + c) - a*d), -1/48*(27*sqrt(2)*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 48*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(31*cos(d*x + c)^3 + 2*cos(d*x + c)^2 - 21*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)^2 - a*d*cos(d*x + c) - a*d)]","B",0
176,1,705,0,0.678078," ","integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{2265 \, \sqrt{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)} \sqrt{a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) - 1}\right) + 3840 \, {\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)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) - 4 \, {\left(2821 \, \cos\left(d x + c\right)^{5} + 278 \, \cos\left(d x + c\right)^{4} - 3964 \, \cos\left(d x + c\right)^{3} - 230 \, \cos\left(d x + c\right)^{2} + 1575 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{7680 \, {\left(a d \cos\left(d x + c\right)^{5} + a d \cos\left(d x + c\right)^{4} - 2 \, a d \cos\left(d x + c\right)^{3} - 2 \, a d \cos\left(d x + c\right)^{2} + a d \cos\left(d x + c\right) + a d\right)}}, \frac{2265 \, \sqrt{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)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 3840 \, {\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)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(2821 \, \cos\left(d x + c\right)^{5} + 278 \, \cos\left(d x + c\right)^{4} - 3964 \, \cos\left(d x + c\right)^{3} - 230 \, \cos\left(d x + c\right)^{2} + 1575 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{3840 \, {\left(a d \cos\left(d x + c\right)^{5} + a d \cos\left(d x + c\right)^{4} - 2 \, a d \cos\left(d x + c\right)^{3} - 2 \, a d \cos\left(d x + c\right)^{2} + a d \cos\left(d x + c\right) + a d\right)}}\right]"," ",0,"[1/7680*(2265*sqrt(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)*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + 3840*(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)*sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(2821*cos(d*x + c)^5 + 278*cos(d*x + c)^4 - 3964*cos(d*x + c)^3 - 230*cos(d*x + c)^2 + 1575*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d), 1/3840*(2265*sqrt(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)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 3840*(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)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(2821*cos(d*x + c)^5 + 278*cos(d*x + c)^4 - 3964*cos(d*x + c)^3 - 230*cos(d*x + c)^2 + 1575*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d)]","A",0
177,1,355,0,0.746392," ","integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[-\frac{315 \, {\left(\cos\left(d x + c\right)^{5} + \cos\left(d x + c\right)^{4}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) - 2 \, {\left(383 \, \cos\left(d x + c\right)^{4} + 34 \, \cos\left(d x + c\right)^{3} - 132 \, \cos\left(d x + c\right)^{2} - 5 \, \cos\left(d x + c\right) + 35\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{315 \, {\left(a d \cos\left(d x + c\right)^{5} + a d \cos\left(d x + c\right)^{4}\right)}}, \frac{2 \, {\left(315 \, {\left(\cos\left(d x + c\right)^{5} + \cos\left(d x + c\right)^{4}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + {\left(383 \, \cos\left(d x + c\right)^{4} + 34 \, \cos\left(d x + c\right)^{3} - 132 \, \cos\left(d x + c\right)^{2} - 5 \, \cos\left(d x + c\right) + 35\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{315 \, {\left(a d \cos\left(d x + c\right)^{5} + a d \cos\left(d x + c\right)^{4}\right)}}\right]"," ",0,"[-1/315*(315*(cos(d*x + c)^5 + cos(d*x + c)^4)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) - 2*(383*cos(d*x + c)^4 + 34*cos(d*x + c)^3 - 132*cos(d*x + c)^2 - 5*cos(d*x + c) + 35)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4), 2/315*(315*(cos(d*x + c)^5 + cos(d*x + c)^4)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (383*cos(d*x + c)^4 + 34*cos(d*x + c)^3 - 132*cos(d*x + c)^2 - 5*cos(d*x + c) + 35)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4)]","A",0
178,1,311,0,0.839671," ","integrate(tan(d*x+c)^4/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) + 2 \, {\left(17 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right) - 3\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{15 \, {\left(a d \cos\left(d x + c\right)^{3} + a d \cos\left(d x + c\right)^{2}\right)}}, -\frac{2 \, {\left(15 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + {\left(17 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right) - 3\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{15 \, {\left(a d \cos\left(d x + c\right)^{3} + a d \cos\left(d x + c\right)^{2}\right)}}\right]"," ",0,"[-1/15*(15*(cos(d*x + c)^3 + cos(d*x + c)^2)*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*(17*cos(d*x + c)^2 + cos(d*x + c) - 3)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)^2), -2/15*(15*(cos(d*x + c)^3 + cos(d*x + c)^2)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (17*cos(d*x + c)^2 + cos(d*x + c) - 3)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)^2)]","A",0
179,1,235,0,0.815703," ","integrate(tan(d*x+c)^2/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a} {\left(\cos\left(d x + c\right) + 1\right)} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) - 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{a d \cos\left(d x + c\right) + a d}, \frac{2 \, {\left(\sqrt{a} {\left(\cos\left(d x + c\right) + 1\right)} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{a d \cos\left(d x + c\right) + a d}\right]"," ",0,"[-(sqrt(-a)*(cos(d*x + c) + 1)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) - 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a*d*cos(d*x + c) + a*d), 2*(sqrt(a)*(cos(d*x + c) + 1)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a*d*cos(d*x + c) + a*d)]","A",0
180,1,503,0,0.575861," ","integrate(cot(d*x+c)^2/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[-\frac{7 \, \sqrt{2} \sqrt{-a} {\left(\cos\left(d x + c\right) + 1\right)} \log\left(\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 3 \, a \cos\left(d x + c\right)^{2} + 2 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 8 \, \sqrt{-a} {\left(\cos\left(d x + c\right) + 1\right)} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 4 \, {\left(3 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{16 \, {\left(a d \cos\left(d x + c\right) + a d\right)} \sin\left(d x + c\right)}, -\frac{7 \, \sqrt{2} \sqrt{a} {\left(\cos\left(d x + c\right) + 1\right)} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 8 \, \sqrt{a} {\left(\cos\left(d x + c\right) + 1\right)} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, {\left(3 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{8 \, {\left(a d \cos\left(d x + c\right) + a d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/16*(7*sqrt(2)*sqrt(-a)*(cos(d*x + c) + 1)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + 3*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 8*sqrt(-a)*(cos(d*x + c) + 1)*log(-(8*a*cos(d*x + c)^3 - 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) + 4*(3*cos(d*x + c)^2 + cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a*d*cos(d*x + c) + a*d)*sin(d*x + c)), -1/8*(7*sqrt(2)*sqrt(a)*(cos(d*x + c) + 1)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 8*sqrt(a)*(cos(d*x + c) + 1)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(3*cos(d*x + c)^2 + cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a*d*cos(d*x + c) + a*d)*sin(d*x + c))]","A",0
181,1,666,0,2.152594," ","integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[-\frac{321 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 3 \, a \cos\left(d x + c\right)^{2} - 2 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 384 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) - 4 \, {\left(205 \, \cos\left(d x + c\right)^{4} + 71 \, \cos\left(d x + c\right)^{3} - 149 \, \cos\left(d x + c\right)^{2} - 63 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{768 \, {\left(a d \cos\left(d x + c\right)^{3} + a d \cos\left(d x + c\right)^{2} - a d \cos\left(d x + c\right) - a d\right)} \sin\left(d x + c\right)}, \frac{321 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 1\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 384 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right) - 1\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, {\left(205 \, \cos\left(d x + c\right)^{4} + 71 \, \cos\left(d x + c\right)^{3} - 149 \, \cos\left(d x + c\right)^{2} - 63 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{384 \, {\left(a d \cos\left(d x + c\right)^{3} + a d \cos\left(d x + c\right)^{2} - a d \cos\left(d x + c\right) - a d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/768*(321*sqrt(2)*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c) + a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 384*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*(205*cos(d*x + c)^4 + 71*cos(d*x + c)^3 - 149*cos(d*x + c)^2 - 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)^2 - a*d*cos(d*x + c) - a*d)*sin(d*x + c)), 1/384*(321*sqrt(2)*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 384*(cos(d*x + c)^3 + cos(d*x + c)^2 - cos(d*x + c) - 1)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(205*cos(d*x + c)^4 + 71*cos(d*x + c)^3 - 149*cos(d*x + c)^2 - 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a*d*cos(d*x + c)^3 + a*d*cos(d*x + c)^2 - a*d*cos(d*x + c) - a*d)*sin(d*x + c))]","A",0
182,1,823,0,0.914736," ","integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[-\frac{12525 \, \sqrt{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)} \sqrt{-a} \log\left(\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 3 \, a \cos\left(d x + c\right)^{2} + 2 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 15360 \, {\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)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 4 \, {\left(9737 \, \cos\left(d x + c\right)^{6} + 3451 \, \cos\left(d x + c\right)^{5} - 14394 \, \cos\left(d x + c\right)^{4} - 6158 \, \cos\left(d x + c\right)^{3} + 6065 \, \cos\left(d x + c\right)^{2} + 2835 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{30720 \, {\left(a d \cos\left(d x + c\right)^{5} + a d \cos\left(d x + c\right)^{4} - 2 \, a d \cos\left(d x + c\right)^{3} - 2 \, a d \cos\left(d x + c\right)^{2} + a d \cos\left(d x + c\right) + a d\right)} \sin\left(d x + c\right)}, -\frac{12525 \, \sqrt{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)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 15360 \, {\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)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, {\left(9737 \, \cos\left(d x + c\right)^{6} + 3451 \, \cos\left(d x + c\right)^{5} - 14394 \, \cos\left(d x + c\right)^{4} - 6158 \, \cos\left(d x + c\right)^{3} + 6065 \, \cos\left(d x + c\right)^{2} + 2835 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{15360 \, {\left(a d \cos\left(d x + c\right)^{5} + a d \cos\left(d x + c\right)^{4} - 2 \, a d \cos\left(d x + c\right)^{3} - 2 \, a d \cos\left(d x + c\right)^{2} + a d \cos\left(d x + c\right) + a d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/30720*(12525*sqrt(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)*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + 3*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 15360*(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)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 - 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) + 4*(9737*cos(d*x + c)^6 + 3451*cos(d*x + c)^5 - 14394*cos(d*x + c)^4 - 6158*cos(d*x + c)^3 + 6065*cos(d*x + c)^2 + 2835*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d)*sin(d*x + c)), -1/15360*(12525*sqrt(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)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 15360*(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)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(9737*cos(d*x + c)^6 + 3451*cos(d*x + c)^5 - 14394*cos(d*x + c)^4 - 6158*cos(d*x + c)^3 + 6065*cos(d*x + c)^2 + 2835*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d)*sin(d*x + c))]","A",0
183,1,261,0,0.591867," ","integrate(tan(d*x+c)^5/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{5 \, \sqrt{a} \cos\left(d x + c\right)^{2} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, {\left(\cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{10 \, a^{2} d \cos\left(d x + c\right)^{2}}, \frac{5 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right)^{2} + 2 \, {\left(\cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{5 \, a^{2} d \cos\left(d x + c\right)^{2}}\right]"," ",0,"[1/10*(5*sqrt(a)*cos(d*x + c)^2*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(cos(d*x + c)^2 - 3*cos(d*x + c) + 1)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^2), 1/5*(5*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c)^2 + 2*(cos(d*x + c)^2 - 3*cos(d*x + c) + 1)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^2)]","A",0
184,1,191,0,0.531101," ","integrate(tan(d*x+c)^3/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{2 \, a^{2} d}, -\frac{\sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{a^{2} d}\right]"," ",0,"[1/2*(sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d), -(sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d)]","A",0
185,1,244,0,0.773765," ","integrate(tan(d*x+c)/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} {\left(\cos\left(d x + c\right) + 1\right)} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, {\left(a^{2} d \cos\left(d x + c\right) + a^{2} d\right)}}, \frac{\sqrt{-a} {\left(\cos\left(d x + c\right) + 1\right)} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) + 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a^{2} d \cos\left(d x + c\right) + a^{2} d}\right]"," ",0,"[1/2*(sqrt(a)*(cos(d*x + c) + 1)*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(a^2*d*cos(d*x + c) + a^2*d), (sqrt(-a)*(cos(d*x + c) + 1)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) + 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(a^2*d*cos(d*x + c) + a^2*d)]","B",0
186,1,485,0,0.706263," ","integrate(cot(d*x+c)/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) - 1}\right) + 12 \, {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) - 4 \, {\left(11 \, \cos\left(d x + c\right)^{2} + 9 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{24 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)}}, \frac{3 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 12 \, {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(11 \, \cos\left(d x + c\right)^{2} + 9 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{12 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)}}\right]"," ",0,"[1/24*(3*sqrt(2)*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + 12*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(11*cos(d*x + c)^2 + 9*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d), 1/12*(3*sqrt(2)*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 12*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(11*cos(d*x + c)^2 + 9*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)]","B",0
187,1,592,0,0.640655," ","integrate(cot(d*x+c)^3/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{165 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right) - 1\right)} \sqrt{a} \log\left(\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + 3 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) - 1}\right) + 480 \, {\left(\cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right) - 1\right)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, {\left(449 \, \cos\left(d x + c\right)^{4} + 351 \, \cos\left(d x + c\right)^{3} - 365 \, \cos\left(d x + c\right)^{2} - 315 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{960 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} + 2 \, a^{2} d \cos\left(d x + c\right)^{3} - 2 \, a^{2} d \cos\left(d x + c\right) - a^{2} d\right)}}, -\frac{165 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 480 \, {\left(\cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(449 \, \cos\left(d x + c\right)^{4} + 351 \, \cos\left(d x + c\right)^{3} - 365 \, \cos\left(d x + c\right)^{2} - 315 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{480 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} + 2 \, a^{2} d \cos\left(d x + c\right)^{3} - 2 \, a^{2} d \cos\left(d x + c\right) - a^{2} d\right)}}\right]"," ",0,"[1/960*(165*sqrt(2)*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(a)*log((2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) + 3*a*cos(d*x + c) + a)/(cos(d*x + c) - 1)) + 480*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(449*cos(d*x + c)^4 + 351*cos(d*x + c)^3 - 365*cos(d*x + c)^2 - 315*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x + c) - a^2*d), -1/480*(165*sqrt(2)*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 480*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(449*cos(d*x + c)^4 + 351*cos(d*x + c)^3 - 365*cos(d*x + c)^2 - 315*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x + c) - a^2*d)]","B",0
188,1,837,0,0.639184," ","integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{4263 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{6} + 2 \, \cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) - 1}\right) + 10752 \, {\left(\cos\left(d x + c\right)^{6} + 2 \, \cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) - 4 \, {\left(10363 \, \cos\left(d x + c\right)^{6} + 8037 \, \cos\left(d x + c\right)^{5} - 16538 \, \cos\left(d x + c\right)^{4} - 14238 \, \cos\left(d x + c\right)^{3} + 7231 \, \cos\left(d x + c\right)^{2} + 6489 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{21504 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} + 2 \, a^{2} d \cos\left(d x + c\right)^{5} - a^{2} d \cos\left(d x + c\right)^{4} - 4 \, a^{2} d \cos\left(d x + c\right)^{3} - a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)}}, \frac{4263 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{6} + 2 \, \cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 10752 \, {\left(\cos\left(d x + c\right)^{6} + 2 \, \cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(10363 \, \cos\left(d x + c\right)^{6} + 8037 \, \cos\left(d x + c\right)^{5} - 16538 \, \cos\left(d x + c\right)^{4} - 14238 \, \cos\left(d x + c\right)^{3} + 7231 \, \cos\left(d x + c\right)^{2} + 6489 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{10752 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} + 2 \, a^{2} d \cos\left(d x + c\right)^{5} - a^{2} d \cos\left(d x + c\right)^{4} - 4 \, a^{2} d \cos\left(d x + c\right)^{3} - a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)}}\right]"," ",0,"[1/21504*(4263*sqrt(2)*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + 10752*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(10363*cos(d*x + c)^6 + 8037*cos(d*x + c)^5 - 16538*cos(d*x + c)^4 - 14238*cos(d*x + c)^3 + 7231*cos(d*x + c)^2 + 6489*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d), 1/10752*(4263*sqrt(2)*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 10752*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(10363*cos(d*x + c)^6 + 8037*cos(d*x + c)^5 - 16538*cos(d*x + c)^4 - 14238*cos(d*x + c)^3 + 7231*cos(d*x + c)^2 + 6489*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)]","B",0
189,1,343,0,0.584843," ","integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[-\frac{105 \, {\left(\cos\left(d x + c\right)^{4} + \cos\left(d x + c\right)^{3}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) - 2 \, {\left(146 \, \cos\left(d x + c\right)^{3} - 32 \, \cos\left(d x + c\right)^{2} - 24 \, \cos\left(d x + c\right) + 15\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{105 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} + a^{2} d \cos\left(d x + c\right)^{3}\right)}}, \frac{2 \, {\left(105 \, {\left(\cos\left(d x + c\right)^{4} + \cos\left(d x + c\right)^{3}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + {\left(146 \, \cos\left(d x + c\right)^{3} - 32 \, \cos\left(d x + c\right)^{2} - 24 \, \cos\left(d x + c\right) + 15\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{105 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} + a^{2} d \cos\left(d x + c\right)^{3}\right)}}\right]"," ",0,"[-1/105*(105*(cos(d*x + c)^4 + cos(d*x + c)^3)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) - 2*(146*cos(d*x + c)^3 - 32*cos(d*x + c)^2 - 24*cos(d*x + c) + 15)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^2*d*cos(d*x + c)^4 + a^2*d*cos(d*x + c)^3), 2/105*(105*(cos(d*x + c)^4 + cos(d*x + c)^3)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (146*cos(d*x + c)^3 - 32*cos(d*x + c)^2 - 24*cos(d*x + c) + 15)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^2*d*cos(d*x + c)^4 + a^2*d*cos(d*x + c)^3)]","A",0
190,1,295,0,0.484132," ","integrate(tan(d*x+c)^4/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(\cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) + 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} {\left(4 \, \cos\left(d x + c\right) - 1\right)} \sin\left(d x + c\right)}{3 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d \cos\left(d x + c\right)\right)}}, -\frac{2 \, {\left(3 \, {\left(\cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} {\left(4 \, \cos\left(d x + c\right) - 1\right)} \sin\left(d x + c\right)\right)}}{3 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d \cos\left(d x + c\right)\right)}}\right]"," ",0,"[-1/3*(3*(cos(d*x + c)^2 + cos(d*x + c))*sqrt(-a)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(4*cos(d*x + c) - 1)*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 + a^2*d*cos(d*x + c)), -2/3*(3*(cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(4*cos(d*x + c) - 1)*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 + a^2*d*cos(d*x + c))]","A",0
191,1,295,0,0.632038," ","integrate(tan(d*x+c)^2/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} a \sqrt{-\frac{1}{a}} \log\left(-\frac{2 \, \sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sqrt{-\frac{1}{a}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 3 \, \cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) - \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right)}{a^{2} d}, -\frac{2 \, {\left(\sqrt{2} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) - \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right)\right)}}{a^{2} d}\right]"," ",0,"[(sqrt(2)*a*sqrt(-1/a)*log(-(2*sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(-1/a)*cos(d*x + c)*sin(d*x + c) - 3*cos(d*x + c)^2 - 2*cos(d*x + c) + 1)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) - sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)))/(a^2*d), -2*(sqrt(2)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) - sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))))/(a^2*d)]","A",0
192,1,603,0,0.664858," ","integrate(cot(d*x+c)^2/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[-\frac{71 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \log\left(\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 3 \, a \cos\left(d x + c\right)^{2} + 2 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 64 \, {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 4 \, {\left(27 \, \cos\left(d x + c\right)^{3} + 12 \, \cos\left(d x + c\right)^{2} - 7 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{128 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)} \sin\left(d x + c\right)}, -\frac{71 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 64 \, {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, {\left(27 \, \cos\left(d x + c\right)^{3} + 12 \, \cos\left(d x + c\right)^{2} - 7 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{64 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/128*(71*sqrt(2)*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + 3*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 64*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 - 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) + 4*(27*cos(d*x + c)^3 + 12*cos(d*x + c)^2 - 7*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)*sin(d*x + c)), -1/64*(71*sqrt(2)*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 64*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(27*cos(d*x + c)^3 + 12*cos(d*x + c)^2 - 7*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)*sin(d*x + c))]","A",0
193,1,712,0,0.902497," ","integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[-\frac{1599 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 3 \, a \cos\left(d x + c\right)^{2} - 2 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 1536 \, {\left(\cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) - 4 \, {\left(819 \, \cos\left(d x + c\right)^{5} + 492 \, \cos\left(d x + c\right)^{4} - 690 \, \cos\left(d x + c\right)^{3} - 428 \, \cos\left(d x + c\right)^{2} + 63 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{3072 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} + 2 \, a^{2} d \cos\left(d x + c\right)^{3} - 2 \, a^{2} d \cos\left(d x + c\right) - a^{2} d\right)} \sin\left(d x + c\right)}, \frac{1599 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right) - 1\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 1536 \, {\left(\cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right) - 1\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, {\left(819 \, \cos\left(d x + c\right)^{5} + 492 \, \cos\left(d x + c\right)^{4} - 690 \, \cos\left(d x + c\right)^{3} - 428 \, \cos\left(d x + c\right)^{2} + 63 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{1536 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} + 2 \, a^{2} d \cos\left(d x + c\right)^{3} - 2 \, a^{2} d \cos\left(d x + c\right) - a^{2} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/3072*(1599*sqrt(2)*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c) + a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 1536*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*(819*cos(d*x + c)^5 + 492*cos(d*x + c)^4 - 690*cos(d*x + c)^3 - 428*cos(d*x + c)^2 + 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^2*d*cos(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c)), 1/1536*(1599*sqrt(2)*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 1536*(cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c) - 1)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(819*cos(d*x + c)^5 + 492*cos(d*x + c)^4 - 690*cos(d*x + c)^3 - 428*cos(d*x + c)^2 + 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^2*d*cos(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c))]","A",0
194,1,955,0,0.669934," ","integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[-\frac{245445 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{6} + 2 \, \cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \log\left(\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 3 \, a \cos\left(d x + c\right)^{2} + 2 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 245760 \, {\left(\cos\left(d x + c\right)^{6} + 2 \, \cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 4 \, {\left(151041 \, \cos\left(d x + c\right)^{7} + 103524 \, \cos\left(d x + c\right)^{6} - 228999 \, \cos\left(d x + c\right)^{5} - 181256 \, \cos\left(d x + c\right)^{4} + 97611 \, \cos\left(d x + c\right)^{3} + 82340 \, \cos\left(d x + c\right)^{2} + 315 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{491520 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} + 2 \, a^{2} d \cos\left(d x + c\right)^{5} - a^{2} d \cos\left(d x + c\right)^{4} - 4 \, a^{2} d \cos\left(d x + c\right)^{3} - a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)} \sin\left(d x + c\right)}, -\frac{245445 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{6} + 2 \, \cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 245760 \, {\left(\cos\left(d x + c\right)^{6} + 2 \, \cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, {\left(151041 \, \cos\left(d x + c\right)^{7} + 103524 \, \cos\left(d x + c\right)^{6} - 228999 \, \cos\left(d x + c\right)^{5} - 181256 \, \cos\left(d x + c\right)^{4} + 97611 \, \cos\left(d x + c\right)^{3} + 82340 \, \cos\left(d x + c\right)^{2} + 315 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{245760 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} + 2 \, a^{2} d \cos\left(d x + c\right)^{5} - a^{2} d \cos\left(d x + c\right)^{4} - 4 \, a^{2} d \cos\left(d x + c\right)^{3} - a^{2} d \cos\left(d x + c\right)^{2} + 2 \, a^{2} d \cos\left(d x + c\right) + a^{2} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/491520*(245445*sqrt(2)*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + 3*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 245760*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 - 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) + 4*(151041*cos(d*x + c)^7 + 103524*cos(d*x + c)^6 - 228999*cos(d*x + c)^5 - 181256*cos(d*x + c)^4 + 97611*cos(d*x + c)^3 + 82340*cos(d*x + c)^2 + 315*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)*sin(d*x + c)), -1/245760*(245445*sqrt(2)*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 245760*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(151041*cos(d*x + c)^7 + 103524*cos(d*x + c)^6 - 228999*cos(d*x + c)^5 - 181256*cos(d*x + c)^4 + 97611*cos(d*x + c)^3 + 82340*cos(d*x + c)^2 + 315*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)*sin(d*x + c))]","A",0
195,1,241,0,0.475646," ","integrate(tan(d*x+c)^5/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a} \cos\left(d x + c\right) \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) - 4 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} {\left(8 \, \cos\left(d x + c\right) - 1\right)}}{6 \, a^{3} d \cos\left(d x + c\right)}, \frac{3 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) \cos\left(d x + c\right) - 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} {\left(8 \, \cos\left(d x + c\right) - 1\right)}}{3 \, a^{3} d \cos\left(d x + c\right)}\right]"," ",0,"[1/6*(3*sqrt(a)*cos(d*x + c)*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(8*cos(d*x + c) - 1))/(a^3*d*cos(d*x + c)), 1/3*(3*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a))*cos(d*x + c) - 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(8*cos(d*x + c) - 1))/(a^3*d*cos(d*x + c))]","A",0
196,1,245,0,0.668170," ","integrate(tan(d*x+c)^3/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} {\left(\cos\left(d x + c\right) + 1\right)} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) - 8 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, {\left(a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}, -\frac{\sqrt{-a} {\left(\cos\left(d x + c\right) + 1\right)} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) + 4 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a^{3} d \cos\left(d x + c\right) + a^{3} d}\right]"," ",0,"[1/2*(sqrt(a)*(cos(d*x + c) + 1)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 8*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(a^3*d*cos(d*x + c) + a^3*d), -(sqrt(-a)*(cos(d*x + c) + 1)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) + 4*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c))/(a^3*d*cos(d*x + c) + a^3*d)]","B",0
197,1,321,0,0.584149," ","integrate(tan(d*x+c)/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, {\left(4 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{6 \, {\left(a^{3} d \cos\left(d x + c\right)^{2} + 2 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}, \frac{3 \, {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) + 2 \, {\left(4 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{3 \, {\left(a^{3} d \cos\left(d x + c\right)^{2} + 2 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}\right]"," ",0,"[1/6*(3*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(4*cos(d*x + c)^2 + 3*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + a^3*d), 1/3*(3*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) + 2*(4*cos(d*x + c)^2 + 3*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + a^3*d)]","B",0
198,1,573,0,0.636065," ","integrate(cot(d*x+c)/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{5 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) - 1}\right) + 40 \, {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) - 4 \, {\left(49 \, \cos\left(d x + c\right)^{3} + 80 \, \cos\left(d x + c\right)^{2} + 35 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{80 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}, \frac{5 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 40 \, {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(49 \, \cos\left(d x + c\right)^{3} + 80 \, \cos\left(d x + c\right)^{2} + 35 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{40 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}\right]"," ",0,"[1/80*(5*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + 40*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(49*cos(d*x + c)^3 + 80*cos(d*x + c)^2 + 35*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d), 1/40*(5*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 40*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(49*cos(d*x + c)^3 + 80*cos(d*x + c)^2 + 35*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)]","B",0
199,1,748,0,0.607918," ","integrate(cot(d*x+c)^3/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{1365 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) - 1\right)} \sqrt{a} \log\left(\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + 3 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) - 1}\right) + 6720 \, {\left(\cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) - 1\right)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) + 4 \, {\left(8017 \, \cos\left(d x + c\right)^{5} + 12640 \, \cos\left(d x + c\right)^{4} - 1582 \, \cos\left(d x + c\right)^{3} - 12040 \, \cos\left(d x + c\right)^{2} - 5355 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{13440 \, {\left(a^{3} d \cos\left(d x + c\right)^{5} + 3 \, a^{3} d \cos\left(d x + c\right)^{4} + 2 \, a^{3} d \cos\left(d x + c\right)^{3} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} - 3 \, a^{3} d \cos\left(d x + c\right) - a^{3} d\right)}}, -\frac{1365 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 6720 \, {\left(\cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(8017 \, \cos\left(d x + c\right)^{5} + 12640 \, \cos\left(d x + c\right)^{4} - 1582 \, \cos\left(d x + c\right)^{3} - 12040 \, \cos\left(d x + c\right)^{2} - 5355 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{6720 \, {\left(a^{3} d \cos\left(d x + c\right)^{5} + 3 \, a^{3} d \cos\left(d x + c\right)^{4} + 2 \, a^{3} d \cos\left(d x + c\right)^{3} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} - 3 \, a^{3} d \cos\left(d x + c\right) - a^{3} d\right)}}\right]"," ",0,"[1/13440*(1365*sqrt(2)*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*sqrt(a)*log((2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) + 3*a*cos(d*x + c) + a)/(cos(d*x + c) - 1)) + 6720*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 + 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 4*(8017*cos(d*x + c)^5 + 12640*cos(d*x + c)^4 - 1582*cos(d*x + c)^3 - 12040*cos(d*x + c)^2 - 5355*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + 2*a^3*d*cos(d*x + c)^3 - 2*a^3*d*cos(d*x + c)^2 - 3*a^3*d*cos(d*x + c) - a^3*d), -1/6720*(1365*sqrt(2)*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 6720*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(8017*cos(d*x + c)^5 + 12640*cos(d*x + c)^4 - 1582*cos(d*x + c)^3 - 12040*cos(d*x + c)^2 - 5355*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + 2*a^3*d*cos(d*x + c)^3 - 2*a^3*d*cos(d*x + c)^2 - 3*a^3*d*cos(d*x + c) - a^3*d)]","B",0
200,1,905,0,0.690949," ","integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{82845 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{7} + 3 \, \cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5} - 5 \, \cos\left(d x + c\right)^{4} - 5 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 3 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) - 1}\right) + 322560 \, {\left(\cos\left(d x + c\right)^{7} + 3 \, \cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5} - 5 \, \cos\left(d x + c\right)^{4} - 5 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \log\left(-8 \, a \cos\left(d x + c\right)^{2} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} + \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} - 8 \, a \cos\left(d x + c\right) - a\right) - 4 \, {\left(382201 \, \cos\left(d x + c\right)^{7} + 591520 \, \cos\left(d x + c\right)^{6} - 403607 \, \cos\left(d x + c\right)^{5} - 1112040 \, \cos\left(d x + c\right)^{4} - 189189 \, \cos\left(d x + c\right)^{3} + 531720 \, \cos\left(d x + c\right)^{2} + 239715 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{645120 \, {\left(a^{3} d \cos\left(d x + c\right)^{7} + 3 \, a^{3} d \cos\left(d x + c\right)^{6} + a^{3} d \cos\left(d x + c\right)^{5} - 5 \, a^{3} d \cos\left(d x + c\right)^{4} - 5 \, a^{3} d \cos\left(d x + c\right)^{3} + a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}, \frac{82845 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{7} + 3 \, \cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5} - 5 \, \cos\left(d x + c\right)^{4} - 5 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a \cos\left(d x + c\right) + a}\right) - 322560 \, {\left(\cos\left(d x + c\right)^{7} + 3 \, \cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5} - 5 \, \cos\left(d x + c\right)^{4} - 5 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + a}\right) - 2 \, {\left(382201 \, \cos\left(d x + c\right)^{7} + 591520 \, \cos\left(d x + c\right)^{6} - 403607 \, \cos\left(d x + c\right)^{5} - 1112040 \, \cos\left(d x + c\right)^{4} - 189189 \, \cos\left(d x + c\right)^{3} + 531720 \, \cos\left(d x + c\right)^{2} + 239715 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{322560 \, {\left(a^{3} d \cos\left(d x + c\right)^{7} + 3 \, a^{3} d \cos\left(d x + c\right)^{6} + a^{3} d \cos\left(d x + c\right)^{5} - 5 \, a^{3} d \cos\left(d x + c\right)^{4} - 5 \, a^{3} d \cos\left(d x + c\right)^{3} + a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}\right]"," ",0,"[1/645120*(82845*sqrt(2)*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + 322560*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(382201*cos(d*x + c)^7 + 591520*cos(d*x + c)^6 - 403607*cos(d*x + c)^5 - 1112040*cos(d*x + c)^4 - 189189*cos(d*x + c)^3 + 531720*cos(d*x + c)^2 + 239715*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c)^6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d), 1/322560*(82845*sqrt(2)*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(-a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 322560*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(382201*cos(d*x + c)^7 + 591520*cos(d*x + c)^6 - 403607*cos(d*x + c)^5 - 1112040*cos(d*x + c)^4 - 189189*cos(d*x + c)^3 + 531720*cos(d*x + c)^2 + 239715*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c)^6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)]","B",0
201,1,323,0,0.459000," ","integrate(tan(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2}\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) - 2 \, {\left(23 \, \cos\left(d x + c\right)^{2} - 11 \, \cos\left(d x + c\right) + 3\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{15 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + a^{3} d \cos\left(d x + c\right)^{2}\right)}}, \frac{2 \, {\left(15 \, {\left(\cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) + {\left(23 \, \cos\left(d x + c\right)^{2} - 11 \, \cos\left(d x + c\right) + 3\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{15 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + a^{3} d \cos\left(d x + c\right)^{2}\right)}}\right]"," ",0,"[-1/15*(15*(cos(d*x + c)^3 + cos(d*x + c)^2)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) - 2*(23*cos(d*x + c)^2 - 11*cos(d*x + c) + 3)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2), 2/15*(15*(cos(d*x + c)^3 + cos(d*x + c)^2)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + (23*cos(d*x + c)^2 - 11*cos(d*x + c) + 3)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2)]","A",0
202,1,414,0,0.522919," ","integrate(tan(d*x+c)^4/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{2} {\left(a \cos\left(d x + c\right) + a\right)} \sqrt{-\frac{1}{a}} \log\left(\frac{2 \, \sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sqrt{-\frac{1}{a}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 3 \, \cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) - 1}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) - \sqrt{-a} {\left(\cos\left(d x + c\right) + 1\right)} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} + 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) + 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)}{a^{3} d \cos\left(d x + c\right) + a^{3} d}, -\frac{2 \, {\left(\sqrt{a} {\left(\cos\left(d x + c\right) + 1\right)} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) - \frac{2 \, \sqrt{2} {\left(a \cos\left(d x + c\right) + a\right)} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right)}{\sqrt{a}} - \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right)\right)}}{a^{3} d \cos\left(d x + c\right) + a^{3} d}\right]"," ",0,"[(2*sqrt(2)*(a*cos(d*x + c) + a)*sqrt(-1/a)*log((2*sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(-1/a)*cos(d*x + c)*sin(d*x + c) + 3*cos(d*x + c)^2 + 2*cos(d*x + c) - 1)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) - sqrt(-a)*(cos(d*x + c) + 1)*log((2*a*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) + 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^3*d*cos(d*x + c) + a^3*d), -2*(sqrt(a)*(cos(d*x + c) + 1)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) - 2*sqrt(2)*(a*cos(d*x + c) + a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))/sqrt(a) - sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a^3*d*cos(d*x + c) + a^3*d)]","A",0
203,1,492,0,0.718771," ","integrate(tan(d*x+c)^2/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \log\left(\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 3 \, a \cos\left(d x + c\right)^{2} + 2 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + 4 \, {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(d x + c\right)^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right) + 1}\right) - 4 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{4 \, {\left(a^{3} d \cos\left(d x + c\right)^{2} + 2 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}, -\frac{3 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) - 4 \, {\left(\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) - 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, {\left(a^{3} d \cos\left(d x + c\right)^{2} + 2 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)}}\right]"," ",0,"[-1/4*(3*sqrt(2)*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + 3*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 4*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - a)/(cos(d*x + c) + 1)) - 4*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c))/(a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + a^3*d), -1/2*(3*sqrt(2)*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) - 4*(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) - 2*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c))/(a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + a^3*d)]","A",0
204,1,691,0,0.683118," ","integrate(cot(d*x+c)^2/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[-\frac{957 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \log\left(\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 3 \, a \cos\left(d x + c\right)^{2} + 2 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 768 \, {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 4 \, {\left(409 \, \cos\left(d x + c\right)^{4} + 349 \, \cos\left(d x + c\right)^{3} - 185 \, \cos\left(d x + c\right)^{2} - 189 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{1536 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)} \sin\left(d x + c\right)}, -\frac{957 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 768 \, {\left(\cos\left(d x + c\right)^{3} + 3 \, \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, {\left(409 \, \cos\left(d x + c\right)^{4} + 349 \, \cos\left(d x + c\right)^{3} - 185 \, \cos\left(d x + c\right)^{2} - 189 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{768 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/1536*(957*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + 3*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 768*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 - 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) + 4*(409*cos(d*x + c)^4 + 349*cos(d*x + c)^3 - 185*cos(d*x + c)^2 - 189*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c)), -1/768*(957*sqrt(2)*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 768*(cos(d*x + c)^3 + 3*cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(409*cos(d*x + c)^4 + 349*cos(d*x + c)^3 - 185*cos(d*x + c)^2 - 189*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c))]","A",0
205,1,868,0,0.709754," ","integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[-\frac{29049 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \log\left(-\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 3 \, a \cos\left(d x + c\right)^{2} - 2 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 24576 \, {\left(\cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) - 1\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} + 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) - 4 \, {\left(14629 \, \cos\left(d x + c\right)^{6} + 14233 \, \cos\left(d x + c\right)^{5} - 14058 \, \cos\left(d x + c\right)^{4} - 17426 \, \cos\left(d x + c\right)^{3} + 2245 \, \cos\left(d x + c\right)^{2} + 4473 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{49152 \, {\left(a^{3} d \cos\left(d x + c\right)^{5} + 3 \, a^{3} d \cos\left(d x + c\right)^{4} + 2 \, a^{3} d \cos\left(d x + c\right)^{3} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} - 3 \, a^{3} d \cos\left(d x + c\right) - a^{3} d\right)} \sin\left(d x + c\right)}, \frac{29049 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) - 1\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 24576 \, {\left(\cos\left(d x + c\right)^{5} + 3 \, \cos\left(d x + c\right)^{4} + 2 \, \cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)^{2} - 3 \, \cos\left(d x + c\right) - 1\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, {\left(14629 \, \cos\left(d x + c\right)^{6} + 14233 \, \cos\left(d x + c\right)^{5} - 14058 \, \cos\left(d x + c\right)^{4} - 17426 \, \cos\left(d x + c\right)^{3} + 2245 \, \cos\left(d x + c\right)^{2} + 4473 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{24576 \, {\left(a^{3} d \cos\left(d x + c\right)^{5} + 3 \, a^{3} d \cos\left(d x + c\right)^{4} + 2 \, a^{3} d \cos\left(d x + c\right)^{3} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} - 3 \, a^{3} d \cos\left(d x + c\right) - a^{3} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/49152*(29049*sqrt(2)*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c) + a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 24576*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*(14629*cos(d*x + c)^6 + 14233*cos(d*x + c)^5 - 14058*cos(d*x + c)^4 - 17426*cos(d*x + c)^3 + 2245*cos(d*x + c)^2 + 4473*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + 2*a^3*d*cos(d*x + c)^3 - 2*a^3*d*cos(d*x + c)^2 - 3*a^3*d*cos(d*x + c) - a^3*d)*sin(d*x + c)), 1/24576*(29049*sqrt(2)*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 24576*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(14629*cos(d*x + c)^6 + 14233*cos(d*x + c)^5 - 14058*cos(d*x + c)^4 - 17426*cos(d*x + c)^3 + 2245*cos(d*x + c)^2 + 4473*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + 2*a^3*d*cos(d*x + c)^3 - 2*a^3*d*cos(d*x + c)^2 - 3*a^3*d*cos(d*x + c) - a^3*d)*sin(d*x + c))]","A",0
206,1,1023,0,0.812467," ","integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x, algorithm=""fricas"")","\left[-\frac{1116915 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{7} + 3 \, \cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5} - 5 \, \cos\left(d x + c\right)^{4} - 5 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \log\left(\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 3 \, a \cos\left(d x + c\right)^{2} + 2 \, a \cos\left(d x + c\right) - a}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 983040 \, {\left(\cos\left(d x + c\right)^{7} + 3 \, \cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5} - 5 \, \cos\left(d x + c\right)^{4} - 5 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{-a} \log\left(-\frac{8 \, a \cos\left(d x + c\right)^{3} - 4 \, {\left(2 \, \cos\left(d x + c\right)^{2} - \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \sin\left(d x + c\right) - 7 \, a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right) + 1}\right) \sin\left(d x + c\right) + 4 \, {\left(639063 \, \cos\left(d x + c\right)^{8} + 681555 \, \cos\left(d x + c\right)^{7} - 986085 \, \cos\left(d x + c\right)^{6} - 1360025 \, \cos\left(d x + c\right)^{5} + 405445 \, \cos\left(d x + c\right)^{4} + 836921 \, \cos\left(d x + c\right)^{3} + 15305 \, \cos\left(d x + c\right)^{2} - 133875 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{1966080 \, {\left(a^{3} d \cos\left(d x + c\right)^{7} + 3 \, a^{3} d \cos\left(d x + c\right)^{6} + a^{3} d \cos\left(d x + c\right)^{5} - 5 \, a^{3} d \cos\left(d x + c\right)^{4} - 5 \, a^{3} d \cos\left(d x + c\right)^{3} + a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)} \sin\left(d x + c\right)}, -\frac{1116915 \, \sqrt{2} {\left(\cos\left(d x + c\right)^{7} + 3 \, \cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5} - 5 \, \cos\left(d x + c\right)^{4} - 5 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{\sqrt{a} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 983040 \, {\left(\cos\left(d x + c\right)^{7} + 3 \, \cos\left(d x + c\right)^{6} + \cos\left(d x + c\right)^{5} - 5 \, \cos\left(d x + c\right)^{4} - 5 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)^{2} + 3 \, \cos\left(d x + c\right) + 1\right)} \sqrt{a} \arctan\left(\frac{2 \, \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, a \cos\left(d x + c\right)^{2} + a \cos\left(d x + c\right) - a}\right) \sin\left(d x + c\right) + 2 \, {\left(639063 \, \cos\left(d x + c\right)^{8} + 681555 \, \cos\left(d x + c\right)^{7} - 986085 \, \cos\left(d x + c\right)^{6} - 1360025 \, \cos\left(d x + c\right)^{5} + 405445 \, \cos\left(d x + c\right)^{4} + 836921 \, \cos\left(d x + c\right)^{3} + 15305 \, \cos\left(d x + c\right)^{2} - 133875 \, \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + a}{\cos\left(d x + c\right)}}}{983040 \, {\left(a^{3} d \cos\left(d x + c\right)^{7} + 3 \, a^{3} d \cos\left(d x + c\right)^{6} + a^{3} d \cos\left(d x + c\right)^{5} - 5 \, a^{3} d \cos\left(d x + c\right)^{4} - 5 \, a^{3} d \cos\left(d x + c\right)^{3} + a^{3} d \cos\left(d x + c\right)^{2} + 3 \, a^{3} d \cos\left(d x + c\right) + a^{3} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/1966080*(1116915*sqrt(2)*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + 3*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 983040*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 - 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) + 4*(639063*cos(d*x + c)^8 + 681555*cos(d*x + c)^7 - 986085*cos(d*x + c)^6 - 1360025*cos(d*x + c)^5 + 405445*cos(d*x + c)^4 + 836921*cos(d*x + c)^3 + 15305*cos(d*x + c)^2 - 133875*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c)^6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c)), -1/983040*(1116915*sqrt(2)*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 983040*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*arctan(2*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x + c) + 2*(639063*cos(d*x + c)^8 + 681555*cos(d*x + c)^7 - 986085*cos(d*x + c)^6 - 1360025*cos(d*x + c)^5 + 405445*cos(d*x + c)^4 + 836921*cos(d*x + c)^3 + 15305*cos(d*x + c)^2 - 133875*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c)^6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c))]","A",0
207,1,674,0,0.813955," ","integrate(tan(f*x+e)^2/(a+a*sec(f*x+e))^(9/2),x, algorithm=""fricas"")","\left[-\frac{273 \, \sqrt{2} {\left(\cos\left(f x + e\right)^{4} + 4 \, \cos\left(f x + e\right)^{3} + 6 \, \cos\left(f x + e\right)^{2} + 4 \, \cos\left(f x + e\right) + 1\right)} \sqrt{-a} \log\left(\frac{2 \, \sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right) + a}{\cos\left(f x + e\right)}} \cos\left(f x + e\right) \sin\left(f x + e\right) + 3 \, a \cos\left(f x + e\right)^{2} + 2 \, a \cos\left(f x + e\right) - a}{\cos\left(f x + e\right)^{2} + 2 \, \cos\left(f x + e\right) + 1}\right) + 384 \, {\left(\cos\left(f x + e\right)^{4} + 4 \, \cos\left(f x + e\right)^{3} + 6 \, \cos\left(f x + e\right)^{2} + 4 \, \cos\left(f x + e\right) + 1\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(f x + e\right) + a}{\cos\left(f x + e\right)}} \cos\left(f x + e\right) \sin\left(f x + e\right) + a \cos\left(f x + e\right) - a}{\cos\left(f x + e\right) + 1}\right) - 4 \, {\left(157 \, \cos\left(f x + e\right)^{3} + 206 \, \cos\left(f x + e\right)^{2} + 81 \, \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right) + a}{\cos\left(f x + e\right)}} \sin\left(f x + e\right)}{384 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 4 \, a^{5} f \cos\left(f x + e\right)^{3} + 6 \, a^{5} f \cos\left(f x + e\right)^{2} + 4 \, a^{5} f \cos\left(f x + e\right) + a^{5} f\right)}}, -\frac{273 \, \sqrt{2} {\left(\cos\left(f x + e\right)^{4} + 4 \, \cos\left(f x + e\right)^{3} + 6 \, \cos\left(f x + e\right)^{2} + 4 \, \cos\left(f x + e\right) + 1\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \cos\left(f x + e\right) + a}{\cos\left(f x + e\right)}} \cos\left(f x + e\right)}{\sqrt{a} \sin\left(f x + e\right)}\right) - 384 \, {\left(\cos\left(f x + e\right)^{4} + 4 \, \cos\left(f x + e\right)^{3} + 6 \, \cos\left(f x + e\right)^{2} + 4 \, \cos\left(f x + e\right) + 1\right)} \sqrt{a} \arctan\left(\frac{\sqrt{\frac{a \cos\left(f x + e\right) + a}{\cos\left(f x + e\right)}} \cos\left(f x + e\right)}{\sqrt{a} \sin\left(f x + e\right)}\right) - 2 \, {\left(157 \, \cos\left(f x + e\right)^{3} + 206 \, \cos\left(f x + e\right)^{2} + 81 \, \cos\left(f x + e\right)\right)} \sqrt{\frac{a \cos\left(f x + e\right) + a}{\cos\left(f x + e\right)}} \sin\left(f x + e\right)}{192 \, {\left(a^{5} f \cos\left(f x + e\right)^{4} + 4 \, a^{5} f \cos\left(f x + e\right)^{3} + 6 \, a^{5} f \cos\left(f x + e\right)^{2} + 4 \, a^{5} f \cos\left(f x + e\right) + a^{5} f\right)}}\right]"," ",0,"[-1/384*(273*sqrt(2)*(cos(f*x + e)^4 + 4*cos(f*x + e)^3 + 6*cos(f*x + e)^2 + 4*cos(f*x + e) + 1)*sqrt(-a)*log((2*sqrt(2)*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + 3*a*cos(f*x + e)^2 + 2*a*cos(f*x + e) - a)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) + 384*(cos(f*x + e)^4 + 4*cos(f*x + e)^3 + 6*cos(f*x + e)^2 + 4*cos(f*x + e) + 1)*sqrt(-a)*log((2*a*cos(f*x + e)^2 - 2*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + a*cos(f*x + e) - a)/(cos(f*x + e) + 1)) - 4*(157*cos(f*x + e)^3 + 206*cos(f*x + e)^2 + 81*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a^5*f*cos(f*x + e)^4 + 4*a^5*f*cos(f*x + e)^3 + 6*a^5*f*cos(f*x + e)^2 + 4*a^5*f*cos(f*x + e) + a^5*f), -1/192*(273*sqrt(2)*(cos(f*x + e)^4 + 4*cos(f*x + e)^3 + 6*cos(f*x + e)^2 + 4*cos(f*x + e) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - 384*(cos(f*x + e)^4 + 4*cos(f*x + e)^3 + 6*cos(f*x + e)^2 + 4*cos(f*x + e) + 1)*sqrt(a)*arctan(sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - 2*(157*cos(f*x + e)^3 + 206*cos(f*x + e)^2 + 81*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a^5*f*cos(f*x + e)^4 + 4*a^5*f*cos(f*x + e)^3 + 6*a^5*f*cos(f*x + e)^2 + 4*a^5*f*cos(f*x + e) + a^5*f)]","A",0
208,0,0,0,0.565430," ","integrate((a+a*sec(d*x+c))^n*(e*tan(d*x+c))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \left(e \tan\left(d x + c\right)\right)^{m}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*(e*tan(d*x + c))^m, x)","F",0
209,0,0,0,0.498861," ","integrate((a+a*sec(d*x+c))^3*(e*tan(d*x+c))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a^{3} \sec\left(d x + c\right)^{3} + 3 \, a^{3} \sec\left(d x + c\right)^{2} + 3 \, a^{3} \sec\left(d x + c\right) + a^{3}\right)} \left(e \tan\left(d x + c\right)\right)^{m}, x\right)"," ",0,"integral((a^3*sec(d*x + c)^3 + 3*a^3*sec(d*x + c)^2 + 3*a^3*sec(d*x + c) + a^3)*(e*tan(d*x + c))^m, x)","F",0
210,0,0,0,0.525926," ","integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a^{2} \sec\left(d x + c\right)^{2} + 2 \, a^{2} \sec\left(d x + c\right) + a^{2}\right)} \left(e \tan\left(d x + c\right)\right)^{m}, x\right)"," ",0,"integral((a^2*sec(d*x + c)^2 + 2*a^2*sec(d*x + c) + a^2)*(e*tan(d*x + c))^m, x)","F",0
211,0,0,0,0.601955," ","integrate((a+a*sec(d*x+c))*(e*tan(d*x+c))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)} \left(e \tan\left(d x + c\right)\right)^{m}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)*(e*tan(d*x + c))^m, x)","F",0
212,0,0,0,0.529646," ","integrate((e*tan(d*x+c))^m/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(e \tan\left(d x + c\right)\right)^{m}}{a \sec\left(d x + c\right) + a}, x\right)"," ",0,"integral((e*tan(d*x + c))^m/(a*sec(d*x + c) + a), x)","F",0
213,0,0,0,0.474162," ","integrate((e*tan(d*x+c))^m/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(e \tan\left(d x + c\right)\right)^{m}}{a^{2} \sec\left(d x + c\right)^{2} + 2 \, a^{2} \sec\left(d x + c\right) + a^{2}}, x\right)"," ",0,"integral((e*tan(d*x + c))^m/(a^2*sec(d*x + c)^2 + 2*a^2*sec(d*x + c) + a^2), x)","F",0
214,0,0,0,0.610166," ","integrate((e*tan(d*x+c))^m/(a+a*sec(d*x+c))^3,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(e \tan\left(d x + c\right)\right)^{m}}{a^{3} \sec\left(d x + c\right)^{3} + 3 \, a^{3} \sec\left(d x + c\right)^{2} + 3 \, a^{3} \sec\left(d x + c\right) + a^{3}}, x\right)"," ",0,"integral((e*tan(d*x + c))^m/(a^3*sec(d*x + c)^3 + 3*a^3*sec(d*x + c)^2 + 3*a^3*sec(d*x + c) + a^3), x)","F",0
215,0,0,0,0.835615," ","integrate((a+a*sec(d*x+c))^(3/2)*(e*tan(d*x+c))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{\frac{3}{2}} \left(e \tan\left(d x + c\right)\right)^{m}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^(3/2)*(e*tan(d*x + c))^m, x)","F",0
216,0,0,0,0.683058," ","integrate((a+a*sec(d*x+c))^(1/2)*(e*tan(d*x+c))^m,x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{a \sec\left(d x + c\right) + a} \left(e \tan\left(d x + c\right)\right)^{m}, x\right)"," ",0,"integral(sqrt(a*sec(d*x + c) + a)*(e*tan(d*x + c))^m, x)","F",0
217,0,0,0,0.565287," ","integrate((e*tan(d*x+c))^m/(a+a*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(e \tan\left(d x + c\right)\right)^{m}}{\sqrt{a \sec\left(d x + c\right) + a}}, x\right)"," ",0,"integral((e*tan(d*x + c))^m/sqrt(a*sec(d*x + c) + a), x)","F",0
218,0,0,0,0.504327," ","integrate((e*tan(d*x+c))^m/(a+a*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{a \sec\left(d x + c\right) + a} \left(e \tan\left(d x + c\right)\right)^{m}}{a^{2} \sec\left(d x + c\right)^{2} + 2 \, a^{2} \sec\left(d x + c\right) + a^{2}}, x\right)"," ",0,"integral(sqrt(a*sec(d*x + c) + a)*(e*tan(d*x + c))^m/(a^2*sec(d*x + c)^2 + 2*a^2*sec(d*x + c) + a^2), x)","F",0
219,0,0,0,0.538305," ","integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^7,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right)^{7}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*tan(d*x + c)^7, x)","F",0
220,0,0,0,0.533278," ","integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^5,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right)^{5}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*tan(d*x + c)^5, x)","F",0
221,0,0,0,0.513387," ","integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^3,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right)^{3}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*tan(d*x + c)^3, x)","F",0
222,0,0,0,0.484005," ","integrate((a+a*sec(d*x+c))^n*tan(d*x+c),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right), x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*tan(d*x + c), x)","F",0
223,0,0,0,0.621862," ","integrate(cot(d*x+c)*(a+a*sec(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \cot\left(d x + c\right), x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*cot(d*x + c), x)","F",0
224,0,0,0,0.479005," ","integrate(cot(d*x+c)^3*(a+a*sec(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \cot\left(d x + c\right)^{3}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*cot(d*x + c)^3, x)","F",0
225,0,0,0,0.514574," ","integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^4,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right)^{4}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*tan(d*x + c)^4, x)","F",0
226,0,0,0,0.734030," ","integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right)^{2}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*tan(d*x + c)^2, x)","F",0
227,0,0,0,0.633558," ","integrate(cot(d*x+c)^2*(a+a*sec(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \cot\left(d x + c\right)^{2}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*cot(d*x + c)^2, x)","F",0
228,0,0,0,0.495714," ","integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \cot\left(d x + c\right)^{4}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*cot(d*x + c)^4, x)","F",0
229,0,0,0,0.503181," ","integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right)^{\frac{3}{2}}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*tan(d*x + c)^(3/2), x)","F",0
230,0,0,0,0.517961," ","integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \sec\left(d x + c\right) + a\right)}^{n} \sqrt{\tan\left(d x + c\right)}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n*sqrt(tan(d*x + c)), x)","F",0
231,0,0,0,0.503080," ","integrate((a+a*sec(d*x+c))^n/tan(d*x+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a \sec\left(d x + c\right) + a\right)}^{n}}{\sqrt{\tan\left(d x + c\right)}}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n/sqrt(tan(d*x + c)), x)","F",0
232,0,0,0,0.551465," ","integrate((a+a*sec(d*x+c))^n/tan(d*x+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(a \sec\left(d x + c\right) + a\right)}^{n}}{\tan\left(d x + c\right)^{\frac{3}{2}}}, x\right)"," ",0,"integral((a*sec(d*x + c) + a)^n/tan(d*x + c)^(3/2), x)","F",0
233,-1,0,0,0.000000," ","integrate((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
234,-1,0,0,0.000000," ","integrate((e*cot(d*x+c))^(3/2)*(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
235,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))*(e*cot(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
236,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))/(e*cot(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
237,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))/(e*cot(d*x+c))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
238,-1,0,0,0.000000," ","integrate((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
239,-1,0,0,0.000000," ","integrate((e*cot(d*x+c))^(3/2)*(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
240,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))^2*(e*cot(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
241,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))^2/(e*cot(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
242,-1,0,0,0.000000," ","integrate((a+a*sec(d*x+c))^2/(e*cot(d*x+c))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
243,-1,0,0,0.000000," ","integrate((e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
244,-1,0,0,0.000000," ","integrate((e*cot(d*x+c))^(1/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
245,-1,0,0,0.000000," ","integrate(1/(a+a*sec(d*x+c))/(e*cot(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
246,-1,0,0,0.000000," ","integrate(1/(e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
247,-1,0,0,0.000000," ","integrate(1/(e*cot(d*x+c))^(5/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
248,-1,0,0,0.000000," ","integrate(1/(e*cot(d*x+c))^(7/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
249,-1,0,0,0.000000," ","integrate(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
250,-1,0,0,0.000000," ","integrate(1/(a+a*sec(d*x+c))^2/(e*cot(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
251,-1,0,0,0.000000," ","integrate(1/(e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
252,-1,0,0,0.000000," ","integrate(1/(e*cot(d*x+c))^(5/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
253,-1,0,0,0.000000," ","integrate(1/(e*cot(d*x+c))^(7/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
254,-1,0,0,0.000000," ","integrate(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
255,-1,0,0,0.000000," ","integrate(1/(e*cot(d*x+c))^(11/2)/(a+a*sec(d*x+c))^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
256,1,101,0,1.291976," ","integrate((a+b*sec(d*x+c))*tan(d*x+c)^7,x, algorithm=""fricas"")","\frac{420 \, a \cos\left(d x + c\right)^{7} \log\left(-\cos\left(d x + c\right)\right) - 420 \, b \cos\left(d x + c\right)^{6} + 630 \, a \cos\left(d x + c\right)^{5} + 420 \, b \cos\left(d x + c\right)^{4} - 315 \, a \cos\left(d x + c\right)^{3} - 252 \, b \cos\left(d x + c\right)^{2} + 70 \, a \cos\left(d x + c\right) + 60 \, b}{420 \, d \cos\left(d x + c\right)^{7}}"," ",0,"1/420*(420*a*cos(d*x + c)^7*log(-cos(d*x + c)) - 420*b*cos(d*x + c)^6 + 630*a*cos(d*x + c)^5 + 420*b*cos(d*x + c)^4 - 315*a*cos(d*x + c)^3 - 252*b*cos(d*x + c)^2 + 70*a*cos(d*x + c) + 60*b)/(d*cos(d*x + c)^7)","A",0
257,1,79,0,1.133880," ","integrate((a+b*sec(d*x+c))*tan(d*x+c)^5,x, algorithm=""fricas"")","-\frac{60 \, a \cos\left(d x + c\right)^{5} \log\left(-\cos\left(d x + c\right)\right) - 60 \, b \cos\left(d x + c\right)^{4} + 60 \, a \cos\left(d x + c\right)^{3} + 40 \, b \cos\left(d x + c\right)^{2} - 15 \, a \cos\left(d x + c\right) - 12 \, b}{60 \, d \cos\left(d x + c\right)^{5}}"," ",0,"-1/60*(60*a*cos(d*x + c)^5*log(-cos(d*x + c)) - 60*b*cos(d*x + c)^4 + 60*a*cos(d*x + c)^3 + 40*b*cos(d*x + c)^2 - 15*a*cos(d*x + c) - 12*b)/(d*cos(d*x + c)^5)","A",0
258,1,57,0,0.564252," ","integrate((a+b*sec(d*x+c))*tan(d*x+c)^3,x, algorithm=""fricas"")","\frac{6 \, a \cos\left(d x + c\right)^{3} \log\left(-\cos\left(d x + c\right)\right) - 6 \, b \cos\left(d x + c\right)^{2} + 3 \, a \cos\left(d x + c\right) + 2 \, b}{6 \, d \cos\left(d x + c\right)^{3}}"," ",0,"1/6*(6*a*cos(d*x + c)^3*log(-cos(d*x + c)) - 6*b*cos(d*x + c)^2 + 3*a*cos(d*x + c) + 2*b)/(d*cos(d*x + c)^3)","A",0
259,1,34,0,1.021494," ","integrate((a+b*sec(d*x+c))*tan(d*x+c),x, algorithm=""fricas"")","-\frac{a \cos\left(d x + c\right) \log\left(-\cos\left(d x + c\right)\right) - b}{d \cos\left(d x + c\right)}"," ",0,"-(a*cos(d*x + c)*log(-cos(d*x + c)) - b)/(d*cos(d*x + c))","A",0
260,1,38,0,0.816263," ","integrate(cot(d*x+c)*(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\frac{{\left(a - b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left(a + b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{2 \, d}"," ",0,"1/2*((a - b)*log(1/2*cos(d*x + c) + 1/2) + (a + b)*log(-1/2*cos(d*x + c) + 1/2))/d","A",0
261,1,99,0,0.490784," ","integrate(cot(d*x+c)^3*(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\frac{2 \, b \cos\left(d x + c\right) - {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} - 2 \, a + b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - 2 \, a - b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 2 \, a}{4 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"1/4*(2*b*cos(d*x + c) - ((2*a - b)*cos(d*x + c)^2 - 2*a + b)*log(1/2*cos(d*x + c) + 1/2) - ((2*a + b)*cos(d*x + c)^2 - 2*a - b)*log(-1/2*cos(d*x + c) + 1/2) + 2*a)/(d*cos(d*x + c)^2 - d)","A",0
262,1,168,0,0.478027," ","integrate(cot(d*x+c)^5*(a+b*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{10 \, b \cos\left(d x + c\right)^{3} + 16 \, a \cos\left(d x + c\right)^{2} - 6 \, b \cos\left(d x + c\right) - {\left({\left(8 \, a - 3 \, b\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(8 \, a - 3 \, b\right)} \cos\left(d x + c\right)^{2} + 8 \, a - 3 \, b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - {\left({\left(8 \, a + 3 \, b\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(8 \, a + 3 \, b\right)} \cos\left(d x + c\right)^{2} + 8 \, a + 3 \, b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 12 \, a}{16 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} + d\right)}}"," ",0,"-1/16*(10*b*cos(d*x + c)^3 + 16*a*cos(d*x + c)^2 - 6*b*cos(d*x + c) - ((8*a - 3*b)*cos(d*x + c)^4 - 2*(8*a - 3*b)*cos(d*x + c)^2 + 8*a - 3*b)*log(1/2*cos(d*x + c) + 1/2) - ((8*a + 3*b)*cos(d*x + c)^4 - 2*(8*a + 3*b)*cos(d*x + c)^2 + 8*a + 3*b)*log(-1/2*cos(d*x + c) + 1/2) - 12*a)/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)","A",0
263,1,237,0,0.528760," ","integrate(cot(d*x+c)^7*(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\frac{66 \, b \cos\left(d x + c\right)^{5} + 144 \, a \cos\left(d x + c\right)^{4} - 80 \, b \cos\left(d x + c\right)^{3} - 216 \, a \cos\left(d x + c\right)^{2} + 30 \, b \cos\left(d x + c\right) - 3 \, {\left({\left(16 \, a - 5 \, b\right)} \cos\left(d x + c\right)^{6} - 3 \, {\left(16 \, a - 5 \, b\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(16 \, a - 5 \, b\right)} \cos\left(d x + c\right)^{2} - 16 \, a + 5 \, b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 3 \, {\left({\left(16 \, a + 5 \, b\right)} \cos\left(d x + c\right)^{6} - 3 \, {\left(16 \, a + 5 \, b\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(16 \, a + 5 \, b\right)} \cos\left(d x + c\right)^{2} - 16 \, a - 5 \, b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 88 \, a}{96 \, {\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,"1/96*(66*b*cos(d*x + c)^5 + 144*a*cos(d*x + c)^4 - 80*b*cos(d*x + c)^3 - 216*a*cos(d*x + c)^2 + 30*b*cos(d*x + c) - 3*((16*a - 5*b)*cos(d*x + c)^6 - 3*(16*a - 5*b)*cos(d*x + c)^4 + 3*(16*a - 5*b)*cos(d*x + c)^2 - 16*a + 5*b)*log(1/2*cos(d*x + c) + 1/2) - 3*((16*a + 5*b)*cos(d*x + c)^6 - 3*(16*a + 5*b)*cos(d*x + c)^4 + 3*(16*a + 5*b)*cos(d*x + c)^2 - 16*a - 5*b)*log(-1/2*cos(d*x + c) + 1/2) + 88*a)/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)","A",0
264,1,134,0,0.559167," ","integrate((a+b*sec(d*x+c))*tan(d*x+c)^6,x, algorithm=""fricas"")","-\frac{480 \, a d x \cos\left(d x + c\right)^{6} + 75 \, b \cos\left(d x + c\right)^{6} \log\left(\sin\left(d x + c\right) + 1\right) - 75 \, b \cos\left(d x + c\right)^{6} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(368 \, a \cos\left(d x + c\right)^{5} + 165 \, b \cos\left(d x + c\right)^{4} - 176 \, a \cos\left(d x + c\right)^{3} - 130 \, b \cos\left(d x + c\right)^{2} + 48 \, a \cos\left(d x + c\right) + 40 \, b\right)} \sin\left(d x + c\right)}{480 \, d \cos\left(d x + c\right)^{6}}"," ",0,"-1/480*(480*a*d*x*cos(d*x + c)^6 + 75*b*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 75*b*cos(d*x + c)^6*log(-sin(d*x + c) + 1) - 2*(368*a*cos(d*x + c)^5 + 165*b*cos(d*x + c)^4 - 176*a*cos(d*x + c)^3 - 130*b*cos(d*x + c)^2 + 48*a*cos(d*x + c) + 40*b)*sin(d*x + c))/(d*cos(d*x + c)^6)","A",0
265,1,112,0,0.521955," ","integrate((a+b*sec(d*x+c))*tan(d*x+c)^4,x, algorithm=""fricas"")","\frac{48 \, a d x \cos\left(d x + c\right)^{4} + 9 \, b \cos\left(d x + c\right)^{4} \log\left(\sin\left(d x + c\right) + 1\right) - 9 \, b \cos\left(d x + c\right)^{4} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(32 \, a \cos\left(d x + c\right)^{3} + 15 \, b \cos\left(d x + c\right)^{2} - 8 \, a \cos\left(d x + c\right) - 6 \, b\right)} \sin\left(d x + c\right)}{48 \, d \cos\left(d x + c\right)^{4}}"," ",0,"1/48*(48*a*d*x*cos(d*x + c)^4 + 9*b*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 9*b*cos(d*x + c)^4*log(-sin(d*x + c) + 1) - 2*(32*a*cos(d*x + c)^3 + 15*b*cos(d*x + c)^2 - 8*a*cos(d*x + c) - 6*b)*sin(d*x + c))/(d*cos(d*x + c)^4)","A",0
266,1,87,0,0.565282," ","integrate((a+b*sec(d*x+c))*tan(d*x+c)^2,x, algorithm=""fricas"")","-\frac{4 \, a d x \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - b \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(2 \, a \cos\left(d x + c\right) + b\right)} \sin\left(d x + c\right)}{4 \, d \cos\left(d x + c\right)^{2}}"," ",0,"-1/4*(4*a*d*x*cos(d*x + c)^2 + b*cos(d*x + c)^2*log(sin(d*x + c) + 1) - b*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 2*(2*a*cos(d*x + c) + b)*sin(d*x + c))/(d*cos(d*x + c)^2)","B",0
267,1,33,0,0.635151," ","integrate(cot(d*x+c)^2*(a+b*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{a d x \sin\left(d x + c\right) + a \cos\left(d x + c\right) + b}{d \sin\left(d x + c\right)}"," ",0,"-(a*d*x*sin(d*x + c) + a*cos(d*x + c) + b)/(d*sin(d*x + c))","A",0
268,1,87,0,0.507940," ","integrate(cot(d*x+c)^4*(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\frac{4 \, a \cos\left(d x + c\right)^{3} + 3 \, b \cos\left(d x + c\right)^{2} - 3 \, a \cos\left(d x + c\right) + 3 \, {\left(a d x \cos\left(d x + c\right)^{2} - a d x\right)} \sin\left(d x + c\right) - 2 \, b}{3 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}"," ",0,"1/3*(4*a*cos(d*x + c)^3 + 3*b*cos(d*x + c)^2 - 3*a*cos(d*x + c) + 3*(a*d*x*cos(d*x + c)^2 - a*d*x)*sin(d*x + c) - 2*b)/((d*cos(d*x + c)^2 - d)*sin(d*x + c))","A",0
269,1,130,0,0.601542," ","integrate(cot(d*x+c)^6*(a+b*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{23 \, a \cos\left(d x + c\right)^{5} + 15 \, b \cos\left(d x + c\right)^{4} - 35 \, a \cos\left(d x + c\right)^{3} - 20 \, b \cos\left(d x + c\right)^{2} + 15 \, a \cos\left(d x + c\right) + 15 \, {\left(a d x \cos\left(d x + c\right)^{4} - 2 \, a d x \cos\left(d x + c\right)^{2} + a d x\right)} \sin\left(d x + c\right) + 8 \, b}{15 \, {\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,"-1/15*(23*a*cos(d*x + c)^5 + 15*b*cos(d*x + c)^4 - 35*a*cos(d*x + c)^3 - 20*b*cos(d*x + c)^2 + 15*a*cos(d*x + c) + 15*(a*d*x*cos(d*x + c)^4 - 2*a*d*x*cos(d*x + c)^2 + a*d*x)*sin(d*x + c) + 8*b)/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))","A",0
270,1,179,0,0.512348," ","integrate(cot(d*x+c)^8*(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\frac{176 \, a \cos\left(d x + c\right)^{7} + 105 \, b \cos\left(d x + c\right)^{6} - 406 \, a \cos\left(d x + c\right)^{5} - 210 \, b \cos\left(d x + c\right)^{4} + 350 \, a \cos\left(d x + c\right)^{3} + 168 \, b \cos\left(d x + c\right)^{2} - 105 \, a \cos\left(d x + c\right) + 105 \, {\left(a d x \cos\left(d x + c\right)^{6} - 3 \, a d x \cos\left(d x + c\right)^{4} + 3 \, a d x \cos\left(d x + c\right)^{2} - a d x\right)} \sin\left(d x + c\right) - 48 \, b}{105 \, {\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)} \sin\left(d x + c\right)}"," ",0,"1/105*(176*a*cos(d*x + c)^7 + 105*b*cos(d*x + c)^6 - 406*a*cos(d*x + c)^5 - 210*b*cos(d*x + c)^4 + 350*a*cos(d*x + c)^3 + 168*b*cos(d*x + c)^2 - 105*a*cos(d*x + c) + 105*(a*d*x*cos(d*x + c)^6 - 3*a*d*x*cos(d*x + c)^4 + 3*a*d*x*cos(d*x + c)^2 - a*d*x)*sin(d*x + c) - 48*b)/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*sin(d*x + c))","A",0
271,1,181,0,0.606712," ","integrate((a+b*sec(d*x+c))^2*tan(d*x+c)^9,x, algorithm=""fricas"")","-\frac{2520 \, a^{2} \cos\left(d x + c\right)^{10} \log\left(-\cos\left(d x + c\right)\right) - 5040 \, a b \cos\left(d x + c\right)^{9} + 6720 \, a b \cos\left(d x + c\right)^{7} + 1260 \, {\left(4 \, a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{8} - 6048 \, a b \cos\left(d x + c\right)^{5} - 1260 \, {\left(3 \, a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{6} + 2880 \, a b \cos\left(d x + c\right)^{3} + 840 \, {\left(2 \, a^{2} - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 560 \, a b \cos\left(d x + c\right) - 315 \, {\left(a^{2} - 4 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 252 \, b^{2}}{2520 \, d \cos\left(d x + c\right)^{10}}"," ",0,"-1/2520*(2520*a^2*cos(d*x + c)^10*log(-cos(d*x + c)) - 5040*a*b*cos(d*x + c)^9 + 6720*a*b*cos(d*x + c)^7 + 1260*(4*a^2 - b^2)*cos(d*x + c)^8 - 6048*a*b*cos(d*x + c)^5 - 1260*(3*a^2 - 2*b^2)*cos(d*x + c)^6 + 2880*a*b*cos(d*x + c)^3 + 840*(2*a^2 - 3*b^2)*cos(d*x + c)^4 - 560*a*b*cos(d*x + c) - 315*(a^2 - 4*b^2)*cos(d*x + c)^2 - 252*b^2)/(d*cos(d*x + c)^10)","A",0
272,1,146,0,0.568429," ","integrate((a+b*sec(d*x+c))^2*tan(d*x+c)^7,x, algorithm=""fricas"")","\frac{840 \, a^{2} \cos\left(d x + c\right)^{8} \log\left(-\cos\left(d x + c\right)\right) - 1680 \, a b \cos\left(d x + c\right)^{7} + 1680 \, a b \cos\left(d x + c\right)^{5} + 420 \, {\left(3 \, a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{6} - 1008 \, a b \cos\left(d x + c\right)^{3} - 630 \, {\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{4} + 240 \, a b \cos\left(d x + c\right) + 140 \, {\left(a^{2} - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 105 \, b^{2}}{840 \, d \cos\left(d x + c\right)^{8}}"," ",0,"1/840*(840*a^2*cos(d*x + c)^8*log(-cos(d*x + c)) - 1680*a*b*cos(d*x + c)^7 + 1680*a*b*cos(d*x + c)^5 + 420*(3*a^2 - b^2)*cos(d*x + c)^6 - 1008*a*b*cos(d*x + c)^3 - 630*(a^2 - b^2)*cos(d*x + c)^4 + 240*a*b*cos(d*x + c) + 140*(a^2 - 3*b^2)*cos(d*x + c)^2 + 105*b^2)/(d*cos(d*x + c)^8)","A",0
273,1,115,0,0.559486," ","integrate((a+b*sec(d*x+c))^2*tan(d*x+c)^5,x, algorithm=""fricas"")","-\frac{60 \, a^{2} \cos\left(d x + c\right)^{6} \log\left(-\cos\left(d x + c\right)\right) - 120 \, a b \cos\left(d x + c\right)^{5} + 80 \, a b \cos\left(d x + c\right)^{3} + 30 \, {\left(2 \, a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{4} - 24 \, a b \cos\left(d x + c\right) - 15 \, {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 10 \, b^{2}}{60 \, d \cos\left(d x + c\right)^{6}}"," ",0,"-1/60*(60*a^2*cos(d*x + c)^6*log(-cos(d*x + c)) - 120*a*b*cos(d*x + c)^5 + 80*a*b*cos(d*x + c)^3 + 30*(2*a^2 - b^2)*cos(d*x + c)^4 - 24*a*b*cos(d*x + c) - 15*(a^2 - 2*b^2)*cos(d*x + c)^2 - 10*b^2)/(d*cos(d*x + c)^6)","A",0
274,1,82,0,0.526222," ","integrate((a+b*sec(d*x+c))^2*tan(d*x+c)^3,x, algorithm=""fricas"")","\frac{12 \, a^{2} \cos\left(d x + c\right)^{4} \log\left(-\cos\left(d x + c\right)\right) - 24 \, a b \cos\left(d x + c\right)^{3} + 8 \, a b \cos\left(d x + c\right) + 6 \, {\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} + 3 \, b^{2}}{12 \, d \cos\left(d x + c\right)^{4}}"," ",0,"1/12*(12*a^2*cos(d*x + c)^4*log(-cos(d*x + c)) - 24*a*b*cos(d*x + c)^3 + 8*a*b*cos(d*x + c) + 6*(a^2 - b^2)*cos(d*x + c)^2 + 3*b^2)/(d*cos(d*x + c)^4)","A",0
275,1,51,0,0.559058," ","integrate((a+b*sec(d*x+c))^2*tan(d*x+c),x, algorithm=""fricas"")","-\frac{2 \, a^{2} \cos\left(d x + c\right)^{2} \log\left(-\cos\left(d x + c\right)\right) - 4 \, a b \cos\left(d x + c\right) - b^{2}}{2 \, d \cos\left(d x + c\right)^{2}}"," ",0,"-1/2*(2*a^2*cos(d*x + c)^2*log(-cos(d*x + c)) - 4*a*b*cos(d*x + c) - b^2)/(d*cos(d*x + c)^2)","A",0
276,1,68,0,0.900464," ","integrate(cot(d*x+c)*(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{2 \, b^{2} \log\left(-\cos\left(d x + c\right)\right) - {\left(a^{2} - 2 \, a b + b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - {\left(a^{2} + 2 \, a b + b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{2 \, d}"," ",0,"-1/2*(2*b^2*log(-cos(d*x + c)) - (a^2 - 2*a*b + b^2)*log(1/2*cos(d*x + c) + 1/2) - (a^2 + 2*a*b + b^2)*log(-1/2*cos(d*x + c) + 1/2))/d","A",0
277,1,113,0,0.508915," ","integrate(cot(d*x+c)^3*(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{2 \, a b \cos\left(d x + c\right) + a^{2} + b^{2} - {\left({\left(a^{2} - a b\right)} \cos\left(d x + c\right)^{2} - a^{2} + a b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - {\left({\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} - a^{2} - a b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{2 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"1/2*(2*a*b*cos(d*x + c) + a^2 + b^2 - ((a^2 - a*b)*cos(d*x + c)^2 - a^2 + a*b)*log(1/2*cos(d*x + c) + 1/2) - ((a^2 + a*b)*cos(d*x + c)^2 - a^2 - a*b)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^2 - d)","A",0
278,1,203,0,0.514872," ","integrate(cot(d*x+c)^5*(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{10 \, a b \cos\left(d x + c\right)^{3} - 6 \, a b \cos\left(d x + c\right) + 4 \, {\left(2 \, a^{2} + b^{2}\right)} \cos\left(d x + c\right)^{2} - 6 \, a^{2} - 2 \, b^{2} - {\left({\left(4 \, a^{2} - 3 \, a b\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} - 3 \, a b\right)} \cos\left(d x + c\right)^{2} + 4 \, a^{2} - 3 \, a b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - {\left({\left(4 \, a^{2} + 3 \, a b\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 3 \, a b\right)} \cos\left(d x + c\right)^{2} + 4 \, a^{2} + 3 \, a b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{8 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} + d\right)}}"," ",0,"-1/8*(10*a*b*cos(d*x + c)^3 - 6*a*b*cos(d*x + c) + 4*(2*a^2 + b^2)*cos(d*x + c)^2 - 6*a^2 - 2*b^2 - ((4*a^2 - 3*a*b)*cos(d*x + c)^4 - 2*(4*a^2 - 3*a*b)*cos(d*x + c)^2 + 4*a^2 - 3*a*b)*log(1/2*cos(d*x + c) + 1/2) - ((4*a^2 + 3*a*b)*cos(d*x + c)^4 - 2*(4*a^2 + 3*a*b)*cos(d*x + c)^2 + 4*a^2 + 3*a*b)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)","A",0
279,1,184,0,0.507801," ","integrate((a+b*sec(d*x+c))^2*tan(d*x+c)^6,x, algorithm=""fricas"")","-\frac{1680 \, a^{2} d x \cos\left(d x + c\right)^{7} + 525 \, a b \cos\left(d x + c\right)^{7} \log\left(\sin\left(d x + c\right) + 1\right) - 525 \, a b \cos\left(d x + c\right)^{7} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(1155 \, a b \cos\left(d x + c\right)^{5} + 8 \, {\left(161 \, a^{2} - 15 \, b^{2}\right)} \cos\left(d x + c\right)^{6} - 910 \, a b \cos\left(d x + c\right)^{3} - 8 \, {\left(77 \, a^{2} - 45 \, b^{2}\right)} \cos\left(d x + c\right)^{4} + 280 \, a b \cos\left(d x + c\right) + 24 \, {\left(7 \, a^{2} - 15 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 120 \, b^{2}\right)} \sin\left(d x + c\right)}{1680 \, d \cos\left(d x + c\right)^{7}}"," ",0,"-1/1680*(1680*a^2*d*x*cos(d*x + c)^7 + 525*a*b*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 525*a*b*cos(d*x + c)^7*log(-sin(d*x + c) + 1) - 2*(1155*a*b*cos(d*x + c)^5 + 8*(161*a^2 - 15*b^2)*cos(d*x + c)^6 - 910*a*b*cos(d*x + c)^3 - 8*(77*a^2 - 45*b^2)*cos(d*x + c)^4 + 280*a*b*cos(d*x + c) + 24*(7*a^2 - 15*b^2)*cos(d*x + c)^2 + 120*b^2)*sin(d*x + c))/(d*cos(d*x + c)^7)","A",0
280,1,151,0,0.554055," ","integrate((a+b*sec(d*x+c))^2*tan(d*x+c)^4,x, algorithm=""fricas"")","\frac{120 \, a^{2} d x \cos\left(d x + c\right)^{5} + 45 \, a b \cos\left(d x + c\right)^{5} \log\left(\sin\left(d x + c\right) + 1\right) - 45 \, a b \cos\left(d x + c\right)^{5} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(75 \, a b \cos\left(d x + c\right)^{3} + 4 \, {\left(20 \, a^{2} - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 30 \, a b \cos\left(d x + c\right) - 4 \, {\left(5 \, a^{2} - 6 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 12 \, b^{2}\right)} \sin\left(d x + c\right)}{120 \, d \cos\left(d x + c\right)^{5}}"," ",0,"1/120*(120*a^2*d*x*cos(d*x + c)^5 + 45*a*b*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 45*a*b*cos(d*x + c)^5*log(-sin(d*x + c) + 1) - 2*(75*a*b*cos(d*x + c)^3 + 4*(20*a^2 - 3*b^2)*cos(d*x + c)^4 - 30*a*b*cos(d*x + c) - 4*(5*a^2 - 6*b^2)*cos(d*x + c)^2 - 12*b^2)*sin(d*x + c))/(d*cos(d*x + c)^5)","A",0
281,1,115,0,0.634282," ","integrate((a+b*sec(d*x+c))^2*tan(d*x+c)^2,x, algorithm=""fricas"")","-\frac{6 \, a^{2} d x \cos\left(d x + c\right)^{3} + 3 \, a b \cos\left(d x + c\right)^{3} \log\left(\sin\left(d x + c\right) + 1\right) - 3 \, a b \cos\left(d x + c\right)^{3} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(3 \, a b \cos\left(d x + c\right) + {\left(3 \, a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2}\right)} \sin\left(d x + c\right)}{6 \, d \cos\left(d x + c\right)^{3}}"," ",0,"-1/6*(6*a^2*d*x*cos(d*x + c)^3 + 3*a*b*cos(d*x + c)^3*log(sin(d*x + c) + 1) - 3*a*b*cos(d*x + c)^3*log(-sin(d*x + c) + 1) - 2*(3*a*b*cos(d*x + c) + (3*a^2 - b^2)*cos(d*x + c)^2 + b^2)*sin(d*x + c))/(d*cos(d*x + c)^3)","A",0
282,1,44,0,0.564102," ","integrate(cot(d*x+c)^2*(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{a^{2} d x \sin\left(d x + c\right) + 2 \, a b + {\left(a^{2} + b^{2}\right)} \cos\left(d x + c\right)}{d \sin\left(d x + c\right)}"," ",0,"-(a^2*d*x*sin(d*x + c) + 2*a*b + (a^2 + b^2)*cos(d*x + c))/(d*sin(d*x + c))","A",0
283,1,102,0,0.499167," ","integrate(cot(d*x+c)^4*(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{6 \, a b \cos\left(d x + c\right)^{2} + {\left(4 \, a^{2} + b^{2}\right)} \cos\left(d x + c\right)^{3} - 3 \, a^{2} \cos\left(d x + c\right) - 4 \, a b + 3 \, {\left(a^{2} d x \cos\left(d x + c\right)^{2} - a^{2} d x\right)} \sin\left(d x + c\right)}{3 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}"," ",0,"1/3*(6*a*b*cos(d*x + c)^2 + (4*a^2 + b^2)*cos(d*x + c)^3 - 3*a^2*cos(d*x + c) - 4*a*b + 3*(a^2*d*x*cos(d*x + c)^2 - a^2*d*x)*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))","A",0
284,1,152,0,0.480707," ","integrate(cot(d*x+c)^6*(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{30 \, a b \cos\left(d x + c\right)^{4} + {\left(23 \, a^{2} + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{5} - 35 \, a^{2} \cos\left(d x + c\right)^{3} - 40 \, a b \cos\left(d x + c\right)^{2} + 15 \, a^{2} \cos\left(d x + c\right) + 16 \, a b + 15 \, {\left(a^{2} d x \cos\left(d x + c\right)^{4} - 2 \, a^{2} d x \cos\left(d x + c\right)^{2} + a^{2} d x\right)} \sin\left(d x + c\right)}{15 \, {\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,"-1/15*(30*a*b*cos(d*x + c)^4 + (23*a^2 + 3*b^2)*cos(d*x + c)^5 - 35*a^2*cos(d*x + c)^3 - 40*a*b*cos(d*x + c)^2 + 15*a^2*cos(d*x + c) + 16*a*b + 15*(a^2*d*x*cos(d*x + c)^4 - 2*a^2*d*x*cos(d*x + c)^2 + a^2*d*x)*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))","A",0
285,1,206,0,0.499606," ","integrate(cot(d*x+c)^8*(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{210 \, a b \cos\left(d x + c\right)^{6} + {\left(176 \, a^{2} + 15 \, b^{2}\right)} \cos\left(d x + c\right)^{7} - 406 \, a^{2} \cos\left(d x + c\right)^{5} - 420 \, a b \cos\left(d x + c\right)^{4} + 350 \, a^{2} \cos\left(d x + c\right)^{3} + 336 \, a b \cos\left(d x + c\right)^{2} - 105 \, a^{2} \cos\left(d x + c\right) - 96 \, a b + 105 \, {\left(a^{2} d x \cos\left(d x + c\right)^{6} - 3 \, a^{2} d x \cos\left(d x + c\right)^{4} + 3 \, a^{2} d x \cos\left(d x + c\right)^{2} - a^{2} d x\right)} \sin\left(d x + c\right)}{105 \, {\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)} \sin\left(d x + c\right)}"," ",0,"1/105*(210*a*b*cos(d*x + c)^6 + (176*a^2 + 15*b^2)*cos(d*x + c)^7 - 406*a^2*cos(d*x + c)^5 - 420*a*b*cos(d*x + c)^4 + 350*a^2*cos(d*x + c)^3 + 336*a*b*cos(d*x + c)^2 - 105*a^2*cos(d*x + c) - 96*a*b + 105*(a^2*d*x*cos(d*x + c)^6 - 3*a^2*d*x*cos(d*x + c)^4 + 3*a^2*d*x*cos(d*x + c)^2 - a^2*d*x)*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)*sin(d*x + c))","A",0
286,1,293,0,0.646702," ","integrate(tan(d*x+c)^9/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{70 \, a^{2} b^{6} \cos\left(d x + c\right) + 420 \, {\left(a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right)} \cos\left(d x + c\right)^{7} \log\left(a \cos\left(d x + c\right) + b\right) - 420 \, {\left(a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6}\right)} \cos\left(d x + c\right)^{7} \log\left(-\cos\left(d x + c\right)\right) - 60 \, a b^{7} - 420 \, {\left(a^{7} b - 4 \, a^{5} b^{3} + 6 \, a^{3} b^{5} - 4 \, a b^{7}\right)} \cos\left(d x + c\right)^{6} + 210 \, {\left(a^{6} b^{2} - 4 \, a^{4} b^{4} + 6 \, a^{2} b^{6}\right)} \cos\left(d x + c\right)^{5} - 140 \, {\left(a^{5} b^{3} - 4 \, a^{3} b^{5} + 6 \, a b^{7}\right)} \cos\left(d x + c\right)^{4} + 105 \, {\left(a^{4} b^{4} - 4 \, a^{2} b^{6}\right)} \cos\left(d x + c\right)^{3} - 84 \, {\left(a^{3} b^{5} - 4 \, a b^{7}\right)} \cos\left(d x + c\right)^{2}}{420 \, a b^{8} d \cos\left(d x + c\right)^{7}}"," ",0,"-1/420*(70*a^2*b^6*cos(d*x + c) + 420*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*cos(d*x + c)^7*log(a*cos(d*x + c) + b) - 420*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6)*cos(d*x + c)^7*log(-cos(d*x + c)) - 60*a*b^7 - 420*(a^7*b - 4*a^5*b^3 + 6*a^3*b^5 - 4*a*b^7)*cos(d*x + c)^6 + 210*(a^6*b^2 - 4*a^4*b^4 + 6*a^2*b^6)*cos(d*x + c)^5 - 140*(a^5*b^3 - 4*a^3*b^5 + 6*a*b^7)*cos(d*x + c)^4 + 105*(a^4*b^4 - 4*a^2*b^6)*cos(d*x + c)^3 - 84*(a^3*b^5 - 4*a*b^7)*cos(d*x + c)^2)/(a*b^8*d*cos(d*x + c)^7)","A",0
287,1,205,0,0.583565," ","integrate(tan(d*x+c)^7/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{15 \, a^{2} b^{4} \cos\left(d x + c\right) + 60 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} \cos\left(d x + c\right)^{5} \log\left(a \cos\left(d x + c\right) + b\right) - 60 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{5} \log\left(-\cos\left(d x + c\right)\right) - 12 \, a b^{5} - 60 \, {\left(a^{5} b - 3 \, a^{3} b^{3} + 3 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} + 30 \, {\left(a^{4} b^{2} - 3 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} - 20 \, {\left(a^{3} b^{3} - 3 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}}{60 \, a b^{6} d \cos\left(d x + c\right)^{5}}"," ",0,"-1/60*(15*a^2*b^4*cos(d*x + c) + 60*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos(d*x + c)^5*log(a*cos(d*x + c) + b) - 60*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c)^5*log(-cos(d*x + c)) - 12*a*b^5 - 60*(a^5*b - 3*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^4 + 30*(a^4*b^2 - 3*a^2*b^4)*cos(d*x + c)^3 - 20*(a^3*b^3 - 3*a*b^5)*cos(d*x + c)^2)/(a*b^6*d*cos(d*x + c)^5)","A",0
288,1,129,0,0.545811," ","integrate(tan(d*x+c)^5/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{3 \, a^{2} b^{2} \cos\left(d x + c\right) + 6 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{3} \log\left(a \cos\left(d x + c\right) + b\right) - 6 \, {\left(a^{4} - 2 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{3} \log\left(-\cos\left(d x + c\right)\right) - 2 \, a b^{3} - 6 \, {\left(a^{3} b - 2 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}}{6 \, a b^{4} d \cos\left(d x + c\right)^{3}}"," ",0,"-1/6*(3*a^2*b^2*cos(d*x + c) + 6*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^3*log(a*cos(d*x + c) + b) - 6*(a^4 - 2*a^2*b^2)*cos(d*x + c)^3*log(-cos(d*x + c)) - 2*a*b^3 - 6*(a^3*b - 2*a*b^3)*cos(d*x + c)^2)/(a*b^4*d*cos(d*x + c)^3)","A",0
289,1,69,0,0.509402," ","integrate(tan(d*x+c)^3/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\frac{a^{2} \cos\left(d x + c\right) \log\left(-\cos\left(d x + c\right)\right) - {\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right) \log\left(a \cos\left(d x + c\right) + b\right) + a b}{a b^{2} d \cos\left(d x + c\right)}"," ",0,"(a^2*cos(d*x + c)*log(-cos(d*x + c)) - (a^2 - b^2)*cos(d*x + c)*log(a*cos(d*x + c) + b) + a*b)/(a*b^2*d*cos(d*x + c))","A",0
290,1,19,0,0.484396," ","integrate(tan(d*x+c)/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{\log\left(a \cos\left(d x + c\right) + b\right)}{a d}"," ",0,"-log(a*cos(d*x + c) + b)/(a*d)","A",0
291,1,75,0,0.518349," ","integrate(cot(d*x+c)/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","-\frac{2 \, b^{2} \log\left(a \cos\left(d x + c\right) + b\right) - {\left(a^{2} + a b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - {\left(a^{2} - a b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{2 \, {\left(a^{3} - a b^{2}\right)} d}"," ",0,"-1/2*(2*b^2*log(a*cos(d*x + c) + b) - (a^2 + a*b)*log(1/2*cos(d*x + c) + 1/2) - (a^2 - a*b)*log(-1/2*cos(d*x + c) + 1/2))/((a^3 - a*b^2)*d)","A",0
292,1,263,0,0.603475," ","integrate(cot(d*x+c)^3/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\frac{2 \, a^{4} - 2 \, a^{2} b^{2} - 2 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right) - 4 \, {\left(b^{4} \cos\left(d x + c\right)^{2} - b^{4}\right)} \log\left(a \cos\left(d x + c\right) + b\right) + {\left(2 \, a^{4} + a^{3} b - 4 \, a^{2} b^{2} - 3 \, a b^{3} - {\left(2 \, a^{4} + a^{3} b - 4 \, a^{2} b^{2} - 3 \, a 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) + {\left(2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3} - {\left(2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a 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)}{4 \, {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d\right)}}"," ",0,"1/4*(2*a^4 - 2*a^2*b^2 - 2*(a^3*b - a*b^3)*cos(d*x + c) - 4*(b^4*cos(d*x + c)^2 - b^4)*log(a*cos(d*x + c) + b) + (2*a^4 + a^3*b - 4*a^2*b^2 - 3*a*b^3 - (2*a^4 + a^3*b - 4*a^2*b^2 - 3*a*b^3)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + (2*a^4 - a^3*b - 4*a^2*b^2 + 3*a*b^3 - (2*a^4 - a^3*b - 4*a^2*b^2 + 3*a*b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d)","A",0
293,1,576,0,0.767538," ","integrate(cot(d*x+c)^5/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\frac{12 \, a^{6} - 32 \, a^{4} b^{2} + 20 \, a^{2} b^{4} + 2 \, {\left(5 \, a^{5} b - 14 \, a^{3} b^{3} + 9 \, a b^{5}\right)} \cos\left(d x + c\right)^{3} - 8 \, {\left(2 \, a^{6} - 5 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(3 \, a^{5} b - 10 \, a^{3} b^{3} + 7 \, a b^{5}\right)} \cos\left(d x + c\right) - 16 \, {\left(b^{6} \cos\left(d x + c\right)^{4} - 2 \, b^{6} \cos\left(d x + c\right)^{2} + b^{6}\right)} \log\left(a \cos\left(d x + c\right) + b\right) + {\left(8 \, a^{6} + 3 \, a^{5} b - 24 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 15 \, a b^{5} + {\left(8 \, a^{6} + 3 \, a^{5} b - 24 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(8 \, a^{6} + 3 \, a^{5} b - 24 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 15 \, a 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) + {\left(8 \, a^{6} - 3 \, a^{5} b - 24 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} - 15 \, a b^{5} + {\left(8 \, a^{6} - 3 \, a^{5} b - 24 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} - 15 \, a b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(8 \, a^{6} - 3 \, a^{5} b - 24 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} - 15 \, a 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)}{16 \, {\left({\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\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} + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} d\right)}}"," ",0,"1/16*(12*a^6 - 32*a^4*b^2 + 20*a^2*b^4 + 2*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cos(d*x + c)^3 - 8*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c)^2 - 2*(3*a^5*b - 10*a^3*b^3 + 7*a*b^5)*cos(d*x + c) - 16*(b^6*cos(d*x + c)^4 - 2*b^6*cos(d*x + c)^2 + b^6)*log(a*cos(d*x + c) + b) + (8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5 + (8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cos(d*x + c)^4 - 2*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + (8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5 + (8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cos(d*x + c)^4 - 2*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d*cos(d*x + c)^4 - 2*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d)","B",0
294,1,603,0,1.052513," ","integrate(tan(d*x+c)^6/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\left[-\frac{48 \, b^{5} d x \cos\left(d x + c\right)^{4} - 24 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \sqrt{a^{2} - b^{2}} \cos\left(d x + c\right)^{4} \log\left(\frac{2 \, a b \cos\left(d x + c\right) - {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - b^{2}} {\left(b \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) + 2 \, a^{2} - b^{2}}{a^{2} \cos\left(d x + c\right)^{2} + 2 \, a b \cos\left(d x + c\right) + b^{2}}\right) - 3 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(d x + c\right)^{4} \log\left(\sin\left(d x + c\right) + 1\right) + 3 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(d x + c\right)^{4} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(8 \, a^{2} b^{3} \cos\left(d x + c\right) - 6 \, a b^{4} + 8 \, {\left(3 \, a^{4} b - 7 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(4 \, a^{3} b^{2} - 9 \, a b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{48 \, a b^{5} d \cos\left(d x + c\right)^{4}}, -\frac{48 \, b^{5} d x \cos\left(d x + c\right)^{4} + 48 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \cos\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right)}\right) \cos\left(d x + c\right)^{4} - 3 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(d x + c\right)^{4} \log\left(\sin\left(d x + c\right) + 1\right) + 3 \, {\left(8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right)} \cos\left(d x + c\right)^{4} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(8 \, a^{2} b^{3} \cos\left(d x + c\right) - 6 \, a b^{4} + 8 \, {\left(3 \, a^{4} b - 7 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(4 \, a^{3} b^{2} - 9 \, a b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{48 \, a b^{5} d \cos\left(d x + c\right)^{4}}\right]"," ",0,"[-1/48*(48*b^5*d*x*cos(d*x + c)^4 - 24*(a^4 - 2*a^2*b^2 + b^4)*sqrt(a^2 - b^2)*cos(d*x + c)^4*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^4*log(sin(d*x + c) + 1) + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(8*a^2*b^3*cos(d*x + c) - 6*a*b^4 + 8*(3*a^4*b - 7*a^2*b^3)*cos(d*x + c)^3 - 3*(4*a^3*b^2 - 9*a*b^4)*cos(d*x + c)^2)*sin(d*x + c))/(a*b^5*d*cos(d*x + c)^4), -1/48*(48*b^5*d*x*cos(d*x + c)^4 + 48*(a^4 - 2*a^2*b^2 + b^4)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c)))*cos(d*x + c)^4 - 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^4*log(sin(d*x + c) + 1) + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(8*a^2*b^3*cos(d*x + c) - 6*a*b^4 + 8*(3*a^4*b - 7*a^2*b^3)*cos(d*x + c)^3 - 3*(4*a^3*b^2 - 9*a*b^4)*cos(d*x + c)^2)*sin(d*x + c))/(a*b^5*d*cos(d*x + c)^4)]","A",0
295,1,444,0,0.926222," ","integrate(tan(d*x+c)^4/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\left[\frac{4 \, b^{3} d x \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{2} - b^{2}\right)}^{\frac{3}{2}} \cos\left(d x + c\right)^{2} \log\left(\frac{2 \, a b \cos\left(d x + c\right) - {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, \sqrt{a^{2} - b^{2}} {\left(b \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) + 2 \, a^{2} - b^{2}}{a^{2} \cos\left(d x + c\right)^{2} + 2 \, a b \cos\left(d x + c\right) + b^{2}}\right) + {\left(2 \, a^{3} - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(2 \, a^{3} - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(2 \, a^{2} b \cos\left(d x + c\right) - a b^{2}\right)} \sin\left(d x + c\right)}{4 \, a b^{3} d \cos\left(d x + c\right)^{2}}, \frac{4 \, b^{3} d x \cos\left(d x + c\right)^{2} - 4 \, {\left(a^{2} - b^{2}\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \cos\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right)}\right) \cos\left(d x + c\right)^{2} + {\left(2 \, a^{3} - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - {\left(2 \, a^{3} - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(2 \, a^{2} b \cos\left(d x + c\right) - a b^{2}\right)} \sin\left(d x + c\right)}{4 \, a b^{3} d \cos\left(d x + c\right)^{2}}\right]"," ",0,"[1/4*(4*b^3*d*x*cos(d*x + c)^2 - 2*(a^2 - b^2)^(3/2)*cos(d*x + c)^2*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 + 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + (2*a^3 - 3*a*b^2)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (2*a^3 - 3*a*b^2)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 2*(2*a^2*b*cos(d*x + c) - a*b^2)*sin(d*x + c))/(a*b^3*d*cos(d*x + c)^2), 1/4*(4*b^3*d*x*cos(d*x + c)^2 - 4*(a^2 - b^2)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c)))*cos(d*x + c)^2 + (2*a^3 - 3*a*b^2)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (2*a^3 - 3*a*b^2)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 2*(2*a^2*b*cos(d*x + c) - a*b^2)*sin(d*x + c))/(a*b^3*d*cos(d*x + c)^2)]","A",0
296,1,253,0,0.546701," ","integrate(tan(d*x+c)^2/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\left[-\frac{2 \, b d x - a \log\left(\sin\left(d x + c\right) + 1\right) + a \log\left(-\sin\left(d x + c\right) + 1\right) - \sqrt{a^{2} - b^{2}} \log\left(\frac{2 \, a b \cos\left(d x + c\right) - {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - b^{2}} {\left(b \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) + 2 \, a^{2} - b^{2}}{a^{2} \cos\left(d x + c\right)^{2} + 2 \, a b \cos\left(d x + c\right) + b^{2}}\right)}{2 \, a b d}, -\frac{2 \, b d x - a \log\left(\sin\left(d x + c\right) + 1\right) + a \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \cos\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right)}\right)}{2 \, a b d}\right]"," ",0,"[-1/2*(2*b*d*x - a*log(sin(d*x + c) + 1) + a*log(-sin(d*x + c) + 1) - sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)))/(a*b*d), -1/2*(2*b*d*x - a*log(sin(d*x + c) + 1) + a*log(-sin(d*x + c) + 1) + 2*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))))/(a*b*d)]","A",0
297,1,362,0,0.506150," ","integrate(cot(d*x+c)^2/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{a^{2} - b^{2}} b^{3} \log\left(\frac{2 \, a b \cos\left(d x + c\right) - {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, \sqrt{a^{2} - b^{2}} {\left(b \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) + 2 \, a^{2} - b^{2}}{a^{2} \cos\left(d x + c\right)^{2} + 2 \, a b \cos\left(d x + c\right) + b^{2}}\right) \sin\left(d x + c\right) - 2 \, a^{3} b + 2 \, a b^{3} + 2 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d x \sin\left(d x + c\right) + 2 \, {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right)}{2 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d \sin\left(d x + c\right)}, -\frac{\sqrt{-a^{2} + b^{2}} b^{3} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \cos\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) - a^{3} b + a b^{3} + {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d x \sin\left(d x + c\right) + {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right)}{{\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d \sin\left(d x + c\right)}\right]"," ",0,"[-1/2*(sqrt(a^2 - b^2)*b^3*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 + 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2))*sin(d*x + c) - 2*a^3*b + 2*a*b^3 + 2*(a^4 - 2*a^2*b^2 + b^4)*d*x*sin(d*x + c) + 2*(a^4 - a^2*b^2)*cos(d*x + c))/((a^5 - 2*a^3*b^2 + a*b^4)*d*sin(d*x + c)), -(sqrt(-a^2 + b^2)*b^3*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c)))*sin(d*x + c) - a^3*b + a*b^3 + (a^4 - 2*a^2*b^2 + b^4)*d*x*sin(d*x + c) + (a^4 - a^2*b^2)*cos(d*x + c))/((a^5 - 2*a^3*b^2 + a*b^4)*d*sin(d*x + c))]","A",0
298,1,742,0,0.529621," ","integrate(cot(d*x+c)^4/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\left[\frac{4 \, a^{5} b - 14 \, a^{3} b^{3} + 10 \, a b^{5} + 2 \, {\left(4 \, a^{6} - 11 \, a^{4} b^{2} + 7 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(b^{5} \cos\left(d x + c\right)^{2} - b^{5}\right)} \sqrt{a^{2} - b^{2}} \log\left(\frac{2 \, a b \cos\left(d x + c\right) - {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - b^{2}} {\left(b \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) + 2 \, a^{2} - b^{2}}{a^{2} \cos\left(d x + c\right)^{2} + 2 \, a b \cos\left(d x + c\right) + b^{2}}\right) \sin\left(d x + c\right) - 6 \, {\left(a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right)} \cos\left(d x + c\right)^{2} - 6 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 2 \, a^{2} b^{4}\right)} \cos\left(d x + c\right) + 6 \, {\left({\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} d x \cos\left(d x + c\right)^{2} - {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} d x\right)} \sin\left(d x + c\right)}{6 \, {\left({\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} - {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} d\right)} \sin\left(d x + c\right)}, \frac{2 \, a^{5} b - 7 \, a^{3} b^{3} + 5 \, a b^{5} + {\left(4 \, a^{6} - 11 \, a^{4} b^{2} + 7 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(b^{5} \cos\left(d x + c\right)^{2} - b^{5}\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \cos\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) - 3 \, {\left(a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right)} \cos\left(d x + c\right)^{2} - 3 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 2 \, a^{2} b^{4}\right)} \cos\left(d x + c\right) + 3 \, {\left({\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} d x \cos\left(d x + c\right)^{2} - {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} d x\right)} \sin\left(d x + c\right)}{3 \, {\left({\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} - {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[1/6*(4*a^5*b - 14*a^3*b^3 + 10*a*b^5 + 2*(4*a^6 - 11*a^4*b^2 + 7*a^2*b^4)*cos(d*x + c)^3 + 3*(b^5*cos(d*x + c)^2 - b^5)*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2))*sin(d*x + c) - 6*(a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(d*x + c)^2 - 6*(a^6 - 3*a^4*b^2 + 2*a^2*b^4)*cos(d*x + c) + 6*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d*x*cos(d*x + c)^2 - (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d*x)*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 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d)*sin(d*x + c)), 1/3*(2*a^5*b - 7*a^3*b^3 + 5*a*b^5 + (4*a^6 - 11*a^4*b^2 + 7*a^2*b^4)*cos(d*x + c)^3 - 3*(b^5*cos(d*x + c)^2 - b^5)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c)))*sin(d*x + c) - 3*(a^5*b - 3*a^3*b^3 + 2*a*b^5)*cos(d*x + c)^2 - 3*(a^6 - 3*a^4*b^2 + 2*a^2*b^4)*cos(d*x + c) + 3*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d*x*cos(d*x + c)^2 - (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d*x)*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 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d)*sin(d*x + c))]","B",0
299,1,423,0,0.639598," ","integrate(tan(d*x+c)^9/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{14 \, a^{3} b^{6} \cos\left(d x + c\right) - 10 \, a^{2} b^{7} + 60 \, {\left(7 \, a^{8} b - 20 \, a^{6} b^{3} + 18 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right)} \cos\left(d x + c\right)^{6} + 30 \, {\left(7 \, a^{7} b^{2} - 20 \, a^{5} b^{4} + 18 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)^{5} - 10 \, {\left(7 \, a^{6} b^{3} - 20 \, a^{4} b^{5} + 18 \, a^{2} b^{7}\right)} \cos\left(d x + c\right)^{4} + 5 \, {\left(7 \, a^{5} b^{4} - 20 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(7 \, a^{4} b^{5} - 20 \, a^{2} b^{7}\right)} \cos\left(d x + c\right)^{2} - 60 \, {\left({\left(7 \, a^{9} - 20 \, a^{7} b^{2} + 18 \, a^{5} b^{4} - 4 \, a^{3} b^{6} - a b^{8}\right)} \cos\left(d x + c\right)^{7} + {\left(7 \, a^{8} b - 20 \, a^{6} b^{3} + 18 \, a^{4} b^{5} - 4 \, a^{2} b^{7} - b^{9}\right)} \cos\left(d x + c\right)^{6}\right)} \log\left(a \cos\left(d x + c\right) + b\right) + 60 \, {\left({\left(7 \, a^{9} - 20 \, a^{7} b^{2} + 18 \, a^{5} b^{4} - 4 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)^{7} + {\left(7 \, a^{8} b - 20 \, a^{6} b^{3} + 18 \, a^{4} b^{5} - 4 \, a^{2} b^{7}\right)} \cos\left(d x + c\right)^{6}\right)} \log\left(-\cos\left(d x + c\right)\right)}{60 \, {\left(a^{3} b^{8} d \cos\left(d x + c\right)^{7} + a^{2} b^{9} d \cos\left(d x + c\right)^{6}\right)}}"," ",0,"-1/60*(14*a^3*b^6*cos(d*x + c) - 10*a^2*b^7 + 60*(7*a^8*b - 20*a^6*b^3 + 18*a^4*b^5 - 4*a^2*b^7 + b^9)*cos(d*x + c)^6 + 30*(7*a^7*b^2 - 20*a^5*b^4 + 18*a^3*b^6)*cos(d*x + c)^5 - 10*(7*a^6*b^3 - 20*a^4*b^5 + 18*a^2*b^7)*cos(d*x + c)^4 + 5*(7*a^5*b^4 - 20*a^3*b^6)*cos(d*x + c)^3 - 3*(7*a^4*b^5 - 20*a^2*b^7)*cos(d*x + c)^2 - 60*((7*a^9 - 20*a^7*b^2 + 18*a^5*b^4 - 4*a^3*b^6 - a*b^8)*cos(d*x + c)^7 + (7*a^8*b - 20*a^6*b^3 + 18*a^4*b^5 - 4*a^2*b^7 - b^9)*cos(d*x + c)^6)*log(a*cos(d*x + c) + b) + 60*((7*a^9 - 20*a^7*b^2 + 18*a^5*b^4 - 4*a^3*b^6)*cos(d*x + c)^7 + (7*a^8*b - 20*a^6*b^3 + 18*a^4*b^5 - 4*a^2*b^7)*cos(d*x + c)^6)*log(-cos(d*x + c)))/(a^3*b^8*d*cos(d*x + c)^7 + a^2*b^9*d*cos(d*x + c)^6)","A",0
300,1,312,0,0.584044," ","integrate(tan(d*x+c)^7/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{5 \, a^{3} b^{4} \cos\left(d x + c\right) - 3 \, a^{2} b^{5} + 12 \, {\left(5 \, a^{6} b - 9 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{4} + 6 \, {\left(5 \, a^{5} b^{2} - 9 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(5 \, a^{4} b^{3} - 9 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} - 12 \, {\left({\left(5 \, a^{7} - 9 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right)} \cos\left(d x + c\right)^{5} + {\left(5 \, a^{6} b - 9 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right)} \cos\left(d x + c\right)^{4}\right)} \log\left(a \cos\left(d x + c\right) + b\right) + 12 \, {\left({\left(5 \, a^{7} - 9 \, a^{5} b^{2} + 3 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{5} + {\left(5 \, a^{6} b - 9 \, a^{4} b^{3} + 3 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{4}\right)} \log\left(-\cos\left(d x + c\right)\right)}{12 \, {\left(a^{3} b^{6} d \cos\left(d x + c\right)^{5} + a^{2} b^{7} d \cos\left(d x + c\right)^{4}\right)}}"," ",0,"-1/12*(5*a^3*b^4*cos(d*x + c) - 3*a^2*b^5 + 12*(5*a^6*b - 9*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^4 + 6*(5*a^5*b^2 - 9*a^3*b^4)*cos(d*x + c)^3 - 2*(5*a^4*b^3 - 9*a^2*b^5)*cos(d*x + c)^2 - 12*((5*a^7 - 9*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(d*x + c)^5 + (5*a^6*b - 9*a^4*b^3 + 3*a^2*b^5 + b^7)*cos(d*x + c)^4)*log(a*cos(d*x + c) + b) + 12*((5*a^7 - 9*a^5*b^2 + 3*a^3*b^4)*cos(d*x + c)^5 + (5*a^6*b - 9*a^4*b^3 + 3*a^2*b^5)*cos(d*x + c)^4)*log(-cos(d*x + c)))/(a^3*b^6*d*cos(d*x + c)^5 + a^2*b^7*d*cos(d*x + c)^4)","A",0
301,1,219,0,0.545926," ","integrate(tan(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{3 \, a^{3} b^{2} \cos\left(d x + c\right) - a^{2} b^{3} + 2 \, {\left(3 \, a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left({\left(3 \, a^{5} - 2 \, a^{3} b^{2} - a b^{4}\right)} \cos\left(d x + c\right)^{3} + {\left(3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(a \cos\left(d x + c\right) + b\right) + 2 \, {\left({\left(3 \, a^{5} - 2 \, a^{3} b^{2}\right)} \cos\left(d x + c\right)^{3} + {\left(3 \, a^{4} b - 2 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\cos\left(d x + c\right)\right)}{2 \, {\left(a^{3} b^{4} d \cos\left(d x + c\right)^{3} + a^{2} b^{5} d \cos\left(d x + c\right)^{2}\right)}}"," ",0,"-1/2*(3*a^3*b^2*cos(d*x + c) - a^2*b^3 + 2*(3*a^4*b - 2*a^2*b^3 + b^5)*cos(d*x + c)^2 - 2*((3*a^5 - 2*a^3*b^2 - a*b^4)*cos(d*x + c)^3 + (3*a^4*b - 2*a^2*b^3 - b^5)*cos(d*x + c)^2)*log(a*cos(d*x + c) + b) + 2*((3*a^5 - 2*a^3*b^2)*cos(d*x + c)^3 + (3*a^4*b - 2*a^2*b^3)*cos(d*x + c)^2)*log(-cos(d*x + c)))/(a^3*b^4*d*cos(d*x + c)^3 + a^2*b^5*d*cos(d*x + c)^2)","A",0
302,1,102,0,0.509053," ","integrate(tan(d*x+c)^3/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{a^{2} b - b^{3} - {\left(a^{2} b + b^{3} + {\left(a^{3} + a b^{2}\right)} \cos\left(d x + c\right)\right)} \log\left(a \cos\left(d x + c\right) + b\right) + {\left(a^{3} \cos\left(d x + c\right) + a^{2} b\right)} \log\left(-\cos\left(d x + c\right)\right)}{a^{3} b^{2} d \cos\left(d x + c\right) + a^{2} b^{3} d}"," ",0,"-(a^2*b - b^3 - (a^2*b + b^3 + (a^3 + a*b^2)*cos(d*x + c))*log(a*cos(d*x + c) + b) + (a^3*cos(d*x + c) + a^2*b)*log(-cos(d*x + c)))/(a^3*b^2*d*cos(d*x + c) + a^2*b^3*d)","A",0
303,1,46,0,0.506212," ","integrate(tan(d*x+c)/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{{\left(a \cos\left(d x + c\right) + b\right)} \log\left(a \cos\left(d x + c\right) + b\right) + b}{a^{3} d \cos\left(d x + c\right) + a^{2} b d}"," ",0,"-((a*cos(d*x + c) + b)*log(a*cos(d*x + c) + b) + b)/(a^3*d*cos(d*x + c) + a^2*b*d)","A",0
304,1,234,0,0.603297," ","integrate(cot(d*x+c)/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","-\frac{2 \, a^{2} b^{3} - 2 \, b^{5} + 2 \, {\left(3 \, a^{2} b^{3} - b^{5} + {\left(3 \, a^{3} b^{2} - a b^{4}\right)} \cos\left(d x + c\right)\right)} \log\left(a \cos\left(d x + c\right) + b\right) - {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\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^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} d \cos\left(d x + c\right) + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} d\right)}}"," ",0,"-1/2*(2*a^2*b^3 - 2*b^5 + 2*(3*a^2*b^3 - b^5 + (3*a^3*b^2 - a*b^4)*cos(d*x + c))*log(a*cos(d*x + c) + b) - (a^4*b + 2*a^3*b^2 + a^2*b^3 + (a^5 + 2*a^4*b + a^3*b^2)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - (a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d)","A",0
305,1,693,0,0.770057," ","integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{a^{6} b + a^{2} b^{5} - 2 \, b^{7} - 2 \, {\left(a^{6} b - a^{4} b^{3} + a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} \cos\left(d x + c\right) + 2 \, {\left(5 \, a^{2} b^{5} - b^{7} - {\left(5 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(5 \, a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(5 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)\right)} \log\left(a \cos\left(d x + c\right) + b\right) + {\left(a^{6} b + a^{5} b^{2} - 3 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5} - {\left(a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b + a^{5} b^{2} - 3 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left(a^{6} b - a^{5} b^{2} - 3 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5} - {\left(a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - a^{5} b^{2} - 3 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 5 \, a^{4} b^{3} - 2 \, a^{3} b^{4}\right)} \cos\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^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right)^{3} + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right) - {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)}}"," ",0,"1/2*(a^6*b + a^2*b^5 - 2*b^7 - 2*(a^6*b - a^4*b^3 + a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 2*a^5*b^2 + a^3*b^4)*cos(d*x + c) + 2*(5*a^2*b^5 - b^7 - (5*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (5*a^2*b^5 - b^7)*cos(d*x + c)^2 + (5*a^3*b^4 - a*b^6)*cos(d*x + c))*log(a*cos(d*x + c) + b) + (a^6*b + a^5*b^2 - 3*a^4*b^3 - 5*a^3*b^4 - 2*a^2*b^5 - (a^7 + a^6*b - 3*a^5*b^2 - 5*a^4*b^3 - 2*a^3*b^4)*cos(d*x + c)^3 - (a^6*b + a^5*b^2 - 3*a^4*b^3 - 5*a^3*b^4 - 2*a^2*b^5)*cos(d*x + c)^2 + (a^7 + a^6*b - 3*a^5*b^2 - 5*a^4*b^3 - 2*a^3*b^4)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (a^6*b - a^5*b^2 - 3*a^4*b^3 + 5*a^3*b^4 - 2*a^2*b^5 - (a^7 - a^6*b - 3*a^5*b^2 + 5*a^4*b^3 - 2*a^3*b^4)*cos(d*x + c)^3 - (a^6*b - a^5*b^2 - 3*a^4*b^3 + 5*a^3*b^4 - 2*a^2*b^5)*cos(d*x + c)^2 + (a^7 - a^6*b - 3*a^5*b^2 + 5*a^4*b^3 - 2*a^3*b^4)*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d)","B",0
306,1,1378,0,1.101759," ","integrate(cot(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","\frac{6 \, a^{8} b - 18 \, a^{6} b^{3} + 2 \, a^{4} b^{5} + 2 \, a^{2} b^{7} + 8 \, b^{9} + 2 \, {\left(5 \, a^{8} b - 18 \, a^{6} b^{3} + 13 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + 4 \, b^{9}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{9} - 15 \, a^{7} b^{2} + 18 \, a^{5} b^{4} - 7 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(7 \, a^{8} b - 24 \, a^{6} b^{3} + 11 \, a^{4} b^{5} - 2 \, a^{2} b^{7} + 8 \, b^{9}\right)} \cos\left(d x + c\right)^{2} + 6 \, {\left(a^{9} - 4 \, a^{7} b^{2} + 5 \, a^{5} b^{4} - 2 \, a^{3} b^{6}\right)} \cos\left(d x + c\right) - 8 \, {\left(7 \, a^{2} b^{7} - b^{9} + {\left(7 \, a^{3} b^{6} - a b^{8}\right)} \cos\left(d x + c\right)^{5} + {\left(7 \, a^{2} b^{7} - b^{9}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(7 \, a^{3} b^{6} - a b^{8}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(7 \, a^{2} b^{7} - b^{9}\right)} \cos\left(d x + c\right)^{2} + {\left(7 \, a^{3} b^{6} - a b^{8}\right)} \cos\left(d x + c\right)\right)} \log\left(a \cos\left(d x + c\right) + b\right) + {\left(4 \, a^{8} b + 3 \, a^{7} b^{2} - 16 \, a^{6} b^{3} - 14 \, a^{5} b^{4} + 24 \, a^{4} b^{5} + 35 \, a^{3} b^{6} + 12 \, a^{2} b^{7} + {\left(4 \, a^{9} + 3 \, a^{8} b - 16 \, a^{7} b^{2} - 14 \, a^{6} b^{3} + 24 \, a^{5} b^{4} + 35 \, a^{4} b^{5} + 12 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)^{5} + {\left(4 \, a^{8} b + 3 \, a^{7} b^{2} - 16 \, a^{6} b^{3} - 14 \, a^{5} b^{4} + 24 \, a^{4} b^{5} + 35 \, a^{3} b^{6} + 12 \, a^{2} b^{7}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{9} + 3 \, a^{8} b - 16 \, a^{7} b^{2} - 14 \, a^{6} b^{3} + 24 \, a^{5} b^{4} + 35 \, a^{4} b^{5} + 12 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(4 \, a^{8} b + 3 \, a^{7} b^{2} - 16 \, a^{6} b^{3} - 14 \, a^{5} b^{4} + 24 \, a^{4} b^{5} + 35 \, a^{3} b^{6} + 12 \, a^{2} b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{9} + 3 \, a^{8} b - 16 \, a^{7} b^{2} - 14 \, a^{6} b^{3} + 24 \, a^{5} b^{4} + 35 \, a^{4} b^{5} + 12 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left(4 \, a^{8} b - 3 \, a^{7} b^{2} - 16 \, a^{6} b^{3} + 14 \, a^{5} b^{4} + 24 \, a^{4} b^{5} - 35 \, a^{3} b^{6} + 12 \, a^{2} b^{7} + {\left(4 \, a^{9} - 3 \, a^{8} b - 16 \, a^{7} b^{2} + 14 \, a^{6} b^{3} + 24 \, a^{5} b^{4} - 35 \, a^{4} b^{5} + 12 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)^{5} + {\left(4 \, a^{8} b - 3 \, a^{7} b^{2} - 16 \, a^{6} b^{3} + 14 \, a^{5} b^{4} + 24 \, a^{4} b^{5} - 35 \, a^{3} b^{6} + 12 \, a^{2} b^{7}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{9} - 3 \, a^{8} b - 16 \, a^{7} b^{2} + 14 \, a^{6} b^{3} + 24 \, a^{5} b^{4} - 35 \, a^{4} b^{5} + 12 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(4 \, a^{8} b - 3 \, a^{7} b^{2} - 16 \, a^{6} b^{3} + 14 \, a^{5} b^{4} + 24 \, a^{4} b^{5} - 35 \, a^{3} b^{6} + 12 \, a^{2} b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{9} - 3 \, a^{8} b - 16 \, a^{7} b^{2} + 14 \, a^{6} b^{3} + 24 \, a^{5} b^{4} - 35 \, a^{4} b^{5} + 12 \, a^{3} b^{6}\right)} \cos\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{8 \, {\left({\left(a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right)} d \cos\left(d x + c\right)^{5} + {\left(a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right)} d \cos\left(d x + c\right)^{3} - 2 \, {\left(a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right)} d \cos\left(d x + c\right) + {\left(a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right)} d\right)}}"," ",0,"1/8*(6*a^8*b - 18*a^6*b^3 + 2*a^4*b^5 + 2*a^2*b^7 + 8*b^9 + 2*(5*a^8*b - 18*a^6*b^3 + 13*a^4*b^5 - 4*a^2*b^7 + 4*b^9)*cos(d*x + c)^4 - 2*(4*a^9 - 15*a^7*b^2 + 18*a^5*b^4 - 7*a^3*b^6)*cos(d*x + c)^3 - 2*(7*a^8*b - 24*a^6*b^3 + 11*a^4*b^5 - 2*a^2*b^7 + 8*b^9)*cos(d*x + c)^2 + 6*(a^9 - 4*a^7*b^2 + 5*a^5*b^4 - 2*a^3*b^6)*cos(d*x + c) - 8*(7*a^2*b^7 - b^9 + (7*a^3*b^6 - a*b^8)*cos(d*x + c)^5 + (7*a^2*b^7 - b^9)*cos(d*x + c)^4 - 2*(7*a^3*b^6 - a*b^8)*cos(d*x + c)^3 - 2*(7*a^2*b^7 - b^9)*cos(d*x + c)^2 + (7*a^3*b^6 - a*b^8)*cos(d*x + c))*log(a*cos(d*x + c) + b) + (4*a^8*b + 3*a^7*b^2 - 16*a^6*b^3 - 14*a^5*b^4 + 24*a^4*b^5 + 35*a^3*b^6 + 12*a^2*b^7 + (4*a^9 + 3*a^8*b - 16*a^7*b^2 - 14*a^6*b^3 + 24*a^5*b^4 + 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c)^5 + (4*a^8*b + 3*a^7*b^2 - 16*a^6*b^3 - 14*a^5*b^4 + 24*a^4*b^5 + 35*a^3*b^6 + 12*a^2*b^7)*cos(d*x + c)^4 - 2*(4*a^9 + 3*a^8*b - 16*a^7*b^2 - 14*a^6*b^3 + 24*a^5*b^4 + 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c)^3 - 2*(4*a^8*b + 3*a^7*b^2 - 16*a^6*b^3 - 14*a^5*b^4 + 24*a^4*b^5 + 35*a^3*b^6 + 12*a^2*b^7)*cos(d*x + c)^2 + (4*a^9 + 3*a^8*b - 16*a^7*b^2 - 14*a^6*b^3 + 24*a^5*b^4 + 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (4*a^8*b - 3*a^7*b^2 - 16*a^6*b^3 + 14*a^5*b^4 + 24*a^4*b^5 - 35*a^3*b^6 + 12*a^2*b^7 + (4*a^9 - 3*a^8*b - 16*a^7*b^2 + 14*a^6*b^3 + 24*a^5*b^4 - 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c)^5 + (4*a^8*b - 3*a^7*b^2 - 16*a^6*b^3 + 14*a^5*b^4 + 24*a^4*b^5 - 35*a^3*b^6 + 12*a^2*b^7)*cos(d*x + c)^4 - 2*(4*a^9 - 3*a^8*b - 16*a^7*b^2 + 14*a^6*b^3 + 24*a^5*b^4 - 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c)^3 - 2*(4*a^8*b - 3*a^7*b^2 - 16*a^6*b^3 + 14*a^5*b^4 + 24*a^4*b^5 - 35*a^3*b^6 + 12*a^2*b^7)*cos(d*x + c)^2 + (4*a^9 - 3*a^8*b - 16*a^7*b^2 + 14*a^6*b^3 + 24*a^5*b^4 - 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c)^5 + (a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*cos(d*x + c)^4 - 2*(a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c)^3 - 2*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*cos(d*x + c)^2 + (a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c) + (a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d)","B",0
307,1,843,0,0.919746," ","integrate(tan(d*x+c)^6/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","\left[-\frac{6 \, a b^{5} d x \cos\left(d x + c\right)^{4} + 6 \, b^{6} d x \cos\left(d x + c\right)^{3} + 3 \, {\left({\left(4 \, a^{5} - 3 \, a^{3} b^{2} - a b^{4}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{4} b - 3 \, a^{2} b^{3} - b^{5}\right)} \cos\left(d x + c\right)^{3}\right)} \sqrt{a^{2} - b^{2}} \log\left(\frac{2 \, a b \cos\left(d x + c\right) - {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - b^{2}} {\left(b \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) + 2 \, a^{2} - b^{2}}{a^{2} \cos\left(d x + c\right)^{2} + 2 \, a b \cos\left(d x + c\right) + b^{2}}\right) + 3 \, {\left({\left(4 \, a^{6} - 5 \, a^{4} b^{2}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{5} b - 5 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{3}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 3 \, {\left({\left(4 \, a^{6} - 5 \, a^{4} b^{2}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{5} b - 5 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{3}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(2 \, a^{3} b^{3} \cos\left(d x + c\right) - a^{2} b^{4} - {\left(12 \, a^{5} b - 13 \, a^{3} b^{3} + 3 \, a b^{5}\right)} \cos\left(d x + c\right)^{3} - {\left(6 \, a^{4} b^{2} - 7 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{6 \, {\left(a^{3} b^{5} d \cos\left(d x + c\right)^{4} + a^{2} b^{6} d \cos\left(d x + c\right)^{3}\right)}}, -\frac{6 \, a b^{5} d x \cos\left(d x + c\right)^{4} + 6 \, b^{6} d x \cos\left(d x + c\right)^{3} - 6 \, {\left({\left(4 \, a^{5} - 3 \, a^{3} b^{2} - a b^{4}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{4} b - 3 \, a^{2} b^{3} - b^{5}\right)} \cos\left(d x + c\right)^{3}\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \cos\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right)}\right) + 3 \, {\left({\left(4 \, a^{6} - 5 \, a^{4} b^{2}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{5} b - 5 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{3}\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 3 \, {\left({\left(4 \, a^{6} - 5 \, a^{4} b^{2}\right)} \cos\left(d x + c\right)^{4} + {\left(4 \, a^{5} b - 5 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{3}\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(2 \, a^{3} b^{3} \cos\left(d x + c\right) - a^{2} b^{4} - {\left(12 \, a^{5} b - 13 \, a^{3} b^{3} + 3 \, a b^{5}\right)} \cos\left(d x + c\right)^{3} - {\left(6 \, a^{4} b^{2} - 7 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{6 \, {\left(a^{3} b^{5} d \cos\left(d x + c\right)^{4} + a^{2} b^{6} d \cos\left(d x + c\right)^{3}\right)}}\right]"," ",0,"[-1/6*(6*a*b^5*d*x*cos(d*x + c)^4 + 6*b^6*d*x*cos(d*x + c)^3 + 3*((4*a^5 - 3*a^3*b^2 - a*b^4)*cos(d*x + c)^4 + (4*a^4*b - 3*a^2*b^3 - b^5)*cos(d*x + c)^3)*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + 3*((4*a^6 - 5*a^4*b^2)*cos(d*x + c)^4 + (4*a^5*b - 5*a^3*b^3)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) - 3*((4*a^6 - 5*a^4*b^2)*cos(d*x + c)^4 + (4*a^5*b - 5*a^3*b^3)*cos(d*x + c)^3)*log(-sin(d*x + c) + 1) + 2*(2*a^3*b^3*cos(d*x + c) - a^2*b^4 - (12*a^5*b - 13*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^3 - (6*a^4*b^2 - 7*a^2*b^4)*cos(d*x + c)^2)*sin(d*x + c))/(a^3*b^5*d*cos(d*x + c)^4 + a^2*b^6*d*cos(d*x + c)^3), -1/6*(6*a*b^5*d*x*cos(d*x + c)^4 + 6*b^6*d*x*cos(d*x + c)^3 - 6*((4*a^5 - 3*a^3*b^2 - a*b^4)*cos(d*x + c)^4 + (4*a^4*b - 3*a^2*b^3 - b^5)*cos(d*x + c)^3)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + 3*((4*a^6 - 5*a^4*b^2)*cos(d*x + c)^4 + (4*a^5*b - 5*a^3*b^3)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) - 3*((4*a^6 - 5*a^4*b^2)*cos(d*x + c)^4 + (4*a^5*b - 5*a^3*b^3)*cos(d*x + c)^3)*log(-sin(d*x + c) + 1) + 2*(2*a^3*b^3*cos(d*x + c) - a^2*b^4 - (12*a^5*b - 13*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^3 - (6*a^4*b^2 - 7*a^2*b^4)*cos(d*x + c)^2)*sin(d*x + c))/(a^3*b^5*d*cos(d*x + c)^4 + a^2*b^6*d*cos(d*x + c)^3)]","B",0
308,1,584,0,0.654391," ","integrate(tan(d*x+c)^4/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","\left[\frac{2 \, a b^{3} d x \cos\left(d x + c\right)^{2} + 2 \, b^{4} d x \cos\left(d x + c\right) + {\left({\left(2 \, a^{3} + a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(2 \, a^{2} b + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{a^{2} - b^{2}} \log\left(\frac{2 \, a b \cos\left(d x + c\right) - {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, \sqrt{a^{2} - b^{2}} {\left(b \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) + 2 \, a^{2} - b^{2}}{a^{2} \cos\left(d x + c\right)^{2} + 2 \, a b \cos\left(d x + c\right) + b^{2}}\right) - 2 \, {\left(a^{4} \cos\left(d x + c\right)^{2} + a^{3} b \cos\left(d x + c\right)\right)} \log\left(\sin\left(d x + c\right) + 1\right) + 2 \, {\left(a^{4} \cos\left(d x + c\right)^{2} + a^{3} b \cos\left(d x + c\right)\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + 2 \, {\left(a^{2} b^{2} + {\left(2 \, a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{2 \, {\left(a^{3} b^{3} d \cos\left(d x + c\right)^{2} + a^{2} b^{4} d \cos\left(d x + c\right)\right)}}, \frac{a b^{3} d x \cos\left(d x + c\right)^{2} + b^{4} d x \cos\left(d x + c\right) + {\left({\left(2 \, a^{3} + a b^{2}\right)} \cos\left(d x + c\right)^{2} + {\left(2 \, a^{2} b + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \cos\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right)}\right) - {\left(a^{4} \cos\left(d x + c\right)^{2} + a^{3} b \cos\left(d x + c\right)\right)} \log\left(\sin\left(d x + c\right) + 1\right) + {\left(a^{4} \cos\left(d x + c\right)^{2} + a^{3} b \cos\left(d x + c\right)\right)} \log\left(-\sin\left(d x + c\right) + 1\right) + {\left(a^{2} b^{2} + {\left(2 \, a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{a^{3} b^{3} d \cos\left(d x + c\right)^{2} + a^{2} b^{4} d \cos\left(d x + c\right)}\right]"," ",0,"[1/2*(2*a*b^3*d*x*cos(d*x + c)^2 + 2*b^4*d*x*cos(d*x + c) + ((2*a^3 + a*b^2)*cos(d*x + c)^2 + (2*a^2*b + b^3)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 + 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - 2*(a^4*cos(d*x + c)^2 + a^3*b*cos(d*x + c))*log(sin(d*x + c) + 1) + 2*(a^4*cos(d*x + c)^2 + a^3*b*cos(d*x + c))*log(-sin(d*x + c) + 1) + 2*(a^2*b^2 + (2*a^3*b - a*b^3)*cos(d*x + c))*sin(d*x + c))/(a^3*b^3*d*cos(d*x + c)^2 + a^2*b^4*d*cos(d*x + c)), (a*b^3*d*x*cos(d*x + c)^2 + b^4*d*x*cos(d*x + c) + ((2*a^3 + a*b^2)*cos(d*x + c)^2 + (2*a^2*b + b^3)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - (a^4*cos(d*x + c)^2 + a^3*b*cos(d*x + c))*log(sin(d*x + c) + 1) + (a^4*cos(d*x + c)^2 + a^3*b*cos(d*x + c))*log(-sin(d*x + c) + 1) + (a^2*b^2 + (2*a^3*b - a*b^3)*cos(d*x + c))*sin(d*x + c))/(a^3*b^3*d*cos(d*x + c)^2 + a^2*b^4*d*cos(d*x + c))]","A",0
309,1,383,0,0.532290," ","integrate(tan(d*x+c)^2/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a^{3} - a b^{2}\right)} d x \cos\left(d x + c\right) + 2 \, {\left(a^{2} b - b^{3}\right)} d x - {\left(a b \cos\left(d x + c\right) + b^{2}\right)} \sqrt{a^{2} - b^{2}} \log\left(\frac{2 \, a b \cos\left(d x + c\right) - {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, \sqrt{a^{2} - b^{2}} {\left(b \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) + 2 \, a^{2} - b^{2}}{a^{2} \cos\left(d x + c\right)^{2} + 2 \, a b \cos\left(d x + c\right) + b^{2}}\right) - 2 \, {\left(a^{3} - a b^{2}\right)} \sin\left(d x + c\right)}{2 \, {\left({\left(a^{5} - a^{3} b^{2}\right)} d \cos\left(d x + c\right) + {\left(a^{4} b - a^{2} b^{3}\right)} d\right)}}, -\frac{{\left(a^{3} - a b^{2}\right)} d x \cos\left(d x + c\right) + {\left(a^{2} b - b^{3}\right)} d x - {\left(a b \cos\left(d x + c\right) + b^{2}\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \cos\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right)}\right) - {\left(a^{3} - a b^{2}\right)} \sin\left(d x + c\right)}{{\left(a^{5} - a^{3} b^{2}\right)} d \cos\left(d x + c\right) + {\left(a^{4} b - a^{2} b^{3}\right)} d}\right]"," ",0,"[-1/2*(2*(a^3 - a*b^2)*d*x*cos(d*x + c) + 2*(a^2*b - b^3)*d*x - (a*b*cos(d*x + c) + b^2)*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 + 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - 2*(a^3 - a*b^2)*sin(d*x + c))/((a^5 - a^3*b^2)*d*cos(d*x + c) + (a^4*b - a^2*b^3)*d), -((a^3 - a*b^2)*d*x*cos(d*x + c) + (a^2*b - b^3)*d*x - (a*b*cos(d*x + c) + b^2)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - (a^3 - a*b^2)*sin(d*x + c))/((a^5 - a^3*b^2)*d*cos(d*x + c) + (a^4*b - a^2*b^3)*d)]","B",0
310,1,705,0,0.566639," ","integrate(cot(d*x+c)^2/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","\left[\frac{4 \, a^{5} b^{2} - 2 \, a^{3} b^{4} - 2 \, a b^{6} - {\left(4 \, a^{2} b^{4} - b^{6} + {\left(4 \, a^{3} b^{3} - a b^{5}\right)} \cos\left(d x + c\right)\right)} \sqrt{a^{2} - b^{2}} \log\left(\frac{2 \, a b \cos\left(d x + c\right) - {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, \sqrt{a^{2} - b^{2}} {\left(b \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) + 2 \, a^{2} - b^{2}}{a^{2} \cos\left(d x + c\right)^{2} + 2 \, a b \cos\left(d x + c\right) + b^{2}}\right) \sin\left(d x + c\right) - 2 \, {\left(a^{7} - a b^{6}\right)} \cos\left(d x + c\right)^{2} + 2 \, {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(d x + c\right) - 2 \, {\left({\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} d x \cos\left(d x + c\right) + {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} d x\right)} \sin\left(d x + c\right)}{2 \, {\left({\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right) + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)} \sin\left(d x + c\right)}, \frac{2 \, a^{5} b^{2} - a^{3} b^{4} - a b^{6} - {\left(4 \, a^{2} b^{4} - b^{6} + {\left(4 \, a^{3} b^{3} - a b^{5}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \cos\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) - {\left(a^{7} - a b^{6}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(d x + c\right) - {\left({\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} d x \cos\left(d x + c\right) + {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} d x\right)} \sin\left(d x + c\right)}{{\left({\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right) + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[1/2*(4*a^5*b^2 - 2*a^3*b^4 - 2*a*b^6 - (4*a^2*b^4 - b^6 + (4*a^3*b^3 - a*b^5)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 + 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2))*sin(d*x + c) - 2*(a^7 - a*b^6)*cos(d*x + c)^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c) - 2*((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d*x*cos(d*x + c) + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*d*x)*sin(d*x + c))/(((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d)*sin(d*x + c)), (2*a^5*b^2 - a^3*b^4 - a*b^6 - (4*a^2*b^4 - b^6 + (4*a^3*b^3 - a*b^5)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c)))*sin(d*x + c) - (a^7 - a*b^6)*cos(d*x + c)^2 + (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c) - ((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d*x*cos(d*x + c) + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*d*x)*sin(d*x + c))/(((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d)*sin(d*x + c))]","A",0
311,1,1481,0,0.663234," ","integrate(cot(d*x+c)^4/(a+b*sec(d*x+c))^2,x, algorithm=""fricas"")","\left[\frac{8 \, a^{7} b^{2} - 40 \, a^{5} b^{4} + 26 \, a^{3} b^{6} + 6 \, a b^{8} + 2 \, {\left(4 \, a^{9} - 13 \, a^{7} b^{2} + 2 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + 3 \, a b^{8}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(2 \, a^{8} b - 11 \, a^{6} b^{3} + 16 \, a^{4} b^{5} - 7 \, a^{2} b^{7}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(6 \, a^{2} b^{6} - b^{8} - {\left(6 \, a^{3} b^{5} - a b^{7}\right)} \cos\left(d x + c\right)^{3} - {\left(6 \, a^{2} b^{6} - b^{8}\right)} \cos\left(d x + c\right)^{2} + {\left(6 \, a^{3} b^{5} - a b^{7}\right)} \cos\left(d x + c\right)\right)} \sqrt{a^{2} - b^{2}} \log\left(\frac{2 \, a b \cos\left(d x + c\right) - {\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a^{2} - b^{2}} {\left(b \cos\left(d x + c\right) + a\right)} \sin\left(d x + c\right) + 2 \, a^{2} - b^{2}}{a^{2} \cos\left(d x + c\right)^{2} + 2 \, a b \cos\left(d x + c\right) + b^{2}}\right) \sin\left(d x + c\right) - 6 \, {\left(a^{9} - 2 \, a^{7} b^{2} - 7 \, a^{5} b^{4} + 6 \, a^{3} b^{6} + 2 \, a b^{8}\right)} \cos\left(d x + c\right)^{2} + 2 \, {\left(a^{8} b - 8 \, a^{6} b^{3} + 13 \, a^{4} b^{5} - 6 \, a^{2} b^{7}\right)} \cos\left(d x + c\right) + 6 \, {\left({\left(a^{9} - 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} + a b^{8}\right)} d x \cos\left(d x + c\right)^{3} + {\left(a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right)} d x \cos\left(d x + c\right)^{2} - {\left(a^{9} - 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} + a b^{8}\right)} d x \cos\left(d x + c\right) - {\left(a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right)} d x\right)} \sin\left(d x + c\right)}{6 \, {\left({\left(a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right)} d \cos\left(d x + c\right)^{3} + {\left(a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right)} d \cos\left(d x + c\right) - {\left(a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right)} d\right)} \sin\left(d x + c\right)}, \frac{4 \, a^{7} b^{2} - 20 \, a^{5} b^{4} + 13 \, a^{3} b^{6} + 3 \, a b^{8} + {\left(4 \, a^{9} - 13 \, a^{7} b^{2} + 2 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + 3 \, a b^{8}\right)} \cos\left(d x + c\right)^{4} - {\left(2 \, a^{8} b - 11 \, a^{6} b^{3} + 16 \, a^{4} b^{5} - 7 \, a^{2} b^{7}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(6 \, a^{2} b^{6} - b^{8} - {\left(6 \, a^{3} b^{5} - a b^{7}\right)} \cos\left(d x + c\right)^{3} - {\left(6 \, a^{2} b^{6} - b^{8}\right)} \cos\left(d x + c\right)^{2} + {\left(6 \, a^{3} b^{5} - a b^{7}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \cos\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) - 3 \, {\left(a^{9} - 2 \, a^{7} b^{2} - 7 \, a^{5} b^{4} + 6 \, a^{3} b^{6} + 2 \, a b^{8}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{8} b - 8 \, a^{6} b^{3} + 13 \, a^{4} b^{5} - 6 \, a^{2} b^{7}\right)} \cos\left(d x + c\right) + 3 \, {\left({\left(a^{9} - 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} + a b^{8}\right)} d x \cos\left(d x + c\right)^{3} + {\left(a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right)} d x \cos\left(d x + c\right)^{2} - {\left(a^{9} - 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} + a b^{8}\right)} d x \cos\left(d x + c\right) - {\left(a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right)} d x\right)} \sin\left(d x + c\right)}{3 \, {\left({\left(a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right)} d \cos\left(d x + c\right)^{3} + {\left(a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right)} d \cos\left(d x + c\right) - {\left(a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right)} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[1/6*(8*a^7*b^2 - 40*a^5*b^4 + 26*a^3*b^6 + 6*a*b^8 + 2*(4*a^9 - 13*a^7*b^2 + 2*a^5*b^4 + 4*a^3*b^6 + 3*a*b^8)*cos(d*x + c)^4 - 2*(2*a^8*b - 11*a^6*b^3 + 16*a^4*b^5 - 7*a^2*b^7)*cos(d*x + c)^3 - 3*(6*a^2*b^6 - b^8 - (6*a^3*b^5 - a*b^7)*cos(d*x + c)^3 - (6*a^2*b^6 - b^8)*cos(d*x + c)^2 + (6*a^3*b^5 - a*b^7)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2))*sin(d*x + c) - 6*(a^9 - 2*a^7*b^2 - 7*a^5*b^4 + 6*a^3*b^6 + 2*a*b^8)*cos(d*x + c)^2 + 2*(a^8*b - 8*a^6*b^3 + 13*a^4*b^5 - 6*a^2*b^7)*cos(d*x + c) + 6*((a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*x*cos(d*x + c)^3 + (a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*x*cos(d*x + c)^2 - (a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*x*cos(d*x + c) - (a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*x)*sin(d*x + c))/(((a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c)^3 + (a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*cos(d*x + c)^2 - (a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c) - (a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d)*sin(d*x + c)), 1/3*(4*a^7*b^2 - 20*a^5*b^4 + 13*a^3*b^6 + 3*a*b^8 + (4*a^9 - 13*a^7*b^2 + 2*a^5*b^4 + 4*a^3*b^6 + 3*a*b^8)*cos(d*x + c)^4 - (2*a^8*b - 11*a^6*b^3 + 16*a^4*b^5 - 7*a^2*b^7)*cos(d*x + c)^3 + 3*(6*a^2*b^6 - b^8 - (6*a^3*b^5 - a*b^7)*cos(d*x + c)^3 - (6*a^2*b^6 - b^8)*cos(d*x + c)^2 + (6*a^3*b^5 - a*b^7)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c)))*sin(d*x + c) - 3*(a^9 - 2*a^7*b^2 - 7*a^5*b^4 + 6*a^3*b^6 + 2*a*b^8)*cos(d*x + c)^2 + (a^8*b - 8*a^6*b^3 + 13*a^4*b^5 - 6*a^2*b^7)*cos(d*x + c) + 3*((a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*x*cos(d*x + c)^3 + (a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*x*cos(d*x + c)^2 - (a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*x*cos(d*x + c) - (a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*x)*sin(d*x + c))/(((a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c)^3 + (a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*cos(d*x + c)^2 - (a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*cos(d*x + c) - (a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d)*sin(d*x + c))]","B",0
312,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(5/2)/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
313,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(3/2)/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
314,-1,0,0,0.000000," ","integrate((e*tan(d*x+c))^(1/2)/(a+b*sec(d*x+c)),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
315,-1,0,0,0.000000," ","integrate(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
316,-1,0,0,0.000000," ","integrate(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
317,-1,0,0,0.000000," ","integrate(1/(a+b*sec(d*x+c))/(e*tan(d*x+c))^(5/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
318,1,425,0,1.638115," ","integrate((a+b*sec(d*x+c))^(1/2)*tan(d*x+c)^5,x, algorithm=""fricas"")","\left[\frac{315 \, \sqrt{a} b^{4} \cos\left(d x + c\right)^{4} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + 4 \, {\left(5 \, a b^{3} \cos\left(d x + c\right) - {\left(16 \, a^{4} - 84 \, a^{2} b^{2} - 315 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 35 \, b^{4} + 2 \, {\left(4 \, a^{3} b - 21 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} - 6 \, {\left(a^{2} b^{2} + 21 \, b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{630 \, b^{4} d \cos\left(d x + c\right)^{4}}, \frac{315 \, \sqrt{-a} b^{4} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) \cos\left(d x + c\right)^{4} + 2 \, {\left(5 \, a b^{3} \cos\left(d x + c\right) - {\left(16 \, a^{4} - 84 \, a^{2} b^{2} - 315 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 35 \, b^{4} + 2 \, {\left(4 \, a^{3} b - 21 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} - 6 \, {\left(a^{2} b^{2} + 21 \, b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{315 \, b^{4} d \cos\left(d x + c\right)^{4}}\right]"," ",0,"[1/630*(315*sqrt(a)*b^4*cos(d*x + c)^4*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + 4*(5*a*b^3*cos(d*x + c) - (16*a^4 - 84*a^2*b^2 - 315*b^4)*cos(d*x + c)^4 + 35*b^4 + 2*(4*a^3*b - 21*a*b^3)*cos(d*x + c)^3 - 6*(a^2*b^2 + 21*b^4)*cos(d*x + c)^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(b^4*d*cos(d*x + c)^4), 1/315*(315*sqrt(-a)*b^4*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b))*cos(d*x + c)^4 + 2*(5*a*b^3*cos(d*x + c) - (16*a^4 - 84*a^2*b^2 - 315*b^4)*cos(d*x + c)^4 + 35*b^4 + 2*(4*a^3*b - 21*a*b^3)*cos(d*x + c)^3 - 6*(a^2*b^2 + 21*b^4)*cos(d*x + c)^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(b^4*d*cos(d*x + c)^4)]","A",0
319,1,311,0,1.446503," ","integrate((a+b*sec(d*x+c))^(1/2)*tan(d*x+c)^3,x, algorithm=""fricas"")","\left[\frac{15 \, \sqrt{a} b^{2} \cos\left(d x + c\right)^{2} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + 4 \, {\left(a b \cos\left(d x + c\right) - {\left(2 \, a^{2} + 15 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 3 \, b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{30 \, b^{2} d \cos\left(d x + c\right)^{2}}, -\frac{15 \, \sqrt{-a} b^{2} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) \cos\left(d x + c\right)^{2} - 2 \, {\left(a b \cos\left(d x + c\right) - {\left(2 \, a^{2} + 15 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 3 \, b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{15 \, b^{2} d \cos\left(d x + c\right)^{2}}\right]"," ",0,"[1/30*(15*sqrt(a)*b^2*cos(d*x + c)^2*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + 4*(a*b*cos(d*x + c) - (2*a^2 + 15*b^2)*cos(d*x + c)^2 + 3*b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(b^2*d*cos(d*x + c)^2), -1/15*(15*sqrt(-a)*b^2*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b))*cos(d*x + c)^2 - 2*(a*b*cos(d*x + c) - (2*a^2 + 15*b^2)*cos(d*x + c)^2 + 3*b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(b^2*d*cos(d*x + c)^2)]","A",0
320,1,192,0,0.709869," ","integrate((a+b*sec(d*x+c))^(1/2)*tan(d*x+c),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + 4 \, \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{2 \, d}, \frac{\sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + 2 \, \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{d}\right]"," ",0,"[1/2*(sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + 4*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/d, (sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + 2*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/d]","A",0
321,1,2132,0,1.081872," ","integrate(cot(d*x+c)*(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, d}, \frac{2 \, \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) + 2 \, \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, d}, -\frac{2 \, \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - 2 \, \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, d}, -\frac{\sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right)}{2 \, d}, -\frac{4 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) - \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, d}, -\frac{4 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - 2 \, \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, d}, -\frac{4 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + 2 \, \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, d}, -\frac{2 \, \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right)}{2 \, d}\right]"," ",0,"[1/4*(2*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/d, 1/4*(2*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) + 2*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)))/d, -1/4*(2*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - 2*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/d, -1/2*(sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))))/d, -1/4*(4*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) - sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/d, -1/4*(4*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 2*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)))/d, -1/4*(4*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + 2*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/d, -1/2*(2*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)))/d]","B",0
322,1,3523,0,3.138614," ","integrate(cot(d*x+c)^3*(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{2} - b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)^{2} + 8 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - {\left({\left(4 \, a^{2} + a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} - a b + 3 \, b^{2}\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + {\left({\left(4 \, a^{2} - a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} + a b + 3 \, b^{2}\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{16 \, {\left({\left(a^{2} - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - b^{2}\right)} d\right)}}, \frac{8 \, {\left(a^{2} - b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)^{2} - 2 \, {\left({\left(4 \, a^{2} - a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} + a b + 3 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) + 8 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - {\left({\left(4 \, a^{2} + a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} - a b + 3 \, b^{2}\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right)}{16 \, {\left({\left(a^{2} - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - b^{2}\right)} d\right)}}, \frac{8 \, {\left(a^{2} - b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)^{2} + 2 \, {\left({\left(4 \, a^{2} + a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} - a b + 3 \, b^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) + 8 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + {\left({\left(4 \, a^{2} - a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} + a b + 3 \, b^{2}\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{16 \, {\left({\left(a^{2} - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - b^{2}\right)} d\right)}}, \frac{4 \, {\left(a^{2} - b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)^{2} + {\left({\left(4 \, a^{2} + a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} - a b + 3 \, b^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - {\left({\left(4 \, a^{2} - a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} + a b + 3 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) + 4 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right)}{8 \, {\left({\left(a^{2} - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - b^{2}\right)} d\right)}}, \frac{8 \, {\left(a^{2} - b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)^{2} + 16 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - {\left({\left(4 \, a^{2} + a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} - a b + 3 \, b^{2}\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + {\left({\left(4 \, a^{2} - a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} + a b + 3 \, b^{2}\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{16 \, {\left({\left(a^{2} - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - b^{2}\right)} d\right)}}, \frac{8 \, {\left(a^{2} - b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)^{2} + 16 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - 2 \, {\left({\left(4 \, a^{2} - a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} + a b + 3 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - {\left({\left(4 \, a^{2} + a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} - a b + 3 \, b^{2}\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right)}{16 \, {\left({\left(a^{2} - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - b^{2}\right)} d\right)}}, \frac{8 \, {\left(a^{2} - b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)^{2} + 16 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + 2 \, {\left({\left(4 \, a^{2} + a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} - a b + 3 \, b^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) + {\left({\left(4 \, a^{2} - a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} + a b + 3 \, b^{2}\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{16 \, {\left({\left(a^{2} - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - b^{2}\right)} d\right)}}, \frac{4 \, {\left(a^{2} - b^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)^{2} + 8 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + b^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + {\left({\left(4 \, a^{2} + a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} - a b + 3 \, b^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - {\left({\left(4 \, a^{2} - a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, a^{2} + a b + 3 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right)}{8 \, {\left({\left(a^{2} - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - b^{2}\right)} d\right)}}\right]"," ",0,"[1/16*(8*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + 8*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - ((4*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b^2)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + ((4*a^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^2)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/16*(8*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 - 2*((4*a^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) + 8*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - ((4*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b^2)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/16*(8*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + 2*((4*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) + 8*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + ((4*a^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^2)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/8*(4*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + ((4*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - ((4*a^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) + 4*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/16*(8*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + 16*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - ((4*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b^2)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + ((4*a^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^2)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/16*(8*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + 16*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 2*((4*a^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - ((4*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b^2)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/16*(8*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + 16*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + 2*((4*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) + ((4*a^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^2)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d), 1/8*(4*(a^2 - b^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)^2 + 8*((a^2 - b^2)*cos(d*x + c)^2 - a^2 + b^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + ((4*a^2 + a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 - a*b + 3*b^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - ((4*a^2 - a*b - 3*b^2)*cos(d*x + c)^2 - 4*a^2 + a*b + 3*b^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)))/((a^2 - b^2)*d*cos(d*x + c)^2 - (a^2 - b^2)*d)]","B",0
323,0,0,0,22.659237," ","integrate((a+b*sec(d*x+c))^(1/2)*tan(d*x+c)^2,x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sec\left(d x + c\right) + a} \tan\left(d x + c\right)^{2}, x\right)"," ",0,"integral(sqrt(b*sec(d*x + c) + a)*tan(d*x + c)^2, x)","F",0
324,-1,0,0,0.000000," ","integrate((a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
325,-1,0,0,0.000000," ","integrate(cot(d*x+c)^2*(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
326,1,381,0,2.010148," ","integrate(tan(d*x+c)^5/(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{105 \, \sqrt{a} b^{4} \cos\left(d x + c\right)^{3} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - 4 \, {\left(18 \, a^{2} b^{2} \cos\left(d x + c\right) - 15 \, a b^{3} + 4 \, {\left(12 \, a^{4} - 35 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(12 \, a^{3} b - 35 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{210 \, a b^{4} d \cos\left(d x + c\right)^{3}}, \frac{105 \, \sqrt{-a} b^{4} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) \cos\left(d x + c\right)^{3} - 2 \, {\left(18 \, a^{2} b^{2} \cos\left(d x + c\right) - 15 \, a b^{3} + 4 \, {\left(12 \, a^{4} - 35 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(12 \, a^{3} b - 35 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{105 \, a b^{4} d \cos\left(d x + c\right)^{3}}\right]"," ",0,"[1/210*(105*sqrt(a)*b^4*cos(d*x + c)^3*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - 4*(18*a^2*b^2*cos(d*x + c) - 15*a*b^3 + 4*(12*a^4 - 35*a^2*b^2)*cos(d*x + c)^3 - 2*(12*a^3*b - 35*a*b^3)*cos(d*x + c)^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(a*b^4*d*cos(d*x + c)^3), 1/105*(105*sqrt(-a)*b^4*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b))*cos(d*x + c)^3 - 2*(18*a^2*b^2*cos(d*x + c) - 15*a*b^3 + 4*(12*a^4 - 35*a^2*b^2)*cos(d*x + c)^3 - 2*(12*a^3*b - 35*a*b^3)*cos(d*x + c)^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(a*b^4*d*cos(d*x + c)^3)]","A",0
327,1,273,0,1.194303," ","integrate(tan(d*x+c)^3/(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a} b^{2} \cos\left(d x + c\right) \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - 4 \, {\left(2 \, a^{2} \cos\left(d x + c\right) - a b\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{6 \, a b^{2} d \cos\left(d x + c\right)}, -\frac{3 \, \sqrt{-a} b^{2} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) \cos\left(d x + c\right) + 2 \, {\left(2 \, a^{2} \cos\left(d x + c\right) - a b\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{3 \, a b^{2} d \cos\left(d x + c\right)}\right]"," ",0,"[1/6*(3*sqrt(a)*b^2*cos(d*x + c)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - 4*(2*a^2*cos(d*x + c) - a*b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(a*b^2*d*cos(d*x + c)), -1/3*(3*sqrt(-a)*b^2*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b))*cos(d*x + c) + 2*(2*a^2*cos(d*x + c) - a*b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(a*b^2*d*cos(d*x + c))]","A",0
328,1,145,0,1.607414," ","integrate(tan(d*x+c)/(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right)}{2 \, \sqrt{a} d}, \frac{\sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right)}{a d}\right]"," ",0,"[1/2*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(sqrt(a)*d), sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b))/(a*d)]","B",0
329,1,2420,0,4.410357," ","integrate(cot(d*x+c)/(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{2} - b^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + {\left(a^{2} + a b\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + {\left(a^{2} - a b\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left(a^{3} - a b^{2}\right)} d}, -\frac{4 \, {\left(a^{2} - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - {\left(a^{2} + a b\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) - {\left(a^{2} - a b\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left(a^{3} - a b^{2}\right)} d}, -\frac{2 \, {\left(a^{2} + a b\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - 2 \, {\left(a^{2} - b^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - {\left(a^{2} - a b\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left(a^{3} - a b^{2}\right)} d}, -\frac{4 \, {\left(a^{2} - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + 2 \, {\left(a^{2} + a b\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - {\left(a^{2} - a b\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left(a^{3} - a b^{2}\right)} d}, \frac{2 \, {\left(a^{2} - a b\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) + 2 \, {\left(a^{2} - b^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + {\left(a^{2} + a b\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left(a^{3} - a b^{2}\right)} d}, -\frac{4 \, {\left(a^{2} - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - 2 \, {\left(a^{2} - a b\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - {\left(a^{2} + a b\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left(a^{3} - a b^{2}\right)} d}, -\frac{{\left(a^{2} + a b\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - {\left(a^{2} - a b\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - {\left(a^{2} - b^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right)}{2 \, {\left(a^{3} - a b^{2}\right)} d}, -\frac{2 \, {\left(a^{2} - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + {\left(a^{2} + a b\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - {\left(a^{2} - a b\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right)}{2 \, {\left(a^{3} - a b^{2}\right)} d}\right]"," ",0,"[1/4*(2*(a^2 - b^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + (a^2 + a*b)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + (a^2 - a*b)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^3 - a*b^2)*d), -1/4*(4*(a^2 - b^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - (a^2 + a*b)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) - (a^2 - a*b)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^3 - a*b^2)*d), -1/4*(2*(a^2 + a*b)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - 2*(a^2 - b^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - (a^2 - a*b)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^3 - a*b^2)*d), -1/4*(4*(a^2 - b^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + 2*(a^2 + a*b)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - (a^2 - a*b)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^3 - a*b^2)*d), 1/4*(2*(a^2 - a*b)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) + 2*(a^2 - b^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + (a^2 + a*b)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)))/((a^3 - a*b^2)*d), -1/4*(4*(a^2 - b^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 2*(a^2 - a*b)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - (a^2 + a*b)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)))/((a^3 - a*b^2)*d), -1/2*((a^2 + a*b)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - (a^2 - a*b)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - (a^2 - b^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))))/((a^3 - a*b^2)*d), -1/2*(2*(a^2 - b^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + (a^2 + a*b)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - (a^2 - a*b)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)))/((a^3 - a*b^2)*d)]","B",0
330,1,4336,0,42.439469," ","integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3} - {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3} - {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right) - 8 \, {\left({\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d\right)}}, -\frac{16 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3} - {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3} - {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right) - 8 \, {\left({\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d\right)}}, -\frac{2 \, {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3} - {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) + 8 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3} - {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right) - 8 \, {\left({\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d\right)}}, -\frac{16 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + 2 \, {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3} - {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) + {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3} - {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right) - 8 \, {\left({\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d\right)}}, \frac{2 \, {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3} - {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - 8 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3} - {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + 8 \, {\left({\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d\right)}}, -\frac{16 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - 2 \, {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3} - {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) + {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3} - {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) - 8 \, {\left({\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d\right)}}, -\frac{{\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3} - {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3} - {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) + 4 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - 4 \, {\left({\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{8 \, {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d\right)}}, -\frac{8 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3} - {\left(4 \, a^{4} + 3 \, a^{3} b - 6 \, a^{2} b^{2} - 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3} - {\left(4 \, a^{4} - 3 \, a^{3} b - 6 \, a^{2} b^{2} + 5 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - 4 \, {\left({\left(a^{4} - a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{8 \, {\left({\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d\right)}}\right]"," ",0,"[-1/16*(8*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/16*(16*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/16*(2*(4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) + 8*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/16*(16*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + 2*(4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) + (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), 1/16*(2*(4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - 8*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/16*(16*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 2*(4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) + (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) - 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/8*((4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) + 4*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - 4*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/8*(8*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - 4*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d)]","B",0
331,0,0,0,0.834388," ","integrate(tan(d*x+c)^4/(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\tan\left(d x + c\right)^{4}}{\sqrt{b \sec\left(d x + c\right) + a}}, x\right)"," ",0,"integral(tan(d*x + c)^4/sqrt(b*sec(d*x + c) + a), x)","F",0
332,0,0,0,22.980436," ","integrate(tan(d*x+c)^2/(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\tan\left(d x + c\right)^{2}}{\sqrt{b \sec\left(d x + c\right) + a}}, x\right)"," ",0,"integral(tan(d*x + c)^2/sqrt(b*sec(d*x + c) + a), x)","F",0
333,0,0,0,24.122863," ","integrate(1/(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{1}{\sqrt{b \sec\left(d x + c\right) + a}}, x\right)"," ",0,"integral(1/sqrt(b*sec(d*x + c) + a), x)","F",0
334,-1,0,0,0.000000," ","integrate(cot(d*x+c)^2/(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
335,1,467,0,1.417218," ","integrate(tan(d*x+c)^5/(a+b*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{5 \, {\left(a b^{4} \cos\left(d x + c\right)^{3} + b^{5} \cos\left(d x + c\right)^{2}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - 4 \, {\left(2 \, a^{3} b^{2} \cos\left(d x + c\right) - a^{2} b^{3} - {\left(16 \, a^{5} - 20 \, a^{3} b^{2} + 5 \, a b^{4}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(4 \, a^{4} b - 5 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{10 \, {\left(a^{3} b^{4} d \cos\left(d x + c\right)^{3} + a^{2} b^{5} d \cos\left(d x + c\right)^{2}\right)}}, \frac{5 \, {\left(a b^{4} \cos\left(d x + c\right)^{3} + b^{5} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - 2 \, {\left(2 \, a^{3} b^{2} \cos\left(d x + c\right) - a^{2} b^{3} - {\left(16 \, a^{5} - 20 \, a^{3} b^{2} + 5 \, a b^{4}\right)} \cos\left(d x + c\right)^{3} - 2 \, {\left(4 \, a^{4} b - 5 \, a^{2} b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{5 \, {\left(a^{3} b^{4} d \cos\left(d x + c\right)^{3} + a^{2} b^{5} d \cos\left(d x + c\right)^{2}\right)}}\right]"," ",0,"[1/10*(5*(a*b^4*cos(d*x + c)^3 + b^5*cos(d*x + c)^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - 4*(2*a^3*b^2*cos(d*x + c) - a^2*b^3 - (16*a^5 - 20*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^3 - 2*(4*a^4*b - 5*a^2*b^3)*cos(d*x + c)^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(a^3*b^4*d*cos(d*x + c)^3 + a^2*b^5*d*cos(d*x + c)^2), 1/5*(5*(a*b^4*cos(d*x + c)^3 + b^5*cos(d*x + c)^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 2*(2*a^3*b^2*cos(d*x + c) - a^2*b^3 - (16*a^5 - 20*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^3 - 2*(4*a^4*b - 5*a^2*b^3)*cos(d*x + c)^2)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(a^3*b^4*d*cos(d*x + c)^3 + a^2*b^5*d*cos(d*x + c)^2)]","A",0
336,1,317,0,1.173750," ","integrate(tan(d*x+c)^3/(a+b*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a b^{2} \cos\left(d x + c\right) + b^{3}\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + 4 \, {\left(a^{2} b + {\left(2 \, a^{3} - a b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{2 \, {\left(a^{3} b^{2} d \cos\left(d x + c\right) + a^{2} b^{3} d\right)}}, -\frac{{\left(a b^{2} \cos\left(d x + c\right) + b^{3}\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - 2 \, {\left(a^{2} b + {\left(2 \, a^{3} - a b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{a^{3} b^{2} d \cos\left(d x + c\right) + a^{2} b^{3} d}\right]"," ",0,"[1/2*((a*b^2*cos(d*x + c) + b^3)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + 4*(a^2*b + (2*a^3 - a*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(a^3*b^2*d*cos(d*x + c) + a^2*b^3*d), -((a*b^2*cos(d*x + c) + b^3)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 2*(a^2*b + (2*a^3 - a*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/(a^3*b^2*d*cos(d*x + c) + a^2*b^3*d)]","A",0
337,1,260,0,1.037689," ","integrate(tan(d*x+c)/(a+b*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{4 \, a \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + {\left(a \cos\left(d x + c\right) + b\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right)}{2 \, {\left(a^{3} d \cos\left(d x + c\right) + a^{2} b d\right)}}, \frac{{\left(a \cos\left(d x + c\right) + b\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + 2 \, a \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{a^{3} d \cos\left(d x + c\right) + a^{2} b d}\right]"," ",0,"[1/2*(4*a*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c) + (a*cos(d*x + c) + b)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))))/(a^3*d*cos(d*x + c) + a^2*b*d), ((a*cos(d*x + c) + b)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + 2*a*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c))/(a^3*d*cos(d*x + c) + a^2*b*d)]","B",0
338,1,3924,0,38.668844," ","integrate(cot(d*x+c)/(a+b*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{3} b^{2} - a b^{4}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + 2 \, {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left({\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} d \cos\left(d x + c\right) + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} d\right)}}, -\frac{4 \, {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - 8 \, {\left(a^{3} b^{2} - a b^{4}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) - {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left({\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} d \cos\left(d x + c\right) + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} d\right)}}, -\frac{2 \, {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - 8 \, {\left(a^{3} b^{2} - a b^{4}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - 2 \, {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left({\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} d \cos\left(d x + c\right) + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} d\right)}}, -\frac{4 \, {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + 2 \, {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - 8 \, {\left(a^{3} b^{2} - a b^{4}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left({\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} d \cos\left(d x + c\right) + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} d\right)}}, \frac{2 \, {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) + 8 \, {\left(a^{3} b^{2} - a b^{4}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + 2 \, {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left({\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} d \cos\left(d x + c\right) + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} d\right)}}, -\frac{4 \, {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - 2 \, {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - 8 \, {\left(a^{3} b^{2} - a b^{4}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right)}{4 \, {\left({\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} d \cos\left(d x + c\right) + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} d\right)}}, -\frac{{\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - 4 \, {\left(a^{3} b^{2} - a b^{4}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right) - {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} - 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right)}{2 \, {\left({\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} d \cos\left(d x + c\right) + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} d\right)}}, -\frac{2 \, {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5} + {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + {\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - 4 \, {\left(a^{3} b^{2} - a b^{4}\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, {\left({\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} d \cos\left(d x + c\right) + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} d\right)}}\right]"," ",0,"[1/4*(8*(a^3*b^2 - a*b^4)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c) + 2*(a^4*b - 2*a^2*b^3 + b^5 + (a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c))*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - (a^4*b + 2*a^3*b^2 + a^2*b^3 + (a^5 + 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + (a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d), -1/4*(4*(a^4*b - 2*a^2*b^3 + b^5 + (a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 8*(a^3*b^2 - a*b^4)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c) + (a^4*b + 2*a^3*b^2 + a^2*b^3 + (a^5 + 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) - (a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d), -1/4*(2*(a^4*b + 2*a^3*b^2 + a^2*b^3 + (a^5 + 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - 8*(a^3*b^2 - a*b^4)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c) - 2*(a^4*b - 2*a^2*b^3 + b^5 + (a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c))*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - (a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d), -1/4*(4*(a^4*b - 2*a^2*b^3 + b^5 + (a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + 2*(a^4*b + 2*a^3*b^2 + a^2*b^3 + (a^5 + 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - 8*(a^3*b^2 - a*b^4)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c) - (a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d), 1/4*(2*(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) + 8*(a^3*b^2 - a*b^4)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c) + 2*(a^4*b - 2*a^2*b^3 + b^5 + (a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c))*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - (a^4*b + 2*a^3*b^2 + a^2*b^3 + (a^5 + 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d), -1/4*(4*(a^4*b - 2*a^2*b^3 + b^5 + (a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 2*(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - 8*(a^3*b^2 - a*b^4)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c) + (a^4*b + 2*a^3*b^2 + a^2*b^3 + (a^5 + 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d), -1/2*((a^4*b + 2*a^3*b^2 + a^2*b^3 + (a^5 + 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - (a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - 4*(a^3*b^2 - a*b^4)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c) - (a^4*b - 2*a^2*b^3 + b^5 + (a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c))*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d), -1/2*(2*(a^4*b - 2*a^2*b^3 + b^5 + (a^5 - 2*a^3*b^2 + a*b^4)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + (a^4*b + 2*a^3*b^2 + a^2*b^3 + (a^5 + 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - (a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*cos(d*x + c))*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - 4*(a^3*b^2 - a*b^4)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cos(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d)]","B",0
339,1,8098,0,76.758692," ","integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right) - 8 \, {\left({\left(a^{7} + 3 \, a^{3} b^{4} - 4 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right)^{3} + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right) - {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)}}, -\frac{16 \, {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right) - 8 \, {\left({\left(a^{7} + 3 \, a^{3} b^{4} - 4 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right)^{3} + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right) - {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)}}, -\frac{2 \, {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) + 8 \, {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right) - 8 \, {\left({\left(a^{7} + 3 \, a^{3} b^{4} - 4 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right)^{3} + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right) - {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)}}, -\frac{16 \, {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + 2 \, {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) + {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a + b} \log\left(-\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} - 2 \, \cos\left(d x + c\right) + 1}\right) - 8 \, {\left({\left(a^{7} + 3 \, a^{3} b^{4} - 4 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right)^{3} + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right) - {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)}}, \frac{2 \, {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - 8 \, {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) + {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) + 8 \, {\left({\left(a^{7} + 3 \, a^{3} b^{4} - 4 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right)^{3} + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right) - {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)}}, -\frac{16 \, {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) - 2 \, {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{a - b} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + b^{2} - 4 \, {\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)}{\cos\left(d x + c\right)^{2} + 2 \, \cos\left(d x + c\right) + 1}\right) - 8 \, {\left({\left(a^{7} + 3 \, a^{3} b^{4} - 4 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{16 \, {\left({\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right)^{3} + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right) - {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)}}, -\frac{{\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) + 4 \, {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{a} \log\left(-8 \, a^{2} \cos\left(d x + c\right)^{2} - 8 \, a b \cos\left(d x + c\right) - b^{2} + 4 \, {\left(2 \, a \cos\left(d x + c\right)^{2} + b \cos\left(d x + c\right)\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}\right) - 4 \, {\left({\left(a^{7} + 3 \, a^{3} b^{4} - 4 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{8 \, {\left({\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right)^{3} + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right) - {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)}}, -\frac{8 \, {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} - {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right)} \cos\left(d x + c\right)^{2} + {\left(a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \arctan\left(\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{2 \, a \cos\left(d x + c\right) + b}\right) + {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b + 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} + 5 \, a^{6} b - 9 \, a^{5} b^{2} - 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a - b\right)} \cos\left(d x + c\right) + b}\right) - {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5} - {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{6} b - 5 \, a^{5} b^{2} - 9 \, a^{4} b^{3} + 17 \, a^{3} b^{4} - 7 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + {\left(4 \, a^{7} - 5 \, a^{6} b - 9 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 7 \, a^{3} b^{4}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a - b} \arctan\left(\frac{2 \, \sqrt{-a - b} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)}{{\left(2 \, a + b\right)} \cos\left(d x + c\right) + b}\right) - 4 \, {\left({\left(a^{7} + 3 \, a^{3} b^{4} - 4 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{5} b^{2} + a^{3} b^{4} - 2 \, a b^{6}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right) + b}{\cos\left(d x + c\right)}}}{8 \, {\left({\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right)^{3} + {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} d \cos\left(d x + c\right) - {\left(a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)}}\right]"," ",0,"[-1/16*(8*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), -1/16*(16*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), -1/16*(2*(4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) + 8*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), -1/16*(16*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + 2*(4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) + (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), 1/16*(2*(4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - 8*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) + (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 8*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), -1/16*(16*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 2*(4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 - 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) - 8*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), -1/8*((4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) + 4*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))) - 4*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d), -1/8*(8*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b + 5*a^5*b^2 - 9*a^4*b^3 - 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 + 5*a^6*b - 9*a^5*b^2 - 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5 - (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c)^3 - (4*a^6*b - 5*a^5*b^2 - 9*a^4*b^3 + 17*a^3*b^4 - 7*a^2*b^5)*cos(d*x + c)^2 + (4*a^7 - 5*a^6*b - 9*a^5*b^2 + 17*a^4*b^3 - 7*a^3*b^4)*cos(d*x + c))*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - 4*((a^7 + 3*a^3*b^4 - 4*a*b^6)*cos(d*x + c)^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c)^2 - 2*(a^5*b^2 + a^3*b^4 - 2*a*b^6)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c)^3 + (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d*cos(d*x + c)^2 - (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*d*cos(d*x + c) - (a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d)]","B",0
340,0,0,0,23.514393," ","integrate(tan(d*x+c)^4/(a+b*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(d x + c\right) + a} \tan\left(d x + c\right)^{4}}{b^{2} \sec\left(d x + c\right)^{2} + 2 \, a b \sec\left(d x + c\right) + a^{2}}, x\right)"," ",0,"integral(sqrt(b*sec(d*x + c) + a)*tan(d*x + c)^4/(b^2*sec(d*x + c)^2 + 2*a*b*sec(d*x + c) + a^2), x)","F",0
341,0,0,0,0.774163," ","integrate(tan(d*x+c)^2/(a+b*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(d x + c\right) + a} \tan\left(d x + c\right)^{2}}{b^{2} \sec\left(d x + c\right)^{2} + 2 \, a b \sec\left(d x + c\right) + a^{2}}, x\right)"," ",0,"integral(sqrt(b*sec(d*x + c) + a)*tan(d*x + c)^2/(b^2*sec(d*x + c)^2 + 2*a*b*sec(d*x + c) + a^2), x)","F",0
342,0,0,0,23.568994," ","integrate(1/(a+b*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(d x + c\right) + a}}{b^{2} \sec\left(d x + c\right)^{2} + 2 \, a b \sec\left(d x + c\right) + a^{2}}, x\right)"," ",0,"integral(sqrt(b*sec(d*x + c) + a)/(b^2*sec(d*x + c)^2 + 2*a*b*sec(d*x + c) + a^2), x)","F",0
343,-1,0,0,0.000000," ","integrate(cot(d*x+c)^2/(a+b*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
344,0,0,0,0.823431," ","integrate((a+b*sec(f*x+e))^3*(d*tan(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b^{3} \sec\left(f x + e\right)^{3} + 3 \, a b^{2} \sec\left(f x + e\right)^{2} + 3 \, a^{2} b \sec\left(f x + e\right) + a^{3}\right)} \left(d \tan\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral((b^3*sec(f*x + e)^3 + 3*a*b^2*sec(f*x + e)^2 + 3*a^2*b*sec(f*x + e) + a^3)*(d*tan(f*x + e))^n, x)","F",0
345,0,0,0,0.688838," ","integrate((a+b*sec(f*x+e))^2*(d*tan(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b^{2} \sec\left(f x + e\right)^{2} + 2 \, a b \sec\left(f x + e\right) + a^{2}\right)} \left(d \tan\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral((b^2*sec(f*x + e)^2 + 2*a*b*sec(f*x + e) + a^2)*(d*tan(f*x + e))^n, x)","F",0
346,0,0,0,1.418451," ","integrate((a+b*sec(f*x+e))*(d*tan(f*x+e))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(f x + e\right) + a\right)} \left(d \tan\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral((b*sec(f*x + e) + a)*(d*tan(f*x + e))^n, x)","F",0
347,0,0,0,0.933596," ","integrate((d*tan(f*x+e))^n/(a+b*sec(f*x+e)),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(d \tan\left(f x + e\right)\right)^{n}}{b \sec\left(f x + e\right) + a}, x\right)"," ",0,"integral((d*tan(f*x + e))^n/(b*sec(f*x + e) + a), x)","F",0
348,0,0,0,0.607214," ","integrate((a+b*sec(d*x+c))^(3/2)*(e*tan(d*x+c))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{\frac{3}{2}} \left(e \tan\left(d x + c\right)\right)^{m}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^(3/2)*(e*tan(d*x + c))^m, x)","F",0
349,0,0,0,0.607260," ","integrate((a+b*sec(d*x+c))^(1/2)*(e*tan(d*x+c))^m,x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \sec\left(d x + c\right) + a} \left(e \tan\left(d x + c\right)\right)^{m}, x\right)"," ",0,"integral(sqrt(b*sec(d*x + c) + a)*(e*tan(d*x + c))^m, x)","F",0
350,0,0,0,0.632705," ","integrate((e*tan(d*x+c))^m/(a+b*sec(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\left(e \tan\left(d x + c\right)\right)^{m}}{\sqrt{b \sec\left(d x + c\right) + a}}, x\right)"," ",0,"integral((e*tan(d*x + c))^m/sqrt(b*sec(d*x + c) + a), x)","F",0
351,0,0,0,0.902574," ","integrate((e*tan(d*x+c))^m/(a+b*sec(d*x+c))^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{b \sec\left(d x + c\right) + a} \left(e \tan\left(d x + c\right)\right)^{m}}{b^{2} \sec\left(d x + c\right)^{2} + 2 \, a b \sec\left(d x + c\right) + a^{2}}, x\right)"," ",0,"integral(sqrt(b*sec(d*x + c) + a)*(e*tan(d*x + c))^m/(b^2*sec(d*x + c)^2 + 2*a*b*sec(d*x + c) + a^2), x)","F",0
352,0,0,0,0.803809," ","integrate((a+b*sec(d*x+c))^n*(e*tan(d*x+c))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \left(e \tan\left(d x + c\right)\right)^{m}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*(e*tan(d*x + c))^m, x)","F",0
353,0,0,0,0.523295," ","integrate((a+b*sec(d*x+c))^n*tan(d*x+c)^5,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right)^{5}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*tan(d*x + c)^5, x)","F",0
354,0,0,0,0.503958," ","integrate((a+b*sec(d*x+c))^n*tan(d*x+c)^3,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right)^{3}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*tan(d*x + c)^3, x)","F",0
355,0,0,0,0.433558," ","integrate((a+b*sec(d*x+c))^n*tan(d*x+c),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right), x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*tan(d*x + c), x)","F",0
356,0,0,0,0.449903," ","integrate(cot(d*x+c)*(a+b*sec(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \cot\left(d x + c\right), x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*cot(d*x + c), x)","F",0
357,0,0,0,0.737067," ","integrate(cot(d*x+c)^3*(a+b*sec(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \cot\left(d x + c\right)^{3}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*cot(d*x + c)^3, x)","F",0
358,0,0,0,0.515617," ","integrate((a+b*sec(d*x+c))^n*tan(d*x+c)^4,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right)^{4}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*tan(d*x + c)^4, x)","F",0
359,0,0,0,0.469304," ","integrate((a+b*sec(d*x+c))^n*tan(d*x+c)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right)^{2}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*tan(d*x + c)^2, x)","F",0
360,0,0,0,0.561216," ","integrate(cot(d*x+c)^2*(a+b*sec(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \cot\left(d x + c\right)^{2}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*cot(d*x + c)^2, x)","F",0
361,0,0,0,0.945300," ","integrate(cot(d*x+c)^4*(a+b*sec(d*x+c))^n,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \cot\left(d x + c\right)^{4}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*cot(d*x + c)^4, x)","F",0
362,0,0,0,0.631645," ","integrate((a+b*sec(d*x+c))^n*tan(d*x+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \tan\left(d x + c\right)^{\frac{3}{2}}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*tan(d*x + c)^(3/2), x)","F",0
363,0,0,0,0.609937," ","integrate((a+b*sec(d*x+c))^n*tan(d*x+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sec\left(d x + c\right) + a\right)}^{n} \sqrt{\tan\left(d x + c\right)}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n*sqrt(tan(d*x + c)), x)","F",0
364,0,0,0,0.609190," ","integrate((a+b*sec(d*x+c))^n/tan(d*x+c)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(b \sec\left(d x + c\right) + a\right)}^{n}}{\sqrt{\tan\left(d x + c\right)}}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n/sqrt(tan(d*x + c)), x)","F",0
365,0,0,0,0.743773," ","integrate((a+b*sec(d*x+c))^n/tan(d*x+c)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(b \sec\left(d x + c\right) + a\right)}^{n}}{\tan\left(d x + c\right)^{\frac{3}{2}}}, x\right)"," ",0,"integral((b*sec(d*x + c) + a)^n/tan(d*x + c)^(3/2), x)","F",0
