1,1,43,0,0.922287," ","integrate((a*sin(x)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, a^{2} \cos\left(x\right)^{5} - 10 \, a^{2} \cos\left(x\right)^{3} + 15 \, a^{2} \cos\left(x\right)\right)} \sqrt{-a \cos\left(x\right)^{2} + a}}{15 \, \sin\left(x\right)}"," ",0,"-1/15*(3*a^2*cos(x)^5 - 10*a^2*cos(x)^3 + 15*a^2*cos(x))*sqrt(-a*cos(x)^2 + a)/sin(x)","A",0
2,1,29,0,0.814515," ","integrate((a*sin(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{{\left(a \cos\left(x\right)^{3} - 3 \, a \cos\left(x\right)\right)} \sqrt{-a \cos\left(x\right)^{2} + a}}{3 \, \sin\left(x\right)}"," ",0,"1/3*(a*cos(x)^3 - 3*a*cos(x))*sqrt(-a*cos(x)^2 + a)/sin(x)","A",0
3,1,19,0,1.103482," ","integrate((a*sin(x)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{-a \cos\left(x\right)^{2} + a} \cos\left(x\right)}{\sin\left(x\right)}"," ",0,"-sqrt(-a*cos(x)^2 + a)*cos(x)/sin(x)","A",0
4,1,70,0,0.687119," ","integrate(1/(a*sin(x)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a \cos\left(x\right)^{2} + a} \log\left(-\frac{\cos\left(x\right) - 1}{\cos\left(x\right) + 1}\right)}{2 \, a \sin\left(x\right)}, \frac{\sqrt{-a} \arctan\left(\frac{\sqrt{-a \cos\left(x\right)^{2} + a} \sqrt{-a} \cos\left(x\right)}{a \sin\left(x\right)}\right)}{a}\right]"," ",0,"[1/2*sqrt(-a*cos(x)^2 + a)*log(-(cos(x) - 1)/(cos(x) + 1))/(a*sin(x)), sqrt(-a)*arctan(sqrt(-a*cos(x)^2 + a)*sqrt(-a)*cos(x)/(a*sin(x)))/a]","B",0
5,1,58,0,1.052033," ","integrate(1/(a*sin(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{-a \cos\left(x\right)^{2} + a} {\left({\left(\cos\left(x\right)^{2} - 1\right)} \log\left(-\frac{\cos\left(x\right) - 1}{\cos\left(x\right) + 1}\right) + 2 \, \cos\left(x\right)\right)}}{4 \, {\left(a^{2} \cos\left(x\right)^{2} - a^{2}\right)} \sin\left(x\right)}"," ",0,"1/4*sqrt(-a*cos(x)^2 + a)*((cos(x)^2 - 1)*log(-(cos(x) - 1)/(cos(x) + 1)) + 2*cos(x))/((a^2*cos(x)^2 - a^2)*sin(x))","A",0
6,1,78,0,1.157638," ","integrate(1/(a*sin(x)^2)^(5/2),x, algorithm=""fricas"")","\frac{\sqrt{-a \cos\left(x\right)^{2} + a} {\left(6 \, \cos\left(x\right)^{3} + 3 \, {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{2} + 1\right)} \log\left(-\frac{\cos\left(x\right) - 1}{\cos\left(x\right) + 1}\right) - 10 \, \cos\left(x\right)\right)}}{16 \, {\left(a^{3} \cos\left(x\right)^{4} - 2 \, a^{3} \cos\left(x\right)^{2} + a^{3}\right)} \sin\left(x\right)}"," ",0,"1/16*sqrt(-a*cos(x)^2 + a)*(6*cos(x)^3 + 3*(cos(x)^4 - 2*cos(x)^2 + 1)*log(-(cos(x) - 1)/(cos(x) + 1)) - 10*cos(x))/((a^3*cos(x)^4 - 2*a^3*cos(x)^2 + a^3)*sin(x))","A",0
7,0,0,0,0.697153," ","integrate((a*sin(x)^3)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a^{2} \cos\left(x\right)^{6} - 3 \, a^{2} \cos\left(x\right)^{4} + 3 \, a^{2} \cos\left(x\right)^{2} - a^{2}\right)} \sqrt{-{\left(a \cos\left(x\right)^{2} - a\right)} \sin\left(x\right)}, x\right)"," ",0,"integral(-(a^2*cos(x)^6 - 3*a^2*cos(x)^4 + 3*a^2*cos(x)^2 - a^2)*sqrt(-(a*cos(x)^2 - a)*sin(x)), x)","F",0
8,0,0,0,0.681692," ","integrate((a*sin(x)^3)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(a \cos\left(x\right)^{2} - a\right)} \sqrt{-{\left(a \cos\left(x\right)^{2} - a\right)} \sin\left(x\right)} \sin\left(x\right), x\right)"," ",0,"integral(-(a*cos(x)^2 - a)*sqrt(-(a*cos(x)^2 - a)*sin(x))*sin(x), x)","F",0
9,0,0,0,0.696340," ","integrate((a*sin(x)^3)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-{\left(a \cos\left(x\right)^{2} - a\right)} \sin\left(x\right)}, x\right)"," ",0,"integral(sqrt(-(a*cos(x)^2 - a)*sin(x)), x)","F",0
10,0,0,0,0.526842," ","integrate(1/(a*sin(x)^3)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-{\left(a \cos\left(x\right)^{2} - a\right)} \sin\left(x\right)}}{{\left(a \cos\left(x\right)^{2} - a\right)} \sin\left(x\right)}, x\right)"," ",0,"integral(-sqrt(-(a*cos(x)^2 - a)*sin(x))/((a*cos(x)^2 - a)*sin(x)), x)","F",0
11,0,0,0,0.587091," ","integrate(1/(a*sin(x)^3)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-{\left(a \cos\left(x\right)^{2} - a\right)} \sin\left(x\right)}}{a^{2} \cos\left(x\right)^{6} - 3 \, a^{2} \cos\left(x\right)^{4} + 3 \, a^{2} \cos\left(x\right)^{2} - a^{2}}, x\right)"," ",0,"integral(-sqrt(-(a*cos(x)^2 - a)*sin(x))/(a^2*cos(x)^6 - 3*a^2*cos(x)^4 + 3*a^2*cos(x)^2 - a^2), x)","F",0
12,0,0,0,0.722735," ","integrate(1/(a*sin(x)^3)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-{\left(a \cos\left(x\right)^{2} - a\right)} \sin\left(x\right)}}{{\left(a^{3} \cos\left(x\right)^{8} - 4 \, a^{3} \cos\left(x\right)^{6} + 6 \, a^{3} \cos\left(x\right)^{4} - 4 \, a^{3} \cos\left(x\right)^{2} + a^{3}\right)} \sin\left(x\right)}, x\right)"," ",0,"integral(sqrt(-(a*cos(x)^2 - a)*sin(x))/((a^3*cos(x)^8 - 4*a^3*cos(x)^6 + 6*a^3*cos(x)^4 - 4*a^3*cos(x)^2 + a^3)*sin(x)), x)","F",0
13,1,82,0,0.594625," ","integrate((a*sin(x)^4)^(5/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(x\right)^{4} - 2 \, a \cos\left(x\right)^{2} + a} {\left(315 \, a^{2} x - {\left(128 \, a^{2} \cos\left(x\right)^{9} - 656 \, a^{2} \cos\left(x\right)^{7} + 1368 \, a^{2} \cos\left(x\right)^{5} - 1490 \, a^{2} \cos\left(x\right)^{3} + 965 \, a^{2} \cos\left(x\right)\right)} \sin\left(x\right)\right)}}{1280 \, {\left(\cos\left(x\right)^{2} - 1\right)}}"," ",0,"-1/1280*sqrt(a*cos(x)^4 - 2*a*cos(x)^2 + a)*(315*a^2*x - (128*a^2*cos(x)^9 - 656*a^2*cos(x)^7 + 1368*a^2*cos(x)^5 - 1490*a^2*cos(x)^3 + 965*a^2*cos(x))*sin(x))/(cos(x)^2 - 1)","A",0
14,1,56,0,0.612784," ","integrate((a*sin(x)^4)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(x\right)^{4} - 2 \, a \cos\left(x\right)^{2} + a} {\left(15 \, a x - {\left(8 \, a \cos\left(x\right)^{5} - 26 \, a \cos\left(x\right)^{3} + 33 \, a \cos\left(x\right)\right)} \sin\left(x\right)\right)}}{48 \, {\left(\cos\left(x\right)^{2} - 1\right)}}"," ",0,"-1/48*sqrt(a*cos(x)^4 - 2*a*cos(x)^2 + a)*(15*a*x - (8*a*cos(x)^5 - 26*a*cos(x)^3 + 33*a*cos(x))*sin(x))/(cos(x)^2 - 1)","A",0
15,1,36,0,0.588885," ","integrate((a*sin(x)^4)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{a \cos\left(x\right)^{4} - 2 \, a \cos\left(x\right)^{2} + a} {\left(\cos\left(x\right) \sin\left(x\right) - x\right)}}{2 \, {\left(\cos\left(x\right)^{2} - 1\right)}}"," ",0,"1/2*sqrt(a*cos(x)^4 - 2*a*cos(x)^2 + a)*(cos(x)*sin(x) - x)/(cos(x)^2 - 1)","A",0
16,1,36,0,0.683797," ","integrate(1/(a*sin(x)^4)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{a \cos\left(x\right)^{4} - 2 \, a \cos\left(x\right)^{2} + a} \cos\left(x\right)}{{\left(a \cos\left(x\right)^{2} - a\right)} \sin\left(x\right)}"," ",0,"sqrt(a*cos(x)^4 - 2*a*cos(x)^2 + a)*cos(x)/((a*cos(x)^2 - a)*sin(x))","B",0
17,1,74,0,0.576197," ","integrate(1/(a*sin(x)^4)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{a \cos\left(x\right)^{4} - 2 \, a \cos\left(x\right)^{2} + a} {\left(8 \, \cos\left(x\right)^{5} - 20 \, \cos\left(x\right)^{3} + 15 \, \cos\left(x\right)\right)}}{15 \, {\left(a^{2} \cos\left(x\right)^{6} - 3 \, a^{2} \cos\left(x\right)^{4} + 3 \, a^{2} \cos\left(x\right)^{2} - a^{2}\right)} \sin\left(x\right)}"," ",0,"1/15*sqrt(a*cos(x)^4 - 2*a*cos(x)^2 + a)*(8*cos(x)^5 - 20*cos(x)^3 + 15*cos(x))/((a^2*cos(x)^6 - 3*a^2*cos(x)^4 + 3*a^2*cos(x)^2 - a^2)*sin(x))","A",0
18,1,104,0,0.581186," ","integrate(1/(a*sin(x)^4)^(5/2),x, algorithm=""fricas"")","\frac{{\left(128 \, \cos\left(x\right)^{9} - 576 \, \cos\left(x\right)^{7} + 1008 \, \cos\left(x\right)^{5} - 840 \, \cos\left(x\right)^{3} + 315 \, \cos\left(x\right)\right)} \sqrt{a \cos\left(x\right)^{4} - 2 \, a \cos\left(x\right)^{2} + a}}{315 \, {\left(a^{3} \cos\left(x\right)^{10} - 5 \, a^{3} \cos\left(x\right)^{8} + 10 \, a^{3} \cos\left(x\right)^{6} - 10 \, a^{3} \cos\left(x\right)^{4} + 5 \, a^{3} \cos\left(x\right)^{2} - a^{3}\right)} \sin\left(x\right)}"," ",0,"1/315*(128*cos(x)^9 - 576*cos(x)^7 + 1008*cos(x)^5 - 840*cos(x)^3 + 315*cos(x))*sqrt(a*cos(x)^4 - 2*a*cos(x)^2 + a)/((a^3*cos(x)^10 - 5*a^3*cos(x)^8 + 10*a^3*cos(x)^6 - 10*a^3*cos(x)^4 + 5*a^3*cos(x)^2 - a^3)*sin(x))","A",0
19,-2,0,0,0.000000," ","integrate((c*sin(b*x+a)^m)^(5/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
20,-2,0,0,0.000000," ","integrate((c*sin(b*x+a)^m)^(3/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
21,-2,0,0,0.000000," ","integrate((c*sin(b*x+a)^m)^(1/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
22,-2,0,0,0.000000," ","integrate(1/(c*sin(b*x+a)^m)^(1/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
23,-2,0,0,0.000000," ","integrate(1/(c*sin(b*x+a)^m)^(3/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
24,-2,0,0,0.000000," ","integrate(1/(c*sin(b*x+a)^m)^(5/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (has polynomial part)","F(-2)",0
25,0,0,0,0.643139," ","integrate((b*sin(d*x+c)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(b \sin\left(d x + c\right)^{n}\right)^{p}, x\right)"," ",0,"integral((b*sin(d*x + c)^n)^p, x)","F",0
26,0,0,0,0.690194," ","integrate((c*sin(b*x+a)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-c \cos\left(b x + a\right)^{2} + c\right)}^{p}, x\right)"," ",0,"integral((-c*cos(b*x + a)^2 + c)^p, x)","F",0
27,0,0,0,0.522634," ","integrate((c*sin(b*x+a)^3)^p,x, algorithm=""fricas"")","{\rm integral}\left(\left(-{\left(c \cos\left(b x + a\right)^{2} - c\right)} \sin\left(b x + a\right)\right)^{p}, x\right)"," ",0,"integral((-(c*cos(b*x + a)^2 - c)*sin(b*x + a))^p, x)","F",0
28,0,0,0,0.732838," ","integrate((c*sin(b*x+a)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(c \cos\left(b x + a\right)^{4} - 2 \, c \cos\left(b x + a\right)^{2} + c\right)}^{p}, x\right)"," ",0,"integral((c*cos(b*x + a)^4 - 2*c*cos(b*x + a)^2 + c)^p, x)","F",0
29,1,16,0,0.622169," ","integrate((c*sin(b*x+a)^n)^(1/n),x, algorithm=""fricas"")","-\frac{c^{\left(\frac{1}{n}\right)} \cos\left(b x + a\right)}{b}"," ",0,"-c^(1/n)*cos(b*x + a)/b","A",0
30,0,0,0,0.805306," ","integrate((a*(b*sin(d*x+c))^p)^n,x, algorithm=""fricas"")","{\rm integral}\left(\left(\left(b \sin\left(d x + c\right)\right)^{p} a\right)^{n}, x\right)"," ",0,"integral(((b*sin(d*x + c))^p*a)^n, x)","F",0
31,1,12,0,0.628623," ","integrate(a-a*sin(x)^2,x, algorithm=""fricas"")","\frac{1}{2} \, a \cos\left(x\right) \sin\left(x\right) + \frac{1}{2} \, a x"," ",0,"1/2*a*cos(x)*sin(x) + 1/2*a*x","A",0
32,1,28,0,0.598511," ","integrate((a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{3}{8} \, a^{2} x + \frac{1}{8} \, {\left(2 \, a^{2} \cos\left(x\right)^{3} + 3 \, a^{2} \cos\left(x\right)\right)} \sin\left(x\right)"," ",0,"3/8*a^2*x + 1/8*(2*a^2*cos(x)^3 + 3*a^2*cos(x))*sin(x)","A",0
33,1,37,0,0.517771," ","integrate((a-a*sin(x)^2)^3,x, algorithm=""fricas"")","\frac{5}{16} \, a^{3} x + \frac{1}{48} \, {\left(8 \, a^{3} \cos\left(x\right)^{5} + 10 \, a^{3} \cos\left(x\right)^{3} + 15 \, a^{3} \cos\left(x\right)\right)} \sin\left(x\right)"," ",0,"5/16*a^3*x + 1/48*(8*a^3*cos(x)^5 + 10*a^3*cos(x)^3 + 15*a^3*cos(x))*sin(x)","A",0
34,1,46,0,0.727588," ","integrate((a-a*sin(x)^2)^4,x, algorithm=""fricas"")","\frac{35}{128} \, a^{4} x + \frac{1}{384} \, {\left(48 \, a^{4} \cos\left(x\right)^{7} + 56 \, a^{4} \cos\left(x\right)^{5} + 70 \, a^{4} \cos\left(x\right)^{3} + 105 \, a^{4} \cos\left(x\right)\right)} \sin\left(x\right)"," ",0,"35/128*a^4*x + 1/384*(48*a^4*cos(x)^7 + 56*a^4*cos(x)^5 + 70*a^4*cos(x)^3 + 105*a^4*cos(x))*sin(x)","A",0
35,1,46,0,0.742583," ","integrate(sin(d*x+c)^7/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{\cos\left(d x + c\right)^{6} - 5 \, \cos\left(d x + c\right)^{4} + 15 \, \cos\left(d x + c\right)^{2} + 5}{5 \, a d \cos\left(d x + c\right)}"," ",0,"1/5*(cos(d*x + c)^6 - 5*cos(d*x + c)^4 + 15*cos(d*x + c)^2 + 5)/(a*d*cos(d*x + c))","A",0
36,1,36,0,0.732128," ","integrate(sin(d*x+c)^5/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{\cos\left(d x + c\right)^{4} - 6 \, \cos\left(d x + c\right)^{2} - 3}{3 \, a d \cos\left(d x + c\right)}"," ",0,"-1/3*(cos(d*x + c)^4 - 6*cos(d*x + c)^2 - 3)/(a*d*cos(d*x + c))","A",0
37,1,25,0,0.674026," ","integrate(sin(d*x+c)^3/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{\cos\left(d x + c\right)^{2} + 1}{a d \cos\left(d x + c\right)}"," ",0,"(cos(d*x + c)^2 + 1)/(a*d*cos(d*x + c))","A",0
38,1,15,0,0.597478," ","integrate(sin(d*x+c)/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{1}{a d \cos\left(d x + c\right)}"," ",0,"1/(a*d*cos(d*x + c))","A",0
39,1,55,0,0.666987," ","integrate(csc(d*x+c)/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{\cos\left(d x + c\right) \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - \cos\left(d x + c\right) \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 2}{2 \, a d \cos\left(d x + c\right)}"," ",0,"-1/2*(cos(d*x + c)*log(1/2*cos(d*x + c) + 1/2) - cos(d*x + c)*log(-1/2*cos(d*x + c) + 1/2) - 2)/(a*d*cos(d*x + c))","A",0
40,1,98,0,0.612064," ","integrate(csc(d*x+c)^3/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{6 \, \cos\left(d x + c\right)^{2} - 3 \, {\left(\cos\left(d x + c\right)^{3} - \cos\left(d x + c\right)\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} - \cos\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 4}{4 \, {\left(a d \cos\left(d x + c\right)^{3} - a d \cos\left(d x + c\right)\right)}}"," ",0,"1/4*(6*cos(d*x + c)^2 - 3*(cos(d*x + c)^3 - cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 3*(cos(d*x + c)^3 - cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 4)/(a*d*cos(d*x + c)^3 - a*d*cos(d*x + c))","A",0
41,1,135,0,0.737116," ","integrate(csc(d*x+c)^5/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{30 \, \cos\left(d x + c\right)^{4} - 50 \, \cos\left(d x + c\right)^{2} - 15 \, {\left(\cos\left(d x + c\right)^{5} - 2 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 15 \, {\left(\cos\left(d x + c\right)^{5} - 2 \, \cos\left(d x + c\right)^{3} + \cos\left(d x + c\right)\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 16}{16 \, {\left(a d \cos\left(d x + c\right)^{5} - 2 \, a d \cos\left(d x + c\right)^{3} + a d \cos\left(d x + c\right)\right)}}"," ",0,"1/16*(30*cos(d*x + c)^4 - 50*cos(d*x + c)^2 - 15*(cos(d*x + c)^5 - 2*cos(d*x + c)^3 + cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 15*(cos(d*x + c)^5 - 2*cos(d*x + c)^3 + cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 16)/(a*d*cos(d*x + c)^5 - 2*a*d*cos(d*x + c)^3 + a*d*cos(d*x + c))","A",0
42,1,56,0,0.678059," ","integrate(sin(d*x+c)^6/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{15 \, d x \cos\left(d x + c\right) + {\left(2 \, \cos\left(d x + c\right)^{4} - 9 \, \cos\left(d x + c\right)^{2} - 8\right)} \sin\left(d x + c\right)}{8 \, a d \cos\left(d x + c\right)}"," ",0,"-1/8*(15*d*x*cos(d*x + c) + (2*cos(d*x + c)^4 - 9*cos(d*x + c)^2 - 8)*sin(d*x + c))/(a*d*cos(d*x + c))","A",0
43,1,45,0,0.625635," ","integrate(sin(d*x+c)^4/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{3 \, d x \cos\left(d x + c\right) - {\left(\cos\left(d x + c\right)^{2} + 2\right)} \sin\left(d x + c\right)}{2 \, a d \cos\left(d x + c\right)}"," ",0,"-1/2*(3*d*x*cos(d*x + c) - (cos(d*x + c)^2 + 2)*sin(d*x + c))/(a*d*cos(d*x + c))","A",0
44,1,34,0,0.677963," ","integrate(sin(d*x+c)^2/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{d x \cos\left(d x + c\right) - \sin\left(d x + c\right)}{a d \cos\left(d x + c\right)}"," ",0,"-(d*x*cos(d*x + c) - sin(d*x + c))/(a*d*cos(d*x + c))","A",0
45,1,21,0,0.628512," ","integrate(1/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{\sin\left(d x + c\right)}{a d \cos\left(d x + c\right)}"," ",0,"sin(d*x + c)/(a*d*cos(d*x + c))","A",0
46,1,36,0,0.635508," ","integrate(csc(d*x+c)^2/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{2 \, \cos\left(d x + c\right)^{2} - 1}{a d \cos\left(d x + c\right) \sin\left(d x + c\right)}"," ",0,"-(2*cos(d*x + c)^2 - 1)/(a*d*cos(d*x + c)*sin(d*x + c))","A",0
47,1,56,0,0.864641," ","integrate(csc(d*x+c)^4/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{8 \, \cos\left(d x + c\right)^{4} - 12 \, \cos\left(d x + c\right)^{2} + 3}{3 \, {\left(a d \cos\left(d x + c\right)^{3} - a d \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}"," ",0,"-1/3*(8*cos(d*x + c)^4 - 12*cos(d*x + c)^2 + 3)/((a*d*cos(d*x + c)^3 - a*d*cos(d*x + c))*sin(d*x + c))","A",0
48,1,77,0,0.572995," ","integrate(csc(d*x+c)^6/(a-a*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{16 \, \cos\left(d x + c\right)^{6} - 40 \, \cos\left(d x + c\right)^{4} + 30 \, \cos\left(d x + c\right)^{2} - 5}{5 \, {\left(a d \cos\left(d x + c\right)^{5} - 2 \, a d \cos\left(d x + c\right)^{3} + a d \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}"," ",0,"-1/5*(16*cos(d*x + c)^6 - 40*cos(d*x + c)^4 + 30*cos(d*x + c)^2 - 5)/((a*d*cos(d*x + c)^5 - 2*a*d*cos(d*x + c)^3 + a*d*cos(d*x + c))*sin(d*x + c))","A",0
49,1,46,0,0.690594," ","integrate(sin(d*x+c)^7/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{\cos\left(d x + c\right)^{6} - 9 \, \cos\left(d x + c\right)^{4} - 9 \, \cos\left(d x + c\right)^{2} + 1}{3 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"1/3*(cos(d*x + c)^6 - 9*cos(d*x + c)^4 - 9*cos(d*x + c)^2 + 1)/(a^2*d*cos(d*x + c)^3)","A",0
50,1,38,0,0.511924," ","integrate(sin(d*x+c)^5/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","-\frac{3 \, \cos\left(d x + c\right)^{4} + 6 \, \cos\left(d x + c\right)^{2} - 1}{3 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"-1/3*(3*cos(d*x + c)^4 + 6*cos(d*x + c)^2 - 1)/(a^2*d*cos(d*x + c)^3)","A",0
51,1,28,0,0.787458," ","integrate(sin(d*x+c)^3/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","-\frac{3 \, \cos\left(d x + c\right)^{2} - 1}{3 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"-1/3*(3*cos(d*x + c)^2 - 1)/(a^2*d*cos(d*x + c)^3)","A",0
52,1,16,0,0.684515," ","integrate(sin(d*x+c)/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{1}{3 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"1/3/(a^2*d*cos(d*x + c)^3)","A",0
53,1,70,0,0.658129," ","integrate(csc(d*x+c)/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","-\frac{3 \, \cos\left(d x + c\right)^{3} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 3 \, \cos\left(d x + c\right)^{3} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 6 \, \cos\left(d x + c\right)^{2} - 2}{6 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"-1/6*(3*cos(d*x + c)^3*log(1/2*cos(d*x + c) + 1/2) - 3*cos(d*x + c)^3*log(-1/2*cos(d*x + c) + 1/2) - 6*cos(d*x + c)^2 - 2)/(a^2*d*cos(d*x + c)^3)","A",0
54,1,118,0,0.661033," ","integrate(csc(d*x+c)^3/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{30 \, \cos\left(d x + c\right)^{4} - 20 \, \cos\left(d x + c\right)^{2} - 15 \, {\left(\cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{3}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 15 \, {\left(\cos\left(d x + c\right)^{5} - \cos\left(d x + c\right)^{3}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 4}{12 \, {\left(a^{2} d \cos\left(d x + c\right)^{5} - a^{2} d \cos\left(d x + c\right)^{3}\right)}}"," ",0,"1/12*(30*cos(d*x + c)^4 - 20*cos(d*x + c)^2 - 15*(cos(d*x + c)^5 - cos(d*x + c)^3)*log(1/2*cos(d*x + c) + 1/2) + 15*(cos(d*x + c)^5 - cos(d*x + c)^3)*log(-1/2*cos(d*x + c) + 1/2) - 4)/(a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^3)","A",0
55,1,59,0,0.547964," ","integrate(sin(d*x+c)^6/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{15 \, d x \cos\left(d x + c\right)^{3} - {\left(3 \, \cos\left(d x + c\right)^{4} + 14 \, \cos\left(d x + c\right)^{2} - 2\right)} \sin\left(d x + c\right)}{6 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"1/6*(15*d*x*cos(d*x + c)^3 - (3*cos(d*x + c)^4 + 14*cos(d*x + c)^2 - 2)*sin(d*x + c))/(a^2*d*cos(d*x + c)^3)","A",0
56,1,49,0,0.666086," ","integrate(sin(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{3 \, d x \cos\left(d x + c\right)^{3} - {\left(4 \, \cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right)}{3 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"1/3*(3*d*x*cos(d*x + c)^3 - (4*cos(d*x + c)^2 - 1)*sin(d*x + c))/(a^2*d*cos(d*x + c)^3)","A",0
57,1,32,0,0.804668," ","integrate(sin(d*x+c)^2/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","-\frac{{\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right)}{3 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"-1/3*(cos(d*x + c)^2 - 1)*sin(d*x + c)/(a^2*d*cos(d*x + c)^3)","A",0
58,1,34,0,0.520144," ","integrate(1/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{{\left(2 \, \cos\left(d x + c\right)^{2} + 1\right)} \sin\left(d x + c\right)}{3 \, a^{2} d \cos\left(d x + c\right)^{3}}"," ",0,"1/3*(2*cos(d*x + c)^2 + 1)*sin(d*x + c)/(a^2*d*cos(d*x + c)^3)","A",0
59,1,46,0,0.623697," ","integrate(csc(d*x+c)^2/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","-\frac{8 \, \cos\left(d x + c\right)^{4} - 4 \, \cos\left(d x + c\right)^{2} - 1}{3 \, a^{2} d \cos\left(d x + c\right)^{3} \sin\left(d x + c\right)}"," ",0,"-1/3*(8*cos(d*x + c)^4 - 4*cos(d*x + c)^2 - 1)/(a^2*d*cos(d*x + c)^3*sin(d*x + c))","A",0
60,1,72,0,0.762180," ","integrate(csc(d*x+c)^4/(a-a*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","-\frac{16 \, \cos\left(d x + c\right)^{6} - 24 \, \cos\left(d x + c\right)^{4} + 6 \, \cos\left(d x + c\right)^{2} + 1}{3 \, {\left(a^{2} d \cos\left(d x + c\right)^{5} - a^{2} d \cos\left(d x + c\right)^{3}\right)} \sin\left(d x + c\right)}"," ",0,"-1/3*(16*cos(d*x + c)^6 - 24*cos(d*x + c)^4 + 6*cos(d*x + c)^2 + 1)/((a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^3)*sin(d*x + c))","A",0
61,1,25,0,0.527776," ","integrate(1/(a-a*sin(x)^2)^3,x, algorithm=""fricas"")","\frac{{\left(8 \, \cos\left(x\right)^{4} + 4 \, \cos\left(x\right)^{2} + 3\right)} \sin\left(x\right)}{15 \, a^{3} \cos\left(x\right)^{5}}"," ",0,"1/15*(8*cos(x)^4 + 4*cos(x)^2 + 3)*sin(x)/(a^3*cos(x)^5)","A",0
62,1,31,0,0.590148," ","integrate(1/(a-a*sin(x)^2)^4,x, algorithm=""fricas"")","\frac{{\left(16 \, \cos\left(x\right)^{6} + 8 \, \cos\left(x\right)^{4} + 6 \, \cos\left(x\right)^{2} + 5\right)} \sin\left(x\right)}{35 \, a^{4} \cos\left(x\right)^{7}}"," ",0,"1/35*(16*cos(x)^6 + 8*cos(x)^4 + 6*cos(x)^2 + 5)*sin(x)/(a^4*cos(x)^7)","A",0
63,1,37,0,0.768209," ","integrate(1/(a-a*sin(x)^2)^5,x, algorithm=""fricas"")","\frac{{\left(128 \, \cos\left(x\right)^{8} + 64 \, \cos\left(x\right)^{6} + 48 \, \cos\left(x\right)^{4} + 40 \, \cos\left(x\right)^{2} + 35\right)} \sin\left(x\right)}{315 \, a^{5} \cos\left(x\right)^{9}}"," ",0,"1/315*(128*cos(x)^8 + 64*cos(x)^6 + 48*cos(x)^4 + 40*cos(x)^2 + 35)*sin(x)/(a^5*cos(x)^9)","A",0
64,1,43,0,0.707633," ","integrate(sin(d*x+c)^3*(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{3 \, b \cos\left(d x + c\right)^{5} - 5 \, {\left(a + 2 \, b\right)} \cos\left(d x + c\right)^{3} + 15 \, {\left(a + b\right)} \cos\left(d x + c\right)}{15 \, d}"," ",0,"-1/15*(3*b*cos(d*x + c)^5 - 5*(a + 2*b)*cos(d*x + c)^3 + 15*(a + b)*cos(d*x + c))/d","A",0
65,1,27,0,0.550619," ","integrate(sin(d*x+c)*(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{b \cos\left(d x + c\right)^{3} - 3 \, {\left(a + b\right)} \cos\left(d x + c\right)}{3 \, d}"," ",0,"1/3*(b*cos(d*x + c)^3 - 3*(a + b)*cos(d*x + c))/d","A",0
66,1,42,0,0.762578," ","integrate(csc(d*x+c)*(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{2 \, b \cos\left(d x + c\right) + a \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - a \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{2 \, d}"," ",0,"-1/2*(2*b*cos(d*x + c) + a*log(1/2*cos(d*x + c) + 1/2) - a*log(-1/2*cos(d*x + c) + 1/2))/d","A",0
67,1,95,0,0.680899," ","integrate(csc(d*x+c)^3*(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{2 \, a \cos\left(d x + c\right) - {\left({\left(a + 2 \, b\right)} \cos\left(d x + c\right)^{2} - a - 2 \, b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left({\left(a + 2 \, b\right)} \cos\left(d x + c\right)^{2} - a - 2 \, b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{4 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"1/4*(2*a*cos(d*x + c) - ((a + 2*b)*cos(d*x + c)^2 - a - 2*b)*log(1/2*cos(d*x + c) + 1/2) + ((a + 2*b)*cos(d*x + c)^2 - a - 2*b)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^2 - d)","B",0
68,1,69,0,0.542622," ","integrate(sin(d*x+c)^4*(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{3 \, {\left(6 \, a + 5 \, b\right)} d x - {\left(8 \, b \cos\left(d x + c\right)^{5} - 2 \, {\left(6 \, a + 13 \, b\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(10 \, a + 11 \, b\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{48 \, d}"," ",0,"1/48*(3*(6*a + 5*b)*d*x - (8*b*cos(d*x + c)^5 - 2*(6*a + 13*b)*cos(d*x + c)^3 + 3*(10*a + 11*b)*cos(d*x + c))*sin(d*x + c))/d","A",0
69,1,50,0,0.657031," ","integrate(sin(d*x+c)^2*(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left(4 \, a + 3 \, b\right)} d x + {\left(2 \, b \cos\left(d x + c\right)^{3} - {\left(4 \, a + 5 \, b\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, d}"," ",0,"1/8*((4*a + 3*b)*d*x + (2*b*cos(d*x + c)^3 - (4*a + 5*b)*cos(d*x + c))*sin(d*x + c))/d","A",0
70,1,29,0,0.669596," ","integrate(a+b*sin(d*x+c)^2,x, algorithm=""fricas"")","\frac{{\left(2 \, a + b\right)} d x - b \cos\left(d x + c\right) \sin\left(d x + c\right)}{2 \, d}"," ",0,"1/2*((2*a + b)*d*x - b*cos(d*x + c)*sin(d*x + c))/d","A",0
71,1,32,0,0.700572," ","integrate(csc(d*x+c)^2*(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{b d x \sin\left(d x + c\right) - a \cos\left(d x + c\right)}{d \sin\left(d x + c\right)}"," ",0,"(b*d*x*sin(d*x + c) - a*cos(d*x + c))/(d*sin(d*x + c))","A",0
72,1,54,0,0.580691," ","integrate(csc(d*x+c)^4*(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{{\left(2 \, a + 3 \, b\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(a + b\right)} \cos\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*((2*a + 3*b)*cos(d*x + c)^3 - 3*(a + b)*cos(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))","A",0
73,1,81,0,0.560291," ","integrate(csc(d*x+c)^6*(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{2 \, {\left(4 \, a + 5 \, b\right)} \cos\left(d x + c\right)^{5} - 5 \, {\left(4 \, a + 5 \, b\right)} \cos\left(d x + c\right)^{3} + 15 \, {\left(a + b\right)} \cos\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*(2*(4*a + 5*b)*cos(d*x + c)^5 - 5*(4*a + 5*b)*cos(d*x + c)^3 + 15*(a + b)*cos(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))","A",0
74,1,16,0,0.773478," ","integrate(a+b*sin(x)^2,x, algorithm=""fricas"")","-\frac{1}{2} \, b \cos\left(x\right) \sin\left(x\right) + \frac{1}{2} \, {\left(2 \, a + b\right)} x"," ",0,"-1/2*b*cos(x)*sin(x) + 1/2*(2*a + b)*x","A",0
75,1,47,0,0.563909," ","integrate((a+b*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{1}{8} \, {\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} x + \frac{1}{8} \, {\left(2 \, b^{2} \cos\left(x\right)^{3} - {\left(8 \, a b + 5 \, b^{2}\right)} \cos\left(x\right)\right)} \sin\left(x\right)"," ",0,"1/8*(8*a^2 + 8*a*b + 3*b^2)*x + 1/8*(2*b^2*cos(x)^3 - (8*a*b + 5*b^2)*cos(x))*sin(x)","A",0
76,1,81,0,0.529929," ","integrate((a+b*sin(x)^2)^3,x, algorithm=""fricas"")","\frac{1}{16} \, {\left(16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right)} x - \frac{1}{48} \, {\left(8 \, b^{3} \cos\left(x\right)^{5} - 2 \, {\left(18 \, a b^{2} + 13 \, b^{3}\right)} \cos\left(x\right)^{3} + 3 \, {\left(24 \, a^{2} b + 30 \, a b^{2} + 11 \, b^{3}\right)} \cos\left(x\right)\right)} \sin\left(x\right)"," ",0,"1/16*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*x - 1/48*(8*b^3*cos(x)^5 - 2*(18*a*b^2 + 13*b^3)*cos(x)^3 + 3*(24*a^2*b + 30*a*b^2 + 11*b^3)*cos(x))*sin(x)","A",0
77,1,123,0,0.577819," ","integrate((a+b*sin(x)^2)^4,x, algorithm=""fricas"")","\frac{1}{128} \, {\left(128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right)} x + \frac{1}{384} \, {\left(48 \, b^{4} \cos\left(x\right)^{7} - 8 \, {\left(32 \, a b^{3} + 25 \, b^{4}\right)} \cos\left(x\right)^{5} + 2 \, {\left(288 \, a^{2} b^{2} + 416 \, a b^{3} + 163 \, b^{4}\right)} \cos\left(x\right)^{3} - 3 \, {\left(256 \, a^{3} b + 480 \, a^{2} b^{2} + 352 \, a b^{3} + 93 \, b^{4}\right)} \cos\left(x\right)\right)} \sin\left(x\right)"," ",0,"1/128*(128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*x + 1/384*(48*b^4*cos(x)^7 - 8*(32*a*b^3 + 25*b^4)*cos(x)^5 + 2*(288*a^2*b^2 + 416*a*b^3 + 163*b^4)*cos(x)^3 - 3*(256*a^3*b + 480*a^2*b^2 + 352*a*b^3 + 93*b^4)*cos(x))*sin(x)","A",0
78,1,272,0,0.808769," ","integrate(sin(d*x+c)^7/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{6 \, {\left(a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{5} - 15 \, \sqrt{a b + b^{2}} a^{3} \log\left(\frac{b \cos\left(d x + c\right)^{2} + 2 \, \sqrt{a b + b^{2}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) + 10 \, {\left(a^{2} b^{2} - a b^{3} - 2 \, b^{4}\right)} \cos\left(d x + c\right)^{3} + 30 \, {\left(a^{3} b + b^{4}\right)} \cos\left(d x + c\right)}{30 \, {\left(a b^{4} + b^{5}\right)} d}, -\frac{3 \, {\left(a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{5} + 15 \, \sqrt{-a b - b^{2}} a^{3} \arctan\left(\frac{\sqrt{-a b - b^{2}} \cos\left(d x + c\right)}{a + b}\right) + 5 \, {\left(a^{2} b^{2} - a b^{3} - 2 \, b^{4}\right)} \cos\left(d x + c\right)^{3} + 15 \, {\left(a^{3} b + b^{4}\right)} \cos\left(d x + c\right)}{15 \, {\left(a b^{4} + b^{5}\right)} d}\right]"," ",0,"[-1/30*(6*(a*b^3 + b^4)*cos(d*x + c)^5 - 15*sqrt(a*b + b^2)*a^3*log((b*cos(d*x + c)^2 + 2*sqrt(a*b + b^2)*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) + 10*(a^2*b^2 - a*b^3 - 2*b^4)*cos(d*x + c)^3 + 30*(a^3*b + b^4)*cos(d*x + c))/((a*b^4 + b^5)*d), -1/15*(3*(a*b^3 + b^4)*cos(d*x + c)^5 + 15*sqrt(-a*b - b^2)*a^3*arctan(sqrt(-a*b - b^2)*cos(d*x + c)/(a + b)) + 5*(a^2*b^2 - a*b^3 - 2*b^4)*cos(d*x + c)^3 + 15*(a^3*b + b^4)*cos(d*x + c))/((a*b^4 + b^5)*d)]","A",0
79,1,218,0,0.682387," ","integrate(sin(d*x+c)^5/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{3} + 3 \, \sqrt{a b + b^{2}} a^{2} \log\left(-\frac{b \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a b + b^{2}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) + 6 \, {\left(a^{2} b - b^{3}\right)} \cos\left(d x + c\right)}{6 \, {\left(a b^{3} + b^{4}\right)} d}, \frac{{\left(a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{3} + 3 \, \sqrt{-a b - b^{2}} a^{2} \arctan\left(\frac{\sqrt{-a b - b^{2}} \cos\left(d x + c\right)}{a + b}\right) + 3 \, {\left(a^{2} b - b^{3}\right)} \cos\left(d x + c\right)}{3 \, {\left(a b^{3} + b^{4}\right)} d}\right]"," ",0,"[1/6*(2*(a*b^2 + b^3)*cos(d*x + c)^3 + 3*sqrt(a*b + b^2)*a^2*log(-(b*cos(d*x + c)^2 - 2*sqrt(a*b + b^2)*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) + 6*(a^2*b - b^3)*cos(d*x + c))/((a*b^3 + b^4)*d), 1/3*((a*b^2 + b^3)*cos(d*x + c)^3 + 3*sqrt(-a*b - b^2)*a^2*arctan(sqrt(-a*b - b^2)*cos(d*x + c)/(a + b)) + 3*(a^2*b - b^3)*cos(d*x + c))/((a*b^3 + b^4)*d)]","A",0
80,1,165,0,0.724123," ","integrate(sin(d*x+c)^3/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a b + b^{2}} a \log\left(\frac{b \cos\left(d x + c\right)^{2} + 2 \, \sqrt{a b + b^{2}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)}{2 \, {\left(a b^{2} + b^{3}\right)} d}, -\frac{\sqrt{-a b - b^{2}} a \arctan\left(\frac{\sqrt{-a b - b^{2}} \cos\left(d x + c\right)}{a + b}\right) + {\left(a b + b^{2}\right)} \cos\left(d x + c\right)}{{\left(a b^{2} + b^{3}\right)} d}\right]"," ",0,"[1/2*(sqrt(a*b + b^2)*a*log((b*cos(d*x + c)^2 + 2*sqrt(a*b + b^2)*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) - 2*(a*b + b^2)*cos(d*x + c))/((a*b^2 + b^3)*d), -(sqrt(-a*b - b^2)*a*arctan(sqrt(-a*b - b^2)*cos(d*x + c)/(a + b)) + (a*b + b^2)*cos(d*x + c))/((a*b^2 + b^3)*d)]","A",0
81,1,117,0,0.610825," ","integrate(sin(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{\log\left(-\frac{b \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a b + b^{2}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right)}{2 \, \sqrt{a b + b^{2}} d}, \frac{\sqrt{-a b - b^{2}} \arctan\left(\frac{\sqrt{-a b - b^{2}} \cos\left(d x + c\right)}{a + b}\right)}{{\left(a b + b^{2}\right)} d}\right]"," ",0,"[1/2*log(-(b*cos(d*x + c)^2 - 2*sqrt(a*b + b^2)*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b))/(sqrt(a*b + b^2)*d), sqrt(-a*b - b^2)*arctan(sqrt(-a*b - b^2)*cos(d*x + c)/(a + b))/((a*b + b^2)*d)]","A",0
82,1,161,0,0.512460," ","integrate(csc(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{\frac{b}{a + b}} \log\left(\frac{b \cos\left(d x + c\right)^{2} + 2 \, {\left(a + b\right)} \sqrt{\frac{b}{a + b}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) - \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{2 \, a d}, -\frac{2 \, \sqrt{-\frac{b}{a + b}} \arctan\left(\sqrt{-\frac{b}{a + b}} \cos\left(d x + c\right)\right) + \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{2 \, a d}\right]"," ",0,"[1/2*(sqrt(b/(a + b))*log((b*cos(d*x + c)^2 + 2*(a + b)*sqrt(b/(a + b))*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) - log(1/2*cos(d*x + c) + 1/2) + log(-1/2*cos(d*x + c) + 1/2))/(a*d), -1/2*(2*sqrt(-b/(a + b))*arctan(sqrt(-b/(a + b))*cos(d*x + c)) + log(1/2*cos(d*x + c) + 1/2) - log(-1/2*cos(d*x + c) + 1/2))/(a*d)]","A",0
83,1,327,0,0.822056," ","integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(b \cos\left(d x + c\right)^{2} - b\right)} \sqrt{\frac{b}{a + b}} \log\left(-\frac{b \cos\left(d x + c\right)^{2} - 2 \, {\left(a + b\right)} \sqrt{\frac{b}{a + b}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) + 2 \, a \cos\left(d x + c\right) - {\left({\left(a - 2 \, b\right)} \cos\left(d x + c\right)^{2} - a + 2 \, b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left({\left(a - 2 \, b\right)} \cos\left(d x + c\right)^{2} - a + 2 \, b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{4 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)}}, \frac{4 \, {\left(b \cos\left(d x + c\right)^{2} - b\right)} \sqrt{-\frac{b}{a + b}} \arctan\left(\sqrt{-\frac{b}{a + b}} \cos\left(d x + c\right)\right) + 2 \, a \cos\left(d x + c\right) - {\left({\left(a - 2 \, b\right)} \cos\left(d x + c\right)^{2} - a + 2 \, b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left({\left(a - 2 \, b\right)} \cos\left(d x + c\right)^{2} - a + 2 \, b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{4 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)}}\right]"," ",0,"[1/4*(2*(b*cos(d*x + c)^2 - b)*sqrt(b/(a + b))*log(-(b*cos(d*x + c)^2 - 2*(a + b)*sqrt(b/(a + b))*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) + 2*a*cos(d*x + c) - ((a - 2*b)*cos(d*x + c)^2 - a + 2*b)*log(1/2*cos(d*x + c) + 1/2) + ((a - 2*b)*cos(d*x + c)^2 - a + 2*b)*log(-1/2*cos(d*x + c) + 1/2))/(a^2*d*cos(d*x + c)^2 - a^2*d), 1/4*(4*(b*cos(d*x + c)^2 - b)*sqrt(-b/(a + b))*arctan(sqrt(-b/(a + b))*cos(d*x + c)) + 2*a*cos(d*x + c) - ((a - 2*b)*cos(d*x + c)^2 - a + 2*b)*log(1/2*cos(d*x + c) + 1/2) + ((a - 2*b)*cos(d*x + c)^2 - a + 2*b)*log(-1/2*cos(d*x + c) + 1/2))/(a^2*d*cos(d*x + c)^2 - a^2*d)]","A",0
84,1,612,0,0.706170," ","integrate(csc(d*x+c)^5/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \cos\left(d x + c\right)^{3} + 8 \, {\left(b^{2} \cos\left(d x + c\right)^{4} - 2 \, b^{2} \cos\left(d x + c\right)^{2} + b^{2}\right)} \sqrt{\frac{b}{a + b}} \log\left(\frac{b \cos\left(d x + c\right)^{2} + 2 \, {\left(a + b\right)} \sqrt{\frac{b}{a + b}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) - 2 \, {\left(5 \, a^{2} - 4 \, a b\right)} \cos\left(d x + c\right) - {\left({\left(3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left({\left(3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{16 \, {\left(a^{3} d \cos\left(d x + c\right)^{4} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} + a^{3} d\right)}}, \frac{2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \cos\left(d x + c\right)^{3} - 16 \, {\left(b^{2} \cos\left(d x + c\right)^{4} - 2 \, b^{2} \cos\left(d x + c\right)^{2} + b^{2}\right)} \sqrt{-\frac{b}{a + b}} \arctan\left(\sqrt{-\frac{b}{a + b}} \cos\left(d x + c\right)\right) - 2 \, {\left(5 \, a^{2} - 4 \, a b\right)} \cos\left(d x + c\right) - {\left({\left(3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left({\left(3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{16 \, {\left(a^{3} d \cos\left(d x + c\right)^{4} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} + a^{3} d\right)}}\right]"," ",0,"[1/16*(2*(3*a^2 - 4*a*b)*cos(d*x + c)^3 + 8*(b^2*cos(d*x + c)^4 - 2*b^2*cos(d*x + c)^2 + b^2)*sqrt(b/(a + b))*log((b*cos(d*x + c)^2 + 2*(a + b)*sqrt(b/(a + b))*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) - 2*(5*a^2 - 4*a*b)*cos(d*x + c) - ((3*a^2 - 4*a*b + 8*b^2)*cos(d*x + c)^4 - 2*(3*a^2 - 4*a*b + 8*b^2)*cos(d*x + c)^2 + 3*a^2 - 4*a*b + 8*b^2)*log(1/2*cos(d*x + c) + 1/2) + ((3*a^2 - 4*a*b + 8*b^2)*cos(d*x + c)^4 - 2*(3*a^2 - 4*a*b + 8*b^2)*cos(d*x + c)^2 + 3*a^2 - 4*a*b + 8*b^2)*log(-1/2*cos(d*x + c) + 1/2))/(a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d), 1/16*(2*(3*a^2 - 4*a*b)*cos(d*x + c)^3 - 16*(b^2*cos(d*x + c)^4 - 2*b^2*cos(d*x + c)^2 + b^2)*sqrt(-b/(a + b))*arctan(sqrt(-b/(a + b))*cos(d*x + c)) - 2*(5*a^2 - 4*a*b)*cos(d*x + c) - ((3*a^2 - 4*a*b + 8*b^2)*cos(d*x + c)^4 - 2*(3*a^2 - 4*a*b + 8*b^2)*cos(d*x + c)^2 + 3*a^2 - 4*a*b + 8*b^2)*log(1/2*cos(d*x + c) + 1/2) + ((3*a^2 - 4*a*b + 8*b^2)*cos(d*x + c)^4 - 2*(3*a^2 - 4*a*b + 8*b^2)*cos(d*x + c)^2 + 3*a^2 - 4*a*b + 8*b^2)*log(-1/2*cos(d*x + c) + 1/2))/(a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)]","B",0
85,1,453,0,0.834531," ","integrate(sin(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{12 \, a^{3} \sqrt{-\frac{a}{a + b}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 3 \, {\left(16 \, a^{3} - 8 \, a^{2} b + 6 \, a b^{2} - 5 \, b^{3}\right)} d x - {\left(8 \, b^{3} \cos\left(d x + c\right)^{5} + 2 \, {\left(6 \, a b^{2} - 13 \, b^{3}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(8 \, a^{2} b - 10 \, a b^{2} + 11 \, b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{48 \, b^{4} d}, -\frac{24 \, a^{3} \sqrt{\frac{a}{a + b}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a}{a + b}}}{2 \, a \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) + 3 \, {\left(16 \, a^{3} - 8 \, a^{2} b + 6 \, a b^{2} - 5 \, b^{3}\right)} d x + {\left(8 \, b^{3} \cos\left(d x + c\right)^{5} + 2 \, {\left(6 \, a b^{2} - 13 \, b^{3}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(8 \, a^{2} b - 10 \, a b^{2} + 11 \, b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{48 \, b^{4} d}\right]"," ",0,"[1/48*(12*a^3*sqrt(-a/(a + b))*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 3*(16*a^3 - 8*a^2*b + 6*a*b^2 - 5*b^3)*d*x - (8*b^3*cos(d*x + c)^5 + 2*(6*a*b^2 - 13*b^3)*cos(d*x + c)^3 + 3*(8*a^2*b - 10*a*b^2 + 11*b^3)*cos(d*x + c))*sin(d*x + c))/(b^4*d), -1/48*(24*a^3*sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c))) + 3*(16*a^3 - 8*a^2*b + 6*a*b^2 - 5*b^3)*d*x + (8*b^3*cos(d*x + c)^5 + 2*(6*a*b^2 - 13*b^3)*cos(d*x + c)^3 + 3*(8*a^2*b - 10*a*b^2 + 11*b^3)*cos(d*x + c))*sin(d*x + c))/(b^4*d)]","A",0
86,1,372,0,0.932267," ","integrate(sin(d*x+c)^6/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{2 \, a^{2} \sqrt{-\frac{a}{a + b}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) + {\left(8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right)} d x + {\left(2 \, b^{2} \cos\left(d x + c\right)^{3} + {\left(4 \, a b - 5 \, b^{2}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, b^{3} d}, \frac{4 \, a^{2} \sqrt{\frac{a}{a + b}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a}{a + b}}}{2 \, a \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) + {\left(8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right)} d x + {\left(2 \, b^{2} \cos\left(d x + c\right)^{3} + {\left(4 \, a b - 5 \, b^{2}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, b^{3} d}\right]"," ",0,"[1/8*(2*a^2*sqrt(-a/(a + b))*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) + (8*a^2 - 4*a*b + 3*b^2)*d*x + (2*b^2*cos(d*x + c)^3 + (4*a*b - 5*b^2)*cos(d*x + c))*sin(d*x + c))/(b^3*d), 1/8*(4*a^2*sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c))) + (8*a^2 - 4*a*b + 3*b^2)*d*x + (2*b^2*cos(d*x + c)^3 + (4*a*b - 5*b^2)*cos(d*x + c))*sin(d*x + c))/(b^3*d)]","A",0
87,1,305,0,0.787229," ","integrate(sin(d*x+c)^4/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(2 \, a - b\right)} d x + 2 \, b \cos\left(d x + c\right) \sin\left(d x + c\right) - a \sqrt{-\frac{a}{a + b}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right)}{4 \, b^{2} d}, -\frac{{\left(2 \, a - b\right)} d x + b \cos\left(d x + c\right) \sin\left(d x + c\right) + a \sqrt{\frac{a}{a + b}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a}{a + b}}}{2 \, a \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{2 \, b^{2} d}\right]"," ",0,"[-1/4*(2*(2*a - b)*d*x + 2*b*cos(d*x + c)*sin(d*x + c) - a*sqrt(-a/(a + b))*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)))/(b^2*d), -1/2*((2*a - b)*d*x + b*cos(d*x + c)*sin(d*x + c) + a*sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c))))/(b^2*d)]","A",0
88,1,260,0,0.747432," ","integrate(sin(d*x+c)^2/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{4 \, d x + \sqrt{-\frac{a}{a + b}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right)}{4 \, b d}, \frac{2 \, d x + \sqrt{\frac{a}{a + b}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a}{a + b}}}{2 \, a \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{2 \, b d}\right]"," ",0,"[1/4*(4*d*x + sqrt(-a/(a + b))*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)))/(b*d), 1/2*(2*d*x + sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c))))/(b*d)]","A",0
89,1,236,0,0.780979," ","integrate(1/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right)}{4 \, {\left(a^{2} + a b\right)} d}, -\frac{\arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{2 \, \sqrt{a^{2} + a b} d}\right]"," ",0,"[-1/4*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))/((a^2 + a*b)*d), -1/2*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c)))/(sqrt(a^2 + a*b)*d)]","B",0
90,1,313,0,0.712426," ","integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} - a b} b \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \sin\left(d x + c\right) + 4 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)}{4 \, {\left(a^{3} + a^{2} b\right)} d \sin\left(d x + c\right)}, \frac{\sqrt{a^{2} + a b} b \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) - 2 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)}{2 \, {\left(a^{3} + a^{2} b\right)} d \sin\left(d x + c\right)}\right]"," ",0,"[-1/4*(sqrt(-a^2 - a*b)*b*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c) + 4*(a^2 + a*b)*cos(d*x + c))/((a^3 + a^2*b)*d*sin(d*x + c)), 1/2*(sqrt(a^2 + a*b)*b*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) - 2*(a^2 + a*b)*cos(d*x + c))/((a^3 + a^2*b)*d*sin(d*x + c))]","B",0
91,1,451,0,0.647669," ","integrate(csc(d*x+c)^4/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(2 \, a^{3} - a^{2} b - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(b^{2} \cos\left(d x + c\right)^{2} - b^{2}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \sin\left(d x + c\right) - 12 \, {\left(a^{3} - a b^{2}\right)} \cos\left(d x + c\right)}{12 \, {\left({\left(a^{4} + a^{3} b\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} + a^{3} b\right)} d\right)} \sin\left(d x + c\right)}, -\frac{2 \, {\left(2 \, a^{3} - a^{2} b - 3 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(b^{2} \cos\left(d x + c\right)^{2} - b^{2}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) - 6 \, {\left(a^{3} - a b^{2}\right)} \cos\left(d x + c\right)}{6 \, {\left({\left(a^{4} + a^{3} b\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} + a^{3} b\right)} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/12*(4*(2*a^3 - a^2*b - 3*a*b^2)*cos(d*x + c)^3 + 3*(b^2*cos(d*x + c)^2 - b^2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c) - 12*(a^3 - a*b^2)*cos(d*x + c))/(((a^4 + a^3*b)*d*cos(d*x + c)^2 - (a^4 + a^3*b)*d)*sin(d*x + c)), -1/6*(2*(2*a^3 - a^2*b - 3*a*b^2)*cos(d*x + c)^3 + 3*(b^2*cos(d*x + c)^2 - b^2)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) - 6*(a^3 - a*b^2)*cos(d*x + c))/(((a^4 + a^3*b)*d*cos(d*x + c)^2 - (a^4 + a^3*b)*d)*sin(d*x + c))]","B",0
92,1,595,0,0.951212," ","integrate(csc(d*x+c)^6/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(8 \, a^{4} - 2 \, a^{3} b + 5 \, a^{2} b^{2} + 15 \, a b^{3}\right)} \cos\left(d x + c\right)^{5} - 20 \, {\left(4 \, a^{4} - a^{3} b + a^{2} b^{2} + 6 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} + 15 \, {\left(b^{3} \cos\left(d x + c\right)^{4} - 2 \, b^{3} \cos\left(d x + c\right)^{2} + b^{3}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \sin\left(d x + c\right) + 60 \, {\left(a^{4} + a b^{3}\right)} \cos\left(d x + c\right)}{60 \, {\left({\left(a^{5} + a^{4} b\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{5} + a^{4} b\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} + a^{4} b\right)} d\right)} \sin\left(d x + c\right)}, -\frac{2 \, {\left(8 \, a^{4} - 2 \, a^{3} b + 5 \, a^{2} b^{2} + 15 \, a b^{3}\right)} \cos\left(d x + c\right)^{5} - 10 \, {\left(4 \, a^{4} - a^{3} b + a^{2} b^{2} + 6 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} - 15 \, {\left(b^{3} \cos\left(d x + c\right)^{4} - 2 \, b^{3} \cos\left(d x + c\right)^{2} + b^{3}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 30 \, {\left(a^{4} + a b^{3}\right)} \cos\left(d x + c\right)}{30 \, {\left({\left(a^{5} + a^{4} b\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{5} + a^{4} b\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} + a^{4} b\right)} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/60*(4*(8*a^4 - 2*a^3*b + 5*a^2*b^2 + 15*a*b^3)*cos(d*x + c)^5 - 20*(4*a^4 - a^3*b + a^2*b^2 + 6*a*b^3)*cos(d*x + c)^3 + 15*(b^3*cos(d*x + c)^4 - 2*b^3*cos(d*x + c)^2 + b^3)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c) + 60*(a^4 + a*b^3)*cos(d*x + c))/(((a^5 + a^4*b)*d*cos(d*x + c)^4 - 2*(a^5 + a^4*b)*d*cos(d*x + c)^2 + (a^5 + a^4*b)*d)*sin(d*x + c)), -1/30*(2*(8*a^4 - 2*a^3*b + 5*a^2*b^2 + 15*a*b^3)*cos(d*x + c)^5 - 10*(4*a^4 - a^3*b + a^2*b^2 + 6*a*b^3)*cos(d*x + c)^3 - 15*(b^3*cos(d*x + c)^4 - 2*b^3*cos(d*x + c)^2 + b^3)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) + 30*(a^4 + a*b^3)*cos(d*x + c))/(((a^5 + a^4*b)*d*cos(d*x + c)^4 - 2*(a^5 + a^4*b)*d*cos(d*x + c)^2 + (a^5 + a^4*b)*d)*sin(d*x + c))]","B",0
93,1,789,0,0.813185," ","integrate(csc(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(48 \, a^{5} - 8 \, a^{4} b + 14 \, a^{3} b^{2} - 35 \, a^{2} b^{3} - 105 \, a b^{4}\right)} \cos\left(d x + c\right)^{7} - 28 \, {\left(24 \, a^{5} - 4 \, a^{4} b + 7 \, a^{3} b^{2} - 10 \, a^{2} b^{3} - 45 \, a b^{4}\right)} \cos\left(d x + c\right)^{5} + 140 \, {\left(6 \, a^{5} - a^{4} b + a^{3} b^{2} - a^{2} b^{3} - 9 \, a b^{4}\right)} \cos\left(d x + c\right)^{3} + 105 \, {\left(b^{4} \cos\left(d x + c\right)^{6} - 3 \, b^{4} \cos\left(d x + c\right)^{4} + 3 \, b^{4} \cos\left(d x + c\right)^{2} - b^{4}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \sin\left(d x + c\right) - 420 \, {\left(a^{5} - a b^{4}\right)} \cos\left(d x + c\right)}{420 \, {\left({\left(a^{6} + a^{5} b\right)} d \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{6} + a^{5} b\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{6} + a^{5} b\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{6} + a^{5} b\right)} d\right)} \sin\left(d x + c\right)}, -\frac{2 \, {\left(48 \, a^{5} - 8 \, a^{4} b + 14 \, a^{3} b^{2} - 35 \, a^{2} b^{3} - 105 \, a b^{4}\right)} \cos\left(d x + c\right)^{7} - 14 \, {\left(24 \, a^{5} - 4 \, a^{4} b + 7 \, a^{3} b^{2} - 10 \, a^{2} b^{3} - 45 \, a b^{4}\right)} \cos\left(d x + c\right)^{5} + 70 \, {\left(6 \, a^{5} - a^{4} b + a^{3} b^{2} - a^{2} b^{3} - 9 \, a b^{4}\right)} \cos\left(d x + c\right)^{3} + 105 \, {\left(b^{4} \cos\left(d x + c\right)^{6} - 3 \, b^{4} \cos\left(d x + c\right)^{4} + 3 \, b^{4} \cos\left(d x + c\right)^{2} - b^{4}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) - 210 \, {\left(a^{5} - a b^{4}\right)} \cos\left(d x + c\right)}{210 \, {\left({\left(a^{6} + a^{5} b\right)} d \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{6} + a^{5} b\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{6} + a^{5} b\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{6} + a^{5} b\right)} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/420*(4*(48*a^5 - 8*a^4*b + 14*a^3*b^2 - 35*a^2*b^3 - 105*a*b^4)*cos(d*x + c)^7 - 28*(24*a^5 - 4*a^4*b + 7*a^3*b^2 - 10*a^2*b^3 - 45*a*b^4)*cos(d*x + c)^5 + 140*(6*a^5 - a^4*b + a^3*b^2 - a^2*b^3 - 9*a*b^4)*cos(d*x + c)^3 + 105*(b^4*cos(d*x + c)^6 - 3*b^4*cos(d*x + c)^4 + 3*b^4*cos(d*x + c)^2 - b^4)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c) - 420*(a^5 - a*b^4)*cos(d*x + c))/(((a^6 + a^5*b)*d*cos(d*x + c)^6 - 3*(a^6 + a^5*b)*d*cos(d*x + c)^4 + 3*(a^6 + a^5*b)*d*cos(d*x + c)^2 - (a^6 + a^5*b)*d)*sin(d*x + c)), -1/210*(2*(48*a^5 - 8*a^4*b + 14*a^3*b^2 - 35*a^2*b^3 - 105*a*b^4)*cos(d*x + c)^7 - 14*(24*a^5 - 4*a^4*b + 7*a^3*b^2 - 10*a^2*b^3 - 45*a*b^4)*cos(d*x + c)^5 + 70*(6*a^5 - a^4*b + a^3*b^2 - a^2*b^3 - 9*a*b^4)*cos(d*x + c)^3 + 105*(b^4*cos(d*x + c)^6 - 3*b^4*cos(d*x + c)^4 + 3*b^4*cos(d*x + c)^2 - b^4)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) - 210*(a^5 - a*b^4)*cos(d*x + c))/(((a^6 + a^5*b)*d*cos(d*x + c)^6 - 3*(a^6 + a^5*b)*d*cos(d*x + c)^4 + 3*(a^6 + a^5*b)*d*cos(d*x + c)^2 - (a^6 + a^5*b)*d)*sin(d*x + c))]","B",0
94,1,529,0,0.776614," ","integrate(sin(d*x+c)^7/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right)} \cos\left(d x + c\right)^{5} + 4 \, {\left(5 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 3 \, a b^{4} - 4 \, b^{5}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(5 \, a^{4} + 11 \, a^{3} b + 6 \, a^{2} b^{2} - {\left(5 \, a^{3} b + 6 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a b + b^{2}} \log\left(-\frac{b \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a b + b^{2}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) - 6 \, {\left(5 \, a^{4} b + 11 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 2 \, a b^{4} - 2 \, b^{5}\right)} \cos\left(d x + c\right)}{12 \, {\left({\left(a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} d\right)}}, \frac{2 \, {\left(a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right)} \cos\left(d x + c\right)^{5} + 2 \, {\left(5 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 3 \, a b^{4} - 4 \, b^{5}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(5 \, a^{4} + 11 \, a^{3} b + 6 \, a^{2} b^{2} - {\left(5 \, a^{3} b + 6 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a b - b^{2}} \arctan\left(\frac{\sqrt{-a b - b^{2}} \cos\left(d x + c\right)}{a + b}\right) - 3 \, {\left(5 \, a^{4} b + 11 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 2 \, a b^{4} - 2 \, b^{5}\right)} \cos\left(d x + c\right)}{6 \, {\left({\left(a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} d\right)}}\right]"," ",0,"[1/12*(4*(a^2*b^3 + 2*a*b^4 + b^5)*cos(d*x + c)^5 + 4*(5*a^3*b^2 + 6*a^2*b^3 - 3*a*b^4 - 4*b^5)*cos(d*x + c)^3 - 3*(5*a^4 + 11*a^3*b + 6*a^2*b^2 - (5*a^3*b + 6*a^2*b^2)*cos(d*x + c)^2)*sqrt(a*b + b^2)*log(-(b*cos(d*x + c)^2 - 2*sqrt(a*b + b^2)*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) - 6*(5*a^4*b + 11*a^3*b^2 + 6*a^2*b^3 - 2*a*b^4 - 2*b^5)*cos(d*x + c))/((a^2*b^5 + 2*a*b^6 + b^7)*d*cos(d*x + c)^2 - (a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + b^7)*d), 1/6*(2*(a^2*b^3 + 2*a*b^4 + b^5)*cos(d*x + c)^5 + 2*(5*a^3*b^2 + 6*a^2*b^3 - 3*a*b^4 - 4*b^5)*cos(d*x + c)^3 - 3*(5*a^4 + 11*a^3*b + 6*a^2*b^2 - (5*a^3*b + 6*a^2*b^2)*cos(d*x + c)^2)*sqrt(-a*b - b^2)*arctan(sqrt(-a*b - b^2)*cos(d*x + c)/(a + b)) - 3*(5*a^4*b + 11*a^3*b^2 + 6*a^2*b^3 - 2*a*b^4 - 2*b^5)*cos(d*x + c))/((a^2*b^5 + 2*a*b^6 + b^7)*d*cos(d*x + c)^2 - (a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + b^7)*d)]","B",0
95,1,427,0,0.646294," ","integrate(sin(d*x+c)^5/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{3} + {\left(3 \, a^{3} + 7 \, a^{2} b + 4 \, a b^{2} - {\left(3 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a b + b^{2}} \log\left(\frac{b \cos\left(d x + c\right)^{2} + 2 \, \sqrt{a b + b^{2}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) - 2 \, {\left(3 \, a^{3} b + 7 \, a^{2} b^{2} + 6 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(d x + c\right)}{4 \, {\left({\left(a^{2} b^{4} + 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6}\right)} d\right)}}, -\frac{2 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(3 \, a^{3} + 7 \, a^{2} b + 4 \, a b^{2} - {\left(3 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a b - b^{2}} \arctan\left(\frac{\sqrt{-a b - b^{2}} \cos\left(d x + c\right)}{a + b}\right) - {\left(3 \, a^{3} b + 7 \, a^{2} b^{2} + 6 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(d x + c\right)}{2 \, {\left({\left(a^{2} b^{4} + 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} b^{3} + 3 \, a^{2} b^{4} + 3 \, a b^{5} + b^{6}\right)} d\right)}}\right]"," ",0,"[-1/4*(4*(a^2*b^2 + 2*a*b^3 + b^4)*cos(d*x + c)^3 + (3*a^3 + 7*a^2*b + 4*a*b^2 - (3*a^2*b + 4*a*b^2)*cos(d*x + c)^2)*sqrt(a*b + b^2)*log((b*cos(d*x + c)^2 + 2*sqrt(a*b + b^2)*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) - 2*(3*a^3*b + 7*a^2*b^2 + 6*a*b^3 + 2*b^4)*cos(d*x + c))/((a^2*b^4 + 2*a*b^5 + b^6)*d*cos(d*x + c)^2 - (a^3*b^3 + 3*a^2*b^4 + 3*a*b^5 + b^6)*d), -1/2*(2*(a^2*b^2 + 2*a*b^3 + b^4)*cos(d*x + c)^3 - (3*a^3 + 7*a^2*b + 4*a*b^2 - (3*a^2*b + 4*a*b^2)*cos(d*x + c)^2)*sqrt(-a*b - b^2)*arctan(sqrt(-a*b - b^2)*cos(d*x + c)/(a + b)) - (3*a^3*b + 7*a^2*b^2 + 6*a*b^3 + 2*b^4)*cos(d*x + c))/((a^2*b^4 + 2*a*b^5 + b^6)*d*cos(d*x + c)^2 - (a^3*b^3 + 3*a^2*b^4 + 3*a*b^5 + b^6)*d)]","B",0
96,1,327,0,0.945806," ","integrate(sin(d*x+c)^3/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left({\left(a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} - 3 \, a b - 2 \, b^{2}\right)} \sqrt{a b + b^{2}} \log\left(-\frac{b \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a b + b^{2}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) - 2 \, {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)}{4 \, {\left({\left(a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} d\right)}}, \frac{{\left({\left(a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} - 3 \, a b - 2 \, b^{2}\right)} \sqrt{-a b - b^{2}} \arctan\left(\frac{\sqrt{-a b - b^{2}} \cos\left(d x + c\right)}{a + b}\right) - {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)}{2 \, {\left({\left(a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} d\right)}}\right]"," ",0,"[1/4*(((a*b + 2*b^2)*cos(d*x + c)^2 - a^2 - 3*a*b - 2*b^2)*sqrt(a*b + b^2)*log(-(b*cos(d*x + c)^2 - 2*sqrt(a*b + b^2)*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) - 2*(a^2*b + a*b^2)*cos(d*x + c))/((a^2*b^3 + 2*a*b^4 + b^5)*d*cos(d*x + c)^2 - (a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*d), 1/2*(((a*b + 2*b^2)*cos(d*x + c)^2 - a^2 - 3*a*b - 2*b^2)*sqrt(-a*b - b^2)*arctan(sqrt(-a*b - b^2)*cos(d*x + c)/(a + b)) - (a^2*b + a*b^2)*cos(d*x + c))/((a^2*b^3 + 2*a*b^4 + b^5)*d*cos(d*x + c)^2 - (a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*d)]","B",0
97,1,282,0,0.807791," ","integrate(sin(d*x+c)/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left(b \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{a b + b^{2}} \log\left(-\frac{b \cos\left(d x + c\right)^{2} - 2 \, \sqrt{a b + b^{2}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) + 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)}{4 \, {\left({\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} d\right)}}, \frac{{\left(b \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{-a b - b^{2}} \arctan\left(\frac{\sqrt{-a b - b^{2}} \cos\left(d x + c\right)}{a + b}\right) + {\left(a b + b^{2}\right)} \cos\left(d x + c\right)}{2 \, {\left({\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} d\right)}}\right]"," ",0,"[1/4*((b*cos(d*x + c)^2 - a - b)*sqrt(a*b + b^2)*log(-(b*cos(d*x + c)^2 - 2*sqrt(a*b + b^2)*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) + 2*(a*b + b^2)*cos(d*x + c))/((a^2*b^2 + 2*a*b^3 + b^4)*d*cos(d*x + c)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d), 1/2*((b*cos(d*x + c)^2 - a - b)*sqrt(-a*b - b^2)*arctan(sqrt(-a*b - b^2)*cos(d*x + c)/(a + b)) + (a*b + b^2)*cos(d*x + c))/((a^2*b^2 + 2*a*b^3 + b^4)*d*cos(d*x + c)^2 - (a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d)]","A",0
98,1,455,0,0.737450," ","integrate(csc(d*x+c)/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[-\frac{2 \, a b \cos\left(d x + c\right) - {\left({\left(3 \, a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 3 \, a^{2} - 5 \, a b - 2 \, b^{2}\right)} \sqrt{\frac{b}{a + b}} \log\left(\frac{b \cos\left(d x + c\right)^{2} + 2 \, {\left(a + b\right)} \sqrt{\frac{b}{a + b}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) + 2 \, {\left({\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 2 \, {\left({\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{4 \, {\left({\left(a^{3} b + a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)}}, -\frac{a b \cos\left(d x + c\right) + {\left({\left(3 \, a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 3 \, a^{2} - 5 \, a b - 2 \, b^{2}\right)} \sqrt{-\frac{b}{a + b}} \arctan\left(\sqrt{-\frac{b}{a + b}} \cos\left(d x + c\right)\right) + {\left({\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - {\left({\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{2 \, {\left({\left(a^{3} b + a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)}}\right]"," ",0,"[-1/4*(2*a*b*cos(d*x + c) - ((3*a*b + 2*b^2)*cos(d*x + c)^2 - 3*a^2 - 5*a*b - 2*b^2)*sqrt(b/(a + b))*log((b*cos(d*x + c)^2 + 2*(a + b)*sqrt(b/(a + b))*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) + 2*((a*b + b^2)*cos(d*x + c)^2 - a^2 - 2*a*b - b^2)*log(1/2*cos(d*x + c) + 1/2) - 2*((a*b + b^2)*cos(d*x + c)^2 - a^2 - 2*a*b - b^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^3*b + a^2*b^2)*d*cos(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d), -1/2*(a*b*cos(d*x + c) + ((3*a*b + 2*b^2)*cos(d*x + c)^2 - 3*a^2 - 5*a*b - 2*b^2)*sqrt(-b/(a + b))*arctan(sqrt(-b/(a + b))*cos(d*x + c)) + ((a*b + b^2)*cos(d*x + c)^2 - a^2 - 2*a*b - b^2)*log(1/2*cos(d*x + c) + 1/2) - ((a*b + b^2)*cos(d*x + c)^2 - a^2 - 2*a*b - b^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^3*b + a^2*b^2)*d*cos(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d)]","B",0
99,1,838,0,0.960606," ","integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{2} b + 2 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} + {\left({\left(5 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(d x + c\right)^{4} + 5 \, a^{2} b + 9 \, a b^{2} + 4 \, b^{3} - {\left(5 \, a^{2} b + 14 \, a b^{2} + 8 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \log\left(-\frac{b \cos\left(d x + c\right)^{2} - 2 \, {\left(a + b\right)} \sqrt{\frac{b}{a + b}} \cos\left(d x + c\right) + a + b}{b \cos\left(d x + c\right)^{2} - a - b}\right) - 2 \, {\left(a^{3} + 2 \, a^{2} b + 2 \, a b^{2}\right)} \cos\left(d x + c\right) - {\left({\left(a^{2} b - 3 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{4} + a^{3} - 2 \, a^{2} b - 7 \, a b^{2} - 4 \, b^{3} - {\left(a^{3} - a^{2} b - 10 \, a b^{2} - 8 \, 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({\left(a^{2} b - 3 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{4} + a^{3} - 2 \, a^{2} b - 7 \, a b^{2} - 4 \, b^{3} - {\left(a^{3} - a^{2} b - 10 \, a b^{2} - 8 \, 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^{4} b + a^{3} b^{2}\right)} d \cos\left(d x + c\right)^{4} - {\left(a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} d\right)}}, \frac{2 \, {\left(a^{2} b + 2 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} + 2 \, {\left({\left(5 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(d x + c\right)^{4} + 5 \, a^{2} b + 9 \, a b^{2} + 4 \, b^{3} - {\left(5 \, a^{2} b + 14 \, a b^{2} + 8 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \arctan\left(\sqrt{-\frac{b}{a + b}} \cos\left(d x + c\right)\right) - 2 \, {\left(a^{3} + 2 \, a^{2} b + 2 \, a b^{2}\right)} \cos\left(d x + c\right) - {\left({\left(a^{2} b - 3 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{4} + a^{3} - 2 \, a^{2} b - 7 \, a b^{2} - 4 \, b^{3} - {\left(a^{3} - a^{2} b - 10 \, a b^{2} - 8 \, 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({\left(a^{2} b - 3 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{4} + a^{3} - 2 \, a^{2} b - 7 \, a b^{2} - 4 \, b^{3} - {\left(a^{3} - a^{2} b - 10 \, a b^{2} - 8 \, 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^{4} b + a^{3} b^{2}\right)} d \cos\left(d x + c\right)^{4} - {\left(a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} d\right)}}\right]"," ",0,"[1/4*(2*(a^2*b + 2*a*b^2)*cos(d*x + c)^3 + ((5*a*b^2 + 4*b^3)*cos(d*x + c)^4 + 5*a^2*b + 9*a*b^2 + 4*b^3 - (5*a^2*b + 14*a*b^2 + 8*b^3)*cos(d*x + c)^2)*sqrt(b/(a + b))*log(-(b*cos(d*x + c)^2 - 2*(a + b)*sqrt(b/(a + b))*cos(d*x + c) + a + b)/(b*cos(d*x + c)^2 - a - b)) - 2*(a^3 + 2*a^2*b + 2*a*b^2)*cos(d*x + c) - ((a^2*b - 3*a*b^2 - 4*b^3)*cos(d*x + c)^4 + a^3 - 2*a^2*b - 7*a*b^2 - 4*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + ((a^2*b - 3*a*b^2 - 4*b^3)*cos(d*x + c)^4 + a^3 - 2*a^2*b - 7*a*b^2 - 4*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^4*b + a^3*b^2)*d*cos(d*x + c)^4 - (a^5 + 3*a^4*b + 2*a^3*b^2)*d*cos(d*x + c)^2 + (a^5 + 2*a^4*b + a^3*b^2)*d), 1/4*(2*(a^2*b + 2*a*b^2)*cos(d*x + c)^3 + 2*((5*a*b^2 + 4*b^3)*cos(d*x + c)^4 + 5*a^2*b + 9*a*b^2 + 4*b^3 - (5*a^2*b + 14*a*b^2 + 8*b^3)*cos(d*x + c)^2)*sqrt(-b/(a + b))*arctan(sqrt(-b/(a + b))*cos(d*x + c)) - 2*(a^3 + 2*a^2*b + 2*a*b^2)*cos(d*x + c) - ((a^2*b - 3*a*b^2 - 4*b^3)*cos(d*x + c)^4 + a^3 - 2*a^2*b - 7*a*b^2 - 4*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) + ((a^2*b - 3*a*b^2 - 4*b^3)*cos(d*x + c)^4 + a^3 - 2*a^2*b - 7*a*b^2 - 4*b^3 - (a^3 - a^2*b - 10*a*b^2 - 8*b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^4*b + a^3*b^2)*d*cos(d*x + c)^4 - (a^5 + 3*a^4*b + 2*a^3*b^2)*d*cos(d*x + c)^2 + (a^5 + 2*a^4*b + a^3*b^2)*d)]","B",0
100,1,623,0,0.842135," ","integrate(sin(d*x+c)^6/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(4 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d x \cos\left(d x + c\right)^{2} - 4 \, {\left(4 \, a^{3} + 7 \, a^{2} b + 2 \, a b^{2} - b^{3}\right)} d x + {\left(4 \, a^{3} + 9 \, a^{2} b + 5 \, a b^{2} - {\left(4 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-\frac{a}{a + b}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) + 4 \, {\left({\left(a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{3} - {\left(2 \, a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, {\left({\left(a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right)} d\right)}}, -\frac{2 \, {\left(4 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d x \cos\left(d x + c\right)^{2} - 2 \, {\left(4 \, a^{3} + 7 \, a^{2} b + 2 \, a b^{2} - b^{3}\right)} d x - {\left(4 \, a^{3} + 9 \, a^{2} b + 5 \, a b^{2} - {\left(4 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a}{a + b}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a}{a + b}}}{2 \, a \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) + 2 \, {\left({\left(a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{3} - {\left(2 \, a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{4 \, {\left({\left(a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right)} d\right)}}\right]"," ",0,"[-1/8*(4*(4*a^2*b + 3*a*b^2 - b^3)*d*x*cos(d*x + c)^2 - 4*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*d*x + (4*a^3 + 9*a^2*b + 5*a*b^2 - (4*a^2*b + 5*a*b^2)*cos(d*x + c)^2)*sqrt(-a/(a + b))*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) + 4*((a*b^2 + b^3)*cos(d*x + c)^3 - (2*a^2*b + 2*a*b^2 + b^3)*cos(d*x + c))*sin(d*x + c))/((a*b^4 + b^5)*d*cos(d*x + c)^2 - (a^2*b^3 + 2*a*b^4 + b^5)*d), -1/4*(2*(4*a^2*b + 3*a*b^2 - b^3)*d*x*cos(d*x + c)^2 - 2*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*d*x - (4*a^3 + 9*a^2*b + 5*a*b^2 - (4*a^2*b + 5*a*b^2)*cos(d*x + c)^2)*sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c))) + 2*((a*b^2 + b^3)*cos(d*x + c)^3 - (2*a^2*b + 2*a*b^2 + b^3)*cos(d*x + c))*sin(d*x + c))/((a*b^4 + b^5)*d*cos(d*x + c)^2 - (a^2*b^3 + 2*a*b^4 + b^5)*d)]","A",0
101,1,492,0,0.982371," ","integrate(sin(d*x+c)^4/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a b + b^{2}\right)} d x \cos\left(d x + c\right)^{2} - 4 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right) - 8 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} d x + {\left({\left(2 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, a^{2} - 5 \, a b - 3 \, b^{2}\right)} \sqrt{-\frac{a}{a + b}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right)}{8 \, {\left({\left(a b^{3} + b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} d\right)}}, \frac{4 \, {\left(a b + b^{2}\right)} d x \cos\left(d x + c\right)^{2} - 2 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right) - 4 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} d x + {\left({\left(2 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, a^{2} - 5 \, a b - 3 \, b^{2}\right)} \sqrt{\frac{a}{a + b}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a}{a + b}}}{2 \, a \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{4 \, {\left({\left(a b^{3} + b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} d\right)}}\right]"," ",0,"[1/8*(8*(a*b + b^2)*d*x*cos(d*x + c)^2 - 4*a*b*cos(d*x + c)*sin(d*x + c) - 8*(a^2 + 2*a*b + b^2)*d*x + ((2*a*b + 3*b^2)*cos(d*x + c)^2 - 2*a^2 - 5*a*b - 3*b^2)*sqrt(-a/(a + b))*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)))/((a*b^3 + b^4)*d*cos(d*x + c)^2 - (a^2*b^2 + 2*a*b^3 + b^4)*d), 1/4*(4*(a*b + b^2)*d*x*cos(d*x + c)^2 - 2*a*b*cos(d*x + c)*sin(d*x + c) - 4*(a^2 + 2*a*b + b^2)*d*x + ((2*a*b + 3*b^2)*cos(d*x + c)^2 - 2*a^2 - 5*a*b - 3*b^2)*sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c))))/((a*b^3 + b^4)*d*cos(d*x + c)^2 - (a^2*b^2 + 2*a*b^3 + b^4)*d)]","B",0
102,1,419,0,0.700471," ","integrate(sin(d*x+c)^2/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(b \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right)}{8 \, {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} d\right)}}, \frac{2 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(b \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{4 \, {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} d\right)}}\right]"," ",0,"[1/8*(4*(a^2 + a*b)*cos(d*x + c)*sin(d*x + c) - (b*cos(d*x + c)^2 - a - b)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)))/((a^3*b + 2*a^2*b^2 + a*b^3)*d*cos(d*x + c)^2 - (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d), 1/4*(2*(a^2 + a*b)*cos(d*x + c)*sin(d*x + c) - (b*cos(d*x + c)^2 - a - b)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c))))/((a^3*b + 2*a^2*b^2 + a*b^3)*d*cos(d*x + c)^2 - (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d)]","B",0
103,1,463,0,0.764409," ","integrate(1/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left({\left(2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, a^{2} - 3 \, a b - b^{2}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right)}{8 \, {\left({\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} d\right)}}, -\frac{2 \, {\left(a^{2} b + a b^{2}\right)} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left({\left(2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, a^{2} - 3 \, a b - b^{2}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{4 \, {\left({\left(a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right)} d\right)}}\right]"," ",0,"[-1/8*(4*(a^2*b + a*b^2)*cos(d*x + c)*sin(d*x + c) + ((2*a*b + b^2)*cos(d*x + c)^2 - 2*a^2 - 3*a*b - b^2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)))/((a^4*b + 2*a^3*b^2 + a^2*b^3)*d*cos(d*x + c)^2 - (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d), -1/4*(2*(a^2*b + a*b^2)*cos(d*x + c)*sin(d*x + c) + ((2*a*b + b^2)*cos(d*x + c)^2 - 2*a^2 - 3*a*b - b^2)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c))))/((a^4*b + 2*a^3*b^2 + a^2*b^3)*d*cos(d*x + c)^2 - (a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*d)]","B",0
104,1,588,0,0.737971," ","integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(2 \, a^{3} b + 5 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{2} b + 7 \, a b^{2} + 3 \, b^{3} - {\left(4 \, a b^{2} + 3 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \sin\left(d x + c\right) - 4 \, {\left(2 \, a^{4} + 6 \, a^{3} b + 7 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(d x + c\right)}{8 \, {\left({\left(a^{5} b + 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} d\right)} \sin\left(d x + c\right)}, -\frac{2 \, {\left(2 \, a^{3} b + 5 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} + {\left(4 \, a^{2} b + 7 \, a b^{2} + 3 \, b^{3} - {\left(4 \, a b^{2} + 3 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) - 2 \, {\left(2 \, a^{4} + 6 \, a^{3} b + 7 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(d x + c\right)}{4 \, {\left({\left(a^{5} b + 2 \, a^{4} b^{2} + a^{3} b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/8*(4*(2*a^3*b + 5*a^2*b^2 + 3*a*b^3)*cos(d*x + c)^3 - (4*a^2*b + 7*a*b^2 + 3*b^3 - (4*a*b^2 + 3*b^3)*cos(d*x + c)^2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c) - 4*(2*a^4 + 6*a^3*b + 7*a^2*b^2 + 3*a*b^3)*cos(d*x + c))/(((a^5*b + 2*a^4*b^2 + a^3*b^3)*d*cos(d*x + c)^2 - (a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d)*sin(d*x + c)), -1/4*(2*(2*a^3*b + 5*a^2*b^2 + 3*a*b^3)*cos(d*x + c)^3 + (4*a^2*b + 7*a*b^2 + 3*b^3 - (4*a*b^2 + 3*b^3)*cos(d*x + c)^2)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) - 2*(2*a^4 + 6*a^3*b + 7*a^2*b^2 + 3*a*b^3)*cos(d*x + c))/(((a^5*b + 2*a^4*b^2 + a^3*b^3)*d*cos(d*x + c)^2 - (a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d)*sin(d*x + c))]","B",0
105,1,843,0,0.925680," ","integrate(csc(d*x+c)^4/(a+b*sin(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(4 \, a^{4} b - 4 \, a^{3} b^{2} - 23 \, a^{2} b^{3} - 15 \, a b^{4}\right)} \cos\left(d x + c\right)^{5} - 8 \, {\left(2 \, a^{5} + 3 \, a^{4} b - 12 \, a^{3} b^{2} - 28 \, a^{2} b^{3} - 15 \, a b^{4}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left({\left(6 \, a b^{3} + 5 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 6 \, a^{2} b^{2} + 11 \, a b^{3} + 5 \, b^{4} - {\left(6 \, a^{2} b^{2} + 17 \, a b^{3} + 10 \, b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \sin\left(d x + c\right) + 12 \, {\left(2 \, a^{5} + 2 \, a^{4} b - 6 \, a^{3} b^{2} - 11 \, a^{2} b^{3} - 5 \, a b^{4}\right)} \cos\left(d x + c\right)}{24 \, {\left({\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} d \cos\left(d x + c\right)^{4} - {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} + 2 \, a^{4} b^{3}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} d\right)} \sin\left(d x + c\right)}, -\frac{2 \, {\left(4 \, a^{4} b - 4 \, a^{3} b^{2} - 23 \, a^{2} b^{3} - 15 \, a b^{4}\right)} \cos\left(d x + c\right)^{5} - 4 \, {\left(2 \, a^{5} + 3 \, a^{4} b - 12 \, a^{3} b^{2} - 28 \, a^{2} b^{3} - 15 \, a b^{4}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left({\left(6 \, a b^{3} + 5 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 6 \, a^{2} b^{2} + 11 \, a b^{3} + 5 \, b^{4} - {\left(6 \, a^{2} b^{2} + 17 \, a b^{3} + 10 \, b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 6 \, {\left(2 \, a^{5} + 2 \, a^{4} b - 6 \, a^{3} b^{2} - 11 \, a^{2} b^{3} - 5 \, a b^{4}\right)} \cos\left(d x + c\right)}{12 \, {\left({\left(a^{6} b + 2 \, a^{5} b^{2} + a^{4} b^{3}\right)} d \cos\left(d x + c\right)^{4} - {\left(a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} + 2 \, a^{4} b^{3}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/24*(4*(4*a^4*b - 4*a^3*b^2 - 23*a^2*b^3 - 15*a*b^4)*cos(d*x + c)^5 - 8*(2*a^5 + 3*a^4*b - 12*a^3*b^2 - 28*a^2*b^3 - 15*a*b^4)*cos(d*x + c)^3 + 3*((6*a*b^3 + 5*b^4)*cos(d*x + c)^4 + 6*a^2*b^2 + 11*a*b^3 + 5*b^4 - (6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cos(d*x + c)^2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c) + 12*(2*a^5 + 2*a^4*b - 6*a^3*b^2 - 11*a^2*b^3 - 5*a*b^4)*cos(d*x + c))/(((a^6*b + 2*a^5*b^2 + a^4*b^3)*d*cos(d*x + c)^4 - (a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cos(d*x + c)^2 + (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d)*sin(d*x + c)), -1/12*(2*(4*a^4*b - 4*a^3*b^2 - 23*a^2*b^3 - 15*a*b^4)*cos(d*x + c)^5 - 4*(2*a^5 + 3*a^4*b - 12*a^3*b^2 - 28*a^2*b^3 - 15*a*b^4)*cos(d*x + c)^3 + 3*((6*a*b^3 + 5*b^4)*cos(d*x + c)^4 + 6*a^2*b^2 + 11*a*b^3 + 5*b^4 - (6*a^2*b^2 + 17*a*b^3 + 10*b^4)*cos(d*x + c)^2)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) + 6*(2*a^5 + 2*a^4*b - 6*a^3*b^2 - 11*a^2*b^3 - 5*a*b^4)*cos(d*x + c))/(((a^6*b + 2*a^5*b^2 + a^4*b^3)*d*cos(d*x + c)^4 - (a^7 + 4*a^6*b + 5*a^5*b^2 + 2*a^4*b^3)*d*cos(d*x + c)^2 + (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d)*sin(d*x + c))]","B",0
106,1,950,0,0.801752," ","integrate(sin(d*x+c)^6/(a+b*sin(d*x+c)^2)^3,x, algorithm=""fricas"")","\left[\frac{32 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} d x \cos\left(d x + c\right)^{4} - 64 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} d x \cos\left(d x + c\right)^{2} + 32 \, {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} d x + {\left({\left(8 \, a^{2} b^{2} + 20 \, a b^{3} + 15 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 8 \, a^{4} + 36 \, a^{3} b + 63 \, a^{2} b^{2} + 50 \, a b^{3} + 15 \, b^{4} - 2 \, {\left(8 \, a^{3} b + 28 \, a^{2} b^{2} + 35 \, a b^{3} + 15 \, b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-\frac{a}{a + b}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, {\left(3 \, {\left(2 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{3} b + 13 \, a^{2} b^{2} + 9 \, a b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{32 \, {\left({\left(a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b^{3} + 4 \, a^{3} b^{4} + 6 \, a^{2} b^{5} + 4 \, a b^{6} + b^{7}\right)} d\right)}}, \frac{16 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} d x \cos\left(d x + c\right)^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} d x \cos\left(d x + c\right)^{2} + 16 \, {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} d x + {\left({\left(8 \, a^{2} b^{2} + 20 \, a b^{3} + 15 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 8 \, a^{4} + 36 \, a^{3} b + 63 \, a^{2} b^{2} + 50 \, a b^{3} + 15 \, b^{4} - 2 \, {\left(8 \, a^{3} b + 28 \, a^{2} b^{2} + 35 \, a b^{3} + 15 \, b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a}{a + b}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a}{a + b}}}{2 \, a \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) - 2 \, {\left(3 \, {\left(2 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{3} b + 13 \, a^{2} b^{2} + 9 \, a b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{16 \, {\left({\left(a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b^{3} + 4 \, a^{3} b^{4} + 6 \, a^{2} b^{5} + 4 \, a b^{6} + b^{7}\right)} d\right)}}\right]"," ",0,"[1/32*(32*(a^2*b^2 + 2*a*b^3 + b^4)*d*x*cos(d*x + c)^4 - 64*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x*cos(d*x + c)^2 + 32*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x + ((8*a^2*b^2 + 20*a*b^3 + 15*b^4)*cos(d*x + c)^4 + 8*a^4 + 36*a^3*b + 63*a^2*b^2 + 50*a*b^3 + 15*b^4 - 2*(8*a^3*b + 28*a^2*b^2 + 35*a*b^3 + 15*b^4)*cos(d*x + c)^2)*sqrt(-a/(a + b))*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*(3*(2*a^2*b^2 + 3*a*b^3)*cos(d*x + c)^3 - (4*a^3*b + 13*a^2*b^2 + 9*a*b^3)*cos(d*x + c))*sin(d*x + c))/((a^2*b^5 + 2*a*b^6 + b^7)*d*cos(d*x + c)^4 - 2*(a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + b^7)*d*cos(d*x + c)^2 + (a^4*b^3 + 4*a^3*b^4 + 6*a^2*b^5 + 4*a*b^6 + b^7)*d), 1/16*(16*(a^2*b^2 + 2*a*b^3 + b^4)*d*x*cos(d*x + c)^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*d*x*cos(d*x + c)^2 + 16*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x + ((8*a^2*b^2 + 20*a*b^3 + 15*b^4)*cos(d*x + c)^4 + 8*a^4 + 36*a^3*b + 63*a^2*b^2 + 50*a*b^3 + 15*b^4 - 2*(8*a^3*b + 28*a^2*b^2 + 35*a*b^3 + 15*b^4)*cos(d*x + c)^2)*sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c))) - 2*(3*(2*a^2*b^2 + 3*a*b^3)*cos(d*x + c)^3 - (4*a^3*b + 13*a^2*b^2 + 9*a*b^3)*cos(d*x + c))*sin(d*x + c))/((a^2*b^5 + 2*a*b^6 + b^7)*d*cos(d*x + c)^4 - 2*(a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + b^7)*d*cos(d*x + c)^2 + (a^4*b^3 + 4*a^3*b^4 + 6*a^2*b^5 + 4*a*b^6 + b^7)*d)]","B",0
107,1,683,0,0.798036," ","integrate(sin(d*x+c)^4/(a+b*sin(d*x+c)^2)^3,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, {\left({\left(2 \, a^{3} + 7 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} - 5 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{32 \, {\left({\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{5} b + 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} + 4 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} + 5 \, a^{5} b + 10 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + a b^{5}\right)} d\right)}}, -\frac{3 \, {\left(b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) - 2 \, {\left({\left(2 \, a^{3} + 7 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(d x + c\right)^{3} - 5 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{16 \, {\left({\left(a^{4} b^{2} + 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{5} b + 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} + 4 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} + 5 \, a^{5} b + 10 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + a b^{5}\right)} d\right)}}\right]"," ",0,"[-1/32*(3*(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*((2*a^3 + 7*a^2*b + 5*a*b^2)*cos(d*x + c)^3 - 5*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c))*sin(d*x + c))/((a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*d*cos(d*x + c)^4 - 2*(a^5*b + 4*a^4*b^2 + 6*a^3*b^3 + 4*a^2*b^4 + a*b^5)*d*cos(d*x + c)^2 + (a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*d), -1/16*(3*(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c))) - 2*((2*a^3 + 7*a^2*b + 5*a*b^2)*cos(d*x + c)^3 - 5*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c))*sin(d*x + c))/((a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 + a*b^5)*d*cos(d*x + c)^4 - 2*(a^5*b + 4*a^4*b^2 + 6*a^3*b^3 + 4*a^2*b^4 + a*b^5)*d*cos(d*x + c)^2 + (a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*d)]","B",0
108,1,771,0,0.607058," ","integrate(sin(d*x+c)^2/(a+b*sin(d*x+c)^2)^3,x, algorithm=""fricas"")","\left[-\frac{{\left({\left(4 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{4} + 4 \, a^{3} + 9 \, a^{2} b + 6 \, a b^{2} + b^{3} - 2 \, {\left(4 \, a^{2} b + 5 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, {\left({\left(2 \, a^{3} b + a^{2} b^{2} - a b^{3}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{4} + 7 \, a^{3} b + 2 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{32 \, {\left({\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} d\right)}}, -\frac{{\left({\left(4 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{4} + 4 \, a^{3} + 9 \, a^{2} b + 6 \, a b^{2} + b^{3} - 2 \, {\left(4 \, a^{2} b + 5 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) - 2 \, {\left({\left(2 \, a^{3} b + a^{2} b^{2} - a b^{3}\right)} \cos\left(d x + c\right)^{3} - {\left(4 \, a^{4} + 7 \, a^{3} b + 2 \, a^{2} b^{2} - a b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{16 \, {\left({\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} d\right)}}\right]"," ",0,"[-1/32*(((4*a*b^2 + b^3)*cos(d*x + c)^4 + 4*a^3 + 9*a^2*b + 6*a*b^2 + b^3 - 2*(4*a^2*b + 5*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*((2*a^3*b + a^2*b^2 - a*b^3)*cos(d*x + c)^3 - (4*a^4 + 7*a^3*b + 2*a^2*b^2 - a*b^3)*cos(d*x + c))*sin(d*x + c))/((a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*d*cos(d*x + c)^4 - 2*(a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*d*cos(d*x + c)^2 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*d), -1/16*(((4*a*b^2 + b^3)*cos(d*x + c)^4 + 4*a^3 + 9*a^2*b + 6*a*b^2 + b^3 - 2*(4*a^2*b + 5*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c))) - 2*((2*a^3*b + a^2*b^2 - a*b^3)*cos(d*x + c)^3 - (4*a^4 + 7*a^3*b + 2*a^2*b^2 - a*b^3)*cos(d*x + c))*sin(d*x + c))/((a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*d*cos(d*x + c)^4 - 2*(a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*d*cos(d*x + c)^2 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*d)]","B",0
109,1,843,0,0.820063," ","integrate(1/(a+b*sin(d*x+c)^2)^3,x, algorithm=""fricas"")","\left[-\frac{{\left({\left(8 \, a^{2} b^{2} + 8 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 8 \, a^{4} + 24 \, a^{3} b + 27 \, a^{2} b^{2} + 14 \, a b^{3} + 3 \, b^{4} - 2 \, {\left(8 \, a^{3} b + 16 \, a^{2} b^{2} + 11 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) + 4 \, {\left(3 \, {\left(2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(8 \, a^{4} b + 19 \, a^{3} b^{2} + 14 \, a^{2} b^{3} + 3 \, a b^{4}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{32 \, {\left({\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{7} b + 4 \, a^{6} b^{2} + 6 \, a^{5} b^{3} + 4 \, a^{4} b^{4} + a^{3} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{8} + 5 \, a^{7} b + 10 \, a^{6} b^{2} + 10 \, a^{5} b^{3} + 5 \, a^{4} b^{4} + a^{3} b^{5}\right)} d\right)}}, -\frac{{\left({\left(8 \, a^{2} b^{2} + 8 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + 8 \, a^{4} + 24 \, a^{3} b + 27 \, a^{2} b^{2} + 14 \, a b^{3} + 3 \, b^{4} - 2 \, {\left(8 \, a^{3} b + 16 \, a^{2} b^{2} + 11 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) + 2 \, {\left(3 \, {\left(2 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(d x + c\right)^{3} - {\left(8 \, a^{4} b + 19 \, a^{3} b^{2} + 14 \, a^{2} b^{3} + 3 \, a b^{4}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{16 \, {\left({\left(a^{6} b^{2} + 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{7} b + 4 \, a^{6} b^{2} + 6 \, a^{5} b^{3} + 4 \, a^{4} b^{4} + a^{3} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{8} + 5 \, a^{7} b + 10 \, a^{6} b^{2} + 10 \, a^{5} b^{3} + 5 \, a^{4} b^{4} + a^{3} b^{5}\right)} d\right)}}\right]"," ",0,"[-1/32*(((8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cos(d*x + c)^4 + 8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4 - 2*(8*a^3*b + 16*a^2*b^2 + 11*a*b^3 + 3*b^4)*cos(d*x + c)^2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) + 4*(3*(2*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(d*x + c)^3 - (8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cos(d*x + c))*sin(d*x + c))/((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cos(d*x + c)^4 - 2*(a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cos(d*x + c)^2 + (a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d), -1/16*(((8*a^2*b^2 + 8*a*b^3 + 3*b^4)*cos(d*x + c)^4 + 8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4 - 2*(8*a^3*b + 16*a^2*b^2 + 11*a*b^3 + 3*b^4)*cos(d*x + c)^2)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c))) + 2*(3*(2*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(d*x + c)^3 - (8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cos(d*x + c))*sin(d*x + c))/((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cos(d*x + c)^4 - 2*(a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cos(d*x + c)^2 + (a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d)]","B",0
110,1,1003,0,0.767023," ","integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^2)^3,x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(8 \, a^{4} b^{2} + 34 \, a^{3} b^{3} + 41 \, a^{2} b^{4} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)^{5} - 4 \, {\left(16 \, a^{5} b + 76 \, a^{4} b^{2} + 137 \, a^{3} b^{3} + 107 \, a^{2} b^{4} + 30 \, a b^{5}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(8 \, a^{4} b + 28 \, a^{3} b^{2} + 37 \, a^{2} b^{3} + 22 \, a b^{4} + 5 \, b^{5} + {\left(8 \, a^{2} b^{3} + 12 \, a b^{4} + 5 \, b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(8 \, a^{3} b^{2} + 20 \, a^{2} b^{3} + 17 \, a b^{4} + 5 \, b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \sin\left(d x + c\right) + 4 \, {\left(8 \, a^{6} + 40 \, a^{5} b + 92 \, a^{4} b^{2} + 111 \, a^{3} b^{3} + 66 \, a^{2} b^{4} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)}{32 \, {\left({\left(a^{7} b^{2} + 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} + a^{4} b^{5}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{8} b + 4 \, a^{7} b^{2} + 6 \, a^{6} b^{3} + 4 \, a^{5} b^{4} + a^{4} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{9} + 5 \, a^{8} b + 10 \, a^{7} b^{2} + 10 \, a^{6} b^{3} + 5 \, a^{5} b^{4} + a^{4} b^{5}\right)} d\right)} \sin\left(d x + c\right)}, -\frac{2 \, {\left(8 \, a^{4} b^{2} + 34 \, a^{3} b^{3} + 41 \, a^{2} b^{4} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)^{5} - 2 \, {\left(16 \, a^{5} b + 76 \, a^{4} b^{2} + 137 \, a^{3} b^{3} + 107 \, a^{2} b^{4} + 30 \, a b^{5}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(8 \, a^{4} b + 28 \, a^{3} b^{2} + 37 \, a^{2} b^{3} + 22 \, a b^{4} + 5 \, b^{5} + {\left(8 \, a^{2} b^{3} + 12 \, a b^{4} + 5 \, b^{5}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(8 \, a^{3} b^{2} + 20 \, a^{2} b^{3} + 17 \, a b^{4} + 5 \, b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 2 \, {\left(8 \, a^{6} + 40 \, a^{5} b + 92 \, a^{4} b^{2} + 111 \, a^{3} b^{3} + 66 \, a^{2} b^{4} + 15 \, a b^{5}\right)} \cos\left(d x + c\right)}{16 \, {\left({\left(a^{7} b^{2} + 3 \, a^{6} b^{3} + 3 \, a^{5} b^{4} + a^{4} b^{5}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{8} b + 4 \, a^{7} b^{2} + 6 \, a^{6} b^{3} + 4 \, a^{5} b^{4} + a^{4} b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{9} + 5 \, a^{8} b + 10 \, a^{7} b^{2} + 10 \, a^{6} b^{3} + 5 \, a^{5} b^{4} + a^{4} b^{5}\right)} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/32*(4*(8*a^4*b^2 + 34*a^3*b^3 + 41*a^2*b^4 + 15*a*b^5)*cos(d*x + c)^5 - 4*(16*a^5*b + 76*a^4*b^2 + 137*a^3*b^3 + 107*a^2*b^4 + 30*a*b^5)*cos(d*x + c)^3 + 3*(8*a^4*b + 28*a^3*b^2 + 37*a^2*b^3 + 22*a*b^4 + 5*b^5 + (8*a^2*b^3 + 12*a*b^4 + 5*b^5)*cos(d*x + c)^4 - 2*(8*a^3*b^2 + 20*a^2*b^3 + 17*a*b^4 + 5*b^5)*cos(d*x + c)^2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c) + 4*(8*a^6 + 40*a^5*b + 92*a^4*b^2 + 111*a^3*b^3 + 66*a^2*b^4 + 15*a*b^5)*cos(d*x + c))/(((a^7*b^2 + 3*a^6*b^3 + 3*a^5*b^4 + a^4*b^5)*d*cos(d*x + c)^4 - 2*(a^8*b + 4*a^7*b^2 + 6*a^6*b^3 + 4*a^5*b^4 + a^4*b^5)*d*cos(d*x + c)^2 + (a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d)*sin(d*x + c)), -1/16*(2*(8*a^4*b^2 + 34*a^3*b^3 + 41*a^2*b^4 + 15*a*b^5)*cos(d*x + c)^5 - 2*(16*a^5*b + 76*a^4*b^2 + 137*a^3*b^3 + 107*a^2*b^4 + 30*a*b^5)*cos(d*x + c)^3 - 3*(8*a^4*b + 28*a^3*b^2 + 37*a^2*b^3 + 22*a*b^4 + 5*b^5 + (8*a^2*b^3 + 12*a*b^4 + 5*b^5)*cos(d*x + c)^4 - 2*(8*a^3*b^2 + 20*a^2*b^3 + 17*a*b^4 + 5*b^5)*cos(d*x + c)^2)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) + 2*(8*a^6 + 40*a^5*b + 92*a^4*b^2 + 111*a^3*b^3 + 66*a^2*b^4 + 15*a*b^5)*cos(d*x + c))/(((a^7*b^2 + 3*a^6*b^3 + 3*a^5*b^4 + a^4*b^5)*d*cos(d*x + c)^4 - 2*(a^8*b + 4*a^7*b^2 + 6*a^6*b^3 + 4*a^5*b^4 + a^4*b^5)*d*cos(d*x + c)^2 + (a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d)*sin(d*x + c))]","B",0
111,1,1361,0,0.915097," ","integrate(1/(a+b*sin(d*x+c)^2)^4,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(16 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 18 \, a b^{5} + 5 \, b^{6}\right)} \cos\left(d x + c\right)^{6} - 16 \, a^{6} - 72 \, a^{5} b - 138 \, a^{4} b^{2} - 147 \, a^{3} b^{3} - 93 \, a^{2} b^{4} - 33 \, a b^{5} - 5 \, b^{6} - 3 \, {\left(16 \, a^{4} b^{2} + 40 \, a^{3} b^{3} + 42 \, a^{2} b^{4} + 23 \, a b^{5} + 5 \, b^{6}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(16 \, a^{5} b + 56 \, a^{4} b^{2} + 82 \, a^{3} b^{3} + 65 \, a^{2} b^{4} + 28 \, a b^{5} + 5 \, b^{6}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) + 4 \, {\left({\left(44 \, a^{4} b^{3} + 88 \, a^{3} b^{4} + 59 \, a^{2} b^{5} + 15 \, a b^{6}\right)} \cos\left(d x + c\right)^{5} - 2 \, {\left(54 \, a^{5} b^{2} + 157 \, a^{4} b^{3} + 167 \, a^{3} b^{4} + 79 \, a^{2} b^{5} + 15 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(24 \, a^{6} b + 90 \, a^{5} b^{2} + 131 \, a^{4} b^{3} + 93 \, a^{3} b^{4} + 33 \, a^{2} b^{5} + 5 \, a b^{6}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{192 \, {\left({\left(a^{8} b^{3} + 4 \, a^{7} b^{4} + 6 \, a^{6} b^{5} + 4 \, a^{5} b^{6} + a^{4} b^{7}\right)} d \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{9} b^{2} + 5 \, a^{8} b^{3} + 10 \, a^{7} b^{4} + 10 \, a^{6} b^{5} + 5 \, a^{5} b^{6} + a^{4} b^{7}\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{10} b + 6 \, a^{9} b^{2} + 15 \, a^{8} b^{3} + 20 \, a^{7} b^{4} + 15 \, a^{6} b^{5} + 6 \, a^{5} b^{6} + a^{4} b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{11} + 7 \, a^{10} b + 21 \, a^{9} b^{2} + 35 \, a^{8} b^{3} + 35 \, a^{7} b^{4} + 21 \, a^{6} b^{5} + 7 \, a^{5} b^{6} + a^{4} b^{7}\right)} d\right)}}, -\frac{3 \, {\left({\left(16 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 18 \, a b^{5} + 5 \, b^{6}\right)} \cos\left(d x + c\right)^{6} - 16 \, a^{6} - 72 \, a^{5} b - 138 \, a^{4} b^{2} - 147 \, a^{3} b^{3} - 93 \, a^{2} b^{4} - 33 \, a b^{5} - 5 \, b^{6} - 3 \, {\left(16 \, a^{4} b^{2} + 40 \, a^{3} b^{3} + 42 \, a^{2} b^{4} + 23 \, a b^{5} + 5 \, b^{6}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(16 \, a^{5} b + 56 \, a^{4} b^{2} + 82 \, a^{3} b^{3} + 65 \, a^{2} b^{4} + 28 \, a b^{5} + 5 \, b^{6}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) + 2 \, {\left({\left(44 \, a^{4} b^{3} + 88 \, a^{3} b^{4} + 59 \, a^{2} b^{5} + 15 \, a b^{6}\right)} \cos\left(d x + c\right)^{5} - 2 \, {\left(54 \, a^{5} b^{2} + 157 \, a^{4} b^{3} + 167 \, a^{3} b^{4} + 79 \, a^{2} b^{5} + 15 \, a b^{6}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(24 \, a^{6} b + 90 \, a^{5} b^{2} + 131 \, a^{4} b^{3} + 93 \, a^{3} b^{4} + 33 \, a^{2} b^{5} + 5 \, a b^{6}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{96 \, {\left({\left(a^{8} b^{3} + 4 \, a^{7} b^{4} + 6 \, a^{6} b^{5} + 4 \, a^{5} b^{6} + a^{4} b^{7}\right)} d \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{9} b^{2} + 5 \, a^{8} b^{3} + 10 \, a^{7} b^{4} + 10 \, a^{6} b^{5} + 5 \, a^{5} b^{6} + a^{4} b^{7}\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{10} b + 6 \, a^{9} b^{2} + 15 \, a^{8} b^{3} + 20 \, a^{7} b^{4} + 15 \, a^{6} b^{5} + 6 \, a^{5} b^{6} + a^{4} b^{7}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{11} + 7 \, a^{10} b + 21 \, a^{9} b^{2} + 35 \, a^{8} b^{3} + 35 \, a^{7} b^{4} + 21 \, a^{6} b^{5} + 7 \, a^{5} b^{6} + a^{4} b^{7}\right)} d\right)}}\right]"," ",0,"[-1/192*(3*((16*a^3*b^3 + 24*a^2*b^4 + 18*a*b^5 + 5*b^6)*cos(d*x + c)^6 - 16*a^6 - 72*a^5*b - 138*a^4*b^2 - 147*a^3*b^3 - 93*a^2*b^4 - 33*a*b^5 - 5*b^6 - 3*(16*a^4*b^2 + 40*a^3*b^3 + 42*a^2*b^4 + 23*a*b^5 + 5*b^6)*cos(d*x + c)^4 + 3*(16*a^5*b + 56*a^4*b^2 + 82*a^3*b^3 + 65*a^2*b^4 + 28*a*b^5 + 5*b^6)*cos(d*x + c)^2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) + 4*((44*a^4*b^3 + 88*a^3*b^4 + 59*a^2*b^5 + 15*a*b^6)*cos(d*x + c)^5 - 2*(54*a^5*b^2 + 157*a^4*b^3 + 167*a^3*b^4 + 79*a^2*b^5 + 15*a*b^6)*cos(d*x + c)^3 + 3*(24*a^6*b + 90*a^5*b^2 + 131*a^4*b^3 + 93*a^3*b^4 + 33*a^2*b^5 + 5*a*b^6)*cos(d*x + c))*sin(d*x + c))/((a^8*b^3 + 4*a^7*b^4 + 6*a^6*b^5 + 4*a^5*b^6 + a^4*b^7)*d*cos(d*x + c)^6 - 3*(a^9*b^2 + 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6*b^5 + 5*a^5*b^6 + a^4*b^7)*d*cos(d*x + c)^4 + 3*(a^10*b + 6*a^9*b^2 + 15*a^8*b^3 + 20*a^7*b^4 + 15*a^6*b^5 + 6*a^5*b^6 + a^4*b^7)*d*cos(d*x + c)^2 - (a^11 + 7*a^10*b + 21*a^9*b^2 + 35*a^8*b^3 + 35*a^7*b^4 + 21*a^6*b^5 + 7*a^5*b^6 + a^4*b^7)*d), -1/96*(3*((16*a^3*b^3 + 24*a^2*b^4 + 18*a*b^5 + 5*b^6)*cos(d*x + c)^6 - 16*a^6 - 72*a^5*b - 138*a^4*b^2 - 147*a^3*b^3 - 93*a^2*b^4 - 33*a*b^5 - 5*b^6 - 3*(16*a^4*b^2 + 40*a^3*b^3 + 42*a^2*b^4 + 23*a*b^5 + 5*b^6)*cos(d*x + c)^4 + 3*(16*a^5*b + 56*a^4*b^2 + 82*a^3*b^3 + 65*a^2*b^4 + 28*a*b^5 + 5*b^6)*cos(d*x + c)^2)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c))) + 2*((44*a^4*b^3 + 88*a^3*b^4 + 59*a^2*b^5 + 15*a*b^6)*cos(d*x + c)^5 - 2*(54*a^5*b^2 + 157*a^4*b^3 + 167*a^3*b^4 + 79*a^2*b^5 + 15*a*b^6)*cos(d*x + c)^3 + 3*(24*a^6*b + 90*a^5*b^2 + 131*a^4*b^3 + 93*a^3*b^4 + 33*a^2*b^5 + 5*a*b^6)*cos(d*x + c))*sin(d*x + c))/((a^8*b^3 + 4*a^7*b^4 + 6*a^6*b^5 + 4*a^5*b^6 + a^4*b^7)*d*cos(d*x + c)^6 - 3*(a^9*b^2 + 5*a^8*b^3 + 10*a^7*b^4 + 10*a^6*b^5 + 5*a^5*b^6 + a^4*b^7)*d*cos(d*x + c)^4 + 3*(a^10*b + 6*a^9*b^2 + 15*a^8*b^3 + 20*a^7*b^4 + 15*a^6*b^5 + 6*a^5*b^6 + a^4*b^7)*d*cos(d*x + c)^2 - (a^11 + 7*a^10*b + 21*a^9*b^2 + 35*a^8*b^3 + 35*a^7*b^4 + 21*a^6*b^5 + 7*a^5*b^6 + a^4*b^7)*d)]","B",0
112,1,2017,0,1.077396," ","integrate(1/(a+b*sin(d*x+c)^2)^5,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(128 \, a^{4} b^{4} + 256 \, a^{3} b^{5} + 288 \, a^{2} b^{6} + 160 \, a b^{7} + 35 \, b^{8}\right)} \cos\left(d x + c\right)^{8} + 128 \, a^{8} + 768 \, a^{7} b + 2080 \, a^{6} b^{2} + 3360 \, a^{5} b^{3} + 3555 \, a^{4} b^{4} + 2508 \, a^{3} b^{5} + 1138 \, a^{2} b^{6} + 300 \, a b^{7} + 35 \, b^{8} - 4 \, {\left(128 \, a^{5} b^{3} + 384 \, a^{4} b^{4} + 544 \, a^{3} b^{5} + 448 \, a^{2} b^{6} + 195 \, a b^{7} + 35 \, b^{8}\right)} \cos\left(d x + c\right)^{6} + 6 \, {\left(128 \, a^{6} b^{2} + 512 \, a^{5} b^{3} + 928 \, a^{4} b^{4} + 992 \, a^{3} b^{5} + 643 \, a^{2} b^{6} + 230 \, a b^{7} + 35 \, b^{8}\right)} \cos\left(d x + c\right)^{4} - 4 \, {\left(128 \, a^{7} b + 640 \, a^{6} b^{2} + 1440 \, a^{5} b^{3} + 1920 \, a^{4} b^{4} + 1635 \, a^{3} b^{5} + 873 \, a^{2} b^{6} + 265 \, a b^{7} + 35 \, b^{8}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{-a^{2} - a b} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) + 4 \, {\left(5 \, {\left(80 \, a^{5} b^{4} + 200 \, a^{4} b^{5} + 202 \, a^{3} b^{6} + 103 \, a^{2} b^{7} + 21 \, a b^{8}\right)} \cos\left(d x + c\right)^{7} - {\left(1408 \, a^{6} b^{3} + 4824 \, a^{5} b^{4} + 6724 \, a^{4} b^{5} + 4923 \, a^{3} b^{6} + 1930 \, a^{2} b^{7} + 315 \, a b^{8}\right)} \cos\left(d x + c\right)^{5} + {\left(1728 \, a^{7} b^{2} + 7456 \, a^{6} b^{3} + 13370 \, a^{5} b^{4} + 12969 \, a^{4} b^{5} + 7327 \, a^{3} b^{6} + 2315 \, a^{2} b^{7} + 315 \, a b^{8}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(256 \, a^{8} b + 1312 \, a^{7} b^{2} + 2848 \, a^{6} b^{3} + 3427 \, a^{5} b^{4} + 2508 \, a^{4} b^{5} + 1138 \, a^{3} b^{6} + 300 \, a^{2} b^{7} + 35 \, a b^{8}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{1536 \, {\left({\left(a^{10} b^{4} + 5 \, a^{9} b^{5} + 10 \, a^{8} b^{6} + 10 \, a^{7} b^{7} + 5 \, a^{6} b^{8} + a^{5} b^{9}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{11} b^{3} + 6 \, a^{10} b^{4} + 15 \, a^{9} b^{5} + 20 \, a^{8} b^{6} + 15 \, a^{7} b^{7} + 6 \, a^{6} b^{8} + a^{5} b^{9}\right)} d \cos\left(d x + c\right)^{6} + 6 \, {\left(a^{12} b^{2} + 7 \, a^{11} b^{3} + 21 \, a^{10} b^{4} + 35 \, a^{9} b^{5} + 35 \, a^{8} b^{6} + 21 \, a^{7} b^{7} + 7 \, a^{6} b^{8} + a^{5} b^{9}\right)} d \cos\left(d x + c\right)^{4} - 4 \, {\left(a^{13} b + 8 \, a^{12} b^{2} + 28 \, a^{11} b^{3} + 56 \, a^{10} b^{4} + 70 \, a^{9} b^{5} + 56 \, a^{8} b^{6} + 28 \, a^{7} b^{7} + 8 \, a^{6} b^{8} + a^{5} b^{9}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{14} + 9 \, a^{13} b + 36 \, a^{12} b^{2} + 84 \, a^{11} b^{3} + 126 \, a^{10} b^{4} + 126 \, a^{9} b^{5} + 84 \, a^{8} b^{6} + 36 \, a^{7} b^{7} + 9 \, a^{6} b^{8} + a^{5} b^{9}\right)} d\right)}}, -\frac{3 \, {\left({\left(128 \, a^{4} b^{4} + 256 \, a^{3} b^{5} + 288 \, a^{2} b^{6} + 160 \, a b^{7} + 35 \, b^{8}\right)} \cos\left(d x + c\right)^{8} + 128 \, a^{8} + 768 \, a^{7} b + 2080 \, a^{6} b^{2} + 3360 \, a^{5} b^{3} + 3555 \, a^{4} b^{4} + 2508 \, a^{3} b^{5} + 1138 \, a^{2} b^{6} + 300 \, a b^{7} + 35 \, b^{8} - 4 \, {\left(128 \, a^{5} b^{3} + 384 \, a^{4} b^{4} + 544 \, a^{3} b^{5} + 448 \, a^{2} b^{6} + 195 \, a b^{7} + 35 \, b^{8}\right)} \cos\left(d x + c\right)^{6} + 6 \, {\left(128 \, a^{6} b^{2} + 512 \, a^{5} b^{3} + 928 \, a^{4} b^{4} + 992 \, a^{3} b^{5} + 643 \, a^{2} b^{6} + 230 \, a b^{7} + 35 \, b^{8}\right)} \cos\left(d x + c\right)^{4} - 4 \, {\left(128 \, a^{7} b + 640 \, a^{6} b^{2} + 1440 \, a^{5} b^{3} + 1920 \, a^{4} b^{4} + 1635 \, a^{3} b^{5} + 873 \, a^{2} b^{6} + 265 \, a b^{7} + 35 \, b^{8}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) + 2 \, {\left(5 \, {\left(80 \, a^{5} b^{4} + 200 \, a^{4} b^{5} + 202 \, a^{3} b^{6} + 103 \, a^{2} b^{7} + 21 \, a b^{8}\right)} \cos\left(d x + c\right)^{7} - {\left(1408 \, a^{6} b^{3} + 4824 \, a^{5} b^{4} + 6724 \, a^{4} b^{5} + 4923 \, a^{3} b^{6} + 1930 \, a^{2} b^{7} + 315 \, a b^{8}\right)} \cos\left(d x + c\right)^{5} + {\left(1728 \, a^{7} b^{2} + 7456 \, a^{6} b^{3} + 13370 \, a^{5} b^{4} + 12969 \, a^{4} b^{5} + 7327 \, a^{3} b^{6} + 2315 \, a^{2} b^{7} + 315 \, a b^{8}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(256 \, a^{8} b + 1312 \, a^{7} b^{2} + 2848 \, a^{6} b^{3} + 3427 \, a^{5} b^{4} + 2508 \, a^{4} b^{5} + 1138 \, a^{3} b^{6} + 300 \, a^{2} b^{7} + 35 \, a b^{8}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{768 \, {\left({\left(a^{10} b^{4} + 5 \, a^{9} b^{5} + 10 \, a^{8} b^{6} + 10 \, a^{7} b^{7} + 5 \, a^{6} b^{8} + a^{5} b^{9}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{11} b^{3} + 6 \, a^{10} b^{4} + 15 \, a^{9} b^{5} + 20 \, a^{8} b^{6} + 15 \, a^{7} b^{7} + 6 \, a^{6} b^{8} + a^{5} b^{9}\right)} d \cos\left(d x + c\right)^{6} + 6 \, {\left(a^{12} b^{2} + 7 \, a^{11} b^{3} + 21 \, a^{10} b^{4} + 35 \, a^{9} b^{5} + 35 \, a^{8} b^{6} + 21 \, a^{7} b^{7} + 7 \, a^{6} b^{8} + a^{5} b^{9}\right)} d \cos\left(d x + c\right)^{4} - 4 \, {\left(a^{13} b + 8 \, a^{12} b^{2} + 28 \, a^{11} b^{3} + 56 \, a^{10} b^{4} + 70 \, a^{9} b^{5} + 56 \, a^{8} b^{6} + 28 \, a^{7} b^{7} + 8 \, a^{6} b^{8} + a^{5} b^{9}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{14} + 9 \, a^{13} b + 36 \, a^{12} b^{2} + 84 \, a^{11} b^{3} + 126 \, a^{10} b^{4} + 126 \, a^{9} b^{5} + 84 \, a^{8} b^{6} + 36 \, a^{7} b^{7} + 9 \, a^{6} b^{8} + a^{5} b^{9}\right)} d\right)}}\right]"," ",0,"[-1/1536*(3*((128*a^4*b^4 + 256*a^3*b^5 + 288*a^2*b^6 + 160*a*b^7 + 35*b^8)*cos(d*x + c)^8 + 128*a^8 + 768*a^7*b + 2080*a^6*b^2 + 3360*a^5*b^3 + 3555*a^4*b^4 + 2508*a^3*b^5 + 1138*a^2*b^6 + 300*a*b^7 + 35*b^8 - 4*(128*a^5*b^3 + 384*a^4*b^4 + 544*a^3*b^5 + 448*a^2*b^6 + 195*a*b^7 + 35*b^8)*cos(d*x + c)^6 + 6*(128*a^6*b^2 + 512*a^5*b^3 + 928*a^4*b^4 + 992*a^3*b^5 + 643*a^2*b^6 + 230*a*b^7 + 35*b^8)*cos(d*x + c)^4 - 4*(128*a^7*b + 640*a^6*b^2 + 1440*a^5*b^3 + 1920*a^4*b^4 + 1635*a^3*b^5 + 873*a^2*b^6 + 265*a*b^7 + 35*b^8)*cos(d*x + c)^2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a + b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) + 4*(5*(80*a^5*b^4 + 200*a^4*b^5 + 202*a^3*b^6 + 103*a^2*b^7 + 21*a*b^8)*cos(d*x + c)^7 - (1408*a^6*b^3 + 4824*a^5*b^4 + 6724*a^4*b^5 + 4923*a^3*b^6 + 1930*a^2*b^7 + 315*a*b^8)*cos(d*x + c)^5 + (1728*a^7*b^2 + 7456*a^6*b^3 + 13370*a^5*b^4 + 12969*a^4*b^5 + 7327*a^3*b^6 + 2315*a^2*b^7 + 315*a*b^8)*cos(d*x + c)^3 - 3*(256*a^8*b + 1312*a^7*b^2 + 2848*a^6*b^3 + 3427*a^5*b^4 + 2508*a^4*b^5 + 1138*a^3*b^6 + 300*a^2*b^7 + 35*a*b^8)*cos(d*x + c))*sin(d*x + c))/((a^10*b^4 + 5*a^9*b^5 + 10*a^8*b^6 + 10*a^7*b^7 + 5*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^8 - 4*(a^11*b^3 + 6*a^10*b^4 + 15*a^9*b^5 + 20*a^8*b^6 + 15*a^7*b^7 + 6*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^6 + 6*(a^12*b^2 + 7*a^11*b^3 + 21*a^10*b^4 + 35*a^9*b^5 + 35*a^8*b^6 + 21*a^7*b^7 + 7*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^4 - 4*(a^13*b + 8*a^12*b^2 + 28*a^11*b^3 + 56*a^10*b^4 + 70*a^9*b^5 + 56*a^8*b^6 + 28*a^7*b^7 + 8*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^2 + (a^14 + 9*a^13*b + 36*a^12*b^2 + 84*a^11*b^3 + 126*a^10*b^4 + 126*a^9*b^5 + 84*a^8*b^6 + 36*a^7*b^7 + 9*a^6*b^8 + a^5*b^9)*d), -1/768*(3*((128*a^4*b^4 + 256*a^3*b^5 + 288*a^2*b^6 + 160*a*b^7 + 35*b^8)*cos(d*x + c)^8 + 128*a^8 + 768*a^7*b + 2080*a^6*b^2 + 3360*a^5*b^3 + 3555*a^4*b^4 + 2508*a^3*b^5 + 1138*a^2*b^6 + 300*a*b^7 + 35*b^8 - 4*(128*a^5*b^3 + 384*a^4*b^4 + 544*a^3*b^5 + 448*a^2*b^6 + 195*a*b^7 + 35*b^8)*cos(d*x + c)^6 + 6*(128*a^6*b^2 + 512*a^5*b^3 + 928*a^4*b^4 + 992*a^3*b^5 + 643*a^2*b^6 + 230*a*b^7 + 35*b^8)*cos(d*x + c)^4 - 4*(128*a^7*b + 640*a^6*b^2 + 1440*a^5*b^3 + 1920*a^4*b^4 + 1635*a^3*b^5 + 873*a^2*b^6 + 265*a*b^7 + 35*b^8)*cos(d*x + c)^2)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)/(sqrt(a^2 + a*b)*cos(d*x + c)*sin(d*x + c))) + 2*(5*(80*a^5*b^4 + 200*a^4*b^5 + 202*a^3*b^6 + 103*a^2*b^7 + 21*a*b^8)*cos(d*x + c)^7 - (1408*a^6*b^3 + 4824*a^5*b^4 + 6724*a^4*b^5 + 4923*a^3*b^6 + 1930*a^2*b^7 + 315*a*b^8)*cos(d*x + c)^5 + (1728*a^7*b^2 + 7456*a^6*b^3 + 13370*a^5*b^4 + 12969*a^4*b^5 + 7327*a^3*b^6 + 2315*a^2*b^7 + 315*a*b^8)*cos(d*x + c)^3 - 3*(256*a^8*b + 1312*a^7*b^2 + 2848*a^6*b^3 + 3427*a^5*b^4 + 2508*a^4*b^5 + 1138*a^3*b^6 + 300*a^2*b^7 + 35*a*b^8)*cos(d*x + c))*sin(d*x + c))/((a^10*b^4 + 5*a^9*b^5 + 10*a^8*b^6 + 10*a^7*b^7 + 5*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^8 - 4*(a^11*b^3 + 6*a^10*b^4 + 15*a^9*b^5 + 20*a^8*b^6 + 15*a^7*b^7 + 6*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^6 + 6*(a^12*b^2 + 7*a^11*b^3 + 21*a^10*b^4 + 35*a^9*b^5 + 35*a^8*b^6 + 21*a^7*b^7 + 7*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^4 - 4*(a^13*b + 8*a^12*b^2 + 28*a^11*b^3 + 56*a^10*b^4 + 70*a^9*b^5 + 56*a^8*b^6 + 28*a^7*b^7 + 8*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^2 + (a^14 + 9*a^13*b + 36*a^12*b^2 + 84*a^11*b^3 + 126*a^10*b^4 + 126*a^9*b^5 + 84*a^8*b^6 + 36*a^7*b^7 + 9*a^6*b^8 + a^5*b^9)*d)]","B",0
113,1,57,0,0.541121," ","integrate(sin(x)/(1+sin(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{2} \, \arctan\left(-\frac{\cos\left(x\right) \sin\left(x\right) - {\left(\cos\left(x\right)^{3} - \cos\left(x\right)\right)} \sqrt{-\cos\left(x\right)^{2} + 2}}{\cos\left(x\right)^{4} - 3 \, \cos\left(x\right)^{2} + 1}\right) - \frac{1}{2} \, \arctan\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)"," ",0,"1/2*arctan(-(cos(x)*sin(x) - (cos(x)^3 - cos(x))*sqrt(-cos(x)^2 + 2))/(cos(x)^4 - 3*cos(x)^2 + 1)) - 1/2*arctan(sin(x)/cos(x))","B",0
114,1,71,0,1.105653," ","integrate(sin(x)*(1+sin(x)^2)^(1/2),x, algorithm=""fricas"")","-\frac{1}{2} \, \sqrt{-\cos\left(x\right)^{2} + 2} \cos\left(x\right) + \frac{1}{2} \, \arctan\left(-\frac{\cos\left(x\right) \sin\left(x\right) - {\left(\cos\left(x\right)^{3} - \cos\left(x\right)\right)} \sqrt{-\cos\left(x\right)^{2} + 2}}{\cos\left(x\right)^{4} - 3 \, \cos\left(x\right)^{2} + 1}\right) - \frac{1}{2} \, \arctan\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)"," ",0,"-1/2*sqrt(-cos(x)^2 + 2)*cos(x) + 1/2*arctan(-(cos(x)*sin(x) - (cos(x)^3 - cos(x))*sqrt(-cos(x)^2 + 2))/(cos(x)^4 - 3*cos(x)^2 + 1)) - 1/2*arctan(sin(x)/cos(x))","B",0
115,1,94,0,0.740604," ","integrate(sin(7+3*x)/(3+sin(7+3*x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{6} \, \arctan\left(-\frac{4 \, \cos\left(3 \, x + 7\right) \sin\left(3 \, x + 7\right) - {\left(\cos\left(3 \, x + 7\right)^{3} - 2 \, \cos\left(3 \, x + 7\right)\right)} \sqrt{-\cos\left(3 \, x + 7\right)^{2} + 4}}{\cos\left(3 \, x + 7\right)^{4} - 8 \, \cos\left(3 \, x + 7\right)^{2} + 4}\right) - \frac{1}{6} \, \arctan\left(\frac{\sin\left(3 \, x + 7\right)}{\cos\left(3 \, x + 7\right)}\right)"," ",0,"1/6*arctan(-(4*cos(3*x + 7)*sin(3*x + 7) - (cos(3*x + 7)^3 - 2*cos(3*x + 7))*sqrt(-cos(3*x + 7)^2 + 4))/(cos(3*x + 7)^4 - 8*cos(3*x + 7)^2 + 4)) - 1/6*arctan(sin(3*x + 7)/cos(3*x + 7))","B",0
116,1,40,0,0.735443," ","integrate((a-a*sin(x)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left(3 \, a^{2} \cos\left(x\right)^{4} + 4 \, a^{2} \cos\left(x\right)^{2} + 8 \, a^{2}\right)} \sqrt{a \cos\left(x\right)^{2}} \sin\left(x\right)}{15 \, \cos\left(x\right)}"," ",0,"1/15*(3*a^2*cos(x)^4 + 4*a^2*cos(x)^2 + 8*a^2)*sqrt(a*cos(x)^2)*sin(x)/cos(x)","A",0
117,1,26,0,0.621446," ","integrate((a-a*sin(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{{\left(a \cos\left(x\right)^{2} + 2 \, a\right)} \sqrt{a \cos\left(x\right)^{2}} \sin\left(x\right)}{3 \, \cos\left(x\right)}"," ",0,"1/3*(a*cos(x)^2 + 2*a)*sqrt(a*cos(x)^2)*sin(x)/cos(x)","A",0
118,1,15,0,0.653266," ","integrate((a-a*sin(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{a \cos\left(x\right)^{2}} \sin\left(x\right)}{\cos\left(x\right)}"," ",0,"sqrt(a*cos(x)^2)*sin(x)/cos(x)","A",0
119,1,65,0,0.952772," ","integrate(1/(a-a*sin(x)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{a \cos\left(x\right)^{2}} \log\left(-\frac{\sin\left(x\right) - 1}{\sin\left(x\right) + 1}\right)}{2 \, a \cos\left(x\right)}, -\frac{\sqrt{-a} \arctan\left(\frac{\sqrt{a \cos\left(x\right)^{2}} \sqrt{-a} \sin\left(x\right)}{a \cos\left(x\right)}\right)}{a}\right]"," ",0,"[-1/2*sqrt(a*cos(x)^2)*log(-(sin(x) - 1)/(sin(x) + 1))/(a*cos(x)), -sqrt(-a)*arctan(sqrt(a*cos(x)^2)*sqrt(-a)*sin(x)/(a*cos(x)))/a]","B",0
120,1,40,0,0.669479," ","integrate(1/(a-a*sin(x)^2)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(x\right)^{2}} {\left(\cos\left(x\right)^{2} \log\left(-\frac{\sin\left(x\right) - 1}{\sin\left(x\right) + 1}\right) - 2 \, \sin\left(x\right)\right)}}{4 \, a^{2} \cos\left(x\right)^{3}}"," ",0,"-1/4*sqrt(a*cos(x)^2)*(cos(x)^2*log(-(sin(x) - 1)/(sin(x) + 1)) - 2*sin(x))/(a^2*cos(x)^3)","A",0
121,1,49,0,0.710249," ","integrate(1/(a-a*sin(x)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, \cos\left(x\right)^{4} \log\left(-\frac{\sin\left(x\right) - 1}{\sin\left(x\right) + 1}\right) - 2 \, {\left(3 \, \cos\left(x\right)^{2} + 2\right)} \sin\left(x\right)\right)} \sqrt{a \cos\left(x\right)^{2}}}{16 \, a^{3} \cos\left(x\right)^{5}}"," ",0,"-1/16*(3*cos(x)^4*log(-(sin(x) - 1)/(sin(x) + 1)) - 2*(3*cos(x)^2 + 2)*sin(x))*sqrt(a*cos(x)^2)/(a^3*cos(x)^5)","A",0
122,1,501,0,1.243348," ","integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} - 2 \, a b - 3 \, b^{2}\right)} \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right) + 8 \, {\left(2 \, b^{2} \cos\left(f x + e\right)^{3} - {\left(a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{64 \, b^{2} f}, -\frac{{\left(a^{2} - 2 \, a b - 3 \, b^{2}\right)} \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left(2 \, b^{2} \cos\left(f x + e\right)^{3} - {\left(a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{32 \, b^{2} f}\right]"," ",0,"[1/64*((a^2 - 2*a*b - 3*b^2)*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)) + 8*(2*b^2*cos(f*x + e)^3 - (a*b + 5*b^2)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(b^2*f), -1/32*((a^2 - 2*a*b - 3*b^2)*sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))) - 4*(2*b^2*cos(f*x + e)^3 - (a*b + 5*b^2)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(b^2*f)]","A",0
123,1,433,0,0.878443," ","integrate(sin(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{8 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \cos\left(f x + e\right) + {\left(a + b\right)} \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right)}{16 \, b f}, \frac{{\left(a + b\right)} \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \cos\left(f x + e\right)}{8 \, b f}\right]"," ",0,"[-1/16*(8*sqrt(-b*cos(f*x + e)^2 + a + b)*b*cos(f*x + e) + (a + b)*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)))/(b*f), 1/8*((a + b)*sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))) - 4*sqrt(-b*cos(f*x + e)^2 + a + b)*b*cos(f*x + e))/(b*f)]","B",0
124,1,1158,0,1.440420," ","integrate(csc(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right) + 2 \, \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, f}, \frac{4 \, \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) + \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right)}{8 \, f}, \frac{\sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{4 \, f}, \frac{2 \, \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) + \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right)}{4 \, f}\right]"," ",0,"[1/8*(sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)) + 2*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/f, 1/8*(4*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) + sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)))/f, 1/4*(sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))) + sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/f, 1/4*(2*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) + sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))))/f]","B",0
125,1,338,0,0.880862," ","integrate(csc(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a \cos\left(f x + e\right) + {\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left(a f \cos\left(f x + e\right)^{2} - a f\right)}}, \frac{{\left({\left(a + b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a \cos\left(f x + e\right)}{4 \, {\left(a f \cos\left(f x + e\right)^{2} - a f\right)}}\right]"," ",0,"[1/8*(4*sqrt(-b*cos(f*x + e)^2 + a + b)*a*cos(f*x + e) + ((a + b)*cos(f*x + e)^2 - a - b)*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/(a*f*cos(f*x + e)^2 - a*f), 1/4*(((a + b)*cos(f*x + e)^2 - a - b)*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*a*cos(f*x + e))/(a*f*cos(f*x + e)^2 - a*f)]","A",0
126,1,520,0,0.868150," ","integrate(csc(f*x+e)^5*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 2 \, a b - b^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left({\left(3 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{32 \, {\left(a^{2} f \cos\left(f x + e\right)^{4} - 2 \, a^{2} f \cos\left(f x + e\right)^{2} + a^{2} f\right)}}, \frac{{\left({\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 2 \, a b - b^{2}\right)} \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) + 2 \, {\left({\left(3 \, a^{2} + a b\right)} \cos\left(f x + e\right)^{3} - {\left(5 \, a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{16 \, {\left(a^{2} f \cos\left(f x + e\right)^{4} - 2 \, a^{2} f \cos\left(f x + e\right)^{2} + a^{2} f\right)}}\right]"," ",0,"[-1/32*(((3*a^2 + 2*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 + 2*a*b - b^2)*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 4*((3*a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + a*b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f), 1/16*(((3*a^2 + 2*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 + 2*a*b - b^2)*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) + 2*((3*a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + a*b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f)]","A",0
127,0,0,0,0.858789," ","integrate(sin(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}, x\right)"," ",0,"integral((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(-b*cos(f*x + e)^2 + a + b), x)","F",0
128,0,0,0,0.730561," ","integrate(sin(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(\cos\left(f x + e\right)^{2} - 1\right)}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*(cos(f*x + e)^2 - 1), x)","F",0
129,0,0,0,0.707098," ","integrate((a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b), x)","F",0
130,0,0,0,0.774286," ","integrate(csc(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \csc\left(f x + e\right)^{2}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*csc(f*x + e)^2, x)","F",0
131,0,0,0,0.742239," ","integrate(csc(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \csc\left(f x + e\right)^{4}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*csc(f*x + e)^4, x)","F",0
132,1,579,0,2.943905," ","integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right)} \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right) - 8 \, {\left(8 \, b^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(7 \, a b^{2} + 13 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{2} b + 12 \, a b^{2} + 11 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{384 \, b^{2} f}, -\frac{3 \, {\left(a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right)} \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(8 \, b^{3} \cos\left(f x + e\right)^{5} - 2 \, {\left(7 \, a b^{2} + 13 \, b^{3}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(a^{2} b + 12 \, a b^{2} + 11 \, b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{192 \, b^{2} f}\right]"," ",0,"[1/384*(3*(a^3 - 3*a^2*b - 9*a*b^2 - 5*b^3)*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)) - 8*(8*b^3*cos(f*x + e)^5 - 2*(7*a*b^2 + 13*b^3)*cos(f*x + e)^3 + 3*(a^2*b + 12*a*b^2 + 11*b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(b^2*f), -1/192*(3*(a^3 - 3*a^2*b - 9*a*b^2 - 5*b^3)*sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))) + 4*(8*b^3*cos(f*x + e)^5 - 2*(7*a*b^2 + 13*b^3)*cos(f*x + e)^3 + 3*(a^2*b + 12*a*b^2 + 11*b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(b^2*f)]","A",0
133,1,495,0,1.253279," ","integrate(sin(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right) - 8 \, {\left(2 \, b^{2} \cos\left(f x + e\right)^{3} - 5 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{64 \, b f}, \frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, {\left(2 \, b^{2} \cos\left(f x + e\right)^{3} - 5 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{32 \, b f}\right]"," ",0,"[-1/64*(3*(a^2 + 2*a*b + b^2)*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)) - 8*(2*b^2*cos(f*x + e)^3 - 5*(a*b + b^2)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(b*f), 1/32*(3*(a^2 + 2*a*b + b^2)*sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))) + 4*(2*b^2*cos(f*x + e)^3 - 5*(a*b + b^2)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(b*f)]","A",0
134,1,1282,0,1.200255," ","integrate(csc(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \cos\left(f x + e\right) - {\left(3 \, a + b\right)} \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right) - 4 \, a^{\frac{3}{2}} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{16 \, f}, \frac{8 \, \sqrt{-a} a \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) - 8 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \cos\left(f x + e\right) + {\left(3 \, a + b\right)} \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right)}{16 \, f}, \frac{{\left(3 \, a + b\right)} \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \cos\left(f x + e\right) + 2 \, a^{\frac{3}{2}} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, f}, \frac{4 \, \sqrt{-a} a \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) + {\left(3 \, a + b\right)} \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \cos\left(f x + e\right)}{8 \, f}\right]"," ",0,"[-1/16*(8*sqrt(-b*cos(f*x + e)^2 + a + b)*b*cos(f*x + e) - (3*a + b)*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)) - 4*a^(3/2)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/f, 1/16*(8*sqrt(-a)*a*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) - 8*sqrt(-b*cos(f*x + e)^2 + a + b)*b*cos(f*x + e) + (3*a + b)*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)))/f, 1/8*((3*a + b)*sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))) - 4*sqrt(-b*cos(f*x + e)^2 + a + b)*b*cos(f*x + e) + 2*a^(3/2)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/f, 1/8*(4*sqrt(-a)*a*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) + (3*a + b)*sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))) - 4*sqrt(-b*cos(f*x + e)^2 + a + b)*b*cos(f*x + e))/f]","B",0
135,1,1449,0,1.392757," ","integrate(csc(f*x+e)^3*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a \cos\left(f x + e\right) + {\left(b \cos\left(f x + e\right)^{2} - b\right)} \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right) + {\left({\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{2} - a - 3 \, b\right)} \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}, \frac{2 \, {\left({\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{2} - a - 3 \, b\right)} \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a \cos\left(f x + e\right) + {\left(b \cos\left(f x + e\right)^{2} - b\right)} \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right)}{8 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}, \frac{2 \, {\left(b \cos\left(f x + e\right)^{2} - b\right)} \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a \cos\left(f x + e\right) + {\left({\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{2} - a - 3 \, b\right)} \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}, \frac{{\left({\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{2} - a - 3 \, b\right)} \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) + {\left(b \cos\left(f x + e\right)^{2} - b\right)} \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a \cos\left(f x + e\right)}{4 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}\right]"," ",0,"[1/8*(4*sqrt(-b*cos(f*x + e)^2 + a + b)*a*cos(f*x + e) + (b*cos(f*x + e)^2 - b)*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)) + ((a + 3*b)*cos(f*x + e)^2 - a - 3*b)*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/(f*cos(f*x + e)^2 - f), 1/8*(2*((a + 3*b)*cos(f*x + e)^2 - a - 3*b)*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) + 4*sqrt(-b*cos(f*x + e)^2 + a + b)*a*cos(f*x + e) + (b*cos(f*x + e)^2 - b)*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)))/(f*cos(f*x + e)^2 - f), 1/8*(2*(b*cos(f*x + e)^2 - b)*sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))) + 4*sqrt(-b*cos(f*x + e)^2 + a + b)*a*cos(f*x + e) + ((a + 3*b)*cos(f*x + e)^2 - a - 3*b)*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/(f*cos(f*x + e)^2 - f), 1/4*(((a + 3*b)*cos(f*x + e)^2 - a - 3*b)*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) + (b*cos(f*x + e)^2 - b)*sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))) + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*a*cos(f*x + e))/(f*cos(f*x + e)^2 - f)]","B",0
136,1,484,0,0.985977," ","integrate(csc(f*x+e)^5*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) + 4 \, {\left({\left(3 \, a^{2} + 5 \, a b\right)} \cos\left(f x + e\right)^{3} - 5 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{32 \, {\left(a f \cos\left(f x + e\right)^{4} - 2 \, a f \cos\left(f x + e\right)^{2} + a f\right)}}, \frac{3 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) + 2 \, {\left({\left(3 \, a^{2} + 5 \, a b\right)} \cos\left(f x + e\right)^{3} - 5 \, {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{16 \, {\left(a f \cos\left(f x + e\right)^{4} - 2 \, a f \cos\left(f x + e\right)^{2} + a f\right)}}\right]"," ",0,"[1/32*(3*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) + 4*((3*a^2 + 5*a*b)*cos(f*x + e)^3 - 5*(a^2 + a*b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(a*f*cos(f*x + e)^4 - 2*a*f*cos(f*x + e)^2 + a*f), 1/16*(3*((a^2 + 2*a*b + b^2)*cos(f*x + e)^4 - 2*(a^2 + 2*a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) + 2*((3*a^2 + 5*a*b)*cos(f*x + e)^3 - 5*(a^2 + a*b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(a*f*cos(f*x + e)^4 - 2*a*f*cos(f*x + e)^2 + a*f)]","A",0
137,1,752,0,2.923829," ","integrate(csc(f*x+e)^7*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{6} - 3 \, {\left(5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - 5 \, a^{3} - 9 \, a^{2} b - 3 \, a b^{2} + b^{3} + 3 \, {\left(5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left({\left(15 \, a^{3} + 22 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(20 \, a^{3} + 29 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(11 \, a^{3} + 12 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{192 \, {\left(a^{2} f \cos\left(f x + e\right)^{6} - 3 \, a^{2} f \cos\left(f x + e\right)^{4} + 3 \, a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)}}, \frac{3 \, {\left({\left(5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{6} - 3 \, {\left(5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - 5 \, a^{3} - 9 \, a^{2} b - 3 \, a b^{2} + b^{3} + 3 \, {\left(5 \, a^{3} + 9 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) + 2 \, {\left({\left(15 \, a^{3} + 22 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{5} - 2 \, {\left(20 \, a^{3} + 29 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(11 \, a^{3} + 12 \, a^{2} b + a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{96 \, {\left(a^{2} f \cos\left(f x + e\right)^{6} - 3 \, a^{2} f \cos\left(f x + e\right)^{4} + 3 \, a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)}}\right]"," ",0,"[-1/192*(3*((5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^6 - 3*(5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - 5*a^3 - 9*a^2*b - 3*a*b^2 + b^3 + 3*(5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^2)*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 4*((15*a^3 + 22*a^2*b + 3*a*b^2)*cos(f*x + e)^5 - 2*(20*a^3 + 29*a^2*b + 3*a*b^2)*cos(f*x + e)^3 + 3*(11*a^3 + 12*a^2*b + a*b^2)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^2*f*cos(f*x + e)^6 - 3*a^2*f*cos(f*x + e)^4 + 3*a^2*f*cos(f*x + e)^2 - a^2*f), 1/96*(3*((5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^6 - 3*(5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^4 - 5*a^3 - 9*a^2*b - 3*a*b^2 + b^3 + 3*(5*a^3 + 9*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^2)*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) + 2*((15*a^3 + 22*a^2*b + 3*a*b^2)*cos(f*x + e)^5 - 2*(20*a^3 + 29*a^2*b + 3*a*b^2)*cos(f*x + e)^3 + 3*(11*a^3 + 12*a^2*b + a*b^2)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^2*f*cos(f*x + e)^6 - 3*a^2*f*cos(f*x + e)^4 + 3*a^2*f*cos(f*x + e)^2 - a^2*f)]","A",0
138,0,0,0,0.873433," ","integrate(sin(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b \cos\left(f x + e\right)^{6} - {\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{4} + {\left(2 \, a + 3 \, b\right)} \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^6 - (a + 3*b)*cos(f*x + e)^4 + (2*a + 3*b)*cos(f*x + e)^2 - a - b)*sqrt(-b*cos(f*x + e)^2 + a + b), x)","F",0
139,0,0,0,0.828320," ","integrate(sin(f*x+e)^2*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - {\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}, x\right)"," ",0,"integral((b*cos(f*x + e)^4 - (a + 2*b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b), x)","F",0
140,0,0,0,0.649260," ","integrate((a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{\frac{3}{2}}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^(3/2), x)","F",0
141,0,0,0,0.596253," ","integrate(csc(f*x+e)^2*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \csc\left(f x + e\right)^{2}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^2 - a - b)*sqrt(-b*cos(f*x + e)^2 + a + b)*csc(f*x + e)^2, x)","F",0
142,0,0,0,0.849960," ","integrate(csc(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \csc\left(f x + e\right)^{4}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^2 - a - b)*sqrt(-b*cos(f*x + e)^2 + a + b)*csc(f*x + e)^4, x)","F",0
143,0,0,0,0.978707," ","integrate((a+b*sin(d*x+c)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(d x + c\right)^{2} + a + b}, x\right)"," ",0,"integral((b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(d*x + c)^2 + a + b), x)","F",0
144,1,438,0,0.902475," ","integrate(sin(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{8 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \cos\left(f x + e\right) - {\left(a - b\right)} \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right)}{16 \, b^{2} f}, -\frac{{\left(a - b\right)} \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right) + 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \cos\left(f x + e\right)}{8 \, b^{2} f}\right]"," ",0,"[-1/16*(8*sqrt(-b*cos(f*x + e)^2 + a + b)*b*cos(f*x + e) - (a - b)*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)))/(b^2*f), -1/8*((a - b)*sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))) + 4*sqrt(-b*cos(f*x + e)^2 + a + b)*b*cos(f*x + e))/(b^2*f)]","B",0
145,1,370,0,0.816921," ","integrate(sin(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right)}{8 \, b f}, \frac{\arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right)}{4 \, \sqrt{b} f}\right]"," ",0,"[-1/8*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b))/(b*f), 1/4*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)))/(sqrt(b)*f)]","B",0
146,1,219,0,0.798370," ","integrate(csc(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{4 \, \sqrt{a} f}, \frac{\sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right)}{2 \, a f}\right]"," ",0,"[1/4*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1))/(sqrt(a)*f), 1/2*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e)))/(a*f)]","B",0
147,1,347,0,0.785858," ","integrate(csc(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a \cos\left(f x + e\right) - {\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - a + b\right)} \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{8 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)}}, \frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} - a + b\right)} \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a \cos\left(f x + e\right)}{4 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)}}\right]"," ",0,"[1/8*(4*sqrt(-b*cos(f*x + e)^2 + a + b)*a*cos(f*x + e) - ((a - b)*cos(f*x + e)^2 - a + b)*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/(a^2*f*cos(f*x + e)^2 - a^2*f), 1/4*(((a - b)*cos(f*x + e)^2 - a + b)*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*a*cos(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f)]","A",0
148,0,0,0,0.946135," ","integrate(sin(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(-b*cos(f*x + e)^2 + a + b)/(b*cos(f*x + e)^2 - a - b), x)","F",0
149,0,0,0,0.755991," ","integrate(sin(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(\cos\left(f x + e\right)^{2} - 1\right)}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*(cos(f*x + e)^2 - 1)/(b*cos(f*x + e)^2 - a - b), x)","F",0
150,0,0,0,0.495167," ","integrate(1/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)/(b*cos(f*x + e)^2 - a - b), x)","F",0
151,0,0,0,0.480896," ","integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \csc\left(f x + e\right)^{2}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*csc(f*x + e)^2/(b*cos(f*x + e)^2 - a - b), x)","F",0
152,0,0,0,0.603625," ","integrate(csc(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \csc\left(f x + e\right)^{4}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*csc(f*x + e)^4/(b*cos(f*x + e)^2 - a - b), x)","F",0
153,1,564,0,0.859185," ","integrate(sin(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a b \cos\left(f x + e\right) + {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right)}{8 \, {\left({\left(a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f\right)}}, -\frac{4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a b \cos\left(f x + e\right) - {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right)}{4 \, {\left({\left(a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f\right)}}\right]"," ",0,"[-1/8*(8*sqrt(-b*cos(f*x + e)^2 + a + b)*a*b*cos(f*x + e) + ((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)))/((a*b^3 + b^4)*f*cos(f*x + e)^2 - (a^2*b^2 + 2*a*b^3 + b^4)*f), -1/4*(4*sqrt(-b*cos(f*x + e)^2 + a + b)*a*b*cos(f*x + e) - ((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))))/((a*b^3 + b^4)*f*cos(f*x + e)^2 - (a^2*b^2 + 2*a*b^3 + b^4)*f)]","B",0
154,1,57,0,0.547417," ","integrate(sin(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cos\left(f x + e\right)}{{\left(a b + b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{2} + 2 \, a b + b^{2}\right)} f}"," ",0,"sqrt(-b*cos(f*x + e)^2 + a + b)*cos(f*x + e)/((a*b + b^2)*f*cos(f*x + e)^2 - (a^2 + 2*a*b + b^2)*f)","A",0
155,1,422,0,0.808865," ","integrate(csc(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a b \cos\left(f x + e\right) - {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right)}{4 \, {\left({\left(a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f\right)}}, -\frac{2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a b \cos\left(f x + e\right) - {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right)}{2 \, {\left({\left(a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f\right)}}\right]"," ",0,"[-1/4*(4*sqrt(-b*cos(f*x + e)^2 + a + b)*a*b*cos(f*x + e) - ((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)))/((a^3*b + a^2*b^2)*f*cos(f*x + e)^2 - (a^4 + 2*a^3*b + a^2*b^2)*f), -1/2*(2*sqrt(-b*cos(f*x + e)^2 + a + b)*a*b*cos(f*x + e) - ((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))))/((a^3*b + a^2*b^2)*f*cos(f*x + e)^2 - (a^4 + 2*a^3*b + a^2*b^2)*f)]","B",0
156,1,634,0,1.398089," ","integrate(csc(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(a^{2} b - 2 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{3} - a^{2} b - 5 \, a b^{2} - 3 \, b^{3} - {\left(a^{3} - 7 \, a b^{2} - 6 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left({\left(a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 2 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{8 \, {\left({\left(a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f\right)}}, \frac{{\left({\left(a^{2} b - 2 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + a^{3} - a^{2} b - 5 \, a b^{2} - 3 \, b^{3} - {\left(a^{3} - 7 \, a b^{2} - 6 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) + 2 \, {\left({\left(a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 2 \, a^{2} b + 3 \, a b^{2}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{4 \, {\left({\left(a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f\right)}}\right]"," ",0,"[-1/8*(((a^2*b - 2*a*b^2 - 3*b^3)*cos(f*x + e)^4 + a^3 - a^2*b - 5*a*b^2 - 3*b^3 - (a^3 - 7*a*b^2 - 6*b^3)*cos(f*x + e)^2)*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 + 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 4*((a^2*b + 3*a*b^2)*cos(f*x + e)^3 - (a^3 + 2*a^2*b + 3*a*b^2)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^4*b + a^3*b^2)*f*cos(f*x + e)^4 - (a^5 + 3*a^4*b + 2*a^3*b^2)*f*cos(f*x + e)^2 + (a^5 + 2*a^4*b + a^3*b^2)*f), 1/4*(((a^2*b - 2*a*b^2 - 3*b^3)*cos(f*x + e)^4 + a^3 - a^2*b - 5*a*b^2 - 3*b^3 - (a^3 - 7*a*b^2 - 6*b^3)*cos(f*x + e)^2)*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) + 2*((a^2*b + 3*a*b^2)*cos(f*x + e)^3 - (a^3 + 2*a^2*b + 3*a*b^2)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^4*b + a^3*b^2)*f*cos(f*x + e)^4 - (a^5 + 3*a^4*b + 2*a^3*b^2)*f*cos(f*x + e)^2 + (a^5 + 2*a^4*b + a^3*b^2)*f)]","B",0
157,0,0,0,0.681840," ","integrate(sin(f*x+e)^6/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(\cos\left(f x + e\right)^{6} - 3 \, \cos\left(f x + e\right)^{4} + 3 \, \cos\left(f x + e\right)^{2} - 1\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(-(cos(f*x + e)^6 - 3*cos(f*x + e)^4 + 3*cos(f*x + e)^2 - 1)*sqrt(-b*cos(f*x + e)^2 + a + b)/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
158,0,0,0,0.725871," ","integrate(sin(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(-b*cos(f*x + e)^2 + a + b)/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
159,0,0,0,0.555331," ","integrate(sin(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(\cos\left(f x + e\right)^{2} - 1\right)}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*(cos(f*x + e)^2 - 1)/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
160,0,0,0,0.550299," ","integrate(1/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
161,0,0,0,0.696452," ","integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \csc\left(f x + e\right)^{2}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*csc(f*x + e)^2/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
162,1,885,0,1.517911," ","integrate(sin(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 2 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{6} + 160 \, {\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{7} - 24 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{5} + 10 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}\right) + 8 \, {\left(2 \, {\left(2 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{24 \, {\left({\left(a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{3} + 4 \, a^{3} b^{4} + 6 \, a^{2} b^{5} + 4 \, a b^{6} + b^{7}\right)} f\right)}}, \frac{3 \, {\left({\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 2 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{5} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{3} + {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)\right)}}\right) - 4 \, {\left(2 \, {\left(2 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 2 \, a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{12 \, {\left({\left(a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{3} + 4 \, a^{3} b^{4} + 6 \, a^{2} b^{5} + 4 \, a b^{6} + b^{7}\right)} f\right)}}\right]"," ",0,"[-1/24*(3*((a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(-b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + b^4)*cos(f*x + e)^6 + 160*(a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^7 - 24*(a*b^2 + b^3)*cos(f*x + e)^5 + 10*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)) + 8*(2*(2*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^3 - 3*(a^3*b + 3*a^2*b^2 + 2*a*b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^2*b^5 + 2*a*b^6 + b^7)*f*cos(f*x + e)^4 - 2*(a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + b^7)*f*cos(f*x + e)^2 + (a^4*b^3 + 4*a^3*b^4 + 6*a^2*b^5 + 4*a*b^6 + b^7)*f), 1/12*(3*((a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)/(2*b^3*cos(f*x + e)^5 - 3*(a*b^2 + b^3)*cos(f*x + e)^3 + (a^2*b + 2*a*b^2 + b^3)*cos(f*x + e))) - 4*(2*(2*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^3 - 3*(a^3*b + 3*a^2*b^2 + 2*a*b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^2*b^5 + 2*a*b^6 + b^7)*f*cos(f*x + e)^4 - 2*(a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + b^7)*f*cos(f*x + e)^2 + (a^4*b^3 + 4*a^3*b^4 + 6*a^2*b^5 + 4*a*b^6 + b^7)*f)]","B",0
163,1,137,0,1.090309," ","integrate(sin(f*x+e)^3/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left({\left(a + 3 \, b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{3 \, {\left({\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} f\right)}}"," ",0,"1/3*((a + 3*b)*cos(f*x + e)^3 - 3*(a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)/((a^2*b^2 + 2*a*b^3 + b^4)*f*cos(f*x + e)^4 - 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f*cos(f*x + e)^2 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*f)","A",0
164,1,134,0,0.791438," ","integrate(sin(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left(2 \, b \cos\left(f x + e\right)^{3} - 3 \, {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{3 \, {\left({\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} f\right)}}"," ",0,"1/3*(2*b*cos(f*x + e)^3 - 3*(a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)/((a^2*b^2 + 2*a*b^3 + b^4)*f*cos(f*x + e)^4 - 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f*cos(f*x + e)^2 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*f)","B",0
165,1,752,0,1.283494," ","integrate(csc(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 2 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(3 \, a^{2} + 2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a - b\right)} \cos\left(f x + e\right)^{3} + {\left(a + b\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1}\right) - 4 \, {\left({\left(5 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{12 \, {\left({\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}, \frac{3 \, {\left({\left(a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 2 \, {\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{2 \, {\left(a b \cos\left(f x + e\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(f x + e\right)\right)}}\right) - 2 \, {\left({\left(5 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(f x + e\right)\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{6 \, {\left({\left(a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} f\right)}}\right]"," ",0,"[1/12*(3*((a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(a)*log(2*((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 2*(3*a^2 + 2*a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3 + (a + b)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) + a^2 + 2*a*b + b^2)/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)) - 4*((5*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^3 - 3*(2*a^3*b + 3*a^2*b^2 + a*b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^4 - 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^2 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f), 1/6*(3*((a^2*b^2 + 2*a*b^3 + b^4)*cos(f*x + e)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 2*(a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(-a)*arctan(-1/2*((a - b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(a*b*cos(f*x + e)^3 - (a^2 + a*b)*cos(f*x + e))) - 2*((5*a^2*b^2 + 3*a*b^3)*cos(f*x + e)^3 - 3*(2*a^3*b + 3*a^2*b^2 + a*b^3)*cos(f*x + e))*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^5*b^2 + 2*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^4 - 2*(a^6*b + 3*a^5*b^2 + 3*a^4*b^3 + a^3*b^4)*f*cos(f*x + e)^2 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*f)]","B",0
166,0,0,0,0.764988," ","integrate(sin(f*x+e)^6/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(\cos\left(f x + e\right)^{6} - 3 \, \cos\left(f x + e\right)^{4} + 3 \, \cos\left(f x + e\right)^{2} - 1\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral((cos(f*x + e)^6 - 3*cos(f*x + e)^4 + 3*cos(f*x + e)^2 - 1)*sqrt(-b*cos(f*x + e)^2 + a + b)/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
167,0,0,0,0.624349," ","integrate(sin(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sqrt(-b*cos(f*x + e)^2 + a + b)/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
168,0,0,0,0.778943," ","integrate(sin(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(\cos\left(f x + e\right)^{2} - 1\right)}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*(cos(f*x + e)^2 - 1)/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
169,0,0,0,0.722593," ","integrate(1/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
170,0,0,0,0.864307," ","integrate(csc(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \csc\left(f x + e\right)^{2}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*csc(f*x + e)^2/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
171,0,0,0,0.708216," ","integrate((d*sin(f*x+e))^m*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \left(d \sin\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*(d*sin(f*x + e))^m, x)","F",0
172,0,0,0,0.646351," ","integrate(sin(f*x+e)^5*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1\right)} {\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \sin\left(f x + e\right), x\right)"," ",0,"integral((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*(-b*cos(f*x + e)^2 + a + b)^p*sin(f*x + e), x)","F",0
173,0,0,0,0.775338," ","integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(f x + e\right)^{2} - 1\right)} {\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \sin\left(f x + e\right), x\right)"," ",0,"integral(-(cos(f*x + e)^2 - 1)*(-b*cos(f*x + e)^2 + a + b)^p*sin(f*x + e), x)","F",0
174,0,0,0,0.686602," ","integrate(sin(f*x+e)*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \sin\left(f x + e\right), x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*sin(f*x + e), x)","F",0
175,0,0,0,0.778621," ","integrate(csc(f*x+e)*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \csc\left(f x + e\right), x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*csc(f*x + e), x)","F",0
176,0,0,0,0.677135," ","integrate(csc(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \csc\left(f x + e\right)^{3}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*csc(f*x + e)^3, x)","F",0
177,0,0,0,0.737101," ","integrate(csc(f*x+e)^5*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \csc\left(f x + e\right)^{5}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*csc(f*x + e)^5, x)","F",0
178,0,0,0,0.559762," ","integrate(sin(f*x+e)^4*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(\cos\left(f x + e\right)^{4} - 2 \, \cos\left(f x + e\right)^{2} + 1\right)} {\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p}, x\right)"," ",0,"integral((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*(-b*cos(f*x + e)^2 + a + b)^p, x)","F",0
179,0,0,0,0.709259," ","integrate(sin(f*x+e)^2*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(f x + e\right)^{2} - 1\right)} {\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p}, x\right)"," ",0,"integral(-(cos(f*x + e)^2 - 1)*(-b*cos(f*x + e)^2 + a + b)^p, x)","F",0
180,0,0,0,0.937088," ","integrate(csc(f*x+e)^2*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \csc\left(f x + e\right)^{2}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*csc(f*x + e)^2, x)","F",0
181,0,0,0,0.718611," ","integrate(csc(f*x+e)^4*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \csc\left(f x + e\right)^{4}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*csc(f*x + e)^4, x)","F",0
182,-1,0,0,0.000000," ","integrate(sin(d*x+c)^7/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
183,-1,0,0,0.000000," ","integrate(sin(d*x+c)^5/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
184,-1,0,0,0.000000," ","integrate(sin(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
185,-1,0,0,0.000000," ","integrate(sin(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
186,-1,0,0,0.000000," ","integrate(csc(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
187,-1,0,0,0.000000," ","integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
188,-1,0,0,0.000000," ","integrate(csc(d*x+c)^5/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
189,-1,0,0,0.000000," ","integrate(sin(d*x+c)^6/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
190,-1,0,0,0.000000," ","integrate(sin(d*x+c)^4/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
191,-1,0,0,0.000000," ","integrate(sin(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
192,-1,0,0,0.000000," ","integrate(1/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
193,-1,0,0,0.000000," ","integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
194,-1,0,0,0.000000," ","integrate(csc(d*x+c)^4/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
195,1,872,0,1.345027," ","integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{12 \, b \cos\left(d x + c\right)^{5} - 15 \, b^{2} d \sqrt{-\frac{{\left(a b^{4} - b^{5}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{2} + a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}} \log\left(a^{5} \cos\left(d x + c\right) + {\left(a^{4} b^{2} d - {\left(a b^{7} - b^{8}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{3}\right)} \sqrt{-\frac{{\left(a b^{4} - b^{5}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{2} + a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}}\right) + 15 \, b^{2} d \sqrt{\frac{{\left(a b^{4} - b^{5}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{2} - a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}} \log\left(a^{5} \cos\left(d x + c\right) - {\left(a^{4} b^{2} d + {\left(a b^{7} - b^{8}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{3}\right)} \sqrt{\frac{{\left(a b^{4} - b^{5}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{2} - a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}}\right) + 15 \, b^{2} d \sqrt{-\frac{{\left(a b^{4} - b^{5}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{2} + a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}} \log\left(-a^{5} \cos\left(d x + c\right) + {\left(a^{4} b^{2} d - {\left(a b^{7} - b^{8}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{3}\right)} \sqrt{-\frac{{\left(a b^{4} - b^{5}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{2} + a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}}\right) - 15 \, b^{2} d \sqrt{\frac{{\left(a b^{4} - b^{5}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{2} - a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}} \log\left(-a^{5} \cos\left(d x + c\right) - {\left(a^{4} b^{2} d + {\left(a b^{7} - b^{8}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{3}\right)} \sqrt{\frac{{\left(a b^{4} - b^{5}\right)} \sqrt{\frac{a^{7}}{{\left(a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right)} d^{4}}} d^{2} - a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}}\right) - 40 \, b \cos\left(d x + c\right)^{3} + 60 \, {\left(a + b\right)} \cos\left(d x + c\right)}{60 \, b^{2} d}"," ",0,"1/60*(12*b*cos(d*x + c)^5 - 15*b^2*d*sqrt(-((a*b^4 - b^5)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^2 + a^3)/((a*b^4 - b^5)*d^2))*log(a^5*cos(d*x + c) + (a^4*b^2*d - (a*b^7 - b^8)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^3)*sqrt(-((a*b^4 - b^5)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^2 + a^3)/((a*b^4 - b^5)*d^2))) + 15*b^2*d*sqrt(((a*b^4 - b^5)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^2 - a^3)/((a*b^4 - b^5)*d^2))*log(a^5*cos(d*x + c) - (a^4*b^2*d + (a*b^7 - b^8)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^3)*sqrt(((a*b^4 - b^5)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^2 - a^3)/((a*b^4 - b^5)*d^2))) + 15*b^2*d*sqrt(-((a*b^4 - b^5)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^2 + a^3)/((a*b^4 - b^5)*d^2))*log(-a^5*cos(d*x + c) + (a^4*b^2*d - (a*b^7 - b^8)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^3)*sqrt(-((a*b^4 - b^5)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^2 + a^3)/((a*b^4 - b^5)*d^2))) - 15*b^2*d*sqrt(((a*b^4 - b^5)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^2 - a^3)/((a*b^4 - b^5)*d^2))*log(-a^5*cos(d*x + c) - (a^4*b^2*d + (a*b^7 - b^8)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^3)*sqrt(((a*b^4 - b^5)*sqrt(a^7/((a^2*b^9 - 2*a*b^10 + b^11)*d^4))*d^2 - a^3)/((a*b^4 - b^5)*d^2))) - 40*b*cos(d*x + c)^3 + 60*(a + b)*cos(d*x + c))/(b^2*d)","B",0
196,1,849,0,1.239032," ","integrate(sin(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{3 \, b d \sqrt{-\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} + a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}} \log\left(a^{3} \cos\left(d x + c\right) + {\left(a^{2} b^{2} d - {\left(a b^{5} - b^{6}\right)} d^{3} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}}\right)} \sqrt{-\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} + a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}}\right) - 3 \, b d \sqrt{\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} - a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}} \log\left(a^{3} \cos\left(d x + c\right) - {\left(a^{2} b^{2} d + {\left(a b^{5} - b^{6}\right)} d^{3} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}}\right)} \sqrt{\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} - a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}}\right) - 3 \, b d \sqrt{-\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} + a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}} \log\left(-a^{3} \cos\left(d x + c\right) + {\left(a^{2} b^{2} d - {\left(a b^{5} - b^{6}\right)} d^{3} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}}\right)} \sqrt{-\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} + a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}}\right) + 3 \, b d \sqrt{\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} - a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}} \log\left(-a^{3} \cos\left(d x + c\right) - {\left(a^{2} b^{2} d + {\left(a b^{5} - b^{6}\right)} d^{3} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}}\right)} \sqrt{\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} - a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}}\right) - 4 \, \cos\left(d x + c\right)^{3} + 12 \, \cos\left(d x + c\right)}{12 \, b d}"," ",0,"1/12*(3*b*d*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(a^3*cos(d*x + c) + (a^2*b^2*d - (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) - 3*b*d*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(a^3*cos(d*x + c) - (a^2*b^2*d + (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))) - 3*b*d*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(-a^3*cos(d*x + c) + (a^2*b^2*d - (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) + 3*b*d*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(-a^3*cos(d*x + c) - (a^2*b^2*d + (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))) - 4*cos(d*x + c)^3 + 12*cos(d*x + c))/(b*d)","B",0
197,1,815,0,1.389029," ","integrate(sin(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{b d \sqrt{-\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} \log\left(a^{2} \cos\left(d x + c\right) - {\left({\left(a b^{4} - b^{5}\right)} d^{3} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - a^{2} b d\right)} \sqrt{-\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + a}{{\left(a b^{2} - b^{3}\right)} d^{2}}}\right) - b d \sqrt{\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} \log\left(a^{2} \cos\left(d x + c\right) - {\left({\left(a b^{4} - b^{5}\right)} d^{3} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + a^{2} b d\right)} \sqrt{\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - a}{{\left(a b^{2} - b^{3}\right)} d^{2}}}\right) - b d \sqrt{-\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} \log\left(-a^{2} \cos\left(d x + c\right) - {\left({\left(a b^{4} - b^{5}\right)} d^{3} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - a^{2} b d\right)} \sqrt{-\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + a}{{\left(a b^{2} - b^{3}\right)} d^{2}}}\right) + b d \sqrt{\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} \log\left(-a^{2} \cos\left(d x + c\right) - {\left({\left(a b^{4} - b^{5}\right)} d^{3} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + a^{2} b d\right)} \sqrt{\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - a}{{\left(a b^{2} - b^{3}\right)} d^{2}}}\right) - 4 \, \cos\left(d x + c\right)}{4 \, b d}"," ",0,"-1/4*(b*d*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a)/((a*b^2 - b^3)*d^2))*log(a^2*cos(d*x + c) - ((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2*b*d)*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a)/((a*b^2 - b^3)*d^2))) - b*d*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a)/((a*b^2 - b^3)*d^2))*log(a^2*cos(d*x + c) - ((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2*b*d)*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a)/((a*b^2 - b^3)*d^2))) - b*d*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a)/((a*b^2 - b^3)*d^2))*log(-a^2*cos(d*x + c) - ((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2*b*d)*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a)/((a*b^2 - b^3)*d^2))) + b*d*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a)/((a*b^2 - b^3)*d^2))*log(-a^2*cos(d*x + c) - ((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2*b*d)*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a)/((a*b^2 - b^3)*d^2))) - 4*cos(d*x + c))/(b*d)","B",0
198,1,703,0,1.292116," ","integrate(sin(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{1}{4} \, \sqrt{-\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + 1}{{\left(a b - b^{2}\right)} d^{2}}} \log\left(-{\left({\left(a b^{2} - b^{3}\right)} d^{3} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - b d\right)} \sqrt{-\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + 1}{{\left(a b - b^{2}\right)} d^{2}}} + \cos\left(d x + c\right)\right) - \frac{1}{4} \, \sqrt{-\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + 1}{{\left(a b - b^{2}\right)} d^{2}}} \log\left(-{\left({\left(a b^{2} - b^{3}\right)} d^{3} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - b d\right)} \sqrt{-\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + 1}{{\left(a b - b^{2}\right)} d^{2}}} - \cos\left(d x + c\right)\right) - \frac{1}{4} \, \sqrt{\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - 1}{{\left(a b - b^{2}\right)} d^{2}}} \log\left(-{\left({\left(a b^{2} - b^{3}\right)} d^{3} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + b d\right)} \sqrt{\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - 1}{{\left(a b - b^{2}\right)} d^{2}}} + \cos\left(d x + c\right)\right) + \frac{1}{4} \, \sqrt{\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - 1}{{\left(a b - b^{2}\right)} d^{2}}} \log\left(-{\left({\left(a b^{2} - b^{3}\right)} d^{3} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + b d\right)} \sqrt{\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - 1}{{\left(a b - b^{2}\right)} d^{2}}} - \cos\left(d x + c\right)\right)"," ",0,"1/4*sqrt(-((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2))*log(-((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - b*d)*sqrt(-((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2)) + cos(d*x + c)) - 1/4*sqrt(-((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2))*log(-((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - b*d)*sqrt(-((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2)) - cos(d*x + c)) - 1/4*sqrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1)/((a*b - b^2)*d^2))*log(-((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + b*d)*sqrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1)/((a*b - b^2)*d^2)) + cos(d*x + c)) + 1/4*sqrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1)/((a*b - b^2)*d^2))*log(-((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + b*d)*sqrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1)/((a*b - b^2)*d^2)) - cos(d*x + c))","B",0
199,1,703,0,0.992487," ","integrate(sin(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{1}{4} \, \sqrt{-\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + 1}{{\left(a^{2} - a b\right)} d^{2}}} \log\left(-{\left({\left(a^{2} b - a b^{2}\right)} d^{3} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - a d\right)} \sqrt{-\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + 1}{{\left(a^{2} - a b\right)} d^{2}}} + \cos\left(d x + c\right)\right) + \frac{1}{4} \, \sqrt{-\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + 1}{{\left(a^{2} - a b\right)} d^{2}}} \log\left(-{\left({\left(a^{2} b - a b^{2}\right)} d^{3} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - a d\right)} \sqrt{-\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + 1}{{\left(a^{2} - a b\right)} d^{2}}} - \cos\left(d x + c\right)\right) + \frac{1}{4} \, \sqrt{\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - 1}{{\left(a^{2} - a b\right)} d^{2}}} \log\left(-{\left({\left(a^{2} b - a b^{2}\right)} d^{3} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + a d\right)} \sqrt{\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - 1}{{\left(a^{2} - a b\right)} d^{2}}} + \cos\left(d x + c\right)\right) - \frac{1}{4} \, \sqrt{\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - 1}{{\left(a^{2} - a b\right)} d^{2}}} \log\left(-{\left({\left(a^{2} b - a b^{2}\right)} d^{3} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + a d\right)} \sqrt{\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - 1}{{\left(a^{2} - a b\right)} d^{2}}} - \cos\left(d x + c\right)\right)"," ",0,"-1/4*sqrt(-((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1)/((a^2 - a*b)*d^2))*log(-((a^2*b - a*b^2)*d^3*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - a*d)*sqrt(-((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1)/((a^2 - a*b)*d^2)) + cos(d*x + c)) + 1/4*sqrt(-((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1)/((a^2 - a*b)*d^2))*log(-((a^2*b - a*b^2)*d^3*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - a*d)*sqrt(-((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1)/((a^2 - a*b)*d^2)) - cos(d*x + c)) + 1/4*sqrt(((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1)/((a^2 - a*b)*d^2))*log(-((a^2*b - a*b^2)*d^3*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + a*d)*sqrt(((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1)/((a^2 - a*b)*d^2)) + cos(d*x + c)) - 1/4*sqrt(((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1)/((a^2 - a*b)*d^2))*log(-((a^2*b - a*b^2)*d^3*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + a*d)*sqrt(((a^2 - a*b)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1)/((a^2 - a*b)*d^2)) - cos(d*x + c))","B",0
200,1,773,0,1.287642," ","integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{a d \sqrt{-\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + b}{{\left(a^{3} - a^{2} b\right)} d^{2}}} \log\left(b \cos\left(d x + c\right) - {\left({\left(a^{4} - a^{3} b\right)} d^{3} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - a b d\right)} \sqrt{-\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + b}{{\left(a^{3} - a^{2} b\right)} d^{2}}}\right) - a d \sqrt{\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - b}{{\left(a^{3} - a^{2} b\right)} d^{2}}} \log\left(b \cos\left(d x + c\right) - {\left({\left(a^{4} - a^{3} b\right)} d^{3} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + a b d\right)} \sqrt{\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - b}{{\left(a^{3} - a^{2} b\right)} d^{2}}}\right) - a d \sqrt{-\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + b}{{\left(a^{3} - a^{2} b\right)} d^{2}}} \log\left(-b \cos\left(d x + c\right) - {\left({\left(a^{4} - a^{3} b\right)} d^{3} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - a b d\right)} \sqrt{-\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + b}{{\left(a^{3} - a^{2} b\right)} d^{2}}}\right) + a d \sqrt{\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - b}{{\left(a^{3} - a^{2} b\right)} d^{2}}} \log\left(-b \cos\left(d x + c\right) - {\left({\left(a^{4} - a^{3} b\right)} d^{3} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + a b d\right)} \sqrt{\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - b}{{\left(a^{3} - a^{2} b\right)} d^{2}}}\right) - 2 \, \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + 2 \, \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{4 \, a d}"," ",0,"1/4*(a*d*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))*log(b*cos(d*x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - a*b*d)*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))) - a*d*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))*log(b*cos(d*x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + a*b*d)*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))) - a*d*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))*log(-b*cos(d*x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - a*b*d)*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))) + a*d*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))*log(-b*cos(d*x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + a*b*d)*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))) - 2*log(1/2*cos(d*x + c) + 1/2) + 2*log(-1/2*cos(d*x + c) + 1/2))/(a*d)","B",0
201,1,924,0,1.894295," ","integrate(csc(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{{\left(a d \cos\left(d x + c\right)^{2} - a d\right)} \sqrt{-\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}} \log\left(b^{2} \cos\left(d x + c\right) - {\left({\left(a^{5} - a^{4} b\right)} d^{3} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - a^{2} b d\right)} \sqrt{-\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}}\right) - {\left(a d \cos\left(d x + c\right)^{2} - a d\right)} \sqrt{\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}} \log\left(b^{2} \cos\left(d x + c\right) - {\left({\left(a^{5} - a^{4} b\right)} d^{3} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + a^{2} b d\right)} \sqrt{\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}}\right) - {\left(a d \cos\left(d x + c\right)^{2} - a d\right)} \sqrt{-\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}} \log\left(-b^{2} \cos\left(d x + c\right) - {\left({\left(a^{5} - a^{4} b\right)} d^{3} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - a^{2} b d\right)} \sqrt{-\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}}\right) + {\left(a d \cos\left(d x + c\right)^{2} - a d\right)} \sqrt{\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}} \log\left(-b^{2} \cos\left(d x + c\right) - {\left({\left(a^{5} - a^{4} b\right)} d^{3} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + a^{2} b d\right)} \sqrt{\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}}\right) + {\left(\cos\left(d x + c\right)^{2} - 1\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - {\left(\cos\left(d x + c\right)^{2} - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 2 \, \cos\left(d x + c\right)}{4 \, {\left(a d \cos\left(d x + c\right)^{2} - a d\right)}}"," ",0,"-1/4*((a*d*cos(d*x + c)^2 - a*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cos(d*x + c) - ((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - a^2*b*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))) - (a*d*cos(d*x + c)^2 - a*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(b^2*cos(d*x + c) - ((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + a^2*b*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))) - (a*d*cos(d*x + c)^2 - a*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(-b^2*cos(d*x + c) - ((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - a^2*b*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))) + (a*d*cos(d*x + c)^2 - a*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(-b^2*cos(d*x + c) - ((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + a^2*b*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))) + (cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2) - (cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c) + 1/2) - 2*cos(d*x + c))/(a*d*cos(d*x + c)^2 - a*d)","B",0
202,1,1089,0,1.719885," ","integrate(csc(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{6 \, a \cos\left(d x + c\right)^{3} + 4 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)} \sqrt{-\frac{{\left(a^{5} - a^{4} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} + b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}} \log\left(b^{4} \cos\left(d x + c\right) + {\left(a^{2} b^{3} d - {\left(a^{7} - a^{6} b\right)} d^{3} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}}\right)} \sqrt{-\frac{{\left(a^{5} - a^{4} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} + b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}}\right) - 4 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)} \sqrt{\frac{{\left(a^{5} - a^{4} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} - b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}} \log\left(b^{4} \cos\left(d x + c\right) - {\left(a^{2} b^{3} d + {\left(a^{7} - a^{6} b\right)} d^{3} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}}\right)} \sqrt{\frac{{\left(a^{5} - a^{4} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} - b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}}\right) - 4 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)} \sqrt{-\frac{{\left(a^{5} - a^{4} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} + b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}} \log\left(-b^{4} \cos\left(d x + c\right) + {\left(a^{2} b^{3} d - {\left(a^{7} - a^{6} b\right)} d^{3} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}}\right)} \sqrt{-\frac{{\left(a^{5} - a^{4} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} + b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}}\right) + 4 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)} \sqrt{\frac{{\left(a^{5} - a^{4} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} - b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}} \log\left(-b^{4} \cos\left(d x + c\right) - {\left(a^{2} b^{3} d + {\left(a^{7} - a^{6} b\right)} d^{3} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}}\right)} \sqrt{\frac{{\left(a^{5} - a^{4} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} - b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}}\right) - 10 \, a \cos\left(d x + c\right) - {\left({\left(3 \, a + 8 \, b\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a + 8 \, b\right)} \cos\left(d x + c\right)^{2} + 3 \, a + 8 \, b\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) + {\left({\left(3 \, a + 8 \, b\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(3 \, a + 8 \, b\right)} \cos\left(d x + c\right)^{2} + 3 \, a + 8 \, b\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{16 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)}}"," ",0,"1/16*(6*a*cos(d*x + c)^3 + 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(-((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))*log(b^4*cos(d*x + c) + (a^2*b^3*d - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(-((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))) - 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))*log(b^4*cos(d*x + c) - (a^2*b^3*d + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))) - 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(-((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))*log(-b^4*cos(d*x + c) + (a^2*b^3*d - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(-((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^3)/((a^5 - a^4*b)*d^2))) + 4*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))*log(-b^4*cos(d*x + c) - (a^2*b^3*d + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)))*sqrt(((a^5 - a^4*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^3)/((a^5 - a^4*b)*d^2))) - 10*a*cos(d*x + c) - ((3*a + 8*b)*cos(d*x + c)^4 - 2*(3*a + 8*b)*cos(d*x + c)^2 + 3*a + 8*b)*log(1/2*cos(d*x + c) + 1/2) + ((3*a + 8*b)*cos(d*x + c)^4 - 2*(3*a + 8*b)*cos(d*x + c)^2 + 3*a + 8*b)*log(-1/2*cos(d*x + c) + 1/2))/(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)","B",0
203,1,1311,0,1.599580," ","integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{b^{2} d \sqrt{-\frac{{\left(a b^{4} - b^{5}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} + a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}} \log\left(\frac{1}{4} \, a^{3} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, a^{3} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} b^{3} - a b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{3} - a b^{4}\right)} d^{2}\right)} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} + \frac{1}{2} \, {\left(a^{2} b^{2} d \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a b^{5} - b^{6}\right)} d^{3} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a b^{4} - b^{5}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} + a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}}\right) - b^{2} d \sqrt{-\frac{{\left(a b^{4} - b^{5}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} + a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}} \log\left(\frac{1}{4} \, a^{3} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, a^{3} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} b^{3} - a b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{3} - a b^{4}\right)} d^{2}\right)} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} - \frac{1}{2} \, {\left(a^{2} b^{2} d \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a b^{5} - b^{6}\right)} d^{3} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a b^{4} - b^{5}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} + a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}}\right) - b^{2} d \sqrt{\frac{{\left(a b^{4} - b^{5}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} - a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}} \log\left(-\frac{1}{4} \, a^{3} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, a^{3} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} b^{3} - a b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{3} - a b^{4}\right)} d^{2}\right)} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} + \frac{1}{2} \, {\left(a^{2} b^{2} d \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a b^{5} - b^{6}\right)} d^{3} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a b^{4} - b^{5}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} - a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}}\right) + b^{2} d \sqrt{\frac{{\left(a b^{4} - b^{5}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} - a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}} \log\left(-\frac{1}{4} \, a^{3} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, a^{3} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} b^{3} - a b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{3} - a b^{4}\right)} d^{2}\right)} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} - \frac{1}{2} \, {\left(a^{2} b^{2} d \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a b^{5} - b^{6}\right)} d^{3} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a b^{4} - b^{5}\right)} d^{2} \sqrt{\frac{a^{5}}{{\left(a^{2} b^{7} - 2 \, a b^{8} + b^{9}\right)} d^{4}}} - a^{3}}{{\left(a b^{4} - b^{5}\right)} d^{2}}}\right) + {\left(8 \, a + 3 \, b\right)} d x + {\left(2 \, b \cos\left(d x + c\right)^{3} - 5 \, b \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, b^{2} d}"," ",0,"-1/8*(b^2*d*sqrt(-((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^3)/((a*b^4 - b^5)*d^2))*log(1/4*a^3*cos(d*x + c)^2 - 1/4*a^3 - 1/4*(2*(a^2*b^3 - a*b^4)*d^2*cos(d*x + c)^2 - (a^2*b^3 - a*b^4)*d^2)*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + 1/2*(a^2*b^2*d*cos(d*x + c)*sin(d*x + c) - (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^3)/((a*b^4 - b^5)*d^2))) - b^2*d*sqrt(-((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^3)/((a*b^4 - b^5)*d^2))*log(1/4*a^3*cos(d*x + c)^2 - 1/4*a^3 - 1/4*(2*(a^2*b^3 - a*b^4)*d^2*cos(d*x + c)^2 - (a^2*b^3 - a*b^4)*d^2)*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - 1/2*(a^2*b^2*d*cos(d*x + c)*sin(d*x + c) - (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + a^3)/((a*b^4 - b^5)*d^2))) - b^2*d*sqrt(((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^3)/((a*b^4 - b^5)*d^2))*log(-1/4*a^3*cos(d*x + c)^2 + 1/4*a^3 - 1/4*(2*(a^2*b^3 - a*b^4)*d^2*cos(d*x + c)^2 - (a^2*b^3 - a*b^4)*d^2)*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) + 1/2*(a^2*b^2*d*cos(d*x + c)*sin(d*x + c) + (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^3)/((a*b^4 - b^5)*d^2))) + b^2*d*sqrt(((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^3)/((a*b^4 - b^5)*d^2))*log(-1/4*a^3*cos(d*x + c)^2 + 1/4*a^3 - 1/4*(2*(a^2*b^3 - a*b^4)*d^2*cos(d*x + c)^2 - (a^2*b^3 - a*b^4)*d^2)*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - 1/2*(a^2*b^2*d*cos(d*x + c)*sin(d*x + c) + (a*b^5 - b^6)*d^3*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b^4 - b^5)*d^2*sqrt(a^5/((a^2*b^7 - 2*a*b^8 + b^9)*d^4)) - a^3)/((a*b^4 - b^5)*d^2))) + (8*a + 3*b)*d*x + (2*b*cos(d*x + c)^3 - 5*b*cos(d*x + c))*sin(d*x + c))/(b^2*d)","B",0
204,1,1275,0,1.577535," ","integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{b d \sqrt{-\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}} \log\left(\frac{1}{4} \, a^{2} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, a^{2} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} b^{2} - a b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - a b^{3}\right)} d^{2}\right)} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + \frac{1}{2} \, {\left({\left(a b^{4} - b^{5}\right)} d^{3} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - a^{2} b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}}\right) - b d \sqrt{-\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}} \log\left(\frac{1}{4} \, a^{2} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, a^{2} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} b^{2} - a b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - a b^{3}\right)} d^{2}\right)} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - \frac{1}{2} \, {\left({\left(a b^{4} - b^{5}\right)} d^{3} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - a^{2} b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}}\right) + b d \sqrt{\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}} \log\left(-\frac{1}{4} \, a^{2} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, a^{2} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} b^{2} - a b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - a b^{3}\right)} d^{2}\right)} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} + \frac{1}{2} \, {\left({\left(a b^{4} - b^{5}\right)} d^{3} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a^{2} b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}}\right) - b d \sqrt{\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}} \log\left(-\frac{1}{4} \, a^{2} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, a^{2} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} b^{2} - a b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - a b^{3}\right)} d^{2}\right)} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - \frac{1}{2} \, {\left({\left(a b^{4} - b^{5}\right)} d^{3} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a^{2} b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a b^{3} - b^{4}\right)} d^{2} \sqrt{\frac{a^{3}}{{\left(a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right)} d^{4}}} - a^{2}}{{\left(a b^{3} - b^{4}\right)} d^{2}}}\right) + 4 \, d x - 4 \, \cos\left(d x + c\right) \sin\left(d x + c\right)}{8 \, b d}"," ",0,"-1/8*(b*d*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(1/4*a^2*cos(d*x + c)^2 - 1/4*a^2 - 1/4*(2*(a^2*b^2 - a*b^3)*d^2*cos(d*x + c)^2 - (a^2*b^2 - a*b^3)*d^2)*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + 1/2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4))*cos(d*x + c)*sin(d*x + c) - a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) - b*d*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))*log(1/4*a^2*cos(d*x + c)^2 - 1/4*a^2 - 1/4*(2*(a^2*b^2 - a*b^3)*d^2*cos(d*x + c)^2 - (a^2*b^2 - a*b^3)*d^2)*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - 1/2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4))*cos(d*x + c)*sin(d*x + c) - a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a^2)/((a*b^3 - b^4)*d^2))) + b*d*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(-1/4*a^2*cos(d*x + c)^2 + 1/4*a^2 - 1/4*(2*(a^2*b^2 - a*b^3)*d^2*cos(d*x + c)^2 - (a^2*b^2 - a*b^3)*d^2)*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + 1/2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4))*cos(d*x + c)*sin(d*x + c) + a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))) - b*d*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))*log(-1/4*a^2*cos(d*x + c)^2 + 1/4*a^2 - 1/4*(2*(a^2*b^2 - a*b^3)*d^2*cos(d*x + c)^2 - (a^2*b^2 - a*b^3)*d^2)*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - 1/2*((a*b^4 - b^5)*d^3*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4))*cos(d*x + c)*sin(d*x + c) + a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b^3 - b^4)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a^2)/((a*b^3 - b^4)*d^2))) + 4*d*x - 4*cos(d*x + c)*sin(d*x + c))/(b*d)","B",0
205,1,1125,0,1.292325," ","integrate(sin(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{b \sqrt{-\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} \log\left(\frac{1}{4} \, \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a b^{2} - b^{3}\right)} d^{3} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a b - b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a b - b^{2}\right)} d^{2}\right)} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - \frac{1}{4}\right) - b \sqrt{-\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} \log\left(\frac{1}{4} \, \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a b^{2} - b^{3}\right)} d^{3} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a b - b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a b - b^{2}\right)} d^{2}\right)} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - \frac{1}{4}\right) + b \sqrt{\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} \log\left(-\frac{1}{4} \, \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a b^{2} - b^{3}\right)} d^{3} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a b - b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a b - b^{2}\right)} d^{2}\right)} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + \frac{1}{4}\right) - b \sqrt{\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} \log\left(-\frac{1}{4} \, \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a b^{2} - b^{3}\right)} d^{3} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a b^{2} - b^{3}\right)} d^{2} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} - a}{{\left(a b^{2} - b^{3}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a b - b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a b - b^{2}\right)} d^{2}\right)} \sqrt{\frac{a}{{\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d^{4}}} + \frac{1}{4}\right) - 8 \, x}{8 \, b}"," ",0,"1/8*(b*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d^2))*log(1/4*cos(d*x + c)^2 + 1/2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4))*cos(d*x + c)*sin(d*x + c) - b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d^2)) - 1/4*(2*(a*b - b^2)*d^2*cos(d*x + c)^2 - (a*b - b^2)*d^2)*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1/4) - b*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d^2))*log(1/4*cos(d*x + c)^2 - 1/2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4))*cos(d*x + c)*sin(d*x + c) - b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + a)/((a*b^2 - b^3)*d^2)) - 1/4*(2*(a*b - b^2)*d^2*cos(d*x + c)^2 - (a*b - b^2)*d^2)*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1/4) + b*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2))*log(-1/4*cos(d*x + c)^2 + 1/2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4))*cos(d*x + c)*sin(d*x + c) + b*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2)) - 1/4*(2*(a*b - b^2)*d^2*cos(d*x + c)^2 - (a*b - b^2)*d^2)*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1/4) - b*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2))*log(-1/4*cos(d*x + c)^2 - 1/2*((a*b^2 - b^3)*d^3*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4))*cos(d*x + c)*sin(d*x + c) + b*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b^2 - b^3)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - a)/((a*b^2 - b^3)*d^2)) - 1/4*(2*(a*b - b^2)*d^2*cos(d*x + c)^2 - (a*b - b^2)*d^2)*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1/4) - 8*x)/b","B",0
206,1,1087,0,1.530780," ","integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{1}{8} \, \sqrt{-\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + 1}{{\left(a b - b^{2}\right)} d^{2}}} \log\left(\frac{1}{4} \, \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a^{2} b - a b^{2}\right)} d^{3} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - a d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + 1}{{\left(a b - b^{2}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} - a b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} - a b\right)} d^{2}\right)} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - \frac{1}{4}\right) + \frac{1}{8} \, \sqrt{-\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + 1}{{\left(a b - b^{2}\right)} d^{2}}} \log\left(\frac{1}{4} \, \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a^{2} b - a b^{2}\right)} d^{3} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - a d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + 1}{{\left(a b - b^{2}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} - a b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} - a b\right)} d^{2}\right)} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - \frac{1}{4}\right) - \frac{1}{8} \, \sqrt{\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - 1}{{\left(a b - b^{2}\right)} d^{2}}} \log\left(-\frac{1}{4} \, \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a^{2} b - a b^{2}\right)} d^{3} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - 1}{{\left(a b - b^{2}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} - a b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} - a b\right)} d^{2}\right)} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + \frac{1}{4}\right) + \frac{1}{8} \, \sqrt{\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - 1}{{\left(a b - b^{2}\right)} d^{2}}} \log\left(-\frac{1}{4} \, \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a^{2} b - a b^{2}\right)} d^{3} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a b - b^{2}\right)} d^{2} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} - 1}{{\left(a b - b^{2}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{2} - a b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{2} - a b\right)} d^{2}\right)} \sqrt{\frac{1}{{\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} d^{4}}} + \frac{1}{4}\right)"," ",0,"-1/8*sqrt(-((a*b - b^2)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1)/((a*b - b^2)*d^2))*log(1/4*cos(d*x + c)^2 + 1/2*((a^2*b - a*b^2)*d^3*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4))*cos(d*x + c)*sin(d*x + c) - a*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b - b^2)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1)/((a*b - b^2)*d^2)) - 1/4*(2*(a^2 - a*b)*d^2*cos(d*x + c)^2 - (a^2 - a*b)*d^2)*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1/4) + 1/8*sqrt(-((a*b - b^2)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1)/((a*b - b^2)*d^2))*log(1/4*cos(d*x + c)^2 - 1/2*((a^2*b - a*b^2)*d^3*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4))*cos(d*x + c)*sin(d*x + c) - a*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a*b - b^2)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1)/((a*b - b^2)*d^2)) - 1/4*(2*(a^2 - a*b)*d^2*cos(d*x + c)^2 - (a^2 - a*b)*d^2)*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1/4) - 1/8*sqrt(((a*b - b^2)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1)/((a*b - b^2)*d^2))*log(-1/4*cos(d*x + c)^2 + 1/2*((a^2*b - a*b^2)*d^3*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4))*cos(d*x + c)*sin(d*x + c) + a*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b - b^2)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1)/((a*b - b^2)*d^2)) - 1/4*(2*(a^2 - a*b)*d^2*cos(d*x + c)^2 - (a^2 - a*b)*d^2)*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1/4) + 1/8*sqrt(((a*b - b^2)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1)/((a*b - b^2)*d^2))*log(-1/4*cos(d*x + c)^2 - 1/2*((a^2*b - a*b^2)*d^3*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4))*cos(d*x + c)*sin(d*x + c) + a*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a*b - b^2)*d^2*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) - 1)/((a*b - b^2)*d^2)) - 1/4*(2*(a^2 - a*b)*d^2*cos(d*x + c)^2 - (a^2 - a*b)*d^2)*sqrt(1/((a^3*b - 2*a^2*b^2 + a*b^3)*d^4)) + 1/4)","B",0
207,1,1079,0,1.338085," ","integrate(1/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{1}{8} \, \sqrt{-\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + 1}{{\left(a^{2} - a b\right)} d^{2}}} \log\left(\frac{1}{4} \, b \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a^{4} - a^{3} b\right)} d^{3} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - a b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + 1}{{\left(a^{2} - a b\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{3} - a^{2} b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{3} - a^{2} b\right)} d^{2}\right)} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - \frac{1}{4} \, b\right) - \frac{1}{8} \, \sqrt{-\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + 1}{{\left(a^{2} - a b\right)} d^{2}}} \log\left(\frac{1}{4} \, b \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a^{4} - a^{3} b\right)} d^{3} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - a b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + 1}{{\left(a^{2} - a b\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{3} - a^{2} b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{3} - a^{2} b\right)} d^{2}\right)} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - \frac{1}{4} \, b\right) + \frac{1}{8} \, \sqrt{\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - 1}{{\left(a^{2} - a b\right)} d^{2}}} \log\left(-\frac{1}{4} \, b \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a^{4} - a^{3} b\right)} d^{3} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - 1}{{\left(a^{2} - a b\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{3} - a^{2} b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{3} - a^{2} b\right)} d^{2}\right)} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + \frac{1}{4} \, b\right) - \frac{1}{8} \, \sqrt{\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - 1}{{\left(a^{2} - a b\right)} d^{2}}} \log\left(-\frac{1}{4} \, b \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a^{4} - a^{3} b\right)} d^{3} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{2} - a b\right)} d^{2} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} - 1}{{\left(a^{2} - a b\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{3} - a^{2} b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{3} - a^{2} b\right)} d^{2}\right)} \sqrt{\frac{b}{{\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{4}}} + \frac{1}{4} \, b\right)"," ",0,"1/8*sqrt(-((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + 1)/((a^2 - a*b)*d^2))*log(1/4*b*cos(d*x + c)^2 + 1/2*((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4))*cos(d*x + c)*sin(d*x + c) - a*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + 1)/((a^2 - a*b)*d^2)) - 1/4*(2*(a^3 - a^2*b)*d^2*cos(d*x + c)^2 - (a^3 - a^2*b)*d^2)*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - 1/4*b) - 1/8*sqrt(-((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + 1)/((a^2 - a*b)*d^2))*log(1/4*b*cos(d*x + c)^2 - 1/2*((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4))*cos(d*x + c)*sin(d*x + c) - a*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + 1)/((a^2 - a*b)*d^2)) - 1/4*(2*(a^3 - a^2*b)*d^2*cos(d*x + c)^2 - (a^3 - a^2*b)*d^2)*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - 1/4*b) + 1/8*sqrt(((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - 1)/((a^2 - a*b)*d^2))*log(-1/4*b*cos(d*x + c)^2 + 1/2*((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4))*cos(d*x + c)*sin(d*x + c) + a*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - 1)/((a^2 - a*b)*d^2)) - 1/4*(2*(a^3 - a^2*b)*d^2*cos(d*x + c)^2 - (a^3 - a^2*b)*d^2)*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + 1/4*b) - 1/8*sqrt(((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - 1)/((a^2 - a*b)*d^2))*log(-1/4*b*cos(d*x + c)^2 - 1/2*((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4))*cos(d*x + c)*sin(d*x + c) + a*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - 1)/((a^2 - a*b)*d^2)) - 1/4*(2*(a^3 - a^2*b)*d^2*cos(d*x + c)^2 - (a^3 - a^2*b)*d^2)*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + 1/4*b)","B",0
208,1,1229,0,1.691183," ","integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{a d \sqrt{-\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + b}{{\left(a^{3} - a^{2} b\right)} d^{2}}} \log\left(\frac{1}{4} \, b^{2} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, b^{2} - \frac{1}{4} \, {\left(2 \, {\left(a^{4} - a^{3} b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{4} - a^{3} b\right)} d^{2}\right)} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + \frac{1}{2} \, {\left({\left(a^{5} - a^{4} b\right)} d^{3} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - a^{2} b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + b}{{\left(a^{3} - a^{2} b\right)} d^{2}}}\right) \sin\left(d x + c\right) - a d \sqrt{-\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + b}{{\left(a^{3} - a^{2} b\right)} d^{2}}} \log\left(\frac{1}{4} \, b^{2} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, b^{2} - \frac{1}{4} \, {\left(2 \, {\left(a^{4} - a^{3} b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{4} - a^{3} b\right)} d^{2}\right)} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - \frac{1}{2} \, {\left({\left(a^{5} - a^{4} b\right)} d^{3} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - a^{2} b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + b}{{\left(a^{3} - a^{2} b\right)} d^{2}}}\right) \sin\left(d x + c\right) + a d \sqrt{\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - b}{{\left(a^{3} - a^{2} b\right)} d^{2}}} \log\left(-\frac{1}{4} \, b^{2} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, b^{2} - \frac{1}{4} \, {\left(2 \, {\left(a^{4} - a^{3} b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{4} - a^{3} b\right)} d^{2}\right)} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} + \frac{1}{2} \, {\left({\left(a^{5} - a^{4} b\right)} d^{3} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a^{2} b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - b}{{\left(a^{3} - a^{2} b\right)} d^{2}}}\right) \sin\left(d x + c\right) - a d \sqrt{\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - b}{{\left(a^{3} - a^{2} b\right)} d^{2}}} \log\left(-\frac{1}{4} \, b^{2} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, b^{2} - \frac{1}{4} \, {\left(2 \, {\left(a^{4} - a^{3} b\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{4} - a^{3} b\right)} d^{2}\right)} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - \frac{1}{2} \, {\left({\left(a^{5} - a^{4} b\right)} d^{3} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + a^{2} b d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{3} - a^{2} b\right)} d^{2} \sqrt{\frac{b^{3}}{{\left(a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right)} d^{4}}} - b}{{\left(a^{3} - a^{2} b\right)} d^{2}}}\right) \sin\left(d x + c\right) + 8 \, \cos\left(d x + c\right)}{8 \, a d \sin\left(d x + c\right)}"," ",0,"-1/8*(a*d*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))*log(1/4*b^2*cos(d*x + c)^2 - 1/4*b^2 - 1/4*(2*(a^4 - a^3*b)*d^2*cos(d*x + c)^2 - (a^4 - a^3*b)*d^2)*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + 1/2*((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4))*cos(d*x + c)*sin(d*x + c) - a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2)))*sin(d*x + c) - a*d*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))*log(1/4*b^2*cos(d*x + c)^2 - 1/4*b^2 - 1/4*(2*(a^4 - a^3*b)*d^2*cos(d*x + c)^2 - (a^4 - a^3*b)*d^2)*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - 1/2*((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4))*cos(d*x + c)*sin(d*x + c) - a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2)))*sin(d*x + c) + a*d*sqrt(((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))*log(-1/4*b^2*cos(d*x + c)^2 + 1/4*b^2 - 1/4*(2*(a^4 - a^3*b)*d^2*cos(d*x + c)^2 - (a^4 - a^3*b)*d^2)*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + 1/2*((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4))*cos(d*x + c)*sin(d*x + c) + a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2)))*sin(d*x + c) - a*d*sqrt(((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))*log(-1/4*b^2*cos(d*x + c)^2 + 1/4*b^2 - 1/4*(2*(a^4 - a^3*b)*d^2*cos(d*x + c)^2 - (a^4 - a^3*b)*d^2)*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - 1/2*((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4))*cos(d*x + c)*sin(d*x + c) + a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2)))*sin(d*x + c) + 8*cos(d*x + c))/(a*d*sin(d*x + c))","B",0
209,1,1365,0,1.493146," ","integrate(csc(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{3 \, {\left(a d \cos\left(d x + c\right)^{2} - a d\right)} \sqrt{-\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} + b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}} \log\left(\frac{1}{4} \, b^{4} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, b^{4} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} b - a^{4} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} b - a^{4} b^{2}\right)} d^{2}\right)} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} + \frac{1}{2} \, {\left(a^{2} b^{3} d \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a^{7} - a^{6} b\right)} d^{3} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} + b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}}\right) \sin\left(d x + c\right) - 3 \, {\left(a d \cos\left(d x + c\right)^{2} - a d\right)} \sqrt{-\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} + b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}} \log\left(\frac{1}{4} \, b^{4} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, b^{4} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} b - a^{4} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} b - a^{4} b^{2}\right)} d^{2}\right)} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} - \frac{1}{2} \, {\left(a^{2} b^{3} d \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a^{7} - a^{6} b\right)} d^{3} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} + b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}}\right) \sin\left(d x + c\right) - 3 \, {\left(a d \cos\left(d x + c\right)^{2} - a d\right)} \sqrt{\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} - b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}} \log\left(-\frac{1}{4} \, b^{4} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, b^{4} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} b - a^{4} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} b - a^{4} b^{2}\right)} d^{2}\right)} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} + \frac{1}{2} \, {\left(a^{2} b^{3} d \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a^{7} - a^{6} b\right)} d^{3} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} - b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}}\right) \sin\left(d x + c\right) + 3 \, {\left(a d \cos\left(d x + c\right)^{2} - a d\right)} \sqrt{\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} - b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}} \log\left(-\frac{1}{4} \, b^{4} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, b^{4} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} b - a^{4} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} b - a^{4} b^{2}\right)} d^{2}\right)} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} - \frac{1}{2} \, {\left(a^{2} b^{3} d \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a^{7} - a^{6} b\right)} d^{3} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{4} - a^{3} b\right)} d^{2} \sqrt{\frac{b^{5}}{{\left(a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right)} d^{4}}} - b^{2}}{{\left(a^{4} - a^{3} b\right)} d^{2}}}\right) \sin\left(d x + c\right) + 16 \, \cos\left(d x + c\right)^{3} - 24 \, \cos\left(d x + c\right)}{24 \, {\left(a d \cos\left(d x + c\right)^{2} - a d\right)} \sin\left(d x + c\right)}"," ",0,"-1/24*(3*(a*d*cos(d*x + c)^2 - a*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(1/4*b^4*cos(d*x + c)^2 - 1/4*b^4 - 1/4*(2*(a^5*b - a^4*b^2)*d^2*cos(d*x + c)^2 - (a^5*b - a^4*b^2)*d^2)*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + 1/2*(a^2*b^3*d*cos(d*x + c)*sin(d*x + c) - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2)))*sin(d*x + c) - 3*(a*d*cos(d*x + c)^2 - a*d)*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2))*log(1/4*b^4*cos(d*x + c)^2 - 1/4*b^4 - 1/4*(2*(a^5*b - a^4*b^2)*d^2*cos(d*x + c)^2 - (a^5*b - a^4*b^2)*d^2)*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - 1/2*(a^2*b^3*d*cos(d*x + c)*sin(d*x + c) - (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + b^2)/((a^4 - a^3*b)*d^2)))*sin(d*x + c) - 3*(a*d*cos(d*x + c)^2 - a*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(-1/4*b^4*cos(d*x + c)^2 + 1/4*b^4 - 1/4*(2*(a^5*b - a^4*b^2)*d^2*cos(d*x + c)^2 - (a^5*b - a^4*b^2)*d^2)*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) + 1/2*(a^2*b^3*d*cos(d*x + c)*sin(d*x + c) + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2)))*sin(d*x + c) + 3*(a*d*cos(d*x + c)^2 - a*d)*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2))*log(-1/4*b^4*cos(d*x + c)^2 + 1/4*b^4 - 1/4*(2*(a^5*b - a^4*b^2)*d^2*cos(d*x + c)^2 - (a^5*b - a^4*b^2)*d^2)*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - 1/2*(a^2*b^3*d*cos(d*x + c)*sin(d*x + c) + (a^7 - a^6*b)*d^3*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4))*cos(d*x + c)*sin(d*x + c))*sqrt(((a^4 - a^3*b)*d^2*sqrt(b^5/((a^9 - 2*a^8*b + a^7*b^2)*d^4)) - b^2)/((a^4 - a^3*b)*d^2)))*sin(d*x + c) + 16*cos(d*x + c)^3 - 24*cos(d*x + c))/((a*d*cos(d*x + c)^2 - a*d)*sin(d*x + c))","B",0
210,1,1477,0,1.474685," ","integrate(csc(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{8 \, {\left(8 \, a + 15 \, b\right)} \cos\left(d x + c\right)^{5} - 80 \, {\left(2 \, a + 3 \, b\right)} \cos\left(d x + c\right)^{3} - 15 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)} \sqrt{-\frac{{\left(a^{5} - a^{4} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{2} + b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}} \log\left(\frac{1}{4} \, b^{5} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, b^{5} - \frac{1}{4} \, {\left(2 \, {\left(a^{6} b - a^{5} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{6} b - a^{5} b^{2}\right)} d^{2}\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} + \frac{1}{2} \, {\left(a^{3} b^{3} d \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a^{8} - a^{7} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{3} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{5} - a^{4} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{2} + b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}}\right) \sin\left(d x + c\right) + 15 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)} \sqrt{-\frac{{\left(a^{5} - a^{4} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{2} + b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}} \log\left(\frac{1}{4} \, b^{5} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, b^{5} - \frac{1}{4} \, {\left(2 \, {\left(a^{6} b - a^{5} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{6} b - a^{5} b^{2}\right)} d^{2}\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} - \frac{1}{2} \, {\left(a^{3} b^{3} d \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a^{8} - a^{7} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{3} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{5} - a^{4} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{2} + b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}}\right) \sin\left(d x + c\right) + 15 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)} \sqrt{\frac{{\left(a^{5} - a^{4} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{2} - b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}} \log\left(-\frac{1}{4} \, b^{5} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, b^{5} - \frac{1}{4} \, {\left(2 \, {\left(a^{6} b - a^{5} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{6} b - a^{5} b^{2}\right)} d^{2}\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} + \frac{1}{2} \, {\left(a^{3} b^{3} d \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a^{8} - a^{7} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{3} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{5} - a^{4} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{2} - b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}}\right) \sin\left(d x + c\right) - 15 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)} \sqrt{\frac{{\left(a^{5} - a^{4} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{2} - b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}} \log\left(-\frac{1}{4} \, b^{5} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, b^{5} - \frac{1}{4} \, {\left(2 \, {\left(a^{6} b - a^{5} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{6} b - a^{5} b^{2}\right)} d^{2}\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} - \frac{1}{2} \, {\left(a^{3} b^{3} d \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a^{8} - a^{7} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{3} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{5} - a^{4} b\right)} \sqrt{\frac{b^{7}}{{\left(a^{11} - 2 \, a^{10} b + a^{9} b^{2}\right)} d^{4}}} d^{2} - b^{3}}{{\left(a^{5} - a^{4} b\right)} d^{2}}}\right) \sin\left(d x + c\right) + 120 \, {\left(a + b\right)} \cos\left(d x + c\right)}{120 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)} \sin\left(d x + c\right)}"," ",0,"-1/120*(8*(8*a + 15*b)*cos(d*x + c)^5 - 80*(2*a + 3*b)*cos(d*x + c)^3 - 15*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(-((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 - a^4*b)*d^2))*log(1/4*b^5*cos(d*x + c)^2 - 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^6*b - a^5*b^2)*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) + 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) - (a^8 - a^7*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) + 15*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(-((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 - a^4*b)*d^2))*log(1/4*b^5*cos(d*x + c)^2 - 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^6*b - a^5*b^2)*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) - 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) - (a^8 - a^7*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) + 15*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/((a^5 - a^4*b)*d^2))*log(-1/4*b^5*cos(d*x + c)^2 + 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^6*b - a^5*b^2)*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) + 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) + (a^8 - a^7*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) - 15*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/((a^5 - a^4*b)*d^2))*log(-1/4*b^5*cos(d*x + c)^2 + 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^6*b - a^5*b^2)*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) - 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) + (a^8 - a^7*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) + 120*(a + b)*cos(d*x + c))/((a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sin(d*x + c))","B",0
211,1,1585,0,1.676595," ","integrate(csc(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{16 \, {\left(24 \, a + 35 \, b\right)} \cos\left(d x + c\right)^{7} - 56 \, {\left(24 \, a + 35 \, b\right)} \cos\left(d x + c\right)^{5} + 560 \, {\left(3 \, a + 4 \, b\right)} \cos\left(d x + c\right)^{3} + 105 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} - 3 \, a^{2} d \cos\left(d x + c\right)^{4} + 3 \, a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)} \sqrt{-\frac{b^{4} + {\left(a^{6} - a^{5} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{2}}{{\left(a^{6} - a^{5} b\right)} d^{2}}} \log\left(\frac{1}{4} \, b^{7} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, b^{7} - \frac{1}{4} \, {\left(2 \, {\left(a^{7} b^{2} - a^{6} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{7} b^{2} - a^{6} b^{3}\right)} d^{2}\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} + \frac{1}{2} \, {\left(a^{3} b^{5} d \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a^{10} - a^{9} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{3} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{b^{4} + {\left(a^{6} - a^{5} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{2}}{{\left(a^{6} - a^{5} b\right)} d^{2}}}\right) \sin\left(d x + c\right) - 105 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} - 3 \, a^{2} d \cos\left(d x + c\right)^{4} + 3 \, a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)} \sqrt{-\frac{b^{4} + {\left(a^{6} - a^{5} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{2}}{{\left(a^{6} - a^{5} b\right)} d^{2}}} \log\left(\frac{1}{4} \, b^{7} \cos\left(d x + c\right)^{2} - \frac{1}{4} \, b^{7} - \frac{1}{4} \, {\left(2 \, {\left(a^{7} b^{2} - a^{6} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{7} b^{2} - a^{6} b^{3}\right)} d^{2}\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} - \frac{1}{2} \, {\left(a^{3} b^{5} d \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(a^{10} - a^{9} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{3} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{b^{4} + {\left(a^{6} - a^{5} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{2}}{{\left(a^{6} - a^{5} b\right)} d^{2}}}\right) \sin\left(d x + c\right) - 105 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} - 3 \, a^{2} d \cos\left(d x + c\right)^{4} + 3 \, a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)} \sqrt{-\frac{b^{4} - {\left(a^{6} - a^{5} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{2}}{{\left(a^{6} - a^{5} b\right)} d^{2}}} \log\left(-\frac{1}{4} \, b^{7} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, b^{7} - \frac{1}{4} \, {\left(2 \, {\left(a^{7} b^{2} - a^{6} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{7} b^{2} - a^{6} b^{3}\right)} d^{2}\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} + \frac{1}{2} \, {\left(a^{3} b^{5} d \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a^{10} - a^{9} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{3} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{b^{4} - {\left(a^{6} - a^{5} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{2}}{{\left(a^{6} - a^{5} b\right)} d^{2}}}\right) \sin\left(d x + c\right) + 105 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} - 3 \, a^{2} d \cos\left(d x + c\right)^{4} + 3 \, a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)} \sqrt{-\frac{b^{4} - {\left(a^{6} - a^{5} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{2}}{{\left(a^{6} - a^{5} b\right)} d^{2}}} \log\left(-\frac{1}{4} \, b^{7} \cos\left(d x + c\right)^{2} + \frac{1}{4} \, b^{7} - \frac{1}{4} \, {\left(2 \, {\left(a^{7} b^{2} - a^{6} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{7} b^{2} - a^{6} b^{3}\right)} d^{2}\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} - \frac{1}{2} \, {\left(a^{3} b^{5} d \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(a^{10} - a^{9} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{3} \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{b^{4} - {\left(a^{6} - a^{5} b\right)} \sqrt{\frac{b^{9}}{{\left(a^{13} - 2 \, a^{12} b + a^{11} b^{2}\right)} d^{4}}} d^{2}}{{\left(a^{6} - a^{5} b\right)} d^{2}}}\right) \sin\left(d x + c\right) - 840 \, {\left(a + b\right)} \cos\left(d x + c\right)}{840 \, {\left(a^{2} d \cos\left(d x + c\right)^{6} - 3 \, a^{2} d \cos\left(d x + c\right)^{4} + 3 \, a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)} \sin\left(d x + c\right)}"," ",0,"-1/840*(16*(24*a + 35*b)*cos(d*x + c)^7 - 56*(24*a + 35*b)*cos(d*x + c)^5 + 560*(3*a + 4*b)*cos(d*x + c)^3 + 105*(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sqrt(-(b^4 + (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2))*log(1/4*b^7*cos(d*x + c)^2 - 1/4*b^7 - 1/4*(2*(a^7*b^2 - a^6*b^3)*d^2*cos(d*x + c)^2 - (a^7*b^2 - a^6*b^3)*d^2)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4)) + 1/2*(a^3*b^5*d*cos(d*x + c)*sin(d*x + c) - (a^10 - a^9*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-(b^4 + (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2)))*sin(d*x + c) - 105*(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sqrt(-(b^4 + (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2))*log(1/4*b^7*cos(d*x + c)^2 - 1/4*b^7 - 1/4*(2*(a^7*b^2 - a^6*b^3)*d^2*cos(d*x + c)^2 - (a^7*b^2 - a^6*b^3)*d^2)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4)) - 1/2*(a^3*b^5*d*cos(d*x + c)*sin(d*x + c) - (a^10 - a^9*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-(b^4 + (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2)))*sin(d*x + c) - 105*(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sqrt(-(b^4 - (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2))*log(-1/4*b^7*cos(d*x + c)^2 + 1/4*b^7 - 1/4*(2*(a^7*b^2 - a^6*b^3)*d^2*cos(d*x + c)^2 - (a^7*b^2 - a^6*b^3)*d^2)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4)) + 1/2*(a^3*b^5*d*cos(d*x + c)*sin(d*x + c) + (a^10 - a^9*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-(b^4 - (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2)))*sin(d*x + c) + 105*(a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sqrt(-(b^4 - (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2))*log(-1/4*b^7*cos(d*x + c)^2 + 1/4*b^7 - 1/4*(2*(a^7*b^2 - a^6*b^3)*d^2*cos(d*x + c)^2 - (a^7*b^2 - a^6*b^3)*d^2)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4)) - 1/2*(a^3*b^5*d*cos(d*x + c)*sin(d*x + c) + (a^10 - a^9*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-(b^4 - (a^6 - a^5*b)*sqrt(b^9/((a^13 - 2*a^12*b + a^11*b^2)*d^4))*d^2)/((a^6 - a^5*b)*d^2)))*sin(d*x + c) - 840*(a + b)*cos(d*x + c))/((a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))","B",0
212,1,2649,0,2.418332," ","integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","-\frac{16 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{5} - 4 \, {\left(7 \, a b - 8 \, b^{2}\right)} \cos\left(d x + c\right)^{3} + {\left({\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} + 15 \, a^{3} - 47 \, a^{2} b + 36 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} \log\left({\left(625 \, a^{5} - 2625 \, a^{4} b + 3684 \, a^{3} b^{2} - 1728 \, a^{2} b^{3}\right)} \cos\left(d x + c\right) + {\left(2 \, {\left(2 \, a^{4} b^{7} - 9 \, a^{3} b^{8} + 15 \, a^{2} b^{9} - 11 \, a b^{10} + 3 \, b^{11}\right)} d^{3} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} - {\left(125 \, a^{5} b^{2} - 520 \, a^{4} b^{3} + 723 \, a^{3} b^{4} - 336 \, a^{2} b^{5}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} + 15 \, a^{3} - 47 \, a^{2} b + 36 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}}\right) - {\left({\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} - 15 \, a^{3} + 47 \, a^{2} b - 36 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} \log\left({\left(625 \, a^{5} - 2625 \, a^{4} b + 3684 \, a^{3} b^{2} - 1728 \, a^{2} b^{3}\right)} \cos\left(d x + c\right) + {\left(2 \, {\left(2 \, a^{4} b^{7} - 9 \, a^{3} b^{8} + 15 \, a^{2} b^{9} - 11 \, a b^{10} + 3 \, b^{11}\right)} d^{3} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} + {\left(125 \, a^{5} b^{2} - 520 \, a^{4} b^{3} + 723 \, a^{3} b^{4} - 336 \, a^{2} b^{5}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} - 15 \, a^{3} + 47 \, a^{2} b - 36 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}}\right) - {\left({\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} + 15 \, a^{3} - 47 \, a^{2} b + 36 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} \log\left(-{\left(625 \, a^{5} - 2625 \, a^{4} b + 3684 \, a^{3} b^{2} - 1728 \, a^{2} b^{3}\right)} \cos\left(d x + c\right) + {\left(2 \, {\left(2 \, a^{4} b^{7} - 9 \, a^{3} b^{8} + 15 \, a^{2} b^{9} - 11 \, a b^{10} + 3 \, b^{11}\right)} d^{3} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} - {\left(125 \, a^{5} b^{2} - 520 \, a^{4} b^{3} + 723 \, a^{3} b^{4} - 336 \, a^{2} b^{5}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} + 15 \, a^{3} - 47 \, a^{2} b + 36 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}}\right) + {\left({\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} - 15 \, a^{3} + 47 \, a^{2} b - 36 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} \log\left(-{\left(625 \, a^{5} - 2625 \, a^{4} b + 3684 \, a^{3} b^{2} - 1728 \, a^{2} b^{3}\right)} \cos\left(d x + c\right) + {\left(2 \, {\left(2 \, a^{4} b^{7} - 9 \, a^{3} b^{8} + 15 \, a^{2} b^{9} - 11 \, a b^{10} + 3 \, b^{11}\right)} d^{3} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} + {\left(125 \, a^{5} b^{2} - 520 \, a^{4} b^{3} + 723 \, a^{3} b^{4} - 336 \, a^{2} b^{5}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{625 \, a^{7} - 3450 \, a^{6} b + 7161 \, a^{5} b^{2} - 6624 \, a^{4} b^{3} + 2304 \, a^{3} b^{4}}{{\left(a^{6} b^{9} - 6 \, a^{5} b^{10} + 15 \, a^{4} b^{11} - 20 \, a^{3} b^{12} + 15 \, a^{2} b^{13} - 6 \, a b^{14} + b^{15}\right)} d^{4}}} - 15 \, a^{3} + 47 \, a^{2} b - 36 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}}\right) - 4 \, {\left(5 \, a^{2} - 7 \, a b + 4 \, b^{2}\right)} \cos\left(d x + c\right)}{16 \, {\left({\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} d\right)}}"," ",0,"-1/16*(16*(a*b - b^2)*cos(d*x + c)^5 - 4*(7*a*b - 8*b^2)*cos(d*x + c)^3 + ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + 15*a^3 - 47*a^2*b + 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log((625*a^5 - 2625*a^4*b + 3684*a^3*b^2 - 1728*a^2*b^3)*cos(d*x + c) + (2*(2*a^4*b^7 - 9*a^3*b^8 + 15*a^2*b^9 - 11*a*b^10 + 3*b^11)*d^3*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - (125*a^5*b^2 - 520*a^4*b^3 + 723*a^3*b^4 - 336*a^2*b^5)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + 15*a^3 - 47*a^2*b + 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))) - ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - 15*a^3 + 47*a^2*b - 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log((625*a^5 - 2625*a^4*b + 3684*a^3*b^2 - 1728*a^2*b^3)*cos(d*x + c) + (2*(2*a^4*b^7 - 9*a^3*b^8 + 15*a^2*b^9 - 11*a*b^10 + 3*b^11)*d^3*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + (125*a^5*b^2 - 520*a^4*b^3 + 723*a^3*b^4 - 336*a^2*b^5)*d)*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - 15*a^3 + 47*a^2*b - 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))) - ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + 15*a^3 - 47*a^2*b + 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(-(625*a^5 - 2625*a^4*b + 3684*a^3*b^2 - 1728*a^2*b^3)*cos(d*x + c) + (2*(2*a^4*b^7 - 9*a^3*b^8 + 15*a^2*b^9 - 11*a*b^10 + 3*b^11)*d^3*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - (125*a^5*b^2 - 520*a^4*b^3 + 723*a^3*b^4 - 336*a^2*b^5)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + 15*a^3 - 47*a^2*b + 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))) + ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - 15*a^3 + 47*a^2*b - 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(-(625*a^5 - 2625*a^4*b + 3684*a^3*b^2 - 1728*a^2*b^3)*cos(d*x + c) + (2*(2*a^4*b^7 - 9*a^3*b^8 + 15*a^2*b^9 - 11*a*b^10 + 3*b^11)*d^3*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) + (125*a^5*b^2 - 520*a^4*b^3 + 723*a^3*b^4 - 336*a^2*b^5)*d)*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((625*a^7 - 3450*a^6*b + 7161*a^5*b^2 - 6624*a^4*b^3 + 2304*a^3*b^4)/((a^6*b^9 - 6*a^5*b^10 + 15*a^4*b^11 - 20*a^3*b^12 + 15*a^2*b^13 - 6*a*b^14 + b^15)*d^4)) - 15*a^3 + 47*a^2*b - 36*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))) - 4*(5*a^2 - 7*a*b + 4*b^2)*cos(d*x + c))/((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)","B",0
213,1,2507,0,1.724180," ","integrate(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","-\frac{4 \, a \cos\left(d x + c\right)^{3} - {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} + 3 \, a^{2} - 15 \, a b + 16 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} \log\left({\left(81 \, a^{3} - 405 \, a^{2} b + 680 \, a b^{2} - 384 \, b^{3}\right)} \cos\left(d x + c\right) + {\left({\left(3 \, a^{4} b^{5} - 14 \, a^{3} b^{6} + 24 \, a^{2} b^{7} - 18 \, a b^{8} + 5 \, b^{9}\right)} d^{3} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} - 2 \, {\left(9 \, a^{3} b^{2} - 47 \, a^{2} b^{3} + 82 \, a b^{4} - 48 \, b^{5}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} + 3 \, a^{2} - 15 \, a b + 16 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}}\right) + {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} - 3 \, a^{2} + 15 \, a b - 16 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} \log\left({\left(81 \, a^{3} - 405 \, a^{2} b + 680 \, a b^{2} - 384 \, b^{3}\right)} \cos\left(d x + c\right) + {\left({\left(3 \, a^{4} b^{5} - 14 \, a^{3} b^{6} + 24 \, a^{2} b^{7} - 18 \, a b^{8} + 5 \, b^{9}\right)} d^{3} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} + 2 \, {\left(9 \, a^{3} b^{2} - 47 \, a^{2} b^{3} + 82 \, a b^{4} - 48 \, b^{5}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} - 3 \, a^{2} + 15 \, a b - 16 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}}\right) + {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} + 3 \, a^{2} - 15 \, a b + 16 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} \log\left(-{\left(81 \, a^{3} - 405 \, a^{2} b + 680 \, a b^{2} - 384 \, b^{3}\right)} \cos\left(d x + c\right) + {\left({\left(3 \, a^{4} b^{5} - 14 \, a^{3} b^{6} + 24 \, a^{2} b^{7} - 18 \, a b^{8} + 5 \, b^{9}\right)} d^{3} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} - 2 \, {\left(9 \, a^{3} b^{2} - 47 \, a^{2} b^{3} + 82 \, a b^{4} - 48 \, b^{5}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} + 3 \, a^{2} - 15 \, a b + 16 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}}\right) - {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} - 3 \, a^{2} + 15 \, a b - 16 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} \log\left(-{\left(81 \, a^{3} - 405 \, a^{2} b + 680 \, a b^{2} - 384 \, b^{3}\right)} \cos\left(d x + c\right) + {\left({\left(3 \, a^{4} b^{5} - 14 \, a^{3} b^{6} + 24 \, a^{2} b^{7} - 18 \, a b^{8} + 5 \, b^{9}\right)} d^{3} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} + 2 \, {\left(9 \, a^{3} b^{2} - 47 \, a^{2} b^{3} + 82 \, a b^{4} - 48 \, b^{5}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{81 \, a^{5} - 522 \, a^{4} b + 1273 \, a^{3} b^{2} - 1392 \, a^{2} b^{3} + 576 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} - 3 \, a^{2} + 15 \, a b - 16 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}}\right) - 8 \, a \cos\left(d x + c\right)}{16 \, {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)}}"," ",0,"-1/16*(4*a*cos(d*x + c)^3 - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 3*a^2 - 15*a*b + 16*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))*log((81*a^3 - 405*a^2*b + 680*a*b^2 - 384*b^3)*cos(d*x + c) + ((3*a^4*b^5 - 14*a^3*b^6 + 24*a^2*b^7 - 18*a*b^8 + 5*b^9)*d^3*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 2*(9*a^3*b^2 - 47*a^2*b^3 + 82*a*b^4 - 48*b^5)*d)*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 3*a^2 - 15*a*b + 16*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))) + ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 3*a^2 + 15*a*b - 16*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))*log((81*a^3 - 405*a^2*b + 680*a*b^2 - 384*b^3)*cos(d*x + c) + ((3*a^4*b^5 - 14*a^3*b^6 + 24*a^2*b^7 - 18*a*b^8 + 5*b^9)*d^3*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 2*(9*a^3*b^2 - 47*a^2*b^3 + 82*a*b^4 - 48*b^5)*d)*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 3*a^2 + 15*a*b - 16*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))) + ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 3*a^2 - 15*a*b + 16*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))*log(-(81*a^3 - 405*a^2*b + 680*a*b^2 - 384*b^3)*cos(d*x + c) + ((3*a^4*b^5 - 14*a^3*b^6 + 24*a^2*b^7 - 18*a*b^8 + 5*b^9)*d^3*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 2*(9*a^3*b^2 - 47*a^2*b^3 + 82*a*b^4 - 48*b^5)*d)*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 3*a^2 - 15*a*b + 16*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))) - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 3*a^2 + 15*a*b - 16*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))*log(-(81*a^3 - 405*a^2*b + 680*a*b^2 - 384*b^3)*cos(d*x + c) + ((3*a^4*b^5 - 14*a^3*b^6 + 24*a^2*b^7 - 18*a*b^8 + 5*b^9)*d^3*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 2*(9*a^3*b^2 - 47*a^2*b^3 + 82*a*b^4 - 48*b^5)*d)*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((81*a^5 - 522*a^4*b + 1273*a^3*b^2 - 1392*a^2*b^3 + 576*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 3*a^2 + 15*a*b - 16*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))) - 8*a*cos(d*x + c))/((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)","B",0
214,1,2507,0,1.956825," ","integrate(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","-\frac{4 \, b \cos\left(d x + c\right)^{3} - {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{\frac{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} + a^{2} - a b - 4 \, b^{2}}{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} \log\left({\left(a^{3} - 9 \, a^{2} b + 28 \, a b^{2} - 32 \, b^{3}\right)} \cos\left(d x + c\right) - {\left(2 \, {\left(a^{4} b^{5} - 3 \, a^{3} b^{6} + 3 \, a^{2} b^{7} - a b^{8}\right)} d^{3} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} - {\left(a^{4} b - 8 \, a^{3} b^{2} + 23 \, a^{2} b^{3} - 24 \, a b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} + a^{2} - a b - 4 \, b^{2}}{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}}\right) + {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} - a^{2} + a b + 4 \, b^{2}}{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} \log\left({\left(a^{3} - 9 \, a^{2} b + 28 \, a b^{2} - 32 \, b^{3}\right)} \cos\left(d x + c\right) - {\left(2 \, {\left(a^{4} b^{5} - 3 \, a^{3} b^{6} + 3 \, a^{2} b^{7} - a b^{8}\right)} d^{3} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} + {\left(a^{4} b - 8 \, a^{3} b^{2} + 23 \, a^{2} b^{3} - 24 \, a b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} - a^{2} + a b + 4 \, b^{2}}{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}}\right) + {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{\frac{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} + a^{2} - a b - 4 \, b^{2}}{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} \log\left(-{\left(a^{3} - 9 \, a^{2} b + 28 \, a b^{2} - 32 \, b^{3}\right)} \cos\left(d x + c\right) - {\left(2 \, {\left(a^{4} b^{5} - 3 \, a^{3} b^{6} + 3 \, a^{2} b^{7} - a b^{8}\right)} d^{3} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} - {\left(a^{4} b - 8 \, a^{3} b^{2} + 23 \, a^{2} b^{3} - 24 \, a b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} + a^{2} - a b - 4 \, b^{2}}{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}}\right) - {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} - a^{2} + a b + 4 \, b^{2}}{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} \log\left(-{\left(a^{3} - 9 \, a^{2} b + 28 \, a b^{2} - 32 \, b^{3}\right)} \cos\left(d x + c\right) - {\left(2 \, {\left(a^{4} b^{5} - 3 \, a^{3} b^{6} + 3 \, a^{2} b^{7} - a b^{8}\right)} d^{3} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} + {\left(a^{4} b - 8 \, a^{3} b^{2} + 23 \, a^{2} b^{3} - 24 \, a b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{a^{4} - 10 \, a^{3} b + 41 \, a^{2} b^{2} - 80 \, a b^{3} + 64 \, b^{4}}{{\left(a^{7} b^{5} - 6 \, a^{6} b^{6} + 15 \, a^{5} b^{7} - 20 \, a^{4} b^{8} + 15 \, a^{3} b^{9} - 6 \, a^{2} b^{10} + a b^{11}\right)} d^{4}}} - a^{2} + a b + 4 \, b^{2}}{{\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}}\right) - 4 \, {\left(a + b\right)} \cos\left(d x + c\right)}{16 \, {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)}}"," ",0,"-1/16*(4*b*cos(d*x + c)^3 - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) + a^2 - a*b - 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))*log((a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x + c) - (2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - (a^4*b - 8*a^3*b^2 + 23*a^2*b^3 - 24*a*b^4)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) + a^2 - a*b - 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))) + ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))*log((a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x + c) - (2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) + (a^4*b - 8*a^3*b^2 + 23*a^2*b^3 - 24*a*b^4)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))) + ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) + a^2 - a*b - 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))*log(-(a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x + c) - (2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - (a^4*b - 8*a^3*b^2 + 23*a^2*b^3 - 24*a*b^4)*d)*sqrt(((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) + a^2 - a*b - 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))) - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))*log(-(a^3 - 9*a^2*b + 28*a*b^2 - 32*b^3)*cos(d*x + c) - (2*(a^4*b^5 - 3*a^3*b^6 + 3*a^2*b^7 - a*b^8)*d^3*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) + (a^4*b - 8*a^3*b^2 + 23*a^2*b^3 - 24*a*b^4)*d)*sqrt(-((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2*sqrt((a^4 - 10*a^3*b + 41*a^2*b^2 - 80*a*b^3 + 64*b^4)/((a^7*b^5 - 6*a^6*b^6 + 15*a^5*b^7 - 20*a^4*b^8 + 15*a^3*b^9 - 6*a^2*b^10 + a*b^11)*d^4)) - a^2 + a*b + 4*b^2)/((a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d^2))) - 4*(a + b)*cos(d*x + c))/((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)","B",0
215,1,2049,0,1.561879," ","integrate(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","-\frac{4 \, \cos\left(d x + c\right)^{3} - {\left({\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} + 3 \, a + b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} \log\left({\left(a + 3 \, b\right)} \cos\left(d x + c\right) - {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + 2 \, a^{2} b^{5} - a b^{6}\right)} d^{3} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} - 2 \, {\left(a^{2} b + 3 \, a b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} + 3 \, a + b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}}\right) + {\left({\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} - 3 \, a - b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} \log\left({\left(a + 3 \, b\right)} \cos\left(d x + c\right) - {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + 2 \, a^{2} b^{5} - a b^{6}\right)} d^{3} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} + 2 \, {\left(a^{2} b + 3 \, a b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} - 3 \, a - b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}}\right) + {\left({\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} + 3 \, a + b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} \log\left(-{\left(a + 3 \, b\right)} \cos\left(d x + c\right) - {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + 2 \, a^{2} b^{5} - a b^{6}\right)} d^{3} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} - 2 \, {\left(a^{2} b + 3 \, a b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} + 3 \, a + b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}}\right) - {\left({\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} - 3 \, a - b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} \log\left(-{\left(a + 3 \, b\right)} \cos\left(d x + c\right) - {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + 2 \, a^{2} b^{5} - a b^{6}\right)} d^{3} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} + 2 \, {\left(a^{2} b + 3 \, a b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{a^{2} + 6 \, a b + 9 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} - 3 \, a - b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}}\right) - 8 \, \cos\left(d x + c\right)}{16 \, {\left({\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} d\right)}}"," ",0,"-1/16*(4*cos(d*x + c)^3 - ((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log((a + 3*b)*cos(d*x + c) - ((a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*d^3*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 2*(a^2*b + 3*a*b^2)*d)*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))) + ((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 3*a - b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log((a + 3*b)*cos(d*x + c) - ((a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*d^3*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 2*(a^2*b + 3*a*b^2)*d)*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 3*a - b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))) + ((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log(-(a + 3*b)*cos(d*x + c) - ((a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*d^3*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 2*(a^2*b + 3*a*b^2)*d)*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 3*a + b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))) - ((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 3*a - b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log(-(a + 3*b)*cos(d*x + c) - ((a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*d^3*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 2*(a^2*b + 3*a*b^2)*d)*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((a^2 + 6*a*b + 9*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 3*a - b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))) - 8*cos(d*x + c))/((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)","B",0
216,1,2269,0,2.079005," ","integrate(sin(d*x+c)/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","-\frac{4 \, b \cos\left(d x + c\right)^{3} - {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + 15 \, a^{2} - 15 \, a b + 4 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} \log\left({\left(81 \, a^{2} - 81 \, a b + 20 \, b^{2}\right)} \cos\left(d x + c\right) + {\left(2 \, {\left(2 \, a^{7} b - 7 \, a^{6} b^{2} + 9 \, a^{5} b^{3} - 5 \, a^{4} b^{4} + a^{3} b^{5}\right)} d^{3} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - {\left(27 \, a^{4} - 24 \, a^{3} b + 5 \, a^{2} b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + 15 \, a^{2} - 15 \, a b + 4 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}}\right) + {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - 15 \, a^{2} + 15 \, a b - 4 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} \log\left({\left(81 \, a^{2} - 81 \, a b + 20 \, b^{2}\right)} \cos\left(d x + c\right) + {\left(2 \, {\left(2 \, a^{7} b - 7 \, a^{6} b^{2} + 9 \, a^{5} b^{3} - 5 \, a^{4} b^{4} + a^{3} b^{5}\right)} d^{3} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + {\left(27 \, a^{4} - 24 \, a^{3} b + 5 \, a^{2} b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - 15 \, a^{2} + 15 \, a b - 4 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}}\right) + {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + 15 \, a^{2} - 15 \, a b + 4 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} \log\left(-{\left(81 \, a^{2} - 81 \, a b + 20 \, b^{2}\right)} \cos\left(d x + c\right) + {\left(2 \, {\left(2 \, a^{7} b - 7 \, a^{6} b^{2} + 9 \, a^{5} b^{3} - 5 \, a^{4} b^{4} + a^{3} b^{5}\right)} d^{3} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - {\left(27 \, a^{4} - 24 \, a^{3} b + 5 \, a^{2} b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + 15 \, a^{2} - 15 \, a b + 4 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}}\right) - {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - 15 \, a^{2} + 15 \, a b - 4 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} \log\left(-{\left(81 \, a^{2} - 81 \, a b + 20 \, b^{2}\right)} \cos\left(d x + c\right) + {\left(2 \, {\left(2 \, a^{7} b - 7 \, a^{6} b^{2} + 9 \, a^{5} b^{3} - 5 \, a^{4} b^{4} + a^{3} b^{5}\right)} d^{3} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + {\left(27 \, a^{4} - 24 \, a^{3} b + 5 \, a^{2} b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{81 \, a^{2} - 90 \, a b + 25 \, b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - 15 \, a^{2} + 15 \, a b - 4 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}}\right) - 4 \, {\left(a + b\right)} \cos\left(d x + c\right)}{16 \, {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)}}"," ",0,"-1/16*(4*b*cos(d*x + c)^3 - ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + 15*a^2 - 15*a*b + 4*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log((81*a^2 - 81*a*b + 20*b^2)*cos(d*x + c) + (2*(2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d^3*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - (27*a^4 - 24*a^3*b + 5*a^2*b^2)*d)*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + 15*a^2 - 15*a*b + 4*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))) + ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - 15*a^2 + 15*a*b - 4*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log((81*a^2 - 81*a*b + 20*b^2)*cos(d*x + c) + (2*(2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d^3*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + (27*a^4 - 24*a^3*b + 5*a^2*b^2)*d)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - 15*a^2 + 15*a*b - 4*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))) + ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + 15*a^2 - 15*a*b + 4*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(-(81*a^2 - 81*a*b + 20*b^2)*cos(d*x + c) + (2*(2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d^3*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - (27*a^4 - 24*a^3*b + 5*a^2*b^2)*d)*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + 15*a^2 - 15*a*b + 4*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))) - ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - 15*a^2 + 15*a*b - 4*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(-(81*a^2 - 81*a*b + 20*b^2)*cos(d*x + c) + (2*(2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d^3*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + (27*a^4 - 24*a^3*b + 5*a^2*b^2)*d)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - 15*a^2 + 15*a*b - 4*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))) - 4*(a + b)*cos(d*x + c))/((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)","B",0
217,1,2711,0,3.447029," ","integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","-\frac{4 \, a b \cos\left(d x + c\right)^{3} - 8 \, a b \cos\left(d x + c\right) + {\left({\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} + 35 \, a^{2} b - 47 \, a b^{2} + 16 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} \log\left({\left(625 \, a^{3} b - 1125 \, a^{2} b^{2} + 664 \, a b^{3} - 128 \, b^{4}\right)} \cos\left(d x + c\right) + {\left({\left(5 \, a^{10} - 18 \, a^{9} b + 24 \, a^{8} b^{2} - 14 \, a^{7} b^{3} + 3 \, a^{6} b^{4}\right)} d^{3} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} - 2 \, {\left(75 \, a^{5} b - 137 \, a^{4} b^{2} + 82 \, a^{3} b^{3} - 16 \, a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} + 35 \, a^{2} b - 47 \, a b^{2} + 16 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}}\right) - {\left({\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} - 35 \, a^{2} b + 47 \, a b^{2} - 16 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} \log\left({\left(625 \, a^{3} b - 1125 \, a^{2} b^{2} + 664 \, a b^{3} - 128 \, b^{4}\right)} \cos\left(d x + c\right) + {\left({\left(5 \, a^{10} - 18 \, a^{9} b + 24 \, a^{8} b^{2} - 14 \, a^{7} b^{3} + 3 \, a^{6} b^{4}\right)} d^{3} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} + 2 \, {\left(75 \, a^{5} b - 137 \, a^{4} b^{2} + 82 \, a^{3} b^{3} - 16 \, a^{2} b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} - 35 \, a^{2} b + 47 \, a b^{2} - 16 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}}\right) - {\left({\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} + 35 \, a^{2} b - 47 \, a b^{2} + 16 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} \log\left(-{\left(625 \, a^{3} b - 1125 \, a^{2} b^{2} + 664 \, a b^{3} - 128 \, b^{4}\right)} \cos\left(d x + c\right) + {\left({\left(5 \, a^{10} - 18 \, a^{9} b + 24 \, a^{8} b^{2} - 14 \, a^{7} b^{3} + 3 \, a^{6} b^{4}\right)} d^{3} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} - 2 \, {\left(75 \, a^{5} b - 137 \, a^{4} b^{2} + 82 \, a^{3} b^{3} - 16 \, a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} + 35 \, a^{2} b - 47 \, a b^{2} + 16 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}}\right) + {\left({\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} - 35 \, a^{2} b + 47 \, a b^{2} - 16 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} \log\left(-{\left(625 \, a^{3} b - 1125 \, a^{2} b^{2} + 664 \, a b^{3} - 128 \, b^{4}\right)} \cos\left(d x + c\right) + {\left({\left(5 \, a^{10} - 18 \, a^{9} b + 24 \, a^{8} b^{2} - 14 \, a^{7} b^{3} + 3 \, a^{6} b^{4}\right)} d^{3} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} + 2 \, {\left(75 \, a^{5} b - 137 \, a^{4} b^{2} + 82 \, a^{3} b^{3} - 16 \, a^{2} b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{625 \, a^{4} b - 1450 \, a^{3} b^{2} + 1241 \, a^{2} b^{3} - 464 \, a b^{4} + 64 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} - 35 \, a^{2} b + 47 \, a b^{2} - 16 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}}\right) + 8 \, {\left({\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + 2 \, a b - b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 8 \, {\left({\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} \cos\left(d x + c\right)^{2} - a^{2} + 2 \, a b - b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{16 \, {\left({\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)}}"," ",0,"-1/16*(4*a*b*cos(d*x + c)^3 - 8*a*b*cos(d*x + c) + ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 35*a^2*b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log((625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*cos(d*x + c) + ((5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b^3 + 3*a^6*b^4)*d^3*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 2*(75*a^5*b - 137*a^4*b^2 + 82*a^3*b^3 - 16*a^2*b^4)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 35*a^2*b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) - ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 35*a^2*b + 47*a*b^2 - 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log((625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*cos(d*x + c) + ((5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b^3 + 3*a^6*b^4)*d^3*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 2*(75*a^5*b - 137*a^4*b^2 + 82*a^3*b^3 - 16*a^2*b^4)*d)*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 35*a^2*b + 47*a*b^2 - 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) - ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 35*a^2*b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(-(625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*cos(d*x + c) + ((5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b^3 + 3*a^6*b^4)*d^3*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 2*(75*a^5*b - 137*a^4*b^2 + 82*a^3*b^3 - 16*a^2*b^4)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 35*a^2*b - 47*a*b^2 + 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) + ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 35*a^2*b + 47*a*b^2 - 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(-(625*a^3*b - 1125*a^2*b^2 + 664*a*b^3 - 128*b^4)*cos(d*x + c) + ((5*a^10 - 18*a^9*b + 24*a^8*b^2 - 14*a^7*b^3 + 3*a^6*b^4)*d^3*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 2*(75*a^5*b - 137*a^4*b^2 + 82*a^3*b^3 - 16*a^2*b^4)*d)*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((625*a^4*b - 1450*a^3*b^2 + 1241*a^2*b^3 - 464*a*b^4 + 64*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 35*a^2*b + 47*a*b^2 - 16*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))) + 8*((a*b - b^2)*cos(d*x + c)^4 - 2*(a*b - b^2)*cos(d*x + c)^2 - a^2 + 2*a*b - b^2)*log(1/2*cos(d*x + c) + 1/2) - 8*((a*b - b^2)*cos(d*x + c)^4 - 2*(a*b - b^2)*cos(d*x + c)^2 - a^2 + 2*a*b - b^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)","B",0
218,1,3544,0,3.330872," ","integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","\frac{32 \, {\left(a b - b^{2}\right)} d x \cos\left(d x + c\right)^{4} - 64 \, {\left(a b - b^{2}\right)} d x \cos\left(d x + c\right)^{2} - 32 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} d x + {\left({\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} + 16 \, a^{3} - 47 \, a^{2} b + 35 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} \log\left(32 \, a^{3} - 166 \, a^{2} b + \frac{1125}{4} \, a b^{2} - \frac{625}{4} \, b^{3} - \frac{1}{4} \, {\left(128 \, a^{3} - 664 \, a^{2} b + 1125 \, a b^{2} - 625 \, b^{3}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(2 \, {\left(2 \, a^{4} b^{5} - 9 \, a^{3} b^{6} + 15 \, a^{2} b^{7} - 11 \, a b^{8} + 3 \, b^{9}\right)} d^{3} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(24 \, a^{3} b^{2} - 127 \, a^{2} b^{3} + 220 \, a b^{4} - 125 \, b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} + 16 \, a^{3} - 47 \, a^{2} b + 35 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{4} b^{3} - 73 \, a^{3} b^{4} + 123 \, a^{2} b^{5} - 91 \, a b^{6} + 25 \, b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{4} b^{3} - 73 \, a^{3} b^{4} + 123 \, a^{2} b^{5} - 91 \, a b^{6} + 25 \, b^{7}\right)} d^{2}\right)} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}}\right) - {\left({\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} + 16 \, a^{3} - 47 \, a^{2} b + 35 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} \log\left(32 \, a^{3} - 166 \, a^{2} b + \frac{1125}{4} \, a b^{2} - \frac{625}{4} \, b^{3} - \frac{1}{4} \, {\left(128 \, a^{3} - 664 \, a^{2} b + 1125 \, a b^{2} - 625 \, b^{3}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(2 \, {\left(2 \, a^{4} b^{5} - 9 \, a^{3} b^{6} + 15 \, a^{2} b^{7} - 11 \, a b^{8} + 3 \, b^{9}\right)} d^{3} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(24 \, a^{3} b^{2} - 127 \, a^{2} b^{3} + 220 \, a b^{4} - 125 \, b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} + 16 \, a^{3} - 47 \, a^{2} b + 35 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{4} b^{3} - 73 \, a^{3} b^{4} + 123 \, a^{2} b^{5} - 91 \, a b^{6} + 25 \, b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{4} b^{3} - 73 \, a^{3} b^{4} + 123 \, a^{2} b^{5} - 91 \, a b^{6} + 25 \, b^{7}\right)} d^{2}\right)} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}}\right) + {\left({\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} - 16 \, a^{3} + 47 \, a^{2} b - 35 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} \log\left(-32 \, a^{3} + 166 \, a^{2} b - \frac{1125}{4} \, a b^{2} + \frac{625}{4} \, b^{3} + \frac{1}{4} \, {\left(128 \, a^{3} - 664 \, a^{2} b + 1125 \, a b^{2} - 625 \, b^{3}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(2 \, {\left(2 \, a^{4} b^{5} - 9 \, a^{3} b^{6} + 15 \, a^{2} b^{7} - 11 \, a b^{8} + 3 \, b^{9}\right)} d^{3} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(24 \, a^{3} b^{2} - 127 \, a^{2} b^{3} + 220 \, a b^{4} - 125 \, b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} - 16 \, a^{3} + 47 \, a^{2} b - 35 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{4} b^{3} - 73 \, a^{3} b^{4} + 123 \, a^{2} b^{5} - 91 \, a b^{6} + 25 \, b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{4} b^{3} - 73 \, a^{3} b^{4} + 123 \, a^{2} b^{5} - 91 \, a b^{6} + 25 \, b^{7}\right)} d^{2}\right)} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}}\right) - {\left({\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} - 16 \, a^{3} + 47 \, a^{2} b - 35 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} \log\left(-32 \, a^{3} + 166 \, a^{2} b - \frac{1125}{4} \, a b^{2} + \frac{625}{4} \, b^{3} + \frac{1}{4} \, {\left(128 \, a^{3} - 664 \, a^{2} b + 1125 \, a b^{2} - 625 \, b^{3}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(2 \, {\left(2 \, a^{4} b^{5} - 9 \, a^{3} b^{6} + 15 \, a^{2} b^{7} - 11 \, a b^{8} + 3 \, b^{9}\right)} d^{3} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(24 \, a^{3} b^{2} - 127 \, a^{2} b^{3} + 220 \, a b^{4} - 125 \, b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}} - 16 \, a^{3} + 47 \, a^{2} b - 35 \, a b^{2}}{{\left(a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{4} b^{3} - 73 \, a^{3} b^{4} + 123 \, a^{2} b^{5} - 91 \, a b^{6} + 25 \, b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{4} b^{3} - 73 \, a^{3} b^{4} + 123 \, a^{2} b^{5} - 91 \, a b^{6} + 25 \, b^{7}\right)} d^{2}\right)} \sqrt{\frac{64 \, a^{5} - 464 \, a^{4} b + 1241 \, a^{3} b^{2} - 1450 \, a^{2} b^{3} + 625 \, a b^{4}}{{\left(a^{6} b^{7} - 6 \, a^{5} b^{8} + 15 \, a^{4} b^{9} - 20 \, a^{3} b^{10} + 15 \, a^{2} b^{11} - 6 \, a b^{12} + b^{13}\right)} d^{4}}}\right) - 8 \, {\left(a b \cos\left(d x + c\right)^{3} - 2 \, a b \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{32 \, {\left({\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{3} - b^{4}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} d\right)}}"," ",0,"1/32*(32*(a*b - b^2)*d*x*cos(d*x + c)^4 - 64*(a*b - b^2)*d*x*cos(d*x + c)^2 - 32*(a^2 - 2*a*b + b^2)*d*x + ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(32*a^3 - 166*a^2*b + 1125/4*a*b^2 - 625/4*b^3 - 1/4*(128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*cos(d*x + c)^2 + 1/2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*a*b^8 + 3*b^9)*d^3*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (24*a^3*b^2 - 127*a^2*b^3 + 220*a*b^4 - 125*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2)) + 1/4*(2*(16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2*cos(d*x + c)^2 - (16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2)*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))) - ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(32*a^3 - 166*a^2*b + 1125/4*a*b^2 - 625/4*b^3 - 1/4*(128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*cos(d*x + c)^2 - 1/2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*a*b^8 + 3*b^9)*d^3*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (24*a^3*b^2 - 127*a^2*b^3 + 220*a*b^4 - 125*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2)) + 1/4*(2*(16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2*cos(d*x + c)^2 - (16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2)*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))) + ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 16*a^3 + 47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(-32*a^3 + 166*a^2*b - 1125/4*a*b^2 + 625/4*b^3 + 1/4*(128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*cos(d*x + c)^2 + 1/2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*a*b^8 + 3*b^9)*d^3*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))*cos(d*x + c)*sin(d*x + c) + (24*a^3*b^2 - 127*a^2*b^3 + 220*a*b^4 - 125*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 16*a^3 + 47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2)) + 1/4*(2*(16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2*cos(d*x + c)^2 - (16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2)*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))) - ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 16*a^3 + 47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(-32*a^3 + 166*a^2*b - 1125/4*a*b^2 + 625/4*b^3 + 1/4*(128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*cos(d*x + c)^2 - 1/2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*a*b^8 + 3*b^9)*d^3*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))*cos(d*x + c)*sin(d*x + c) + (24*a^3*b^2 - 127*a^2*b^3 + 220*a*b^4 - 125*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 16*a^3 + 47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2)) + 1/4*(2*(16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2*cos(d*x + c)^2 - (16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2)*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))) - 8*(a*b*cos(d*x + c)^3 - 2*a*b*cos(d*x + c))*sin(d*x + c))/((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)","B",0
219,1,3135,0,3.197978," ","integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","-\frac{{\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} + 4 \, a^{2} - 15 \, a b + 15 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} \log\left(\frac{1}{4} \, {\left(20 \, a^{2} - 81 \, a b + 81 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 5 \, a^{2} + \frac{81}{4} \, a b - \frac{81}{4} \, b^{2} + \frac{1}{2} \, {\left({\left(a^{5} b^{3} - 6 \, a^{4} b^{4} + 12 \, a^{3} b^{5} - 10 \, a^{2} b^{6} + 3 \, a b^{7}\right)} d^{3} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 2 \, {\left(5 \, a^{3} b - 19 \, a^{2} b^{2} + 18 \, a b^{3}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} + 4 \, a^{2} - 15 \, a b + 15 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{5} b - 21 \, a^{4} b^{2} + 39 \, a^{3} b^{3} - 31 \, a^{2} b^{4} + 9 \, a b^{5}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{5} b - 21 \, a^{4} b^{2} + 39 \, a^{3} b^{3} - 31 \, a^{2} b^{4} + 9 \, a b^{5}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}}\right) - {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} + 4 \, a^{2} - 15 \, a b + 15 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} \log\left(\frac{1}{4} \, {\left(20 \, a^{2} - 81 \, a b + 81 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 5 \, a^{2} + \frac{81}{4} \, a b - \frac{81}{4} \, b^{2} - \frac{1}{2} \, {\left({\left(a^{5} b^{3} - 6 \, a^{4} b^{4} + 12 \, a^{3} b^{5} - 10 \, a^{2} b^{6} + 3 \, a b^{7}\right)} d^{3} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 2 \, {\left(5 \, a^{3} b - 19 \, a^{2} b^{2} + 18 \, a b^{3}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} + 4 \, a^{2} - 15 \, a b + 15 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{5} b - 21 \, a^{4} b^{2} + 39 \, a^{3} b^{3} - 31 \, a^{2} b^{4} + 9 \, a b^{5}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{5} b - 21 \, a^{4} b^{2} + 39 \, a^{3} b^{3} - 31 \, a^{2} b^{4} + 9 \, a b^{5}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}}\right) + {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} - 4 \, a^{2} + 15 \, a b - 15 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} \log\left(-\frac{1}{4} \, {\left(20 \, a^{2} - 81 \, a b + 81 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 5 \, a^{2} - \frac{81}{4} \, a b + \frac{81}{4} \, b^{2} + \frac{1}{2} \, {\left({\left(a^{5} b^{3} - 6 \, a^{4} b^{4} + 12 \, a^{3} b^{5} - 10 \, a^{2} b^{6} + 3 \, a b^{7}\right)} d^{3} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 2 \, {\left(5 \, a^{3} b - 19 \, a^{2} b^{2} + 18 \, a b^{3}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} - 4 \, a^{2} + 15 \, a b - 15 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{5} b - 21 \, a^{4} b^{2} + 39 \, a^{3} b^{3} - 31 \, a^{2} b^{4} + 9 \, a b^{5}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{5} b - 21 \, a^{4} b^{2} + 39 \, a^{3} b^{3} - 31 \, a^{2} b^{4} + 9 \, a b^{5}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}}\right) - {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)} \sqrt{\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} - 4 \, a^{2} + 15 \, a b - 15 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} \log\left(-\frac{1}{4} \, {\left(20 \, a^{2} - 81 \, a b + 81 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 5 \, a^{2} - \frac{81}{4} \, a b + \frac{81}{4} \, b^{2} - \frac{1}{2} \, {\left({\left(a^{5} b^{3} - 6 \, a^{4} b^{4} + 12 \, a^{3} b^{5} - 10 \, a^{2} b^{6} + 3 \, a b^{7}\right)} d^{3} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 2 \, {\left(5 \, a^{3} b - 19 \, a^{2} b^{2} + 18 \, a b^{3}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}} - 4 \, a^{2} + 15 \, a b - 15 \, b^{2}}{{\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{5} b - 21 \, a^{4} b^{2} + 39 \, a^{3} b^{3} - 31 \, a^{2} b^{4} + 9 \, a b^{5}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{5} b - 21 \, a^{4} b^{2} + 39 \, a^{3} b^{3} - 31 \, a^{2} b^{4} + 9 \, a b^{5}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{2} - 90 \, a b + 81 \, b^{2}}{{\left(a^{7} b^{3} - 6 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 20 \, a^{4} b^{6} + 15 \, a^{3} b^{7} - 6 \, a^{2} b^{8} + a b^{9}\right)} d^{4}}}\right) + 8 \, {\left(b \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{32 \, {\left({\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} d\right)}}"," ",0,"-1/32*(((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 4*a^2 - 15*a*b + 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))*log(1/4*(20*a^2 - 81*a*b + 81*b^2)*cos(d*x + c)^2 - 5*a^2 + 81/4*a*b - 81/4*b^2 + 1/2*((a^5*b^3 - 6*a^4*b^4 + 12*a^3*b^5 - 10*a^2*b^6 + 3*a*b^7)*d^3*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))*cos(d*x + c)*sin(d*x + c) + 2*(5*a^3*b - 19*a^2*b^2 + 18*a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 4*a^2 - 15*a*b + 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2)) + 1/4*(2*(4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2*cos(d*x + c)^2 - (4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2)*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))) - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 4*a^2 - 15*a*b + 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))*log(1/4*(20*a^2 - 81*a*b + 81*b^2)*cos(d*x + c)^2 - 5*a^2 + 81/4*a*b - 81/4*b^2 - 1/2*((a^5*b^3 - 6*a^4*b^4 + 12*a^3*b^5 - 10*a^2*b^6 + 3*a*b^7)*d^3*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))*cos(d*x + c)*sin(d*x + c) + 2*(5*a^3*b - 19*a^2*b^2 + 18*a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) + 4*a^2 - 15*a*b + 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2)) + 1/4*(2*(4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2*cos(d*x + c)^2 - (4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2)*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))) + ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))*log(-1/4*(20*a^2 - 81*a*b + 81*b^2)*cos(d*x + c)^2 + 5*a^2 - 81/4*a*b + 81/4*b^2 + 1/2*((a^5*b^3 - 6*a^4*b^4 + 12*a^3*b^5 - 10*a^2*b^6 + 3*a*b^7)*d^3*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))*cos(d*x + c)*sin(d*x + c) - 2*(5*a^3*b - 19*a^2*b^2 + 18*a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2)) + 1/4*(2*(4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2*cos(d*x + c)^2 - (4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2)*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))) - ((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2))*log(-1/4*(20*a^2 - 81*a*b + 81*b^2)*cos(d*x + c)^2 + 5*a^2 - 81/4*a*b + 81/4*b^2 - 1/2*((a^5*b^3 - 6*a^4*b^4 + 12*a^3*b^5 - 10*a^2*b^6 + 3*a*b^7)*d^3*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))*cos(d*x + c)*sin(d*x + c) - 2*(5*a^3*b - 19*a^2*b^2 + 18*a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4)) - 4*a^2 + 15*a*b - 15*b^2)/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d^2)) + 1/4*(2*(4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2*cos(d*x + c)^2 - (4*a^5*b - 21*a^4*b^2 + 39*a^3*b^3 - 31*a^2*b^4 + 9*a*b^5)*d^2)*sqrt((25*a^2 - 90*a*b + 81*b^2)/((a^7*b^3 - 6*a^6*b^4 + 15*a^5*b^5 - 20*a^4*b^6 + 15*a^3*b^7 - 6*a^2*b^8 + a*b^9)*d^4))) + 8*(b*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sin(d*x + c))/((a*b^2 - b^3)*d*cos(d*x + c)^4 - 2*(a*b^2 - b^3)*d*cos(d*x + c)^2 - (a^2*b - 2*a*b^2 + b^3)*d)","B",0
220,1,2796,0,2.115972," ","integrate(sin(d*x+c)^4/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","-\frac{{\left({\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + a + 3 \, b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} \log\left(\frac{1}{4} \, {\left(3 \, a + b\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d^{3} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(3 \, a^{3} + 4 \, a^{2} b + a b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + a + 3 \, b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - \frac{3}{4} \, a - \frac{1}{4} \, b\right) - {\left({\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + a + 3 \, b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} \log\left(\frac{1}{4} \, {\left(3 \, a + b\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d^{3} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(3 \, a^{3} + 4 \, a^{2} b + a b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + a + 3 \, b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - \frac{3}{4} \, a - \frac{1}{4} \, b\right) + {\left({\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - a - 3 \, b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} \log\left(-\frac{1}{4} \, {\left(3 \, a + b\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d^{3} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(3 \, a^{3} + 4 \, a^{2} b + a b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - a - 3 \, b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + \frac{3}{4} \, a + \frac{1}{4} \, b\right) - {\left({\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - a - 3 \, b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} \log\left(-\frac{1}{4} \, {\left(3 \, a + b\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(2 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d^{3} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(3 \, a^{3} + 4 \, a^{2} b + a b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} - a - 3 \, b}{{\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{{\left(a^{9} b - 6 \, a^{8} b^{2} + 15 \, a^{7} b^{3} - 20 \, a^{6} b^{4} + 15 \, a^{5} b^{5} - 6 \, a^{4} b^{6} + a^{3} b^{7}\right)} d^{4}}} + \frac{3}{4} \, a + \frac{1}{4} \, b\right) + 8 \, {\left(\cos\left(d x + c\right)^{3} - 2 \, \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{32 \, {\left({\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a b - b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} d\right)}}"," ",0,"-1/32*(((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + a + 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log(1/4*(3*a + b)*cos(d*x + c)^2 + 1/2*(2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d^3*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) - (3*a^3 + 4*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + a + 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - 3/4*a - 1/4*b) - ((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + a + 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log(1/4*(3*a + b)*cos(d*x + c)^2 - 1/2*(2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d^3*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) - (3*a^3 + 4*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + a + 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - 3/4*a - 1/4*b) + ((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - a - 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log(-1/4*(3*a + b)*cos(d*x + c)^2 + 1/2*(2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d^3*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) + (3*a^3 + 4*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - a - 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + 3/4*a + 1/4*b) - ((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - a - 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2))*log(-1/4*(3*a + b)*cos(d*x + c)^2 - 1/2*(2*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d^3*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) + (3*a^3 + 4*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) - a - 3*b)/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + 3/4*a + 1/4*b) + 8*(cos(d*x + c)^3 - 2*cos(d*x + c))*sin(d*x + c))/((a*b - b^2)*d*cos(d*x + c)^4 - 2*(a*b - b^2)*d*cos(d*x + c)^2 - (a^2 - 2*a*b + b^2)*d)","B",0
221,1,3445,0,3.093592," ","integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","\frac{{\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} + 4 \, a^{2} + a b - b^{2}}{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2}}} \log\left(8 \, a^{3} - 7 \, a^{2} b + \frac{9}{4} \, a b^{2} - \frac{1}{4} \, b^{3} - \frac{1}{4} \, {\left(32 \, a^{3} - 28 \, a^{2} b + 9 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(3 \, a^{8} b - 10 \, a^{7} b^{2} + 12 \, a^{6} b^{3} - 6 \, a^{5} b^{4} + a^{4} b^{5}\right)} d^{3} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 2 \, {\left(8 \, a^{5} - 5 \, a^{4} b + a^{3} b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} + 4 \, a^{2} + a b - b^{2}}{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{7} - 13 \, a^{6} b + 15 \, a^{5} b^{2} - 7 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{7} - 13 \, a^{6} b + 15 \, a^{5} b^{2} - 7 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}\right)} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}}\right) - {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} + 4 \, a^{2} + a b - b^{2}}{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2}}} \log\left(8 \, a^{3} - 7 \, a^{2} b + \frac{9}{4} \, a b^{2} - \frac{1}{4} \, b^{3} - \frac{1}{4} \, {\left(32 \, a^{3} - 28 \, a^{2} b + 9 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(3 \, a^{8} b - 10 \, a^{7} b^{2} + 12 \, a^{6} b^{3} - 6 \, a^{5} b^{4} + a^{4} b^{5}\right)} d^{3} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 2 \, {\left(8 \, a^{5} - 5 \, a^{4} b + a^{3} b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} + 4 \, a^{2} + a b - b^{2}}{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{7} - 13 \, a^{6} b + 15 \, a^{5} b^{2} - 7 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{7} - 13 \, a^{6} b + 15 \, a^{5} b^{2} - 7 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}\right)} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}}\right) + {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} - 4 \, a^{2} - a b + b^{2}}{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2}}} \log\left(-8 \, a^{3} + 7 \, a^{2} b - \frac{9}{4} \, a b^{2} + \frac{1}{4} \, b^{3} + \frac{1}{4} \, {\left(32 \, a^{3} - 28 \, a^{2} b + 9 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(3 \, a^{8} b - 10 \, a^{7} b^{2} + 12 \, a^{6} b^{3} - 6 \, a^{5} b^{4} + a^{4} b^{5}\right)} d^{3} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 2 \, {\left(8 \, a^{5} - 5 \, a^{4} b + a^{3} b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} - 4 \, a^{2} - a b + b^{2}}{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{7} - 13 \, a^{6} b + 15 \, a^{5} b^{2} - 7 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{7} - 13 \, a^{6} b + 15 \, a^{5} b^{2} - 7 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}\right)} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}}\right) - {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} - 4 \, a^{2} - a b + b^{2}}{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2}}} \log\left(-8 \, a^{3} + 7 \, a^{2} b - \frac{9}{4} \, a b^{2} + \frac{1}{4} \, b^{3} + \frac{1}{4} \, {\left(32 \, a^{3} - 28 \, a^{2} b + 9 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(3 \, a^{8} b - 10 \, a^{7} b^{2} + 12 \, a^{6} b^{3} - 6 \, a^{5} b^{4} + a^{4} b^{5}\right)} d^{3} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 2 \, {\left(8 \, a^{5} - 5 \, a^{4} b + a^{3} b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}} - 4 \, a^{2} - a b + b^{2}}{{\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{7} - 13 \, a^{6} b + 15 \, a^{5} b^{2} - 7 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{7} - 13 \, a^{6} b + 15 \, a^{5} b^{2} - 7 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{2}\right)} \sqrt{\frac{64 \, a^{4} - 80 \, a^{3} b + 41 \, a^{2} b^{2} - 10 \, a b^{3} + b^{4}}{{\left(a^{11} b - 6 \, a^{10} b^{2} + 15 \, a^{9} b^{3} - 20 \, a^{8} b^{4} + 15 \, a^{7} b^{5} - 6 \, a^{6} b^{6} + a^{5} b^{7}\right)} d^{4}}}\right) - 8 \, {\left(b \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{32 \, {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)}}"," ",0,"1/32*(((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(-((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4)) + 4*a^2 + a*b - b^2)/((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2))*log(8*a^3 - 7*a^2*b + 9/4*a*b^2 - 1/4*b^3 - 1/4*(32*a^3 - 28*a^2*b + 9*a*b^2 - b^3)*cos(d*x + c)^2 + 1/2*((3*a^8*b - 10*a^7*b^2 + 12*a^6*b^3 - 6*a^5*b^4 + a^4*b^5)*d^3*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) - 2*(8*a^5 - 5*a^4*b + a^3*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4)) + 4*a^2 + a*b - b^2)/((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2)) + 1/4*(2*(4*a^7 - 13*a^6*b + 15*a^5*b^2 - 7*a^4*b^3 + a^3*b^4)*d^2*cos(d*x + c)^2 - (4*a^7 - 13*a^6*b + 15*a^5*b^2 - 7*a^4*b^3 + a^3*b^4)*d^2)*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4))) - ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(-((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4)) + 4*a^2 + a*b - b^2)/((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2))*log(8*a^3 - 7*a^2*b + 9/4*a*b^2 - 1/4*b^3 - 1/4*(32*a^3 - 28*a^2*b + 9*a*b^2 - b^3)*cos(d*x + c)^2 - 1/2*((3*a^8*b - 10*a^7*b^2 + 12*a^6*b^3 - 6*a^5*b^4 + a^4*b^5)*d^3*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) - 2*(8*a^5 - 5*a^4*b + a^3*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4)) + 4*a^2 + a*b - b^2)/((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2)) + 1/4*(2*(4*a^7 - 13*a^6*b + 15*a^5*b^2 - 7*a^4*b^3 + a^3*b^4)*d^2*cos(d*x + c)^2 - (4*a^7 - 13*a^6*b + 15*a^5*b^2 - 7*a^4*b^3 + a^3*b^4)*d^2)*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4))) + ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4)) - 4*a^2 - a*b + b^2)/((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2))*log(-8*a^3 + 7*a^2*b - 9/4*a*b^2 + 1/4*b^3 + 1/4*(32*a^3 - 28*a^2*b + 9*a*b^2 - b^3)*cos(d*x + c)^2 + 1/2*((3*a^8*b - 10*a^7*b^2 + 12*a^6*b^3 - 6*a^5*b^4 + a^4*b^5)*d^3*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) + 2*(8*a^5 - 5*a^4*b + a^3*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4)) - 4*a^2 - a*b + b^2)/((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2)) + 1/4*(2*(4*a^7 - 13*a^6*b + 15*a^5*b^2 - 7*a^4*b^3 + a^3*b^4)*d^2*cos(d*x + c)^2 - (4*a^7 - 13*a^6*b + 15*a^5*b^2 - 7*a^4*b^3 + a^3*b^4)*d^2)*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4))) - ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4)) - 4*a^2 - a*b + b^2)/((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2))*log(-8*a^3 + 7*a^2*b - 9/4*a*b^2 + 1/4*b^3 + 1/4*(32*a^3 - 28*a^2*b + 9*a*b^2 - b^3)*cos(d*x + c)^2 - 1/2*((3*a^8*b - 10*a^7*b^2 + 12*a^6*b^3 - 6*a^5*b^4 + a^4*b^5)*d^3*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) + 2*(8*a^5 - 5*a^4*b + a^3*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4)) - 4*a^2 - a*b + b^2)/((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2)) + 1/4*(2*(4*a^7 - 13*a^6*b + 15*a^5*b^2 - 7*a^4*b^3 + a^3*b^4)*d^2*cos(d*x + c)^2 - (4*a^7 - 13*a^6*b + 15*a^5*b^2 - 7*a^4*b^3 + a^3*b^4)*d^2)*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4))) - 8*(b*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sin(d*x + c))/((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)","B",0
222,1,3477,0,3.166187," ","integrate(1/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","-\frac{{\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} + 16 \, a^{2} - 15 \, a b + 3 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} \log\left(96 \, a^{3} b - 170 \, a^{2} b^{2} + \frac{405}{4} \, a b^{3} - \frac{81}{4} \, b^{4} - \frac{1}{4} \, {\left(384 \, a^{3} b - 680 \, a^{2} b^{2} + 405 \, a b^{3} - 81 \, b^{4}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(2 \, {\left(2 \, a^{10} - 7 \, a^{9} b + 9 \, a^{8} b^{2} - 5 \, a^{7} b^{3} + a^{6} b^{4}\right)} d^{3} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(120 \, a^{5} b - 217 \, a^{4} b^{2} + 132 \, a^{3} b^{3} - 27 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} + 16 \, a^{2} - 15 \, a b + 3 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{8} - 57 \, a^{7} b + 75 \, a^{6} b^{2} - 43 \, a^{5} b^{3} + 9 \, a^{4} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{8} - 57 \, a^{7} b + 75 \, a^{6} b^{2} - 43 \, a^{5} b^{3} + 9 \, a^{4} b^{4}\right)} d^{2}\right)} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}}\right) - {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} + 16 \, a^{2} - 15 \, a b + 3 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} \log\left(96 \, a^{3} b - 170 \, a^{2} b^{2} + \frac{405}{4} \, a b^{3} - \frac{81}{4} \, b^{4} - \frac{1}{4} \, {\left(384 \, a^{3} b - 680 \, a^{2} b^{2} + 405 \, a b^{3} - 81 \, b^{4}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(2 \, {\left(2 \, a^{10} - 7 \, a^{9} b + 9 \, a^{8} b^{2} - 5 \, a^{7} b^{3} + a^{6} b^{4}\right)} d^{3} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(120 \, a^{5} b - 217 \, a^{4} b^{2} + 132 \, a^{3} b^{3} - 27 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} + 16 \, a^{2} - 15 \, a b + 3 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{8} - 57 \, a^{7} b + 75 \, a^{6} b^{2} - 43 \, a^{5} b^{3} + 9 \, a^{4} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{8} - 57 \, a^{7} b + 75 \, a^{6} b^{2} - 43 \, a^{5} b^{3} + 9 \, a^{4} b^{4}\right)} d^{2}\right)} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}}\right) + {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} - 16 \, a^{2} + 15 \, a b - 3 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} \log\left(-96 \, a^{3} b + 170 \, a^{2} b^{2} - \frac{405}{4} \, a b^{3} + \frac{81}{4} \, b^{4} + \frac{1}{4} \, {\left(384 \, a^{3} b - 680 \, a^{2} b^{2} + 405 \, a b^{3} - 81 \, b^{4}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(2 \, {\left(2 \, a^{10} - 7 \, a^{9} b + 9 \, a^{8} b^{2} - 5 \, a^{7} b^{3} + a^{6} b^{4}\right)} d^{3} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(120 \, a^{5} b - 217 \, a^{4} b^{2} + 132 \, a^{3} b^{3} - 27 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} - 16 \, a^{2} + 15 \, a b - 3 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{8} - 57 \, a^{7} b + 75 \, a^{6} b^{2} - 43 \, a^{5} b^{3} + 9 \, a^{4} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{8} - 57 \, a^{7} b + 75 \, a^{6} b^{2} - 43 \, a^{5} b^{3} + 9 \, a^{4} b^{4}\right)} d^{2}\right)} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}}\right) - {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} - 16 \, a^{2} + 15 \, a b - 3 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} \log\left(-96 \, a^{3} b + 170 \, a^{2} b^{2} - \frac{405}{4} \, a b^{3} + \frac{81}{4} \, b^{4} + \frac{1}{4} \, {\left(384 \, a^{3} b - 680 \, a^{2} b^{2} + 405 \, a b^{3} - 81 \, b^{4}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(2 \, {\left(2 \, a^{10} - 7 \, a^{9} b + 9 \, a^{8} b^{2} - 5 \, a^{7} b^{3} + a^{6} b^{4}\right)} d^{3} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(120 \, a^{5} b - 217 \, a^{4} b^{2} + 132 \, a^{3} b^{3} - 27 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}} - 16 \, a^{2} + 15 \, a b - 3 \, b^{2}}{{\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{8} - 57 \, a^{7} b + 75 \, a^{6} b^{2} - 43 \, a^{5} b^{3} + 9 \, a^{4} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{8} - 57 \, a^{7} b + 75 \, a^{6} b^{2} - 43 \, a^{5} b^{3} + 9 \, a^{4} b^{4}\right)} d^{2}\right)} \sqrt{\frac{576 \, a^{4} b - 1392 \, a^{3} b^{2} + 1273 \, a^{2} b^{3} - 522 \, a b^{4} + 81 \, b^{5}}{{\left(a^{13} - 6 \, a^{12} b + 15 \, a^{11} b^{2} - 20 \, a^{10} b^{3} + 15 \, a^{9} b^{4} - 6 \, a^{8} b^{5} + a^{7} b^{6}\right)} d^{4}}}\right) + 8 \, {\left(b \cos\left(d x + c\right)^{3} - 2 \, b \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{32 \, {\left({\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} b - a b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d\right)}}"," ",0,"-1/32*(((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(96*a^3*b - 170*a^2*b^2 + 405/4*a*b^3 - 81/4*b^4 - 1/4*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cos(d*x + c)^2 + 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - (120*a^5*b - 217*a^4*b^2 + 132*a^3*b^3 - 27*a^2*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2)) + 1/4*(2*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*cos(d*x + c)^2 - (16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))) - ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(96*a^3*b - 170*a^2*b^2 + 405/4*a*b^3 - 81/4*b^4 - 1/4*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cos(d*x + c)^2 - 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - (120*a^5*b - 217*a^4*b^2 + 132*a^3*b^3 - 27*a^2*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2)) + 1/4*(2*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*cos(d*x + c)^2 - (16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))) + ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(-96*a^3*b + 170*a^2*b^2 - 405/4*a*b^3 + 81/4*b^4 + 1/4*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cos(d*x + c)^2 + 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + (120*a^5*b - 217*a^4*b^2 + 132*a^3*b^3 - 27*a^2*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2)) + 1/4*(2*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*cos(d*x + c)^2 - (16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))) - ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(-96*a^3*b + 170*a^2*b^2 - 405/4*a*b^3 + 81/4*b^4 + 1/4*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cos(d*x + c)^2 - 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + (120*a^5*b - 217*a^4*b^2 + 132*a^3*b^3 - 27*a^2*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2)) + 1/4*(2*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*cos(d*x + c)^2 - (16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))) + 8*(b*cos(d*x + c)^3 - 2*b*cos(d*x + c))*sin(d*x + c))/((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)","B",0
223,1,3648,0,3.385493," ","integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^2,x, algorithm=""fricas"")","-\frac{8 \, {\left(4 \, a b - 5 \, b^{2}\right)} \cos\left(d x + c\right)^{5} - 8 \, {\left(7 \, a b - 10 \, b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left({\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} + 36 \, a^{2} b - 47 \, a b^{2} + 15 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} \log\left(432 \, a^{3} b^{2} - 921 \, a^{2} b^{3} + \frac{2625}{4} \, a b^{4} - \frac{625}{4} \, b^{5} - \frac{1}{4} \, {\left(1728 \, a^{3} b^{2} - 3684 \, a^{2} b^{3} + 2625 \, a b^{4} - 625 \, b^{5}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(7 \, a^{11} - 26 \, a^{10} b + 36 \, a^{9} b^{2} - 22 \, a^{8} b^{3} + 5 \, a^{7} b^{4}\right)} d^{3} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 2 \, {\left(144 \, a^{6} b - 303 \, a^{5} b^{2} + 213 \, a^{4} b^{3} - 50 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} + 36 \, a^{2} b - 47 \, a b^{2} + 15 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(36 \, a^{9} - 133 \, a^{8} b + 183 \, a^{7} b^{2} - 111 \, a^{6} b^{3} + 25 \, a^{5} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(36 \, a^{9} - 133 \, a^{8} b + 183 \, a^{7} b^{2} - 111 \, a^{6} b^{3} + 25 \, a^{5} b^{4}\right)} d^{2}\right)} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}}\right) \sin\left(d x + c\right) + {\left({\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} + 36 \, a^{2} b - 47 \, a b^{2} + 15 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} \log\left(432 \, a^{3} b^{2} - 921 \, a^{2} b^{3} + \frac{2625}{4} \, a b^{4} - \frac{625}{4} \, b^{5} - \frac{1}{4} \, {\left(1728 \, a^{3} b^{2} - 3684 \, a^{2} b^{3} + 2625 \, a b^{4} - 625 \, b^{5}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(7 \, a^{11} - 26 \, a^{10} b + 36 \, a^{9} b^{2} - 22 \, a^{8} b^{3} + 5 \, a^{7} b^{4}\right)} d^{3} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - 2 \, {\left(144 \, a^{6} b - 303 \, a^{5} b^{2} + 213 \, a^{4} b^{3} - 50 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} + 36 \, a^{2} b - 47 \, a b^{2} + 15 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(36 \, a^{9} - 133 \, a^{8} b + 183 \, a^{7} b^{2} - 111 \, a^{6} b^{3} + 25 \, a^{5} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(36 \, a^{9} - 133 \, a^{8} b + 183 \, a^{7} b^{2} - 111 \, a^{6} b^{3} + 25 \, a^{5} b^{4}\right)} d^{2}\right)} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}}\right) \sin\left(d x + c\right) - {\left({\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} - 36 \, a^{2} b + 47 \, a b^{2} - 15 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} \log\left(-432 \, a^{3} b^{2} + 921 \, a^{2} b^{3} - \frac{2625}{4} \, a b^{4} + \frac{625}{4} \, b^{5} + \frac{1}{4} \, {\left(1728 \, a^{3} b^{2} - 3684 \, a^{2} b^{3} + 2625 \, a b^{4} - 625 \, b^{5}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(7 \, a^{11} - 26 \, a^{10} b + 36 \, a^{9} b^{2} - 22 \, a^{8} b^{3} + 5 \, a^{7} b^{4}\right)} d^{3} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 2 \, {\left(144 \, a^{6} b - 303 \, a^{5} b^{2} + 213 \, a^{4} b^{3} - 50 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} - 36 \, a^{2} b + 47 \, a b^{2} - 15 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(36 \, a^{9} - 133 \, a^{8} b + 183 \, a^{7} b^{2} - 111 \, a^{6} b^{3} + 25 \, a^{5} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(36 \, a^{9} - 133 \, a^{8} b + 183 \, a^{7} b^{2} - 111 \, a^{6} b^{3} + 25 \, a^{5} b^{4}\right)} d^{2}\right)} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}}\right) \sin\left(d x + c\right) + {\left({\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} - 36 \, a^{2} b + 47 \, a b^{2} - 15 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} \log\left(-432 \, a^{3} b^{2} + 921 \, a^{2} b^{3} - \frac{2625}{4} \, a b^{4} + \frac{625}{4} \, b^{5} + \frac{1}{4} \, {\left(1728 \, a^{3} b^{2} - 3684 \, a^{2} b^{3} + 2625 \, a b^{4} - 625 \, b^{5}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(7 \, a^{11} - 26 \, a^{10} b + 36 \, a^{9} b^{2} - 22 \, a^{8} b^{3} + 5 \, a^{7} b^{4}\right)} d^{3} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + 2 \, {\left(144 \, a^{6} b - 303 \, a^{5} b^{2} + 213 \, a^{4} b^{3} - 50 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}} - 36 \, a^{2} b + 47 \, a b^{2} - 15 \, b^{3}}{{\left(a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(36 \, a^{9} - 133 \, a^{8} b + 183 \, a^{7} b^{2} - 111 \, a^{6} b^{3} + 25 \, a^{5} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(36 \, a^{9} - 133 \, a^{8} b + 183 \, a^{7} b^{2} - 111 \, a^{6} b^{3} + 25 \, a^{5} b^{4}\right)} d^{2}\right)} \sqrt{\frac{2304 \, a^{4} b^{3} - 6624 \, a^{3} b^{4} + 7161 \, a^{2} b^{5} - 3450 \, a b^{6} + 625 \, b^{7}}{{\left(a^{15} - 6 \, a^{14} b + 15 \, a^{13} b^{2} - 20 \, a^{12} b^{3} + 15 \, a^{11} b^{4} - 6 \, a^{10} b^{5} + a^{9} b^{6}\right)} d^{4}}}\right) \sin\left(d x + c\right) - 8 \, {\left(4 \, a^{2} - 7 \, a b + 5 \, b^{2}\right)} \cos\left(d x + c\right)}{32 \, {\left({\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)} \sin\left(d x + c\right)}"," ",0,"-1/32*(8*(4*a*b - 5*b^2)*cos(d*x + c)^5 - 8*(7*a*b - 10*b^2)*cos(d*x + c)^3 - ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(432*a^3*b^2 - 921*a^2*b^3 + 2625/4*a*b^4 - 625/4*b^5 - 1/4*(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cos(d*x + c)^2 + 1/2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - 2*(144*a^6*b - 303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2)) + 1/4*(2*(36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*cos(d*x + c)^2 - (36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2)*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)))*sin(d*x + c) + ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(432*a^3*b^2 - 921*a^2*b^3 + 2625/4*a*b^4 - 625/4*b^5 - 1/4*(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cos(d*x + c)^2 - 1/2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - 2*(144*a^6*b - 303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2)) + 1/4*(2*(36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*cos(d*x + c)^2 - (36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2)*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)))*sin(d*x + c) - ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b + 47*a*b^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(-432*a^3*b^2 + 921*a^2*b^3 - 2625/4*a*b^4 + 625/4*b^5 + 1/4*(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cos(d*x + c)^2 + 1/2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + 2*(144*a^6*b - 303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b + 47*a*b^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2)) + 1/4*(2*(36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*cos(d*x + c)^2 - (36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2)*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)))*sin(d*x + c) + ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b + 47*a*b^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(-432*a^3*b^2 + 921*a^2*b^3 - 2625/4*a*b^4 + 625/4*b^5 + 1/4*(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cos(d*x + c)^2 - 1/2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + 2*(144*a^6*b - 303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b + 47*a*b^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2)) + 1/4*(2*(36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*cos(d*x + c)^2 - (36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2)*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)))*sin(d*x + c) - 8*(4*a^2 - 7*a*b + 5*b^2)*cos(d*x + c))/(((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sin(d*x + c))","B",0
224,1,4640,0,3.092409," ","integrate(sin(d*x+c)^9/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","\frac{8 \, {\left(2 \, a b^{2} - 5 \, b^{3}\right)} \cos\left(d x + c\right)^{7} - 12 \, {\left(3 \, a^{2} b - a b^{2} - 10 \, b^{3}\right)} \cos\left(d x + c\right)^{5} + 24 \, {\left(3 \, a^{2} b - 2 \, a b^{2} - 5 \, b^{3}\right)} \cos\left(d x + c\right)^{3} + {\left({\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{3} - 5 \, a^{2} b^{4} + 7 \, a b^{5} - 3 \, b^{6}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 94 \, a^{3} b + 155 \, a^{2} b^{2} - 76 \, a b^{3} - 144 \, b^{4} + {\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}}}{{\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2}}} \log\left({\left(625 \, a^{6} - 5250 \, a^{5} b + 22509 \, a^{4} b^{2} - 57820 \, a^{3} b^{3} + 96336 \, a^{2} b^{4} - 98304 \, a b^{5} + 55296 \, b^{6}\right)} \cos\left(d x + c\right) - {\left({\left(a^{8} b^{7} - 6 \, a^{7} b^{8} + 27 \, a^{6} b^{9} - 80 \, a^{5} b^{10} + 135 \, a^{4} b^{11} - 126 \, a^{3} b^{12} + 61 \, a^{2} b^{13} - 12 \, a b^{14}\right)} d^{3} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}} + {\left(125 \, a^{7} b^{2} - 1045 \, a^{6} b^{3} + 4305 \, a^{5} b^{4} - 10583 \, a^{4} b^{5} + 16798 \, a^{3} b^{6} - 16320 \, a^{2} b^{7} + 8448 \, a b^{8}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 94 \, a^{3} b + 155 \, a^{2} b^{2} - 76 \, a b^{3} - 144 \, b^{4} + {\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}}}{{\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2}}}\right) - {\left({\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{3} - 5 \, a^{2} b^{4} + 7 \, a b^{5} - 3 \, b^{6}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 94 \, a^{3} b + 155 \, a^{2} b^{2} - 76 \, a b^{3} - 144 \, b^{4} - {\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}}}{{\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2}}} \log\left({\left(625 \, a^{6} - 5250 \, a^{5} b + 22509 \, a^{4} b^{2} - 57820 \, a^{3} b^{3} + 96336 \, a^{2} b^{4} - 98304 \, a b^{5} + 55296 \, b^{6}\right)} \cos\left(d x + c\right) - {\left({\left(a^{8} b^{7} - 6 \, a^{7} b^{8} + 27 \, a^{6} b^{9} - 80 \, a^{5} b^{10} + 135 \, a^{4} b^{11} - 126 \, a^{3} b^{12} + 61 \, a^{2} b^{13} - 12 \, a b^{14}\right)} d^{3} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}} - {\left(125 \, a^{7} b^{2} - 1045 \, a^{6} b^{3} + 4305 \, a^{5} b^{4} - 10583 \, a^{4} b^{5} + 16798 \, a^{3} b^{6} - 16320 \, a^{2} b^{7} + 8448 \, a b^{8}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 94 \, a^{3} b + 155 \, a^{2} b^{2} - 76 \, a b^{3} - 144 \, b^{4} - {\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}}}{{\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2}}}\right) - {\left({\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{3} - 5 \, a^{2} b^{4} + 7 \, a b^{5} - 3 \, b^{6}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 94 \, a^{3} b + 155 \, a^{2} b^{2} - 76 \, a b^{3} - 144 \, b^{4} + {\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}}}{{\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2}}} \log\left(-{\left(625 \, a^{6} - 5250 \, a^{5} b + 22509 \, a^{4} b^{2} - 57820 \, a^{3} b^{3} + 96336 \, a^{2} b^{4} - 98304 \, a b^{5} + 55296 \, b^{6}\right)} \cos\left(d x + c\right) - {\left({\left(a^{8} b^{7} - 6 \, a^{7} b^{8} + 27 \, a^{6} b^{9} - 80 \, a^{5} b^{10} + 135 \, a^{4} b^{11} - 126 \, a^{3} b^{12} + 61 \, a^{2} b^{13} - 12 \, a b^{14}\right)} d^{3} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}} + {\left(125 \, a^{7} b^{2} - 1045 \, a^{6} b^{3} + 4305 \, a^{5} b^{4} - 10583 \, a^{4} b^{5} + 16798 \, a^{3} b^{6} - 16320 \, a^{2} b^{7} + 8448 \, a b^{8}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 94 \, a^{3} b + 155 \, a^{2} b^{2} - 76 \, a b^{3} - 144 \, b^{4} + {\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}}}{{\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2}}}\right) + {\left({\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{3} - 5 \, a^{2} b^{4} + 7 \, a b^{5} - 3 \, b^{6}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 94 \, a^{3} b + 155 \, a^{2} b^{2} - 76 \, a b^{3} - 144 \, b^{4} - {\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}}}{{\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2}}} \log\left(-{\left(625 \, a^{6} - 5250 \, a^{5} b + 22509 \, a^{4} b^{2} - 57820 \, a^{3} b^{3} + 96336 \, a^{2} b^{4} - 98304 \, a b^{5} + 55296 \, b^{6}\right)} \cos\left(d x + c\right) - {\left({\left(a^{8} b^{7} - 6 \, a^{7} b^{8} + 27 \, a^{6} b^{9} - 80 \, a^{5} b^{10} + 135 \, a^{4} b^{11} - 126 \, a^{3} b^{12} + 61 \, a^{2} b^{13} - 12 \, a b^{14}\right)} d^{3} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}} - {\left(125 \, a^{7} b^{2} - 1045 \, a^{6} b^{3} + 4305 \, a^{5} b^{4} - 10583 \, a^{4} b^{5} + 16798 \, a^{3} b^{6} - 16320 \, a^{2} b^{7} + 8448 \, a b^{8}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 94 \, a^{3} b + 155 \, a^{2} b^{2} - 76 \, a b^{3} - 144 \, b^{4} - {\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2} \sqrt{\frac{625 \, a^{8} - 6700 \, a^{7} b + 35406 \, a^{6} b^{2} - 117532 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 437952 \, a^{3} b^{5} + 498432 \, a^{2} b^{6} - 368640 \, a b^{7} + 147456 \, b^{8}}{{\left(a^{11} b^{9} - 10 \, a^{10} b^{10} + 45 \, a^{9} b^{11} - 120 \, a^{8} b^{12} + 210 \, a^{7} b^{13} - 252 \, a^{6} b^{14} + 210 \, a^{5} b^{15} - 120 \, a^{4} b^{16} + 45 \, a^{3} b^{17} - 10 \, a^{2} b^{18} + a b^{19}\right)} d^{4}}}}{{\left(a^{6} b^{4} - 5 \, a^{5} b^{5} + 10 \, a^{4} b^{6} - 10 \, a^{3} b^{7} + 5 \, a^{2} b^{8} - a b^{9}\right)} d^{2}}}\right) + 20 \, {\left(a^{3} - 4 \, a^{2} b + a b^{2} + 2 \, b^{3}\right)} \cos\left(d x + c\right)}{128 \, {\left({\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{3} - 5 \, a^{2} b^{4} + 7 \, a b^{5} - 3 \, b^{6}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{3} - 3 \, a^{2} b^{4} + 3 \, a b^{5} - b^{6}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b^{2} - 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} - 4 \, a b^{5} + b^{6}\right)} d\right)}}"," ",0,"1/128*(8*(2*a*b^2 - 5*b^3)*cos(d*x + c)^7 - 12*(3*a^2*b - a*b^2 - 10*b^3)*cos(d*x + c)^5 + 24*(3*a^2*b - 2*a*b^2 - 5*b^3)*cos(d*x + c)^3 + ((a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^8 - 4*(a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^6 - 2*(a^3*b^3 - 5*a^2*b^4 + 7*a*b^5 - 3*b^6)*d*cos(d*x + c)^4 + 4*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cos(d*x + c)^2 + (a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 144*b^4 + (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))*log((625*a^6 - 5250*a^5*b + 22509*a^4*b^2 - 57820*a^3*b^3 + 96336*a^2*b^4 - 98304*a*b^5 + 55296*b^6)*cos(d*x + c) - ((a^8*b^7 - 6*a^7*b^8 + 27*a^6*b^9 - 80*a^5*b^10 + 135*a^4*b^11 - 126*a^3*b^12 + 61*a^2*b^13 - 12*a*b^14)*d^3*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)) + (125*a^7*b^2 - 1045*a^6*b^3 + 4305*a^5*b^4 - 10583*a^4*b^5 + 16798*a^3*b^6 - 16320*a^2*b^7 + 8448*a*b^8)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 144*b^4 + (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))) - ((a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^8 - 4*(a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^6 - 2*(a^3*b^3 - 5*a^2*b^4 + 7*a*b^5 - 3*b^6)*d*cos(d*x + c)^4 + 4*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cos(d*x + c)^2 + (a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 144*b^4 - (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))*log((625*a^6 - 5250*a^5*b + 22509*a^4*b^2 - 57820*a^3*b^3 + 96336*a^2*b^4 - 98304*a*b^5 + 55296*b^6)*cos(d*x + c) - ((a^8*b^7 - 6*a^7*b^8 + 27*a^6*b^9 - 80*a^5*b^10 + 135*a^4*b^11 - 126*a^3*b^12 + 61*a^2*b^13 - 12*a*b^14)*d^3*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)) - (125*a^7*b^2 - 1045*a^6*b^3 + 4305*a^5*b^4 - 10583*a^4*b^5 + 16798*a^3*b^6 - 16320*a^2*b^7 + 8448*a*b^8)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 144*b^4 - (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))) - ((a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^8 - 4*(a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^6 - 2*(a^3*b^3 - 5*a^2*b^4 + 7*a*b^5 - 3*b^6)*d*cos(d*x + c)^4 + 4*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cos(d*x + c)^2 + (a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 144*b^4 + (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))*log(-(625*a^6 - 5250*a^5*b + 22509*a^4*b^2 - 57820*a^3*b^3 + 96336*a^2*b^4 - 98304*a*b^5 + 55296*b^6)*cos(d*x + c) - ((a^8*b^7 - 6*a^7*b^8 + 27*a^6*b^9 - 80*a^5*b^10 + 135*a^4*b^11 - 126*a^3*b^12 + 61*a^2*b^13 - 12*a*b^14)*d^3*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)) + (125*a^7*b^2 - 1045*a^6*b^3 + 4305*a^5*b^4 - 10583*a^4*b^5 + 16798*a^3*b^6 - 16320*a^2*b^7 + 8448*a*b^8)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 144*b^4 + (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))) + ((a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^8 - 4*(a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^6 - 2*(a^3*b^3 - 5*a^2*b^4 + 7*a*b^5 - 3*b^6)*d*cos(d*x + c)^4 + 4*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cos(d*x + c)^2 + (a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 144*b^4 - (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))*log(-(625*a^6 - 5250*a^5*b + 22509*a^4*b^2 - 57820*a^3*b^3 + 96336*a^2*b^4 - 98304*a*b^5 + 55296*b^6)*cos(d*x + c) - ((a^8*b^7 - 6*a^7*b^8 + 27*a^6*b^9 - 80*a^5*b^10 + 135*a^4*b^11 - 126*a^3*b^12 + 61*a^2*b^13 - 12*a*b^14)*d^3*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)) - (125*a^7*b^2 - 1045*a^6*b^3 + 4305*a^5*b^4 - 10583*a^4*b^5 + 16798*a^3*b^6 - 16320*a^2*b^7 + 8448*a*b^8)*d)*sqrt((15*a^4 - 94*a^3*b + 155*a^2*b^2 - 76*a*b^3 - 144*b^4 - (a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2*sqrt((625*a^8 - 6700*a^7*b + 35406*a^6*b^2 - 117532*a^5*b^3 + 269641*a^4*b^4 - 437952*a^3*b^5 + 498432*a^2*b^6 - 368640*a*b^7 + 147456*b^8)/((a^11*b^9 - 10*a^10*b^10 + 45*a^9*b^11 - 120*a^8*b^12 + 210*a^7*b^13 - 252*a^6*b^14 + 210*a^5*b^15 - 120*a^4*b^16 + 45*a^3*b^17 - 10*a^2*b^18 + a*b^19)*d^4)))/((a^6*b^4 - 5*a^5*b^5 + 10*a^4*b^6 - 10*a^3*b^7 + 5*a^2*b^8 - a*b^9)*d^2))) + 20*(a^3 - 4*a^2*b + a*b^2 + 2*b^3)*cos(d*x + c))/((a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^8 - 4*(a^2*b^4 - 2*a*b^5 + b^6)*d*cos(d*x + c)^6 - 2*(a^3*b^3 - 5*a^2*b^4 + 7*a*b^5 - 3*b^6)*d*cos(d*x + c)^4 + 4*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cos(d*x + c)^2 + (a^4*b^2 - 4*a^3*b^3 + 6*a^2*b^4 - 4*a*b^5 + b^6)*d)","B",0
225,1,4185,0,2.788548," ","integrate(sin(d*x+c)^7/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","\frac{12 \, {\left(a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{7} - 4 \, {\left(11 \, a b - 35 \, b^{2}\right)} \cos\left(d x + c\right)^{5} + 4 \, {\left(a^{2} + 18 \, a b - 43 \, b^{2}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{2} - 5 \, a^{2} b^{3} + 7 \, a b^{4} - 3 \, b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} + a^{3} - 10 \, a^{2} b + 21 \, a b^{2} + 4 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} \log\left(27 \, {\left(a^{4} - 10 \, a^{3} b + 29 \, a^{2} b^{2} - 4 \, a b^{3} - 64 \, b^{4}\right)} \cos\left(d x + c\right) + 27 \, {\left({\left(a^{8} b^{5} - 8 \, a^{7} b^{6} + 23 \, a^{6} b^{7} - 30 \, a^{5} b^{8} + 15 \, a^{4} b^{9} + 4 \, a^{3} b^{10} - 7 \, a^{2} b^{11} + 2 \, a b^{12}\right)} d^{3} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} - {\left(a^{5} b^{2} - 11 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 9 \, a^{2} b^{5} - 80 \, a b^{6}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} + a^{3} - 10 \, a^{2} b + 21 \, a b^{2} + 4 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}}\right) - 3 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{2} - 5 \, a^{2} b^{3} + 7 \, a b^{4} - 3 \, b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} d\right)} \sqrt{\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} - a^{3} + 10 \, a^{2} b - 21 \, a b^{2} - 4 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} \log\left(27 \, {\left(a^{4} - 10 \, a^{3} b + 29 \, a^{2} b^{2} - 4 \, a b^{3} - 64 \, b^{4}\right)} \cos\left(d x + c\right) + 27 \, {\left({\left(a^{8} b^{5} - 8 \, a^{7} b^{6} + 23 \, a^{6} b^{7} - 30 \, a^{5} b^{8} + 15 \, a^{4} b^{9} + 4 \, a^{3} b^{10} - 7 \, a^{2} b^{11} + 2 \, a b^{12}\right)} d^{3} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} + {\left(a^{5} b^{2} - 11 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 9 \, a^{2} b^{5} - 80 \, a b^{6}\right)} d\right)} \sqrt{\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} - a^{3} + 10 \, a^{2} b - 21 \, a b^{2} - 4 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}}\right) - 3 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{2} - 5 \, a^{2} b^{3} + 7 \, a b^{4} - 3 \, b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} + a^{3} - 10 \, a^{2} b + 21 \, a b^{2} + 4 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} \log\left(-27 \, {\left(a^{4} - 10 \, a^{3} b + 29 \, a^{2} b^{2} - 4 \, a b^{3} - 64 \, b^{4}\right)} \cos\left(d x + c\right) + 27 \, {\left({\left(a^{8} b^{5} - 8 \, a^{7} b^{6} + 23 \, a^{6} b^{7} - 30 \, a^{5} b^{8} + 15 \, a^{4} b^{9} + 4 \, a^{3} b^{10} - 7 \, a^{2} b^{11} + 2 \, a b^{12}\right)} d^{3} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} - {\left(a^{5} b^{2} - 11 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 9 \, a^{2} b^{5} - 80 \, a b^{6}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} + a^{3} - 10 \, a^{2} b + 21 \, a b^{2} + 4 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}}\right) + 3 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{2} - 5 \, a^{2} b^{3} + 7 \, a b^{4} - 3 \, b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} d\right)} \sqrt{\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} - a^{3} + 10 \, a^{2} b - 21 \, a b^{2} - 4 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} \log\left(-27 \, {\left(a^{4} - 10 \, a^{3} b + 29 \, a^{2} b^{2} - 4 \, a b^{3} - 64 \, b^{4}\right)} \cos\left(d x + c\right) + 27 \, {\left({\left(a^{8} b^{5} - 8 \, a^{7} b^{6} + 23 \, a^{6} b^{7} - 30 \, a^{5} b^{8} + 15 \, a^{4} b^{9} + 4 \, a^{3} b^{10} - 7 \, a^{2} b^{11} + 2 \, a b^{12}\right)} d^{3} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} + {\left(a^{5} b^{2} - 11 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 9 \, a^{2} b^{5} - 80 \, a b^{6}\right)} d\right)} \sqrt{\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{a^{6} - 12 \, a^{5} b + 46 \, a^{4} b^{2} - 28 \, a^{3} b^{3} - 167 \, a^{2} b^{4} + 160 \, a b^{5} + 256 \, b^{6}}{{\left(a^{11} b^{7} - 10 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 252 \, a^{6} b^{12} + 210 \, a^{5} b^{13} - 120 \, a^{4} b^{14} + 45 \, a^{3} b^{15} - 10 \, a^{2} b^{16} + a b^{17}\right)} d^{4}}} - a^{3} + 10 \, a^{2} b - 21 \, a b^{2} - 4 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}}\right) - 4 \, {\left(3 \, a^{2} + 14 \, a b - 17 \, b^{2}\right)} \cos\left(d x + c\right)}{128 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{2} - 5 \, a^{2} b^{3} + 7 \, a b^{4} - 3 \, b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} d\right)}}"," ",0,"1/128*(12*(a*b - 3*b^2)*cos(d*x + c)^7 - 4*(11*a*b - 35*b^2)*cos(d*x + c)^5 + 4*(a^2 + 18*a*b - 43*b^2)*cos(d*x + c)^3 + 3*((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + a^3 - 10*a^2*b + 21*a*b^2 + 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(27*(a^4 - 10*a^3*b + 29*a^2*b^2 - 4*a*b^3 - 64*b^4)*cos(d*x + c) + 27*((a^8*b^5 - 8*a^7*b^6 + 23*a^6*b^7 - 30*a^5*b^8 + 15*a^4*b^9 + 4*a^3*b^10 - 7*a^2*b^11 + 2*a*b^12)*d^3*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - (a^5*b^2 - 11*a^4*b^3 + 35*a^3*b^4 - 9*a^2*b^5 - 80*a*b^6)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + a^3 - 10*a^2*b + 21*a*b^2 + 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))) - 3*((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - a^3 + 10*a^2*b - 21*a*b^2 - 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(27*(a^4 - 10*a^3*b + 29*a^2*b^2 - 4*a*b^3 - 64*b^4)*cos(d*x + c) + 27*((a^8*b^5 - 8*a^7*b^6 + 23*a^6*b^7 - 30*a^5*b^8 + 15*a^4*b^9 + 4*a^3*b^10 - 7*a^2*b^11 + 2*a*b^12)*d^3*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + (a^5*b^2 - 11*a^4*b^3 + 35*a^3*b^4 - 9*a^2*b^5 - 80*a*b^6)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - a^3 + 10*a^2*b - 21*a*b^2 - 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))) - 3*((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + a^3 - 10*a^2*b + 21*a*b^2 + 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(-27*(a^4 - 10*a^3*b + 29*a^2*b^2 - 4*a*b^3 - 64*b^4)*cos(d*x + c) + 27*((a^8*b^5 - 8*a^7*b^6 + 23*a^6*b^7 - 30*a^5*b^8 + 15*a^4*b^9 + 4*a^3*b^10 - 7*a^2*b^11 + 2*a*b^12)*d^3*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - (a^5*b^2 - 11*a^4*b^3 + 35*a^3*b^4 - 9*a^2*b^5 - 80*a*b^6)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + a^3 - 10*a^2*b + 21*a*b^2 + 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))) + 3*((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - a^3 + 10*a^2*b - 21*a*b^2 - 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(-27*(a^4 - 10*a^3*b + 29*a^2*b^2 - 4*a*b^3 - 64*b^4)*cos(d*x + c) + 27*((a^8*b^5 - 8*a^7*b^6 + 23*a^6*b^7 - 30*a^5*b^8 + 15*a^4*b^9 + 4*a^3*b^10 - 7*a^2*b^11 + 2*a*b^12)*d^3*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) + (a^5*b^2 - 11*a^4*b^3 + 35*a^3*b^4 - 9*a^2*b^5 - 80*a*b^6)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((a^6 - 12*a^5*b + 46*a^4*b^2 - 28*a^3*b^3 - 167*a^2*b^4 + 160*a*b^5 + 256*b^6)/((a^11*b^7 - 10*a^10*b^8 + 45*a^9*b^9 - 120*a^8*b^10 + 210*a^7*b^11 - 252*a^6*b^12 + 210*a^5*b^13 - 120*a^4*b^14 + 45*a^3*b^15 - 10*a^2*b^16 + a*b^17)*d^4)) - a^3 + 10*a^2*b - 21*a*b^2 - 4*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))) - 4*(3*a^2 + 14*a*b - 17*b^2)*cos(d*x + c))/((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)","B",0
226,1,4524,0,3.338232," ","integrate(sin(d*x+c)^5/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","-\frac{8 \, {\left(2 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{7} + 4 \, {\left(a^{2} b - 19 \, a b^{2} - 6 \, b^{3}\right)} \cos\left(d x + c\right)^{5} - 8 \, {\left(5 \, a^{2} b - 14 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(d x + c\right)^{3} + {\left({\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} + 7 \, a^{2} b^{4} - 3 \, a b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 30 \, a^{3} b - 229 \, a^{2} b^{2} + 116 \, a b^{3} - 16 \, b^{4} + {\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}}}{{\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2}}} \log\left({\left(81 \, a^{5} - 1458 \, a^{4} b + 9389 \, a^{3} b^{2} - 24972 \, a^{2} b^{3} + 10896 \, a b^{4} - 1280 \, b^{5}\right)} \cos\left(d x + c\right) + {\left({\left(a^{10} b^{4} + 10 \, a^{9} b^{5} - 69 \, a^{8} b^{6} + 160 \, a^{7} b^{7} - 185 \, a^{6} b^{8} + 114 \, a^{5} b^{9} - 35 \, a^{4} b^{10} + 4 \, a^{3} b^{11}\right)} d^{3} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}} - {\left(27 \, a^{7} b - 411 \, a^{6} b^{2} + 2383 \, a^{5} b^{3} - 5529 \, a^{4} b^{4} + 1962 \, a^{3} b^{5} - 160 \, a^{2} b^{6}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 30 \, a^{3} b - 229 \, a^{2} b^{2} + 116 \, a b^{3} - 16 \, b^{4} + {\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}}}{{\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2}}}\right) - {\left({\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} + 7 \, a^{2} b^{4} - 3 \, a b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 30 \, a^{3} b - 229 \, a^{2} b^{2} + 116 \, a b^{3} - 16 \, b^{4} - {\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}}}{{\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2}}} \log\left({\left(81 \, a^{5} - 1458 \, a^{4} b + 9389 \, a^{3} b^{2} - 24972 \, a^{2} b^{3} + 10896 \, a b^{4} - 1280 \, b^{5}\right)} \cos\left(d x + c\right) + {\left({\left(a^{10} b^{4} + 10 \, a^{9} b^{5} - 69 \, a^{8} b^{6} + 160 \, a^{7} b^{7} - 185 \, a^{6} b^{8} + 114 \, a^{5} b^{9} - 35 \, a^{4} b^{10} + 4 \, a^{3} b^{11}\right)} d^{3} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}} + {\left(27 \, a^{7} b - 411 \, a^{6} b^{2} + 2383 \, a^{5} b^{3} - 5529 \, a^{4} b^{4} + 1962 \, a^{3} b^{5} - 160 \, a^{2} b^{6}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 30 \, a^{3} b - 229 \, a^{2} b^{2} + 116 \, a b^{3} - 16 \, b^{4} - {\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}}}{{\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2}}}\right) - {\left({\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} + 7 \, a^{2} b^{4} - 3 \, a b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 30 \, a^{3} b - 229 \, a^{2} b^{2} + 116 \, a b^{3} - 16 \, b^{4} + {\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}}}{{\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2}}} \log\left(-{\left(81 \, a^{5} - 1458 \, a^{4} b + 9389 \, a^{3} b^{2} - 24972 \, a^{2} b^{3} + 10896 \, a b^{4} - 1280 \, b^{5}\right)} \cos\left(d x + c\right) + {\left({\left(a^{10} b^{4} + 10 \, a^{9} b^{5} - 69 \, a^{8} b^{6} + 160 \, a^{7} b^{7} - 185 \, a^{6} b^{8} + 114 \, a^{5} b^{9} - 35 \, a^{4} b^{10} + 4 \, a^{3} b^{11}\right)} d^{3} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}} - {\left(27 \, a^{7} b - 411 \, a^{6} b^{2} + 2383 \, a^{5} b^{3} - 5529 \, a^{4} b^{4} + 1962 \, a^{3} b^{5} - 160 \, a^{2} b^{6}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 30 \, a^{3} b - 229 \, a^{2} b^{2} + 116 \, a b^{3} - 16 \, b^{4} + {\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}}}{{\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2}}}\right) + {\left({\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} + 7 \, a^{2} b^{4} - 3 \, a b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 30 \, a^{3} b - 229 \, a^{2} b^{2} + 116 \, a b^{3} - 16 \, b^{4} - {\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}}}{{\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2}}} \log\left(-{\left(81 \, a^{5} - 1458 \, a^{4} b + 9389 \, a^{3} b^{2} - 24972 \, a^{2} b^{3} + 10896 \, a b^{4} - 1280 \, b^{5}\right)} \cos\left(d x + c\right) + {\left({\left(a^{10} b^{4} + 10 \, a^{9} b^{5} - 69 \, a^{8} b^{6} + 160 \, a^{7} b^{7} - 185 \, a^{6} b^{8} + 114 \, a^{5} b^{9} - 35 \, a^{4} b^{10} + 4 \, a^{3} b^{11}\right)} d^{3} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}} + {\left(27 \, a^{7} b - 411 \, a^{6} b^{2} + 2383 \, a^{5} b^{3} - 5529 \, a^{4} b^{4} + 1962 \, a^{3} b^{5} - 160 \, a^{2} b^{6}\right)} d\right)} \sqrt{\frac{15 \, a^{4} - 30 \, a^{3} b - 229 \, a^{2} b^{2} + 116 \, a b^{3} - 16 \, b^{4} - {\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2} \sqrt{\frac{81 \, a^{6} - 1548 \, a^{5} b + 12814 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 104361 \, a^{2} b^{4} - 48160 \, a b^{5} + 6400 \, b^{6}}{{\left(a^{13} b^{5} - 10 \, a^{12} b^{6} + 45 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 252 \, a^{8} b^{10} + 210 \, a^{7} b^{11} - 120 \, a^{6} b^{12} + 45 \, a^{5} b^{13} - 10 \, a^{4} b^{14} + a^{3} b^{15}\right)} d^{4}}}}{{\left(a^{8} b^{2} - 5 \, a^{7} b^{3} + 10 \, a^{6} b^{4} - 10 \, a^{5} b^{5} + 5 \, a^{4} b^{6} - a^{3} b^{7}\right)} d^{2}}}\right) + 4 \, {\left(3 \, a^{3} + 12 \, a^{2} b - 13 \, a b^{2} - 2 \, b^{3}\right)} \cos\left(d x + c\right)}{128 \, {\left({\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} + 7 \, a^{2} b^{4} - 3 \, a b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)}}"," ",0,"-1/128*(8*(2*a*b^2 + b^3)*cos(d*x + c)^7 + 4*(a^2*b - 19*a*b^2 - 6*b^3)*cos(d*x + c)^5 - 8*(5*a^2*b - 14*a*b^2 - 3*b^3)*cos(d*x + c)^3 + ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 + (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))*log((81*a^5 - 1458*a^4*b + 9389*a^3*b^2 - 24972*a^2*b^3 + 10896*a*b^4 - 1280*b^5)*cos(d*x + c) + ((a^10*b^4 + 10*a^9*b^5 - 69*a^8*b^6 + 160*a^7*b^7 - 185*a^6*b^8 + 114*a^5*b^9 - 35*a^4*b^10 + 4*a^3*b^11)*d^3*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)) - (27*a^7*b - 411*a^6*b^2 + 2383*a^5*b^3 - 5529*a^4*b^4 + 1962*a^3*b^5 - 160*a^2*b^6)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 + (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))) - ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 - (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))*log((81*a^5 - 1458*a^4*b + 9389*a^3*b^2 - 24972*a^2*b^3 + 10896*a*b^4 - 1280*b^5)*cos(d*x + c) + ((a^10*b^4 + 10*a^9*b^5 - 69*a^8*b^6 + 160*a^7*b^7 - 185*a^6*b^8 + 114*a^5*b^9 - 35*a^4*b^10 + 4*a^3*b^11)*d^3*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)) + (27*a^7*b - 411*a^6*b^2 + 2383*a^5*b^3 - 5529*a^4*b^4 + 1962*a^3*b^5 - 160*a^2*b^6)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 - (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))) - ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 + (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))*log(-(81*a^5 - 1458*a^4*b + 9389*a^3*b^2 - 24972*a^2*b^3 + 10896*a*b^4 - 1280*b^5)*cos(d*x + c) + ((a^10*b^4 + 10*a^9*b^5 - 69*a^8*b^6 + 160*a^7*b^7 - 185*a^6*b^8 + 114*a^5*b^9 - 35*a^4*b^10 + 4*a^3*b^11)*d^3*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)) - (27*a^7*b - 411*a^6*b^2 + 2383*a^5*b^3 - 5529*a^4*b^4 + 1962*a^3*b^5 - 160*a^2*b^6)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 + (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))) + ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 - (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))*log(-(81*a^5 - 1458*a^4*b + 9389*a^3*b^2 - 24972*a^2*b^3 + 10896*a*b^4 - 1280*b^5)*cos(d*x + c) + ((a^10*b^4 + 10*a^9*b^5 - 69*a^8*b^6 + 160*a^7*b^7 - 185*a^6*b^8 + 114*a^5*b^9 - 35*a^4*b^10 + 4*a^3*b^11)*d^3*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)) + (27*a^7*b - 411*a^6*b^2 + 2383*a^5*b^3 - 5529*a^4*b^4 + 1962*a^3*b^5 - 160*a^2*b^6)*d)*sqrt((15*a^4 - 30*a^3*b - 229*a^2*b^2 + 116*a*b^3 - 16*b^4 - (a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2*sqrt((81*a^6 - 1548*a^5*b + 12814*a^4*b^2 - 53212*a^3*b^3 + 104361*a^2*b^4 - 48160*a*b^5 + 6400*b^6)/((a^13*b^5 - 10*a^12*b^6 + 45*a^11*b^7 - 120*a^10*b^8 + 210*a^9*b^9 - 252*a^8*b^10 + 210*a^7*b^11 - 120*a^6*b^12 + 45*a^5*b^13 - 10*a^4*b^14 + a^3*b^15)*d^4)))/((a^8*b^2 - 5*a^7*b^3 + 10*a^6*b^4 - 10*a^5*b^5 + 5*a^4*b^6 - a^3*b^7)*d^2))) + 4*(3*a^3 + 12*a^2*b - 13*a*b^2 - 2*b^3)*cos(d*x + c))/((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)","B",0
227,1,4050,0,3.297475," ","integrate(sin(d*x+c)^3/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","-\frac{4 \, {\left(5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{7} - 12 \, {\left(7 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{5} - 12 \, {\left(3 \, a^{2} - 10 \, a b - b^{2}\right)} \cos\left(d x + c\right)^{3} + {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b - 5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 3 \, a b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} + 105 \, a^{3} + 70 \, a^{2} b - 35 \, a b^{2} + 4 \, b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} \log\left({\left(625 \, a^{3} + 3750 \, a^{2} b - 1491 \, a b^{2} + 140 \, b^{3}\right)} \cos\left(d x + c\right) + {\left({\left(5 \, a^{10} b^{2} - 16 \, a^{9} b^{3} + 3 \, a^{8} b^{4} + 50 \, a^{7} b^{5} - 85 \, a^{6} b^{6} + 60 \, a^{5} b^{7} - 19 \, a^{4} b^{8} + 2 \, a^{3} b^{9}\right)} d^{3} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} - {\left(325 \, a^{5} b + 1977 \, a^{4} b^{2} - 609 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} + 105 \, a^{3} + 70 \, a^{2} b - 35 \, a b^{2} + 4 \, b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}}\right) - {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b - 5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 3 \, a b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} - 105 \, a^{3} - 70 \, a^{2} b + 35 \, a b^{2} - 4 \, b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} \log\left({\left(625 \, a^{3} + 3750 \, a^{2} b - 1491 \, a b^{2} + 140 \, b^{3}\right)} \cos\left(d x + c\right) + {\left({\left(5 \, a^{10} b^{2} - 16 \, a^{9} b^{3} + 3 \, a^{8} b^{4} + 50 \, a^{7} b^{5} - 85 \, a^{6} b^{6} + 60 \, a^{5} b^{7} - 19 \, a^{4} b^{8} + 2 \, a^{3} b^{9}\right)} d^{3} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} + {\left(325 \, a^{5} b + 1977 \, a^{4} b^{2} - 609 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} - 105 \, a^{3} - 70 \, a^{2} b + 35 \, a b^{2} - 4 \, b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}}\right) - {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b - 5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 3 \, a b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} + 105 \, a^{3} + 70 \, a^{2} b - 35 \, a b^{2} + 4 \, b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} \log\left(-{\left(625 \, a^{3} + 3750 \, a^{2} b - 1491 \, a b^{2} + 140 \, b^{3}\right)} \cos\left(d x + c\right) + {\left({\left(5 \, a^{10} b^{2} - 16 \, a^{9} b^{3} + 3 \, a^{8} b^{4} + 50 \, a^{7} b^{5} - 85 \, a^{6} b^{6} + 60 \, a^{5} b^{7} - 19 \, a^{4} b^{8} + 2 \, a^{3} b^{9}\right)} d^{3} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} - {\left(325 \, a^{5} b + 1977 \, a^{4} b^{2} - 609 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} + 105 \, a^{3} + 70 \, a^{2} b - 35 \, a b^{2} + 4 \, b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}}\right) + {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b - 5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 3 \, a b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} - 105 \, a^{3} - 70 \, a^{2} b + 35 \, a b^{2} - 4 \, b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} \log\left(-{\left(625 \, a^{3} + 3750 \, a^{2} b - 1491 \, a b^{2} + 140 \, b^{3}\right)} \cos\left(d x + c\right) + {\left({\left(5 \, a^{10} b^{2} - 16 \, a^{9} b^{3} + 3 \, a^{8} b^{4} + 50 \, a^{7} b^{5} - 85 \, a^{6} b^{6} + 60 \, a^{5} b^{7} - 19 \, a^{4} b^{8} + 2 \, a^{3} b^{9}\right)} d^{3} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} + {\left(325 \, a^{5} b + 1977 \, a^{4} b^{2} - 609 \, a^{3} b^{3} + 35 \, a^{2} b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{625 \, a^{4} + 7700 \, a^{3} b + 21966 \, a^{2} b^{2} - 10780 \, a b^{3} + 1225 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} - 105 \, a^{3} - 70 \, a^{2} b + 35 \, a b^{2} - 4 \, b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}}\right) + 4 \, {\left(19 \, a^{2} - 18 \, a b - b^{2}\right)} \cos\left(d x + c\right)}{128 \, {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b - 5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 3 \, a b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d\right)}}"," ",0,"-1/128*(4*(5*a*b + b^2)*cos(d*x + c)^7 - 12*(7*a*b + b^2)*cos(d*x + c)^5 - 12*(3*a^2 - 10*a*b - b^2)*cos(d*x + c)^3 + ((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 105*a^3 + 70*a^2*b - 35*a*b^2 + 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log((625*a^3 + 3750*a^2*b - 1491*a*b^2 + 140*b^3)*cos(d*x + c) + ((5*a^10*b^2 - 16*a^9*b^3 + 3*a^8*b^4 + 50*a^7*b^5 - 85*a^6*b^6 + 60*a^5*b^7 - 19*a^4*b^8 + 2*a^3*b^9)*d^3*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - (325*a^5*b + 1977*a^4*b^2 - 609*a^3*b^3 + 35*a^2*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 105*a^3 + 70*a^2*b - 35*a*b^2 + 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))) - ((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 105*a^3 - 70*a^2*b + 35*a*b^2 - 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log((625*a^3 + 3750*a^2*b - 1491*a*b^2 + 140*b^3)*cos(d*x + c) + ((5*a^10*b^2 - 16*a^9*b^3 + 3*a^8*b^4 + 50*a^7*b^5 - 85*a^6*b^6 + 60*a^5*b^7 - 19*a^4*b^8 + 2*a^3*b^9)*d^3*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + (325*a^5*b + 1977*a^4*b^2 - 609*a^3*b^3 + 35*a^2*b^4)*d)*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 105*a^3 - 70*a^2*b + 35*a*b^2 - 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))) - ((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 105*a^3 + 70*a^2*b - 35*a*b^2 + 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log(-(625*a^3 + 3750*a^2*b - 1491*a*b^2 + 140*b^3)*cos(d*x + c) + ((5*a^10*b^2 - 16*a^9*b^3 + 3*a^8*b^4 + 50*a^7*b^5 - 85*a^6*b^6 + 60*a^5*b^7 - 19*a^4*b^8 + 2*a^3*b^9)*d^3*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - (325*a^5*b + 1977*a^4*b^2 - 609*a^3*b^3 + 35*a^2*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 105*a^3 + 70*a^2*b - 35*a*b^2 + 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))) + ((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 105*a^3 - 70*a^2*b + 35*a*b^2 - 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log(-(625*a^3 + 3750*a^2*b - 1491*a*b^2 + 140*b^3)*cos(d*x + c) + ((5*a^10*b^2 - 16*a^9*b^3 + 3*a^8*b^4 + 50*a^7*b^5 - 85*a^6*b^6 + 60*a^5*b^7 - 19*a^4*b^8 + 2*a^3*b^9)*d^3*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + (325*a^5*b + 1977*a^4*b^2 - 609*a^3*b^3 + 35*a^2*b^4)*d)*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((625*a^4 + 7700*a^3*b + 21966*a^2*b^2 - 10780*a*b^3 + 1225*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 105*a^3 - 70*a^2*b + 35*a*b^2 - 4*b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))) + 4*(19*a^2 - 18*a*b - b^2)*cos(d*x + c))/((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)","B",0
228,1,4160,0,3.593699," ","integrate(sin(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","-\frac{24 \, {\left(2 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)^{7} - 4 \, {\left(7 \, a^{2} b + 35 \, a b^{2} - 18 \, b^{3}\right)} \cos\left(d x + c\right)^{5} - 8 \, {\left(a^{2} b - 22 \, a b^{2} + 9 \, b^{3}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{105 \, a^{4} - 210 \, a^{3} b + 189 \, a^{2} b^{2} - 84 \, a b^{3} + 16 \, b^{4} + {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} \log\left(27 \, {\left(2401 \, a^{4} - 4802 \, a^{3} b + 4189 \, a^{2} b^{2} - 1788 \, a b^{3} + 336 \, b^{4}\right)} \cos\left(d x + c\right) - 27 \, {\left({\left(11 \, a^{12} b - 66 \, a^{11} b^{2} + 169 \, a^{10} b^{3} - 240 \, a^{9} b^{4} + 205 \, a^{8} b^{5} - 106 \, a^{7} b^{6} + 31 \, a^{6} b^{7} - 4 \, a^{5} b^{8}\right)} d^{3} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}} - {\left(343 \, a^{7} - 623 \, a^{6} b + 515 \, a^{5} b^{2} - 213 \, a^{4} b^{3} + 42 \, a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{105 \, a^{4} - 210 \, a^{3} b + 189 \, a^{2} b^{2} - 84 \, a b^{3} + 16 \, b^{4} + {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}}\right) - 3 \, {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{105 \, a^{4} - 210 \, a^{3} b + 189 \, a^{2} b^{2} - 84 \, a b^{3} + 16 \, b^{4} - {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} \log\left(27 \, {\left(2401 \, a^{4} - 4802 \, a^{3} b + 4189 \, a^{2} b^{2} - 1788 \, a b^{3} + 336 \, b^{4}\right)} \cos\left(d x + c\right) - 27 \, {\left({\left(11 \, a^{12} b - 66 \, a^{11} b^{2} + 169 \, a^{10} b^{3} - 240 \, a^{9} b^{4} + 205 \, a^{8} b^{5} - 106 \, a^{7} b^{6} + 31 \, a^{6} b^{7} - 4 \, a^{5} b^{8}\right)} d^{3} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}} + {\left(343 \, a^{7} - 623 \, a^{6} b + 515 \, a^{5} b^{2} - 213 \, a^{4} b^{3} + 42 \, a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{105 \, a^{4} - 210 \, a^{3} b + 189 \, a^{2} b^{2} - 84 \, a b^{3} + 16 \, b^{4} - {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}}\right) - 3 \, {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{105 \, a^{4} - 210 \, a^{3} b + 189 \, a^{2} b^{2} - 84 \, a b^{3} + 16 \, b^{4} + {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} \log\left(-27 \, {\left(2401 \, a^{4} - 4802 \, a^{3} b + 4189 \, a^{2} b^{2} - 1788 \, a b^{3} + 336 \, b^{4}\right)} \cos\left(d x + c\right) - 27 \, {\left({\left(11 \, a^{12} b - 66 \, a^{11} b^{2} + 169 \, a^{10} b^{3} - 240 \, a^{9} b^{4} + 205 \, a^{8} b^{5} - 106 \, a^{7} b^{6} + 31 \, a^{6} b^{7} - 4 \, a^{5} b^{8}\right)} d^{3} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}} - {\left(343 \, a^{7} - 623 \, a^{6} b + 515 \, a^{5} b^{2} - 213 \, a^{4} b^{3} + 42 \, a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{105 \, a^{4} - 210 \, a^{3} b + 189 \, a^{2} b^{2} - 84 \, a b^{3} + 16 \, b^{4} + {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}}\right) + 3 \, {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{105 \, a^{4} - 210 \, a^{3} b + 189 \, a^{2} b^{2} - 84 \, a b^{3} + 16 \, b^{4} - {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} \log\left(-27 \, {\left(2401 \, a^{4} - 4802 \, a^{3} b + 4189 \, a^{2} b^{2} - 1788 \, a b^{3} + 336 \, b^{4}\right)} \cos\left(d x + c\right) - 27 \, {\left({\left(11 \, a^{12} b - 66 \, a^{11} b^{2} + 169 \, a^{10} b^{3} - 240 \, a^{9} b^{4} + 205 \, a^{8} b^{5} - 106 \, a^{7} b^{6} + 31 \, a^{6} b^{7} - 4 \, a^{5} b^{8}\right)} d^{3} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}} + {\left(343 \, a^{7} - 623 \, a^{6} b + 515 \, a^{5} b^{2} - 213 \, a^{4} b^{3} + 42 \, a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{105 \, a^{4} - 210 \, a^{3} b + 189 \, a^{2} b^{2} - 84 \, a b^{3} + 16 \, b^{4} - {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{2401 \, a^{4} - 5292 \, a^{3} b + 4974 \, a^{2} b^{2} - 2268 \, a b^{3} + 441 \, b^{4}}{{\left(a^{15} b - 10 \, a^{14} b^{2} + 45 \, a^{13} b^{3} - 120 \, a^{12} b^{4} + 210 \, a^{11} b^{5} - 252 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 120 \, a^{8} b^{8} + 45 \, a^{7} b^{9} - 10 \, a^{6} b^{10} + a^{5} b^{11}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}}\right) + 4 \, {\left(11 \, a^{3} + 4 \, a^{2} b - 21 \, a b^{2} + 6 \, b^{3}\right)} \cos\left(d x + c\right)}{128 \, {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)}}"," ",0,"-1/128*(24*(2*a*b^2 - b^3)*cos(d*x + c)^7 - 4*(7*a^2*b + 35*a*b^2 - 18*b^3)*cos(d*x + c)^5 - 8*(a^2*b - 22*a*b^2 + 9*b^3)*cos(d*x + c)^3 + 3*((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(27*(2401*a^4 - 4802*a^3*b + 4189*a^2*b^2 - 1788*a*b^3 + 336*b^4)*cos(d*x + c) - 27*((11*a^12*b - 66*a^11*b^2 + 169*a^10*b^3 - 240*a^9*b^4 + 205*a^8*b^5 - 106*a^7*b^6 + 31*a^6*b^7 - 4*a^5*b^8)*d^3*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)) - (343*a^7 - 623*a^6*b + 515*a^5*b^2 - 213*a^4*b^3 + 42*a^3*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))) - 3*((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(27*(2401*a^4 - 4802*a^3*b + 4189*a^2*b^2 - 1788*a*b^3 + 336*b^4)*cos(d*x + c) - 27*((11*a^12*b - 66*a^11*b^2 + 169*a^10*b^3 - 240*a^9*b^4 + 205*a^8*b^5 - 106*a^7*b^6 + 31*a^6*b^7 - 4*a^5*b^8)*d^3*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)) + (343*a^7 - 623*a^6*b + 515*a^5*b^2 - 213*a^4*b^3 + 42*a^3*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))) - 3*((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(-27*(2401*a^4 - 4802*a^3*b + 4189*a^2*b^2 - 1788*a*b^3 + 336*b^4)*cos(d*x + c) - 27*((11*a^12*b - 66*a^11*b^2 + 169*a^10*b^3 - 240*a^9*b^4 + 205*a^8*b^5 - 106*a^7*b^6 + 31*a^6*b^7 - 4*a^5*b^8)*d^3*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)) - (343*a^7 - 623*a^6*b + 515*a^5*b^2 - 213*a^4*b^3 + 42*a^3*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))) + 3*((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(-27*(2401*a^4 - 4802*a^3*b + 4189*a^2*b^2 - 1788*a*b^3 + 336*b^4)*cos(d*x + c) - 27*((11*a^12*b - 66*a^11*b^2 + 169*a^10*b^3 - 240*a^9*b^4 + 205*a^8*b^5 - 106*a^7*b^6 + 31*a^6*b^7 - 4*a^5*b^8)*d^3*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)) + (343*a^7 - 623*a^6*b + 515*a^5*b^2 - 213*a^4*b^3 + 42*a^3*b^4)*d)*sqrt(-(105*a^4 - 210*a^3*b + 189*a^2*b^2 - 84*a*b^3 + 16*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((2401*a^4 - 5292*a^3*b + 4974*a^2*b^2 - 2268*a*b^3 + 441*b^4)/((a^15*b - 10*a^14*b^2 + 45*a^13*b^3 - 120*a^12*b^4 + 210*a^11*b^5 - 252*a^10*b^6 + 210*a^9*b^7 - 120*a^8*b^8 + 45*a^7*b^9 - 10*a^6*b^10 + a^5*b^11)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))) + 4*(11*a^3 + 4*a^2*b - 21*a*b^2 + 6*b^3)*cos(d*x + c))/((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)","B",0
229,1,5020,0,7.194122," ","integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","-\frac{4 \, {\left(13 \, a^{2} b^{2} - 7 \, a b^{3}\right)} \cos\left(d x + c\right)^{7} - 4 \, {\left(53 \, a^{2} b^{2} - 29 \, a b^{3}\right)} \cos\left(d x + c\right)^{5} - 4 \, {\left(17 \, a^{3} b - 78 \, a^{2} b^{2} + 37 \, a b^{3}\right)} \cos\left(d x + c\right)^{3} - {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 7 \, a^{4} b^{3} - 3 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{3465 \, a^{4} b - 9306 \, a^{3} b^{2} + 10045 \, a^{2} b^{3} - 5084 \, a b^{4} + 1024 \, b^{5} + {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} \log\left({\left(4100625 \, a^{6} b - 14762250 \, a^{5} b^{2} + 23227949 \, a^{4} b^{3} - 20354340 \, a^{3} b^{4} + 10504896 \, a^{2} b^{5} - 3044864 \, a b^{6} + 393216 \, b^{7}\right)} \cos\left(d x + c\right) - {\left({\left(45 \, a^{16} - 280 \, a^{15} b + 747 \, a^{14} b^{2} - 1110 \, a^{13} b^{3} + 995 \, a^{12} b^{4} - 540 \, a^{11} b^{5} + 165 \, a^{10} b^{6} - 22 \, a^{9} b^{7}\right)} d^{3} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}} - {\left(123525 \, a^{9} b - 450359 \, a^{8} b^{2} + 715183 \, a^{7} b^{3} - 630957 \, a^{6} b^{4} + 327152 \, a^{5} b^{5} - 95104 \, a^{4} b^{6} + 12288 \, a^{3} b^{7}\right)} d\right)} \sqrt{-\frac{3465 \, a^{4} b - 9306 \, a^{3} b^{2} + 10045 \, a^{2} b^{3} - 5084 \, a b^{4} + 1024 \, b^{5} + {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}}\right) + {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 7 \, a^{4} b^{3} - 3 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{3465 \, a^{4} b - 9306 \, a^{3} b^{2} + 10045 \, a^{2} b^{3} - 5084 \, a b^{4} + 1024 \, b^{5} - {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} \log\left({\left(4100625 \, a^{6} b - 14762250 \, a^{5} b^{2} + 23227949 \, a^{4} b^{3} - 20354340 \, a^{3} b^{4} + 10504896 \, a^{2} b^{5} - 3044864 \, a b^{6} + 393216 \, b^{7}\right)} \cos\left(d x + c\right) - {\left({\left(45 \, a^{16} - 280 \, a^{15} b + 747 \, a^{14} b^{2} - 1110 \, a^{13} b^{3} + 995 \, a^{12} b^{4} - 540 \, a^{11} b^{5} + 165 \, a^{10} b^{6} - 22 \, a^{9} b^{7}\right)} d^{3} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}} + {\left(123525 \, a^{9} b - 450359 \, a^{8} b^{2} + 715183 \, a^{7} b^{3} - 630957 \, a^{6} b^{4} + 327152 \, a^{5} b^{5} - 95104 \, a^{4} b^{6} + 12288 \, a^{3} b^{7}\right)} d\right)} \sqrt{-\frac{3465 \, a^{4} b - 9306 \, a^{3} b^{2} + 10045 \, a^{2} b^{3} - 5084 \, a b^{4} + 1024 \, b^{5} - {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}}\right) + {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 7 \, a^{4} b^{3} - 3 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{3465 \, a^{4} b - 9306 \, a^{3} b^{2} + 10045 \, a^{2} b^{3} - 5084 \, a b^{4} + 1024 \, b^{5} + {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} \log\left(-{\left(4100625 \, a^{6} b - 14762250 \, a^{5} b^{2} + 23227949 \, a^{4} b^{3} - 20354340 \, a^{3} b^{4} + 10504896 \, a^{2} b^{5} - 3044864 \, a b^{6} + 393216 \, b^{7}\right)} \cos\left(d x + c\right) - {\left({\left(45 \, a^{16} - 280 \, a^{15} b + 747 \, a^{14} b^{2} - 1110 \, a^{13} b^{3} + 995 \, a^{12} b^{4} - 540 \, a^{11} b^{5} + 165 \, a^{10} b^{6} - 22 \, a^{9} b^{7}\right)} d^{3} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}} - {\left(123525 \, a^{9} b - 450359 \, a^{8} b^{2} + 715183 \, a^{7} b^{3} - 630957 \, a^{6} b^{4} + 327152 \, a^{5} b^{5} - 95104 \, a^{4} b^{6} + 12288 \, a^{3} b^{7}\right)} d\right)} \sqrt{-\frac{3465 \, a^{4} b - 9306 \, a^{3} b^{2} + 10045 \, a^{2} b^{3} - 5084 \, a b^{4} + 1024 \, b^{5} + {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}}\right) - {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 7 \, a^{4} b^{3} - 3 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{3465 \, a^{4} b - 9306 \, a^{3} b^{2} + 10045 \, a^{2} b^{3} - 5084 \, a b^{4} + 1024 \, b^{5} - {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} \log\left(-{\left(4100625 \, a^{6} b - 14762250 \, a^{5} b^{2} + 23227949 \, a^{4} b^{3} - 20354340 \, a^{3} b^{4} + 10504896 \, a^{2} b^{5} - 3044864 \, a b^{6} + 393216 \, b^{7}\right)} \cos\left(d x + c\right) - {\left({\left(45 \, a^{16} - 280 \, a^{15} b + 747 \, a^{14} b^{2} - 1110 \, a^{13} b^{3} + 995 \, a^{12} b^{4} - 540 \, a^{11} b^{5} + 165 \, a^{10} b^{6} - 22 \, a^{9} b^{7}\right)} d^{3} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}} + {\left(123525 \, a^{9} b - 450359 \, a^{8} b^{2} + 715183 \, a^{7} b^{3} - 630957 \, a^{6} b^{4} + 327152 \, a^{5} b^{5} - 95104 \, a^{4} b^{6} + 12288 \, a^{3} b^{7}\right)} d\right)} \sqrt{-\frac{3465 \, a^{4} b - 9306 \, a^{3} b^{2} + 10045 \, a^{2} b^{3} - 5084 \, a b^{4} + 1024 \, b^{5} - {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{4100625 \, a^{8} b - 19010700 \, a^{7} b^{2} + 39971086 \, a^{6} b^{3} - 49679452 \, a^{5} b^{4} + 39947241 \, a^{4} b^{5} - 21320992 \, a^{3} b^{6} + 7401472 \, a^{2} b^{7} - 1536000 \, a b^{8} + 147456 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}}\right) + 20 \, {\left(7 \, a^{3} b - 10 \, a^{2} b^{2} + 3 \, a b^{3}\right)} \cos\left(d x + c\right) + 64 \, {\left({\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b - 5 \, a^{2} b^{2} + 7 \, a b^{3} - 3 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4} + 4 \, {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - 64 \, {\left({\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b - 5 \, a^{2} b^{2} + 7 \, a b^{3} - 3 \, b^{4}\right)} \cos\left(d x + c\right)^{4} + a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4} + 4 \, {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{128 \, {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 7 \, a^{4} b^{3} - 3 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)}}"," ",0,"-1/128*(4*(13*a^2*b^2 - 7*a*b^3)*cos(d*x + c)^7 - 4*(53*a^2*b^2 - 29*a*b^3)*cos(d*x + c)^5 - 4*(17*a^3*b - 78*a^2*b^2 + 37*a*b^3)*cos(d*x + c)^3 - ((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log((4100625*a^6*b - 14762250*a^5*b^2 + 23227949*a^4*b^3 - 20354340*a^3*b^4 + 10504896*a^2*b^5 - 3044864*a*b^6 + 393216*b^7)*cos(d*x + c) - ((45*a^16 - 280*a^15*b + 747*a^14*b^2 - 1110*a^13*b^3 + 995*a^12*b^4 - 540*a^11*b^5 + 165*a^10*b^6 - 22*a^9*b^7)*d^3*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)) - (123525*a^9*b - 450359*a^8*b^2 + 715183*a^7*b^3 - 630957*a^6*b^4 + 327152*a^5*b^5 - 95104*a^4*b^6 + 12288*a^3*b^7)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))) + ((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log((4100625*a^6*b - 14762250*a^5*b^2 + 23227949*a^4*b^3 - 20354340*a^3*b^4 + 10504896*a^2*b^5 - 3044864*a*b^6 + 393216*b^7)*cos(d*x + c) - ((45*a^16 - 280*a^15*b + 747*a^14*b^2 - 1110*a^13*b^3 + 995*a^12*b^4 - 540*a^11*b^5 + 165*a^10*b^6 - 22*a^9*b^7)*d^3*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)) + (123525*a^9*b - 450359*a^8*b^2 + 715183*a^7*b^3 - 630957*a^6*b^4 + 327152*a^5*b^5 - 95104*a^4*b^6 + 12288*a^3*b^7)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))) + ((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log(-(4100625*a^6*b - 14762250*a^5*b^2 + 23227949*a^4*b^3 - 20354340*a^3*b^4 + 10504896*a^2*b^5 - 3044864*a*b^6 + 393216*b^7)*cos(d*x + c) - ((45*a^16 - 280*a^15*b + 747*a^14*b^2 - 1110*a^13*b^3 + 995*a^12*b^4 - 540*a^11*b^5 + 165*a^10*b^6 - 22*a^9*b^7)*d^3*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)) - (123525*a^9*b - 450359*a^8*b^2 + 715183*a^7*b^3 - 630957*a^6*b^4 + 327152*a^5*b^5 - 95104*a^4*b^6 + 12288*a^3*b^7)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))) - ((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log(-(4100625*a^6*b - 14762250*a^5*b^2 + 23227949*a^4*b^3 - 20354340*a^3*b^4 + 10504896*a^2*b^5 - 3044864*a*b^6 + 393216*b^7)*cos(d*x + c) - ((45*a^16 - 280*a^15*b + 747*a^14*b^2 - 1110*a^13*b^3 + 995*a^12*b^4 - 540*a^11*b^5 + 165*a^10*b^6 - 22*a^9*b^7)*d^3*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)) + (123525*a^9*b - 450359*a^8*b^2 + 715183*a^7*b^3 - 630957*a^6*b^4 + 327152*a^5*b^5 - 95104*a^4*b^6 + 12288*a^3*b^7)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))) + 20*(7*a^3*b - 10*a^2*b^2 + 3*a*b^3)*cos(d*x + c) + 64*((a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x + c)^8 - 4*(a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x + c)^6 - 2*(a^3*b - 5*a^2*b^2 + 7*a*b^3 - 3*b^4)*cos(d*x + c)^4 + a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 + 4*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - 64*((a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x + c)^8 - 4*(a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x + c)^6 - 2*(a^3*b - 5*a^2*b^2 + 7*a*b^3 - 3*b^4)*cos(d*x + c)^4 + a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 + 4*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)","B",0
230,1,5219,0,4.046345," ","integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","-\frac{{\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{2} - 5 \, a^{2} b^{3} + 7 \, a b^{4} - 3 \, b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} d\right)} \sqrt{\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} - 4 \, a^{3} + 35 \, a^{2} b - 70 \, a b^{2} - 105 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} \log\left(35 \, a^{3} - \frac{1491}{4} \, a^{2} b + \frac{1875}{2} \, a b^{2} + \frac{625}{4} \, b^{3} - \frac{1}{4} \, {\left(140 \, a^{3} - 1491 \, a^{2} b + 3750 \, a b^{2} + 625 \, b^{3}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a^{9} b^{3} - 18 \, a^{8} b^{4} + 75 \, a^{7} b^{5} - 140 \, a^{6} b^{6} + 135 \, a^{5} b^{7} - 66 \, a^{4} b^{8} + 13 \, a^{3} b^{9}\right)} d^{3} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(70 \, a^{5} b - 623 \, a^{4} b^{2} + 1161 \, a^{3} b^{3} + 995 \, a^{2} b^{4} + 125 \, a b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} - 4 \, a^{3} + 35 \, a^{2} b - 70 \, a b^{2} - 105 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{8} b - 45 \, a^{7} b^{2} + 165 \, a^{6} b^{3} - 290 \, a^{5} b^{4} + 270 \, a^{4} b^{5} - 129 \, a^{3} b^{6} + 25 \, a^{2} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{8} b - 45 \, a^{7} b^{2} + 165 \, a^{6} b^{3} - 290 \, a^{5} b^{4} + 270 \, a^{4} b^{5} - 129 \, a^{3} b^{6} + 25 \, a^{2} b^{7}\right)} d^{2}\right)} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}}\right) - {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{2} - 5 \, a^{2} b^{3} + 7 \, a b^{4} - 3 \, b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} d\right)} \sqrt{\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} - 4 \, a^{3} + 35 \, a^{2} b - 70 \, a b^{2} - 105 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} \log\left(35 \, a^{3} - \frac{1491}{4} \, a^{2} b + \frac{1875}{2} \, a b^{2} + \frac{625}{4} \, b^{3} - \frac{1}{4} \, {\left(140 \, a^{3} - 1491 \, a^{2} b + 3750 \, a b^{2} + 625 \, b^{3}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a^{9} b^{3} - 18 \, a^{8} b^{4} + 75 \, a^{7} b^{5} - 140 \, a^{6} b^{6} + 135 \, a^{5} b^{7} - 66 \, a^{4} b^{8} + 13 \, a^{3} b^{9}\right)} d^{3} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(70 \, a^{5} b - 623 \, a^{4} b^{2} + 1161 \, a^{3} b^{3} + 995 \, a^{2} b^{4} + 125 \, a b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} - 4 \, a^{3} + 35 \, a^{2} b - 70 \, a b^{2} - 105 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{8} b - 45 \, a^{7} b^{2} + 165 \, a^{6} b^{3} - 290 \, a^{5} b^{4} + 270 \, a^{4} b^{5} - 129 \, a^{3} b^{6} + 25 \, a^{2} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{8} b - 45 \, a^{7} b^{2} + 165 \, a^{6} b^{3} - 290 \, a^{5} b^{4} + 270 \, a^{4} b^{5} - 129 \, a^{3} b^{6} + 25 \, a^{2} b^{7}\right)} d^{2}\right)} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}}\right) + {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{2} - 5 \, a^{2} b^{3} + 7 \, a b^{4} - 3 \, b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} + 4 \, a^{3} - 35 \, a^{2} b + 70 \, a b^{2} + 105 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} \log\left(-35 \, a^{3} + \frac{1491}{4} \, a^{2} b - \frac{1875}{2} \, a b^{2} - \frac{625}{4} \, b^{3} + \frac{1}{4} \, {\left(140 \, a^{3} - 1491 \, a^{2} b + 3750 \, a b^{2} + 625 \, b^{3}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a^{9} b^{3} - 18 \, a^{8} b^{4} + 75 \, a^{7} b^{5} - 140 \, a^{6} b^{6} + 135 \, a^{5} b^{7} - 66 \, a^{4} b^{8} + 13 \, a^{3} b^{9}\right)} d^{3} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(70 \, a^{5} b - 623 \, a^{4} b^{2} + 1161 \, a^{3} b^{3} + 995 \, a^{2} b^{4} + 125 \, a b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} + 4 \, a^{3} - 35 \, a^{2} b + 70 \, a b^{2} + 105 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{8} b - 45 \, a^{7} b^{2} + 165 \, a^{6} b^{3} - 290 \, a^{5} b^{4} + 270 \, a^{4} b^{5} - 129 \, a^{3} b^{6} + 25 \, a^{2} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{8} b - 45 \, a^{7} b^{2} + 165 \, a^{6} b^{3} - 290 \, a^{5} b^{4} + 270 \, a^{4} b^{5} - 129 \, a^{3} b^{6} + 25 \, a^{2} b^{7}\right)} d^{2}\right)} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}}\right) - {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{2} - 5 \, a^{2} b^{3} + 7 \, a b^{4} - 3 \, b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} d\right)} \sqrt{-\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} + 4 \, a^{3} - 35 \, a^{2} b + 70 \, a b^{2} + 105 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} \log\left(-35 \, a^{3} + \frac{1491}{4} \, a^{2} b - \frac{1875}{2} \, a b^{2} - \frac{625}{4} \, b^{3} + \frac{1}{4} \, {\left(140 \, a^{3} - 1491 \, a^{2} b + 3750 \, a b^{2} + 625 \, b^{3}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a^{9} b^{3} - 18 \, a^{8} b^{4} + 75 \, a^{7} b^{5} - 140 \, a^{6} b^{6} + 135 \, a^{5} b^{7} - 66 \, a^{4} b^{8} + 13 \, a^{3} b^{9}\right)} d^{3} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(70 \, a^{5} b - 623 \, a^{4} b^{2} + 1161 \, a^{3} b^{3} + 995 \, a^{2} b^{4} + 125 \, a b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}} + 4 \, a^{3} - 35 \, a^{2} b + 70 \, a b^{2} + 105 \, b^{3}}{{\left(a^{6} b^{3} - 5 \, a^{5} b^{4} + 10 \, a^{4} b^{5} - 10 \, a^{3} b^{6} + 5 \, a^{2} b^{7} - a b^{8}\right)} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(4 \, a^{8} b - 45 \, a^{7} b^{2} + 165 \, a^{6} b^{3} - 290 \, a^{5} b^{4} + 270 \, a^{4} b^{5} - 129 \, a^{3} b^{6} + 25 \, a^{2} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{8} b - 45 \, a^{7} b^{2} + 165 \, a^{6} b^{3} - 290 \, a^{5} b^{4} + 270 \, a^{4} b^{5} - 129 \, a^{3} b^{6} + 25 \, a^{2} b^{7}\right)} d^{2}\right)} \sqrt{\frac{1225 \, a^{4} - 10780 \, a^{3} b + 21966 \, a^{2} b^{2} + 7700 \, a b^{3} + 625 \, b^{4}}{{\left(a^{13} b^{3} - 10 \, a^{12} b^{4} + 45 \, a^{11} b^{5} - 120 \, a^{10} b^{6} + 210 \, a^{9} b^{7} - 252 \, a^{8} b^{8} + 210 \, a^{7} b^{9} - 120 \, a^{6} b^{10} + 45 \, a^{5} b^{11} - 10 \, a^{4} b^{12} + a^{3} b^{13}\right)} d^{4}}}\right) - 8 \, {\left(2 \, {\left(2 \, a b - 5 \, b^{2}\right)} \cos\left(d x + c\right)^{7} - 3 \, {\left(5 \, a b - 13 \, b^{2}\right)} \cos\left(d x + c\right)^{5} + 24 \, {\left(a b - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 18 \, a b - 19 \, b^{2}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{256 \, {\left({\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{3} b^{2} - 5 \, a^{2} b^{3} + 7 \, a b^{4} - 3 \, b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right)} d\right)}}"," ",0,"-1/256*(((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(35*a^3 - 1491/4*a^2*b + 1875/2*a*b^2 + 625/4*b^3 - 1/4*(140*a^3 - 1491*a^2*b + 3750*a*b^2 + 625*b^3)*cos(d*x + c)^2 + 1/2*((a^9*b^3 - 18*a^8*b^4 + 75*a^7*b^5 - 140*a^6*b^6 + 135*a^5*b^7 - 66*a^4*b^8 + 13*a^3*b^9)*d^3*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (70*a^5*b - 623*a^4*b^2 + 1161*a^3*b^3 + 995*a^2*b^4 + 125*a*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2)) + 1/4*(2*(4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2*cos(d*x + c)^2 - (4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2)*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))) - ((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(35*a^3 - 1491/4*a^2*b + 1875/2*a*b^2 + 625/4*b^3 - 1/4*(140*a^3 - 1491*a^2*b + 3750*a*b^2 + 625*b^3)*cos(d*x + c)^2 - 1/2*((a^9*b^3 - 18*a^8*b^4 + 75*a^7*b^5 - 140*a^6*b^6 + 135*a^5*b^7 - 66*a^4*b^8 + 13*a^3*b^9)*d^3*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (70*a^5*b - 623*a^4*b^2 + 1161*a^3*b^3 + 995*a^2*b^4 + 125*a*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2)) + 1/4*(2*(4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2*cos(d*x + c)^2 - (4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2)*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))) + ((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 4*a^3 - 35*a^2*b + 70*a*b^2 + 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(-35*a^3 + 1491/4*a^2*b - 1875/2*a*b^2 - 625/4*b^3 + 1/4*(140*a^3 - 1491*a^2*b + 3750*a*b^2 + 625*b^3)*cos(d*x + c)^2 + 1/2*((a^9*b^3 - 18*a^8*b^4 + 75*a^7*b^5 - 140*a^6*b^6 + 135*a^5*b^7 - 66*a^4*b^8 + 13*a^3*b^9)*d^3*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) + (70*a^5*b - 623*a^4*b^2 + 1161*a^3*b^3 + 995*a^2*b^4 + 125*a*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 4*a^3 - 35*a^2*b + 70*a*b^2 + 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2)) + 1/4*(2*(4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2*cos(d*x + c)^2 - (4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2)*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))) - ((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 4*a^3 - 35*a^2*b + 70*a*b^2 + 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(-35*a^3 + 1491/4*a^2*b - 1875/2*a*b^2 - 625/4*b^3 + 1/4*(140*a^3 - 1491*a^2*b + 3750*a*b^2 + 625*b^3)*cos(d*x + c)^2 - 1/2*((a^9*b^3 - 18*a^8*b^4 + 75*a^7*b^5 - 140*a^6*b^6 + 135*a^5*b^7 - 66*a^4*b^8 + 13*a^3*b^9)*d^3*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) + (70*a^5*b - 623*a^4*b^2 + 1161*a^3*b^3 + 995*a^2*b^4 + 125*a*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 4*a^3 - 35*a^2*b + 70*a*b^2 + 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2)) + 1/4*(2*(4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2*cos(d*x + c)^2 - (4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2)*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))) - 8*(2*(2*a*b - 5*b^2)*cos(d*x + c)^7 - 3*(5*a*b - 13*b^2)*cos(d*x + c)^5 + 24*(a*b - 2*b^2)*cos(d*x + c)^3 - (a^2 + 18*a*b - 19*b^2)*cos(d*x + c))*sin(d*x + c))/((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)","B",0
231,1,5961,0,6.273175," ","integrate(sin(d*x+c)^6/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","\frac{{\left({\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} + 7 \, a^{2} b^{4} - 3 \, a b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)} \sqrt{-\frac{16 \, a^{4} - 116 \, a^{3} b + 229 \, a^{2} b^{2} + 30 \, a b^{3} - 15 \, b^{4} + {\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}}{{\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}}} \log\left(320 \, a^{5} - 2724 \, a^{4} b + 6243 \, a^{3} b^{2} - \frac{9389}{4} \, a^{2} b^{3} + \frac{729}{2} \, a b^{4} - \frac{81}{4} \, b^{5} - \frac{1}{4} \, {\left(1280 \, a^{5} - 10896 \, a^{4} b + 24972 \, a^{3} b^{2} - 9389 \, a^{2} b^{3} + 1458 \, a b^{4} - 81 \, b^{5}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(2 \, a^{11} b^{3} - 27 \, a^{10} b^{4} + 108 \, a^{9} b^{5} - 205 \, a^{8} b^{6} + 210 \, a^{7} b^{7} - 117 \, a^{6} b^{8} + 32 \, a^{5} b^{9} - 3 \, a^{4} b^{10}\right)} d^{3} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(320 \, a^{7} b - 2404 \, a^{6} b^{2} + 4779 \, a^{5} b^{3} - 1025 \, a^{4} b^{4} + 49 \, a^{3} b^{5} + 9 \, a^{2} b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{16 \, a^{4} - 116 \, a^{3} b + 229 \, a^{2} b^{2} + 30 \, a b^{3} - 15 \, b^{4} + {\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}}{{\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{10} b - 156 \, a^{9} b^{2} + 549 \, a^{8} b^{3} - 965 \, a^{7} b^{4} + 930 \, a^{6} b^{5} - 486 \, a^{5} b^{6} + 121 \, a^{4} b^{7} - 9 \, a^{3} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{10} b - 156 \, a^{9} b^{2} + 549 \, a^{8} b^{3} - 965 \, a^{7} b^{4} + 930 \, a^{6} b^{5} - 486 \, a^{5} b^{6} + 121 \, a^{4} b^{7} - 9 \, a^{3} b^{8}\right)} d^{2}\right)} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}\right) - {\left({\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} + 7 \, a^{2} b^{4} - 3 \, a b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)} \sqrt{-\frac{16 \, a^{4} - 116 \, a^{3} b + 229 \, a^{2} b^{2} + 30 \, a b^{3} - 15 \, b^{4} + {\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}}{{\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}}} \log\left(320 \, a^{5} - 2724 \, a^{4} b + 6243 \, a^{3} b^{2} - \frac{9389}{4} \, a^{2} b^{3} + \frac{729}{2} \, a b^{4} - \frac{81}{4} \, b^{5} - \frac{1}{4} \, {\left(1280 \, a^{5} - 10896 \, a^{4} b + 24972 \, a^{3} b^{2} - 9389 \, a^{2} b^{3} + 1458 \, a b^{4} - 81 \, b^{5}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(2 \, a^{11} b^{3} - 27 \, a^{10} b^{4} + 108 \, a^{9} b^{5} - 205 \, a^{8} b^{6} + 210 \, a^{7} b^{7} - 117 \, a^{6} b^{8} + 32 \, a^{5} b^{9} - 3 \, a^{4} b^{10}\right)} d^{3} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(320 \, a^{7} b - 2404 \, a^{6} b^{2} + 4779 \, a^{5} b^{3} - 1025 \, a^{4} b^{4} + 49 \, a^{3} b^{5} + 9 \, a^{2} b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{16 \, a^{4} - 116 \, a^{3} b + 229 \, a^{2} b^{2} + 30 \, a b^{3} - 15 \, b^{4} + {\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}}{{\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{10} b - 156 \, a^{9} b^{2} + 549 \, a^{8} b^{3} - 965 \, a^{7} b^{4} + 930 \, a^{6} b^{5} - 486 \, a^{5} b^{6} + 121 \, a^{4} b^{7} - 9 \, a^{3} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{10} b - 156 \, a^{9} b^{2} + 549 \, a^{8} b^{3} - 965 \, a^{7} b^{4} + 930 \, a^{6} b^{5} - 486 \, a^{5} b^{6} + 121 \, a^{4} b^{7} - 9 \, a^{3} b^{8}\right)} d^{2}\right)} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}\right) + {\left({\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} + 7 \, a^{2} b^{4} - 3 \, a b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)} \sqrt{-\frac{16 \, a^{4} - 116 \, a^{3} b + 229 \, a^{2} b^{2} + 30 \, a b^{3} - 15 \, b^{4} - {\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}}{{\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}}} \log\left(-320 \, a^{5} + 2724 \, a^{4} b - 6243 \, a^{3} b^{2} + \frac{9389}{4} \, a^{2} b^{3} - \frac{729}{2} \, a b^{4} + \frac{81}{4} \, b^{5} + \frac{1}{4} \, {\left(1280 \, a^{5} - 10896 \, a^{4} b + 24972 \, a^{3} b^{2} - 9389 \, a^{2} b^{3} + 1458 \, a b^{4} - 81 \, b^{5}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(2 \, a^{11} b^{3} - 27 \, a^{10} b^{4} + 108 \, a^{9} b^{5} - 205 \, a^{8} b^{6} + 210 \, a^{7} b^{7} - 117 \, a^{6} b^{8} + 32 \, a^{5} b^{9} - 3 \, a^{4} b^{10}\right)} d^{3} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(320 \, a^{7} b - 2404 \, a^{6} b^{2} + 4779 \, a^{5} b^{3} - 1025 \, a^{4} b^{4} + 49 \, a^{3} b^{5} + 9 \, a^{2} b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{16 \, a^{4} - 116 \, a^{3} b + 229 \, a^{2} b^{2} + 30 \, a b^{3} - 15 \, b^{4} - {\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}}{{\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{10} b - 156 \, a^{9} b^{2} + 549 \, a^{8} b^{3} - 965 \, a^{7} b^{4} + 930 \, a^{6} b^{5} - 486 \, a^{5} b^{6} + 121 \, a^{4} b^{7} - 9 \, a^{3} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{10} b - 156 \, a^{9} b^{2} + 549 \, a^{8} b^{3} - 965 \, a^{7} b^{4} + 930 \, a^{6} b^{5} - 486 \, a^{5} b^{6} + 121 \, a^{4} b^{7} - 9 \, a^{3} b^{8}\right)} d^{2}\right)} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}\right) - {\left({\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} + 7 \, a^{2} b^{4} - 3 \, a b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)} \sqrt{-\frac{16 \, a^{4} - 116 \, a^{3} b + 229 \, a^{2} b^{2} + 30 \, a b^{3} - 15 \, b^{4} - {\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}}{{\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}}} \log\left(-320 \, a^{5} + 2724 \, a^{4} b - 6243 \, a^{3} b^{2} + \frac{9389}{4} \, a^{2} b^{3} - \frac{729}{2} \, a b^{4} + \frac{81}{4} \, b^{5} + \frac{1}{4} \, {\left(1280 \, a^{5} - 10896 \, a^{4} b + 24972 \, a^{3} b^{2} - 9389 \, a^{2} b^{3} + 1458 \, a b^{4} - 81 \, b^{5}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(2 \, a^{11} b^{3} - 27 \, a^{10} b^{4} + 108 \, a^{9} b^{5} - 205 \, a^{8} b^{6} + 210 \, a^{7} b^{7} - 117 \, a^{6} b^{8} + 32 \, a^{5} b^{9} - 3 \, a^{4} b^{10}\right)} d^{3} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(320 \, a^{7} b - 2404 \, a^{6} b^{2} + 4779 \, a^{5} b^{3} - 1025 \, a^{4} b^{4} + 49 \, a^{3} b^{5} + 9 \, a^{2} b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{16 \, a^{4} - 116 \, a^{3} b + 229 \, a^{2} b^{2} + 30 \, a b^{3} - 15 \, b^{4} - {\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}}{{\left(a^{7} b^{3} - 5 \, a^{6} b^{4} + 10 \, a^{5} b^{5} - 10 \, a^{4} b^{6} + 5 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(16 \, a^{10} b - 156 \, a^{9} b^{2} + 549 \, a^{8} b^{3} - 965 \, a^{7} b^{4} + 930 \, a^{6} b^{5} - 486 \, a^{5} b^{6} + 121 \, a^{4} b^{7} - 9 \, a^{3} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(16 \, a^{10} b - 156 \, a^{9} b^{2} + 549 \, a^{8} b^{3} - 965 \, a^{7} b^{4} + 930 \, a^{6} b^{5} - 486 \, a^{5} b^{6} + 121 \, a^{4} b^{7} - 9 \, a^{3} b^{8}\right)} d^{2}\right)} \sqrt{\frac{6400 \, a^{6} - 48160 \, a^{5} b + 104361 \, a^{4} b^{2} - 53212 \, a^{3} b^{3} + 12814 \, a^{2} b^{4} - 1548 \, a b^{5} + 81 \, b^{6}}{{\left(a^{15} b^{3} - 10 \, a^{14} b^{4} + 45 \, a^{13} b^{5} - 120 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 252 \, a^{10} b^{8} + 210 \, a^{9} b^{9} - 120 \, a^{8} b^{10} + 45 \, a^{7} b^{11} - 10 \, a^{6} b^{12} + a^{5} b^{13}\right)} d^{4}}}\right) - 8 \, {\left(3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{7} + {\left(2 \, a^{2} b - 17 \, a b^{2} - 9 \, b^{3}\right)} \cos\left(d x + c\right)^{5} - {\left(11 \, a^{2} b - 26 \, a b^{2} - 9 \, b^{3}\right)} \cos\left(d x + c\right)^{3} + {\left(2 \, a^{3} + 13 \, a^{2} b - 12 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{256 \, {\left({\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b^{2} - 5 \, a^{3} b^{3} + 7 \, a^{2} b^{4} - 3 \, a b^{5}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} b - 4 \, a^{4} b^{2} + 6 \, a^{3} b^{3} - 4 \, a^{2} b^{4} + a b^{5}\right)} d\right)}}"," ",0,"1/256*(((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 + (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2))*log(320*a^5 - 2724*a^4*b + 6243*a^3*b^2 - 9389/4*a^2*b^3 + 729/2*a*b^4 - 81/4*b^5 - 1/4*(1280*a^5 - 10896*a^4*b + 24972*a^3*b^2 - 9389*a^2*b^3 + 1458*a*b^4 - 81*b^5)*cos(d*x + c)^2 + 1/2*((2*a^11*b^3 - 27*a^10*b^4 + 108*a^9*b^5 - 205*a^8*b^6 + 210*a^7*b^7 - 117*a^6*b^8 + 32*a^5*b^9 - 3*a^4*b^10)*d^3*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) + (320*a^7*b - 2404*a^6*b^2 + 4779*a^5*b^3 - 1025*a^4*b^4 + 49*a^3*b^5 + 9*a^2*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 + (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2)) - 1/4*(2*(16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2*cos(d*x + c)^2 - (16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2)*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))) - ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 + (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2))*log(320*a^5 - 2724*a^4*b + 6243*a^3*b^2 - 9389/4*a^2*b^3 + 729/2*a*b^4 - 81/4*b^5 - 1/4*(1280*a^5 - 10896*a^4*b + 24972*a^3*b^2 - 9389*a^2*b^3 + 1458*a*b^4 - 81*b^5)*cos(d*x + c)^2 - 1/2*((2*a^11*b^3 - 27*a^10*b^4 + 108*a^9*b^5 - 205*a^8*b^6 + 210*a^7*b^7 - 117*a^6*b^8 + 32*a^5*b^9 - 3*a^4*b^10)*d^3*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) + (320*a^7*b - 2404*a^6*b^2 + 4779*a^5*b^3 - 1025*a^4*b^4 + 49*a^3*b^5 + 9*a^2*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 + (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2)) - 1/4*(2*(16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2*cos(d*x + c)^2 - (16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2)*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))) + ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 - (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2))*log(-320*a^5 + 2724*a^4*b - 6243*a^3*b^2 + 9389/4*a^2*b^3 - 729/2*a*b^4 + 81/4*b^5 + 1/4*(1280*a^5 - 10896*a^4*b + 24972*a^3*b^2 - 9389*a^2*b^3 + 1458*a*b^4 - 81*b^5)*cos(d*x + c)^2 + 1/2*((2*a^11*b^3 - 27*a^10*b^4 + 108*a^9*b^5 - 205*a^8*b^6 + 210*a^7*b^7 - 117*a^6*b^8 + 32*a^5*b^9 - 3*a^4*b^10)*d^3*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (320*a^7*b - 2404*a^6*b^2 + 4779*a^5*b^3 - 1025*a^4*b^4 + 49*a^3*b^5 + 9*a^2*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 - (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2)) - 1/4*(2*(16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2*cos(d*x + c)^2 - (16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2)*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))) - ((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 - (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2))*log(-320*a^5 + 2724*a^4*b - 6243*a^3*b^2 + 9389/4*a^2*b^3 - 729/2*a*b^4 + 81/4*b^5 + 1/4*(1280*a^5 - 10896*a^4*b + 24972*a^3*b^2 - 9389*a^2*b^3 + 1458*a*b^4 - 81*b^5)*cos(d*x + c)^2 - 1/2*((2*a^11*b^3 - 27*a^10*b^4 + 108*a^9*b^5 - 205*a^8*b^6 + 210*a^7*b^7 - 117*a^6*b^8 + 32*a^5*b^9 - 3*a^4*b^10)*d^3*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (320*a^7*b - 2404*a^6*b^2 + 4779*a^5*b^3 - 1025*a^4*b^4 + 49*a^3*b^5 + 9*a^2*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(16*a^4 - 116*a^3*b + 229*a^2*b^2 + 30*a*b^3 - 15*b^4 - (a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4)))/((a^7*b^3 - 5*a^6*b^4 + 10*a^5*b^5 - 10*a^4*b^6 + 5*a^3*b^7 - a^2*b^8)*d^2)) - 1/4*(2*(16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2*cos(d*x + c)^2 - (16*a^10*b - 156*a^9*b^2 + 549*a^8*b^3 - 965*a^7*b^4 + 930*a^6*b^5 - 486*a^5*b^6 + 121*a^4*b^7 - 9*a^3*b^8)*d^2)*sqrt((6400*a^6 - 48160*a^5*b + 104361*a^4*b^2 - 53212*a^3*b^3 + 12814*a^2*b^4 - 1548*a*b^5 + 81*b^6)/((a^15*b^3 - 10*a^14*b^4 + 45*a^13*b^5 - 120*a^12*b^6 + 210*a^11*b^7 - 252*a^10*b^8 + 210*a^9*b^9 - 120*a^8*b^10 + 45*a^7*b^11 - 10*a^6*b^12 + a^5*b^13)*d^4))) - 8*(3*(a*b^2 + b^3)*cos(d*x + c)^7 + (2*a^2*b - 17*a*b^2 - 9*b^3)*cos(d*x + c)^5 - (11*a^2*b - 26*a*b^2 - 9*b^3)*cos(d*x + c)^3 + (2*a^3 + 13*a^2*b - 12*a*b^2 - 3*b^3)*cos(d*x + c))*sin(d*x + c))/((a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^8 - 4*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*d*cos(d*x + c)^6 - 2*(a^4*b^2 - 5*a^3*b^3 + 7*a^2*b^4 - 3*a*b^5)*d*cos(d*x + c)^4 + 4*(a^4*b^2 - 3*a^3*b^3 + 3*a^2*b^4 - a*b^5)*d*cos(d*x + c)^2 + (a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5)*d)","B",0
232,1,5510,0,3.809170," ","integrate(sin(d*x+c)^4/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","\frac{3 \, {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b - 5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 3 \, a b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} + 4 \, a^{3} + 21 \, a^{2} b - 10 \, a b^{2} + b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} \log\left(432 \, a^{4} + 27 \, a^{3} b - \frac{783}{4} \, a^{2} b^{2} + \frac{135}{2} \, a b^{3} - \frac{27}{4} \, b^{4} - \frac{27}{4} \, {\left(64 \, a^{4} + 4 \, a^{3} b - 29 \, a^{2} b^{2} + 10 \, a b^{3} - b^{4}\right)} \cos\left(d x + c\right)^{2} + \frac{27}{2} \, {\left({\left(5 \, a^{12} b - 26 \, a^{11} b^{2} + 55 \, a^{10} b^{3} - 60 \, a^{9} b^{4} + 35 \, a^{8} b^{5} - 10 \, a^{7} b^{6} + a^{6} b^{7}\right)} d^{3} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(32 \, a^{7} + 58 \, a^{6} b - 13 \, a^{5} b^{2} - 21 \, a^{4} b^{3} + 9 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} + 4 \, a^{3} + 21 \, a^{2} b - 10 \, a b^{2} + b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} + \frac{27}{4} \, {\left(2 \, {\left(4 \, a^{10} - 21 \, a^{9} b + 45 \, a^{8} b^{2} - 50 \, a^{7} b^{3} + 30 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{10} - 21 \, a^{9} b + 45 \, a^{8} b^{2} - 50 \, a^{7} b^{3} + 30 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6}\right)} d^{2}\right)} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}}\right) - 3 \, {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b - 5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 3 \, a b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} + 4 \, a^{3} + 21 \, a^{2} b - 10 \, a b^{2} + b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} \log\left(432 \, a^{4} + 27 \, a^{3} b - \frac{783}{4} \, a^{2} b^{2} + \frac{135}{2} \, a b^{3} - \frac{27}{4} \, b^{4} - \frac{27}{4} \, {\left(64 \, a^{4} + 4 \, a^{3} b - 29 \, a^{2} b^{2} + 10 \, a b^{3} - b^{4}\right)} \cos\left(d x + c\right)^{2} - \frac{27}{2} \, {\left({\left(5 \, a^{12} b - 26 \, a^{11} b^{2} + 55 \, a^{10} b^{3} - 60 \, a^{9} b^{4} + 35 \, a^{8} b^{5} - 10 \, a^{7} b^{6} + a^{6} b^{7}\right)} d^{3} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(32 \, a^{7} + 58 \, a^{6} b - 13 \, a^{5} b^{2} - 21 \, a^{4} b^{3} + 9 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} + 4 \, a^{3} + 21 \, a^{2} b - 10 \, a b^{2} + b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} + \frac{27}{4} \, {\left(2 \, {\left(4 \, a^{10} - 21 \, a^{9} b + 45 \, a^{8} b^{2} - 50 \, a^{7} b^{3} + 30 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{10} - 21 \, a^{9} b + 45 \, a^{8} b^{2} - 50 \, a^{7} b^{3} + 30 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6}\right)} d^{2}\right)} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}}\right) + 3 \, {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b - 5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 3 \, a b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} - 4 \, a^{3} - 21 \, a^{2} b + 10 \, a b^{2} - b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} \log\left(-432 \, a^{4} - 27 \, a^{3} b + \frac{783}{4} \, a^{2} b^{2} - \frac{135}{2} \, a b^{3} + \frac{27}{4} \, b^{4} + \frac{27}{4} \, {\left(64 \, a^{4} + 4 \, a^{3} b - 29 \, a^{2} b^{2} + 10 \, a b^{3} - b^{4}\right)} \cos\left(d x + c\right)^{2} + \frac{27}{2} \, {\left({\left(5 \, a^{12} b - 26 \, a^{11} b^{2} + 55 \, a^{10} b^{3} - 60 \, a^{9} b^{4} + 35 \, a^{8} b^{5} - 10 \, a^{7} b^{6} + a^{6} b^{7}\right)} d^{3} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(32 \, a^{7} + 58 \, a^{6} b - 13 \, a^{5} b^{2} - 21 \, a^{4} b^{3} + 9 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} - 4 \, a^{3} - 21 \, a^{2} b + 10 \, a b^{2} - b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} + \frac{27}{4} \, {\left(2 \, {\left(4 \, a^{10} - 21 \, a^{9} b + 45 \, a^{8} b^{2} - 50 \, a^{7} b^{3} + 30 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{10} - 21 \, a^{9} b + 45 \, a^{8} b^{2} - 50 \, a^{7} b^{3} + 30 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6}\right)} d^{2}\right)} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}}\right) - 3 \, {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b - 5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 3 \, a b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} - 4 \, a^{3} - 21 \, a^{2} b + 10 \, a b^{2} - b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} \log\left(-432 \, a^{4} - 27 \, a^{3} b + \frac{783}{4} \, a^{2} b^{2} - \frac{135}{2} \, a b^{3} + \frac{27}{4} \, b^{4} + \frac{27}{4} \, {\left(64 \, a^{4} + 4 \, a^{3} b - 29 \, a^{2} b^{2} + 10 \, a b^{3} - b^{4}\right)} \cos\left(d x + c\right)^{2} - \frac{27}{2} \, {\left({\left(5 \, a^{12} b - 26 \, a^{11} b^{2} + 55 \, a^{10} b^{3} - 60 \, a^{9} b^{4} + 35 \, a^{8} b^{5} - 10 \, a^{7} b^{6} + a^{6} b^{7}\right)} d^{3} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(32 \, a^{7} + 58 \, a^{6} b - 13 \, a^{5} b^{2} - 21 \, a^{4} b^{3} + 9 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}} - 4 \, a^{3} - 21 \, a^{2} b + 10 \, a b^{2} - b^{3}}{{\left(a^{8} b - 5 \, a^{7} b^{2} + 10 \, a^{6} b^{3} - 10 \, a^{5} b^{4} + 5 \, a^{4} b^{5} - a^{3} b^{6}\right)} d^{2}}} + \frac{27}{4} \, {\left(2 \, {\left(4 \, a^{10} - 21 \, a^{9} b + 45 \, a^{8} b^{2} - 50 \, a^{7} b^{3} + 30 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(4 \, a^{10} - 21 \, a^{9} b + 45 \, a^{8} b^{2} - 50 \, a^{7} b^{3} + 30 \, a^{6} b^{4} - 9 \, a^{5} b^{5} + a^{4} b^{6}\right)} d^{2}\right)} \sqrt{\frac{256 \, a^{6} + 160 \, a^{5} b - 167 \, a^{4} b^{2} - 28 \, a^{3} b^{3} + 46 \, a^{2} b^{4} - 12 \, a b^{5} + b^{6}}{{\left(a^{17} b - 10 \, a^{16} b^{2} + 45 \, a^{15} b^{3} - 120 \, a^{14} b^{4} + 210 \, a^{13} b^{5} - 252 \, a^{12} b^{6} + 210 \, a^{11} b^{7} - 120 \, a^{10} b^{8} + 45 \, a^{9} b^{9} - 10 \, a^{8} b^{10} + a^{7} b^{11}\right)} d^{4}}}\right) - 8 \, {\left(2 \, {\left(2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{7} - {\left(17 \, a b + 7 \, b^{2}\right)} \cos\left(d x + c\right)^{5} - 8 \, {\left(a^{2} - 3 \, a b - b^{2}\right)} \cos\left(d x + c\right)^{3} + {\left(17 \, a^{2} - 14 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{256 \, {\left({\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{4} b - 5 \, a^{3} b^{2} + 7 \, a^{2} b^{3} - 3 \, a b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d\right)}}"," ",0,"1/256*(3*((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) + 4*a^3 + 21*a^2*b - 10*a*b^2 + b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log(432*a^4 + 27*a^3*b - 783/4*a^2*b^2 + 135/2*a*b^3 - 27/4*b^4 - 27/4*(64*a^4 + 4*a^3*b - 29*a^2*b^2 + 10*a*b^3 - b^4)*cos(d*x + c)^2 + 27/2*((5*a^12*b - 26*a^11*b^2 + 55*a^10*b^3 - 60*a^9*b^4 + 35*a^8*b^5 - 10*a^7*b^6 + a^6*b^7)*d^3*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) - (32*a^7 + 58*a^6*b - 13*a^5*b^2 - 21*a^4*b^3 + 9*a^3*b^4 - a^2*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) + 4*a^3 + 21*a^2*b - 10*a*b^2 + b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2)) + 27/4*(2*(4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2*cos(d*x + c)^2 - (4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2)*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))) - 3*((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) + 4*a^3 + 21*a^2*b - 10*a*b^2 + b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log(432*a^4 + 27*a^3*b - 783/4*a^2*b^2 + 135/2*a*b^3 - 27/4*b^4 - 27/4*(64*a^4 + 4*a^3*b - 29*a^2*b^2 + 10*a*b^3 - b^4)*cos(d*x + c)^2 - 27/2*((5*a^12*b - 26*a^11*b^2 + 55*a^10*b^3 - 60*a^9*b^4 + 35*a^8*b^5 - 10*a^7*b^6 + a^6*b^7)*d^3*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) - (32*a^7 + 58*a^6*b - 13*a^5*b^2 - 21*a^4*b^3 + 9*a^3*b^4 - a^2*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) + 4*a^3 + 21*a^2*b - 10*a*b^2 + b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2)) + 27/4*(2*(4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2*cos(d*x + c)^2 - (4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2)*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))) + 3*((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) - 4*a^3 - 21*a^2*b + 10*a*b^2 - b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log(-432*a^4 - 27*a^3*b + 783/4*a^2*b^2 - 135/2*a*b^3 + 27/4*b^4 + 27/4*(64*a^4 + 4*a^3*b - 29*a^2*b^2 + 10*a*b^3 - b^4)*cos(d*x + c)^2 + 27/2*((5*a^12*b - 26*a^11*b^2 + 55*a^10*b^3 - 60*a^9*b^4 + 35*a^8*b^5 - 10*a^7*b^6 + a^6*b^7)*d^3*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) + (32*a^7 + 58*a^6*b - 13*a^5*b^2 - 21*a^4*b^3 + 9*a^3*b^4 - a^2*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) - 4*a^3 - 21*a^2*b + 10*a*b^2 - b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2)) + 27/4*(2*(4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2*cos(d*x + c)^2 - (4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2)*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))) - 3*((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) - 4*a^3 - 21*a^2*b + 10*a*b^2 - b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2))*log(-432*a^4 - 27*a^3*b + 783/4*a^2*b^2 - 135/2*a*b^3 + 27/4*b^4 + 27/4*(64*a^4 + 4*a^3*b - 29*a^2*b^2 + 10*a*b^3 - b^4)*cos(d*x + c)^2 - 27/2*((5*a^12*b - 26*a^11*b^2 + 55*a^10*b^3 - 60*a^9*b^4 + 35*a^8*b^5 - 10*a^7*b^6 + a^6*b^7)*d^3*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) + (32*a^7 + 58*a^6*b - 13*a^5*b^2 - 21*a^4*b^3 + 9*a^3*b^4 - a^2*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4)) - 4*a^3 - 21*a^2*b + 10*a*b^2 - b^3)/((a^8*b - 5*a^7*b^2 + 10*a^6*b^3 - 10*a^5*b^4 + 5*a^4*b^5 - a^3*b^6)*d^2)) + 27/4*(2*(4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2*cos(d*x + c)^2 - (4*a^10 - 21*a^9*b + 45*a^8*b^2 - 50*a^7*b^3 + 30*a^6*b^4 - 9*a^5*b^5 + a^4*b^6)*d^2)*sqrt((256*a^6 + 160*a^5*b - 167*a^4*b^2 - 28*a^3*b^3 + 46*a^2*b^4 - 12*a*b^5 + b^6)/((a^17*b - 10*a^16*b^2 + 45*a^15*b^3 - 120*a^14*b^4 + 210*a^13*b^5 - 252*a^12*b^6 + 210*a^11*b^7 - 120*a^10*b^8 + 45*a^9*b^9 - 10*a^8*b^10 + a^7*b^11)*d^4))) - 8*(2*(2*a*b + b^2)*cos(d*x + c)^7 - (17*a*b + 7*b^2)*cos(d*x + c)^5 - 8*(a^2 - 3*a*b - b^2)*cos(d*x + c)^3 + (17*a^2 - 14*a*b - 3*b^2)*cos(d*x + c))*sin(d*x + c))/((a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^8 - 4*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*d*cos(d*x + c)^6 - 2*(a^4*b - 5*a^3*b^2 + 7*a^2*b^3 - 3*a*b^4)*d*cos(d*x + c)^4 + 4*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cos(d*x + c)^2 + (a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d)","B",0
233,1,6215,0,6.672303," ","integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","-\frac{{\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{144 \, a^{4} + 76 \, a^{3} b - 155 \, a^{2} b^{2} + 94 \, a b^{3} - 15 \, b^{4} - {\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}}{{\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2}}} \log\left(13824 \, a^{6} - 24576 \, a^{5} b + 24084 \, a^{4} b^{2} - 14455 \, a^{3} b^{3} + \frac{22509}{4} \, a^{2} b^{4} - \frac{2625}{2} \, a b^{5} + \frac{625}{4} \, b^{6} - \frac{1}{4} \, {\left(55296 \, a^{6} - 98304 \, a^{5} b + 96336 \, a^{4} b^{2} - 57820 \, a^{3} b^{3} + 22509 \, a^{2} b^{4} - 5250 \, a b^{5} + 625 \, b^{6}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(22 \, a^{14} b - 125 \, a^{13} b^{2} + 300 \, a^{12} b^{3} - 395 \, a^{11} b^{4} + 310 \, a^{10} b^{5} - 147 \, a^{9} b^{6} + 40 \, a^{8} b^{7} - 5 \, a^{7} b^{8}\right)} d^{3} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(4608 \, a^{9} - 6144 \, a^{8} b + 5052 \, a^{7} b^{2} - 2437 \, a^{6} b^{3} + 783 \, a^{5} b^{4} - 159 \, a^{4} b^{5} + 25 \, a^{3} b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{144 \, a^{4} + 76 \, a^{3} b - 155 \, a^{2} b^{2} + 94 \, a b^{3} - 15 \, b^{4} - {\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}}{{\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(144 \, a^{12} - 796 \, a^{11} b + 1845 \, a^{10} b^{2} - 2325 \, a^{9} b^{3} + 1730 \, a^{8} b^{4} - 774 \, a^{7} b^{5} + 201 \, a^{6} b^{6} - 25 \, a^{5} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(144 \, a^{12} - 796 \, a^{11} b + 1845 \, a^{10} b^{2} - 2325 \, a^{9} b^{3} + 1730 \, a^{8} b^{4} - 774 \, a^{7} b^{5} + 201 \, a^{6} b^{6} - 25 \, a^{5} b^{7}\right)} d^{2}\right)} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}\right) - {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{144 \, a^{4} + 76 \, a^{3} b - 155 \, a^{2} b^{2} + 94 \, a b^{3} - 15 \, b^{4} - {\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}}{{\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2}}} \log\left(13824 \, a^{6} - 24576 \, a^{5} b + 24084 \, a^{4} b^{2} - 14455 \, a^{3} b^{3} + \frac{22509}{4} \, a^{2} b^{4} - \frac{2625}{2} \, a b^{5} + \frac{625}{4} \, b^{6} - \frac{1}{4} \, {\left(55296 \, a^{6} - 98304 \, a^{5} b + 96336 \, a^{4} b^{2} - 57820 \, a^{3} b^{3} + 22509 \, a^{2} b^{4} - 5250 \, a b^{5} + 625 \, b^{6}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(22 \, a^{14} b - 125 \, a^{13} b^{2} + 300 \, a^{12} b^{3} - 395 \, a^{11} b^{4} + 310 \, a^{10} b^{5} - 147 \, a^{9} b^{6} + 40 \, a^{8} b^{7} - 5 \, a^{7} b^{8}\right)} d^{3} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(4608 \, a^{9} - 6144 \, a^{8} b + 5052 \, a^{7} b^{2} - 2437 \, a^{6} b^{3} + 783 \, a^{5} b^{4} - 159 \, a^{4} b^{5} + 25 \, a^{3} b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{144 \, a^{4} + 76 \, a^{3} b - 155 \, a^{2} b^{2} + 94 \, a b^{3} - 15 \, b^{4} - {\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}}{{\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(144 \, a^{12} - 796 \, a^{11} b + 1845 \, a^{10} b^{2} - 2325 \, a^{9} b^{3} + 1730 \, a^{8} b^{4} - 774 \, a^{7} b^{5} + 201 \, a^{6} b^{6} - 25 \, a^{5} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(144 \, a^{12} - 796 \, a^{11} b + 1845 \, a^{10} b^{2} - 2325 \, a^{9} b^{3} + 1730 \, a^{8} b^{4} - 774 \, a^{7} b^{5} + 201 \, a^{6} b^{6} - 25 \, a^{5} b^{7}\right)} d^{2}\right)} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}\right) + {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{144 \, a^{4} + 76 \, a^{3} b - 155 \, a^{2} b^{2} + 94 \, a b^{3} - 15 \, b^{4} + {\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}}{{\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2}}} \log\left(-13824 \, a^{6} + 24576 \, a^{5} b - 24084 \, a^{4} b^{2} + 14455 \, a^{3} b^{3} - \frac{22509}{4} \, a^{2} b^{4} + \frac{2625}{2} \, a b^{5} - \frac{625}{4} \, b^{6} + \frac{1}{4} \, {\left(55296 \, a^{6} - 98304 \, a^{5} b + 96336 \, a^{4} b^{2} - 57820 \, a^{3} b^{3} + 22509 \, a^{2} b^{4} - 5250 \, a b^{5} + 625 \, b^{6}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(22 \, a^{14} b - 125 \, a^{13} b^{2} + 300 \, a^{12} b^{3} - 395 \, a^{11} b^{4} + 310 \, a^{10} b^{5} - 147 \, a^{9} b^{6} + 40 \, a^{8} b^{7} - 5 \, a^{7} b^{8}\right)} d^{3} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(4608 \, a^{9} - 6144 \, a^{8} b + 5052 \, a^{7} b^{2} - 2437 \, a^{6} b^{3} + 783 \, a^{5} b^{4} - 159 \, a^{4} b^{5} + 25 \, a^{3} b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{144 \, a^{4} + 76 \, a^{3} b - 155 \, a^{2} b^{2} + 94 \, a b^{3} - 15 \, b^{4} + {\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}}{{\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(144 \, a^{12} - 796 \, a^{11} b + 1845 \, a^{10} b^{2} - 2325 \, a^{9} b^{3} + 1730 \, a^{8} b^{4} - 774 \, a^{7} b^{5} + 201 \, a^{6} b^{6} - 25 \, a^{5} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(144 \, a^{12} - 796 \, a^{11} b + 1845 \, a^{10} b^{2} - 2325 \, a^{9} b^{3} + 1730 \, a^{8} b^{4} - 774 \, a^{7} b^{5} + 201 \, a^{6} b^{6} - 25 \, a^{5} b^{7}\right)} d^{2}\right)} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}\right) - {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{144 \, a^{4} + 76 \, a^{3} b - 155 \, a^{2} b^{2} + 94 \, a b^{3} - 15 \, b^{4} + {\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}}{{\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2}}} \log\left(-13824 \, a^{6} + 24576 \, a^{5} b - 24084 \, a^{4} b^{2} + 14455 \, a^{3} b^{3} - \frac{22509}{4} \, a^{2} b^{4} + \frac{2625}{2} \, a b^{5} - \frac{625}{4} \, b^{6} + \frac{1}{4} \, {\left(55296 \, a^{6} - 98304 \, a^{5} b + 96336 \, a^{4} b^{2} - 57820 \, a^{3} b^{3} + 22509 \, a^{2} b^{4} - 5250 \, a b^{5} + 625 \, b^{6}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(22 \, a^{14} b - 125 \, a^{13} b^{2} + 300 \, a^{12} b^{3} - 395 \, a^{11} b^{4} + 310 \, a^{10} b^{5} - 147 \, a^{9} b^{6} + 40 \, a^{8} b^{7} - 5 \, a^{7} b^{8}\right)} d^{3} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(4608 \, a^{9} - 6144 \, a^{8} b + 5052 \, a^{7} b^{2} - 2437 \, a^{6} b^{3} + 783 \, a^{5} b^{4} - 159 \, a^{4} b^{5} + 25 \, a^{3} b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{144 \, a^{4} + 76 \, a^{3} b - 155 \, a^{2} b^{2} + 94 \, a b^{3} - 15 \, b^{4} + {\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}}{{\left(a^{9} b - 5 \, a^{8} b^{2} + 10 \, a^{7} b^{3} - 10 \, a^{6} b^{4} + 5 \, a^{5} b^{5} - a^{4} b^{6}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(144 \, a^{12} - 796 \, a^{11} b + 1845 \, a^{10} b^{2} - 2325 \, a^{9} b^{3} + 1730 \, a^{8} b^{4} - 774 \, a^{7} b^{5} + 201 \, a^{6} b^{6} - 25 \, a^{5} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(144 \, a^{12} - 796 \, a^{11} b + 1845 \, a^{10} b^{2} - 2325 \, a^{9} b^{3} + 1730 \, a^{8} b^{4} - 774 \, a^{7} b^{5} + 201 \, a^{6} b^{6} - 25 \, a^{5} b^{7}\right)} d^{2}\right)} \sqrt{\frac{147456 \, a^{8} - 368640 \, a^{7} b + 498432 \, a^{6} b^{2} - 437952 \, a^{5} b^{3} + 269641 \, a^{4} b^{4} - 117532 \, a^{3} b^{5} + 35406 \, a^{2} b^{6} - 6700 \, a b^{7} + 625 \, b^{8}}{{\left(a^{19} b - 10 \, a^{18} b^{2} + 45 \, a^{17} b^{3} - 120 \, a^{16} b^{4} + 210 \, a^{15} b^{5} - 252 \, a^{14} b^{6} + 210 \, a^{13} b^{7} - 120 \, a^{12} b^{8} + 45 \, a^{11} b^{9} - 10 \, a^{10} b^{10} + a^{9} b^{11}\right)} d^{4}}}\right) + 8 \, {\left({\left(11 \, a b^{2} - 5 \, b^{3}\right)} \cos\left(d x + c\right)^{7} - 3 \, {\left(2 \, a^{2} b + 11 \, a b^{2} - 5 \, b^{3}\right)} \cos\left(d x + c\right)^{5} - 3 \, {\left(a^{2} b - 14 \, a b^{2} + 5 \, b^{3}\right)} \cos\left(d x + c\right)^{3} + 5 \, {\left(2 \, a^{3} + a^{2} b - 4 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{256 \, {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)}}"," ",0,"-1/256*(((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2))*log(13824*a^6 - 24576*a^5*b + 24084*a^4*b^2 - 14455*a^3*b^3 + 22509/4*a^2*b^4 - 2625/2*a*b^5 + 625/4*b^6 - 1/4*(55296*a^6 - 98304*a^5*b + 96336*a^4*b^2 - 57820*a^3*b^3 + 22509*a^2*b^4 - 5250*a*b^5 + 625*b^6)*cos(d*x + c)^2 + 1/2*((22*a^14*b - 125*a^13*b^2 + 300*a^12*b^3 - 395*a^11*b^4 + 310*a^10*b^5 - 147*a^9*b^6 + 40*a^8*b^7 - 5*a^7*b^8)*d^3*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) + (4608*a^9 - 6144*a^8*b + 5052*a^7*b^2 - 2437*a^6*b^3 + 783*a^5*b^4 - 159*a^4*b^5 + 25*a^3*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2)) - 1/4*(2*(144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2*cos(d*x + c)^2 - (144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2)*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))) - ((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2))*log(13824*a^6 - 24576*a^5*b + 24084*a^4*b^2 - 14455*a^3*b^3 + 22509/4*a^2*b^4 - 2625/2*a*b^5 + 625/4*b^6 - 1/4*(55296*a^6 - 98304*a^5*b + 96336*a^4*b^2 - 57820*a^3*b^3 + 22509*a^2*b^4 - 5250*a*b^5 + 625*b^6)*cos(d*x + c)^2 - 1/2*((22*a^14*b - 125*a^13*b^2 + 300*a^12*b^3 - 395*a^11*b^4 + 310*a^10*b^5 - 147*a^9*b^6 + 40*a^8*b^7 - 5*a^7*b^8)*d^3*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) + (4608*a^9 - 6144*a^8*b + 5052*a^7*b^2 - 2437*a^6*b^3 + 783*a^5*b^4 - 159*a^4*b^5 + 25*a^3*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2)) - 1/4*(2*(144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2*cos(d*x + c)^2 - (144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2)*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))) + ((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 + (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2))*log(-13824*a^6 + 24576*a^5*b - 24084*a^4*b^2 + 14455*a^3*b^3 - 22509/4*a^2*b^4 + 2625/2*a*b^5 - 625/4*b^6 + 1/4*(55296*a^6 - 98304*a^5*b + 96336*a^4*b^2 - 57820*a^3*b^3 + 22509*a^2*b^4 - 5250*a*b^5 + 625*b^6)*cos(d*x + c)^2 + 1/2*((22*a^14*b - 125*a^13*b^2 + 300*a^12*b^3 - 395*a^11*b^4 + 310*a^10*b^5 - 147*a^9*b^6 + 40*a^8*b^7 - 5*a^7*b^8)*d^3*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) - (4608*a^9 - 6144*a^8*b + 5052*a^7*b^2 - 2437*a^6*b^3 + 783*a^5*b^4 - 159*a^4*b^5 + 25*a^3*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 + (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2)) - 1/4*(2*(144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2*cos(d*x + c)^2 - (144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2)*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))) - ((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 + (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2))*log(-13824*a^6 + 24576*a^5*b - 24084*a^4*b^2 + 14455*a^3*b^3 - 22509/4*a^2*b^4 + 2625/2*a*b^5 - 625/4*b^6 + 1/4*(55296*a^6 - 98304*a^5*b + 96336*a^4*b^2 - 57820*a^3*b^3 + 22509*a^2*b^4 - 5250*a*b^5 + 625*b^6)*cos(d*x + c)^2 - 1/2*((22*a^14*b - 125*a^13*b^2 + 300*a^12*b^3 - 395*a^11*b^4 + 310*a^10*b^5 - 147*a^9*b^6 + 40*a^8*b^7 - 5*a^7*b^8)*d^3*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) - (4608*a^9 - 6144*a^8*b + 5052*a^7*b^2 - 2437*a^6*b^3 + 783*a^5*b^4 - 159*a^4*b^5 + 25*a^3*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 + (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2)) - 1/4*(2*(144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2*cos(d*x + c)^2 - (144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2)*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))) + 8*((11*a*b^2 - 5*b^3)*cos(d*x + c)^7 - 3*(2*a^2*b + 11*a*b^2 - 5*b^3)*cos(d*x + c)^5 - 3*(a^2*b - 14*a*b^2 + 5*b^3)*cos(d*x + c)^3 + 5*(2*a^3 + a^2*b - 4*a*b^2 + b^3)*cos(d*x + c))*sin(d*x + c))/((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)","B",0
234,1,6152,0,7.338248," ","integrate(1/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","\frac{{\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{1024 \, a^{4} - 1916 \, a^{3} b + 1501 \, a^{2} b^{2} - 570 \, a b^{3} + 105 \, b^{4} - {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} \log\left(491520 \, a^{6} b - 1742720 \, a^{5} b^{2} + 2747904 \, a^{4} b^{3} - 2435877 \, a^{3} b^{4} + \frac{5106989}{4} \, a^{2} b^{5} - \frac{750141}{2} \, a b^{6} + \frac{194481}{4} \, b^{7} - \frac{1}{4} \, {\left(1966080 \, a^{6} b - 6970880 \, a^{5} b^{2} + 10991616 \, a^{4} b^{3} - 9743508 \, a^{3} b^{4} + 5106989 \, a^{2} b^{5} - 1500282 \, a b^{6} + 194481 \, b^{7}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(32 \, a^{16} - 193 \, a^{15} b + 498 \, a^{14} b^{2} - 715 \, a^{13} b^{3} + 620 \, a^{12} b^{4} - 327 \, a^{11} b^{5} + 98 \, a^{10} b^{6} - 13 \, a^{9} b^{7}\right)} d^{3} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(88320 \, a^{9} b - 319040 \, a^{8} b^{2} + 510294 \, a^{7} b^{3} - 457551 \, a^{6} b^{4} + 241865 \, a^{5} b^{5} - 71421 \, a^{4} b^{6} + 9261 \, a^{3} b^{7}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{1024 \, a^{4} - 1916 \, a^{3} b + 1501 \, a^{2} b^{2} - 570 \, a b^{3} + 105 \, b^{4} - {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(1024 \, a^{13} - 6276 \, a^{12} b + 16461 \, a^{11} b^{2} - 24005 \, a^{10} b^{3} + 21090 \, a^{9} b^{4} - 11214 \, a^{8} b^{5} + 3361 \, a^{7} b^{6} - 441 \, a^{6} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(1024 \, a^{13} - 6276 \, a^{12} b + 16461 \, a^{11} b^{2} - 24005 \, a^{10} b^{3} + 21090 \, a^{9} b^{4} - 11214 \, a^{8} b^{5} + 3361 \, a^{7} b^{6} - 441 \, a^{6} b^{7}\right)} d^{2}\right)} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}\right) - {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{1024 \, a^{4} - 1916 \, a^{3} b + 1501 \, a^{2} b^{2} - 570 \, a b^{3} + 105 \, b^{4} - {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} \log\left(491520 \, a^{6} b - 1742720 \, a^{5} b^{2} + 2747904 \, a^{4} b^{3} - 2435877 \, a^{3} b^{4} + \frac{5106989}{4} \, a^{2} b^{5} - \frac{750141}{2} \, a b^{6} + \frac{194481}{4} \, b^{7} - \frac{1}{4} \, {\left(1966080 \, a^{6} b - 6970880 \, a^{5} b^{2} + 10991616 \, a^{4} b^{3} - 9743508 \, a^{3} b^{4} + 5106989 \, a^{2} b^{5} - 1500282 \, a b^{6} + 194481 \, b^{7}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(32 \, a^{16} - 193 \, a^{15} b + 498 \, a^{14} b^{2} - 715 \, a^{13} b^{3} + 620 \, a^{12} b^{4} - 327 \, a^{11} b^{5} + 98 \, a^{10} b^{6} - 13 \, a^{9} b^{7}\right)} d^{3} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(88320 \, a^{9} b - 319040 \, a^{8} b^{2} + 510294 \, a^{7} b^{3} - 457551 \, a^{6} b^{4} + 241865 \, a^{5} b^{5} - 71421 \, a^{4} b^{6} + 9261 \, a^{3} b^{7}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{1024 \, a^{4} - 1916 \, a^{3} b + 1501 \, a^{2} b^{2} - 570 \, a b^{3} + 105 \, b^{4} - {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(1024 \, a^{13} - 6276 \, a^{12} b + 16461 \, a^{11} b^{2} - 24005 \, a^{10} b^{3} + 21090 \, a^{9} b^{4} - 11214 \, a^{8} b^{5} + 3361 \, a^{7} b^{6} - 441 \, a^{6} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(1024 \, a^{13} - 6276 \, a^{12} b + 16461 \, a^{11} b^{2} - 24005 \, a^{10} b^{3} + 21090 \, a^{9} b^{4} - 11214 \, a^{8} b^{5} + 3361 \, a^{7} b^{6} - 441 \, a^{6} b^{7}\right)} d^{2}\right)} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}\right) + {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{1024 \, a^{4} - 1916 \, a^{3} b + 1501 \, a^{2} b^{2} - 570 \, a b^{3} + 105 \, b^{4} + {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} \log\left(-491520 \, a^{6} b + 1742720 \, a^{5} b^{2} - 2747904 \, a^{4} b^{3} + 2435877 \, a^{3} b^{4} - \frac{5106989}{4} \, a^{2} b^{5} + \frac{750141}{2} \, a b^{6} - \frac{194481}{4} \, b^{7} + \frac{1}{4} \, {\left(1966080 \, a^{6} b - 6970880 \, a^{5} b^{2} + 10991616 \, a^{4} b^{3} - 9743508 \, a^{3} b^{4} + 5106989 \, a^{2} b^{5} - 1500282 \, a b^{6} + 194481 \, b^{7}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(32 \, a^{16} - 193 \, a^{15} b + 498 \, a^{14} b^{2} - 715 \, a^{13} b^{3} + 620 \, a^{12} b^{4} - 327 \, a^{11} b^{5} + 98 \, a^{10} b^{6} - 13 \, a^{9} b^{7}\right)} d^{3} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(88320 \, a^{9} b - 319040 \, a^{8} b^{2} + 510294 \, a^{7} b^{3} - 457551 \, a^{6} b^{4} + 241865 \, a^{5} b^{5} - 71421 \, a^{4} b^{6} + 9261 \, a^{3} b^{7}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{1024 \, a^{4} - 1916 \, a^{3} b + 1501 \, a^{2} b^{2} - 570 \, a b^{3} + 105 \, b^{4} + {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(1024 \, a^{13} - 6276 \, a^{12} b + 16461 \, a^{11} b^{2} - 24005 \, a^{10} b^{3} + 21090 \, a^{9} b^{4} - 11214 \, a^{8} b^{5} + 3361 \, a^{7} b^{6} - 441 \, a^{6} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(1024 \, a^{13} - 6276 \, a^{12} b + 16461 \, a^{11} b^{2} - 24005 \, a^{10} b^{3} + 21090 \, a^{9} b^{4} - 11214 \, a^{8} b^{5} + 3361 \, a^{7} b^{6} - 441 \, a^{6} b^{7}\right)} d^{2}\right)} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}\right) - {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)} \sqrt{-\frac{1024 \, a^{4} - 1916 \, a^{3} b + 1501 \, a^{2} b^{2} - 570 \, a b^{3} + 105 \, b^{4} + {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} \log\left(-491520 \, a^{6} b + 1742720 \, a^{5} b^{2} - 2747904 \, a^{4} b^{3} + 2435877 \, a^{3} b^{4} - \frac{5106989}{4} \, a^{2} b^{5} + \frac{750141}{2} \, a b^{6} - \frac{194481}{4} \, b^{7} + \frac{1}{4} \, {\left(1966080 \, a^{6} b - 6970880 \, a^{5} b^{2} + 10991616 \, a^{4} b^{3} - 9743508 \, a^{3} b^{4} + 5106989 \, a^{2} b^{5} - 1500282 \, a b^{6} + 194481 \, b^{7}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(32 \, a^{16} - 193 \, a^{15} b + 498 \, a^{14} b^{2} - 715 \, a^{13} b^{3} + 620 \, a^{12} b^{4} - 327 \, a^{11} b^{5} + 98 \, a^{10} b^{6} - 13 \, a^{9} b^{7}\right)} d^{3} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(88320 \, a^{9} b - 319040 \, a^{8} b^{2} + 510294 \, a^{7} b^{3} - 457551 \, a^{6} b^{4} + 241865 \, a^{5} b^{5} - 71421 \, a^{4} b^{6} + 9261 \, a^{3} b^{7}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{1024 \, a^{4} - 1916 \, a^{3} b + 1501 \, a^{2} b^{2} - 570 \, a b^{3} + 105 \, b^{4} + {\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}}{{\left(a^{10} - 5 \, a^{9} b + 10 \, a^{8} b^{2} - 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} - a^{5} b^{5}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(1024 \, a^{13} - 6276 \, a^{12} b + 16461 \, a^{11} b^{2} - 24005 \, a^{10} b^{3} + 21090 \, a^{9} b^{4} - 11214 \, a^{8} b^{5} + 3361 \, a^{7} b^{6} - 441 \, a^{6} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(1024 \, a^{13} - 6276 \, a^{12} b + 16461 \, a^{11} b^{2} - 24005 \, a^{10} b^{3} + 21090 \, a^{9} b^{4} - 11214 \, a^{8} b^{5} + 3361 \, a^{7} b^{6} - 441 \, a^{6} b^{7}\right)} d^{2}\right)} \sqrt{\frac{3686400 \, a^{8} b - 17817600 \, a^{7} b^{2} + 39458560 \, a^{6} b^{3} - 51952960 \, a^{5} b^{4} + 44335881 \, a^{4} b^{5} - 25065628 \, a^{3} b^{6} + 9162574 \, a^{2} b^{7} - 1980972 \, a b^{8} + 194481 \, b^{9}}{{\left(a^{21} - 10 \, a^{20} b + 45 \, a^{19} b^{2} - 120 \, a^{18} b^{3} + 210 \, a^{17} b^{4} - 252 \, a^{16} b^{5} + 210 \, a^{15} b^{6} - 120 \, a^{14} b^{7} + 45 \, a^{13} b^{8} - 10 \, a^{12} b^{9} + a^{11} b^{10}\right)} d^{4}}}\right) - 8 \, {\left(6 \, {\left(2 \, a b^{2} - b^{3}\right)} \cos\left(d x + c\right)^{7} - {\left(49 \, a b^{2} - 25 \, b^{3}\right)} \cos\left(d x + c\right)^{5} - 8 \, {\left(2 \, a^{2} b - 9 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(d x + c\right)^{3} + {\left(33 \, a^{2} b - 46 \, a b^{2} + 13 \, b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{256 \, {\left({\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{5} b - 5 \, a^{4} b^{2} + 7 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} - a^{2} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{6} - 4 \, a^{5} b + 6 \, a^{4} b^{2} - 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d\right)}}"," ",0,"1/256*(((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(491520*a^6*b - 1742720*a^5*b^2 + 2747904*a^4*b^3 - 2435877*a^3*b^4 + 5106989/4*a^2*b^5 - 750141/2*a*b^6 + 194481/4*b^7 - 1/4*(1966080*a^6*b - 6970880*a^5*b^2 + 10991616*a^4*b^3 - 9743508*a^3*b^4 + 5106989*a^2*b^5 - 1500282*a*b^6 + 194481*b^7)*cos(d*x + c)^2 + 1/2*((32*a^16 - 193*a^15*b + 498*a^14*b^2 - 715*a^13*b^3 + 620*a^12*b^4 - 327*a^11*b^5 + 98*a^10*b^6 - 13*a^9*b^7)*d^3*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) + (88320*a^9*b - 319040*a^8*b^2 + 510294*a^7*b^3 - 457551*a^6*b^4 + 241865*a^5*b^5 - 71421*a^4*b^6 + 9261*a^3*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2)) - 1/4*(2*(1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2*cos(d*x + c)^2 - (1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2)*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))) - ((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(491520*a^6*b - 1742720*a^5*b^2 + 2747904*a^4*b^3 - 2435877*a^3*b^4 + 5106989/4*a^2*b^5 - 750141/2*a*b^6 + 194481/4*b^7 - 1/4*(1966080*a^6*b - 6970880*a^5*b^2 + 10991616*a^4*b^3 - 9743508*a^3*b^4 + 5106989*a^2*b^5 - 1500282*a*b^6 + 194481*b^7)*cos(d*x + c)^2 - 1/2*((32*a^16 - 193*a^15*b + 498*a^14*b^2 - 715*a^13*b^3 + 620*a^12*b^4 - 327*a^11*b^5 + 98*a^10*b^6 - 13*a^9*b^7)*d^3*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) + (88320*a^9*b - 319040*a^8*b^2 + 510294*a^7*b^3 - 457551*a^6*b^4 + 241865*a^5*b^5 - 71421*a^4*b^6 + 9261*a^3*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2)) - 1/4*(2*(1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2*cos(d*x + c)^2 - (1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2)*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))) + ((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(-491520*a^6*b + 1742720*a^5*b^2 - 2747904*a^4*b^3 + 2435877*a^3*b^4 - 5106989/4*a^2*b^5 + 750141/2*a*b^6 - 194481/4*b^7 + 1/4*(1966080*a^6*b - 6970880*a^5*b^2 + 10991616*a^4*b^3 - 9743508*a^3*b^4 + 5106989*a^2*b^5 - 1500282*a*b^6 + 194481*b^7)*cos(d*x + c)^2 + 1/2*((32*a^16 - 193*a^15*b + 498*a^14*b^2 - 715*a^13*b^3 + 620*a^12*b^4 - 327*a^11*b^5 + 98*a^10*b^6 - 13*a^9*b^7)*d^3*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) - (88320*a^9*b - 319040*a^8*b^2 + 510294*a^7*b^3 - 457551*a^6*b^4 + 241865*a^5*b^5 - 71421*a^4*b^6 + 9261*a^3*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2)) - 1/4*(2*(1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2*cos(d*x + c)^2 - (1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2)*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))) - ((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(-491520*a^6*b + 1742720*a^5*b^2 - 2747904*a^4*b^3 + 2435877*a^3*b^4 - 5106989/4*a^2*b^5 + 750141/2*a*b^6 - 194481/4*b^7 + 1/4*(1966080*a^6*b - 6970880*a^5*b^2 + 10991616*a^4*b^3 - 9743508*a^3*b^4 + 5106989*a^2*b^5 - 1500282*a*b^6 + 194481*b^7)*cos(d*x + c)^2 - 1/2*((32*a^16 - 193*a^15*b + 498*a^14*b^2 - 715*a^13*b^3 + 620*a^12*b^4 - 327*a^11*b^5 + 98*a^10*b^6 - 13*a^9*b^7)*d^3*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) - (88320*a^9*b - 319040*a^8*b^2 + 510294*a^7*b^3 - 457551*a^6*b^4 + 241865*a^5*b^5 - 71421*a^4*b^6 + 9261*a^3*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2)) - 1/4*(2*(1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2*cos(d*x + c)^2 - (1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2)*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))) - 8*(6*(2*a*b^2 - b^3)*cos(d*x + c)^7 - (49*a*b^2 - 25*b^3)*cos(d*x + c)^5 - 8*(2*a^2*b - 9*a*b^2 + 4*b^3)*cos(d*x + c)^3 + (33*a^2*b - 46*a*b^2 + 13*b^3)*cos(d*x + c))*sin(d*x + c))/((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)","B",0
235,1,6323,0,7.467638," ","integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm=""fricas"")","-\frac{8 \, {\left(32 \, a^{2} b^{2} - 83 \, a b^{3} + 45 \, b^{4}\right)} \cos\left(d x + c\right)^{9} - 48 \, {\left(19 \, a^{2} b^{2} - 54 \, a b^{3} + 30 \, b^{4}\right)} \cos\left(d x + c\right)^{7} - 8 \, {\left(64 \, a^{3} b - 301 \, a^{2} b^{2} + 555 \, a b^{3} - 270 \, b^{4}\right)} \cos\left(d x + c\right)^{5} + 16 \, {\left(55 \, a^{3} b - 188 \, a^{2} b^{2} + 235 \, a b^{3} - 90 \, b^{4}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 7 \, a^{4} b^{3} - 3 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{400 \, a^{4} b - 1044 \, a^{3} b^{2} + 1085 \, a^{2} b^{3} - 530 \, a b^{4} + 105 \, b^{5} - {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} \log\left(1728000 \, a^{6} b^{2} - 7369920 \, a^{5} b^{3} + 13507020 \, a^{4} b^{4} - 13573305 \, a^{3} b^{5} + \frac{31519503}{4} \, a^{2} b^{6} - \frac{5011875}{2} \, a b^{7} + \frac{1366875}{4} \, b^{8} - \frac{27}{4} \, {\left(256000 \, a^{6} b^{2} - 1091840 \, a^{5} b^{3} + 2001040 \, a^{4} b^{4} - 2010860 \, a^{3} b^{5} + 1167389 \, a^{2} b^{6} - 371250 \, a b^{7} + 50625 \, b^{8}\right)} \cos\left(d x + c\right)^{2} + \frac{27}{2} \, {\left({\left(26 \, a^{17} - 167 \, a^{16} b + 460 \, a^{15} b^{2} - 705 \, a^{14} b^{3} + 650 \, a^{13} b^{4} - 361 \, a^{12} b^{5} + 112 \, a^{11} b^{6} - 15 \, a^{10} b^{7}\right)} d^{3} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(12800 \, a^{10} b - 54080 \, a^{9} b^{2} + 98420 \, a^{8} b^{3} - 98415 \, a^{7} b^{4} + 56973 \, a^{6} b^{5} - 18109 \, a^{5} b^{6} + 2475 \, a^{4} b^{7}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{400 \, a^{4} b - 1044 \, a^{3} b^{2} + 1085 \, a^{2} b^{3} - 530 \, a b^{4} + 105 \, b^{5} - {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} - \frac{27}{4} \, {\left(2 \, {\left(400 \, a^{14} - 2556 \, a^{13} b + 7005 \, a^{12} b^{2} - 10685 \, a^{11} b^{3} + 9810 \, a^{10} b^{4} - 5430 \, a^{9} b^{5} + 1681 \, a^{8} b^{6} - 225 \, a^{7} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(400 \, a^{14} - 2556 \, a^{13} b + 7005 \, a^{12} b^{2} - 10685 \, a^{11} b^{3} + 9810 \, a^{10} b^{4} - 5430 \, a^{9} b^{5} + 1681 \, a^{8} b^{6} - 225 \, a^{7} b^{7}\right)} d^{2}\right)} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}\right) \sin\left(d x + c\right) - 3 \, {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 7 \, a^{4} b^{3} - 3 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{400 \, a^{4} b - 1044 \, a^{3} b^{2} + 1085 \, a^{2} b^{3} - 530 \, a b^{4} + 105 \, b^{5} - {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} \log\left(1728000 \, a^{6} b^{2} - 7369920 \, a^{5} b^{3} + 13507020 \, a^{4} b^{4} - 13573305 \, a^{3} b^{5} + \frac{31519503}{4} \, a^{2} b^{6} - \frac{5011875}{2} \, a b^{7} + \frac{1366875}{4} \, b^{8} - \frac{27}{4} \, {\left(256000 \, a^{6} b^{2} - 1091840 \, a^{5} b^{3} + 2001040 \, a^{4} b^{4} - 2010860 \, a^{3} b^{5} + 1167389 \, a^{2} b^{6} - 371250 \, a b^{7} + 50625 \, b^{8}\right)} \cos\left(d x + c\right)^{2} - \frac{27}{2} \, {\left({\left(26 \, a^{17} - 167 \, a^{16} b + 460 \, a^{15} b^{2} - 705 \, a^{14} b^{3} + 650 \, a^{13} b^{4} - 361 \, a^{12} b^{5} + 112 \, a^{11} b^{6} - 15 \, a^{10} b^{7}\right)} d^{3} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(12800 \, a^{10} b - 54080 \, a^{9} b^{2} + 98420 \, a^{8} b^{3} - 98415 \, a^{7} b^{4} + 56973 \, a^{6} b^{5} - 18109 \, a^{5} b^{6} + 2475 \, a^{4} b^{7}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{400 \, a^{4} b - 1044 \, a^{3} b^{2} + 1085 \, a^{2} b^{3} - 530 \, a b^{4} + 105 \, b^{5} - {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} - \frac{27}{4} \, {\left(2 \, {\left(400 \, a^{14} - 2556 \, a^{13} b + 7005 \, a^{12} b^{2} - 10685 \, a^{11} b^{3} + 9810 \, a^{10} b^{4} - 5430 \, a^{9} b^{5} + 1681 \, a^{8} b^{6} - 225 \, a^{7} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(400 \, a^{14} - 2556 \, a^{13} b + 7005 \, a^{12} b^{2} - 10685 \, a^{11} b^{3} + 9810 \, a^{10} b^{4} - 5430 \, a^{9} b^{5} + 1681 \, a^{8} b^{6} - 225 \, a^{7} b^{7}\right)} d^{2}\right)} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}\right) \sin\left(d x + c\right) + 3 \, {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 7 \, a^{4} b^{3} - 3 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{400 \, a^{4} b - 1044 \, a^{3} b^{2} + 1085 \, a^{2} b^{3} - 530 \, a b^{4} + 105 \, b^{5} + {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} \log\left(-1728000 \, a^{6} b^{2} + 7369920 \, a^{5} b^{3} - 13507020 \, a^{4} b^{4} + 13573305 \, a^{3} b^{5} - \frac{31519503}{4} \, a^{2} b^{6} + \frac{5011875}{2} \, a b^{7} - \frac{1366875}{4} \, b^{8} + \frac{27}{4} \, {\left(256000 \, a^{6} b^{2} - 1091840 \, a^{5} b^{3} + 2001040 \, a^{4} b^{4} - 2010860 \, a^{3} b^{5} + 1167389 \, a^{2} b^{6} - 371250 \, a b^{7} + 50625 \, b^{8}\right)} \cos\left(d x + c\right)^{2} + \frac{27}{2} \, {\left({\left(26 \, a^{17} - 167 \, a^{16} b + 460 \, a^{15} b^{2} - 705 \, a^{14} b^{3} + 650 \, a^{13} b^{4} - 361 \, a^{12} b^{5} + 112 \, a^{11} b^{6} - 15 \, a^{10} b^{7}\right)} d^{3} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(12800 \, a^{10} b - 54080 \, a^{9} b^{2} + 98420 \, a^{8} b^{3} - 98415 \, a^{7} b^{4} + 56973 \, a^{6} b^{5} - 18109 \, a^{5} b^{6} + 2475 \, a^{4} b^{7}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{400 \, a^{4} b - 1044 \, a^{3} b^{2} + 1085 \, a^{2} b^{3} - 530 \, a b^{4} + 105 \, b^{5} + {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} - \frac{27}{4} \, {\left(2 \, {\left(400 \, a^{14} - 2556 \, a^{13} b + 7005 \, a^{12} b^{2} - 10685 \, a^{11} b^{3} + 9810 \, a^{10} b^{4} - 5430 \, a^{9} b^{5} + 1681 \, a^{8} b^{6} - 225 \, a^{7} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(400 \, a^{14} - 2556 \, a^{13} b + 7005 \, a^{12} b^{2} - 10685 \, a^{11} b^{3} + 9810 \, a^{10} b^{4} - 5430 \, a^{9} b^{5} + 1681 \, a^{8} b^{6} - 225 \, a^{7} b^{7}\right)} d^{2}\right)} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}\right) \sin\left(d x + c\right) - 3 \, {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 7 \, a^{4} b^{3} - 3 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)} \sqrt{-\frac{400 \, a^{4} b - 1044 \, a^{3} b^{2} + 1085 \, a^{2} b^{3} - 530 \, a b^{4} + 105 \, b^{5} + {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} \log\left(-1728000 \, a^{6} b^{2} + 7369920 \, a^{5} b^{3} - 13507020 \, a^{4} b^{4} + 13573305 \, a^{3} b^{5} - \frac{31519503}{4} \, a^{2} b^{6} + \frac{5011875}{2} \, a b^{7} - \frac{1366875}{4} \, b^{8} + \frac{27}{4} \, {\left(256000 \, a^{6} b^{2} - 1091840 \, a^{5} b^{3} + 2001040 \, a^{4} b^{4} - 2010860 \, a^{3} b^{5} + 1167389 \, a^{2} b^{6} - 371250 \, a b^{7} + 50625 \, b^{8}\right)} \cos\left(d x + c\right)^{2} - \frac{27}{2} \, {\left({\left(26 \, a^{17} - 167 \, a^{16} b + 460 \, a^{15} b^{2} - 705 \, a^{14} b^{3} + 650 \, a^{13} b^{4} - 361 \, a^{12} b^{5} + 112 \, a^{11} b^{6} - 15 \, a^{10} b^{7}\right)} d^{3} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(12800 \, a^{10} b - 54080 \, a^{9} b^{2} + 98420 \, a^{8} b^{3} - 98415 \, a^{7} b^{4} + 56973 \, a^{6} b^{5} - 18109 \, a^{5} b^{6} + 2475 \, a^{4} b^{7}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{400 \, a^{4} b - 1044 \, a^{3} b^{2} + 1085 \, a^{2} b^{3} - 530 \, a b^{4} + 105 \, b^{5} + {\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}}{{\left(a^{11} - 5 \, a^{10} b + 10 \, a^{9} b^{2} - 10 \, a^{8} b^{3} + 5 \, a^{7} b^{4} - a^{6} b^{5}\right)} d^{2}}} - \frac{27}{4} \, {\left(2 \, {\left(400 \, a^{14} - 2556 \, a^{13} b + 7005 \, a^{12} b^{2} - 10685 \, a^{11} b^{3} + 9810 \, a^{10} b^{4} - 5430 \, a^{9} b^{5} + 1681 \, a^{8} b^{6} - 225 \, a^{7} b^{7}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(400 \, a^{14} - 2556 \, a^{13} b + 7005 \, a^{12} b^{2} - 10685 \, a^{11} b^{3} + 9810 \, a^{10} b^{4} - 5430 \, a^{9} b^{5} + 1681 \, a^{8} b^{6} - 225 \, a^{7} b^{7}\right)} d^{2}\right)} \sqrt{\frac{409600 \, a^{8} b^{3} - 2355200 \, a^{7} b^{4} + 6054400 \, a^{6} b^{5} - 9073120 \, a^{5} b^{6} + 8661145 \, a^{4} b^{7} - 5389980 \, a^{3} b^{8} + 2135086 \, a^{2} b^{9} - 492300 \, a b^{10} + 50625 \, b^{11}}{{\left(a^{23} - 10 \, a^{22} b + 45 \, a^{21} b^{2} - 120 \, a^{20} b^{3} + 210 \, a^{19} b^{4} - 252 \, a^{18} b^{5} + 210 \, a^{17} b^{6} - 120 \, a^{16} b^{7} + 45 \, a^{15} b^{8} - 10 \, a^{14} b^{9} + a^{13} b^{10}\right)} d^{4}}}\right) \sin\left(d x + c\right) + 8 \, {\left(32 \, a^{4} - 110 \, a^{3} b + 189 \, a^{2} b^{2} - 156 \, a b^{3} + 45 \, b^{4}\right)} \cos\left(d x + c\right)}{256 \, {\left({\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{8} - 4 \, {\left(a^{5} b^{2} - 2 \, a^{4} b^{3} + a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{6} - 2 \, {\left(a^{6} b - 5 \, a^{5} b^{2} + 7 \, a^{4} b^{3} - 3 \, a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 4 \, {\left(a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)} \sin\left(d x + c\right)}"," ",0,"-1/256*(8*(32*a^2*b^2 - 83*a*b^3 + 45*b^4)*cos(d*x + c)^9 - 48*(19*a^2*b^2 - 54*a*b^3 + 30*b^4)*cos(d*x + c)^7 - 8*(64*a^3*b - 301*a^2*b^2 + 555*a*b^3 - 270*b^4)*cos(d*x + c)^5 + 16*(55*a^3*b - 188*a^2*b^2 + 235*a*b^3 - 90*b^4)*cos(d*x + c)^3 + 3*((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log(1728000*a^6*b^2 - 7369920*a^5*b^3 + 13507020*a^4*b^4 - 13573305*a^3*b^5 + 31519503/4*a^2*b^6 - 5011875/2*a*b^7 + 1366875/4*b^8 - 27/4*(256000*a^6*b^2 - 1091840*a^5*b^3 + 2001040*a^4*b^4 - 2010860*a^3*b^5 + 1167389*a^2*b^6 - 371250*a*b^7 + 50625*b^8)*cos(d*x + c)^2 + 27/2*((26*a^17 - 167*a^16*b + 460*a^15*b^2 - 705*a^14*b^3 + 650*a^13*b^4 - 361*a^12*b^5 + 112*a^11*b^6 - 15*a^10*b^7)*d^3*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) + (12800*a^10*b - 54080*a^9*b^2 + 98420*a^8*b^3 - 98415*a^7*b^4 + 56973*a^6*b^5 - 18109*a^5*b^6 + 2475*a^4*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2)) - 27/4*(2*(400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2*cos(d*x + c)^2 - (400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2)*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))*sin(d*x + c) - 3*((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log(1728000*a^6*b^2 - 7369920*a^5*b^3 + 13507020*a^4*b^4 - 13573305*a^3*b^5 + 31519503/4*a^2*b^6 - 5011875/2*a*b^7 + 1366875/4*b^8 - 27/4*(256000*a^6*b^2 - 1091840*a^5*b^3 + 2001040*a^4*b^4 - 2010860*a^3*b^5 + 1167389*a^2*b^6 - 371250*a*b^7 + 50625*b^8)*cos(d*x + c)^2 - 27/2*((26*a^17 - 167*a^16*b + 460*a^15*b^2 - 705*a^14*b^3 + 650*a^13*b^4 - 361*a^12*b^5 + 112*a^11*b^6 - 15*a^10*b^7)*d^3*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) + (12800*a^10*b - 54080*a^9*b^2 + 98420*a^8*b^3 - 98415*a^7*b^4 + 56973*a^6*b^5 - 18109*a^5*b^6 + 2475*a^4*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2)) - 27/4*(2*(400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2*cos(d*x + c)^2 - (400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2)*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))*sin(d*x + c) + 3*((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log(-1728000*a^6*b^2 + 7369920*a^5*b^3 - 13507020*a^4*b^4 + 13573305*a^3*b^5 - 31519503/4*a^2*b^6 + 5011875/2*a*b^7 - 1366875/4*b^8 + 27/4*(256000*a^6*b^2 - 1091840*a^5*b^3 + 2001040*a^4*b^4 - 2010860*a^3*b^5 + 1167389*a^2*b^6 - 371250*a*b^7 + 50625*b^8)*cos(d*x + c)^2 + 27/2*((26*a^17 - 167*a^16*b + 460*a^15*b^2 - 705*a^14*b^3 + 650*a^13*b^4 - 361*a^12*b^5 + 112*a^11*b^6 - 15*a^10*b^7)*d^3*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) - (12800*a^10*b - 54080*a^9*b^2 + 98420*a^8*b^3 - 98415*a^7*b^4 + 56973*a^6*b^5 - 18109*a^5*b^6 + 2475*a^4*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2)) - 27/4*(2*(400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2*cos(d*x + c)^2 - (400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2)*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))*sin(d*x + c) - 3*((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log(-1728000*a^6*b^2 + 7369920*a^5*b^3 - 13507020*a^4*b^4 + 13573305*a^3*b^5 - 31519503/4*a^2*b^6 + 5011875/2*a*b^7 - 1366875/4*b^8 + 27/4*(256000*a^6*b^2 - 1091840*a^5*b^3 + 2001040*a^4*b^4 - 2010860*a^3*b^5 + 1167389*a^2*b^6 - 371250*a*b^7 + 50625*b^8)*cos(d*x + c)^2 - 27/2*((26*a^17 - 167*a^16*b + 460*a^15*b^2 - 705*a^14*b^3 + 650*a^13*b^4 - 361*a^12*b^5 + 112*a^11*b^6 - 15*a^10*b^7)*d^3*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) - (12800*a^10*b - 54080*a^9*b^2 + 98420*a^8*b^3 - 98415*a^7*b^4 + 56973*a^6*b^5 - 18109*a^5*b^6 + 2475*a^4*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2)) - 27/4*(2*(400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2*cos(d*x + c)^2 - (400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2)*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))*sin(d*x + c) + 8*(32*a^4 - 110*a^3*b + 189*a^2*b^2 - 156*a*b^3 + 45*b^4)*cos(d*x + c))/(((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sin(d*x + c))","B",0
236,1,43,0,0.951142," ","integrate(1/(1-sin(x)^4),x, algorithm=""fricas"")","-\frac{\sqrt{2} \arctan\left(\frac{3 \, \sqrt{2} \cos\left(x\right)^{2} - 2 \, \sqrt{2}}{4 \, \cos\left(x\right) \sin\left(x\right)}\right) \cos\left(x\right) - 4 \, \sin\left(x\right)}{8 \, \cos\left(x\right)}"," ",0,"-1/8*(sqrt(2)*arctan(1/4*(3*sqrt(2)*cos(x)^2 - 2*sqrt(2))/(cos(x)*sin(x)))*cos(x) - 4*sin(x))/cos(x)","B",0
237,1,823,0,1.061999," ","integrate(1/(a+b*sin(x)^4),x, algorithm=""fricas"")","-\frac{1}{8} \, \sqrt{-\frac{{\left(a^{2} + a b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log\left(\frac{1}{4} \, b \cos\left(x\right)^{2} + \frac{1}{2} \, {\left(a b \cos\left(x\right) \sin\left(x\right) + {\left(a^{4} + a^{3} b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos\left(x\right) \sin\left(x\right)\right)} \sqrt{-\frac{{\left(a^{2} + a b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} - \frac{1}{4} \, {\left(a^{3} + a^{2} b - 2 \, {\left(a^{3} + a^{2} b\right)} \cos\left(x\right)^{2}\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - \frac{1}{4} \, b\right) + \frac{1}{8} \, \sqrt{-\frac{{\left(a^{2} + a b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log\left(\frac{1}{4} \, b \cos\left(x\right)^{2} - \frac{1}{2} \, {\left(a b \cos\left(x\right) \sin\left(x\right) + {\left(a^{4} + a^{3} b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos\left(x\right) \sin\left(x\right)\right)} \sqrt{-\frac{{\left(a^{2} + a b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} - \frac{1}{4} \, {\left(a^{3} + a^{2} b - 2 \, {\left(a^{3} + a^{2} b\right)} \cos\left(x\right)^{2}\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - \frac{1}{4} \, b\right) + \frac{1}{8} \, \sqrt{\frac{{\left(a^{2} + a b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log\left(-\frac{1}{4} \, b \cos\left(x\right)^{2} + \frac{1}{2} \, {\left(a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{4} + a^{3} b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos\left(x\right) \sin\left(x\right)\right)} \sqrt{\frac{{\left(a^{2} + a b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - \frac{1}{4} \, {\left(a^{3} + a^{2} b - 2 \, {\left(a^{3} + a^{2} b\right)} \cos\left(x\right)^{2}\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + \frac{1}{4} \, b\right) - \frac{1}{8} \, \sqrt{\frac{{\left(a^{2} + a b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log\left(-\frac{1}{4} \, b \cos\left(x\right)^{2} - \frac{1}{2} \, {\left(a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{4} + a^{3} b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos\left(x\right) \sin\left(x\right)\right)} \sqrt{\frac{{\left(a^{2} + a b\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - \frac{1}{4} \, {\left(a^{3} + a^{2} b - 2 \, {\left(a^{3} + a^{2} b\right)} \cos\left(x\right)^{2}\right)} \sqrt{-\frac{b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + \frac{1}{4} \, b\right)"," ",0,"-1/8*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b))*log(1/4*b*cos(x)^2 + 1/2*(a*b*cos(x)*sin(x) + (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b)) - 1/4*(a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1/4*b) + 1/8*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b))*log(1/4*b*cos(x)^2 - 1/2*(a*b*cos(x)*sin(x) + (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b)) - 1/4*(a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1/4*b) + 1/8*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b))*log(-1/4*b*cos(x)^2 + 1/2*(a*b*cos(x)*sin(x) - (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b)) - 1/4*(a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1/4*b) - 1/8*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b))*log(-1/4*b*cos(x)^2 - 1/2*(a*b*cos(x)*sin(x) - (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b)) - 1/4*(a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1/4*b)","B",0
238,1,3830,0,63.408167," ","integrate(1/(1+sin(x)^4),x, algorithm=""fricas"")","-\frac{1}{32} \cdot 2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} {\left(\sqrt{2} - 1\right)} \log\left(-{\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 2 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} + {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 2\right) + \frac{1}{32} \cdot 2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} {\left(\sqrt{2} - 1\right)} \log\left(-{\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 2 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} - {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 2\right) - \frac{1}{16} \cdot 2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} \arctan\left(\frac{32 \, {\left(\sqrt{2} {\left(3 \, \sqrt{2} + 2\right)} - 2 \, \sqrt{2} - 6\right)} \cos\left(x\right)^{16} - 16 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} + 10\right)} - 24 \, \sqrt{2} - 44\right)} \cos\left(x\right)^{14} + 16 \, {\left(\sqrt{2} {\left(51 \, \sqrt{2} - 4\right)} - 52 \, \sqrt{2} - 46\right)} \cos\left(x\right)^{12} - 16 \, {\left(\sqrt{2} {\left(41 \, \sqrt{2} - 36\right)} - 54 \, \sqrt{2} + 15\right)} \cos\left(x\right)^{10} + 8 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} - 90\right)} - 58 \, \sqrt{2} + 132\right)} \cos\left(x\right)^{8} - 4 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 98\right)} - 32 \, \sqrt{2} + 216\right)} \cos\left(x\right)^{6} - 4 \, {\left(\sqrt{2} {\left(\sqrt{2} + 24\right)} + 4 \, \sqrt{2} - 82\right)} \cos\left(x\right)^{4} + 4 \, {\left(2 \, \sqrt{2} - 15\right)} \cos\left(x\right)^{2} + 2 \, {\left(8 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 1\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(3 \, \sqrt{2} + 2\right)}\right)} \cos\left(x\right)^{15} - 8 \, {\left(2^{\frac{3}{4}} {\left(11 \, \sqrt{2} - 9\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(13 \, \sqrt{2} + 4\right)}\right)} \cos\left(x\right)^{13} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(21 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(79 \, \sqrt{2} - 14\right)}\right)} \cos\left(x\right)^{11} - 8 \, {\left(2^{\frac{3}{4}} {\left(19 \, \sqrt{2} - 27\right)} - 2^{\frac{1}{4}} {\left(27 \, \sqrt{2} - 31\right)}\right)} \cos\left(x\right)^{9} + 2 \, {\left(2^{\frac{3}{4}} {\left(36 \, \sqrt{2} - 65\right)} - 32 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 4\right)}\right)} \cos\left(x\right)^{7} - 2 \, {\left(2^{\frac{3}{4}} {\left(9 \, \sqrt{2} - 19\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 30\right)}\right)} \cos\left(x\right)^{5} + {\left(2 \cdot 2^{\frac{3}{4}} {\left(\sqrt{2} - 2\right)} + 2^{\frac{1}{4}} {\left(\sqrt{2} + 26\right)}\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + {\left(16 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{14} - 56 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{12} + 8 \, {\left(\sqrt{2} {\left(49 \, \sqrt{2} - 62\right)} - 76 \, \sqrt{2} + 54\right)} \cos\left(x\right)^{10} - 40 \, {\left(\sqrt{2} {\left(7 \, \sqrt{2} - 10\right)} - 10 \, \sqrt{2} + 13\right)} \cos\left(x\right)^{8} + 4 \, {\left(\sqrt{2} {\left(27 \, \sqrt{2} - 46\right)} - 32 \, \sqrt{2} + 92\right)} \cos\left(x\right)^{6} - 2 \, {\left(11 \, \sqrt{2} {\left(\sqrt{2} - 2\right)} - 8 \, \sqrt{2} + 72\right)} \cos\left(x\right)^{4} + 2 \, {\left(\sqrt{2} {\left(\sqrt{2} - 2\right)} + 14\right)} \cos\left(x\right)^{2} + {\left(8 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{13} - 24 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{11} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(28 \, \sqrt{2} - 39\right)} - 2^{\frac{1}{4}} {\left(73 \, \sqrt{2} - 94\right)}\right)} \cos\left(x\right)^{9} - 8 \, {\left(2^{\frac{3}{4}} {\left(16 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(23 \, \sqrt{2} - 34\right)}\right)} \cos\left(x\right)^{7} + 2 \, {\left(9 \cdot 2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 8 \cdot 2^{\frac{1}{4}} {\left(4 \, \sqrt{2} - 7\right)}\right)} \cos\left(x\right)^{5} - 2 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 6 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)}\right)} \cos\left(x\right)^{3} - 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - 2\right)} \sqrt{-4 \, {\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 8 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} + 4 \, {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 8} + 4}{4 \, {\left(112 \, \cos\left(x\right)^{16} - 448 \, \cos\left(x\right)^{14} + 608 \, \cos\left(x\right)^{12} - 256 \, \cos\left(x\right)^{10} - 152 \, \cos\left(x\right)^{8} + 208 \, \cos\left(x\right)^{6} - 88 \, \cos\left(x\right)^{4} + 16 \, \cos\left(x\right)^{2} - 1\right)}}\right) + \frac{1}{16} \cdot 2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} \arctan\left(-\frac{32 \, {\left(\sqrt{2} {\left(3 \, \sqrt{2} + 2\right)} - 2 \, \sqrt{2} - 6\right)} \cos\left(x\right)^{16} - 16 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} + 10\right)} - 24 \, \sqrt{2} - 44\right)} \cos\left(x\right)^{14} + 16 \, {\left(\sqrt{2} {\left(51 \, \sqrt{2} - 4\right)} - 52 \, \sqrt{2} - 46\right)} \cos\left(x\right)^{12} - 16 \, {\left(\sqrt{2} {\left(41 \, \sqrt{2} - 36\right)} - 54 \, \sqrt{2} + 15\right)} \cos\left(x\right)^{10} + 8 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} - 90\right)} - 58 \, \sqrt{2} + 132\right)} \cos\left(x\right)^{8} - 4 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 98\right)} - 32 \, \sqrt{2} + 216\right)} \cos\left(x\right)^{6} - 4 \, {\left(\sqrt{2} {\left(\sqrt{2} + 24\right)} + 4 \, \sqrt{2} - 82\right)} \cos\left(x\right)^{4} + 4 \, {\left(2 \, \sqrt{2} - 15\right)} \cos\left(x\right)^{2} + 2 \, {\left(8 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 1\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(3 \, \sqrt{2} + 2\right)}\right)} \cos\left(x\right)^{15} - 8 \, {\left(2^{\frac{3}{4}} {\left(11 \, \sqrt{2} - 9\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(13 \, \sqrt{2} + 4\right)}\right)} \cos\left(x\right)^{13} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(21 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(79 \, \sqrt{2} - 14\right)}\right)} \cos\left(x\right)^{11} - 8 \, {\left(2^{\frac{3}{4}} {\left(19 \, \sqrt{2} - 27\right)} - 2^{\frac{1}{4}} {\left(27 \, \sqrt{2} - 31\right)}\right)} \cos\left(x\right)^{9} + 2 \, {\left(2^{\frac{3}{4}} {\left(36 \, \sqrt{2} - 65\right)} - 32 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 4\right)}\right)} \cos\left(x\right)^{7} - 2 \, {\left(2^{\frac{3}{4}} {\left(9 \, \sqrt{2} - 19\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 30\right)}\right)} \cos\left(x\right)^{5} + {\left(2 \cdot 2^{\frac{3}{4}} {\left(\sqrt{2} - 2\right)} + 2^{\frac{1}{4}} {\left(\sqrt{2} + 26\right)}\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - {\left(16 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{14} - 56 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{12} + 8 \, {\left(\sqrt{2} {\left(49 \, \sqrt{2} - 62\right)} - 76 \, \sqrt{2} + 54\right)} \cos\left(x\right)^{10} - 40 \, {\left(\sqrt{2} {\left(7 \, \sqrt{2} - 10\right)} - 10 \, \sqrt{2} + 13\right)} \cos\left(x\right)^{8} + 4 \, {\left(\sqrt{2} {\left(27 \, \sqrt{2} - 46\right)} - 32 \, \sqrt{2} + 92\right)} \cos\left(x\right)^{6} - 2 \, {\left(11 \, \sqrt{2} {\left(\sqrt{2} - 2\right)} - 8 \, \sqrt{2} + 72\right)} \cos\left(x\right)^{4} + 2 \, {\left(\sqrt{2} {\left(\sqrt{2} - 2\right)} + 14\right)} \cos\left(x\right)^{2} + {\left(8 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{13} - 24 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{11} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(28 \, \sqrt{2} - 39\right)} - 2^{\frac{1}{4}} {\left(73 \, \sqrt{2} - 94\right)}\right)} \cos\left(x\right)^{9} - 8 \, {\left(2^{\frac{3}{4}} {\left(16 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(23 \, \sqrt{2} - 34\right)}\right)} \cos\left(x\right)^{7} + 2 \, {\left(9 \cdot 2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 8 \cdot 2^{\frac{1}{4}} {\left(4 \, \sqrt{2} - 7\right)}\right)} \cos\left(x\right)^{5} - 2 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 6 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)}\right)} \cos\left(x\right)^{3} - 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - 2\right)} \sqrt{-4 \, {\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 8 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} + 4 \, {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 8} + 4}{4 \, {\left(112 \, \cos\left(x\right)^{16} - 448 \, \cos\left(x\right)^{14} + 608 \, \cos\left(x\right)^{12} - 256 \, \cos\left(x\right)^{10} - 152 \, \cos\left(x\right)^{8} + 208 \, \cos\left(x\right)^{6} - 88 \, \cos\left(x\right)^{4} + 16 \, \cos\left(x\right)^{2} - 1\right)}}\right) - \frac{1}{16} \cdot 2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} \arctan\left(-\frac{32 \, {\left(\sqrt{2} {\left(3 \, \sqrt{2} + 2\right)} - 2 \, \sqrt{2} - 6\right)} \cos\left(x\right)^{16} - 16 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} + 10\right)} - 24 \, \sqrt{2} - 44\right)} \cos\left(x\right)^{14} + 16 \, {\left(\sqrt{2} {\left(51 \, \sqrt{2} - 4\right)} - 52 \, \sqrt{2} - 46\right)} \cos\left(x\right)^{12} - 16 \, {\left(\sqrt{2} {\left(41 \, \sqrt{2} - 36\right)} - 54 \, \sqrt{2} + 15\right)} \cos\left(x\right)^{10} + 8 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} - 90\right)} - 58 \, \sqrt{2} + 132\right)} \cos\left(x\right)^{8} - 4 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 98\right)} - 32 \, \sqrt{2} + 216\right)} \cos\left(x\right)^{6} - 4 \, {\left(\sqrt{2} {\left(\sqrt{2} + 24\right)} + 4 \, \sqrt{2} - 82\right)} \cos\left(x\right)^{4} + 4 \, {\left(2 \, \sqrt{2} - 15\right)} \cos\left(x\right)^{2} - 2 \, {\left(8 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 1\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(3 \, \sqrt{2} + 2\right)}\right)} \cos\left(x\right)^{15} - 8 \, {\left(2^{\frac{3}{4}} {\left(11 \, \sqrt{2} - 9\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(13 \, \sqrt{2} + 4\right)}\right)} \cos\left(x\right)^{13} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(21 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(79 \, \sqrt{2} - 14\right)}\right)} \cos\left(x\right)^{11} - 8 \, {\left(2^{\frac{3}{4}} {\left(19 \, \sqrt{2} - 27\right)} - 2^{\frac{1}{4}} {\left(27 \, \sqrt{2} - 31\right)}\right)} \cos\left(x\right)^{9} + 2 \, {\left(2^{\frac{3}{4}} {\left(36 \, \sqrt{2} - 65\right)} - 32 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 4\right)}\right)} \cos\left(x\right)^{7} - 2 \, {\left(2^{\frac{3}{4}} {\left(9 \, \sqrt{2} - 19\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 30\right)}\right)} \cos\left(x\right)^{5} + {\left(2 \cdot 2^{\frac{3}{4}} {\left(\sqrt{2} - 2\right)} + 2^{\frac{1}{4}} {\left(\sqrt{2} + 26\right)}\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + {\left(16 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{14} - 56 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{12} + 8 \, {\left(\sqrt{2} {\left(49 \, \sqrt{2} - 62\right)} - 76 \, \sqrt{2} + 54\right)} \cos\left(x\right)^{10} - 40 \, {\left(\sqrt{2} {\left(7 \, \sqrt{2} - 10\right)} - 10 \, \sqrt{2} + 13\right)} \cos\left(x\right)^{8} + 4 \, {\left(\sqrt{2} {\left(27 \, \sqrt{2} - 46\right)} - 32 \, \sqrt{2} + 92\right)} \cos\left(x\right)^{6} - 2 \, {\left(11 \, \sqrt{2} {\left(\sqrt{2} - 2\right)} - 8 \, \sqrt{2} + 72\right)} \cos\left(x\right)^{4} + 2 \, {\left(\sqrt{2} {\left(\sqrt{2} - 2\right)} + 14\right)} \cos\left(x\right)^{2} - {\left(8 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{13} - 24 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{11} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(28 \, \sqrt{2} - 39\right)} - 2^{\frac{1}{4}} {\left(73 \, \sqrt{2} - 94\right)}\right)} \cos\left(x\right)^{9} - 8 \, {\left(2^{\frac{3}{4}} {\left(16 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(23 \, \sqrt{2} - 34\right)}\right)} \cos\left(x\right)^{7} + 2 \, {\left(9 \cdot 2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 8 \cdot 2^{\frac{1}{4}} {\left(4 \, \sqrt{2} - 7\right)}\right)} \cos\left(x\right)^{5} - 2 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 6 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)}\right)} \cos\left(x\right)^{3} - 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - 2\right)} \sqrt{-4 \, {\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 8 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} - 4 \, {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 8} + 4}{4 \, {\left(112 \, \cos\left(x\right)^{16} - 448 \, \cos\left(x\right)^{14} + 608 \, \cos\left(x\right)^{12} - 256 \, \cos\left(x\right)^{10} - 152 \, \cos\left(x\right)^{8} + 208 \, \cos\left(x\right)^{6} - 88 \, \cos\left(x\right)^{4} + 16 \, \cos\left(x\right)^{2} - 1\right)}}\right) + \frac{1}{16} \cdot 2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} \arctan\left(\frac{32 \, {\left(\sqrt{2} {\left(3 \, \sqrt{2} + 2\right)} - 2 \, \sqrt{2} - 6\right)} \cos\left(x\right)^{16} - 16 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} + 10\right)} - 24 \, \sqrt{2} - 44\right)} \cos\left(x\right)^{14} + 16 \, {\left(\sqrt{2} {\left(51 \, \sqrt{2} - 4\right)} - 52 \, \sqrt{2} - 46\right)} \cos\left(x\right)^{12} - 16 \, {\left(\sqrt{2} {\left(41 \, \sqrt{2} - 36\right)} - 54 \, \sqrt{2} + 15\right)} \cos\left(x\right)^{10} + 8 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} - 90\right)} - 58 \, \sqrt{2} + 132\right)} \cos\left(x\right)^{8} - 4 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 98\right)} - 32 \, \sqrt{2} + 216\right)} \cos\left(x\right)^{6} - 4 \, {\left(\sqrt{2} {\left(\sqrt{2} + 24\right)} + 4 \, \sqrt{2} - 82\right)} \cos\left(x\right)^{4} + 4 \, {\left(2 \, \sqrt{2} - 15\right)} \cos\left(x\right)^{2} - 2 \, {\left(8 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 1\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(3 \, \sqrt{2} + 2\right)}\right)} \cos\left(x\right)^{15} - 8 \, {\left(2^{\frac{3}{4}} {\left(11 \, \sqrt{2} - 9\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(13 \, \sqrt{2} + 4\right)}\right)} \cos\left(x\right)^{13} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(21 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(79 \, \sqrt{2} - 14\right)}\right)} \cos\left(x\right)^{11} - 8 \, {\left(2^{\frac{3}{4}} {\left(19 \, \sqrt{2} - 27\right)} - 2^{\frac{1}{4}} {\left(27 \, \sqrt{2} - 31\right)}\right)} \cos\left(x\right)^{9} + 2 \, {\left(2^{\frac{3}{4}} {\left(36 \, \sqrt{2} - 65\right)} - 32 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 4\right)}\right)} \cos\left(x\right)^{7} - 2 \, {\left(2^{\frac{3}{4}} {\left(9 \, \sqrt{2} - 19\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 30\right)}\right)} \cos\left(x\right)^{5} + {\left(2 \cdot 2^{\frac{3}{4}} {\left(\sqrt{2} - 2\right)} + 2^{\frac{1}{4}} {\left(\sqrt{2} + 26\right)}\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - {\left(16 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{14} - 56 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{12} + 8 \, {\left(\sqrt{2} {\left(49 \, \sqrt{2} - 62\right)} - 76 \, \sqrt{2} + 54\right)} \cos\left(x\right)^{10} - 40 \, {\left(\sqrt{2} {\left(7 \, \sqrt{2} - 10\right)} - 10 \, \sqrt{2} + 13\right)} \cos\left(x\right)^{8} + 4 \, {\left(\sqrt{2} {\left(27 \, \sqrt{2} - 46\right)} - 32 \, \sqrt{2} + 92\right)} \cos\left(x\right)^{6} - 2 \, {\left(11 \, \sqrt{2} {\left(\sqrt{2} - 2\right)} - 8 \, \sqrt{2} + 72\right)} \cos\left(x\right)^{4} + 2 \, {\left(\sqrt{2} {\left(\sqrt{2} - 2\right)} + 14\right)} \cos\left(x\right)^{2} - {\left(8 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{13} - 24 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{11} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(28 \, \sqrt{2} - 39\right)} - 2^{\frac{1}{4}} {\left(73 \, \sqrt{2} - 94\right)}\right)} \cos\left(x\right)^{9} - 8 \, {\left(2^{\frac{3}{4}} {\left(16 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(23 \, \sqrt{2} - 34\right)}\right)} \cos\left(x\right)^{7} + 2 \, {\left(9 \cdot 2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 8 \cdot 2^{\frac{1}{4}} {\left(4 \, \sqrt{2} - 7\right)}\right)} \cos\left(x\right)^{5} - 2 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 6 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)}\right)} \cos\left(x\right)^{3} - 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - 2\right)} \sqrt{-4 \, {\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 8 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} - 4 \, {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 8} + 4}{4 \, {\left(112 \, \cos\left(x\right)^{16} - 448 \, \cos\left(x\right)^{14} + 608 \, \cos\left(x\right)^{12} - 256 \, \cos\left(x\right)^{10} - 152 \, \cos\left(x\right)^{8} + 208 \, \cos\left(x\right)^{6} - 88 \, \cos\left(x\right)^{4} + 16 \, \cos\left(x\right)^{2} - 1\right)}}\right)"," ",0,"-1/32*2^(1/4)*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 1)*log(-(4*sqrt(2) - 5)*cos(x)^4 + 2*(2*sqrt(2) - 3)*cos(x)^2 + (2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 2) + 1/32*2^(1/4)*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 1)*log(-(4*sqrt(2) - 5)*cos(x)^4 + 2*(2*sqrt(2) - 3)*cos(x)^2 - (2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 2) - 1/16*2^(1/4)*sqrt(2*sqrt(2) + 4)*arctan(1/4*(32*(sqrt(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(29*sqrt(2) + 10) - 24*sqrt(2) - 44)*cos(x)^14 + 16*(sqrt(2)*(51*sqrt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*sqrt(2) - 36) - 54*sqrt(2) + 15)*cos(x)^10 + 8*(sqrt(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sqrt(2)*(5*sqrt(2) - 98) - 32*sqrt(2) + 216)*cos(x)^6 - 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4*(2*sqrt(2) - 15)*cos(x)^2 + 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(11*sqrt(2) - 9) - 2*2^(1/4)*(13*sqrt(2) + 4))*cos(x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(2) - 14))*cos(x)^11 - 8*(2^(3/4)*(19*sqrt(2) - 27) - 2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt(2) - 65) - 32*2^(1/4)*(sqrt(2) - 4))*cos(x)^7 - 2*(2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x)^5 + (2*2^(3/4)*(sqrt(2) - 2) + 2^(1/4)*(sqrt(2) + 26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + (16*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^14 - 56*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 + 8*(sqrt(2)*(49*sqrt(2) - 62) - 76*sqrt(2) + 54)*cos(x)^10 - 40*(sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*cos(x)^8 + 4*(sqrt(2)*(27*sqrt(2) - 46) - 32*sqrt(2) + 92)*cos(x)^6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) + 72)*cos(x)^4 + 2*(sqrt(2)*(sqrt(2) - 2) + 14)*cos(x)^2 + (8*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^13 - 24*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(73*sqrt(2) - 94))*cos(x)^9 - 8*(2^(3/4)*(16*sqrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos(x)^7 + 2*(9*2^(3/4)*(2*sqrt(2) - 3) - 8*2^(1/4)*(4*sqrt(2) - 7))*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^(1/4)*(sqrt(2) - 2))*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 8*(2*sqrt(2) - 3)*cos(x)^2 + 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 8) + 4)/(112*cos(x)^16 - 448*cos(x)^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos(x)^8 + 208*cos(x)^6 - 88*cos(x)^4 + 16*cos(x)^2 - 1)) + 1/16*2^(1/4)*sqrt(2*sqrt(2) + 4)*arctan(-1/4*(32*(sqrt(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(29*sqrt(2) + 10) - 24*sqrt(2) - 44)*cos(x)^14 + 16*(sqrt(2)*(51*sqrt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*sqrt(2) - 36) - 54*sqrt(2) + 15)*cos(x)^10 + 8*(sqrt(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sqrt(2)*(5*sqrt(2) - 98) - 32*sqrt(2) + 216)*cos(x)^6 - 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4*(2*sqrt(2) - 15)*cos(x)^2 + 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(11*sqrt(2) - 9) - 2*2^(1/4)*(13*sqrt(2) + 4))*cos(x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(2) - 14))*cos(x)^11 - 8*(2^(3/4)*(19*sqrt(2) - 27) - 2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt(2) - 65) - 32*2^(1/4)*(sqrt(2) - 4))*cos(x)^7 - 2*(2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x)^5 + (2*2^(3/4)*(sqrt(2) - 2) + 2^(1/4)*(sqrt(2) + 26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - (16*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^14 - 56*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 + 8*(sqrt(2)*(49*sqrt(2) - 62) - 76*sqrt(2) + 54)*cos(x)^10 - 40*(sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*cos(x)^8 + 4*(sqrt(2)*(27*sqrt(2) - 46) - 32*sqrt(2) + 92)*cos(x)^6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) + 72)*cos(x)^4 + 2*(sqrt(2)*(sqrt(2) - 2) + 14)*cos(x)^2 + (8*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^13 - 24*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(73*sqrt(2) - 94))*cos(x)^9 - 8*(2^(3/4)*(16*sqrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos(x)^7 + 2*(9*2^(3/4)*(2*sqrt(2) - 3) - 8*2^(1/4)*(4*sqrt(2) - 7))*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^(1/4)*(sqrt(2) - 2))*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 8*(2*sqrt(2) - 3)*cos(x)^2 + 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 8) + 4)/(112*cos(x)^16 - 448*cos(x)^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos(x)^8 + 208*cos(x)^6 - 88*cos(x)^4 + 16*cos(x)^2 - 1)) - 1/16*2^(1/4)*sqrt(2*sqrt(2) + 4)*arctan(-1/4*(32*(sqrt(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(29*sqrt(2) + 10) - 24*sqrt(2) - 44)*cos(x)^14 + 16*(sqrt(2)*(51*sqrt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*sqrt(2) - 36) - 54*sqrt(2) + 15)*cos(x)^10 + 8*(sqrt(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sqrt(2)*(5*sqrt(2) - 98) - 32*sqrt(2) + 216)*cos(x)^6 - 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4*(2*sqrt(2) - 15)*cos(x)^2 - 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(11*sqrt(2) - 9) - 2*2^(1/4)*(13*sqrt(2) + 4))*cos(x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(2) - 14))*cos(x)^11 - 8*(2^(3/4)*(19*sqrt(2) - 27) - 2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt(2) - 65) - 32*2^(1/4)*(sqrt(2) - 4))*cos(x)^7 - 2*(2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x)^5 + (2*2^(3/4)*(sqrt(2) - 2) + 2^(1/4)*(sqrt(2) + 26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + (16*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^14 - 56*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 + 8*(sqrt(2)*(49*sqrt(2) - 62) - 76*sqrt(2) + 54)*cos(x)^10 - 40*(sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*cos(x)^8 + 4*(sqrt(2)*(27*sqrt(2) - 46) - 32*sqrt(2) + 92)*cos(x)^6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) + 72)*cos(x)^4 + 2*(sqrt(2)*(sqrt(2) - 2) + 14)*cos(x)^2 - (8*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^13 - 24*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(73*sqrt(2) - 94))*cos(x)^9 - 8*(2^(3/4)*(16*sqrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos(x)^7 + 2*(9*2^(3/4)*(2*sqrt(2) - 3) - 8*2^(1/4)*(4*sqrt(2) - 7))*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^(1/4)*(sqrt(2) - 2))*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 8*(2*sqrt(2) - 3)*cos(x)^2 - 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 8) + 4)/(112*cos(x)^16 - 448*cos(x)^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos(x)^8 + 208*cos(x)^6 - 88*cos(x)^4 + 16*cos(x)^2 - 1)) + 1/16*2^(1/4)*sqrt(2*sqrt(2) + 4)*arctan(1/4*(32*(sqrt(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(29*sqrt(2) + 10) - 24*sqrt(2) - 44)*cos(x)^14 + 16*(sqrt(2)*(51*sqrt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*sqrt(2) - 36) - 54*sqrt(2) + 15)*cos(x)^10 + 8*(sqrt(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sqrt(2)*(5*sqrt(2) - 98) - 32*sqrt(2) + 216)*cos(x)^6 - 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4*(2*sqrt(2) - 15)*cos(x)^2 - 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(11*sqrt(2) - 9) - 2*2^(1/4)*(13*sqrt(2) + 4))*cos(x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(2) - 14))*cos(x)^11 - 8*(2^(3/4)*(19*sqrt(2) - 27) - 2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt(2) - 65) - 32*2^(1/4)*(sqrt(2) - 4))*cos(x)^7 - 2*(2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x)^5 + (2*2^(3/4)*(sqrt(2) - 2) + 2^(1/4)*(sqrt(2) + 26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - (16*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^14 - 56*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 + 8*(sqrt(2)*(49*sqrt(2) - 62) - 76*sqrt(2) + 54)*cos(x)^10 - 40*(sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*cos(x)^8 + 4*(sqrt(2)*(27*sqrt(2) - 46) - 32*sqrt(2) + 92)*cos(x)^6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) + 72)*cos(x)^4 + 2*(sqrt(2)*(sqrt(2) - 2) + 14)*cos(x)^2 - (8*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^13 - 24*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(73*sqrt(2) - 94))*cos(x)^9 - 8*(2^(3/4)*(16*sqrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos(x)^7 + 2*(9*2^(3/4)*(2*sqrt(2) - 3) - 8*2^(1/4)*(4*sqrt(2) - 7))*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^(1/4)*(sqrt(2) - 2))*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 8*(2*sqrt(2) - 3)*cos(x)^2 - 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 8) + 4)/(112*cos(x)^16 - 448*cos(x)^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos(x)^8 + 208*cos(x)^6 - 88*cos(x)^4 + 16*cos(x)^2 - 1))","B",0
239,0,0,0,0.717113," ","integrate(sin(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} \sin\left(d x + c\right), x\right)"," ",0,"integral(sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*sin(d*x + c), x)","F",0
240,0,0,0,0.704208," ","integrate(csc(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} \csc\left(d x + c\right), x\right)"," ",0,"integral(sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*csc(d*x + c), x)","F",0
241,0,0,0,0.531223," ","integrate(sin(d*x+c)^5/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{{\left(\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1\right)} \sin\left(d x + c\right)}{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}, x\right)"," ",0,"integral((cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*sin(d*x + c)/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b), x)","F",0
242,0,0,0,0.820034," ","integrate(sin(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(\cos\left(d x + c\right)^{2} - 1\right)} \sin\left(d x + c\right)}{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}, x\right)"," ",0,"integral(-(cos(d*x + c)^2 - 1)*sin(d*x + c)/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b), x)","F",0
243,0,0,0,0.786294," ","integrate(sin(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sin\left(d x + c\right)}{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}, x\right)"," ",0,"integral(sin(d*x + c)/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b), x)","F",0
244,-1,0,0,0.000000," ","integrate(csc(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
245,0,0,0,0.937101," ","integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\csc\left(d x + c\right)^{3}}{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}, x\right)"," ",0,"integral(csc(d*x + c)^3/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b), x)","F",0
246,0,0,0,1.513635," ","integrate(sin(d*x+c)^2/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\cos\left(d x + c\right)^{2} - 1}{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}, x\right)"," ",0,"integral(-(cos(d*x + c)^2 - 1)/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b), x)","F",0
247,0,0,0,1.072535," ","integrate(1/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{1}{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}, x\right)"," ",0,"integral(1/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b), x)","F",0
248,0,0,0,0.535660," ","integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\csc\left(d x + c\right)^{2}}{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}, x\right)"," ",0,"integral(csc(d*x + c)^2/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b), x)","F",0
249,-2,0,0,0.000000," ","integrate(1/(a+b*sin(x)^5),x, algorithm=""fricas"")","\text{Exception raised: RuntimeError}"," ",0,"Exception raised: RuntimeError >> no explicit roots found","F(-2)",0
250,-1,0,0,0.000000," ","integrate(1/(a+b*sin(x)^6),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
251,-1,0,0,0.000000," ","integrate(1/(a+b*sin(x)^8),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
252,-2,0,0,0.000000," ","integrate(1/(a-b*sin(x)^5),x, algorithm=""fricas"")","\text{Exception raised: RuntimeError}"," ",0,"Exception raised: RuntimeError >> no explicit roots found","F(-2)",0
253,-1,0,0,0.000000," ","integrate(1/(a-b*sin(x)^6),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
254,-1,0,0,0.000000," ","integrate(1/(a-b*sin(x)^8),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
255,-1,0,0,0.000000," ","integrate(1/(1+sin(x)^5),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
256,1,138,0,1.009156," ","integrate(1/(1+sin(x)^6),x, algorithm=""fricas"")","\frac{1}{12} \, \sqrt{3} \arctan\left(\frac{4 \, \sqrt{3} \cos\left(x\right) \sin\left(x\right) + \sqrt{3}}{3 \, {\left(2 \, \cos\left(x\right)^{2} - 1\right)}}\right) + \frac{1}{12} \, \sqrt{3} \arctan\left(\frac{4 \, \sqrt{3} \cos\left(x\right) \sin\left(x\right) - \sqrt{3}}{3 \, {\left(2 \, \cos\left(x\right)^{2} - 1\right)}}\right) - \frac{1}{12} \, \sqrt{2} \arctan\left(\frac{3 \, \sqrt{2} \cos\left(x\right)^{2} - 2 \, \sqrt{2}}{4 \, \cos\left(x\right) \sin\left(x\right)}\right) + \frac{1}{24} \, \log\left(-\cos\left(x\right)^{4} + \cos\left(x\right)^{2} + 2 \, \cos\left(x\right) \sin\left(x\right) + 1\right) - \frac{1}{24} \, \log\left(-\cos\left(x\right)^{4} + \cos\left(x\right)^{2} - 2 \, \cos\left(x\right) \sin\left(x\right) + 1\right)"," ",0,"1/12*sqrt(3)*arctan(1/3*(4*sqrt(3)*cos(x)*sin(x) + sqrt(3))/(2*cos(x)^2 - 1)) + 1/12*sqrt(3)*arctan(1/3*(4*sqrt(3)*cos(x)*sin(x) - sqrt(3))/(2*cos(x)^2 - 1)) - 1/12*sqrt(2)*arctan(1/4*(3*sqrt(2)*cos(x)^2 - 2*sqrt(2))/(cos(x)*sin(x))) + 1/24*log(-cos(x)^4 + cos(x)^2 + 2*cos(x)*sin(x) + 1) - 1/24*log(-cos(x)^4 + cos(x)^2 - 2*cos(x)*sin(x) + 1)","A",0
257,-1,0,0,0.000000," ","integrate(1/(1+sin(x)^8),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
258,-1,0,0,0.000000," ","integrate(1/(1-sin(x)^5),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
259,-1,0,0,0.000000," ","integrate(1/(1-sin(x)^6),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
260,1,3884,0,80.588456," ","integrate(1/(1-sin(x)^8),x, algorithm=""fricas"")","-\frac{2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} {\left(\sqrt{2} - 1\right)} \cos\left(x\right) \log\left(-{\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 2 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} + {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 2\right) - 2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} {\left(\sqrt{2} - 1\right)} \cos\left(x\right) \log\left(-{\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 2 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} - {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 2\right) + 2 \cdot 2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} \arctan\left(\frac{32 \, {\left(\sqrt{2} {\left(3 \, \sqrt{2} + 2\right)} - 2 \, \sqrt{2} - 6\right)} \cos\left(x\right)^{16} - 16 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} + 10\right)} - 24 \, \sqrt{2} - 44\right)} \cos\left(x\right)^{14} + 16 \, {\left(\sqrt{2} {\left(51 \, \sqrt{2} - 4\right)} - 52 \, \sqrt{2} - 46\right)} \cos\left(x\right)^{12} - 16 \, {\left(\sqrt{2} {\left(41 \, \sqrt{2} - 36\right)} - 54 \, \sqrt{2} + 15\right)} \cos\left(x\right)^{10} + 8 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} - 90\right)} - 58 \, \sqrt{2} + 132\right)} \cos\left(x\right)^{8} - 4 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 98\right)} - 32 \, \sqrt{2} + 216\right)} \cos\left(x\right)^{6} - 4 \, {\left(\sqrt{2} {\left(\sqrt{2} + 24\right)} + 4 \, \sqrt{2} - 82\right)} \cos\left(x\right)^{4} + 4 \, {\left(2 \, \sqrt{2} - 15\right)} \cos\left(x\right)^{2} + 2 \, {\left(8 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 1\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(3 \, \sqrt{2} + 2\right)}\right)} \cos\left(x\right)^{15} - 8 \, {\left(2^{\frac{3}{4}} {\left(11 \, \sqrt{2} - 9\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(13 \, \sqrt{2} + 4\right)}\right)} \cos\left(x\right)^{13} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(21 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(79 \, \sqrt{2} - 14\right)}\right)} \cos\left(x\right)^{11} - 8 \, {\left(2^{\frac{3}{4}} {\left(19 \, \sqrt{2} - 27\right)} - 2^{\frac{1}{4}} {\left(27 \, \sqrt{2} - 31\right)}\right)} \cos\left(x\right)^{9} + 2 \, {\left(2^{\frac{3}{4}} {\left(36 \, \sqrt{2} - 65\right)} - 32 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 4\right)}\right)} \cos\left(x\right)^{7} - 2 \, {\left(2^{\frac{3}{4}} {\left(9 \, \sqrt{2} - 19\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 30\right)}\right)} \cos\left(x\right)^{5} + {\left(2 \cdot 2^{\frac{3}{4}} {\left(\sqrt{2} - 2\right)} + 2^{\frac{1}{4}} {\left(\sqrt{2} + 26\right)}\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + {\left(16 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{14} - 56 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{12} + 8 \, {\left(\sqrt{2} {\left(49 \, \sqrt{2} - 62\right)} - 76 \, \sqrt{2} + 54\right)} \cos\left(x\right)^{10} - 40 \, {\left(\sqrt{2} {\left(7 \, \sqrt{2} - 10\right)} - 10 \, \sqrt{2} + 13\right)} \cos\left(x\right)^{8} + 4 \, {\left(\sqrt{2} {\left(27 \, \sqrt{2} - 46\right)} - 32 \, \sqrt{2} + 92\right)} \cos\left(x\right)^{6} - 2 \, {\left(11 \, \sqrt{2} {\left(\sqrt{2} - 2\right)} - 8 \, \sqrt{2} + 72\right)} \cos\left(x\right)^{4} + 2 \, {\left(\sqrt{2} {\left(\sqrt{2} - 2\right)} + 14\right)} \cos\left(x\right)^{2} + {\left(8 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{13} - 24 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{11} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(28 \, \sqrt{2} - 39\right)} - 2^{\frac{1}{4}} {\left(73 \, \sqrt{2} - 94\right)}\right)} \cos\left(x\right)^{9} - 8 \, {\left(2^{\frac{3}{4}} {\left(16 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(23 \, \sqrt{2} - 34\right)}\right)} \cos\left(x\right)^{7} + 2 \, {\left(9 \cdot 2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 8 \cdot 2^{\frac{1}{4}} {\left(4 \, \sqrt{2} - 7\right)}\right)} \cos\left(x\right)^{5} - 2 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 6 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)}\right)} \cos\left(x\right)^{3} - 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - 2\right)} \sqrt{-4 \, {\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 8 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} + 4 \, {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 8} + 4}{4 \, {\left(112 \, \cos\left(x\right)^{16} - 448 \, \cos\left(x\right)^{14} + 608 \, \cos\left(x\right)^{12} - 256 \, \cos\left(x\right)^{10} - 152 \, \cos\left(x\right)^{8} + 208 \, \cos\left(x\right)^{6} - 88 \, \cos\left(x\right)^{4} + 16 \, \cos\left(x\right)^{2} - 1\right)}}\right) \cos\left(x\right) - 2 \cdot 2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} \arctan\left(-\frac{32 \, {\left(\sqrt{2} {\left(3 \, \sqrt{2} + 2\right)} - 2 \, \sqrt{2} - 6\right)} \cos\left(x\right)^{16} - 16 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} + 10\right)} - 24 \, \sqrt{2} - 44\right)} \cos\left(x\right)^{14} + 16 \, {\left(\sqrt{2} {\left(51 \, \sqrt{2} - 4\right)} - 52 \, \sqrt{2} - 46\right)} \cos\left(x\right)^{12} - 16 \, {\left(\sqrt{2} {\left(41 \, \sqrt{2} - 36\right)} - 54 \, \sqrt{2} + 15\right)} \cos\left(x\right)^{10} + 8 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} - 90\right)} - 58 \, \sqrt{2} + 132\right)} \cos\left(x\right)^{8} - 4 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 98\right)} - 32 \, \sqrt{2} + 216\right)} \cos\left(x\right)^{6} - 4 \, {\left(\sqrt{2} {\left(\sqrt{2} + 24\right)} + 4 \, \sqrt{2} - 82\right)} \cos\left(x\right)^{4} + 4 \, {\left(2 \, \sqrt{2} - 15\right)} \cos\left(x\right)^{2} + 2 \, {\left(8 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 1\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(3 \, \sqrt{2} + 2\right)}\right)} \cos\left(x\right)^{15} - 8 \, {\left(2^{\frac{3}{4}} {\left(11 \, \sqrt{2} - 9\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(13 \, \sqrt{2} + 4\right)}\right)} \cos\left(x\right)^{13} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(21 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(79 \, \sqrt{2} - 14\right)}\right)} \cos\left(x\right)^{11} - 8 \, {\left(2^{\frac{3}{4}} {\left(19 \, \sqrt{2} - 27\right)} - 2^{\frac{1}{4}} {\left(27 \, \sqrt{2} - 31\right)}\right)} \cos\left(x\right)^{9} + 2 \, {\left(2^{\frac{3}{4}} {\left(36 \, \sqrt{2} - 65\right)} - 32 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 4\right)}\right)} \cos\left(x\right)^{7} - 2 \, {\left(2^{\frac{3}{4}} {\left(9 \, \sqrt{2} - 19\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 30\right)}\right)} \cos\left(x\right)^{5} + {\left(2 \cdot 2^{\frac{3}{4}} {\left(\sqrt{2} - 2\right)} + 2^{\frac{1}{4}} {\left(\sqrt{2} + 26\right)}\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - {\left(16 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{14} - 56 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{12} + 8 \, {\left(\sqrt{2} {\left(49 \, \sqrt{2} - 62\right)} - 76 \, \sqrt{2} + 54\right)} \cos\left(x\right)^{10} - 40 \, {\left(\sqrt{2} {\left(7 \, \sqrt{2} - 10\right)} - 10 \, \sqrt{2} + 13\right)} \cos\left(x\right)^{8} + 4 \, {\left(\sqrt{2} {\left(27 \, \sqrt{2} - 46\right)} - 32 \, \sqrt{2} + 92\right)} \cos\left(x\right)^{6} - 2 \, {\left(11 \, \sqrt{2} {\left(\sqrt{2} - 2\right)} - 8 \, \sqrt{2} + 72\right)} \cos\left(x\right)^{4} + 2 \, {\left(\sqrt{2} {\left(\sqrt{2} - 2\right)} + 14\right)} \cos\left(x\right)^{2} + {\left(8 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{13} - 24 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{11} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(28 \, \sqrt{2} - 39\right)} - 2^{\frac{1}{4}} {\left(73 \, \sqrt{2} - 94\right)}\right)} \cos\left(x\right)^{9} - 8 \, {\left(2^{\frac{3}{4}} {\left(16 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(23 \, \sqrt{2} - 34\right)}\right)} \cos\left(x\right)^{7} + 2 \, {\left(9 \cdot 2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 8 \cdot 2^{\frac{1}{4}} {\left(4 \, \sqrt{2} - 7\right)}\right)} \cos\left(x\right)^{5} - 2 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 6 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)}\right)} \cos\left(x\right)^{3} - 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - 2\right)} \sqrt{-4 \, {\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 8 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} + 4 \, {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 8} + 4}{4 \, {\left(112 \, \cos\left(x\right)^{16} - 448 \, \cos\left(x\right)^{14} + 608 \, \cos\left(x\right)^{12} - 256 \, \cos\left(x\right)^{10} - 152 \, \cos\left(x\right)^{8} + 208 \, \cos\left(x\right)^{6} - 88 \, \cos\left(x\right)^{4} + 16 \, \cos\left(x\right)^{2} - 1\right)}}\right) \cos\left(x\right) + 2 \cdot 2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} \arctan\left(-\frac{32 \, {\left(\sqrt{2} {\left(3 \, \sqrt{2} + 2\right)} - 2 \, \sqrt{2} - 6\right)} \cos\left(x\right)^{16} - 16 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} + 10\right)} - 24 \, \sqrt{2} - 44\right)} \cos\left(x\right)^{14} + 16 \, {\left(\sqrt{2} {\left(51 \, \sqrt{2} - 4\right)} - 52 \, \sqrt{2} - 46\right)} \cos\left(x\right)^{12} - 16 \, {\left(\sqrt{2} {\left(41 \, \sqrt{2} - 36\right)} - 54 \, \sqrt{2} + 15\right)} \cos\left(x\right)^{10} + 8 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} - 90\right)} - 58 \, \sqrt{2} + 132\right)} \cos\left(x\right)^{8} - 4 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 98\right)} - 32 \, \sqrt{2} + 216\right)} \cos\left(x\right)^{6} - 4 \, {\left(\sqrt{2} {\left(\sqrt{2} + 24\right)} + 4 \, \sqrt{2} - 82\right)} \cos\left(x\right)^{4} + 4 \, {\left(2 \, \sqrt{2} - 15\right)} \cos\left(x\right)^{2} - 2 \, {\left(8 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 1\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(3 \, \sqrt{2} + 2\right)}\right)} \cos\left(x\right)^{15} - 8 \, {\left(2^{\frac{3}{4}} {\left(11 \, \sqrt{2} - 9\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(13 \, \sqrt{2} + 4\right)}\right)} \cos\left(x\right)^{13} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(21 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(79 \, \sqrt{2} - 14\right)}\right)} \cos\left(x\right)^{11} - 8 \, {\left(2^{\frac{3}{4}} {\left(19 \, \sqrt{2} - 27\right)} - 2^{\frac{1}{4}} {\left(27 \, \sqrt{2} - 31\right)}\right)} \cos\left(x\right)^{9} + 2 \, {\left(2^{\frac{3}{4}} {\left(36 \, \sqrt{2} - 65\right)} - 32 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 4\right)}\right)} \cos\left(x\right)^{7} - 2 \, {\left(2^{\frac{3}{4}} {\left(9 \, \sqrt{2} - 19\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 30\right)}\right)} \cos\left(x\right)^{5} + {\left(2 \cdot 2^{\frac{3}{4}} {\left(\sqrt{2} - 2\right)} + 2^{\frac{1}{4}} {\left(\sqrt{2} + 26\right)}\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + {\left(16 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{14} - 56 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{12} + 8 \, {\left(\sqrt{2} {\left(49 \, \sqrt{2} - 62\right)} - 76 \, \sqrt{2} + 54\right)} \cos\left(x\right)^{10} - 40 \, {\left(\sqrt{2} {\left(7 \, \sqrt{2} - 10\right)} - 10 \, \sqrt{2} + 13\right)} \cos\left(x\right)^{8} + 4 \, {\left(\sqrt{2} {\left(27 \, \sqrt{2} - 46\right)} - 32 \, \sqrt{2} + 92\right)} \cos\left(x\right)^{6} - 2 \, {\left(11 \, \sqrt{2} {\left(\sqrt{2} - 2\right)} - 8 \, \sqrt{2} + 72\right)} \cos\left(x\right)^{4} + 2 \, {\left(\sqrt{2} {\left(\sqrt{2} - 2\right)} + 14\right)} \cos\left(x\right)^{2} - {\left(8 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{13} - 24 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{11} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(28 \, \sqrt{2} - 39\right)} - 2^{\frac{1}{4}} {\left(73 \, \sqrt{2} - 94\right)}\right)} \cos\left(x\right)^{9} - 8 \, {\left(2^{\frac{3}{4}} {\left(16 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(23 \, \sqrt{2} - 34\right)}\right)} \cos\left(x\right)^{7} + 2 \, {\left(9 \cdot 2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 8 \cdot 2^{\frac{1}{4}} {\left(4 \, \sqrt{2} - 7\right)}\right)} \cos\left(x\right)^{5} - 2 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 6 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)}\right)} \cos\left(x\right)^{3} - 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - 2\right)} \sqrt{-4 \, {\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 8 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} - 4 \, {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 8} + 4}{4 \, {\left(112 \, \cos\left(x\right)^{16} - 448 \, \cos\left(x\right)^{14} + 608 \, \cos\left(x\right)^{12} - 256 \, \cos\left(x\right)^{10} - 152 \, \cos\left(x\right)^{8} + 208 \, \cos\left(x\right)^{6} - 88 \, \cos\left(x\right)^{4} + 16 \, \cos\left(x\right)^{2} - 1\right)}}\right) \cos\left(x\right) - 2 \cdot 2^{\frac{1}{4}} \sqrt{2 \, \sqrt{2} + 4} \arctan\left(\frac{32 \, {\left(\sqrt{2} {\left(3 \, \sqrt{2} + 2\right)} - 2 \, \sqrt{2} - 6\right)} \cos\left(x\right)^{16} - 16 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} + 10\right)} - 24 \, \sqrt{2} - 44\right)} \cos\left(x\right)^{14} + 16 \, {\left(\sqrt{2} {\left(51 \, \sqrt{2} - 4\right)} - 52 \, \sqrt{2} - 46\right)} \cos\left(x\right)^{12} - 16 \, {\left(\sqrt{2} {\left(41 \, \sqrt{2} - 36\right)} - 54 \, \sqrt{2} + 15\right)} \cos\left(x\right)^{10} + 8 \, {\left(\sqrt{2} {\left(29 \, \sqrt{2} - 90\right)} - 58 \, \sqrt{2} + 132\right)} \cos\left(x\right)^{8} - 4 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 98\right)} - 32 \, \sqrt{2} + 216\right)} \cos\left(x\right)^{6} - 4 \, {\left(\sqrt{2} {\left(\sqrt{2} + 24\right)} + 4 \, \sqrt{2} - 82\right)} \cos\left(x\right)^{4} + 4 \, {\left(2 \, \sqrt{2} - 15\right)} \cos\left(x\right)^{2} - 2 \, {\left(8 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 1\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(3 \, \sqrt{2} + 2\right)}\right)} \cos\left(x\right)^{15} - 8 \, {\left(2^{\frac{3}{4}} {\left(11 \, \sqrt{2} - 9\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(13 \, \sqrt{2} + 4\right)}\right)} \cos\left(x\right)^{13} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(21 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(79 \, \sqrt{2} - 14\right)}\right)} \cos\left(x\right)^{11} - 8 \, {\left(2^{\frac{3}{4}} {\left(19 \, \sqrt{2} - 27\right)} - 2^{\frac{1}{4}} {\left(27 \, \sqrt{2} - 31\right)}\right)} \cos\left(x\right)^{9} + 2 \, {\left(2^{\frac{3}{4}} {\left(36 \, \sqrt{2} - 65\right)} - 32 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 4\right)}\right)} \cos\left(x\right)^{7} - 2 \, {\left(2^{\frac{3}{4}} {\left(9 \, \sqrt{2} - 19\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 30\right)}\right)} \cos\left(x\right)^{5} + {\left(2 \cdot 2^{\frac{3}{4}} {\left(\sqrt{2} - 2\right)} + 2^{\frac{1}{4}} {\left(\sqrt{2} + 26\right)}\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - {\left(16 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{14} - 56 \, {\left(\sqrt{2} {\left(5 \, \sqrt{2} - 6\right)} - 8 \, \sqrt{2} + 4\right)} \cos\left(x\right)^{12} + 8 \, {\left(\sqrt{2} {\left(49 \, \sqrt{2} - 62\right)} - 76 \, \sqrt{2} + 54\right)} \cos\left(x\right)^{10} - 40 \, {\left(\sqrt{2} {\left(7 \, \sqrt{2} - 10\right)} - 10 \, \sqrt{2} + 13\right)} \cos\left(x\right)^{8} + 4 \, {\left(\sqrt{2} {\left(27 \, \sqrt{2} - 46\right)} - 32 \, \sqrt{2} + 92\right)} \cos\left(x\right)^{6} - 2 \, {\left(11 \, \sqrt{2} {\left(\sqrt{2} - 2\right)} - 8 \, \sqrt{2} + 72\right)} \cos\left(x\right)^{4} + 2 \, {\left(\sqrt{2} {\left(\sqrt{2} - 2\right)} + 14\right)} \cos\left(x\right)^{2} - {\left(8 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{13} - 24 \, {\left(2^{\frac{3}{4}} {\left(8 \, \sqrt{2} - 11\right)} - 2 \cdot 2^{\frac{1}{4}} {\left(5 \, \sqrt{2} - 6\right)}\right)} \cos\left(x\right)^{11} + 4 \, {\left(2 \cdot 2^{\frac{3}{4}} {\left(28 \, \sqrt{2} - 39\right)} - 2^{\frac{1}{4}} {\left(73 \, \sqrt{2} - 94\right)}\right)} \cos\left(x\right)^{9} - 8 \, {\left(2^{\frac{3}{4}} {\left(16 \, \sqrt{2} - 23\right)} - 2^{\frac{1}{4}} {\left(23 \, \sqrt{2} - 34\right)}\right)} \cos\left(x\right)^{7} + 2 \, {\left(9 \cdot 2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 8 \cdot 2^{\frac{1}{4}} {\left(4 \, \sqrt{2} - 7\right)}\right)} \cos\left(x\right)^{5} - 2 \, {\left(2^{\frac{3}{4}} {\left(2 \, \sqrt{2} - 3\right)} - 6 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)}\right)} \cos\left(x\right)^{3} - 2^{\frac{1}{4}} {\left(\sqrt{2} - 2\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) - 2\right)} \sqrt{-4 \, {\left(4 \, \sqrt{2} - 5\right)} \cos\left(x\right)^{4} + 8 \, {\left(2 \, \sqrt{2} - 3\right)} \cos\left(x\right)^{2} - 4 \, {\left(2^{\frac{1}{4}} {\left(3 \, \sqrt{2} - 4\right)} \cos\left(x\right)^{3} - 2 \cdot 2^{\frac{1}{4}} {\left(\sqrt{2} - 1\right)} \cos\left(x\right)\right)} \sqrt{2 \, \sqrt{2} + 4} \sin\left(x\right) + 8} + 4}{4 \, {\left(112 \, \cos\left(x\right)^{16} - 448 \, \cos\left(x\right)^{14} + 608 \, \cos\left(x\right)^{12} - 256 \, \cos\left(x\right)^{10} - 152 \, \cos\left(x\right)^{8} + 208 \, \cos\left(x\right)^{6} - 88 \, \cos\left(x\right)^{4} + 16 \, \cos\left(x\right)^{2} - 1\right)}}\right) \cos\left(x\right) + 4 \, \sqrt{2} \arctan\left(\frac{3 \, \sqrt{2} \cos\left(x\right)^{2} - 2 \, \sqrt{2}}{4 \, \cos\left(x\right) \sin\left(x\right)}\right) \cos\left(x\right) - 16 \, \sin\left(x\right)}{64 \, \cos\left(x\right)}"," ",0,"-1/64*(2^(1/4)*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 1)*cos(x)*log(-(4*sqrt(2) - 5)*cos(x)^4 + 2*(2*sqrt(2) - 3)*cos(x)^2 + (2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 2) - 2^(1/4)*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 1)*cos(x)*log(-(4*sqrt(2) - 5)*cos(x)^4 + 2*(2*sqrt(2) - 3)*cos(x)^2 - (2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 2) + 2*2^(1/4)*sqrt(2*sqrt(2) + 4)*arctan(1/4*(32*(sqrt(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(29*sqrt(2) + 10) - 24*sqrt(2) - 44)*cos(x)^14 + 16*(sqrt(2)*(51*sqrt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*sqrt(2) - 36) - 54*sqrt(2) + 15)*cos(x)^10 + 8*(sqrt(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sqrt(2)*(5*sqrt(2) - 98) - 32*sqrt(2) + 216)*cos(x)^6 - 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4*(2*sqrt(2) - 15)*cos(x)^2 + 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(11*sqrt(2) - 9) - 2*2^(1/4)*(13*sqrt(2) + 4))*cos(x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(2) - 14))*cos(x)^11 - 8*(2^(3/4)*(19*sqrt(2) - 27) - 2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt(2) - 65) - 32*2^(1/4)*(sqrt(2) - 4))*cos(x)^7 - 2*(2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x)^5 + (2*2^(3/4)*(sqrt(2) - 2) + 2^(1/4)*(sqrt(2) + 26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + (16*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^14 - 56*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 + 8*(sqrt(2)*(49*sqrt(2) - 62) - 76*sqrt(2) + 54)*cos(x)^10 - 40*(sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*cos(x)^8 + 4*(sqrt(2)*(27*sqrt(2) - 46) - 32*sqrt(2) + 92)*cos(x)^6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) + 72)*cos(x)^4 + 2*(sqrt(2)*(sqrt(2) - 2) + 14)*cos(x)^2 + (8*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^13 - 24*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(73*sqrt(2) - 94))*cos(x)^9 - 8*(2^(3/4)*(16*sqrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos(x)^7 + 2*(9*2^(3/4)*(2*sqrt(2) - 3) - 8*2^(1/4)*(4*sqrt(2) - 7))*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^(1/4)*(sqrt(2) - 2))*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 8*(2*sqrt(2) - 3)*cos(x)^2 + 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 8) + 4)/(112*cos(x)^16 - 448*cos(x)^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos(x)^8 + 208*cos(x)^6 - 88*cos(x)^4 + 16*cos(x)^2 - 1))*cos(x) - 2*2^(1/4)*sqrt(2*sqrt(2) + 4)*arctan(-1/4*(32*(sqrt(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(29*sqrt(2) + 10) - 24*sqrt(2) - 44)*cos(x)^14 + 16*(sqrt(2)*(51*sqrt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*sqrt(2) - 36) - 54*sqrt(2) + 15)*cos(x)^10 + 8*(sqrt(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sqrt(2)*(5*sqrt(2) - 98) - 32*sqrt(2) + 216)*cos(x)^6 - 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4*(2*sqrt(2) - 15)*cos(x)^2 + 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(11*sqrt(2) - 9) - 2*2^(1/4)*(13*sqrt(2) + 4))*cos(x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(2) - 14))*cos(x)^11 - 8*(2^(3/4)*(19*sqrt(2) - 27) - 2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt(2) - 65) - 32*2^(1/4)*(sqrt(2) - 4))*cos(x)^7 - 2*(2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x)^5 + (2*2^(3/4)*(sqrt(2) - 2) + 2^(1/4)*(sqrt(2) + 26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - (16*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^14 - 56*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 + 8*(sqrt(2)*(49*sqrt(2) - 62) - 76*sqrt(2) + 54)*cos(x)^10 - 40*(sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*cos(x)^8 + 4*(sqrt(2)*(27*sqrt(2) - 46) - 32*sqrt(2) + 92)*cos(x)^6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) + 72)*cos(x)^4 + 2*(sqrt(2)*(sqrt(2) - 2) + 14)*cos(x)^2 + (8*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^13 - 24*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(73*sqrt(2) - 94))*cos(x)^9 - 8*(2^(3/4)*(16*sqrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos(x)^7 + 2*(9*2^(3/4)*(2*sqrt(2) - 3) - 8*2^(1/4)*(4*sqrt(2) - 7))*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^(1/4)*(sqrt(2) - 2))*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 8*(2*sqrt(2) - 3)*cos(x)^2 + 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 8) + 4)/(112*cos(x)^16 - 448*cos(x)^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos(x)^8 + 208*cos(x)^6 - 88*cos(x)^4 + 16*cos(x)^2 - 1))*cos(x) + 2*2^(1/4)*sqrt(2*sqrt(2) + 4)*arctan(-1/4*(32*(sqrt(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(29*sqrt(2) + 10) - 24*sqrt(2) - 44)*cos(x)^14 + 16*(sqrt(2)*(51*sqrt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*sqrt(2) - 36) - 54*sqrt(2) + 15)*cos(x)^10 + 8*(sqrt(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sqrt(2)*(5*sqrt(2) - 98) - 32*sqrt(2) + 216)*cos(x)^6 - 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4*(2*sqrt(2) - 15)*cos(x)^2 - 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(11*sqrt(2) - 9) - 2*2^(1/4)*(13*sqrt(2) + 4))*cos(x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(2) - 14))*cos(x)^11 - 8*(2^(3/4)*(19*sqrt(2) - 27) - 2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt(2) - 65) - 32*2^(1/4)*(sqrt(2) - 4))*cos(x)^7 - 2*(2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x)^5 + (2*2^(3/4)*(sqrt(2) - 2) + 2^(1/4)*(sqrt(2) + 26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + (16*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^14 - 56*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 + 8*(sqrt(2)*(49*sqrt(2) - 62) - 76*sqrt(2) + 54)*cos(x)^10 - 40*(sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*cos(x)^8 + 4*(sqrt(2)*(27*sqrt(2) - 46) - 32*sqrt(2) + 92)*cos(x)^6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) + 72)*cos(x)^4 + 2*(sqrt(2)*(sqrt(2) - 2) + 14)*cos(x)^2 - (8*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^13 - 24*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(73*sqrt(2) - 94))*cos(x)^9 - 8*(2^(3/4)*(16*sqrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos(x)^7 + 2*(9*2^(3/4)*(2*sqrt(2) - 3) - 8*2^(1/4)*(4*sqrt(2) - 7))*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^(1/4)*(sqrt(2) - 2))*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 8*(2*sqrt(2) - 3)*cos(x)^2 - 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 8) + 4)/(112*cos(x)^16 - 448*cos(x)^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos(x)^8 + 208*cos(x)^6 - 88*cos(x)^4 + 16*cos(x)^2 - 1))*cos(x) - 2*2^(1/4)*sqrt(2*sqrt(2) + 4)*arctan(1/4*(32*(sqrt(2)*(3*sqrt(2) + 2) - 2*sqrt(2) - 6)*cos(x)^16 - 16*(sqrt(2)*(29*sqrt(2) + 10) - 24*sqrt(2) - 44)*cos(x)^14 + 16*(sqrt(2)*(51*sqrt(2) - 4) - 52*sqrt(2) - 46)*cos(x)^12 - 16*(sqrt(2)*(41*sqrt(2) - 36) - 54*sqrt(2) + 15)*cos(x)^10 + 8*(sqrt(2)*(29*sqrt(2) - 90) - 58*sqrt(2) + 132)*cos(x)^8 - 4*(sqrt(2)*(5*sqrt(2) - 98) - 32*sqrt(2) + 216)*cos(x)^6 - 4*(sqrt(2)*(sqrt(2) + 24) + 4*sqrt(2) - 82)*cos(x)^4 + 4*(2*sqrt(2) - 15)*cos(x)^2 - 2*(8*(2^(3/4)*(2*sqrt(2) - 1) - 2*2^(1/4)*(3*sqrt(2) + 2))*cos(x)^15 - 8*(2^(3/4)*(11*sqrt(2) - 9) - 2*2^(1/4)*(13*sqrt(2) + 4))*cos(x)^13 + 4*(2*2^(3/4)*(21*sqrt(2) - 23) - 2^(1/4)*(79*sqrt(2) - 14))*cos(x)^11 - 8*(2^(3/4)*(19*sqrt(2) - 27) - 2^(1/4)*(27*sqrt(2) - 31))*cos(x)^9 + 2*(2^(3/4)*(36*sqrt(2) - 65) - 32*2^(1/4)*(sqrt(2) - 4))*cos(x)^7 - 2*(2^(3/4)*(9*sqrt(2) - 19) - 2*2^(1/4)*(sqrt(2) - 30))*cos(x)^5 + (2*2^(3/4)*(sqrt(2) - 2) + 2^(1/4)*(sqrt(2) + 26))*cos(x)^3 - 2*2^(1/4)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - (16*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^14 - 56*(sqrt(2)*(5*sqrt(2) - 6) - 8*sqrt(2) + 4)*cos(x)^12 + 8*(sqrt(2)*(49*sqrt(2) - 62) - 76*sqrt(2) + 54)*cos(x)^10 - 40*(sqrt(2)*(7*sqrt(2) - 10) - 10*sqrt(2) + 13)*cos(x)^8 + 4*(sqrt(2)*(27*sqrt(2) - 46) - 32*sqrt(2) + 92)*cos(x)^6 - 2*(11*sqrt(2)*(sqrt(2) - 2) - 8*sqrt(2) + 72)*cos(x)^4 + 2*(sqrt(2)*(sqrt(2) - 2) + 14)*cos(x)^2 - (8*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^13 - 24*(2^(3/4)*(8*sqrt(2) - 11) - 2*2^(1/4)*(5*sqrt(2) - 6))*cos(x)^11 + 4*(2*2^(3/4)*(28*sqrt(2) - 39) - 2^(1/4)*(73*sqrt(2) - 94))*cos(x)^9 - 8*(2^(3/4)*(16*sqrt(2) - 23) - 2^(1/4)*(23*sqrt(2) - 34))*cos(x)^7 + 2*(9*2^(3/4)*(2*sqrt(2) - 3) - 8*2^(1/4)*(4*sqrt(2) - 7))*cos(x)^5 - 2*(2^(3/4)*(2*sqrt(2) - 3) - 6*2^(1/4)*(sqrt(2) - 2))*cos(x)^3 - 2^(1/4)*(sqrt(2) - 2)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) - 2)*sqrt(-4*(4*sqrt(2) - 5)*cos(x)^4 + 8*(2*sqrt(2) - 3)*cos(x)^2 - 4*(2^(1/4)*(3*sqrt(2) - 4)*cos(x)^3 - 2*2^(1/4)*(sqrt(2) - 1)*cos(x))*sqrt(2*sqrt(2) + 4)*sin(x) + 8) + 4)/(112*cos(x)^16 - 448*cos(x)^14 + 608*cos(x)^12 - 256*cos(x)^10 - 152*cos(x)^8 + 208*cos(x)^6 - 88*cos(x)^4 + 16*cos(x)^2 - 1))*cos(x) + 4*sqrt(2)*arctan(1/4*(3*sqrt(2)*cos(x)^2 - 2*sqrt(2))/(cos(x)*sin(x)))*cos(x) - 16*sin(x))/cos(x)","B",0
261,1,27,0,1.101769," ","integrate(cos(x)^9/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{{\left(5 \, \cos\left(x\right)^{6} + 6 \, \cos\left(x\right)^{4} + 8 \, \cos\left(x\right)^{2} + 16\right)} \sin\left(x\right)}{35 \, a}"," ",0,"1/35*(5*cos(x)^6 + 6*cos(x)^4 + 8*cos(x)^2 + 16)*sin(x)/a","A",0
262,1,21,0,0.893535," ","integrate(cos(x)^7/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{{\left(3 \, \cos\left(x\right)^{4} + 4 \, \cos\left(x\right)^{2} + 8\right)} \sin\left(x\right)}{15 \, a}"," ",0,"1/15*(3*cos(x)^4 + 4*cos(x)^2 + 8)*sin(x)/a","A",0
263,1,13,0,0.758887," ","integrate(cos(x)^5/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{{\left(\cos\left(x\right)^{2} + 2\right)} \sin\left(x\right)}{3 \, a}"," ",0,"1/3*(cos(x)^2 + 2)*sin(x)/a","A",0
264,1,6,0,0.917666," ","integrate(cos(x)^3/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{\sin\left(x\right)}{a}"," ",0,"sin(x)/a","A",0
265,1,20,0,1.081033," ","integrate(cos(x)/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{\log\left(\sin\left(x\right) + 1\right) - \log\left(-\sin\left(x\right) + 1\right)}{2 \, a}"," ",0,"1/2*(log(sin(x) + 1) - log(-sin(x) + 1))/a","B",0
266,1,46,0,0.931730," ","integrate(sec(x)^3/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{3 \, \cos\left(x\right)^{4} \log\left(\sin\left(x\right) + 1\right) - 3 \, \cos\left(x\right)^{4} \log\left(-\sin\left(x\right) + 1\right) + 2 \, {\left(3 \, \cos\left(x\right)^{2} + 2\right)} \sin\left(x\right)}{16 \, a \cos\left(x\right)^{4}}"," ",0,"1/16*(3*cos(x)^4*log(sin(x) + 1) - 3*cos(x)^4*log(-sin(x) + 1) + 2*(3*cos(x)^2 + 2)*sin(x))/(a*cos(x)^4)","A",0
267,1,23,0,1.052709," ","integrate(cos(x)^6/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{{\left(2 \, \cos\left(x\right)^{3} + 3 \, \cos\left(x\right)\right)} \sin\left(x\right) + 3 \, x}{8 \, a}"," ",0,"1/8*((2*cos(x)^3 + 3*cos(x))*sin(x) + 3*x)/a","A",0
268,1,12,0,0.634884," ","integrate(cos(x)^4/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{\cos\left(x\right) \sin\left(x\right) + x}{2 \, a}"," ",0,"1/2*(cos(x)*sin(x) + x)/a","A",0
269,1,5,0,0.751711," ","integrate(cos(x)^2/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{x}{a}"," ",0,"x/a","A",0
270,1,37,0,0.588943," ","integrate(sec(x)/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{\cos\left(x\right)^{2} \log\left(\sin\left(x\right) + 1\right) - \cos\left(x\right)^{2} \log\left(-\sin\left(x\right) + 1\right) + 2 \, \sin\left(x\right)}{4 \, a \cos\left(x\right)^{2}}"," ",0,"1/4*(cos(x)^2*log(sin(x) + 1) - cos(x)^2*log(-sin(x) + 1) + 2*sin(x))/(a*cos(x)^2)","B",0
271,1,19,0,0.656655," ","integrate(sec(x)^2/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{{\left(2 \, \cos\left(x\right)^{2} + 1\right)} \sin\left(x\right)}{3 \, a \cos\left(x\right)^{3}}"," ",0,"1/3*(2*cos(x)^2 + 1)*sin(x)/(a*cos(x)^3)","A",0
272,1,25,0,0.988856," ","integrate(sec(x)^4/(a-a*sin(x)^2),x, algorithm=""fricas"")","\frac{{\left(8 \, \cos\left(x\right)^{4} + 4 \, \cos\left(x\right)^{2} + 3\right)} \sin\left(x\right)}{15 \, a \cos\left(x\right)^{5}}"," ",0,"1/15*(8*cos(x)^4 + 4*cos(x)^2 + 3)*sin(x)/(a*cos(x)^5)","A",0
273,1,21,0,0.984941," ","integrate(cos(x)^9/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{{\left(3 \, \cos\left(x\right)^{4} + 4 \, \cos\left(x\right)^{2} + 8\right)} \sin\left(x\right)}{15 \, a^{2}}"," ",0,"1/15*(3*cos(x)^4 + 4*cos(x)^2 + 8)*sin(x)/a^2","A",0
274,1,13,0,0.970793," ","integrate(cos(x)^7/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{{\left(\cos\left(x\right)^{2} + 2\right)} \sin\left(x\right)}{3 \, a^{2}}"," ",0,"1/3*(cos(x)^2 + 2)*sin(x)/a^2","A",0
275,1,6,0,0.844085," ","integrate(cos(x)^5/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{\sin\left(x\right)}{a^{2}}"," ",0,"sin(x)/a^2","A",0
276,1,20,0,0.912022," ","integrate(cos(x)^3/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{\log\left(\sin\left(x\right) + 1\right) - \log\left(-\sin\left(x\right) + 1\right)}{2 \, a^{2}}"," ",0,"1/2*(log(sin(x) + 1) - log(-sin(x) + 1))/a^2","B",0
277,1,37,0,1.014974," ","integrate(cos(x)/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{\cos\left(x\right)^{2} \log\left(\sin\left(x\right) + 1\right) - \cos\left(x\right)^{2} \log\left(-\sin\left(x\right) + 1\right) + 2 \, \sin\left(x\right)}{4 \, a^{2} \cos\left(x\right)^{2}}"," ",0,"1/4*(cos(x)^2*log(sin(x) + 1) - cos(x)^2*log(-sin(x) + 1) + 2*sin(x))/(a^2*cos(x)^2)","B",0
278,1,46,0,0.996894," ","integrate(sec(x)/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{3 \, \cos\left(x\right)^{4} \log\left(\sin\left(x\right) + 1\right) - 3 \, \cos\left(x\right)^{4} \log\left(-\sin\left(x\right) + 1\right) + 2 \, {\left(3 \, \cos\left(x\right)^{2} + 2\right)} \sin\left(x\right)}{16 \, a^{2} \cos\left(x\right)^{4}}"," ",0,"1/16*(3*cos(x)^4*log(sin(x) + 1) - 3*cos(x)^4*log(-sin(x) + 1) + 2*(3*cos(x)^2 + 2)*sin(x))/(a^2*cos(x)^4)","A",0
279,1,23,0,0.893184," ","integrate(cos(x)^8/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{{\left(2 \, \cos\left(x\right)^{3} + 3 \, \cos\left(x\right)\right)} \sin\left(x\right) + 3 \, x}{8 \, a^{2}}"," ",0,"1/8*((2*cos(x)^3 + 3*cos(x))*sin(x) + 3*x)/a^2","A",0
280,1,12,0,0.970342," ","integrate(cos(x)^6/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{\cos\left(x\right) \sin\left(x\right) + x}{2 \, a^{2}}"," ",0,"1/2*(cos(x)*sin(x) + x)/a^2","A",0
281,1,5,0,0.953751," ","integrate(cos(x)^4/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{x}{a^{2}}"," ",0,"x/a^2","A",0
282,1,10,0,0.942947," ","integrate(cos(x)^2/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{\sin\left(x\right)}{a^{2} \cos\left(x\right)}"," ",0,"sin(x)/(a^2*cos(x))","A",0
283,1,25,0,0.936450," ","integrate(sec(x)^2/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{{\left(8 \, \cos\left(x\right)^{4} + 4 \, \cos\left(x\right)^{2} + 3\right)} \sin\left(x\right)}{15 \, a^{2} \cos\left(x\right)^{5}}"," ",0,"1/15*(8*cos(x)^4 + 4*cos(x)^2 + 3)*sin(x)/(a^2*cos(x)^5)","A",0
284,1,31,0,0.891604," ","integrate(sec(x)^4/(a-a*sin(x)^2)^2,x, algorithm=""fricas"")","\frac{{\left(16 \, \cos\left(x\right)^{6} + 8 \, \cos\left(x\right)^{4} + 6 \, \cos\left(x\right)^{2} + 5\right)} \sin\left(x\right)}{35 \, a^{2} \cos\left(x\right)^{7}}"," ",0,"1/35*(16*cos(x)^6 + 8*cos(x)^4 + 6*cos(x)^2 + 5)*sin(x)/(a^2*cos(x)^7)","A",0
285,1,78,0,0.898755," ","integrate(cos(f*x+e)^6*(a+b*sin(f*x+e)^2),x, algorithm=""fricas"")","\frac{15 \, {\left(8 \, a + b\right)} f x - {\left(48 \, b \cos\left(f x + e\right)^{7} - 8 \, {\left(8 \, a + b\right)} \cos\left(f x + e\right)^{5} - 10 \, {\left(8 \, a + b\right)} \cos\left(f x + e\right)^{3} - 15 \, {\left(8 \, a + b\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{384 \, f}"," ",0,"1/384*(15*(8*a + b)*f*x - (48*b*cos(f*x + e)^7 - 8*(8*a + b)*cos(f*x + e)^5 - 10*(8*a + b)*cos(f*x + e)^3 - 15*(8*a + b)*cos(f*x + e))*sin(f*x + e))/f","A",0
286,1,63,0,1.009053," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^2),x, algorithm=""fricas"")","\frac{3 \, {\left(6 \, a + b\right)} f x - {\left(8 \, b \cos\left(f x + e\right)^{5} - 2 \, {\left(6 \, a + b\right)} \cos\left(f x + e\right)^{3} - 3 \, {\left(6 \, a + b\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, f}"," ",0,"1/48*(3*(6*a + b)*f*x - (8*b*cos(f*x + e)^5 - 2*(6*a + b)*cos(f*x + e)^3 - 3*(6*a + b)*cos(f*x + e))*sin(f*x + e))/f","A",0
287,1,47,0,1.031233," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(4 \, a + b\right)} f x - {\left(2 \, b \cos\left(f x + e\right)^{3} - {\left(4 \, a + b\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, f}"," ",0,"1/8*((4*a + b)*f*x - (2*b*cos(f*x + e)^3 - (4*a + b)*cos(f*x + e))*sin(f*x + e))/f","A",0
288,1,29,0,1.056753," ","integrate(a+b*sin(f*x+e)^2,x, algorithm=""fricas"")","\frac{{\left(2 \, a + b\right)} f x - b \cos\left(f x + e\right) \sin\left(f x + e\right)}{2 \, f}"," ",0,"1/2*((2*a + b)*f*x - b*cos(f*x + e)*sin(f*x + e))/f","A",0
289,1,35,0,0.981183," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^2),x, algorithm=""fricas"")","-\frac{b f x \cos\left(f x + e\right) - {\left(a + b\right)} \sin\left(f x + e\right)}{f \cos\left(f x + e\right)}"," ",0,"-(b*f*x*cos(f*x + e) - (a + b)*sin(f*x + e))/(f*cos(f*x + e))","A",0
290,1,38,0,0.937591," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left({\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sin\left(f x + e\right)}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/3*((2*a - b)*cos(f*x + e)^2 + a + b)*sin(f*x + e)/(f*cos(f*x + e)^3)","A",0
291,1,59,0,0.713361," ","integrate(sec(f*x+e)^6*(a+b*sin(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(2 \, {\left(4 \, a - b\right)} \cos\left(f x + e\right)^{4} + {\left(4 \, a - b\right)} \cos\left(f x + e\right)^{2} + 3 \, a + 3 \, b\right)} \sin\left(f x + e\right)}{15 \, f \cos\left(f x + e\right)^{5}}"," ",0,"1/15*(2*(4*a - b)*cos(f*x + e)^4 + (4*a - b)*cos(f*x + e)^2 + 3*a + 3*b)*sin(f*x + e)/(f*cos(f*x + e)^5)","A",0
292,1,77,0,0.965760," ","integrate(sec(f*x+e)^8*(a+b*sin(f*x+e)^2),x, algorithm=""fricas"")","\frac{{\left(8 \, {\left(6 \, a - b\right)} \cos\left(f x + e\right)^{6} + 4 \, {\left(6 \, a - b\right)} \cos\left(f x + e\right)^{4} + 3 \, {\left(6 \, a - b\right)} \cos\left(f x + e\right)^{2} + 15 \, a + 15 \, b\right)} \sin\left(f x + e\right)}{105 \, f \cos\left(f x + e\right)^{7}}"," ",0,"1/105*(8*(6*a - b)*cos(f*x + e)^6 + 4*(6*a - b)*cos(f*x + e)^4 + 3*(6*a - b)*cos(f*x + e)^2 + 15*a + 15*b)*sin(f*x + e)/(f*cos(f*x + e)^7)","A",0
293,1,114,0,1.062244," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, {\left(48 \, a^{2} + 16 \, a b + 3 \, b^{2}\right)} f x + {\left(48 \, b^{2} \cos\left(f x + e\right)^{7} - 8 \, {\left(16 \, a b + 9 \, b^{2}\right)} \cos\left(f x + e\right)^{5} + 2 \, {\left(48 \, a^{2} + 16 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(48 \, a^{2} + 16 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{384 \, f}"," ",0,"1/384*(3*(48*a^2 + 16*a*b + 3*b^2)*f*x + (48*b^2*cos(f*x + e)^7 - 8*(16*a*b + 9*b^2)*cos(f*x + e)^5 + 2*(48*a^2 + 16*a*b + 3*b^2)*cos(f*x + e)^3 + 3*(48*a^2 + 16*a*b + 3*b^2)*cos(f*x + e))*sin(f*x + e))/f","A",0
294,1,85,0,0.827964," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, {\left(8 \, a^{2} + 4 \, a b + b^{2}\right)} f x + {\left(8 \, b^{2} \cos\left(f x + e\right)^{5} - 2 \, {\left(12 \, a b + 7 \, b^{2}\right)} \cos\left(f x + e\right)^{3} + 3 \, {\left(8 \, a^{2} + 4 \, a b + b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{48 \, f}"," ",0,"1/48*(3*(8*a^2 + 4*a*b + b^2)*f*x + (8*b^2*cos(f*x + e)^5 - 2*(12*a*b + 7*b^2)*cos(f*x + e)^3 + 3*(8*a^2 + 4*a*b + b^2)*cos(f*x + e))*sin(f*x + e))/f","A",0
295,1,63,0,0.934446," ","integrate((a+b*sin(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} f x + {\left(2 \, b^{2} \cos\left(f x + e\right)^{3} - {\left(8 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}{8 \, f}"," ",0,"1/8*((8*a^2 + 8*a*b + 3*b^2)*f*x + (2*b^2*cos(f*x + e)^3 - (8*a*b + 5*b^2)*cos(f*x + e))*sin(f*x + e))/f","A",0
296,1,68,0,0.937556," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^2)^2,x, algorithm=""fricas"")","-\frac{{\left(4 \, a b + 3 \, b^{2}\right)} f x \cos\left(f x + e\right) - {\left(b^{2} \cos\left(f x + e\right)^{2} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}\right)} \sin\left(f x + e\right)}{2 \, f \cos\left(f x + e\right)}"," ",0,"-1/2*((4*a*b + 3*b^2)*f*x*cos(f*x + e) - (b^2*cos(f*x + e)^2 + 2*a^2 + 4*a*b + 2*b^2)*sin(f*x + e))/(f*cos(f*x + e))","A",0
297,1,70,0,1.056346," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{3 \, b^{2} f x \cos\left(f x + e\right)^{3} + {\left(2 \, {\left(a^{2} - a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sin\left(f x + e\right)}{3 \, f \cos\left(f x + e\right)^{3}}"," ",0,"1/3*(3*b^2*f*x*cos(f*x + e)^3 + (2*(a^2 - a*b - 2*b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sin(f*x + e))/(f*cos(f*x + e)^3)","A",0
298,1,83,0,0.989604," ","integrate(sec(f*x+e)^6*(a+b*sin(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left({\left(8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 2 \, {\left(2 \, a^{2} - a b - 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}\right)} \sin\left(f x + e\right)}{15 \, f \cos\left(f x + e\right)^{5}}"," ",0,"1/15*((8*a^2 - 4*a*b + 3*b^2)*cos(f*x + e)^4 + 2*(2*a^2 - a*b - 3*b^2)*cos(f*x + e)^2 + 3*a^2 + 6*a*b + 3*b^2)*sin(f*x + e)/(f*cos(f*x + e)^5)","A",0
299,1,108,0,0.825058," ","integrate(sec(f*x+e)^8*(a+b*sin(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(2 \, {\left(24 \, a^{2} - 8 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{6} + {\left(24 \, a^{2} - 8 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} + 6 \, {\left(3 \, a^{2} - a b - 4 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 15 \, a^{2} + 30 \, a b + 15 \, b^{2}\right)} \sin\left(f x + e\right)}{105 \, f \cos\left(f x + e\right)^{7}}"," ",0,"1/105*(2*(24*a^2 - 8*a*b + 3*b^2)*cos(f*x + e)^6 + (24*a^2 - 8*a*b + 3*b^2)*cos(f*x + e)^4 + 6*(3*a^2 - a*b - 4*b^2)*cos(f*x + e)^2 + 15*a^2 + 30*a*b + 15*b^2)*sin(f*x + e)/(f*cos(f*x + e)^7)","A",0
300,1,128,0,0.682205," ","integrate(sec(f*x+e)^10*(a+b*sin(f*x+e)^2)^2,x, algorithm=""fricas"")","\frac{{\left(8 \, {\left(16 \, a^{2} - 4 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{8} + 4 \, {\left(16 \, a^{2} - 4 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{6} + 3 \, {\left(16 \, a^{2} - 4 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + 10 \, {\left(4 \, a^{2} - a b - 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 35 \, a^{2} + 70 \, a b + 35 \, b^{2}\right)} \sin\left(f x + e\right)}{315 \, f \cos\left(f x + e\right)^{9}}"," ",0,"1/315*(8*(16*a^2 - 4*a*b + b^2)*cos(f*x + e)^8 + 4*(16*a^2 - 4*a*b + b^2)*cos(f*x + e)^6 + 3*(16*a^2 - 4*a*b + b^2)*cos(f*x + e)^4 + 10*(4*a^2 - a*b - 5*b^2)*cos(f*x + e)^2 + 35*a^2 + 70*a*b + 35*b^2)*sin(f*x + e)/(f*cos(f*x + e)^9)","A",0
301,1,233,0,0.911595," ","integrate(cos(x)^7/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \sqrt{-a b} \log\left(-\frac{b \cos\left(x\right)^{2} + 2 \, \sqrt{-a b} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right) + 2 \, {\left(3 \, a b^{3} \cos\left(x\right)^{4} + 15 \, a^{3} b + 40 \, a^{2} b^{2} + 33 \, a b^{3} + {\left(5 \, a^{2} b^{2} + 9 \, a b^{3}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{30 \, a b^{4}}, \frac{15 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \sqrt{a b} \arctan\left(\frac{\sqrt{a b} \sin\left(x\right)}{a}\right) - {\left(3 \, a b^{3} \cos\left(x\right)^{4} + 15 \, a^{3} b + 40 \, a^{2} b^{2} + 33 \, a b^{3} + {\left(5 \, a^{2} b^{2} + 9 \, a b^{3}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{15 \, a b^{4}}\right]"," ",0,"[-1/30*(15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(-a*b)*log(-(b*cos(x)^2 + 2*sqrt(-a*b)*sin(x) + a - b)/(b*cos(x)^2 - a - b)) + 2*(3*a*b^3*cos(x)^4 + 15*a^3*b + 40*a^2*b^2 + 33*a*b^3 + (5*a^2*b^2 + 9*a*b^3)*cos(x)^2)*sin(x))/(a*b^4), 1/15*(15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(a*b)*arctan(sqrt(a*b)*sin(x)/a) - (3*a*b^3*cos(x)^4 + 15*a^3*b + 40*a^2*b^2 + 33*a*b^3 + (5*a^2*b^2 + 9*a*b^3)*cos(x)^2)*sin(x))/(a*b^4)]","A",0
302,1,312,0,0.972319," ","integrate(cos(x)^6/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-\frac{a + b}{a}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(x\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(x\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + a b\right)} \cos\left(x\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(x\right)\right)} \sqrt{-\frac{a + b}{a}} \sin\left(x\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(x\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - {\left(8 \, a^{2} + 20 \, a b + 15 \, b^{2}\right)} x - {\left(2 \, b^{2} \cos\left(x\right)^{3} + {\left(4 \, a b + 7 \, b^{2}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{8 \, b^{3}}, -\frac{4 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{\frac{a + b}{a}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(x\right)^{2} - a - b\right)} \sqrt{\frac{a + b}{a}}}{2 \, {\left(a + b\right)} \cos\left(x\right) \sin\left(x\right)}\right) + {\left(8 \, a^{2} + 20 \, a b + 15 \, b^{2}\right)} x + {\left(2 \, b^{2} \cos\left(x\right)^{3} + {\left(4 \, a b + 7 \, b^{2}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{8 \, b^{3}}\right]"," ",0,"[1/8*(2*(a^2 + 2*a*b + b^2)*sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 - 4*((2*a^2 + a*b)*cos(x)^3 - (a^2 + a*b)*cos(x))*sqrt(-(a + b)/a)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)) - (8*a^2 + 20*a*b + 15*b^2)*x - (2*b^2*cos(x)^3 + (4*a*b + 7*b^2)*cos(x))*sin(x))/b^3, -1/8*(4*(a^2 + 2*a*b + b^2)*sqrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(x)*sin(x))) + (8*a^2 + 20*a*b + 15*b^2)*x + (2*b^2*cos(x)^3 + (4*a*b + 7*b^2)*cos(x))*sin(x))/b^3]","A",0
303,1,159,0,1.011987," ","integrate(cos(x)^5/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a b} \log\left(-\frac{b \cos\left(x\right)^{2} + 2 \, \sqrt{-a b} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right) + 2 \, {\left(a b^{2} \cos\left(x\right)^{2} + 3 \, a^{2} b + 5 \, a b^{2}\right)} \sin\left(x\right)}{6 \, a b^{3}}, \frac{3 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a b} \arctan\left(\frac{\sqrt{a b} \sin\left(x\right)}{a}\right) - {\left(a b^{2} \cos\left(x\right)^{2} + 3 \, a^{2} b + 5 \, a b^{2}\right)} \sin\left(x\right)}{3 \, a b^{3}}\right]"," ",0,"[-1/6*(3*(a^2 + 2*a*b + b^2)*sqrt(-a*b)*log(-(b*cos(x)^2 + 2*sqrt(-a*b)*sin(x) + a - b)/(b*cos(x)^2 - a - b)) + 2*(a*b^2*cos(x)^2 + 3*a^2*b + 5*a*b^2)*sin(x))/(a*b^3), 1/3*(3*(a^2 + 2*a*b + b^2)*sqrt(a*b)*arctan(sqrt(a*b)*sin(x)/a) - (a*b^2*cos(x)^2 + 3*a^2*b + 5*a*b^2)*sin(x))/(a*b^3)]","A",0
304,1,239,0,1.214980," ","integrate(cos(x)^4/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[-\frac{2 \, b \cos\left(x\right) \sin\left(x\right) - {\left(a + b\right)} \sqrt{-\frac{a + b}{a}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(x\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(x\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + a b\right)} \cos\left(x\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(x\right)\right)} \sqrt{-\frac{a + b}{a}} \sin\left(x\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(x\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) + 2 \, {\left(2 \, a + 3 \, b\right)} x}{4 \, b^{2}}, -\frac{b \cos\left(x\right) \sin\left(x\right) + {\left(a + b\right)} \sqrt{\frac{a + b}{a}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(x\right)^{2} - a - b\right)} \sqrt{\frac{a + b}{a}}}{2 \, {\left(a + b\right)} \cos\left(x\right) \sin\left(x\right)}\right) + {\left(2 \, a + 3 \, b\right)} x}{2 \, b^{2}}\right]"," ",0,"[-1/4*(2*b*cos(x)*sin(x) - (a + b)*sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 - 4*((2*a^2 + a*b)*cos(x)^3 - (a^2 + a*b)*cos(x))*sqrt(-(a + b)/a)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)) + 2*(2*a + 3*b)*x)/b^2, -1/2*(b*cos(x)*sin(x) + (a + b)*sqrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(x)*sin(x))) + (2*a + 3*b)*x)/b^2]","A",0
305,1,101,0,1.132518," ","integrate(cos(x)^3/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[-\frac{2 \, a b \sin\left(x\right) + \sqrt{-a b} {\left(a + b\right)} \log\left(-\frac{b \cos\left(x\right)^{2} + 2 \, \sqrt{-a b} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right)}{2 \, a b^{2}}, -\frac{a b \sin\left(x\right) - \sqrt{a b} {\left(a + b\right)} \arctan\left(\frac{\sqrt{a b} \sin\left(x\right)}{a}\right)}{a b^{2}}\right]"," ",0,"[-1/2*(2*a*b*sin(x) + sqrt(-a*b)*(a + b)*log(-(b*cos(x)^2 + 2*sqrt(-a*b)*sin(x) + a - b)/(b*cos(x)^2 - a - b)))/(a*b^2), -(a*b*sin(x) - sqrt(a*b)*(a + b)*arctan(sqrt(a*b)*sin(x)/a))/(a*b^2)]","A",0
306,1,206,0,0.986398," ","integrate(cos(x)^2/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-\frac{a + b}{a}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(x\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(x\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + a b\right)} \cos\left(x\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(x\right)\right)} \sqrt{-\frac{a + b}{a}} \sin\left(x\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(x\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, x}{4 \, b}, -\frac{\sqrt{\frac{a + b}{a}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(x\right)^{2} - a - b\right)} \sqrt{\frac{a + b}{a}}}{2 \, {\left(a + b\right)} \cos\left(x\right) \sin\left(x\right)}\right) + 2 \, x}{2 \, b}\right]"," ",0,"[1/4*(sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 - 4*((2*a^2 + a*b)*cos(x)^3 - (a^2 + a*b)*cos(x))*sqrt(-(a + b)/a)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)) - 4*x)/b, -1/2*(sqrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(x)*sin(x))) + 2*x)/b]","A",0
307,1,78,0,1.107719," ","integrate(cos(x)/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a b} \log\left(-\frac{b \cos\left(x\right)^{2} + 2 \, \sqrt{-a b} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right)}{2 \, a b}, \frac{\sqrt{a b} \arctan\left(\frac{\sqrt{a b} \sin\left(x\right)}{a}\right)}{a b}\right]"," ",0,"[-1/2*sqrt(-a*b)*log(-(b*cos(x)^2 + 2*sqrt(-a*b)*sin(x) + a - b)/(b*cos(x)^2 - a - b))/(a*b), sqrt(a*b)*arctan(sqrt(a*b)*sin(x)/a)/(a*b)]","A",0
308,1,116,0,1.032789," ","integrate(sec(x)/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-\frac{b}{a}} \log\left(-\frac{b \cos\left(x\right)^{2} - 2 \, a \sqrt{-\frac{b}{a}} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right) + \log\left(\sin\left(x\right) + 1\right) - \log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a + b\right)}}, \frac{2 \, \sqrt{\frac{b}{a}} \arctan\left(\sqrt{\frac{b}{a}} \sin\left(x\right)\right) + \log\left(\sin\left(x\right) + 1\right) - \log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a + b\right)}}\right]"," ",0,"[1/2*(sqrt(-b/a)*log(-(b*cos(x)^2 - 2*a*sqrt(-b/a)*sin(x) + a - b)/(b*cos(x)^2 - a - b)) + log(sin(x) + 1) - log(-sin(x) + 1))/(a + b), 1/2*(2*sqrt(b/a)*arctan(sqrt(b/a)*sin(x)) + log(sin(x) + 1) - log(-sin(x) + 1))/(a + b)]","A",0
309,1,255,0,0.886781," ","integrate(sec(x)^2/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a^{2} - a b} b \cos\left(x\right) \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(x\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(x\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(x\right)^{3} - {\left(a + b\right)} \cos\left(x\right)\right)} \sqrt{-a^{2} - a b} \sin\left(x\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(x\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, {\left(a^{2} + a b\right)} \sin\left(x\right)}{4 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(x\right)}, -\frac{\sqrt{a^{2} + a b} b \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(x\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(x\right) \sin\left(x\right)}\right) \cos\left(x\right) - 2 \, {\left(a^{2} + a b\right)} \sin\left(x\right)}{2 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(x\right)}\right]"," ",0,"[-1/4*(sqrt(-a^2 - a*b)*b*cos(x)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 + 4*((2*a + b)*cos(x)^3 - (a + b)*cos(x))*sqrt(-a^2 - a*b)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)) - 4*(a^2 + a*b)*sin(x))/((a^3 + 2*a^2*b + a*b^2)*cos(x)), -1/2*(sqrt(a^2 + a*b)*b*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)/(sqrt(a^2 + a*b)*cos(x)*sin(x)))*cos(x) - 2*(a^2 + a*b)*sin(x))/((a^3 + 2*a^2*b + a*b^2)*cos(x))]","B",0
310,1,203,0,1.217751," ","integrate(sec(x)^3/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[\frac{2 \, b \sqrt{-\frac{b}{a}} \cos\left(x\right)^{2} \log\left(-\frac{b \cos\left(x\right)^{2} - 2 \, a \sqrt{-\frac{b}{a}} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right) + {\left(a + 3 \, b\right)} \cos\left(x\right)^{2} \log\left(\sin\left(x\right) + 1\right) - {\left(a + 3 \, b\right)} \cos\left(x\right)^{2} \log\left(-\sin\left(x\right) + 1\right) + 2 \, {\left(a + b\right)} \sin\left(x\right)}{4 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(x\right)^{2}}, \frac{4 \, b \sqrt{\frac{b}{a}} \arctan\left(\sqrt{\frac{b}{a}} \sin\left(x\right)\right) \cos\left(x\right)^{2} + {\left(a + 3 \, b\right)} \cos\left(x\right)^{2} \log\left(\sin\left(x\right) + 1\right) - {\left(a + 3 \, b\right)} \cos\left(x\right)^{2} \log\left(-\sin\left(x\right) + 1\right) + 2 \, {\left(a + b\right)} \sin\left(x\right)}{4 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(x\right)^{2}}\right]"," ",0,"[1/4*(2*b*sqrt(-b/a)*cos(x)^2*log(-(b*cos(x)^2 - 2*a*sqrt(-b/a)*sin(x) + a - b)/(b*cos(x)^2 - a - b)) + (a + 3*b)*cos(x)^2*log(sin(x) + 1) - (a + 3*b)*cos(x)^2*log(-sin(x) + 1) + 2*(a + b)*sin(x))/((a^2 + 2*a*b + b^2)*cos(x)^2), 1/4*(4*b*sqrt(b/a)*arctan(sqrt(b/a)*sin(x))*cos(x)^2 + (a + 3*b)*cos(x)^2*log(sin(x) + 1) - (a + 3*b)*cos(x)^2*log(-sin(x) + 1) + 2*(a + b)*sin(x))/((a^2 + 2*a*b + b^2)*cos(x)^2)]","A",0
311,1,343,0,1.172334," ","integrate(sec(x)^4/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[-\frac{3 \, \sqrt{-a^{2} - a b} b^{2} \cos\left(x\right)^{3} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(x\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(x\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(x\right)^{3} - {\left(a + b\right)} \cos\left(x\right)\right)} \sqrt{-a^{2} - a b} \sin\left(x\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(x\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2} + {\left(2 \, a^{3} + 7 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{12 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(x\right)^{3}}, -\frac{3 \, \sqrt{a^{2} + a b} b^{2} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(x\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(x\right) \sin\left(x\right)}\right) \cos\left(x\right)^{3} - 2 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2} + {\left(2 \, a^{3} + 7 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{6 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(x\right)^{3}}\right]"," ",0,"[-1/12*(3*sqrt(-a^2 - a*b)*b^2*cos(x)^3*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 + 4*((2*a + b)*cos(x)^3 - (a + b)*cos(x))*sqrt(-a^2 - a*b)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)) - 4*(a^3 + 2*a^2*b + a*b^2 + (2*a^3 + 7*a^2*b + 5*a*b^2)*cos(x)^2)*sin(x))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(x)^3), -1/6*(3*sqrt(a^2 + a*b)*b^2*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)/(sqrt(a^2 + a*b)*cos(x)*sin(x)))*cos(x)^3 - 2*(a^3 + 2*a^2*b + a*b^2 + (2*a^3 + 7*a^2*b + 5*a*b^2)*cos(x)^2)*sin(x))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(x)^3)]","B",0
312,1,327,0,1.170175," ","integrate(sec(x)^5/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[\frac{8 \, b^{2} \sqrt{-\frac{b}{a}} \cos\left(x\right)^{4} \log\left(-\frac{b \cos\left(x\right)^{2} - 2 \, a \sqrt{-\frac{b}{a}} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right) + {\left(3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right)} \cos\left(x\right)^{4} \log\left(\sin\left(x\right) + 1\right) - {\left(3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right)} \cos\left(x\right)^{4} \log\left(-\sin\left(x\right) + 1\right) + 2 \, {\left({\left(3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}\right)} \sin\left(x\right)}{16 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(x\right)^{4}}, \frac{16 \, b^{2} \sqrt{\frac{b}{a}} \arctan\left(\sqrt{\frac{b}{a}} \sin\left(x\right)\right) \cos\left(x\right)^{4} + {\left(3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right)} \cos\left(x\right)^{4} \log\left(\sin\left(x\right) + 1\right) - {\left(3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right)} \cos\left(x\right)^{4} \log\left(-\sin\left(x\right) + 1\right) + 2 \, {\left({\left(3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}\right)} \sin\left(x\right)}{16 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(x\right)^{4}}\right]"," ",0,"[1/16*(8*b^2*sqrt(-b/a)*cos(x)^4*log(-(b*cos(x)^2 - 2*a*sqrt(-b/a)*sin(x) + a - b)/(b*cos(x)^2 - a - b)) + (3*a^2 + 10*a*b + 15*b^2)*cos(x)^4*log(sin(x) + 1) - (3*a^2 + 10*a*b + 15*b^2)*cos(x)^4*log(-sin(x) + 1) + 2*((3*a^2 + 10*a*b + 7*b^2)*cos(x)^2 + 2*a^2 + 4*a*b + 2*b^2)*sin(x))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(x)^4), 1/16*(16*b^2*sqrt(b/a)*arctan(sqrt(b/a)*sin(x))*cos(x)^4 + (3*a^2 + 10*a*b + 15*b^2)*cos(x)^4*log(sin(x) + 1) - (3*a^2 + 10*a*b + 15*b^2)*cos(x)^4*log(-sin(x) + 1) + 2*((3*a^2 + 10*a*b + 7*b^2)*cos(x)^2 + 2*a^2 + 4*a*b + 2*b^2)*sin(x))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(x)^4)]","A",0
313,1,459,0,0.808898," ","integrate(sec(x)^6/(a+b*sin(x)^2),x, algorithm=""fricas"")","\left[-\frac{15 \, \sqrt{-a^{2} - a b} b^{3} \cos\left(x\right)^{5} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(x\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(x\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(x\right)^{3} - {\left(a + b\right)} \cos\left(x\right)\right)} \sqrt{-a^{2} - a b} \sin\left(x\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(x\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, {\left({\left(8 \, a^{4} + 34 \, a^{3} b + 59 \, a^{2} b^{2} + 33 \, a b^{3}\right)} \cos\left(x\right)^{4} + 3 \, a^{4} + 9 \, a^{3} b + 9 \, a^{2} b^{2} + 3 \, a b^{3} + {\left(4 \, a^{4} + 17 \, a^{3} b + 22 \, a^{2} b^{2} + 9 \, a b^{3}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{60 \, {\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(x\right)^{5}}, -\frac{15 \, \sqrt{a^{2} + a b} b^{3} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(x\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(x\right) \sin\left(x\right)}\right) \cos\left(x\right)^{5} - 2 \, {\left({\left(8 \, a^{4} + 34 \, a^{3} b + 59 \, a^{2} b^{2} + 33 \, a b^{3}\right)} \cos\left(x\right)^{4} + 3 \, a^{4} + 9 \, a^{3} b + 9 \, a^{2} b^{2} + 3 \, a b^{3} + {\left(4 \, a^{4} + 17 \, a^{3} b + 22 \, a^{2} b^{2} + 9 \, a b^{3}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{30 \, {\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(x\right)^{5}}\right]"," ",0,"[-1/60*(15*sqrt(-a^2 - a*b)*b^3*cos(x)^5*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 + 4*((2*a + b)*cos(x)^3 - (a + b)*cos(x))*sqrt(-a^2 - a*b)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)) - 4*((8*a^4 + 34*a^3*b + 59*a^2*b^2 + 33*a*b^3)*cos(x)^4 + 3*a^4 + 9*a^3*b + 9*a^2*b^2 + 3*a*b^3 + (4*a^4 + 17*a^3*b + 22*a^2*b^2 + 9*a*b^3)*cos(x)^2)*sin(x))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(x)^5), -1/30*(15*sqrt(a^2 + a*b)*b^3*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)/(sqrt(a^2 + a*b)*cos(x)*sin(x)))*cos(x)^5 - 2*((8*a^4 + 34*a^3*b + 59*a^2*b^2 + 33*a*b^3)*cos(x)^4 + 3*a^4 + 9*a^3*b + 9*a^2*b^2 + 3*a*b^3 + (4*a^4 + 17*a^3*b + 22*a^2*b^2 + 9*a*b^3)*cos(x)^2)*sin(x))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(x)^5)]","B",0
314,1,491,0,1.074654," ","integrate(cos(x)^6/(a+b*sin(x)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(4 \, a^{2} b + 5 \, a b^{2}\right)} x \cos\left(x\right)^{2} + {\left(4 \, a^{3} + 7 \, a^{2} b + 2 \, a b^{2} - b^{3} - {\left(4 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(x\right)^{2}\right)} \sqrt{-\frac{a + b}{a}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(x\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(x\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + a b\right)} \cos\left(x\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(x\right)\right)} \sqrt{-\frac{a + b}{a}} \sin\left(x\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(x\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, {\left(4 \, a^{3} + 9 \, a^{2} b + 5 \, a b^{2}\right)} x + 4 \, {\left(a b^{2} \cos\left(x\right)^{3} - {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{8 \, {\left(a b^{4} \cos\left(x\right)^{2} - a^{2} b^{3} - a b^{4}\right)}}, \frac{2 \, {\left(4 \, a^{2} b + 5 \, a b^{2}\right)} x \cos\left(x\right)^{2} - {\left(4 \, a^{3} + 7 \, a^{2} b + 2 \, a b^{2} - b^{3} - {\left(4 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(x\right)^{2}\right)} \sqrt{\frac{a + b}{a}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(x\right)^{2} - a - b\right)} \sqrt{\frac{a + b}{a}}}{2 \, {\left(a + b\right)} \cos\left(x\right) \sin\left(x\right)}\right) - 2 \, {\left(4 \, a^{3} + 9 \, a^{2} b + 5 \, a b^{2}\right)} x + 2 \, {\left(a b^{2} \cos\left(x\right)^{3} - {\left(2 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{4 \, {\left(a b^{4} \cos\left(x\right)^{2} - a^{2} b^{3} - a b^{4}\right)}}\right]"," ",0,"[1/8*(4*(4*a^2*b + 5*a*b^2)*x*cos(x)^2 + (4*a^3 + 7*a^2*b + 2*a*b^2 - b^3 - (4*a^2*b + 3*a*b^2 - b^3)*cos(x)^2)*sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 - 4*((2*a^2 + a*b)*cos(x)^3 - (a^2 + a*b)*cos(x))*sqrt(-(a + b)/a)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)) - 4*(4*a^3 + 9*a^2*b + 5*a*b^2)*x + 4*(a*b^2*cos(x)^3 - (2*a^2*b + 3*a*b^2 + b^3)*cos(x))*sin(x))/(a*b^4*cos(x)^2 - a^2*b^3 - a*b^4), 1/4*(2*(4*a^2*b + 5*a*b^2)*x*cos(x)^2 - (4*a^3 + 7*a^2*b + 2*a*b^2 - b^3 - (4*a^2*b + 3*a*b^2 - b^3)*cos(x)^2)*sqrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(x)*sin(x))) - 2*(4*a^3 + 9*a^2*b + 5*a*b^2)*x + 2*(a*b^2*cos(x)^3 - (2*a^2*b + 3*a*b^2 + b^3)*cos(x))*sin(x))/(a*b^4*cos(x)^2 - a^2*b^3 - a*b^4)]","B",0
315,1,296,0,0.730468," ","integrate(cos(x)^5/(a+b*sin(x)^2)^2,x, algorithm=""fricas"")","\left[-\frac{{\left(3 \, a^{3} + 5 \, a^{2} b + a b^{2} - b^{3} - {\left(3 \, a^{2} b + 2 \, a b^{2} - b^{3}\right)} \cos\left(x\right)^{2}\right)} \sqrt{-a b} \log\left(-\frac{b \cos\left(x\right)^{2} + 2 \, \sqrt{-a b} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right) - 2 \, {\left(2 \, a^{2} b^{2} \cos\left(x\right)^{2} - 3 \, a^{3} b - 4 \, a^{2} b^{2} - a b^{3}\right)} \sin\left(x\right)}{4 \, {\left(a^{2} b^{4} \cos\left(x\right)^{2} - a^{3} b^{3} - a^{2} b^{4}\right)}}, \frac{{\left(3 \, a^{3} + 5 \, a^{2} b + a b^{2} - b^{3} - {\left(3 \, a^{2} b + 2 \, a b^{2} - b^{3}\right)} \cos\left(x\right)^{2}\right)} \sqrt{a b} \arctan\left(\frac{\sqrt{a b} \sin\left(x\right)}{a}\right) + {\left(2 \, a^{2} b^{2} \cos\left(x\right)^{2} - 3 \, a^{3} b - 4 \, a^{2} b^{2} - a b^{3}\right)} \sin\left(x\right)}{2 \, {\left(a^{2} b^{4} \cos\left(x\right)^{2} - a^{3} b^{3} - a^{2} b^{4}\right)}}\right]"," ",0,"[-1/4*((3*a^3 + 5*a^2*b + a*b^2 - b^3 - (3*a^2*b + 2*a*b^2 - b^3)*cos(x)^2)*sqrt(-a*b)*log(-(b*cos(x)^2 + 2*sqrt(-a*b)*sin(x) + a - b)/(b*cos(x)^2 - a - b)) - 2*(2*a^2*b^2*cos(x)^2 - 3*a^3*b - 4*a^2*b^2 - a*b^3)*sin(x))/(a^2*b^4*cos(x)^2 - a^3*b^3 - a^2*b^4), 1/2*((3*a^3 + 5*a^2*b + a*b^2 - b^3 - (3*a^2*b + 2*a*b^2 - b^3)*cos(x)^2)*sqrt(a*b)*arctan(sqrt(a*b)*sin(x)/a) + (2*a^2*b^2*cos(x)^2 - 3*a^3*b - 4*a^2*b^2 - a*b^3)*sin(x))/(a^2*b^4*cos(x)^2 - a^3*b^3 - a^2*b^4)]","B",0
316,1,367,0,0.820426," ","integrate(cos(x)^4/(a+b*sin(x)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, a b x \cos\left(x\right)^{2} - 4 \, {\left(a b + b^{2}\right)} \cos\left(x\right) \sin\left(x\right) - {\left({\left(2 \, a b - b^{2}\right)} \cos\left(x\right)^{2} - 2 \, a^{2} - a b + b^{2}\right)} \sqrt{-\frac{a + b}{a}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(x\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(x\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + a b\right)} \cos\left(x\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(x\right)\right)} \sqrt{-\frac{a + b}{a}} \sin\left(x\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(x\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 8 \, {\left(a^{2} + a b\right)} x}{8 \, {\left(a b^{3} \cos\left(x\right)^{2} - a^{2} b^{2} - a b^{3}\right)}}, \frac{4 \, a b x \cos\left(x\right)^{2} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right) \sin\left(x\right) + {\left({\left(2 \, a b - b^{2}\right)} \cos\left(x\right)^{2} - 2 \, a^{2} - a b + b^{2}\right)} \sqrt{\frac{a + b}{a}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(x\right)^{2} - a - b\right)} \sqrt{\frac{a + b}{a}}}{2 \, {\left(a + b\right)} \cos\left(x\right) \sin\left(x\right)}\right) - 4 \, {\left(a^{2} + a b\right)} x}{4 \, {\left(a b^{3} \cos\left(x\right)^{2} - a^{2} b^{2} - a b^{3}\right)}}\right]"," ",0,"[1/8*(8*a*b*x*cos(x)^2 - 4*(a*b + b^2)*cos(x)*sin(x) - ((2*a*b - b^2)*cos(x)^2 - 2*a^2 - a*b + b^2)*sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 - 4*((2*a^2 + a*b)*cos(x)^3 - (a^2 + a*b)*cos(x))*sqrt(-(a + b)/a)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)) - 8*(a^2 + a*b)*x)/(a*b^3*cos(x)^2 - a^2*b^2 - a*b^3), 1/4*(4*a*b*x*cos(x)^2 - 2*(a*b + b^2)*cos(x)*sin(x) + ((2*a*b - b^2)*cos(x)^2 - 2*a^2 - a*b + b^2)*sqrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(x)*sin(x))) - 4*(a^2 + a*b)*x)/(a*b^3*cos(x)^2 - a^2*b^2 - a*b^3)]","B",0
317,1,206,0,0.708689," ","integrate(cos(x)^3/(a+b*sin(x)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left({\left(a b - b^{2}\right)} \cos\left(x\right)^{2} - a^{2} + b^{2}\right)} \sqrt{-a b} \log\left(-\frac{b \cos\left(x\right)^{2} + 2 \, \sqrt{-a b} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right) - 2 \, {\left(a^{2} b + a b^{2}\right)} \sin\left(x\right)}{4 \, {\left(a^{2} b^{3} \cos\left(x\right)^{2} - a^{3} b^{2} - a^{2} b^{3}\right)}}, -\frac{{\left({\left(a b - b^{2}\right)} \cos\left(x\right)^{2} - a^{2} + b^{2}\right)} \sqrt{a b} \arctan\left(\frac{\sqrt{a b} \sin\left(x\right)}{a}\right) + {\left(a^{2} b + a b^{2}\right)} \sin\left(x\right)}{2 \, {\left(a^{2} b^{3} \cos\left(x\right)^{2} - a^{3} b^{2} - a^{2} b^{3}\right)}}\right]"," ",0,"[1/4*(((a*b - b^2)*cos(x)^2 - a^2 + b^2)*sqrt(-a*b)*log(-(b*cos(x)^2 + 2*sqrt(-a*b)*sin(x) + a - b)/(b*cos(x)^2 - a - b)) - 2*(a^2*b + a*b^2)*sin(x))/(a^2*b^3*cos(x)^2 - a^3*b^2 - a^2*b^3), -1/2*(((a*b - b^2)*cos(x)^2 - a^2 + b^2)*sqrt(a*b)*arctan(sqrt(a*b)*sin(x)/a) + (a^2*b + a*b^2)*sin(x))/(a^2*b^3*cos(x)^2 - a^3*b^2 - a^2*b^3)]","A",0
318,1,313,0,1.050626," ","integrate(cos(x)^2/(a+b*sin(x)^2)^2,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} + a b\right)} \cos\left(x\right) \sin\left(x\right) + {\left(b \cos\left(x\right)^{2} - a - b\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(x\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(x\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(x\right)^{3} - {\left(a + b\right)} \cos\left(x\right)\right)} \sqrt{-a^{2} - a b} \sin\left(x\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(x\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right)}{8 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2} - {\left(a^{3} b + a^{2} b^{2}\right)} \cos\left(x\right)^{2}\right)}}, \frac{2 \, {\left(a^{2} + a b\right)} \cos\left(x\right) \sin\left(x\right) + {\left(b \cos\left(x\right)^{2} - a - b\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(x\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(x\right) \sin\left(x\right)}\right)}{4 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2} - {\left(a^{3} b + a^{2} b^{2}\right)} \cos\left(x\right)^{2}\right)}}\right]"," ",0,"[1/8*(4*(a^2 + a*b)*cos(x)*sin(x) + (b*cos(x)^2 - a - b)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 + 4*((2*a + b)*cos(x)^3 - (a + b)*cos(x))*sqrt(-a^2 - a*b)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)))/(a^4 + 2*a^3*b + a^2*b^2 - (a^3*b + a^2*b^2)*cos(x)^2), 1/4*(2*(a^2 + a*b)*cos(x)*sin(x) + (b*cos(x)^2 - a - b)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)/(sqrt(a^2 + a*b)*cos(x)*sin(x))))/(a^4 + 2*a^3*b + a^2*b^2 - (a^3*b + a^2*b^2)*cos(x)^2)]","B",0
319,1,165,0,0.616132," ","integrate(cos(x)/(a+b*sin(x)^2)^2,x, algorithm=""fricas"")","\left[-\frac{2 \, a b \sin\left(x\right) + {\left(b \cos\left(x\right)^{2} - a - b\right)} \sqrt{-a b} \log\left(-\frac{b \cos\left(x\right)^{2} + 2 \, \sqrt{-a b} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right)}{4 \, {\left(a^{2} b^{2} \cos\left(x\right)^{2} - a^{3} b - a^{2} b^{2}\right)}}, -\frac{a b \sin\left(x\right) - {\left(b \cos\left(x\right)^{2} - a - b\right)} \sqrt{a b} \arctan\left(\frac{\sqrt{a b} \sin\left(x\right)}{a}\right)}{2 \, {\left(a^{2} b^{2} \cos\left(x\right)^{2} - a^{3} b - a^{2} b^{2}\right)}}\right]"," ",0,"[-1/4*(2*a*b*sin(x) + (b*cos(x)^2 - a - b)*sqrt(-a*b)*log(-(b*cos(x)^2 + 2*sqrt(-a*b)*sin(x) + a - b)/(b*cos(x)^2 - a - b)))/(a^2*b^2*cos(x)^2 - a^3*b - a^2*b^2), -1/2*(a*b*sin(x) - (b*cos(x)^2 - a - b)*sqrt(a*b)*arctan(sqrt(a*b)*sin(x)/a))/(a^2*b^2*cos(x)^2 - a^3*b - a^2*b^2)]","A",0
320,1,354,0,1.021291," ","integrate(sec(x)/(a+b*sin(x)^2)^2,x, algorithm=""fricas"")","\left[-\frac{{\left({\left(3 \, a b + b^{2}\right)} \cos\left(x\right)^{2} - 3 \, a^{2} - 4 \, a b - b^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{b \cos\left(x\right)^{2} - 2 \, a \sqrt{-\frac{b}{a}} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right) + 2 \, {\left(a b \cos\left(x\right)^{2} - a^{2} - a b\right)} \log\left(\sin\left(x\right) + 1\right) - 2 \, {\left(a b \cos\left(x\right)^{2} - a^{2} - a b\right)} \log\left(-\sin\left(x\right) + 1\right) - 2 \, {\left(a b + b^{2}\right)} \sin\left(x\right)}{4 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} - {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(x\right)^{2}\right)}}, -\frac{{\left({\left(3 \, a b + b^{2}\right)} \cos\left(x\right)^{2} - 3 \, a^{2} - 4 \, a b - b^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\sqrt{\frac{b}{a}} \sin\left(x\right)\right) + {\left(a b \cos\left(x\right)^{2} - a^{2} - a b\right)} \log\left(\sin\left(x\right) + 1\right) - {\left(a b \cos\left(x\right)^{2} - a^{2} - a b\right)} \log\left(-\sin\left(x\right) + 1\right) - {\left(a b + b^{2}\right)} \sin\left(x\right)}{2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} - {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(x\right)^{2}\right)}}\right]"," ",0,"[-1/4*(((3*a*b + b^2)*cos(x)^2 - 3*a^2 - 4*a*b - b^2)*sqrt(-b/a)*log(-(b*cos(x)^2 - 2*a*sqrt(-b/a)*sin(x) + a - b)/(b*cos(x)^2 - a - b)) + 2*(a*b*cos(x)^2 - a^2 - a*b)*log(sin(x) + 1) - 2*(a*b*cos(x)^2 - a^2 - a*b)*log(-sin(x) + 1) - 2*(a*b + b^2)*sin(x))/(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 - (a^3*b + 2*a^2*b^2 + a*b^3)*cos(x)^2), -1/2*(((3*a*b + b^2)*cos(x)^2 - 3*a^2 - 4*a*b - b^2)*sqrt(b/a)*arctan(sqrt(b/a)*sin(x)) + (a*b*cos(x)^2 - a^2 - a*b)*log(sin(x) + 1) - (a*b*cos(x)^2 - a^2 - a*b)*log(-sin(x) + 1) - (a*b + b^2)*sin(x))/(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 - (a^3*b + 2*a^2*b^2 + a*b^3)*cos(x)^2)]","B",0
321,1,505,0,1.031100," ","integrate(sec(x)^2/(a+b*sin(x)^2)^2,x, algorithm=""fricas"")","\left[-\frac{{\left({\left(4 \, a b^{2} + b^{3}\right)} \cos\left(x\right)^{3} - {\left(4 \, a^{2} b + 5 \, a b^{2} + b^{3}\right)} \cos\left(x\right)\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(x\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(x\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(x\right)^{3} - {\left(a + b\right)} \cos\left(x\right)\right)} \sqrt{-a^{2} - a b} \sin\left(x\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(x\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) + 4 \, {\left(2 \, a^{4} + 4 \, a^{3} b + 2 \, a^{2} b^{2} - {\left(2 \, a^{3} b + a^{2} b^{2} - a b^{3}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{8 \, {\left({\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(x\right)^{3} - {\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(x\right)\right)}}, -\frac{{\left({\left(4 \, a b^{2} + b^{3}\right)} \cos\left(x\right)^{3} - {\left(4 \, a^{2} b + 5 \, a b^{2} + b^{3}\right)} \cos\left(x\right)\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(x\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(x\right) \sin\left(x\right)}\right) + 2 \, {\left(2 \, a^{4} + 4 \, a^{3} b + 2 \, a^{2} b^{2} - {\left(2 \, a^{3} b + a^{2} b^{2} - a b^{3}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{4 \, {\left({\left(a^{5} b + 3 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(x\right)^{3} - {\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(x\right)\right)}}\right]"," ",0,"[-1/8*(((4*a*b^2 + b^3)*cos(x)^3 - (4*a^2*b + 5*a*b^2 + b^3)*cos(x))*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 + 4*((2*a + b)*cos(x)^3 - (a + b)*cos(x))*sqrt(-a^2 - a*b)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)) + 4*(2*a^4 + 4*a^3*b + 2*a^2*b^2 - (2*a^3*b + a^2*b^2 - a*b^3)*cos(x)^2)*sin(x))/((a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*cos(x)^3 - (a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(x)), -1/4*(((4*a*b^2 + b^3)*cos(x)^3 - (4*a^2*b + 5*a*b^2 + b^3)*cos(x))*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)/(sqrt(a^2 + a*b)*cos(x)*sin(x))) + 2*(2*a^4 + 4*a^3*b + 2*a^2*b^2 - (2*a^3*b + a^2*b^2 - a*b^3)*cos(x)^2)*sin(x))/((a^5*b + 3*a^4*b^2 + 3*a^3*b^3 + a^2*b^4)*cos(x)^3 - (a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(x))]","B",0
322,1,560,0,1.127175," ","integrate(sec(x)^3/(a+b*sin(x)^2)^2,x, algorithm=""fricas"")","\left[\frac{{\left({\left(5 \, a b^{2} + b^{3}\right)} \cos\left(x\right)^{4} - {\left(5 \, a^{2} b + 6 \, a b^{2} + b^{3}\right)} \cos\left(x\right)^{2}\right)} \sqrt{-\frac{b}{a}} \log\left(-\frac{b \cos\left(x\right)^{2} - 2 \, a \sqrt{-\frac{b}{a}} \sin\left(x\right) + a - b}{b \cos\left(x\right)^{2} - a - b}\right) + {\left({\left(a^{2} b + 5 \, a b^{2}\right)} \cos\left(x\right)^{4} - {\left(a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(x\right)^{2}\right)} \log\left(\sin\left(x\right) + 1\right) - {\left({\left(a^{2} b + 5 \, a b^{2}\right)} \cos\left(x\right)^{4} - {\left(a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\sin\left(x\right) + 1\right) - 2 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b - b^{3}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{4 \, {\left({\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(x\right)^{4} - {\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(x\right)^{2}\right)}}, \frac{2 \, {\left({\left(5 \, a b^{2} + b^{3}\right)} \cos\left(x\right)^{4} - {\left(5 \, a^{2} b + 6 \, a b^{2} + b^{3}\right)} \cos\left(x\right)^{2}\right)} \sqrt{\frac{b}{a}} \arctan\left(\sqrt{\frac{b}{a}} \sin\left(x\right)\right) + {\left({\left(a^{2} b + 5 \, a b^{2}\right)} \cos\left(x\right)^{4} - {\left(a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(x\right)^{2}\right)} \log\left(\sin\left(x\right) + 1\right) - {\left({\left(a^{2} b + 5 \, a b^{2}\right)} \cos\left(x\right)^{4} - {\left(a^{3} + 6 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\sin\left(x\right) + 1\right) - 2 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b - b^{3}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{4 \, {\left({\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(x\right)^{4} - {\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} \cos\left(x\right)^{2}\right)}}\right]"," ",0,"[1/4*(((5*a*b^2 + b^3)*cos(x)^4 - (5*a^2*b + 6*a*b^2 + b^3)*cos(x)^2)*sqrt(-b/a)*log(-(b*cos(x)^2 - 2*a*sqrt(-b/a)*sin(x) + a - b)/(b*cos(x)^2 - a - b)) + ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2)*cos(x)^2)*log(sin(x) + 1) - ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2)*cos(x)^2)*log(-sin(x) + 1) - 2*(a^3 + 2*a^2*b + a*b^2 - (a^2*b - b^3)*cos(x)^2)*sin(x))/((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(x)^4 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(x)^2), 1/4*(2*((5*a*b^2 + b^3)*cos(x)^4 - (5*a^2*b + 6*a*b^2 + b^3)*cos(x)^2)*sqrt(b/a)*arctan(sqrt(b/a)*sin(x)) + ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2)*cos(x)^2)*log(sin(x) + 1) - ((a^2*b + 5*a*b^2)*cos(x)^4 - (a^3 + 6*a^2*b + 5*a*b^2)*cos(x)^2)*log(-sin(x) + 1) - 2*(a^3 + 2*a^2*b + a*b^2 - (a^2*b - b^3)*cos(x)^2)*sin(x))/((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*cos(x)^4 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(x)^2)]","B",0
323,1,653,0,1.254266," ","integrate(sec(x)^4/(a+b*sin(x)^2)^2,x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(6 \, a b^{3} + b^{4}\right)} \cos\left(x\right)^{5} - {\left(6 \, a^{2} b^{2} + 7 \, a b^{3} + b^{4}\right)} \cos\left(x\right)^{3}\right)} \sqrt{-a^{2} - a b} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(x\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(x\right)^{2} + 4 \, {\left({\left(2 \, a + b\right)} \cos\left(x\right)^{3} - {\left(a + b\right)} \cos\left(x\right)\right)} \sqrt{-a^{2} - a b} \sin\left(x\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(x\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(x\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) + 4 \, {\left(2 \, a^{5} + 6 \, a^{4} b + 6 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - {\left(4 \, a^{4} b + 20 \, a^{3} b^{2} + 13 \, a^{2} b^{3} - 3 \, a b^{4}\right)} \cos\left(x\right)^{4} + 2 \, {\left(2 \, a^{5} + 11 \, a^{4} b + 16 \, a^{3} b^{2} + 7 \, a^{2} b^{3}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{24 \, {\left({\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(x\right)^{5} - {\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(x\right)^{3}\right)}}, -\frac{3 \, {\left({\left(6 \, a b^{3} + b^{4}\right)} \cos\left(x\right)^{5} - {\left(6 \, a^{2} b^{2} + 7 \, a b^{3} + b^{4}\right)} \cos\left(x\right)^{3}\right)} \sqrt{a^{2} + a b} \arctan\left(\frac{{\left(2 \, a + b\right)} \cos\left(x\right)^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos\left(x\right) \sin\left(x\right)}\right) + 2 \, {\left(2 \, a^{5} + 6 \, a^{4} b + 6 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - {\left(4 \, a^{4} b + 20 \, a^{3} b^{2} + 13 \, a^{2} b^{3} - 3 \, a b^{4}\right)} \cos\left(x\right)^{4} + 2 \, {\left(2 \, a^{5} + 11 \, a^{4} b + 16 \, a^{3} b^{2} + 7 \, a^{2} b^{3}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{12 \, {\left({\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(x\right)^{5} - {\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(x\right)^{3}\right)}}\right]"," ",0,"[-1/24*(3*((6*a*b^3 + b^4)*cos(x)^5 - (6*a^2*b^2 + 7*a*b^3 + b^4)*cos(x)^3)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(x)^2 + 4*((2*a + b)*cos(x)^3 - (a + b)*cos(x))*sqrt(-a^2 - a*b)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)) + 4*(2*a^5 + 6*a^4*b + 6*a^3*b^2 + 2*a^2*b^3 - (4*a^4*b + 20*a^3*b^2 + 13*a^2*b^3 - 3*a*b^4)*cos(x)^4 + 2*(2*a^5 + 11*a^4*b + 16*a^3*b^2 + 7*a^2*b^3)*cos(x)^2)*sin(x))/((a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*cos(x)^5 - (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(x)^3), -1/12*(3*((6*a*b^3 + b^4)*cos(x)^5 - (6*a^2*b^2 + 7*a*b^3 + b^4)*cos(x)^3)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)/(sqrt(a^2 + a*b)*cos(x)*sin(x))) + 2*(2*a^5 + 6*a^4*b + 6*a^3*b^2 + 2*a^2*b^3 - (4*a^4*b + 20*a^3*b^2 + 13*a^2*b^3 - 3*a*b^4)*cos(x)^4 + 2*(2*a^5 + 11*a^4*b + 16*a^3*b^2 + 7*a^2*b^3)*cos(x)^2)*sin(x))/((a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*cos(x)^5 - (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(x)^3)]","B",0
324,1,511,0,1.814578," ","integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{2} + 4 \, a b\right)} \sqrt{b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) + 8 \, {\left(2 \, b^{2} \cos\left(f x + e\right)^{2} - a b + 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{64 \, b^{2} f}, -\frac{{\left(a^{2} + 4 \, a b\right)} \sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, {\left(2 \, b^{2} \cos\left(f x + e\right)^{2} - a b + 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{32 \, b^{2} f}\right]"," ",0,"[1/64*((a^2 + 4*a*b)*sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) + 8*(2*b^2*cos(f*x + e)^2 - a*b + 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/(b^2*f), -1/32*((a^2 + 4*a*b)*sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) - 4*(2*b^2*cos(f*x + e)^2 - a*b + 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/(b^2*f)]","A",0
325,1,453,0,1.239721," ","integrate(cos(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{a \sqrt{b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) + 8 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \sin\left(f x + e\right)}{16 \, b f}, -\frac{a \sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \sin\left(f x + e\right)}{8 \, b f}\right]"," ",0,"[1/16*(a*sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) + 8*sqrt(-b*cos(f*x + e)^2 + a + b)*b*sin(f*x + e))/(b*f), -1/8*(a*sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) - 4*sqrt(-b*cos(f*x + e)^2 + a + b)*b*sin(f*x + e))/(b*f)]","B",0
326,1,1246,0,1.315061," ","integrate(sec(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) + 2 \, \sqrt{a + b} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{8 \, f}, -\frac{4 \, \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) - \sqrt{b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right)}{8 \, f}, \frac{\sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + \sqrt{a + b} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{4 \, f}, -\frac{2 \, \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) - \sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, f}\right]"," ",0,"[1/8*(sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) + 2*sqrt(a + b)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4))/f, -1/8*(4*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e))) - sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)))/f, 1/4*(sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + sqrt(a + b)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4))/f, -1/4*(2*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e))) - sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))))/f]","B",0
327,1,337,0,0.813723," ","integrate(sec(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a + b} a \cos\left(f x + e\right)^{2} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)} \sin\left(f x + e\right)}{8 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{2}}, -\frac{a \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)} \sin\left(f x + e\right)}{4 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{2}}\right]"," ",0,"[1/8*(sqrt(a + b)*a*cos(f*x + e)^2*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) + 4*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b)*sin(f*x + e))/((a + b)*f*cos(f*x + e)^2), -1/4*(a*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e)))*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b)*sin(f*x + e))/((a + b)*f*cos(f*x + e)^2)]","B",0
328,1,443,0,1.962195," ","integrate(sec(f*x+e)^5*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a^{2} + 4 \, a b\right)} \sqrt{a + b} \cos\left(f x + e\right)^{4} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left({\left(3 \, a^{2} + 5 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{32 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4}}, -\frac{{\left(3 \, a^{2} + 4 \, a b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{4} - 2 \, {\left({\left(3 \, a^{2} + 5 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{16 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4}}\right]"," ",0,"[1/32*((3*a^2 + 4*a*b)*sqrt(a + b)*cos(f*x + e)^4*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) + 4*((3*a^2 + 5*a*b + 2*b^2)*cos(f*x + e)^2 + 2*a^2 + 4*a*b + 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4), -1/16*((3*a^2 + 4*a*b)*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e)))*cos(f*x + e)^4 - 2*((3*a^2 + 5*a*b + 2*b^2)*cos(f*x + e)^2 + 2*a^2 + 4*a*b + 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4)]","A",0
329,0,0,0,0.892434," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cos\left(f x + e\right)^{4}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*cos(f*x + e)^4, x)","F",0
330,0,0,0,0.833997," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cos\left(f x + e\right)^{2}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*cos(f*x + e)^2, x)","F",0
331,0,0,0,0.905277," ","integrate((a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b), x)","F",0
332,0,0,0,0.648078," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sec\left(f x + e\right)^{2}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*sec(f*x + e)^2, x)","F",0
333,0,0,0,1.032489," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sec\left(f x + e\right)^{4}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*sec(f*x + e)^4, x)","F",0
334,1,577,0,4.412295," ","integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} + 6 \, a^{2} b\right)} \sqrt{b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) - 8 \, {\left(8 \, b^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b - 16 \, a b^{2} - 4 \, b^{3} - 2 \, {\left(7 \, a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{384 \, b^{2} f}, -\frac{3 \, {\left(a^{3} + 6 \, a^{2} b\right)} \sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(8 \, b^{3} \cos\left(f x + e\right)^{4} + 3 \, a^{2} b - 16 \, a b^{2} - 4 \, b^{3} - 2 \, {\left(7 \, a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{192 \, b^{2} f}\right]"," ",0,"[1/384*(3*(a^3 + 6*a^2*b)*sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) - 8*(8*b^3*cos(f*x + e)^4 + 3*a^2*b - 16*a*b^2 - 4*b^3 - 2*(7*a*b^2 + 2*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/(b^2*f), -1/192*(3*(a^3 + 6*a^2*b)*sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(8*b^3*cos(f*x + e)^4 + 3*a^2*b - 16*a*b^2 - 4*b^3 - 2*(7*a*b^2 + 2*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/(b^2*f)]","A",0
335,1,503,0,1.523517," ","integrate(cos(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, a^{2} \sqrt{b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) - 8 \, {\left(2 \, b^{2} \cos\left(f x + e\right)^{2} - 5 \, a b - 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{64 \, b f}, -\frac{3 \, a^{2} \sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(2 \, b^{2} \cos\left(f x + e\right)^{2} - 5 \, a b - 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{32 \, b f}\right]"," ",0,"[1/64*(3*a^2*sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) - 8*(2*b^2*cos(f*x + e)^2 - 5*a*b - 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/(b*f), -1/32*(3*a^2*sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(2*b^2*cos(f*x + e)^2 - 5*a*b - 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/(b*f)]","B",0
336,1,1381,0,1.941074," ","integrate(sec(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(3 \, a + 2 \, b\right)} \sqrt{b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) + 4 \, {\left(a + b\right)}^{\frac{3}{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 8 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \sin\left(f x + e\right)}{16 \, f}, -\frac{8 \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) - {\left(3 \, a + 2 \, b\right)} \sqrt{b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) + 8 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \sin\left(f x + e\right)}{16 \, f}, \frac{{\left(3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 2 \, {\left(a + b\right)}^{\frac{3}{2}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \sin\left(f x + e\right)}{8 \, f}, -\frac{4 \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) - {\left(3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \sin\left(f x + e\right)}{8 \, f}\right]"," ",0,"[1/16*((3*a + 2*b)*sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) + 4*(a + b)^(3/2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) - 8*sqrt(-b*cos(f*x + e)^2 + a + b)*b*sin(f*x + e))/f, -1/16*(8*(a + b)*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e))) - (3*a + 2*b)*sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) + 8*sqrt(-b*cos(f*x + e)^2 + a + b)*b*sin(f*x + e))/f, 1/8*((3*a + 2*b)*sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + 2*(a + b)^(3/2)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) - 4*sqrt(-b*cos(f*x + e)^2 + a + b)*b*sin(f*x + e))/f, -1/8*(4*(a + b)*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e))) - (3*a + 2*b)*sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + 4*sqrt(-b*cos(f*x + e)^2 + a + b)*b*sin(f*x + e))/f]","B",0
337,1,1471,0,2.342124," ","integrate(sec(f*x+e)^3*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{b^{\frac{3}{2}} \cos\left(f x + e\right)^{2} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) - \sqrt{a + b} {\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{2} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)^{2}}, -\frac{2 \, {\left(a - 2 \, b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{2} - b^{\frac{3}{2}} \cos\left(f x + e\right)^{2} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) - 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)^{2}}, -\frac{2 \, \sqrt{-b} b \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{2} + \sqrt{a + b} {\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{2} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)} \sin\left(f x + e\right)}{8 \, f \cos\left(f x + e\right)^{2}}, -\frac{{\left(a - 2 \, b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{2} + \sqrt{-b} b \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)} \sin\left(f x + e\right)}{4 \, f \cos\left(f x + e\right)^{2}}\right]"," ",0,"[1/8*(b^(3/2)*cos(f*x + e)^2*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) - sqrt(a + b)*(a - 2*b)*cos(f*x + e)^2*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) + 4*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b)*sin(f*x + e))/(f*cos(f*x + e)^2), -1/8*(2*(a - 2*b)*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e)))*cos(f*x + e)^2 - b^(3/2)*cos(f*x + e)^2*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) - 4*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b)*sin(f*x + e))/(f*cos(f*x + e)^2), -1/8*(2*sqrt(-b)*b*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^2 + sqrt(a + b)*(a - 2*b)*cos(f*x + e)^2*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 + 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) - 4*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b)*sin(f*x + e))/(f*cos(f*x + e)^2), -1/4*((a - 2*b)*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e)))*cos(f*x + e)^2 + sqrt(-b)*b*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b)*sin(f*x + e))/(f*cos(f*x + e)^2)]","B",0
338,1,413,0,2.002539," ","integrate(sec(f*x+e)^5*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a + b} a^{2} \cos\left(f x + e\right)^{4} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left({\left(3 \, a^{2} + a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{32 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{4}}, -\frac{3 \, a^{2} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{4} - 2 \, {\left({\left(3 \, a^{2} + a b - 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{16 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{4}}\right]"," ",0,"[1/32*(3*sqrt(a + b)*a^2*cos(f*x + e)^4*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) + 4*((3*a^2 + a*b - 2*b^2)*cos(f*x + e)^2 + 2*a^2 + 4*a*b + 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/((a + b)*f*cos(f*x + e)^4), -1/16*(3*a^2*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e)))*cos(f*x + e)^4 - 2*((3*a^2 + a*b - 2*b^2)*cos(f*x + e)^2 + 2*a^2 + 4*a*b + 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/((a + b)*f*cos(f*x + e)^4)]","A",0
339,1,545,0,7.826075," ","integrate(sec(f*x+e)^7*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(5 \, a^{3} + 6 \, a^{2} b\right)} \sqrt{a + b} \cos\left(f x + e\right)^{6} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left({\left(15 \, a^{3} + 23 \, a^{2} b + 4 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{3} + 24 \, a^{2} b + 24 \, a b^{2} + 8 \, b^{3} + 2 \, {\left(5 \, a^{3} + 8 \, a^{2} b + a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{192 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{6}}, -\frac{3 \, {\left(5 \, a^{3} + 6 \, a^{2} b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{6} - 2 \, {\left({\left(15 \, a^{3} + 23 \, a^{2} b + 4 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{3} + 24 \, a^{2} b + 24 \, a b^{2} + 8 \, b^{3} + 2 \, {\left(5 \, a^{3} + 8 \, a^{2} b + a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{96 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{6}}\right]"," ",0,"[1/192*(3*(5*a^3 + 6*a^2*b)*sqrt(a + b)*cos(f*x + e)^6*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) + 4*((15*a^3 + 23*a^2*b + 4*a*b^2 - 4*b^3)*cos(f*x + e)^4 + 8*a^3 + 24*a^2*b + 24*a*b^2 + 8*b^3 + 2*(5*a^3 + 8*a^2*b + a*b^2 - 2*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^6), -1/96*(3*(5*a^3 + 6*a^2*b)*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e)))*cos(f*x + e)^6 - 2*((15*a^3 + 23*a^2*b + 4*a*b^2 - 4*b^3)*cos(f*x + e)^4 + 8*a^3 + 24*a^2*b + 24*a*b^2 + 8*b^3 + 2*(5*a^3 + 8*a^2*b + a*b^2 - 2*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^6)]","A",0
340,0,0,0,0.871910," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b \cos\left(f x + e\right)^{6} - {\left(a + b\right)} \cos\left(f x + e\right)^{4}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^6 - (a + b)*cos(f*x + e)^4)*sqrt(-b*cos(f*x + e)^2 + a + b), x)","F",0
341,0,0,0,1.046329," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b \cos\left(f x + e\right)^{4} - {\left(a + b\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^4 - (a + b)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b), x)","F",0
342,0,0,0,0.590108," ","integrate((a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{\frac{3}{2}}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^(3/2), x)","F",0
343,0,0,0,0.889941," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sec\left(f x + e\right)^{2}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^2 - a - b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sec(f*x + e)^2, x)","F",0
344,0,0,0,0.747567," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sec\left(f x + e\right)^{4}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^2 - a - b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sec(f*x + e)^4, x)","F",0
345,1,461,0,1.268845," ","integrate(cos(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a + 2 \, b\right)} \sqrt{b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) - 8 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \sin\left(f x + e\right)}{16 \, b^{2} f}, -\frac{{\left(a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} b \sin\left(f x + e\right)}{8 \, b^{2} f}\right]"," ",0,"[1/16*((a + 2*b)*sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) - 8*sqrt(-b*cos(f*x + e)^2 + a + b)*b*sin(f*x + e))/(b^2*f), -1/8*((a + 2*b)*sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + 4*sqrt(-b*cos(f*x + e)^2 + a + b)*b*sin(f*x + e))/(b^2*f)]","B",0
346,1,394,0,0.971489," ","integrate(cos(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right)}{8 \, \sqrt{b} f}, -\frac{\sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right)}{4 \, b f}\right]"," ",0,"[1/8*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e))/(sqrt(b)*f), -1/4*sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e)))/(b*f)]","B",0
347,1,240,0,0.863669," ","integrate(sec(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right)}{4 \, \sqrt{a + b} f}, -\frac{\sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right)}{2 \, {\left(a + b\right)} f}\right]"," ",0,"[1/4*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4)/(sqrt(a + b)*f), -1/2*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e)))/((a + b)*f)]","B",0
348,1,361,0,0.893104," ","integrate(sec(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a + 2 \, b\right)} \sqrt{a + b} \cos\left(f x + e\right)^{2} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)} \sin\left(f x + e\right)}{8 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2}}, -\frac{{\left(a + 2 \, b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)} \sin\left(f x + e\right)}{4 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2}}\right]"," ",0,"[1/8*((a + 2*b)*sqrt(a + b)*cos(f*x + e)^2*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) + 4*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b)*sin(f*x + e))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2), -1/4*((a + 2*b)*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e)))*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b)*sin(f*x + e))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2)]","B",0
349,0,0,0,0.657470," ","integrate(cos(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cos\left(f x + e\right)^{4}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*cos(f*x + e)^4/(b*cos(f*x + e)^2 - a - b), x)","F",0
350,0,0,0,0.769404," ","integrate(cos(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cos\left(f x + e\right)^{2}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*cos(f*x + e)^2/(b*cos(f*x + e)^2 - a - b), x)","F",0
351,0,0,0,0.893213," ","integrate(1/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)/(b*cos(f*x + e)^2 - a - b), x)","F",0
352,0,0,0,0.809344," ","integrate(sec(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sec\left(f x + e\right)^{2}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*sec(f*x + e)^2/(b*cos(f*x + e)^2 - a - b), x)","F",0
353,0,0,0,0.797097," ","integrate(sec(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sec\left(f x + e\right)^{4}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*sec(f*x + e)^4/(b*cos(f*x + e)^2 - a - b), x)","F",0
354,1,559,0,1.108458," ","integrate(cos(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a b \cos\left(f x + e\right)^{2} - a^{2} - a b\right)} \sqrt{b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} + 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) - 8 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a b + b^{2}\right)} \sin\left(f x + e\right)}{8 \, {\left(a b^{3} f \cos\left(f x + e\right)^{2} - {\left(a^{2} b^{2} + a b^{3}\right)} f\right)}}, \frac{{\left(a b \cos\left(f x + e\right)^{2} - a^{2} - a b\right)} \sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) - 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a b + b^{2}\right)} \sin\left(f x + e\right)}{4 \, {\left(a b^{3} f \cos\left(f x + e\right)^{2} - {\left(a^{2} b^{2} + a b^{3}\right)} f\right)}}\right]"," ",0,"[1/8*((a*b*cos(f*x + e)^2 - a^2 - a*b)*sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 + 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) - 8*sqrt(-b*cos(f*x + e)^2 + a + b)*(a*b + b^2)*sin(f*x + e))/(a*b^3*f*cos(f*x + e)^2 - (a^2*b^2 + a*b^3)*f), 1/4*((a*b*cos(f*x + e)^2 - a^2 - a*b)*sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) - 4*sqrt(-b*cos(f*x + e)^2 + a + b)*(a*b + b^2)*sin(f*x + e))/(a*b^3*f*cos(f*x + e)^2 - (a^2*b^2 + a*b^3)*f)]","B",0
355,1,49,0,0.874428," ","integrate(cos(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{a b f \cos\left(f x + e\right)^{2} - {\left(a^{2} + a b\right)} f}"," ",0,"-sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e)/(a*b*f*cos(f*x + e)^2 - (a^2 + a*b)*f)","A",0
356,1,453,0,0.982750," ","integrate(sec(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a b \cos\left(f x + e\right)^{2} - a^{2} - a b\right)} \sqrt{a + b} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a b + b^{2}\right)} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}, -\frac{{\left(a b \cos\left(f x + e\right)^{2} - a^{2} - a b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a b + b^{2}\right)} \sin\left(f x + e\right)}{2 \, {\left({\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} f\right)}}\right]"," ",0,"[1/4*((a*b*cos(f*x + e)^2 - a^2 - a*b)*sqrt(a + b)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) - 4*sqrt(-b*cos(f*x + e)^2 + a + b)*(a*b + b^2)*sin(f*x + e))/((a^3*b + 2*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 - (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f), -1/2*((a*b*cos(f*x + e)^2 - a^2 - a*b)*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e))) + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*(a*b + b^2)*sin(f*x + e))/((a^3*b + 2*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 - (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f)]","B",0
357,1,625,0,1.903947," ","integrate(sec(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(a^{3} + 5 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) - 4 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b - a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{8 \, {\left({\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{2}\right)}}, -\frac{{\left({\left(a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(a^{3} + 5 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) + 2 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2} - {\left(a^{2} b - a b^{2} - 2 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{4 \, {\left({\left(a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right)} f \cos\left(f x + e\right)^{2}\right)}}\right]"," ",0,"[1/8*(((a^2*b + 4*a*b^2)*cos(f*x + e)^4 - (a^3 + 5*a^2*b + 4*a*b^2)*cos(f*x + e)^2)*sqrt(a + b)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) - 4*(a^3 + 2*a^2*b + a*b^2 - (a^2*b - a*b^2 - 2*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*f*cos(f*x + e)^4 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e)^2), -1/4*(((a^2*b + 4*a*b^2)*cos(f*x + e)^4 - (a^3 + 5*a^2*b + 4*a*b^2)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e))) + 2*(a^3 + 2*a^2*b + a*b^2 - (a^2*b - a*b^2 - 2*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/((a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*f*cos(f*x + e)^4 - (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*f*cos(f*x + e)^2)]","B",0
358,0,0,0,1.095132," ","integrate(cos(f*x+e)^6/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cos\left(f x + e\right)^{6}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*cos(f*x + e)^6/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
359,0,0,0,0.680298," ","integrate(cos(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cos\left(f x + e\right)^{4}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*cos(f*x + e)^4/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
360,0,0,0,0.955389," ","integrate(cos(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cos\left(f x + e\right)^{2}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*cos(f*x + e)^2/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
361,0,0,0,0.882902," ","integrate(1/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
362,0,0,0,0.881966," ","integrate(sec(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sec\left(f x + e\right)^{2}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*sec(f*x + e)^2/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
363,1,799,0,3.657240," ","integrate(cos(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} b^{2} \cos\left(f x + e\right)^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2} - 2 \, {\left(a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{b} \log\left(128 \, b^{4} \cos\left(f x + e\right)^{8} - 256 \, {\left(a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + 32 \, {\left(5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left(a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right)} \cos\left(f x + e\right)^{2} - 8 \, {\left(16 \, b^{3} \cos\left(f x + e\right)^{6} - 24 \, {\left(a b^{2} + 2 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left(5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{b} \sin\left(f x + e\right)\right) - 8 \, {\left(3 \, a^{3} b + 4 \, a^{2} b^{2} - a b^{3} - 2 \, b^{4} - 2 \, {\left(2 \, a^{2} b^{2} + a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{24 \, {\left(a^{2} b^{5} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{3} + 2 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}, -\frac{3 \, {\left(a^{2} b^{2} \cos\left(f x + e\right)^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2} - 2 \, {\left(a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b} \arctan\left(\frac{{\left(8 \, b^{2} \cos\left(f x + e\right)^{4} - 8 \, {\left(a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-b}}{4 \, {\left(2 \, b^{3} \cos\left(f x + e\right)^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left(3 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sin\left(f x + e\right)}\right) + 4 \, {\left(3 \, a^{3} b + 4 \, a^{2} b^{2} - a b^{3} - 2 \, b^{4} - 2 \, {\left(2 \, a^{2} b^{2} + a b^{3} - b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{12 \, {\left(a^{2} b^{5} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} b^{3} + 2 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}\right]"," ",0,"[1/24*(3*(a^2*b^2*cos(f*x + e)^4 + a^4 + 2*a^3*b + a^2*b^2 - 2*(a^3*b + a^2*b^2)*cos(f*x + e)^2)*sqrt(b)*log(128*b^4*cos(f*x + e)^8 - 256*(a*b^3 + 2*b^4)*cos(f*x + e)^6 + 32*(5*a^2*b^2 + 24*a*b^3 + 24*b^4)*cos(f*x + e)^4 + a^4 + 32*a^3*b + 160*a^2*b^2 + 256*a*b^3 + 128*b^4 - 32*(a^3*b + 10*a^2*b^2 + 24*a*b^3 + 16*b^4)*cos(f*x + e)^2 - 8*(16*b^3*cos(f*x + e)^6 - 24*(a*b^2 + 2*b^3)*cos(f*x + e)^4 - a^3 - 10*a^2*b - 24*a*b^2 - 16*b^3 + 2*(5*a^2*b + 24*a*b^2 + 24*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(b)*sin(f*x + e)) - 8*(3*a^3*b + 4*a^2*b^2 - a*b^3 - 2*b^4 - 2*(2*a^2*b^2 + a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/(a^2*b^5*f*cos(f*x + e)^4 - 2*(a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^2 + (a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*f), -1/12*(3*(a^2*b^2*cos(f*x + e)^4 + a^4 + 2*a^3*b + a^2*b^2 - 2*(a^3*b + a^2*b^2)*cos(f*x + e)^2)*sqrt(-b)*arctan(1/4*(8*b^2*cos(f*x + e)^4 - 8*(a*b + 2*b^2)*cos(f*x + e)^2 + a^2 + 8*a*b + 8*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-b)/((2*b^3*cos(f*x + e)^4 + a^2*b + 3*a*b^2 + 2*b^3 - (3*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sin(f*x + e))) + 4*(3*a^3*b + 4*a^2*b^2 - a*b^3 - 2*b^4 - 2*(2*a^2*b^2 + a*b^3 - b^4)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/(a^2*b^5*f*cos(f*x + e)^4 - 2*(a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^2 + (a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*f)]","B",0
364,1,107,0,1.543274," ","integrate(cos(f*x+e)^3/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\frac{{\left({\left(a - 2 \, b\right)} \cos\left(f x + e\right)^{2} + 2 \, a + 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{3 \, {\left(a^{2} b^{2} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f\right)}}"," ",0,"1/3*((a - 2*b)*cos(f*x + e)^2 + 2*a + 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e)/(a^2*b^2*f*cos(f*x + e)^4 - 2*(a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^4 + 2*a^3*b + a^2*b^2)*f)","A",0
365,1,104,0,1.095789," ","integrate(cos(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","-\frac{{\left(2 \, b \cos\left(f x + e\right)^{2} - 3 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{3 \, {\left(a^{2} b^{2} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{3} b + a^{2} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} f\right)}}"," ",0,"-1/3*(2*b*cos(f*x + e)^2 - 3*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e)/(a^2*b^2*f*cos(f*x + e)^4 - 2*(a^3*b + a^2*b^2)*f*cos(f*x + e)^2 + (a^4 + 2*a^3*b + a^2*b^2)*f)","A",0
366,1,775,0,1.692054," ","integrate(sec(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} b^{2} \cos\left(f x + e\right)^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2} - 2 \, {\left(a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 8 \, {\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, {\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} \sin\left(f x + e\right) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos\left(f x + e\right)^{4}}\right) + 4 \, {\left(6 \, a^{3} b + 14 \, a^{2} b^{2} + 10 \, a b^{3} + 2 \, b^{4} - {\left(5 \, a^{2} b^{2} + 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{12 \, {\left({\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}, -\frac{3 \, {\left(a^{2} b^{2} \cos\left(f x + e\right)^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2} - 2 \, {\left(a^{3} b + a^{2} b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - 2 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{2 \, {\left({\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sin\left(f x + e\right)}\right) - 2 \, {\left(6 \, a^{3} b + 14 \, a^{2} b^{2} + 10 \, a b^{3} + 2 \, b^{4} - {\left(5 \, a^{2} b^{2} + 7 \, a b^{3} + 2 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sin\left(f x + e\right)}{6 \, {\left({\left(a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} f\right)}}\right]"," ",0,"[1/12*(3*(a^2*b^2*cos(f*x + e)^4 + a^4 + 2*a^3*b + a^2*b^2 - 2*(a^3*b + a^2*b^2)*cos(f*x + e)^2)*sqrt(a + b)*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 8*(a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^2 - 4*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b)*sin(f*x + e) + 8*a^2 + 16*a*b + 8*b^2)/cos(f*x + e)^4) + 4*(6*a^3*b + 14*a^2*b^2 + 10*a*b^3 + 2*b^4 - (5*a^2*b^2 + 7*a*b^3 + 2*b^4)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/((a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^4 - 2*(a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^2 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f), -1/6*(3*(a^2*b^2*cos(f*x + e)^4 + a^4 + 2*a^3*b + a^2*b^2 - 2*(a^3*b + a^2*b^2)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - 2*a - 2*b)*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(((a*b + b^2)*cos(f*x + e)^2 - a^2 - 2*a*b - b^2)*sin(f*x + e))) - 2*(6*a^3*b + 14*a^2*b^2 + 10*a*b^3 + 2*b^4 - (5*a^2*b^2 + 7*a*b^3 + 2*b^4)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b)*sin(f*x + e))/((a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^4 - 2*(a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*f*cos(f*x + e)^2 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*f)]","B",0
367,0,0,0,0.999248," ","integrate(cos(f*x+e)^6/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cos\left(f x + e\right)^{6}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*cos(f*x + e)^6/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
368,0,0,0,0.913907," ","integrate(cos(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cos\left(f x + e\right)^{4}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*cos(f*x + e)^4/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
369,0,0,0,1.019785," ","integrate(cos(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cos\left(f x + e\right)^{2}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*cos(f*x + e)^2/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
370,0,0,0,1.192905," ","integrate(1/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
371,0,0,0,1.094111," ","integrate(sec(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sec\left(f x + e\right)^{2}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*sec(f*x + e)^2/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
372,0,0,0,1.384235," ","integrate((d*cos(f*x+e))^m*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \left(d \cos\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*(d*cos(f*x + e))^m, x)","F",0
373,0,0,0,0.944305," ","integrate(cos(f*x+e)^5*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \cos\left(f x + e\right)^{5}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e)^5, x)","F",0
374,0,0,0,0.790119," ","integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \cos\left(f x + e\right)^{3}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e)^3, x)","F",0
375,0,0,0,1.004308," ","integrate(cos(f*x+e)*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \cos\left(f x + e\right), x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e), x)","F",0
376,0,0,0,1.053467," ","integrate(sec(f*x+e)*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \sec\left(f x + e\right), x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*sec(f*x + e), x)","F",0
377,0,0,0,0.978826," ","integrate(sec(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \sec\left(f x + e\right)^{3}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*sec(f*x + e)^3, x)","F",0
378,0,0,0,0.807169," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \cos\left(f x + e\right)^{4}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e)^4, x)","F",0
379,0,0,0,0.982270," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \cos\left(f x + e\right)^{2}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e)^2, x)","F",0
380,0,0,0,0.618953," ","integrate((a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p, x)","F",0
381,0,0,0,1.022197," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \sec\left(f x + e\right)^{2}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*sec(f*x + e)^2, x)","F",0
382,0,0,0,0.957262," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \sec\left(f x + e\right)^{4}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*sec(f*x + e)^4, x)","F",0
383,1,3216,0,3.052232," ","integrate(cos(d*x+c)^5/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","-\frac{2 \, {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)} b d \log\left(\frac{1}{4} \, {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)}^{2} a^{3} b^{3} d^{2} + 2 \, a^{3} b - 2 \, a b^{3} - \frac{1}{2} \, {\left(4 \, a^{3} b^{2} - a b^{4}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)} d + {\left(a^{4} - b^{4}\right)} \sin\left(d x + c\right)\right) + 6 \, \cos\left(d x + c\right)^{2} - {\left({\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)} b d + 3 \, \sqrt{\frac{1}{3}} b d \sqrt{-\frac{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)}^{2} b d - 8 \, \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} - \frac{32}{b d} - \frac{16 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}}{b d}} - 12\right)} \log\left(\frac{1}{4} \, {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)}^{2} a^{3} b^{3} d^{2} + 2 \, a^{3} b - 2 \, a b^{3} - \frac{1}{2} \, {\left(4 \, a^{3} b^{2} - a b^{4}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)} d - \frac{3}{4} \, \sqrt{\frac{1}{3}} {\left({\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)} a^{3} b^{3} d^{2} - 2 \, {\left(2 \, a^{3} b^{2} + a b^{4}\right)} d\right)} \sqrt{-\frac{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)}^{2} b d - 8 \, \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} - \frac{32}{b d} - \frac{16 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}}{b d}} - 2 \, {\left(a^{4} - b^{4}\right)} \sin\left(d x + c\right)\right) - {\left({\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)} b d - 3 \, \sqrt{\frac{1}{3}} b d \sqrt{-\frac{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)}^{2} b d - 8 \, \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} - \frac{32}{b d} - \frac{16 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}}{b d}} - 12\right)} \log\left(-\frac{1}{4} \, {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)}^{2} a^{3} b^{3} d^{2} - 2 \, a^{3} b + 2 \, a b^{3} + \frac{1}{2} \, {\left(4 \, a^{3} b^{2} - a b^{4}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)} d - \frac{3}{4} \, \sqrt{\frac{1}{3}} {\left({\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)} a^{3} b^{3} d^{2} - 2 \, {\left(2 \, a^{3} b^{2} + a b^{4}\right)} d\right)} \sqrt{-\frac{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} + \frac{4}{b d} + \frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}\right)}^{2} b d - 8 \, \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}} - \frac{32}{b d} - \frac{16 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(-i \, \sqrt{3} + 1\right)}}{b^{2} d^{2} {\left(\frac{2}{b^{3} d^{3}} + \frac{a^{4} - 2 \, a^{2} b^{2} + b^{4}}{a^{2} b^{5} d^{3}} - \frac{a^{4} - b^{4}}{a^{2} b^{5} d^{3}}\right)}^{\frac{1}{3}}}}{b d}} + 2 \, {\left(a^{4} - b^{4}\right)} \sin\left(d x + c\right)\right)}{12 \, b d}"," ",0,"-1/12*(2*((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))*b*d*log(1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))^2*a^3*b^3*d^2 + 2*a^3*b - 2*a*b^3 - 1/2*(4*a^3*b^2 - a*b^4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))*d + (a^4 - b^4)*sin(d*x + c)) + 6*cos(d*x + c)^2 - (((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))*b*d + 3*sqrt(1/3)*b*d*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))^2*b*d - 8*(1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) - 32/(b*d) - 16*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))/(b*d)) - 12)*log(1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))^2*a^3*b^3*d^2 + 2*a^3*b - 2*a*b^3 - 1/2*(4*a^3*b^2 - a*b^4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))*d - 3/4*sqrt(1/3)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))*a^3*b^3*d^2 - 2*(2*a^3*b^2 + a*b^4)*d)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))^2*b*d - 8*(1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) - 32/(b*d) - 16*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))/(b*d)) - 2*(a^4 - b^4)*sin(d*x + c)) - (((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))*b*d - 3*sqrt(1/3)*b*d*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))^2*b*d - 8*(1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) - 32/(b*d) - 16*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))/(b*d)) - 12)*log(-1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))^2*a^3*b^3*d^2 - 2*a^3*b + 2*a*b^3 + 1/2*(4*a^3*b^2 - a*b^4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))*d - 3/4*sqrt(1/3)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))*a^3*b^3*d^2 - 2*(2*a^3*b^2 + a*b^4)*d)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) + 4/(b*d) + 2*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))^2*b*d - 8*(1/2)^(1/3)*(I*sqrt(3) + 1)*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3) - 32/(b*d) - 16*(1/2)^(2/3)*(-I*sqrt(3) + 1)/(b^2*d^2*(2/(b^3*d^3) + (a^4 - 2*a^2*b^2 + b^4)/(a^2*b^5*d^3) - (a^4 - b^4)/(a^2*b^5*d^3))^(1/3)))/(b*d)) + 2*(a^4 - b^4)*sin(d*x + c)))/(b*d)","C",0
384,1,2298,0,125.818063," ","integrate(cos(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\frac{6 \, \sqrt{\frac{1}{3}} b d \sqrt{\frac{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} b^{2} d^{2} - 4 \, {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} b d + 4}{b^{2} d^{2}}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{3}} \sqrt{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} a^{2} b^{2} d^{2} - 4 \, b^{2} \cos\left(d x + c\right)^{2} - 4 \, a b \sin\left(d x + c\right) + 2 \, {\left(a b^{2} d \sin\left(d x + c\right) - 2 \, a^{2} b d\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} + 4 \, a^{2} + 4 \, b^{2}} {\left({\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} a b d^{2} - 2 \, a d\right)} \sqrt{\frac{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} b^{2} d^{2} - 4 \, {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} b d + 4}{b^{2} d^{2}}} + \sqrt{\frac{1}{3}} {\left({\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} a^{2} b^{2} d^{3} - 8 \, a b d \sin\left(d x + c\right) + 4 \, a^{2} d + 4 \, {\left(a b^{2} d^{2} \sin\left(d x + c\right) - a^{2} b d^{2}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}\right)} \sqrt{\frac{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} b^{2} d^{2} - 4 \, {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} b d + 4}{b^{2} d^{2}}}}{8 \, b}\right) - 6 \, \sqrt{\frac{1}{3}} b d \sqrt{\frac{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} b^{2} d^{2} - 4 \, {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} b d + 4}{b^{2} d^{2}}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{3}} \sqrt{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} a^{2} b^{2} d^{2} - 4 \, b^{2} \cos\left(d x + c\right)^{2} - 4 \, a b \sin\left(d x + c\right) + 2 \, {\left(a b^{2} d \sin\left(d x + c\right) - 2 \, a^{2} b d\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} + 4 \, a^{2} + 4 \, b^{2}} {\left({\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} a b d^{2} - 2 \, a d\right)} \sqrt{\frac{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} b^{2} d^{2} - 4 \, {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} b d + 4}{b^{2} d^{2}}} - \sqrt{\frac{1}{3}} {\left({\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} a^{2} b^{2} d^{3} - 8 \, a b d \sin\left(d x + c\right) + 4 \, a^{2} d + 4 \, {\left(a b^{2} d^{2} \sin\left(d x + c\right) - a^{2} b d^{2}\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}\right)} \sqrt{\frac{{\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} b^{2} d^{2} - 4 \, {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} b d + 4}{b^{2} d^{2}}}}{8 \, b}\right) - {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} b d \log\left(\frac{1}{4} \, {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} a^{2} b^{2} d^{2} - b^{2} \cos\left(d x + c\right)^{2} + 2 \, a b \sin\left(d x + c\right) - {\left(a b^{2} d \sin\left(d x + c\right) + a^{2} b d\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} + a^{2} + b^{2}\right) + {\left({\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} b d - 6\right)} \log\left({\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)}^{2} a^{2} b^{2} d^{2} - 4 \, b^{2} \cos\left(d x + c\right)^{2} - 4 \, a b \sin\left(d x + c\right) + 2 \, {\left(a b^{2} d \sin\left(d x + c\right) - 2 \, a^{2} b d\right)} {\left(\left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{b^{3} d^{3}} + \frac{1}{a^{2} b d^{3}} - \frac{a^{2} - b^{2}}{a^{2} b^{3} d^{3}}\right)}^{\frac{1}{3}} + \frac{2}{b d}\right)} + 4 \, a^{2} + 4 \, b^{2}\right)}{12 \, b d}"," ",0,"1/12*(6*sqrt(1/3)*b*d*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*b^2*d^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d + 4)/(b^2*d^2))*arctan(-1/8*(2*sqrt(1/3)*sqrt(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*a^2*b^2*d^2 - 4*b^2*cos(d*x + c)^2 - 4*a*b*sin(d*x + c) + 2*(a*b^2*d*sin(d*x + c) - 2*a^2*b*d)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d)) + 4*a^2 + 4*b^2)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*a*b*d^2 - 2*a*d)*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*b^2*d^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d + 4)/(b^2*d^2)) + sqrt(1/3)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*a^2*b^2*d^3 - 8*a*b*d*sin(d*x + c) + 4*a^2*d + 4*(a*b^2*d^2*sin(d*x + c) - a^2*b*d^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d)))*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*b^2*d^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d + 4)/(b^2*d^2)))/b) - 6*sqrt(1/3)*b*d*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*b^2*d^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d + 4)/(b^2*d^2))*arctan(-1/8*(2*sqrt(1/3)*sqrt(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*a^2*b^2*d^2 - 4*b^2*cos(d*x + c)^2 - 4*a*b*sin(d*x + c) + 2*(a*b^2*d*sin(d*x + c) - 2*a^2*b*d)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d)) + 4*a^2 + 4*b^2)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*a*b*d^2 - 2*a*d)*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*b^2*d^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d + 4)/(b^2*d^2)) - sqrt(1/3)*(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*a^2*b^2*d^3 - 8*a*b*d*sin(d*x + c) + 4*a^2*d + 4*(a*b^2*d^2*sin(d*x + c) - a^2*b*d^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d)))*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*b^2*d^2 - 4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d + 4)/(b^2*d^2)))/b) - ((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d*log(1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*a^2*b^2*d^2 - b^2*cos(d*x + c)^2 + 2*a*b*sin(d*x + c) - (a*b^2*d*sin(d*x + c) + a^2*b*d)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d)) + a^2 + b^2) + (((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))*b*d - 6)*log(((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d))^2*a^2*b^2*d^2 - 4*b^2*cos(d*x + c)^2 - 4*a*b*sin(d*x + c) + 2*(a*b^2*d*sin(d*x + c) - 2*a^2*b*d)*((1/2)^(1/3)*(I*sqrt(3) + 1)*(1/(b^3*d^3) + 1/(a^2*b*d^3) - (a^2 - b^2)/(a^2*b^3*d^3))^(1/3) + 2/(b*d)) + 4*a^2 + 4*b^2))/(b*d)","C",0
385,1,401,0,0.672585," ","integrate(cos(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{\frac{1}{3}} a b \sqrt{-\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}} \log\left(-\frac{3 \, \left(a^{2} b\right)^{\frac{1}{3}} a \sin\left(d x + c\right) + a^{2} + 3 \, \sqrt{\frac{1}{3}} {\left(2 \, a b \cos\left(d x + c\right)^{2} - 2 \, a b - \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{1}{3}} a\right)} \sqrt{-\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}} + 2 \, {\left(a b \cos\left(d x + c\right)^{2} - a b\right)} \sin\left(d x + c\right)}{{\left(b \cos\left(d x + c\right)^{2} - b\right)} \sin\left(d x + c\right) - a}\right) - \left(a^{2} b\right)^{\frac{2}{3}} \log\left(-a b \cos\left(d x + c\right)^{2} + a b - \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{1}{3}} a\right) + 2 \, \left(a^{2} b\right)^{\frac{2}{3}} \log\left(a b \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{2}{3}}\right)}{6 \, a^{2} b d}, \frac{6 \, \sqrt{\frac{1}{3}} a b \sqrt{\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}} \arctan\left(\frac{\sqrt{\frac{1}{3}} {\left(2 \, \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) - \left(a^{2} b\right)^{\frac{1}{3}} a\right)} \sqrt{\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}}}{a^{2}}\right) - \left(a^{2} b\right)^{\frac{2}{3}} \log\left(-a b \cos\left(d x + c\right)^{2} + a b - \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{1}{3}} a\right) + 2 \, \left(a^{2} b\right)^{\frac{2}{3}} \log\left(a b \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{2}{3}}\right)}{6 \, a^{2} b d}\right]"," ",0,"[1/6*(3*sqrt(1/3)*a*b*sqrt(-(a^2*b)^(1/3)/b)*log(-(3*(a^2*b)^(1/3)*a*sin(d*x + c) + a^2 + 3*sqrt(1/3)*(2*a*b*cos(d*x + c)^2 - 2*a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b) + 2*(a*b*cos(d*x + c)^2 - a*b)*sin(d*x + c))/((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)) - (a^2*b)^(2/3)*log(-a*b*cos(d*x + c)^2 + a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a) + 2*(a^2*b)^(2/3)*log(a*b*sin(d*x + c) + (a^2*b)^(2/3)))/(a^2*b*d), 1/6*(6*sqrt(1/3)*a*b*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*sin(d*x + c) - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - (a^2*b)^(2/3)*log(-a*b*cos(d*x + c)^2 + a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a) + 2*(a^2*b)^(2/3)*log(a*b*sin(d*x + c) + (a^2*b)^(2/3)))/(a^2*b*d)]","A",0
386,1,4396,0,3.071384," ","integrate(sec(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","-\frac{2 \, {\left(a^{2} - b^{2}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d \log\left(-\frac{1}{36} \, {\left(a^{5} - a^{3} b^{2}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)}^{2} d^{2} + a b^{2} + \frac{1}{6} \, {\left(2 \, a^{3} b + a b^{3}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d - {\left(a^{2} b + b^{3}\right)} \sin\left(d x + c\right)\right) - {\left({\left(a^{2} - b^{2}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d - 3 \, \sqrt{\frac{1}{3}} {\left(a^{2} - b^{2}\right)} d \sqrt{-\frac{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)}^{2} d^{2} - 12 \, {\left(a^{2} b - b^{3}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d - 108 \, b^{2}}{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d^{2}}} - 18 \, b\right)} \log\left(\frac{1}{36} \, {\left(a^{5} - a^{3} b^{2}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)}^{2} d^{2} - a b^{2} - \frac{1}{6} \, {\left(2 \, a^{3} b + a b^{3}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d + \frac{1}{12} \, \sqrt{\frac{1}{3}} {\left({\left(a^{5} - a^{3} b^{2}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d^{2} - 6 \, {\left(a^{3} b - a b^{3}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)}^{2} d^{2} - 12 \, {\left(a^{2} b - b^{3}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d - 108 \, b^{2}}{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d^{2}}} - 2 \, {\left(a^{2} b + b^{3}\right)} \sin\left(d x + c\right)\right) - {\left({\left(a^{2} - b^{2}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d + 3 \, \sqrt{\frac{1}{3}} {\left(a^{2} - b^{2}\right)} d \sqrt{-\frac{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)}^{2} d^{2} - 12 \, {\left(a^{2} b - b^{3}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d - 108 \, b^{2}}{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d^{2}}} - 18 \, b\right)} \log\left(-\frac{1}{36} \, {\left(a^{5} - a^{3} b^{2}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)}^{2} d^{2} + a b^{2} + \frac{1}{6} \, {\left(2 \, a^{3} b + a b^{3}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d + \frac{1}{12} \, \sqrt{\frac{1}{3}} {\left({\left(a^{5} - a^{3} b^{2}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d^{2} - 6 \, {\left(a^{3} b - a b^{3}\right)} d\right)} \sqrt{-\frac{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)}^{2} d^{2} - 12 \, {\left(a^{2} b - b^{3}\right)} {\left(9 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}} + \frac{b^{2} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(a^{2} d - b^{2} d\right)}^{2} {\left(-\frac{b}{54 \, {\left(a^{4} d^{3} - a^{2} b^{2} d^{3}\right)}} - \frac{b^{3}}{27 \, {\left(a^{2} d - b^{2} d\right)}^{3}} + \frac{{\left(a^{2} + b^{2}\right)} b}{54 \, {\left(a^{2} - b^{2}\right)}^{2} a^{2} d^{3}}\right)}^{\frac{1}{3}}} + \frac{6 \, b}{a^{2} d - b^{2} d}\right)} d - 108 \, b^{2}}{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d^{2}}} + 2 \, {\left(a^{2} b + b^{3}\right)} \sin\left(d x + c\right)\right) - 18 \, {\left(a + b\right)} \log\left(\sin\left(d x + c\right) + 1\right) + 18 \, {\left(a - b\right)} \log\left(-\sin\left(d x + c\right) + 1\right)}{36 \, {\left(a^{2} - b^{2}\right)} d}"," ",0,"-1/36*(2*(a^2 - b^2)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d*log(-1/36*(a^5 - a^3*b^2)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))^2*d^2 + a*b^2 + 1/6*(2*a^3*b + a*b^3)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d - (a^2*b + b^3)*sin(d*x + c)) - ((a^2 - b^2)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d - 3*sqrt(1/3)*(a^2 - b^2)*d*sqrt(-((a^4 - 2*a^2*b^2 + b^4)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))^2*d^2 - 12*(a^2*b - b^3)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d - 108*b^2)/((a^4 - 2*a^2*b^2 + b^4)*d^2)) - 18*b)*log(1/36*(a^5 - a^3*b^2)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))^2*d^2 - a*b^2 - 1/6*(2*a^3*b + a*b^3)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d + 1/12*sqrt(1/3)*((a^5 - a^3*b^2)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d^2 - 6*(a^3*b - a*b^3)*d)*sqrt(-((a^4 - 2*a^2*b^2 + b^4)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))^2*d^2 - 12*(a^2*b - b^3)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d - 108*b^2)/((a^4 - 2*a^2*b^2 + b^4)*d^2)) - 2*(a^2*b + b^3)*sin(d*x + c)) - ((a^2 - b^2)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d + 3*sqrt(1/3)*(a^2 - b^2)*d*sqrt(-((a^4 - 2*a^2*b^2 + b^4)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))^2*d^2 - 12*(a^2*b - b^3)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d - 108*b^2)/((a^4 - 2*a^2*b^2 + b^4)*d^2)) - 18*b)*log(-1/36*(a^5 - a^3*b^2)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))^2*d^2 + a*b^2 + 1/6*(2*a^3*b + a*b^3)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d + 1/12*sqrt(1/3)*((a^5 - a^3*b^2)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d^2 - 6*(a^3*b - a*b^3)*d)*sqrt(-((a^4 - 2*a^2*b^2 + b^4)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))^2*d^2 - 12*(a^2*b - b^3)*(9*(I*sqrt(3) + 1)*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3) + b^2*(-I*sqrt(3) + 1)/((a^2*d - b^2*d)^2*(-1/54*b/(a^4*d^3 - a^2*b^2*d^3) - 1/27*b^3/(a^2*d - b^2*d)^3 + 1/54*(a^2 + b^2)*b/((a^2 - b^2)^2*a^2*d^3))^(1/3)) + 6*b/(a^2*d - b^2*d))*d - 108*b^2)/((a^4 - 2*a^2*b^2 + b^4)*d^2)) + 2*(a^2*b + b^3)*sin(d*x + c)) - 18*(a + b)*log(sin(d*x + c) + 1) + 18*(a - b)*log(-sin(d*x + c) + 1))/((a^2 - b^2)*d)","C",0
387,1,10135,0,5.441251," ","integrate(sec(d*x+c)^3/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\frac{2 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d \cos\left(d x + c\right)^{2} \log\left(7 \, a^{3} b^{2} + 2 \, a b^{4} + \frac{3}{4} \, {\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} - \frac{1}{2} \, {\left(10 \, a^{5} b + 16 \, a^{3} b^{3} + a b^{5}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d - {\left(8 \, a^{2} b^{3} + b^{5}\right)} \sin\left(d x + c\right)\right) + 3 \, {\left(a^{3} - 2 \, a^{2} b - 7 \, a b^{2} - 4 \, b^{3}\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) - 3 \, {\left(a^{3} + 2 \, a^{2} b - 7 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 6 \, a^{2} b + 6 \, b^{3} - {\left({\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d \cos\left(d x + c\right)^{2} - 3 \, \sqrt{\frac{1}{3}} {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d \sqrt{-\frac{4 \, a^{4} b^{2} - 80 \, a^{2} b^{4} - 32 \, b^{6} + {\left(a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} - 4 \, {\left(a^{6} b - 3 \, a^{2} b^{5} + 2 \, b^{7}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right)} d^{2}}} \cos\left(d x + c\right)^{2} - 6 \, {\left(a^{2} b + 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(7 \, a^{3} b^{2} + 2 \, a b^{4} + \frac{3}{4} \, {\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} - \frac{1}{2} \, {\left(10 \, a^{5} b + 16 \, a^{3} b^{3} + a b^{5}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d + \frac{3}{4} \, \sqrt{\frac{1}{3}} {\left(3 \, {\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d^{2} + 2 \, {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d\right)} \sqrt{-\frac{4 \, a^{4} b^{2} - 80 \, a^{2} b^{4} - 32 \, b^{6} + {\left(a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} - 4 \, {\left(a^{6} b - 3 \, a^{2} b^{5} + 2 \, b^{7}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right)} d^{2}}} + 2 \, {\left(8 \, a^{2} b^{3} + b^{5}\right)} \sin\left(d x + c\right)\right) - {\left({\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d \cos\left(d x + c\right)^{2} + 3 \, \sqrt{\frac{1}{3}} {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d \sqrt{-\frac{4 \, a^{4} b^{2} - 80 \, a^{2} b^{4} - 32 \, b^{6} + {\left(a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} - 4 \, {\left(a^{6} b - 3 \, a^{2} b^{5} + 2 \, b^{7}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right)} d^{2}}} \cos\left(d x + c\right)^{2} - 6 \, {\left(a^{2} b + 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-7 \, a^{3} b^{2} - 2 \, a b^{4} - \frac{3}{4} \, {\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} + \frac{1}{2} \, {\left(10 \, a^{5} b + 16 \, a^{3} b^{3} + a b^{5}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d + \frac{3}{4} \, \sqrt{\frac{1}{3}} {\left(3 \, {\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d^{2} + 2 \, {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d\right)} \sqrt{-\frac{4 \, a^{4} b^{2} - 80 \, a^{2} b^{4} - 32 \, b^{6} + {\left(a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} - 4 \, {\left(a^{6} b - 3 \, a^{2} b^{5} + 2 \, b^{7}\right)} {\left(\frac{2 \, \left(\frac{1}{2}\right)^{\frac{2}{3}} {\left(\frac{b^{2}}{a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)}^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}}} - \left(\frac{1}{2}\right)^{\frac{1}{3}} {\left(\frac{b^{3}}{a^{6} d^{3} - 2 \, a^{4} b^{2} d^{3} + a^{2} b^{4} d^{3}} - \frac{3 \, {\left(a^{2} b + 2 \, b^{3}\right)} b^{2}}{{\left(a^{4} d^{2} - 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}^{3}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{{\left(8 \, a^{2} + b^{2}\right)} b^{5}}{{\left(a^{2} - b^{2}\right)}^{4} a^{2} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{2 \, {\left(a^{2} b + 2 \, b^{3}\right)}}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right)} d^{2}}} - 2 \, {\left(8 \, a^{2} b^{3} + b^{5}\right)} \sin\left(d x + c\right)\right) + 6 \, {\left(a^{3} - a b^{2}\right)} \sin\left(d x + c\right)}{12 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} d \cos\left(d x + c\right)^{2}}"," ",0,"1/12*(2*(a^4 - 2*a^2*b^2 + b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d*cos(d*x + c)^2*log(7*a^3*b^2 + 2*a*b^4 + 3/4*(a^7 - 2*a^5*b^2 + a^3*b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 - 1/2*(10*a^5*b + 16*a^3*b^3 + a*b^5)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d - (8*a^2*b^3 + b^5)*sin(d*x + c)) + 3*(a^3 - 2*a^2*b - 7*a*b^2 - 4*b^3)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - 3*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 6*a^2*b + 6*b^3 - ((a^4 - 2*a^2*b^2 + b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d*cos(d*x + c)^2 - 3*sqrt(1/3)*(a^4 - 2*a^2*b^2 + b^4)*d*sqrt(-(4*a^4*b^2 - 80*a^2*b^4 - 32*b^6 + (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 - 4*(a^6*b - 3*a^2*b^5 + 2*b^7)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2))*cos(d*x + c)^2 - 6*(a^2*b + 2*b^3)*cos(d*x + c)^2)*log(7*a^3*b^2 + 2*a*b^4 + 3/4*(a^7 - 2*a^5*b^2 + a^3*b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 - 1/2*(10*a^5*b + 16*a^3*b^3 + a*b^5)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d + 3/4*sqrt(1/3)*(3*(a^7 - 2*a^5*b^2 + a^3*b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d^2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d)*sqrt(-(4*a^4*b^2 - 80*a^2*b^4 - 32*b^6 + (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 - 4*(a^6*b - 3*a^2*b^5 + 2*b^7)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2)) + 2*(8*a^2*b^3 + b^5)*sin(d*x + c)) - ((a^4 - 2*a^2*b^2 + b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d*cos(d*x + c)^2 + 3*sqrt(1/3)*(a^4 - 2*a^2*b^2 + b^4)*d*sqrt(-(4*a^4*b^2 - 80*a^2*b^4 - 32*b^6 + (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 - 4*(a^6*b - 3*a^2*b^5 + 2*b^7)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2))*cos(d*x + c)^2 - 6*(a^2*b + 2*b^3)*cos(d*x + c)^2)*log(-7*a^3*b^2 - 2*a*b^4 - 3/4*(a^7 - 2*a^5*b^2 + a^3*b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + 1/2*(10*a^5*b + 16*a^3*b^3 + a*b^5)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d + 3/4*sqrt(1/3)*(3*(a^7 - 2*a^5*b^2 + a^3*b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d^2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*d)*sqrt(-(4*a^4*b^2 - 80*a^2*b^4 - 32*b^6 + (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 - 4*(a^6*b - 3*a^2*b^5 + 2*b^7)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2)) - 2*(8*a^2*b^3 + b^5)*sin(d*x + c)) + 6*(a^3 - a*b^2)*sin(d*x + c))/((a^4 - 2*a^2*b^2 + b^4)*d*cos(d*x + c)^2)","C",0
388,-1,0,0,0.000000," ","integrate(cos(d*x+c)^4/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
389,-1,0,0,0.000000," ","integrate(cos(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
390,-1,0,0,0.000000," ","integrate(1/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
391,-1,0,0,0.000000," ","integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
392,-1,0,0,0.000000," ","integrate(sec(d*x+c)^4/(a+b*sin(d*x+c)^3),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
393,-1,0,0,0.000000," ","integrate(cos(d*x+c)^7/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
394,-1,0,0,0.000000," ","integrate(cos(d*x+c)^5/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
395,1,665,0,1.435407," ","integrate(cos(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")","\left[\frac{3 \, a^{2} b \sin\left(d x + c\right) + 3 \, a^{3} + 3 \, \sqrt{\frac{1}{3}} {\left(a^{2} b - {\left(a b^{2} \cos\left(d x + c\right)^{2} - a b^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{-\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}} \log\left(-\frac{3 \, \left(a^{2} b\right)^{\frac{1}{3}} a \sin\left(d x + c\right) + a^{2} + 3 \, \sqrt{\frac{1}{3}} {\left(2 \, a b \cos\left(d x + c\right)^{2} - 2 \, a b - \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{1}{3}} a\right)} \sqrt{-\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}} + 2 \, {\left(a b \cos\left(d x + c\right)^{2} - a b\right)} \sin\left(d x + c\right)}{{\left(b \cos\left(d x + c\right)^{2} - b\right)} \sin\left(d x + c\right) - a}\right) + \left(a^{2} b\right)^{\frac{2}{3}} {\left({\left(b \cos\left(d x + c\right)^{2} - b\right)} \sin\left(d x + c\right) - a\right)} \log\left(-a b \cos\left(d x + c\right)^{2} + a b - \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{1}{3}} a\right) - 2 \, \left(a^{2} b\right)^{\frac{2}{3}} {\left({\left(b \cos\left(d x + c\right)^{2} - b\right)} \sin\left(d x + c\right) - a\right)} \log\left(a b \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{2}{3}}\right)}{9 \, {\left(a^{4} b d - {\left(a^{3} b^{2} d \cos\left(d x + c\right)^{2} - a^{3} b^{2} d\right)} \sin\left(d x + c\right)\right)}}, \frac{3 \, a^{2} b \sin\left(d x + c\right) + 3 \, a^{3} + 6 \, \sqrt{\frac{1}{3}} {\left(a^{2} b - {\left(a b^{2} \cos\left(d x + c\right)^{2} - a b^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}} \arctan\left(\frac{\sqrt{\frac{1}{3}} {\left(2 \, \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) - \left(a^{2} b\right)^{\frac{1}{3}} a\right)} \sqrt{\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}}}{a^{2}}\right) + \left(a^{2} b\right)^{\frac{2}{3}} {\left({\left(b \cos\left(d x + c\right)^{2} - b\right)} \sin\left(d x + c\right) - a\right)} \log\left(-a b \cos\left(d x + c\right)^{2} + a b - \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{1}{3}} a\right) - 2 \, \left(a^{2} b\right)^{\frac{2}{3}} {\left({\left(b \cos\left(d x + c\right)^{2} - b\right)} \sin\left(d x + c\right) - a\right)} \log\left(a b \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{2}{3}}\right)}{9 \, {\left(a^{4} b d - {\left(a^{3} b^{2} d \cos\left(d x + c\right)^{2} - a^{3} b^{2} d\right)} \sin\left(d x + c\right)\right)}}\right]"," ",0,"[1/9*(3*a^2*b*sin(d*x + c) + 3*a^3 + 3*sqrt(1/3)*(a^2*b - (a*b^2*cos(d*x + c)^2 - a*b^2)*sin(d*x + c))*sqrt(-(a^2*b)^(1/3)/b)*log(-(3*(a^2*b)^(1/3)*a*sin(d*x + c) + a^2 + 3*sqrt(1/3)*(2*a*b*cos(d*x + c)^2 - 2*a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b) + 2*(a*b*cos(d*x + c)^2 - a*b)*sin(d*x + c))/((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)) + (a^2*b)^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(-a*b*cos(d*x + c)^2 + a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a) - 2*(a^2*b)^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(a*b*sin(d*x + c) + (a^2*b)^(2/3)))/(a^4*b*d - (a^3*b^2*d*cos(d*x + c)^2 - a^3*b^2*d)*sin(d*x + c)), 1/9*(3*a^2*b*sin(d*x + c) + 3*a^3 + 6*sqrt(1/3)*(a^2*b - (a*b^2*cos(d*x + c)^2 - a*b^2)*sin(d*x + c))*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*sin(d*x + c) - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) + (a^2*b)^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(-a*b*cos(d*x + c)^2 + a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a) - 2*(a^2*b)^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(a*b*sin(d*x + c) + (a^2*b)^(2/3)))/(a^4*b*d - (a^3*b^2*d*cos(d*x + c)^2 - a^3*b^2*d)*sin(d*x + c))]","A",0
396,1,655,0,1.257766," ","integrate(cos(d*x+c)/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")","\left[\frac{3 \, a^{2} b \sin\left(d x + c\right) + 3 \, \sqrt{\frac{1}{3}} {\left(a^{2} b - {\left(a b^{2} \cos\left(d x + c\right)^{2} - a b^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{-\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}} \log\left(-\frac{3 \, \left(a^{2} b\right)^{\frac{1}{3}} a \sin\left(d x + c\right) + a^{2} + 3 \, \sqrt{\frac{1}{3}} {\left(2 \, a b \cos\left(d x + c\right)^{2} - 2 \, a b - \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{1}{3}} a\right)} \sqrt{-\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}} + 2 \, {\left(a b \cos\left(d x + c\right)^{2} - a b\right)} \sin\left(d x + c\right)}{{\left(b \cos\left(d x + c\right)^{2} - b\right)} \sin\left(d x + c\right) - a}\right) + \left(a^{2} b\right)^{\frac{2}{3}} {\left({\left(b \cos\left(d x + c\right)^{2} - b\right)} \sin\left(d x + c\right) - a\right)} \log\left(-a b \cos\left(d x + c\right)^{2} + a b - \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{1}{3}} a\right) - 2 \, \left(a^{2} b\right)^{\frac{2}{3}} {\left({\left(b \cos\left(d x + c\right)^{2} - b\right)} \sin\left(d x + c\right) - a\right)} \log\left(a b \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{2}{3}}\right)}{9 \, {\left(a^{4} b d - {\left(a^{3} b^{2} d \cos\left(d x + c\right)^{2} - a^{3} b^{2} d\right)} \sin\left(d x + c\right)\right)}}, \frac{3 \, a^{2} b \sin\left(d x + c\right) + 6 \, \sqrt{\frac{1}{3}} {\left(a^{2} b - {\left(a b^{2} \cos\left(d x + c\right)^{2} - a b^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}} \arctan\left(\frac{\sqrt{\frac{1}{3}} {\left(2 \, \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) - \left(a^{2} b\right)^{\frac{1}{3}} a\right)} \sqrt{\frac{\left(a^{2} b\right)^{\frac{1}{3}}}{b}}}{a^{2}}\right) + \left(a^{2} b\right)^{\frac{2}{3}} {\left({\left(b \cos\left(d x + c\right)^{2} - b\right)} \sin\left(d x + c\right) - a\right)} \log\left(-a b \cos\left(d x + c\right)^{2} + a b - \left(a^{2} b\right)^{\frac{2}{3}} \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{1}{3}} a\right) - 2 \, \left(a^{2} b\right)^{\frac{2}{3}} {\left({\left(b \cos\left(d x + c\right)^{2} - b\right)} \sin\left(d x + c\right) - a\right)} \log\left(a b \sin\left(d x + c\right) + \left(a^{2} b\right)^{\frac{2}{3}}\right)}{9 \, {\left(a^{4} b d - {\left(a^{3} b^{2} d \cos\left(d x + c\right)^{2} - a^{3} b^{2} d\right)} \sin\left(d x + c\right)\right)}}\right]"," ",0,"[1/9*(3*a^2*b*sin(d*x + c) + 3*sqrt(1/3)*(a^2*b - (a*b^2*cos(d*x + c)^2 - a*b^2)*sin(d*x + c))*sqrt(-(a^2*b)^(1/3)/b)*log(-(3*(a^2*b)^(1/3)*a*sin(d*x + c) + a^2 + 3*sqrt(1/3)*(2*a*b*cos(d*x + c)^2 - 2*a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b) + 2*(a*b*cos(d*x + c)^2 - a*b)*sin(d*x + c))/((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)) + (a^2*b)^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(-a*b*cos(d*x + c)^2 + a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a) - 2*(a^2*b)^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(a*b*sin(d*x + c) + (a^2*b)^(2/3)))/(a^4*b*d - (a^3*b^2*d*cos(d*x + c)^2 - a^3*b^2*d)*sin(d*x + c)), 1/9*(3*a^2*b*sin(d*x + c) + 6*sqrt(1/3)*(a^2*b - (a*b^2*cos(d*x + c)^2 - a*b^2)*sin(d*x + c))*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*sin(d*x + c) - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) + (a^2*b)^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(-a*b*cos(d*x + c)^2 + a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a) - 2*(a^2*b)^(2/3)*((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)*log(a*b*sin(d*x + c) + (a^2*b)^(2/3)))/(a^4*b*d - (a^3*b^2*d*cos(d*x + c)^2 - a^3*b^2*d)*sin(d*x + c))]","B",0
397,1,10855,0,8.381583," ","integrate(sec(d*x+c)/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")","\frac{216 \, a^{3} b - 216 \, a b^{3} - 108 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(d x + c\right)^{2} - 2 \, {\left({\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d - {\left({\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d\right)} \sin\left(d x + c\right)\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} \log\left(-56 \, a^{5} b^{2} + 20 \, a^{3} b^{4} + \frac{1}{324} \, {\left(2 \, a^{11} - 3 \, a^{9} b^{2} + a^{5} b^{6}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} - \frac{1}{9} \, {\left(12 \, a^{8} b + 22 \, a^{6} b^{3} - 8 \, a^{4} b^{5} + a^{2} b^{7}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d + 4 \, {\left(8 \, a^{6} b + 28 \, a^{4} b^{3} - 10 \, a^{2} b^{5} + b^{7}\right)} \sin\left(d x + c\right)\right) - {\left(324 \, a^{3} b - {\left({\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d - {\left({\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d\right)} \sin\left(d x + c\right)\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} + 3 \, \sqrt{\frac{1}{3}} {\left({\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d - {\left({\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d\right)} \sin\left(d x + c\right)\right)} \sqrt{\frac{29808 \, a^{4} b^{2} + 10368 \, a^{2} b^{4} - 5184 \, b^{6} - {\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} + 216 \, {\left(a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} d^{2}}} - 324 \, {\left(a^{2} b^{2} \cos\left(d x + c\right)^{2} - a^{2} b^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(56 \, a^{5} b^{2} - 20 \, a^{3} b^{4} - \frac{1}{324} \, {\left(2 \, a^{11} - 3 \, a^{9} b^{2} + a^{5} b^{6}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} + \frac{1}{9} \, {\left(12 \, a^{8} b + 22 \, a^{6} b^{3} - 8 \, a^{4} b^{5} + a^{2} b^{7}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d - \frac{1}{108} \, \sqrt{\frac{1}{3}} {\left({\left(2 \, a^{11} - 3 \, a^{9} b^{2} + a^{5} b^{6}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d^{2} - 36 \, {\left(6 \, a^{8} b - 13 \, a^{6} b^{3} + 8 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)} \sqrt{\frac{29808 \, a^{4} b^{2} + 10368 \, a^{2} b^{4} - 5184 \, b^{6} - {\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} + 216 \, {\left(a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} d^{2}}} + 8 \, {\left(8 \, a^{6} b + 28 \, a^{4} b^{3} - 10 \, a^{2} b^{5} + b^{7}\right)} \sin\left(d x + c\right)\right) - {\left(324 \, a^{3} b - {\left({\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d - {\left({\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d\right)} \sin\left(d x + c\right)\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} - 3 \, \sqrt{\frac{1}{3}} {\left({\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d - {\left({\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d\right)} \sin\left(d x + c\right)\right)} \sqrt{\frac{29808 \, a^{4} b^{2} + 10368 \, a^{2} b^{4} - 5184 \, b^{6} - {\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} + 216 \, {\left(a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} d^{2}}} - 324 \, {\left(a^{2} b^{2} \cos\left(d x + c\right)^{2} - a^{2} b^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-56 \, a^{5} b^{2} + 20 \, a^{3} b^{4} + \frac{1}{324} \, {\left(2 \, a^{11} - 3 \, a^{9} b^{2} + a^{5} b^{6}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} - \frac{1}{9} \, {\left(12 \, a^{8} b + 22 \, a^{6} b^{3} - 8 \, a^{4} b^{5} + a^{2} b^{7}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d - \frac{1}{108} \, \sqrt{\frac{1}{3}} {\left({\left(2 \, a^{11} - 3 \, a^{9} b^{2} + a^{5} b^{6}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d^{2} - 36 \, {\left(6 \, a^{8} b - 13 \, a^{6} b^{3} + 8 \, a^{4} b^{5} - a^{2} b^{7}\right)} d\right)} \sqrt{\frac{29808 \, a^{4} b^{2} + 10368 \, a^{2} b^{4} - 5184 \, b^{6} - {\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)}^{2} d^{2} + 216 \, {\left(a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right)} {\left(\frac{4 \, {\left(\frac{9 \, a^{2} b^{2}}{{\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{2}} - \frac{b^{2}}{a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}}\right)} {\left(-i \, \sqrt{3} + 1\right)}}{{\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}}} + 81 \, {\left(-\frac{8 \, a^{3} b^{3}}{27 \, {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}^{3}} + \frac{4 \, a b^{3}}{81 \, {\left(a^{6} d^{2} - 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right)} {\left(a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d\right)}} - \frac{4 \, {\left(8 \, a^{2} b - b^{3}\right)}}{729 \, {\left(a^{9} d^{3} - 2 \, a^{7} b^{2} d^{3} + a^{5} b^{4} d^{3}\right)}} + \frac{4 \, {\left(8 \, a^{6} + 28 \, a^{4} b^{2} - 10 \, a^{2} b^{4} + b^{6}\right)} b}{729 \, {\left(a^{2} - b^{2}\right)}^{4} a^{5} d^{3}}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)} + \frac{108 \, a b}{a^{4} d - 2 \, a^{2} b^{2} d + b^{4} d}\right)} d}{{\left(a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right)} d^{2}}} - 8 \, {\left(8 \, a^{6} b + 28 \, a^{4} b^{3} - 10 \, a^{2} b^{5} + b^{7}\right)} \sin\left(d x + c\right)\right) + 162 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2} + {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3} - {\left(a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(\sin\left(d x + c\right) + 1\right) - 162 \, {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2} + {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3} - {\left(a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \log\left(-\sin\left(d x + c\right) + 1\right) - 108 \, {\left(a^{2} b^{2} - b^{4}\right)} \sin\left(d x + c\right)}{324 \, {\left({\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} d - {\left({\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right)} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/324*(216*a^3*b - 216*a*b^3 - 108*(a^3*b - a*b^3)*cos(d*x + c)^2 - 2*((a^6 - 2*a^4*b^2 + a^2*b^4)*d - ((a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)^2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))*log(-56*a^5*b^2 + 20*a^3*b^4 + 1/324*(2*a^11 - 3*a^9*b^2 + a^5*b^6)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 - 1/9*(12*a^8*b + 22*a^6*b^3 - 8*a^4*b^5 + a^2*b^7)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))*d + 4*(8*a^6*b + 28*a^4*b^3 - 10*a^2*b^5 + b^7)*sin(d*x + c)) - (324*a^3*b - ((a^6 - 2*a^4*b^2 + a^2*b^4)*d - ((a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)^2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d)) + 3*sqrt(1/3)*((a^6 - 2*a^4*b^2 + a^2*b^4)*d - ((a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)^2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))*sqrt((29808*a^4*b^2 + 10368*a^2*b^4 - 5184*b^6 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + 216*(a^7*b - 2*a^5*b^3 + a^3*b^5)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*d^2)) - 324*(a^2*b^2*cos(d*x + c)^2 - a^2*b^2)*sin(d*x + c))*log(56*a^5*b^2 - 20*a^3*b^4 - 1/324*(2*a^11 - 3*a^9*b^2 + a^5*b^6)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + 1/9*(12*a^8*b + 22*a^6*b^3 - 8*a^4*b^5 + a^2*b^7)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))*d - 1/108*sqrt(1/3)*((2*a^11 - 3*a^9*b^2 + a^5*b^6)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))*d^2 - 36*(6*a^8*b - 13*a^6*b^3 + 8*a^4*b^5 - a^2*b^7)*d)*sqrt((29808*a^4*b^2 + 10368*a^2*b^4 - 5184*b^6 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + 216*(a^7*b - 2*a^5*b^3 + a^3*b^5)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*d^2)) + 8*(8*a^6*b + 28*a^4*b^3 - 10*a^2*b^5 + b^7)*sin(d*x + c)) - (324*a^3*b - ((a^6 - 2*a^4*b^2 + a^2*b^4)*d - ((a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)^2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d)) - 3*sqrt(1/3)*((a^6 - 2*a^4*b^2 + a^2*b^4)*d - ((a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)^2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))*sqrt((29808*a^4*b^2 + 10368*a^2*b^4 - 5184*b^6 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + 216*(a^7*b - 2*a^5*b^3 + a^3*b^5)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*d^2)) - 324*(a^2*b^2*cos(d*x + c)^2 - a^2*b^2)*sin(d*x + c))*log(-56*a^5*b^2 + 20*a^3*b^4 + 1/324*(2*a^11 - 3*a^9*b^2 + a^5*b^6)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 - 1/9*(12*a^8*b + 22*a^6*b^3 - 8*a^4*b^5 + a^2*b^7)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))*d - 1/108*sqrt(1/3)*((2*a^11 - 3*a^9*b^2 + a^5*b^6)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))*d^2 - 36*(6*a^8*b - 13*a^6*b^3 + 8*a^4*b^5 - a^2*b^7)*d)*sqrt((29808*a^4*b^2 + 10368*a^2*b^4 - 5184*b^6 - (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 + 216*(a^7*b - 2*a^5*b^3 + a^3*b^5)*(4*(9*a^2*b^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2 - b^2/(a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2))*(-I*sqrt(3) + 1)/(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3) + 81*(-8/27*a^3*b^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + 4/81*a*b^3/((a^6*d^2 - 2*a^4*b^2*d^2 + a^2*b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) - 4/729*(8*a^2*b - b^3)/(a^9*d^3 - 2*a^7*b^2*d^3 + a^5*b^4*d^3) + 4/729*(8*a^6 + 28*a^4*b^2 - 10*a^2*b^4 + b^6)*b/((a^2 - b^2)^4*a^5*d^3))^(1/3)*(I*sqrt(3) + 1) + 108*a*b/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*d^2)) - 8*(8*a^6*b + 28*a^4*b^3 - 10*a^2*b^5 + b^7)*sin(d*x + c)) + 162*(a^4 + 2*a^3*b + a^2*b^2 + (a^3*b + 2*a^2*b^2 + a*b^3 - (a^3*b + 2*a^2*b^2 + a*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(sin(d*x + c) + 1) - 162*(a^4 - 2*a^3*b + a^2*b^2 + (a^3*b - 2*a^2*b^2 + a*b^3 - (a^3*b - 2*a^2*b^2 + a*b^3)*cos(d*x + c)^2)*sin(d*x + c))*log(-sin(d*x + c) + 1) - 108*(a^2*b^2 - b^4)*sin(d*x + c))/((a^6 - 2*a^4*b^2 + a^2*b^4)*d - ((a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)^2 - (a^5*b - 2*a^3*b^3 + a*b^5)*d)*sin(d*x + c))","C",0
398,-1,0,0,0.000000," ","integrate(sec(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
399,-1,0,0,0.000000," ","integrate(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
400,-1,0,0,0.000000," ","integrate(cos(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
401,-1,0,0,0.000000," ","integrate(1/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
402,-1,0,0,0.000000," ","integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
403,-1,0,0,0.000000," ","integrate(sec(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
404,1,1429,0,1.497793," ","integrate(cos(d*x+c)^7/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{3 \, b d \sqrt{-\frac{a b^{3} d^{2} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + 6 \, a^{2} + 20 \, a b + 6 \, b^{2}}{a b^{3} d^{2}}} \log\left(\frac{1}{2} \, {\left(a^{6} + 12 \, a^{5} b - 27 \, a^{4} b^{2} + 27 \, a^{2} b^{4} - 12 \, a b^{5} - b^{6}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{4} b^{5} + 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - {\left(3 \, a^{5} b^{2} + 46 \, a^{4} b^{3} + 60 \, a^{3} b^{4} + 18 \, a^{2} b^{5} + a b^{6}\right)} d\right)} \sqrt{-\frac{a b^{3} d^{2} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + 6 \, a^{2} + 20 \, a b + 6 \, b^{2}}{a b^{3} d^{2}}}\right) - 3 \, b d \sqrt{\frac{a b^{3} d^{2} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - 6 \, a^{2} - 20 \, a b - 6 \, b^{2}}{a b^{3} d^{2}}} \log\left(\frac{1}{2} \, {\left(a^{6} + 12 \, a^{5} b - 27 \, a^{4} b^{2} + 27 \, a^{2} b^{4} - 12 \, a b^{5} - b^{6}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{4} b^{5} + 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + {\left(3 \, a^{5} b^{2} + 46 \, a^{4} b^{3} + 60 \, a^{3} b^{4} + 18 \, a^{2} b^{5} + a b^{6}\right)} d\right)} \sqrt{\frac{a b^{3} d^{2} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - 6 \, a^{2} - 20 \, a b - 6 \, b^{2}}{a b^{3} d^{2}}}\right) - 3 \, b d \sqrt{-\frac{a b^{3} d^{2} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + 6 \, a^{2} + 20 \, a b + 6 \, b^{2}}{a b^{3} d^{2}}} \log\left(-\frac{1}{2} \, {\left(a^{6} + 12 \, a^{5} b - 27 \, a^{4} b^{2} + 27 \, a^{2} b^{4} - 12 \, a b^{5} - b^{6}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{4} b^{5} + 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - {\left(3 \, a^{5} b^{2} + 46 \, a^{4} b^{3} + 60 \, a^{3} b^{4} + 18 \, a^{2} b^{5} + a b^{6}\right)} d\right)} \sqrt{-\frac{a b^{3} d^{2} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + 6 \, a^{2} + 20 \, a b + 6 \, b^{2}}{a b^{3} d^{2}}}\right) + 3 \, b d \sqrt{\frac{a b^{3} d^{2} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - 6 \, a^{2} - 20 \, a b - 6 \, b^{2}}{a b^{3} d^{2}}} \log\left(-\frac{1}{2} \, {\left(a^{6} + 12 \, a^{5} b - 27 \, a^{4} b^{2} + 27 \, a^{2} b^{4} - 12 \, a b^{5} - b^{6}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{4} b^{5} + 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + {\left(3 \, a^{5} b^{2} + 46 \, a^{4} b^{3} + 60 \, a^{3} b^{4} + 18 \, a^{2} b^{5} + a b^{6}\right)} d\right)} \sqrt{\frac{a b^{3} d^{2} \sqrt{\frac{a^{6} + 30 \, a^{5} b + 255 \, a^{4} b^{2} + 452 \, a^{3} b^{3} + 255 \, a^{2} b^{4} + 30 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - 6 \, a^{2} - 20 \, a b - 6 \, b^{2}}{a b^{3} d^{2}}}\right) - 4 \, {\left(\cos\left(d x + c\right)^{2} + 8\right)} \sin\left(d x + c\right)}{12 \, b d}"," ",0,"1/12*(3*b*d*sqrt(-(a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + 6*a^2 + 20*a*b + 6*b^2)/(a*b^3*d^2))*log(1/2*(a^6 + 12*a^5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - (3*a^5*b^2 + 46*a^4*b^3 + 60*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sqrt(-(a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + 6*a^2 + 20*a*b + 6*b^2)/(a*b^3*d^2))) - 3*b*d*sqrt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))*log(1/2*(a^6 + 12*a^5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + (3*a^5*b^2 + 46*a^4*b^3 + 60*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sqrt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))) - 3*b*d*sqrt(-(a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + 6*a^2 + 20*a*b + 6*b^2)/(a*b^3*d^2))*log(-1/2*(a^6 + 12*a^5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - (3*a^5*b^2 + 46*a^4*b^3 + 60*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sqrt(-(a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + 6*a^2 + 20*a*b + 6*b^2)/(a*b^3*d^2))) + 3*b*d*sqrt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))*log(-1/2*(a^6 + 12*a^5*b - 27*a^4*b^2 + 27*a^2*b^4 - 12*a*b^5 - b^6)*sin(d*x + c) + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) + (3*a^5*b^2 + 46*a^4*b^3 + 60*a^3*b^4 + 18*a^2*b^5 + a*b^6)*d)*sqrt((a*b^3*d^2*sqrt((a^6 + 30*a^5*b + 255*a^4*b^2 + 452*a^3*b^3 + 255*a^2*b^4 + 30*a*b^5 + b^6)/(a^3*b^7*d^4)) - 6*a^2 - 20*a*b - 6*b^2)/(a*b^3*d^2))) - 4*(cos(d*x + c)^2 + 8)*sin(d*x + c))/(b*d)","B",0
405,1,1041,0,1.576161," ","integrate(cos(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{b d \sqrt{-\frac{a b^{2} d^{2} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} + 4 \, a + 4 \, b}{a b^{2} d^{2}}} \log\left(\frac{1}{2} \, {\left(a^{4} + 4 \, a^{3} b - 10 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(2 \, a^{3} b^{4} d^{3} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} - {\left(a^{4} b + 7 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{-\frac{a b^{2} d^{2} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} + 4 \, a + 4 \, b}{a b^{2} d^{2}}}\right) - b d \sqrt{\frac{a b^{2} d^{2} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} - 4 \, a - 4 \, b}{a b^{2} d^{2}}} \log\left(\frac{1}{2} \, {\left(a^{4} + 4 \, a^{3} b - 10 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(2 \, a^{3} b^{4} d^{3} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} + {\left(a^{4} b + 7 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{\frac{a b^{2} d^{2} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} - 4 \, a - 4 \, b}{a b^{2} d^{2}}}\right) - b d \sqrt{-\frac{a b^{2} d^{2} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} + 4 \, a + 4 \, b}{a b^{2} d^{2}}} \log\left(-\frac{1}{2} \, {\left(a^{4} + 4 \, a^{3} b - 10 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(2 \, a^{3} b^{4} d^{3} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} - {\left(a^{4} b + 7 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{-\frac{a b^{2} d^{2} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} + 4 \, a + 4 \, b}{a b^{2} d^{2}}}\right) + b d \sqrt{\frac{a b^{2} d^{2} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} - 4 \, a - 4 \, b}{a b^{2} d^{2}}} \log\left(-\frac{1}{2} \, {\left(a^{4} + 4 \, a^{3} b - 10 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(2 \, a^{3} b^{4} d^{3} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} + {\left(a^{4} b + 7 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{\frac{a b^{2} d^{2} \sqrt{\frac{a^{4} + 12 \, a^{3} b + 38 \, a^{2} b^{2} + 12 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} - 4 \, a - 4 \, b}{a b^{2} d^{2}}}\right) + 4 \, \sin\left(d x + c\right)}{4 \, b d}"," ",0,"-1/4*(b*d*sqrt(-(a*b^2*d^2*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) + 4*a + 4*b)/(a*b^2*d^2))*log(1/2*(a^4 + 4*a^3*b - 10*a^2*b^2 + 4*a*b^3 + b^4)*sin(d*x + c) + 1/2*(2*a^3*b^4*d^3*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) - (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 + a*b^4)*d)*sqrt(-(a*b^2*d^2*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) + 4*a + 4*b)/(a*b^2*d^2))) - b*d*sqrt((a*b^2*d^2*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) - 4*a - 4*b)/(a*b^2*d^2))*log(1/2*(a^4 + 4*a^3*b - 10*a^2*b^2 + 4*a*b^3 + b^4)*sin(d*x + c) + 1/2*(2*a^3*b^4*d^3*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) + (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 + a*b^4)*d)*sqrt((a*b^2*d^2*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) - 4*a - 4*b)/(a*b^2*d^2))) - b*d*sqrt(-(a*b^2*d^2*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) + 4*a + 4*b)/(a*b^2*d^2))*log(-1/2*(a^4 + 4*a^3*b - 10*a^2*b^2 + 4*a*b^3 + b^4)*sin(d*x + c) + 1/2*(2*a^3*b^4*d^3*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) - (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 + a*b^4)*d)*sqrt(-(a*b^2*d^2*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) + 4*a + 4*b)/(a*b^2*d^2))) + b*d*sqrt((a*b^2*d^2*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) - 4*a - 4*b)/(a*b^2*d^2))*log(-1/2*(a^4 + 4*a^3*b - 10*a^2*b^2 + 4*a*b^3 + b^4)*sin(d*x + c) + 1/2*(2*a^3*b^4*d^3*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) + (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 + a*b^4)*d)*sqrt((a*b^2*d^2*sqrt((a^4 + 12*a^3*b + 38*a^2*b^2 + 12*a*b^3 + b^4)/(a^3*b^5*d^4)) - 4*a - 4*b)/(a*b^2*d^2))) + 4*sin(d*x + c))/(b*d)","B",0
406,1,631,0,1.164098," ","integrate(cos(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{1}{4} \, \sqrt{-\frac{a b d^{2} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}} \log\left(\frac{1}{2} \, {\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(a^{3} b^{2} d^{3} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - {\left(a^{2} b + a b^{2}\right)} d\right)} \sqrt{-\frac{a b d^{2} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}}\right) - \frac{1}{4} \, \sqrt{\frac{a b d^{2} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}} \log\left(\frac{1}{2} \, {\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(a^{3} b^{2} d^{3} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + {\left(a^{2} b + a b^{2}\right)} d\right)} \sqrt{\frac{a b d^{2} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}}\right) - \frac{1}{4} \, \sqrt{-\frac{a b d^{2} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}} \log\left(-\frac{1}{2} \, {\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(a^{3} b^{2} d^{3} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - {\left(a^{2} b + a b^{2}\right)} d\right)} \sqrt{-\frac{a b d^{2} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + 2}{a b d^{2}}}\right) + \frac{1}{4} \, \sqrt{\frac{a b d^{2} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}} \log\left(-\frac{1}{2} \, {\left(a^{2} - b^{2}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(a^{3} b^{2} d^{3} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + {\left(a^{2} b + a b^{2}\right)} d\right)} \sqrt{\frac{a b d^{2} \sqrt{\frac{a^{2} + 2 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - 2}{a b d^{2}}}\right)"," ",0,"1/4*sqrt(-(a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) + 2)/(a*b*d^2))*log(1/2*(a^2 - b^2)*sin(d*x + c) + 1/2*(a^3*b^2*d^3*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) - (a^2*b + a*b^2)*d)*sqrt(-(a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) + 2)/(a*b*d^2))) - 1/4*sqrt((a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) - 2)/(a*b*d^2))*log(1/2*(a^2 - b^2)*sin(d*x + c) + 1/2*(a^3*b^2*d^3*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) + (a^2*b + a*b^2)*d)*sqrt((a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) - 2)/(a*b*d^2))) - 1/4*sqrt(-(a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) + 2)/(a*b*d^2))*log(-1/2*(a^2 - b^2)*sin(d*x + c) + 1/2*(a^3*b^2*d^3*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) - (a^2*b + a*b^2)*d)*sqrt(-(a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) + 2)/(a*b*d^2))) + 1/4*sqrt((a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) - 2)/(a*b*d^2))*log(-1/2*(a^2 - b^2)*sin(d*x + c) + 1/2*(a^3*b^2*d^3*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) + (a^2*b + a*b^2)*d)*sqrt((a*b*d^2*sqrt((a^2 + 2*a*b + b^2)/(a^3*b^3*d^4)) - 2)/(a*b*d^2)))","B",0
407,-1,0,0,0.000000," ","integrate(cos(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
408,1,1329,0,1.198548," ","integrate(sec(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{{\left(a - b\right)} d \sqrt{\frac{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} + 2 \, b}{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \log\left(\frac{1}{2} \, {\left(a b + b^{2}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{3} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} - {\left(a^{2} b + a b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} + 2 \, b}{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2}}}\right) - {\left(a - b\right)} d \sqrt{-\frac{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} - 2 \, b}{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \log\left(\frac{1}{2} \, {\left(a b + b^{2}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{3} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} + {\left(a^{2} b + a b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} - 2 \, b}{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2}}}\right) - {\left(a - b\right)} d \sqrt{\frac{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} + 2 \, b}{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \log\left(-\frac{1}{2} \, {\left(a b + b^{2}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{3} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} - {\left(a^{2} b + a b^{2}\right)} d\right)} \sqrt{\frac{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} + 2 \, b}{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2}}}\right) + {\left(a - b\right)} d \sqrt{-\frac{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} - 2 \, b}{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2}}} \log\left(-\frac{1}{2} \, {\left(a b + b^{2}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} d^{3} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} + {\left(a^{2} b + a b^{2}\right)} d\right)} \sqrt{-\frac{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2} \sqrt{\frac{a^{2} b + 2 \, a b^{2} + b^{3}}{{\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{4}}} - 2 \, b}{{\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} d^{2}}}\right) - 2 \, \log\left(\sin\left(d x + c\right) + 1\right) + 2 \, \log\left(-\sin\left(d x + c\right) + 1\right)}{4 \, {\left(a - b\right)} d}"," ",0,"-1/4*((a - b)*d*sqrt(((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))*log(1/2*(a*b + b^2)*sin(d*x + c) + 1/2*((a^5 - 2*a^4*b + a^3*b^2)*d^3*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) - (a^2*b + a*b^2)*d)*sqrt(((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))) - (a - b)*d*sqrt(-((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))*log(1/2*(a*b + b^2)*sin(d*x + c) + 1/2*((a^5 - 2*a^4*b + a^3*b^2)*d^3*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + (a^2*b + a*b^2)*d)*sqrt(-((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))) - (a - b)*d*sqrt(((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))*log(-1/2*(a*b + b^2)*sin(d*x + c) + 1/2*((a^5 - 2*a^4*b + a^3*b^2)*d^3*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) - (a^2*b + a*b^2)*d)*sqrt(((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))) + (a - b)*d*sqrt(-((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))*log(-1/2*(a*b + b^2)*sin(d*x + c) + 1/2*((a^5 - 2*a^4*b + a^3*b^2)*d^3*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + (a^2*b + a*b^2)*d)*sqrt(-((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))) - 2*log(sin(d*x + c) + 1) + 2*log(-sin(d*x + c) + 1))/((a - b)*d)","B",0
409,1,2529,0,3.062491," ","integrate(sec(d*x+c)^3/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{{\left(a^{2} - 2 \, a b + b^{2}\right)} d \sqrt{\frac{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} + 4 \, a b^{2} + 4 \, b^{3}}{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2}}} \cos\left(d x + c\right)^{2} \log\left(\frac{1}{2} \, {\left(a^{2} b^{2} + 6 \, a b^{3} + b^{4}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(2 \, {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{3} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} - {\left(a^{4} b + 7 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} + 4 \, a b^{2} + 4 \, b^{3}}{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2}}}\right) - {\left(a^{2} - 2 \, a b + b^{2}\right)} d \sqrt{-\frac{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} - 4 \, a b^{2} - 4 \, b^{3}}{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2}}} \cos\left(d x + c\right)^{2} \log\left(\frac{1}{2} \, {\left(a^{2} b^{2} + 6 \, a b^{3} + b^{4}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(2 \, {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{3} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} + {\left(a^{4} b + 7 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} - 4 \, a b^{2} - 4 \, b^{3}}{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2}}}\right) - {\left(a^{2} - 2 \, a b + b^{2}\right)} d \sqrt{\frac{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} + 4 \, a b^{2} + 4 \, b^{3}}{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2}}} \cos\left(d x + c\right)^{2} \log\left(-\frac{1}{2} \, {\left(a^{2} b^{2} + 6 \, a b^{3} + b^{4}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(2 \, {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{3} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} - {\left(a^{4} b + 7 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{\frac{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} + 4 \, a b^{2} + 4 \, b^{3}}{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2}}}\right) + {\left(a^{2} - 2 \, a b + b^{2}\right)} d \sqrt{-\frac{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} - 4 \, a b^{2} - 4 \, b^{3}}{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2}}} \cos\left(d x + c\right)^{2} \log\left(-\frac{1}{2} \, {\left(a^{2} b^{2} + 6 \, a b^{3} + b^{4}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left(2 \, {\left(a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d^{3} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} + {\left(a^{4} b + 7 \, a^{3} b^{2} + 7 \, a^{2} b^{3} + a b^{4}\right)} d\right)} \sqrt{-\frac{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2} \sqrt{\frac{a^{4} b^{3} + 12 \, a^{3} b^{4} + 38 \, a^{2} b^{5} + 12 \, a b^{6} + b^{7}}{{\left(a^{11} - 8 \, a^{10} b + 28 \, a^{9} b^{2} - 56 \, a^{8} b^{3} + 70 \, a^{7} b^{4} - 56 \, a^{6} b^{5} + 28 \, a^{5} b^{6} - 8 \, a^{4} b^{7} + a^{3} b^{8}\right)} d^{4}}} - 4 \, a b^{2} - 4 \, b^{3}}{{\left(a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right)} d^{2}}}\right) - {\left(a - 5 \, b\right)} \cos\left(d x + c\right)^{2} \log\left(\sin\left(d x + c\right) + 1\right) + {\left(a - 5 \, b\right)} \cos\left(d x + c\right)^{2} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left(a - b\right)} \sin\left(d x + c\right)}{4 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} d \cos\left(d x + c\right)^{2}}"," ",0,"-1/4*((a^2 - 2*a*b + b^2)*d*sqrt(((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) + 4*a*b^2 + 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))*cos(d*x + c)^2*log(1/2*(a^2*b^2 + 6*a*b^3 + b^4)*sin(d*x + c) + 1/2*(2*(a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^3*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) - (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 + a*b^4)*d)*sqrt(((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) + 4*a*b^2 + 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))) - (a^2 - 2*a*b + b^2)*d*sqrt(-((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) - 4*a*b^2 - 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))*cos(d*x + c)^2*log(1/2*(a^2*b^2 + 6*a*b^3 + b^4)*sin(d*x + c) + 1/2*(2*(a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^3*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) + (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 + a*b^4)*d)*sqrt(-((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) - 4*a*b^2 - 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))) - (a^2 - 2*a*b + b^2)*d*sqrt(((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) + 4*a*b^2 + 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))*cos(d*x + c)^2*log(-1/2*(a^2*b^2 + 6*a*b^3 + b^4)*sin(d*x + c) + 1/2*(2*(a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^3*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) - (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 + a*b^4)*d)*sqrt(((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) + 4*a*b^2 + 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))) + (a^2 - 2*a*b + b^2)*d*sqrt(-((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) - 4*a*b^2 - 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))*cos(d*x + c)^2*log(-1/2*(a^2*b^2 + 6*a*b^3 + b^4)*sin(d*x + c) + 1/2*(2*(a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^3*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) + (a^4*b + 7*a^3*b^2 + 7*a^2*b^3 + a*b^4)*d)*sqrt(-((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2*sqrt((a^4*b^3 + 12*a^3*b^4 + 38*a^2*b^5 + 12*a*b^6 + b^7)/((a^11 - 8*a^10*b + 28*a^9*b^2 - 56*a^8*b^3 + 70*a^7*b^4 - 56*a^6*b^5 + 28*a^5*b^6 - 8*a^4*b^7 + a^3*b^8)*d^4)) - 4*a*b^2 - 4*b^3)/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*d^2))) - (a - 5*b)*cos(d*x + c)^2*log(sin(d*x + c) + 1) + (a - 5*b)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 2*(a - b)*sin(d*x + c))/((a^2 - 2*a*b + b^2)*d*cos(d*x + c)^2)","B",0
410,1,3703,0,5.613229," ","integrate(sec(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{4 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d \sqrt{\frac{6 \, a^{2} b^{3} + 20 \, a b^{4} + 6 \, b^{5} + {\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}}}{{\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2}}} \cos\left(d x + c\right)^{4} \log\left(\frac{1}{2} \, {\left(a^{3} b^{4} + 15 \, a^{2} b^{5} + 15 \, a b^{6} + b^{7}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{10} - 3 \, a^{9} b - 3 \, a^{8} b^{2} + 25 \, a^{7} b^{3} - 45 \, a^{6} b^{4} + 39 \, a^{5} b^{5} - 17 \, a^{4} b^{6} + 3 \, a^{3} b^{7}\right)} d^{3} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}} - {\left(3 \, a^{5} b^{3} + 46 \, a^{4} b^{4} + 60 \, a^{3} b^{5} + 18 \, a^{2} b^{6} + a b^{7}\right)} d\right)} \sqrt{\frac{6 \, a^{2} b^{3} + 20 \, a b^{4} + 6 \, b^{5} + {\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}}}{{\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2}}}\right) - 4 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d \sqrt{\frac{6 \, a^{2} b^{3} + 20 \, a b^{4} + 6 \, b^{5} - {\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}}}{{\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2}}} \cos\left(d x + c\right)^{4} \log\left(\frac{1}{2} \, {\left(a^{3} b^{4} + 15 \, a^{2} b^{5} + 15 \, a b^{6} + b^{7}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{10} - 3 \, a^{9} b - 3 \, a^{8} b^{2} + 25 \, a^{7} b^{3} - 45 \, a^{6} b^{4} + 39 \, a^{5} b^{5} - 17 \, a^{4} b^{6} + 3 \, a^{3} b^{7}\right)} d^{3} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}} + {\left(3 \, a^{5} b^{3} + 46 \, a^{4} b^{4} + 60 \, a^{3} b^{5} + 18 \, a^{2} b^{6} + a b^{7}\right)} d\right)} \sqrt{\frac{6 \, a^{2} b^{3} + 20 \, a b^{4} + 6 \, b^{5} - {\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}}}{{\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2}}}\right) - 4 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d \sqrt{\frac{6 \, a^{2} b^{3} + 20 \, a b^{4} + 6 \, b^{5} + {\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}}}{{\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2}}} \cos\left(d x + c\right)^{4} \log\left(-\frac{1}{2} \, {\left(a^{3} b^{4} + 15 \, a^{2} b^{5} + 15 \, a b^{6} + b^{7}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{10} - 3 \, a^{9} b - 3 \, a^{8} b^{2} + 25 \, a^{7} b^{3} - 45 \, a^{6} b^{4} + 39 \, a^{5} b^{5} - 17 \, a^{4} b^{6} + 3 \, a^{3} b^{7}\right)} d^{3} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}} - {\left(3 \, a^{5} b^{3} + 46 \, a^{4} b^{4} + 60 \, a^{3} b^{5} + 18 \, a^{2} b^{6} + a b^{7}\right)} d\right)} \sqrt{\frac{6 \, a^{2} b^{3} + 20 \, a b^{4} + 6 \, b^{5} + {\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}}}{{\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2}}}\right) + 4 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d \sqrt{\frac{6 \, a^{2} b^{3} + 20 \, a b^{4} + 6 \, b^{5} - {\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}}}{{\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2}}} \cos\left(d x + c\right)^{4} \log\left(-\frac{1}{2} \, {\left(a^{3} b^{4} + 15 \, a^{2} b^{5} + 15 \, a b^{6} + b^{7}\right)} \sin\left(d x + c\right) + \frac{1}{2} \, {\left({\left(a^{10} - 3 \, a^{9} b - 3 \, a^{8} b^{2} + 25 \, a^{7} b^{3} - 45 \, a^{6} b^{4} + 39 \, a^{5} b^{5} - 17 \, a^{4} b^{6} + 3 \, a^{3} b^{7}\right)} d^{3} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}} + {\left(3 \, a^{5} b^{3} + 46 \, a^{4} b^{4} + 60 \, a^{3} b^{5} + 18 \, a^{2} b^{6} + a b^{7}\right)} d\right)} \sqrt{\frac{6 \, a^{2} b^{3} + 20 \, a b^{4} + 6 \, b^{5} - {\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2} \sqrt{\frac{a^{6} b^{5} + 30 \, a^{5} b^{6} + 255 \, a^{4} b^{7} + 452 \, a^{3} b^{8} + 255 \, a^{2} b^{9} + 30 \, a b^{10} + b^{11}}{{\left(a^{15} - 12 \, a^{14} b + 66 \, a^{13} b^{2} - 220 \, a^{12} b^{3} + 495 \, a^{11} b^{4} - 792 \, a^{10} b^{5} + 924 \, a^{9} b^{6} - 792 \, a^{8} b^{7} + 495 \, a^{7} b^{8} - 220 \, a^{6} b^{9} + 66 \, a^{5} b^{10} - 12 \, a^{4} b^{11} + a^{3} b^{12}\right)} d^{4}}}}{{\left(a^{7} - 6 \, a^{6} b + 15 \, a^{5} b^{2} - 20 \, a^{4} b^{3} + 15 \, a^{3} b^{4} - 6 \, a^{2} b^{5} + a b^{6}\right)} d^{2}}}\right) - {\left(3 \, a^{2} - 6 \, a b + 35 \, b^{2}\right)} \cos\left(d x + c\right)^{4} \log\left(\sin\left(d x + c\right) + 1\right) + {\left(3 \, a^{2} - 6 \, a b + 35 \, b^{2}\right)} \cos\left(d x + c\right)^{4} \log\left(-\sin\left(d x + c\right) + 1\right) - 2 \, {\left({\left(3 \, a^{2} - 14 \, a b + 11 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, a^{2} - 4 \, a b + 2 \, b^{2}\right)} \sin\left(d x + c\right)}{16 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{4}}"," ",0,"-1/16*(4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(1/2*(a^3*b^4 + 15*a^2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*((a^10 - 3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*a^5*b^5 - 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)) - (3*a^5*b^3 + 46*a^4*b^4 + 60*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))) - 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(1/2*(a^3*b^4 + 15*a^2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*((a^10 - 3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*a^5*b^5 - 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)) + (3*a^5*b^3 + 46*a^4*b^4 + 60*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))) - 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(-1/2*(a^3*b^4 + 15*a^2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*((a^10 - 3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*a^5*b^5 - 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)) - (3*a^5*b^3 + 46*a^4*b^4 + 60*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))) + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(-1/2*(a^3*b^4 + 15*a^2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*((a^10 - 3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*a^5*b^5 - 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)) + (3*a^5*b^3 + 46*a^4*b^4 + 60*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))) - (3*a^2 - 6*a*b + 35*b^2)*cos(d*x + c)^4*log(sin(d*x + c) + 1) + (3*a^2 - 6*a*b + 35*b^2)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) - 2*((3*a^2 - 14*a*b + 11*b^2)*cos(d*x + c)^2 + 2*a^2 - 4*a*b + 2*b^2)*sin(d*x + c))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(d*x + c)^4)","B",0
411,1,2948,0,4.573573," ","integrate(cos(d*x+c)^10/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{6 \, b^{2} d \sqrt{\frac{a b^{5} d^{2} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} - a^{4} - 36 \, a^{3} b - 126 \, a^{2} b^{2} - 84 \, a b^{3} - 9 \, b^{4}}{a b^{5} d^{2}}} \log\left(\frac{9}{4} \, a^{8} + 12 \, a^{7} b - 39 \, a^{6} b^{2} + \frac{143}{2} \, a^{4} b^{4} - 52 \, a^{3} b^{5} - 3 \, a^{2} b^{6} + 8 \, a b^{7} + \frac{1}{4} \, b^{8} - \frac{1}{4} \, {\left(9 \, a^{8} + 48 \, a^{7} b - 156 \, a^{6} b^{2} + 286 \, a^{4} b^{4} - 208 \, a^{3} b^{5} - 12 \, a^{2} b^{6} + 32 \, a b^{7} + b^{8}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(4 \, {\left(a^{4} b^{7} + a^{3} b^{8}\right)} d^{3} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(9 \, a^{7} b^{2} + 138 \, a^{6} b^{3} + 639 \, a^{5} b^{4} + 876 \, a^{4} b^{5} + 343 \, a^{3} b^{6} + 42 \, a^{2} b^{7} + a b^{8}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{a b^{5} d^{2} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} - a^{4} - 36 \, a^{3} b - 126 \, a^{2} b^{2} - 84 \, a b^{3} - 9 \, b^{4}}{a b^{5} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(a^{6} b^{4} - 4 \, a^{5} b^{5} + 6 \, a^{4} b^{6} - 4 \, a^{3} b^{7} + a^{2} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{6} b^{4} - 4 \, a^{5} b^{5} + 6 \, a^{4} b^{6} - 4 \, a^{3} b^{7} + a^{2} b^{8}\right)} d^{2}\right)} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}}\right) - 6 \, b^{2} d \sqrt{\frac{a b^{5} d^{2} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} - a^{4} - 36 \, a^{3} b - 126 \, a^{2} b^{2} - 84 \, a b^{3} - 9 \, b^{4}}{a b^{5} d^{2}}} \log\left(\frac{9}{4} \, a^{8} + 12 \, a^{7} b - 39 \, a^{6} b^{2} + \frac{143}{2} \, a^{4} b^{4} - 52 \, a^{3} b^{5} - 3 \, a^{2} b^{6} + 8 \, a b^{7} + \frac{1}{4} \, b^{8} - \frac{1}{4} \, {\left(9 \, a^{8} + 48 \, a^{7} b - 156 \, a^{6} b^{2} + 286 \, a^{4} b^{4} - 208 \, a^{3} b^{5} - 12 \, a^{2} b^{6} + 32 \, a b^{7} + b^{8}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(4 \, {\left(a^{4} b^{7} + a^{3} b^{8}\right)} d^{3} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(9 \, a^{7} b^{2} + 138 \, a^{6} b^{3} + 639 \, a^{5} b^{4} + 876 \, a^{4} b^{5} + 343 \, a^{3} b^{6} + 42 \, a^{2} b^{7} + a b^{8}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{a b^{5} d^{2} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} - a^{4} - 36 \, a^{3} b - 126 \, a^{2} b^{2} - 84 \, a b^{3} - 9 \, b^{4}}{a b^{5} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(a^{6} b^{4} - 4 \, a^{5} b^{5} + 6 \, a^{4} b^{6} - 4 \, a^{3} b^{7} + a^{2} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{6} b^{4} - 4 \, a^{5} b^{5} + 6 \, a^{4} b^{6} - 4 \, a^{3} b^{7} + a^{2} b^{8}\right)} d^{2}\right)} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}}\right) + 6 \, b^{2} d \sqrt{-\frac{a b^{5} d^{2} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} + a^{4} + 36 \, a^{3} b + 126 \, a^{2} b^{2} + 84 \, a b^{3} + 9 \, b^{4}}{a b^{5} d^{2}}} \log\left(-\frac{9}{4} \, a^{8} - 12 \, a^{7} b + 39 \, a^{6} b^{2} - \frac{143}{2} \, a^{4} b^{4} + 52 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - 8 \, a b^{7} - \frac{1}{4} \, b^{8} + \frac{1}{4} \, {\left(9 \, a^{8} + 48 \, a^{7} b - 156 \, a^{6} b^{2} + 286 \, a^{4} b^{4} - 208 \, a^{3} b^{5} - 12 \, a^{2} b^{6} + 32 \, a b^{7} + b^{8}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(4 \, {\left(a^{4} b^{7} + a^{3} b^{8}\right)} d^{3} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(9 \, a^{7} b^{2} + 138 \, a^{6} b^{3} + 639 \, a^{5} b^{4} + 876 \, a^{4} b^{5} + 343 \, a^{3} b^{6} + 42 \, a^{2} b^{7} + a b^{8}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a b^{5} d^{2} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} + a^{4} + 36 \, a^{3} b + 126 \, a^{2} b^{2} + 84 \, a b^{3} + 9 \, b^{4}}{a b^{5} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(a^{6} b^{4} - 4 \, a^{5} b^{5} + 6 \, a^{4} b^{6} - 4 \, a^{3} b^{7} + a^{2} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{6} b^{4} - 4 \, a^{5} b^{5} + 6 \, a^{4} b^{6} - 4 \, a^{3} b^{7} + a^{2} b^{8}\right)} d^{2}\right)} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}}\right) - 6 \, b^{2} d \sqrt{-\frac{a b^{5} d^{2} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} + a^{4} + 36 \, a^{3} b + 126 \, a^{2} b^{2} + 84 \, a b^{3} + 9 \, b^{4}}{a b^{5} d^{2}}} \log\left(-\frac{9}{4} \, a^{8} - 12 \, a^{7} b + 39 \, a^{6} b^{2} - \frac{143}{2} \, a^{4} b^{4} + 52 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - 8 \, a b^{7} - \frac{1}{4} \, b^{8} + \frac{1}{4} \, {\left(9 \, a^{8} + 48 \, a^{7} b - 156 \, a^{6} b^{2} + 286 \, a^{4} b^{4} - 208 \, a^{3} b^{5} - 12 \, a^{2} b^{6} + 32 \, a b^{7} + b^{8}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(4 \, {\left(a^{4} b^{7} + a^{3} b^{8}\right)} d^{3} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(9 \, a^{7} b^{2} + 138 \, a^{6} b^{3} + 639 \, a^{5} b^{4} + 876 \, a^{4} b^{5} + 343 \, a^{3} b^{6} + 42 \, a^{2} b^{7} + a b^{8}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a b^{5} d^{2} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}} + a^{4} + 36 \, a^{3} b + 126 \, a^{2} b^{2} + 84 \, a b^{3} + 9 \, b^{4}}{a b^{5} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(a^{6} b^{4} - 4 \, a^{5} b^{5} + 6 \, a^{4} b^{6} - 4 \, a^{3} b^{7} + a^{2} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{6} b^{4} - 4 \, a^{5} b^{5} + 6 \, a^{4} b^{6} - 4 \, a^{3} b^{7} + a^{2} b^{8}\right)} d^{2}\right)} \sqrt{\frac{81 \, a^{8} + 1512 \, a^{7} b + 9324 \, a^{6} b^{2} + 21816 \, a^{5} b^{3} + 21942 \, a^{4} b^{4} + 9240 \, a^{3} b^{5} + 1548 \, a^{2} b^{6} + 72 \, a b^{7} + b^{8}}{a^{3} b^{9} d^{4}}}\right) - 9 \, {\left(24 \, a + 35 \, b\right)} d x - {\left(8 \, b \cos\left(d x + c\right)^{5} + 34 \, b \cos\left(d x + c\right)^{3} + 3 \, {\left(8 \, a + 41 \, b\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{48 \, b^{2} d}"," ",0,"1/48*(6*b^2*d*sqrt((a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) - a^4 - 36*a^3*b - 126*a^2*b^2 - 84*a*b^3 - 9*b^4)/(a*b^5*d^2))*log(9/4*a^8 + 12*a^7*b - 39*a^6*b^2 + 143/2*a^4*b^4 - 52*a^3*b^5 - 3*a^2*b^6 + 8*a*b^7 + 1/4*b^8 - 1/4*(9*a^8 + 48*a^7*b - 156*a^6*b^2 + 286*a^4*b^4 - 208*a^3*b^5 - 12*a^2*b^6 + 32*a*b^7 + b^8)*cos(d*x + c)^2 + 1/2*(4*(a^4*b^7 + a^3*b^8)*d^3*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))*cos(d*x + c)*sin(d*x + c) + (9*a^7*b^2 + 138*a^6*b^3 + 639*a^5*b^4 + 876*a^4*b^5 + 343*a^3*b^6 + 42*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) - a^4 - 36*a^3*b - 126*a^2*b^2 - 84*a*b^3 - 9*b^4)/(a*b^5*d^2)) + 1/4*(2*(a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2*cos(d*x + c)^2 - (a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2)*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))) - 6*b^2*d*sqrt((a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) - a^4 - 36*a^3*b - 126*a^2*b^2 - 84*a*b^3 - 9*b^4)/(a*b^5*d^2))*log(9/4*a^8 + 12*a^7*b - 39*a^6*b^2 + 143/2*a^4*b^4 - 52*a^3*b^5 - 3*a^2*b^6 + 8*a*b^7 + 1/4*b^8 - 1/4*(9*a^8 + 48*a^7*b - 156*a^6*b^2 + 286*a^4*b^4 - 208*a^3*b^5 - 12*a^2*b^6 + 32*a*b^7 + b^8)*cos(d*x + c)^2 - 1/2*(4*(a^4*b^7 + a^3*b^8)*d^3*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))*cos(d*x + c)*sin(d*x + c) + (9*a^7*b^2 + 138*a^6*b^3 + 639*a^5*b^4 + 876*a^4*b^5 + 343*a^3*b^6 + 42*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) - a^4 - 36*a^3*b - 126*a^2*b^2 - 84*a*b^3 - 9*b^4)/(a*b^5*d^2)) + 1/4*(2*(a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2*cos(d*x + c)^2 - (a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2)*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))) + 6*b^2*d*sqrt(-(a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) + a^4 + 36*a^3*b + 126*a^2*b^2 + 84*a*b^3 + 9*b^4)/(a*b^5*d^2))*log(-9/4*a^8 - 12*a^7*b + 39*a^6*b^2 - 143/2*a^4*b^4 + 52*a^3*b^5 + 3*a^2*b^6 - 8*a*b^7 - 1/4*b^8 + 1/4*(9*a^8 + 48*a^7*b - 156*a^6*b^2 + 286*a^4*b^4 - 208*a^3*b^5 - 12*a^2*b^6 + 32*a*b^7 + b^8)*cos(d*x + c)^2 + 1/2*(4*(a^4*b^7 + a^3*b^8)*d^3*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))*cos(d*x + c)*sin(d*x + c) - (9*a^7*b^2 + 138*a^6*b^3 + 639*a^5*b^4 + 876*a^4*b^5 + 343*a^3*b^6 + 42*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) + a^4 + 36*a^3*b + 126*a^2*b^2 + 84*a*b^3 + 9*b^4)/(a*b^5*d^2)) + 1/4*(2*(a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2*cos(d*x + c)^2 - (a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2)*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))) - 6*b^2*d*sqrt(-(a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) + a^4 + 36*a^3*b + 126*a^2*b^2 + 84*a*b^3 + 9*b^4)/(a*b^5*d^2))*log(-9/4*a^8 - 12*a^7*b + 39*a^6*b^2 - 143/2*a^4*b^4 + 52*a^3*b^5 + 3*a^2*b^6 - 8*a*b^7 - 1/4*b^8 + 1/4*(9*a^8 + 48*a^7*b - 156*a^6*b^2 + 286*a^4*b^4 - 208*a^3*b^5 - 12*a^2*b^6 + 32*a*b^7 + b^8)*cos(d*x + c)^2 - 1/2*(4*(a^4*b^7 + a^3*b^8)*d^3*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))*cos(d*x + c)*sin(d*x + c) - (9*a^7*b^2 + 138*a^6*b^3 + 639*a^5*b^4 + 876*a^4*b^5 + 343*a^3*b^6 + 42*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^5*d^2*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4)) + a^4 + 36*a^3*b + 126*a^2*b^2 + 84*a*b^3 + 9*b^4)/(a*b^5*d^2)) + 1/4*(2*(a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2*cos(d*x + c)^2 - (a^6*b^4 - 4*a^5*b^5 + 6*a^4*b^6 - 4*a^3*b^7 + a^2*b^8)*d^2)*sqrt((81*a^8 + 1512*a^7*b + 9324*a^6*b^2 + 21816*a^5*b^3 + 21942*a^4*b^4 + 9240*a^3*b^5 + 1548*a^2*b^6 + 72*a*b^7 + b^8)/(a^3*b^9*d^4))) - 9*(24*a + 35*b)*d*x - (8*b*cos(d*x + c)^5 + 34*b*cos(d*x + c)^3 + 3*(8*a + 41*b)*cos(d*x + c))*sin(d*x + c))/(b^2*d)","B",0
412,1,2433,0,2.273684," ","integrate(cos(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{b^{2} d \sqrt{-\frac{a b^{4} d^{2} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + a^{3} + 21 \, a^{2} b + 35 \, a b^{2} + 7 \, b^{3}}{a b^{4} d^{2}}} \log\left(\frac{7}{4} \, a^{6} + \frac{7}{2} \, a^{5} b - \frac{63}{4} \, a^{4} b^{2} + 9 \, a^{3} b^{3} + \frac{25}{4} \, a^{2} b^{4} - \frac{9}{2} \, a b^{5} - \frac{1}{4} \, b^{6} - \frac{1}{4} \, {\left(7 \, a^{6} + 14 \, a^{5} b - 63 \, a^{4} b^{2} + 36 \, a^{3} b^{3} + 25 \, a^{2} b^{4} - 18 \, a b^{5} - b^{6}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a^{4} b^{5} + 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(21 \, a^{5} b^{2} + 112 \, a^{4} b^{3} + 98 \, a^{3} b^{4} + 24 \, a^{2} b^{5} + a b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a b^{4} d^{2} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + a^{3} + 21 \, a^{2} b + 35 \, a b^{2} + 7 \, b^{3}}{a b^{4} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2}\right)} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}}\right) - b^{2} d \sqrt{-\frac{a b^{4} d^{2} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + a^{3} + 21 \, a^{2} b + 35 \, a b^{2} + 7 \, b^{3}}{a b^{4} d^{2}}} \log\left(\frac{7}{4} \, a^{6} + \frac{7}{2} \, a^{5} b - \frac{63}{4} \, a^{4} b^{2} + 9 \, a^{3} b^{3} + \frac{25}{4} \, a^{2} b^{4} - \frac{9}{2} \, a b^{5} - \frac{1}{4} \, b^{6} - \frac{1}{4} \, {\left(7 \, a^{6} + 14 \, a^{5} b - 63 \, a^{4} b^{2} + 36 \, a^{3} b^{3} + 25 \, a^{2} b^{4} - 18 \, a b^{5} - b^{6}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a^{4} b^{5} + 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(21 \, a^{5} b^{2} + 112 \, a^{4} b^{3} + 98 \, a^{3} b^{4} + 24 \, a^{2} b^{5} + a b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a b^{4} d^{2} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} + a^{3} + 21 \, a^{2} b + 35 \, a b^{2} + 7 \, b^{3}}{a b^{4} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2}\right)} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}}\right) + b^{2} d \sqrt{\frac{a b^{4} d^{2} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - a^{3} - 21 \, a^{2} b - 35 \, a b^{2} - 7 \, b^{3}}{a b^{4} d^{2}}} \log\left(-\frac{7}{4} \, a^{6} - \frac{7}{2} \, a^{5} b + \frac{63}{4} \, a^{4} b^{2} - 9 \, a^{3} b^{3} - \frac{25}{4} \, a^{2} b^{4} + \frac{9}{2} \, a b^{5} + \frac{1}{4} \, b^{6} + \frac{1}{4} \, {\left(7 \, a^{6} + 14 \, a^{5} b - 63 \, a^{4} b^{2} + 36 \, a^{3} b^{3} + 25 \, a^{2} b^{4} - 18 \, a b^{5} - b^{6}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a^{4} b^{5} + 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(21 \, a^{5} b^{2} + 112 \, a^{4} b^{3} + 98 \, a^{3} b^{4} + 24 \, a^{2} b^{5} + a b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{a b^{4} d^{2} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - a^{3} - 21 \, a^{2} b - 35 \, a b^{2} - 7 \, b^{3}}{a b^{4} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2}\right)} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}}\right) - b^{2} d \sqrt{\frac{a b^{4} d^{2} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - a^{3} - 21 \, a^{2} b - 35 \, a b^{2} - 7 \, b^{3}}{a b^{4} d^{2}}} \log\left(-\frac{7}{4} \, a^{6} - \frac{7}{2} \, a^{5} b + \frac{63}{4} \, a^{4} b^{2} - 9 \, a^{3} b^{3} - \frac{25}{4} \, a^{2} b^{4} + \frac{9}{2} \, a b^{5} + \frac{1}{4} \, b^{6} + \frac{1}{4} \, {\left(7 \, a^{6} + 14 \, a^{5} b - 63 \, a^{4} b^{2} + 36 \, a^{3} b^{3} + 25 \, a^{2} b^{4} - 18 \, a b^{5} - b^{6}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a^{4} b^{5} + 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(21 \, a^{5} b^{2} + 112 \, a^{4} b^{3} + 98 \, a^{3} b^{4} + 24 \, a^{2} b^{5} + a b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{a b^{4} d^{2} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}} - a^{3} - 21 \, a^{2} b - 35 \, a b^{2} - 7 \, b^{3}}{a b^{4} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2}\right)} \sqrt{\frac{49 \, a^{6} + 490 \, a^{5} b + 1519 \, a^{4} b^{2} + 1484 \, a^{3} b^{3} + 511 \, a^{2} b^{4} + 42 \, a b^{5} + b^{6}}{a^{3} b^{7} d^{4}}}\right) - {\left(8 \, a + 35 \, b\right)} d x - {\left(2 \, b \cos\left(d x + c\right)^{3} + 11 \, b \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{8 \, b^{2} d}"," ",0,"1/8*(b^2*d*sqrt(-(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*b + 35*a*b^2 + 7*b^3)/(a*b^4*d^2))*log(7/4*a^6 + 7/2*a^5*b - 63/4*a^4*b^2 + 9*a^3*b^3 + 25/4*a^2*b^4 - 9/2*a*b^5 - 1/4*b^6 - 1/4*(7*a^6 + 14*a^5*b - 63*a^4*b^2 + 36*a^3*b^3 + 25*a^2*b^4 - 18*a*b^5 - b^6)*cos(d*x + c)^2 + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))*cos(d*x + c)*sin(d*x + c) - (21*a^5*b^2 + 112*a^4*b^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*b + 35*a*b^2 + 7*b^3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))) - b^2*d*sqrt(-(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*b + 35*a*b^2 + 7*b^3)/(a*b^4*d^2))*log(7/4*a^6 + 7/2*a^5*b - 63/4*a^4*b^2 + 9*a^3*b^3 + 25/4*a^2*b^4 - 9/2*a*b^5 - 1/4*b^6 - 1/4*(7*a^6 + 14*a^5*b - 63*a^4*b^2 + 36*a^3*b^3 + 25*a^2*b^4 - 18*a*b^5 - b^6)*cos(d*x + c)^2 - 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))*cos(d*x + c)*sin(d*x + c) - (21*a^5*b^2 + 112*a^4*b^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*b + 35*a*b^2 + 7*b^3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))) + b^2*d*sqrt((a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3)/(a*b^4*d^2))*log(-7/4*a^6 - 7/2*a^5*b + 63/4*a^4*b^2 - 9*a^3*b^3 - 25/4*a^2*b^4 + 9/2*a*b^5 + 1/4*b^6 + 1/4*(7*a^6 + 14*a^5*b - 63*a^4*b^2 + 36*a^3*b^3 + 25*a^2*b^4 - 18*a*b^5 - b^6)*cos(d*x + c)^2 + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))*cos(d*x + c)*sin(d*x + c) + (21*a^5*b^2 + 112*a^4*b^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))) - b^2*d*sqrt((a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3)/(a*b^4*d^2))*log(-7/4*a^6 - 7/2*a^5*b + 63/4*a^4*b^2 - 9*a^3*b^3 - 25/4*a^2*b^4 + 9/2*a*b^5 + 1/4*b^6 + 1/4*(7*a^6 + 14*a^5*b - 63*a^4*b^2 + 36*a^3*b^3 + 25*a^2*b^4 - 18*a*b^5 - b^6)*cos(d*x + c)^2 - 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))*cos(d*x + c)*sin(d*x + c) + (21*a^5*b^2 + 112*a^4*b^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))) - (8*a + 35*b)*d*x - (2*b*cos(d*x + c)^3 + 11*b*cos(d*x + c))*sin(d*x + c))/(b^2*d)","B",0
413,1,1751,0,1.965956," ","integrate(cos(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{b d \sqrt{\frac{a b^{3} d^{2} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} - a^{2} - 10 \, a b - 5 \, b^{2}}{a b^{3} d^{2}}} \log\left(\frac{5}{4} \, a^{4} - \frac{7}{2} \, a^{2} b^{2} + 2 \, a b^{3} + \frac{1}{4} \, b^{4} - \frac{1}{4} \, {\left(5 \, a^{4} - 14 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(2 \, a^{3} b^{4} d^{3} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(5 \, a^{4} b + 15 \, a^{3} b^{2} + 11 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{a b^{3} d^{2} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} - a^{2} - 10 \, a b - 5 \, b^{2}}{a b^{3} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}}\right) - b d \sqrt{\frac{a b^{3} d^{2} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} - a^{2} - 10 \, a b - 5 \, b^{2}}{a b^{3} d^{2}}} \log\left(\frac{5}{4} \, a^{4} - \frac{7}{2} \, a^{2} b^{2} + 2 \, a b^{3} + \frac{1}{4} \, b^{4} - \frac{1}{4} \, {\left(5 \, a^{4} - 14 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(2 \, a^{3} b^{4} d^{3} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(5 \, a^{4} b + 15 \, a^{3} b^{2} + 11 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{a b^{3} d^{2} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} - a^{2} - 10 \, a b - 5 \, b^{2}}{a b^{3} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}}\right) + b d \sqrt{-\frac{a b^{3} d^{2} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} + a^{2} + 10 \, a b + 5 \, b^{2}}{a b^{3} d^{2}}} \log\left(-\frac{5}{4} \, a^{4} + \frac{7}{2} \, a^{2} b^{2} - 2 \, a b^{3} - \frac{1}{4} \, b^{4} + \frac{1}{4} \, {\left(5 \, a^{4} - 14 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(2 \, a^{3} b^{4} d^{3} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(5 \, a^{4} b + 15 \, a^{3} b^{2} + 11 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a b^{3} d^{2} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} + a^{2} + 10 \, a b + 5 \, b^{2}}{a b^{3} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}}\right) - b d \sqrt{-\frac{a b^{3} d^{2} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} + a^{2} + 10 \, a b + 5 \, b^{2}}{a b^{3} d^{2}}} \log\left(-\frac{5}{4} \, a^{4} + \frac{7}{2} \, a^{2} b^{2} - 2 \, a b^{3} - \frac{1}{4} \, b^{4} + \frac{1}{4} \, {\left(5 \, a^{4} - 14 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(2 \, a^{3} b^{4} d^{3} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(5 \, a^{4} b + 15 \, a^{3} b^{2} + 11 \, a^{2} b^{3} + a b^{4}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a b^{3} d^{2} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}} + a^{2} + 10 \, a b + 5 \, b^{2}}{a b^{3} d^{2}}} + \frac{1}{4} \, {\left(2 \, {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{4} + 100 \, a^{3} b + 110 \, a^{2} b^{2} + 20 \, a b^{3} + b^{4}}{a^{3} b^{5} d^{4}}}\right) - 20 \, d x - 4 \, \cos\left(d x + c\right) \sin\left(d x + c\right)}{8 \, b d}"," ",0,"1/8*(b*d*sqrt((a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) - a^2 - 10*a*b - 5*b^2)/(a*b^3*d^2))*log(5/4*a^4 - 7/2*a^2*b^2 + 2*a*b^3 + 1/4*b^4 - 1/4*(5*a^4 - 14*a^2*b^2 + 8*a*b^3 + b^4)*cos(d*x + c)^2 + 1/2*(2*a^3*b^4*d^3*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))*cos(d*x + c)*sin(d*x + c) + (5*a^4*b + 15*a^3*b^2 + 11*a^2*b^3 + a*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) - a^2 - 10*a*b - 5*b^2)/(a*b^3*d^2)) + 1/4*(2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2*cos(d*x + c)^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2)*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))) - b*d*sqrt((a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) - a^2 - 10*a*b - 5*b^2)/(a*b^3*d^2))*log(5/4*a^4 - 7/2*a^2*b^2 + 2*a*b^3 + 1/4*b^4 - 1/4*(5*a^4 - 14*a^2*b^2 + 8*a*b^3 + b^4)*cos(d*x + c)^2 - 1/2*(2*a^3*b^4*d^3*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))*cos(d*x + c)*sin(d*x + c) + (5*a^4*b + 15*a^3*b^2 + 11*a^2*b^3 + a*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) - a^2 - 10*a*b - 5*b^2)/(a*b^3*d^2)) + 1/4*(2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2*cos(d*x + c)^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2)*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))) + b*d*sqrt(-(a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) + a^2 + 10*a*b + 5*b^2)/(a*b^3*d^2))*log(-5/4*a^4 + 7/2*a^2*b^2 - 2*a*b^3 - 1/4*b^4 + 1/4*(5*a^4 - 14*a^2*b^2 + 8*a*b^3 + b^4)*cos(d*x + c)^2 + 1/2*(2*a^3*b^4*d^3*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))*cos(d*x + c)*sin(d*x + c) - (5*a^4*b + 15*a^3*b^2 + 11*a^2*b^3 + a*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) + a^2 + 10*a*b + 5*b^2)/(a*b^3*d^2)) + 1/4*(2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2*cos(d*x + c)^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2)*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))) - b*d*sqrt(-(a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) + a^2 + 10*a*b + 5*b^2)/(a*b^3*d^2))*log(-5/4*a^4 + 7/2*a^2*b^2 - 2*a*b^3 - 1/4*b^4 + 1/4*(5*a^4 - 14*a^2*b^2 + 8*a*b^3 + b^4)*cos(d*x + c)^2 - 1/2*(2*a^3*b^4*d^3*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))*cos(d*x + c)*sin(d*x + c) - (5*a^4*b + 15*a^3*b^2 + 11*a^2*b^3 + a*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^3*d^2*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4)) + a^2 + 10*a*b + 5*b^2)/(a*b^3*d^2)) + 1/4*(2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2*cos(d*x + c)^2 - (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d^2)*sqrt((25*a^4 + 100*a^3*b + 110*a^2*b^2 + 20*a*b^3 + b^4)/(a^3*b^5*d^4))) - 20*d*x - 4*cos(d*x + c)*sin(d*x + c))/(b*d)","B",0
414,1,1197,0,1.446115," ","integrate(cos(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{b \sqrt{\frac{a b^{2} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - a - 3 \, b}{a b^{2} d^{2}}} \log\left(\frac{1}{4} \, {\left(3 \, a^{2} - 2 \, a b - b^{2}\right)} \cos\left(d x + c\right)^{2} - \frac{3}{4} \, a^{2} + \frac{1}{2} \, a b + \frac{1}{4} \, b^{2} + \frac{1}{2} \, {\left(a^{3} b^{2} d^{3} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(3 \, a^{2} b + a b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{a b^{2} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - a - 3 \, b}{a b^{2} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a^{2} b^{2}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}}\right) - b \sqrt{\frac{a b^{2} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - a - 3 \, b}{a b^{2} d^{2}}} \log\left(\frac{1}{4} \, {\left(3 \, a^{2} - 2 \, a b - b^{2}\right)} \cos\left(d x + c\right)^{2} - \frac{3}{4} \, a^{2} + \frac{1}{2} \, a b + \frac{1}{4} \, b^{2} - \frac{1}{2} \, {\left(a^{3} b^{2} d^{3} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(3 \, a^{2} b + a b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{a b^{2} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} - a - 3 \, b}{a b^{2} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a^{2} b^{2}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}}\right) + b \sqrt{-\frac{a b^{2} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + a + 3 \, b}{a b^{2} d^{2}}} \log\left(-\frac{1}{4} \, {\left(3 \, a^{2} - 2 \, a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + \frac{3}{4} \, a^{2} - \frac{1}{2} \, a b - \frac{1}{4} \, b^{2} + \frac{1}{2} \, {\left(a^{3} b^{2} d^{3} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(3 \, a^{2} b + a b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a b^{2} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + a + 3 \, b}{a b^{2} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a^{2} b^{2}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}}\right) - b \sqrt{-\frac{a b^{2} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + a + 3 \, b}{a b^{2} d^{2}}} \log\left(-\frac{1}{4} \, {\left(3 \, a^{2} - 2 \, a b - b^{2}\right)} \cos\left(d x + c\right)^{2} + \frac{3}{4} \, a^{2} - \frac{1}{2} \, a b - \frac{1}{4} \, b^{2} - \frac{1}{2} \, {\left(a^{3} b^{2} d^{3} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(3 \, a^{2} b + a b^{2}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a b^{2} d^{2} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}} + a + 3 \, b}{a b^{2} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{3} b - a^{2} b^{2}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{3} b - a^{2} b^{2}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} + 6 \, a b + b^{2}}{a^{3} b^{3} d^{4}}}\right) - 8 \, x}{8 \, b}"," ",0,"1/8*(b*sqrt((a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) - a - 3*b)/(a*b^2*d^2))*log(1/4*(3*a^2 - 2*a*b - b^2)*cos(d*x + c)^2 - 3/4*a^2 + 1/2*a*b + 1/4*b^2 + 1/2*(a^3*b^2*d^3*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))*cos(d*x + c)*sin(d*x + c) + (3*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) - a - 3*b)/(a*b^2*d^2)) - 1/4*(2*(a^3*b - a^2*b^2)*d^2*cos(d*x + c)^2 - (a^3*b - a^2*b^2)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))) - b*sqrt((a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) - a - 3*b)/(a*b^2*d^2))*log(1/4*(3*a^2 - 2*a*b - b^2)*cos(d*x + c)^2 - 3/4*a^2 + 1/2*a*b + 1/4*b^2 - 1/2*(a^3*b^2*d^3*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))*cos(d*x + c)*sin(d*x + c) + (3*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt((a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) - a - 3*b)/(a*b^2*d^2)) - 1/4*(2*(a^3*b - a^2*b^2)*d^2*cos(d*x + c)^2 - (a^3*b - a^2*b^2)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))) + b*sqrt(-(a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) + a + 3*b)/(a*b^2*d^2))*log(-1/4*(3*a^2 - 2*a*b - b^2)*cos(d*x + c)^2 + 3/4*a^2 - 1/2*a*b - 1/4*b^2 + 1/2*(a^3*b^2*d^3*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))*cos(d*x + c)*sin(d*x + c) - (3*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) + a + 3*b)/(a*b^2*d^2)) - 1/4*(2*(a^3*b - a^2*b^2)*d^2*cos(d*x + c)^2 - (a^3*b - a^2*b^2)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))) - b*sqrt(-(a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) + a + 3*b)/(a*b^2*d^2))*log(-1/4*(3*a^2 - 2*a*b - b^2)*cos(d*x + c)^2 + 3/4*a^2 - 1/2*a*b - 1/4*b^2 - 1/2*(a^3*b^2*d^3*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))*cos(d*x + c)*sin(d*x + c) - (3*a^2*b + a*b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^2*d^2*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4)) + a + 3*b)/(a*b^2*d^2)) - 1/4*(2*(a^3*b - a^2*b^2)*d^2*cos(d*x + c)^2 - (a^3*b - a^2*b^2)*d^2)*sqrt((9*a^2 + 6*a*b + b^2)/(a^3*b^3*d^4))) - 8*x)/b","B",0
415,1,541,0,1.098464," ","integrate(cos(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","-\frac{1}{8} \, \sqrt{-\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \log\left(\frac{1}{2} \, a d \sqrt{-\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + \frac{1}{4} \, \cos\left(d x + c\right)^{2} + \frac{1}{4} \, {\left(2 \, a^{2} d^{2} \cos\left(d x + c\right)^{2} - a^{2} d^{2}\right)} \sqrt{\frac{1}{a^{3} b d^{4}}} - \frac{1}{4}\right) + \frac{1}{8} \, \sqrt{-\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \log\left(-\frac{1}{2} \, a d \sqrt{-\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + \frac{1}{4} \, \cos\left(d x + c\right)^{2} + \frac{1}{4} \, {\left(2 \, a^{2} d^{2} \cos\left(d x + c\right)^{2} - a^{2} d^{2}\right)} \sqrt{\frac{1}{a^{3} b d^{4}}} - \frac{1}{4}\right) + \frac{1}{8} \, \sqrt{\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \log\left(\frac{1}{2} \, a d \sqrt{\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - \frac{1}{4} \, \cos\left(d x + c\right)^{2} + \frac{1}{4} \, {\left(2 \, a^{2} d^{2} \cos\left(d x + c\right)^{2} - a^{2} d^{2}\right)} \sqrt{\frac{1}{a^{3} b d^{4}}} + \frac{1}{4}\right) - \frac{1}{8} \, \sqrt{\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \log\left(-\frac{1}{2} \, a d \sqrt{\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - \frac{1}{4} \, \cos\left(d x + c\right)^{2} + \frac{1}{4} \, {\left(2 \, a^{2} d^{2} \cos\left(d x + c\right)^{2} - a^{2} d^{2}\right)} \sqrt{\frac{1}{a^{3} b d^{4}}} + \frac{1}{4}\right)"," ",0,"-1/8*sqrt(-(a*b*d^2*sqrt(1/(a^3*b*d^4)) + 1)/(a*b*d^2))*log(1/2*a*d*sqrt(-(a*b*d^2*sqrt(1/(a^3*b*d^4)) + 1)/(a*b*d^2))*cos(d*x + c)*sin(d*x + c) + 1/4*cos(d*x + c)^2 + 1/4*(2*a^2*d^2*cos(d*x + c)^2 - a^2*d^2)*sqrt(1/(a^3*b*d^4)) - 1/4) + 1/8*sqrt(-(a*b*d^2*sqrt(1/(a^3*b*d^4)) + 1)/(a*b*d^2))*log(-1/2*a*d*sqrt(-(a*b*d^2*sqrt(1/(a^3*b*d^4)) + 1)/(a*b*d^2))*cos(d*x + c)*sin(d*x + c) + 1/4*cos(d*x + c)^2 + 1/4*(2*a^2*d^2*cos(d*x + c)^2 - a^2*d^2)*sqrt(1/(a^3*b*d^4)) - 1/4) + 1/8*sqrt((a*b*d^2*sqrt(1/(a^3*b*d^4)) - 1)/(a*b*d^2))*log(1/2*a*d*sqrt((a*b*d^2*sqrt(1/(a^3*b*d^4)) - 1)/(a*b*d^2))*cos(d*x + c)*sin(d*x + c) - 1/4*cos(d*x + c)^2 + 1/4*(2*a^2*d^2*cos(d*x + c)^2 - a^2*d^2)*sqrt(1/(a^3*b*d^4)) + 1/4) - 1/8*sqrt((a*b*d^2*sqrt(1/(a^3*b*d^4)) - 1)/(a*b*d^2))*log(-1/2*a*d*sqrt((a*b*d^2*sqrt(1/(a^3*b*d^4)) - 1)/(a*b*d^2))*cos(d*x + c)*sin(d*x + c) - 1/4*cos(d*x + c)^2 + 1/4*(2*a^2*d^2*cos(d*x + c)^2 - a^2*d^2)*sqrt(1/(a^3*b*d^4)) + 1/4)","B",0
416,1,2589,0,2.684511," ","integrate(sec(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{{\left(a - b\right)} d \sqrt{\frac{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} - a b - 3 \, b^{2}}{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2}}} \cos\left(d x + c\right) \log\left(\frac{3}{4} \, a b^{2} + \frac{1}{4} \, b^{3} - \frac{1}{4} \, {\left(3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(2 \, {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{3} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(3 \, a^{3} b + 4 \, a^{2} b^{2} + a b^{3}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} - a b - 3 \, b^{2}}{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}}\right) - {\left(a - b\right)} d \sqrt{\frac{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} - a b - 3 \, b^{2}}{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2}}} \cos\left(d x + c\right) \log\left(\frac{3}{4} \, a b^{2} + \frac{1}{4} \, b^{3} - \frac{1}{4} \, {\left(3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(2 \, {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{3} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(3 \, a^{3} b + 4 \, a^{2} b^{2} + a b^{3}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{\frac{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} - a b - 3 \, b^{2}}{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}}\right) + {\left(a - b\right)} d \sqrt{-\frac{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} + a b + 3 \, b^{2}}{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2}}} \cos\left(d x + c\right) \log\left(-\frac{3}{4} \, a b^{2} - \frac{1}{4} \, b^{3} + \frac{1}{4} \, {\left(3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(2 \, {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{3} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(3 \, a^{3} b + 4 \, a^{2} b^{2} + a b^{3}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} + a b + 3 \, b^{2}}{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}}\right) - {\left(a - b\right)} d \sqrt{-\frac{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} + a b + 3 \, b^{2}}{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2}}} \cos\left(d x + c\right) \log\left(-\frac{3}{4} \, a b^{2} - \frac{1}{4} \, b^{3} + \frac{1}{4} \, {\left(3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(2 \, {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} d^{3} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(3 \, a^{3} b + 4 \, a^{2} b^{2} + a b^{3}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}} + a b + 3 \, b^{2}}{{\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} d^{2}\right)} \sqrt{\frac{9 \, a^{2} b^{3} + 6 \, a b^{4} + b^{5}}{{\left(a^{9} - 6 \, a^{8} b + 15 \, a^{7} b^{2} - 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + a^{3} b^{6}\right)} d^{4}}}\right) + 8 \, \sin\left(d x + c\right)}{8 \, {\left(a - b\right)} d \cos\left(d x + c\right)}"," ",0,"1/8*((a - b)*d*sqrt(((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) - a*b - 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2))*cos(d*x + c)*log(3/4*a*b^2 + 1/4*b^3 - 1/4*(3*a*b^2 + b^3)*cos(d*x + c)^2 + 1/2*(2*(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^3*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + (3*a^3*b + 4*a^2*b^2 + a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) - a*b - 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))) - (a - b)*d*sqrt(((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) - a*b - 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2))*cos(d*x + c)*log(3/4*a*b^2 + 1/4*b^3 - 1/4*(3*a*b^2 + b^3)*cos(d*x + c)^2 - 1/2*(2*(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^3*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + (3*a^3*b + 4*a^2*b^2 + a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) - a*b - 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))) + (a - b)*d*sqrt(-((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) + a*b + 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2))*cos(d*x + c)*log(-3/4*a*b^2 - 1/4*b^3 + 1/4*(3*a*b^2 + b^3)*cos(d*x + c)^2 + 1/2*(2*(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^3*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - (3*a^3*b + 4*a^2*b^2 + a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) + a*b + 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))) - (a - b)*d*sqrt(-((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) + a*b + 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2))*cos(d*x + c)*log(-3/4*a*b^2 - 1/4*b^3 + 1/4*(3*a*b^2 + b^3)*cos(d*x + c)^2 - 1/2*(2*(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^3*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - (3*a^3*b + 4*a^2*b^2 + a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) + a*b + 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))) + 8*sin(d*x + c))/((a - b)*d*cos(d*x + c))","B",0
417,1,4113,0,2.464067," ","integrate(sec(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} d \sqrt{-\frac{a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4} - {\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}}{{\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} \cos\left(d x + c\right)^{3} \log\left(\frac{5}{4} \, a^{2} b^{4} + \frac{5}{2} \, a b^{5} + \frac{1}{4} \, b^{6} - \frac{1}{4} \, {\left(5 \, a^{2} b^{4} + 10 \, a b^{5} + b^{6}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a^{9} - 2 \, a^{8} b - 5 \, a^{7} b^{2} + 20 \, a^{6} b^{3} - 25 \, a^{5} b^{4} + 14 \, a^{4} b^{5} - 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(15 \, a^{4} b^{3} + 35 \, a^{3} b^{4} + 13 \, a^{2} b^{5} + a b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4} - {\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}}{{\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{7} b - 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} - 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{7} b - 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} - 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}\right) - 3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} d \sqrt{-\frac{a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4} - {\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}}{{\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} \cos\left(d x + c\right)^{3} \log\left(\frac{5}{4} \, a^{2} b^{4} + \frac{5}{2} \, a b^{5} + \frac{1}{4} \, b^{6} - \frac{1}{4} \, {\left(5 \, a^{2} b^{4} + 10 \, a b^{5} + b^{6}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a^{9} - 2 \, a^{8} b - 5 \, a^{7} b^{2} + 20 \, a^{6} b^{3} - 25 \, a^{5} b^{4} + 14 \, a^{4} b^{5} - 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(15 \, a^{4} b^{3} + 35 \, a^{3} b^{4} + 13 \, a^{2} b^{5} + a b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4} - {\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}}{{\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{7} b - 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} - 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{7} b - 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} - 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}\right) + 3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} d \sqrt{-\frac{a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4} + {\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}}{{\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} \cos\left(d x + c\right)^{3} \log\left(-\frac{5}{4} \, a^{2} b^{4} - \frac{5}{2} \, a b^{5} - \frac{1}{4} \, b^{6} + \frac{1}{4} \, {\left(5 \, a^{2} b^{4} + 10 \, a b^{5} + b^{6}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left({\left(a^{9} - 2 \, a^{8} b - 5 \, a^{7} b^{2} + 20 \, a^{6} b^{3} - 25 \, a^{5} b^{4} + 14 \, a^{4} b^{5} - 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(15 \, a^{4} b^{3} + 35 \, a^{3} b^{4} + 13 \, a^{2} b^{5} + a b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4} + {\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}}{{\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{7} b - 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} - 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{7} b - 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} - 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}\right) - 3 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} d \sqrt{-\frac{a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4} + {\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}}{{\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} \cos\left(d x + c\right)^{3} \log\left(-\frac{5}{4} \, a^{2} b^{4} - \frac{5}{2} \, a b^{5} - \frac{1}{4} \, b^{6} + \frac{1}{4} \, {\left(5 \, a^{2} b^{4} + 10 \, a b^{5} + b^{6}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left({\left(a^{9} - 2 \, a^{8} b - 5 \, a^{7} b^{2} + 20 \, a^{6} b^{3} - 25 \, a^{5} b^{4} + 14 \, a^{4} b^{5} - 3 \, a^{3} b^{6}\right)} d^{3} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(15 \, a^{4} b^{3} + 35 \, a^{3} b^{4} + 13 \, a^{2} b^{5} + a b^{6}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a^{2} b^{2} + 10 \, a b^{3} + 5 \, b^{4} + {\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}}{{\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{7} b - 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} - 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{7} b - 5 \, a^{6} b^{2} + 10 \, a^{5} b^{3} - 10 \, a^{4} b^{4} + 5 \, a^{3} b^{5} - a^{2} b^{6}\right)} d^{2}\right)} \sqrt{\frac{25 \, a^{4} b^{5} + 100 \, a^{3} b^{6} + 110 \, a^{2} b^{7} + 20 \, a b^{8} + b^{9}}{{\left(a^{13} - 10 \, a^{12} b + 45 \, a^{11} b^{2} - 120 \, a^{10} b^{3} + 210 \, a^{9} b^{4} - 252 \, a^{8} b^{5} + 210 \, a^{7} b^{6} - 120 \, a^{6} b^{7} + 45 \, a^{5} b^{8} - 10 \, a^{4} b^{9} + a^{3} b^{10}\right)} d^{4}}}\right) + 8 \, {\left(2 \, {\left(a - 4 \, b\right)} \cos\left(d x + c\right)^{2} + a - b\right)} \sin\left(d x + c\right)}{24 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} d \cos\left(d x + c\right)^{3}}"," ",0,"1/24*(3*(a^2 - 2*a*b + b^2)*d*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 - (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2))*cos(d*x + c)^3*log(5/4*a^2*b^4 + 5/2*a*b^5 + 1/4*b^6 - 1/4*(5*a^2*b^4 + 10*a*b^5 + b^6)*cos(d*x + c)^2 + 1/2*((a^9 - 2*a^8*b - 5*a^7*b^2 + 20*a^6*b^3 - 25*a^5*b^4 + 14*a^4*b^5 - 3*a^3*b^6)*d^3*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) + (15*a^4*b^3 + 35*a^3*b^4 + 13*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 - (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2)) - 1/4*(2*(a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2)*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))) - 3*(a^2 - 2*a*b + b^2)*d*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 - (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2))*cos(d*x + c)^3*log(5/4*a^2*b^4 + 5/2*a*b^5 + 1/4*b^6 - 1/4*(5*a^2*b^4 + 10*a*b^5 + b^6)*cos(d*x + c)^2 - 1/2*((a^9 - 2*a^8*b - 5*a^7*b^2 + 20*a^6*b^3 - 25*a^5*b^4 + 14*a^4*b^5 - 3*a^3*b^6)*d^3*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) + (15*a^4*b^3 + 35*a^3*b^4 + 13*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 - (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2)) - 1/4*(2*(a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2)*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))) + 3*(a^2 - 2*a*b + b^2)*d*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2))*cos(d*x + c)^3*log(-5/4*a^2*b^4 - 5/2*a*b^5 - 1/4*b^6 + 1/4*(5*a^2*b^4 + 10*a*b^5 + b^6)*cos(d*x + c)^2 + 1/2*((a^9 - 2*a^8*b - 5*a^7*b^2 + 20*a^6*b^3 - 25*a^5*b^4 + 14*a^4*b^5 - 3*a^3*b^6)*d^3*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) - (15*a^4*b^3 + 35*a^3*b^4 + 13*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2)) - 1/4*(2*(a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2)*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))) - 3*(a^2 - 2*a*b + b^2)*d*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2))*cos(d*x + c)^3*log(-5/4*a^2*b^4 - 5/2*a*b^5 - 1/4*b^6 + 1/4*(5*a^2*b^4 + 10*a*b^5 + b^6)*cos(d*x + c)^2 - 1/2*((a^9 - 2*a^8*b - 5*a^7*b^2 + 20*a^6*b^3 - 25*a^5*b^4 + 14*a^4*b^5 - 3*a^3*b^6)*d^3*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) - (15*a^4*b^3 + 35*a^3*b^4 + 13*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2)) - 1/4*(2*(a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2)*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))) + 8*(2*(a - 4*b)*cos(d*x + c)^2 + a - b)*sin(d*x + c))/((a^2 - 2*a*b + b^2)*d*cos(d*x + c)^3)","B",0
418,1,5587,0,3.751497," ","integrate(sec(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm=""fricas"")","\frac{15 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d \sqrt{-\frac{a^{3} b^{3} + 21 \, a^{2} b^{4} + 35 \, a b^{5} + 7 \, b^{6} - {\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}}{{\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2}}} \cos\left(d x + c\right)^{5} \log\left(\frac{7}{4} \, a^{3} b^{5} + \frac{35}{4} \, a^{2} b^{6} + \frac{21}{4} \, a b^{7} + \frac{1}{4} \, b^{8} - \frac{1}{4} \, {\left(7 \, a^{3} b^{5} + 35 \, a^{2} b^{6} + 21 \, a b^{7} + b^{8}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(4 \, {\left(a^{11} - 6 \, a^{10} b + 14 \, a^{9} b^{2} - 14 \, a^{8} b^{3} + 14 \, a^{6} b^{5} - 14 \, a^{5} b^{6} + 6 \, a^{4} b^{7} - a^{3} b^{8}\right)} d^{3} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(7 \, a^{6} b^{3} + 77 \, a^{5} b^{4} + 238 \, a^{4} b^{5} + 162 \, a^{3} b^{6} + 27 \, a^{2} b^{7} + a b^{8}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a^{3} b^{3} + 21 \, a^{2} b^{4} + 35 \, a b^{5} + 7 \, b^{6} - {\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}}{{\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{9} b - 7 \, a^{8} b^{2} + 21 \, a^{7} b^{3} - 35 \, a^{6} b^{4} + 35 \, a^{5} b^{5} - 21 \, a^{4} b^{6} + 7 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{9} b - 7 \, a^{8} b^{2} + 21 \, a^{7} b^{3} - 35 \, a^{6} b^{4} + 35 \, a^{5} b^{5} - 21 \, a^{4} b^{6} + 7 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}\right)} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}\right) - 15 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d \sqrt{-\frac{a^{3} b^{3} + 21 \, a^{2} b^{4} + 35 \, a b^{5} + 7 \, b^{6} - {\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}}{{\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2}}} \cos\left(d x + c\right)^{5} \log\left(\frac{7}{4} \, a^{3} b^{5} + \frac{35}{4} \, a^{2} b^{6} + \frac{21}{4} \, a b^{7} + \frac{1}{4} \, b^{8} - \frac{1}{4} \, {\left(7 \, a^{3} b^{5} + 35 \, a^{2} b^{6} + 21 \, a b^{7} + b^{8}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(4 \, {\left(a^{11} - 6 \, a^{10} b + 14 \, a^{9} b^{2} - 14 \, a^{8} b^{3} + 14 \, a^{6} b^{5} - 14 \, a^{5} b^{6} + 6 \, a^{4} b^{7} - a^{3} b^{8}\right)} d^{3} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left(7 \, a^{6} b^{3} + 77 \, a^{5} b^{4} + 238 \, a^{4} b^{5} + 162 \, a^{3} b^{6} + 27 \, a^{2} b^{7} + a b^{8}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a^{3} b^{3} + 21 \, a^{2} b^{4} + 35 \, a b^{5} + 7 \, b^{6} - {\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}}{{\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{9} b - 7 \, a^{8} b^{2} + 21 \, a^{7} b^{3} - 35 \, a^{6} b^{4} + 35 \, a^{5} b^{5} - 21 \, a^{4} b^{6} + 7 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{9} b - 7 \, a^{8} b^{2} + 21 \, a^{7} b^{3} - 35 \, a^{6} b^{4} + 35 \, a^{5} b^{5} - 21 \, a^{4} b^{6} + 7 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}\right)} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}\right) + 15 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d \sqrt{-\frac{a^{3} b^{3} + 21 \, a^{2} b^{4} + 35 \, a b^{5} + 7 \, b^{6} + {\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}}{{\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2}}} \cos\left(d x + c\right)^{5} \log\left(-\frac{7}{4} \, a^{3} b^{5} - \frac{35}{4} \, a^{2} b^{6} - \frac{21}{4} \, a b^{7} - \frac{1}{4} \, b^{8} + \frac{1}{4} \, {\left(7 \, a^{3} b^{5} + 35 \, a^{2} b^{6} + 21 \, a b^{7} + b^{8}\right)} \cos\left(d x + c\right)^{2} + \frac{1}{2} \, {\left(4 \, {\left(a^{11} - 6 \, a^{10} b + 14 \, a^{9} b^{2} - 14 \, a^{8} b^{3} + 14 \, a^{6} b^{5} - 14 \, a^{5} b^{6} + 6 \, a^{4} b^{7} - a^{3} b^{8}\right)} d^{3} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(7 \, a^{6} b^{3} + 77 \, a^{5} b^{4} + 238 \, a^{4} b^{5} + 162 \, a^{3} b^{6} + 27 \, a^{2} b^{7} + a b^{8}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a^{3} b^{3} + 21 \, a^{2} b^{4} + 35 \, a b^{5} + 7 \, b^{6} + {\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}}{{\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{9} b - 7 \, a^{8} b^{2} + 21 \, a^{7} b^{3} - 35 \, a^{6} b^{4} + 35 \, a^{5} b^{5} - 21 \, a^{4} b^{6} + 7 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{9} b - 7 \, a^{8} b^{2} + 21 \, a^{7} b^{3} - 35 \, a^{6} b^{4} + 35 \, a^{5} b^{5} - 21 \, a^{4} b^{6} + 7 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}\right)} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}\right) - 15 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d \sqrt{-\frac{a^{3} b^{3} + 21 \, a^{2} b^{4} + 35 \, a b^{5} + 7 \, b^{6} + {\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}}{{\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2}}} \cos\left(d x + c\right)^{5} \log\left(-\frac{7}{4} \, a^{3} b^{5} - \frac{35}{4} \, a^{2} b^{6} - \frac{21}{4} \, a b^{7} - \frac{1}{4} \, b^{8} + \frac{1}{4} \, {\left(7 \, a^{3} b^{5} + 35 \, a^{2} b^{6} + 21 \, a b^{7} + b^{8}\right)} \cos\left(d x + c\right)^{2} - \frac{1}{2} \, {\left(4 \, {\left(a^{11} - 6 \, a^{10} b + 14 \, a^{9} b^{2} - 14 \, a^{8} b^{3} + 14 \, a^{6} b^{5} - 14 \, a^{5} b^{6} + 6 \, a^{4} b^{7} - a^{3} b^{8}\right)} d^{3} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left(7 \, a^{6} b^{3} + 77 \, a^{5} b^{4} + 238 \, a^{4} b^{5} + 162 \, a^{3} b^{6} + 27 \, a^{2} b^{7} + a b^{8}\right)} d \cos\left(d x + c\right) \sin\left(d x + c\right)\right)} \sqrt{-\frac{a^{3} b^{3} + 21 \, a^{2} b^{4} + 35 \, a b^{5} + 7 \, b^{6} + {\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}}{{\left(a^{8} - 7 \, a^{7} b + 21 \, a^{6} b^{2} - 35 \, a^{5} b^{3} + 35 \, a^{4} b^{4} - 21 \, a^{3} b^{5} + 7 \, a^{2} b^{6} - a b^{7}\right)} d^{2}}} - \frac{1}{4} \, {\left(2 \, {\left(a^{9} b - 7 \, a^{8} b^{2} + 21 \, a^{7} b^{3} - 35 \, a^{6} b^{4} + 35 \, a^{5} b^{5} - 21 \, a^{4} b^{6} + 7 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2} \cos\left(d x + c\right)^{2} - {\left(a^{9} b - 7 \, a^{8} b^{2} + 21 \, a^{7} b^{3} - 35 \, a^{6} b^{4} + 35 \, a^{5} b^{5} - 21 \, a^{4} b^{6} + 7 \, a^{3} b^{7} - a^{2} b^{8}\right)} d^{2}\right)} \sqrt{\frac{49 \, a^{6} b^{7} + 490 \, a^{5} b^{8} + 1519 \, a^{4} b^{9} + 1484 \, a^{3} b^{10} + 511 \, a^{2} b^{11} + 42 \, a b^{12} + b^{13}}{{\left(a^{17} - 14 \, a^{16} b + 91 \, a^{15} b^{2} - 364 \, a^{14} b^{3} + 1001 \, a^{13} b^{4} - 2002 \, a^{12} b^{5} + 3003 \, a^{11} b^{6} - 3432 \, a^{10} b^{7} + 3003 \, a^{9} b^{8} - 2002 \, a^{8} b^{9} + 1001 \, a^{7} b^{10} - 364 \, a^{6} b^{11} + 91 \, a^{5} b^{12} - 14 \, a^{4} b^{13} + a^{3} b^{14}\right)} d^{4}}}\right) + 8 \, {\left({\left(8 \, a^{2} - 21 \, a b + 73 \, b^{2}\right)} \cos\left(d x + c\right)^{4} + 2 \, {\left(2 \, a^{2} - 9 \, a b + 7 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 3 \, a^{2} - 6 \, a b + 3 \, b^{2}\right)} \sin\left(d x + c\right)}{120 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d \cos\left(d x + c\right)^{5}}"," ",0,"1/120*(15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 - (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2))*cos(d*x + c)^5*log(7/4*a^3*b^5 + 35/4*a^2*b^6 + 21/4*a*b^7 + 1/4*b^8 - 1/4*(7*a^3*b^5 + 35*a^2*b^6 + 21*a*b^7 + b^8)*cos(d*x + c)^2 + 1/2*(4*(a^11 - 6*a^10*b + 14*a^9*b^2 - 14*a^8*b^3 + 14*a^6*b^5 - 14*a^5*b^6 + 6*a^4*b^7 - a^3*b^8)*d^3*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))*cos(d*x + c)*sin(d*x + c) + (7*a^6*b^3 + 77*a^5*b^4 + 238*a^4*b^5 + 162*a^3*b^6 + 27*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 - (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2)) - 1/4*(2*(a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2*cos(d*x + c)^2 - (a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2)*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))) - 15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 - (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2))*cos(d*x + c)^5*log(7/4*a^3*b^5 + 35/4*a^2*b^6 + 21/4*a*b^7 + 1/4*b^8 - 1/4*(7*a^3*b^5 + 35*a^2*b^6 + 21*a*b^7 + b^8)*cos(d*x + c)^2 - 1/2*(4*(a^11 - 6*a^10*b + 14*a^9*b^2 - 14*a^8*b^3 + 14*a^6*b^5 - 14*a^5*b^6 + 6*a^4*b^7 - a^3*b^8)*d^3*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))*cos(d*x + c)*sin(d*x + c) + (7*a^6*b^3 + 77*a^5*b^4 + 238*a^4*b^5 + 162*a^3*b^6 + 27*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 - (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2)) - 1/4*(2*(a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2*cos(d*x + c)^2 - (a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2)*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))) + 15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 + (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2))*cos(d*x + c)^5*log(-7/4*a^3*b^5 - 35/4*a^2*b^6 - 21/4*a*b^7 - 1/4*b^8 + 1/4*(7*a^3*b^5 + 35*a^2*b^6 + 21*a*b^7 + b^8)*cos(d*x + c)^2 + 1/2*(4*(a^11 - 6*a^10*b + 14*a^9*b^2 - 14*a^8*b^3 + 14*a^6*b^5 - 14*a^5*b^6 + 6*a^4*b^7 - a^3*b^8)*d^3*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))*cos(d*x + c)*sin(d*x + c) - (7*a^6*b^3 + 77*a^5*b^4 + 238*a^4*b^5 + 162*a^3*b^6 + 27*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 + (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2)) - 1/4*(2*(a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2*cos(d*x + c)^2 - (a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2)*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))) - 15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 + (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2))*cos(d*x + c)^5*log(-7/4*a^3*b^5 - 35/4*a^2*b^6 - 21/4*a*b^7 - 1/4*b^8 + 1/4*(7*a^3*b^5 + 35*a^2*b^6 + 21*a*b^7 + b^8)*cos(d*x + c)^2 - 1/2*(4*(a^11 - 6*a^10*b + 14*a^9*b^2 - 14*a^8*b^3 + 14*a^6*b^5 - 14*a^5*b^6 + 6*a^4*b^7 - a^3*b^8)*d^3*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))*cos(d*x + c)*sin(d*x + c) - (7*a^6*b^3 + 77*a^5*b^4 + 238*a^4*b^5 + 162*a^3*b^6 + 27*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 + (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2)) - 1/4*(2*(a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2*cos(d*x + c)^2 - (a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2)*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))) + 8*((8*a^2 - 21*a*b + 73*b^2)*cos(d*x + c)^4 + 2*(2*a^2 - 9*a*b + 7*b^2)*cos(d*x + c)^2 + 3*a^2 - 6*a*b + 3*b^2)*sin(d*x + c))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(d*x + c)^5)","B",0
419,0,0,0,1.763084," ","integrate(cos(f*x+e)^m*(a+b*sin(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - 2 \, b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \cos\left(f x + e\right)^{m}, x\right)"," ",0,"integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e)^m, x)","F",0
420,0,0,0,0.856273," ","integrate(cos(f*x+e)^5*(a+b*sin(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - 2 \, b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \cos\left(f x + e\right)^{5}, x\right)"," ",0,"integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e)^5, x)","F",0
421,0,0,0,0.952490," ","integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - 2 \, b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \cos\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e)^3, x)","F",0
422,0,0,0,1.219235," ","integrate(cos(f*x+e)*(a+b*sin(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - 2 \, b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \cos\left(f x + e\right), x\right)"," ",0,"integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e), x)","F",0
423,0,0,0,0.904107," ","integrate(sec(f*x+e)*(a+b*sin(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - 2 \, b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \sec\left(f x + e\right), x\right)"," ",0,"integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p*sec(f*x + e), x)","F",0
424,0,0,0,1.168841," ","integrate(sec(f*x+e)^3*(a+b*sin(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - 2 \, b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \sec\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p*sec(f*x + e)^3, x)","F",0
425,0,0,0,0.783598," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - 2 \, b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \cos\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e)^4, x)","F",0
426,0,0,0,0.516308," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - 2 \, b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \cos\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e)^2, x)","F",0
427,0,0,0,0.985346," ","integrate((a+b*sin(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - 2 \, b \cos\left(f x + e\right)^{2} + a + b\right)}^{p}, x\right)"," ",0,"integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p, x)","F",0
428,0,0,0,1.061180," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - 2 \, b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \sec\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p*sec(f*x + e)^2, x)","F",0
429,0,0,0,1.128997," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(f x + e\right)^{4} - 2 \, b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \sec\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p*sec(f*x + e)^4, x)","F",0
430,0,0,0,0.929534," ","integrate(cos(f*x+e)^m*(a+b*sin(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right)^{m}, x\right)"," ",0,"integral((b*sin(f*x + e)^n + a)^p*cos(f*x + e)^m, x)","F",0
431,0,0,0,0.897816," ","integrate(cos(f*x+e)^5*(a+b*sin(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right)^{5}, x\right)"," ",0,"integral((b*sin(f*x + e)^n + a)^p*cos(f*x + e)^5, x)","F",0
432,0,0,0,0.998871," ","integrate(cos(f*x+e)^3*(a+b*sin(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*sin(f*x + e)^n + a)^p*cos(f*x + e)^3, x)","F",0
433,0,0,0,0.984619," ","integrate(cos(f*x+e)*(a+b*sin(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right), x\right)"," ",0,"integral((b*sin(f*x + e)^n + a)^p*cos(f*x + e), x)","F",0
434,0,0,0,1.138264," ","integrate(sec(f*x+e)*(a+b*sin(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \sec\left(f x + e\right), x\right)"," ",0,"integral((b*sin(f*x + e)^n + a)^p*sec(f*x + e), x)","F",0
435,0,0,0,0.966665," ","integrate(sec(f*x+e)^3*(a+b*sin(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \sec\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*sin(f*x + e)^n + a)^p*sec(f*x + e)^3, x)","F",0
436,0,0,0,0.803309," ","integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*sin(f*x + e)^n + a)^p*cos(f*x + e)^4, x)","F",0
437,0,0,0,0.986709," ","integrate(cos(f*x+e)^2*(a+b*sin(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \cos\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*sin(f*x + e)^n + a)^p*cos(f*x + e)^2, x)","F",0
438,0,0,0,1.518803," ","integrate((a+b*sin(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p}, x\right)"," ",0,"integral((b*sin(f*x + e)^n + a)^p, x)","F",0
439,0,0,0,0.849251," ","integrate(sec(f*x+e)^2*(a+b*sin(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \sec\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*sin(f*x + e)^n + a)^p*sec(f*x + e)^2, x)","F",0
440,0,0,0,1.426049," ","integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(f x + e\right)^{n} + a\right)}^{p} \sec\left(f x + e\right)^{4}, x\right)"," ",0,"integral((b*sin(f*x + e)^n + a)^p*sec(f*x + e)^4, x)","F",0
441,1,179,0,1.326358," ","integrate(tan(d*x+c)^7/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{6 \, a^{3} \cos\left(d x + c\right)^{6} \log\left(-b \cos\left(d x + c\right)^{2} + a + b\right) - 12 \, a^{3} \cos\left(d x + c\right)^{6} \log\left(-\cos\left(d x + c\right)\right) - 6 \, {\left(3 \, a^{3} + 6 \, a^{2} b + 4 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{4} - 2 \, a^{3} - 6 \, a^{2} b - 6 \, a b^{2} - 2 \, b^{3} + 3 \, {\left(3 \, a^{3} + 8 \, a^{2} b + 7 \, a b^{2} + 2 \, b^{3}\right)} \cos\left(d x + c\right)^{2}}{12 \, {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} d \cos\left(d x + c\right)^{6}}"," ",0,"-1/12*(6*a^3*cos(d*x + c)^6*log(-b*cos(d*x + c)^2 + a + b) - 12*a^3*cos(d*x + c)^6*log(-cos(d*x + c)) - 6*(3*a^3 + 6*a^2*b + 4*a*b^2 + b^3)*cos(d*x + c)^4 - 2*a^3 - 6*a^2*b - 6*a*b^2 - 2*b^3 + 3*(3*a^3 + 8*a^2*b + 7*a*b^2 + 2*b^3)*cos(d*x + c)^2)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cos(d*x + c)^6)","A",0
442,1,118,0,0.965297," ","integrate(tan(d*x+c)^5/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{2 \, a^{2} \cos\left(d x + c\right)^{4} \log\left(-b \cos\left(d x + c\right)^{2} + a + b\right) - 4 \, a^{2} \cos\left(d x + c\right)^{4} \log\left(-\cos\left(d x + c\right)\right) - 2 \, {\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}{4 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d \cos\left(d x + c\right)^{4}}"," ",0,"1/4*(2*a^2*cos(d*x + c)^4*log(-b*cos(d*x + c)^2 + a + b) - 4*a^2*cos(d*x + c)^4*log(-cos(d*x + c)) - 2*(2*a^2 + 3*a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cos(d*x + c)^4)","A",0
443,1,78,0,1.110121," ","integrate(tan(d*x+c)^3/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{a \cos\left(d x + c\right)^{2} \log\left(-b \cos\left(d x + c\right)^{2} + a + b\right) - 2 \, a \cos\left(d x + c\right)^{2} \log\left(-\cos\left(d x + c\right)\right) - a - b}{2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} d \cos\left(d x + c\right)^{2}}"," ",0,"-1/2*(a*cos(d*x + c)^2*log(-b*cos(d*x + c)^2 + a + b) - 2*a*cos(d*x + c)^2*log(-cos(d*x + c)) - a - b)/((a^2 + 2*a*b + b^2)*d*cos(d*x + c)^2)","A",0
444,1,37,0,0.814025," ","integrate(tan(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{\log\left(-b \cos\left(d x + c\right)^{2} + a + b\right) - 2 \, \log\left(-\cos\left(d x + c\right)\right)}{2 \, {\left(a + b\right)} d}"," ",0,"1/2*(log(-b*cos(d*x + c)^2 + a + b) - 2*log(-cos(d*x + c)))/((a + b)*d)","A",0
445,1,35,0,1.051078," ","integrate(cot(d*x+c)/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{\log\left(-b \cos\left(d x + c\right)^{2} + a + b\right) - 2 \, \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right)}{2 \, a d}"," ",0,"-1/2*(log(-b*cos(d*x + c)^2 + a + b) - 2*log(1/2*sin(d*x + c)))/(a*d)","A",0
446,1,91,0,0.960345," ","integrate(cot(d*x+c)^3/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{{\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \log\left(-b \cos\left(d x + c\right)^{2} + a + b\right) - 2 \, {\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right) + a}{2 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)}}"," ",0,"1/2*(((a + b)*cos(d*x + c)^2 - a - b)*log(-b*cos(d*x + c)^2 + a + b) - 2*((a + b)*cos(d*x + c)^2 - a - b)*log(1/2*sin(d*x + c)) + a)/(a^2*d*cos(d*x + c)^2 - a^2*d)","A",0
447,1,198,0,0.956789," ","integrate(cot(d*x+c)^5/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","-\frac{2 \, {\left(2 \, a^{2} + a b\right)} \cos\left(d x + c\right)^{2} - 3 \, a^{2} - 2 \, a b + 2 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \log\left(-b \cos\left(d x + c\right)^{2} + a + b\right) - 4 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right)}{4 \, {\left(a^{3} d \cos\left(d x + c\right)^{4} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} + a^{3} d\right)}}"," ",0,"-1/4*(2*(2*a^2 + a*b)*cos(d*x + c)^2 - 3*a^2 - 2*a*b + 2*((a^2 + 2*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 + 2*a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*log(-b*cos(d*x + c)^2 + a + b) - 4*((a^2 + 2*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 + 2*a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*log(1/2*sin(d*x + c)))/(a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)","B",0
448,1,371,0,1.973998," ","integrate(cot(d*x+c)^7/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\frac{6 \, {\left(3 \, a^{3} + 3 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{4} + 11 \, a^{3} + 15 \, a^{2} b + 6 \, a b^{2} - 3 \, {\left(9 \, a^{3} + 11 \, a^{2} b + 4 \, a b^{2}\right)} \cos\left(d x + c\right)^{2} + 6 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-b \cos\left(d x + c\right)^{2} + a + b\right) - 12 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \sin\left(d x + c\right)\right)}{12 \, {\left(a^{4} d \cos\left(d x + c\right)^{6} - 3 \, a^{4} d \cos\left(d x + c\right)^{4} + 3 \, a^{4} d \cos\left(d x + c\right)^{2} - a^{4} d\right)}}"," ",0,"1/12*(6*(3*a^3 + 3*a^2*b + a*b^2)*cos(d*x + c)^4 + 11*a^3 + 15*a^2*b + 6*a*b^2 - 3*(9*a^3 + 11*a^2*b + 4*a*b^2)*cos(d*x + c)^2 + 6*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^6 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^2)*log(-b*cos(d*x + c)^2 + a + b) - 12*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^6 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^2)*log(1/2*sin(d*x + c)))/(a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)","B",0
449,1,602,0,0.839829," ","integrate(tan(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{105 \, a^{3} \sqrt{-\frac{a}{a + b}} \cos\left(d x + c\right)^{7} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, {\left({\left(176 \, a^{3} + 122 \, a^{2} b + 66 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(d x + c\right)^{6} - {\left(122 \, a^{3} + 254 \, a^{2} b + 177 \, a b^{2} + 45 \, b^{3}\right)} \cos\left(d x + c\right)^{4} - 15 \, a^{3} - 45 \, a^{2} b - 45 \, a b^{2} - 15 \, b^{3} + 3 \, {\left(22 \, a^{3} + 59 \, a^{2} b + 52 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{420 \, {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} d \cos\left(d x + c\right)^{7}}, -\frac{105 \, a^{3} \sqrt{\frac{a}{a + b}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a}{a + b}}}{2 \, a \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \cos\left(d x + c\right)^{7} + 2 \, {\left({\left(176 \, a^{3} + 122 \, a^{2} b + 66 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(d x + c\right)^{6} - {\left(122 \, a^{3} + 254 \, a^{2} b + 177 \, a b^{2} + 45 \, b^{3}\right)} \cos\left(d x + c\right)^{4} - 15 \, a^{3} - 45 \, a^{2} b - 45 \, a b^{2} - 15 \, b^{3} + 3 \, {\left(22 \, a^{3} + 59 \, a^{2} b + 52 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)}{210 \, {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} d \cos\left(d x + c\right)^{7}}\right]"," ",0,"[1/420*(105*a^3*sqrt(-a/(a + b))*cos(d*x + c)^7*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*((176*a^3 + 122*a^2*b + 66*a*b^2 + 15*b^3)*cos(d*x + c)^6 - (122*a^3 + 254*a^2*b + 177*a*b^2 + 45*b^3)*cos(d*x + c)^4 - 15*a^3 - 45*a^2*b - 45*a*b^2 - 15*b^3 + 3*(22*a^3 + 59*a^2*b + 52*a*b^2 + 15*b^3)*cos(d*x + c)^2)*sin(d*x + c))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cos(d*x + c)^7), -1/210*(105*a^3*sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c)))*cos(d*x + c)^7 + 2*((176*a^3 + 122*a^2*b + 66*a*b^2 + 15*b^3)*cos(d*x + c)^6 - (122*a^3 + 254*a^2*b + 177*a*b^2 + 45*b^3)*cos(d*x + c)^4 - 15*a^3 - 45*a^2*b - 45*a*b^2 - 15*b^3 + 3*(22*a^3 + 59*a^2*b + 52*a*b^2 + 15*b^3)*cos(d*x + c)^2)*sin(d*x + c))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cos(d*x + c)^7)]","B",0
450,1,472,0,1.096015," ","integrate(tan(d*x+c)^6/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{15 \, a^{2} \sqrt{-\frac{a}{a + b}} \cos\left(d x + c\right)^{5} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) + 4 \, {\left({\left(23 \, a^{2} + 11 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - {\left(11 \, a^{2} + 17 \, a b + 6 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}\right)} \sin\left(d x + c\right)}{60 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d \cos\left(d x + c\right)^{5}}, \frac{15 \, a^{2} \sqrt{\frac{a}{a + b}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a}{a + b}}}{2 \, a \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \cos\left(d x + c\right)^{5} + 2 \, {\left({\left(23 \, a^{2} + 11 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - {\left(11 \, a^{2} + 17 \, a b + 6 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}\right)} \sin\left(d x + c\right)}{30 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} d \cos\left(d x + c\right)^{5}}\right]"," ",0,"[1/60*(15*a^2*sqrt(-a/(a + b))*cos(d*x + c)^5*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) + 4*((23*a^2 + 11*a*b + 3*b^2)*cos(d*x + c)^4 - (11*a^2 + 17*a*b + 6*b^2)*cos(d*x + c)^2 + 3*a^2 + 6*a*b + 3*b^2)*sin(d*x + c))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cos(d*x + c)^5), 1/30*(15*a^2*sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c)))*cos(d*x + c)^5 + 2*((23*a^2 + 11*a*b + 3*b^2)*cos(d*x + c)^4 - (11*a^2 + 17*a*b + 6*b^2)*cos(d*x + c)^2 + 3*a^2 + 6*a*b + 3*b^2)*sin(d*x + c))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cos(d*x + c)^5)]","B",0
451,1,366,0,1.021949," ","integrate(tan(d*x+c)^4/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{3 \, a \sqrt{-\frac{a}{a + b}} \cos\left(d x + c\right)^{3} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, {\left({\left(4 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sin\left(d x + c\right)}{12 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} d \cos\left(d x + c\right)^{3}}, -\frac{3 \, a \sqrt{\frac{a}{a + b}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a}{a + b}}}{2 \, a \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \cos\left(d x + c\right)^{3} + 2 \, {\left({\left(4 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sin\left(d x + c\right)}{6 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} d \cos\left(d x + c\right)^{3}}\right]"," ",0,"[1/12*(3*a*sqrt(-a/(a + b))*cos(d*x + c)^3*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*((4*a + b)*cos(d*x + c)^2 - a - b)*sin(d*x + c))/((a^2 + 2*a*b + b^2)*d*cos(d*x + c)^3), -1/6*(3*a*sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c)))*cos(d*x + c)^3 + 2*((4*a + b)*cos(d*x + c)^2 - a - b)*sin(d*x + c))/((a^2 + 2*a*b + b^2)*d*cos(d*x + c)^3)]","A",0
452,1,300,0,1.086752," ","integrate(tan(d*x+c)^2/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-\frac{a}{a + b}} \cos\left(d x + c\right) \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a^{2} + 3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) + 4 \, \sin\left(d x + c\right)}{4 \, {\left(a + b\right)} d \cos\left(d x + c\right)}, \frac{\sqrt{\frac{a}{a + b}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a}{a + b}}}{2 \, a \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \cos\left(d x + c\right) + 2 \, \sin\left(d x + c\right)}{2 \, {\left(a + b\right)} d \cos\left(d x + c\right)}\right]"," ",0,"[1/4*(sqrt(-a/(a + b))*cos(d*x + c)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a^2 + 3*a*b + b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-a/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) + 4*sin(d*x + c))/((a + b)*d*cos(d*x + c)), 1/2*(sqrt(a/(a + b))*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt(a/(a + b))/(a*cos(d*x + c)*sin(d*x + c)))*cos(d*x + c) + 2*sin(d*x + c))/((a + b)*d*cos(d*x + c))]","B",0
453,1,290,0,1.210144," ","integrate(cot(d*x+c)^2/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-\frac{a + b}{a}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a^{2} + a b\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a + b}{a}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \sin\left(d x + c\right) - 4 \, \cos\left(d x + c\right)}{4 \, a d \sin\left(d x + c\right)}, \frac{\sqrt{\frac{a + b}{a}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a + b}{a}}}{2 \, {\left(a + b\right)} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) - 2 \, \cos\left(d x + c\right)}{2 \, a d \sin\left(d x + c\right)}\right]"," ",0,"[1/4*(sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a^2 + a*b)*cos(d*x + c)^3 - (a^2 + a*b)*cos(d*x + c))*sqrt(-(a + b)/a)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c) - 4*cos(d*x + c))/(a*d*sin(d*x + c)), 1/2*(sqrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) - 2*cos(d*x + c))/(a*d*sin(d*x + c))]","B",0
454,1,402,0,0.851065," ","integrate(cot(d*x+c)^4/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(4 \, a + 3 \, b\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{-\frac{a + b}{a}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + a b\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a + b}{a}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \sin\left(d x + c\right) - 12 \, {\left(a + b\right)} \cos\left(d x + c\right)}{12 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)} \sin\left(d x + c\right)}, \frac{2 \, {\left(4 \, a + 3 \, b\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left({\left(a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a + b}{a}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a + b}{a}}}{2 \, {\left(a + b\right)} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) - 6 \, {\left(a + b\right)} \cos\left(d x + c\right)}{6 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[1/12*(4*(4*a + 3*b)*cos(d*x + c)^3 + 3*((a + b)*cos(d*x + c)^2 - a - b)*sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a^2 + a*b)*cos(d*x + c)^3 - (a^2 + a*b)*cos(d*x + c))*sqrt(-(a + b)/a)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c) - 12*(a + b)*cos(d*x + c))/((a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c)), 1/6*(2*(4*a + 3*b)*cos(d*x + c)^3 - 3*((a + b)*cos(d*x + c)^2 - a - b)*sqrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) - 6*(a + b)*cos(d*x + c))/((a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))]","B",0
455,1,576,0,1.637585," ","integrate(cot(d*x+c)^6/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[-\frac{4 \, {\left(23 \, a^{2} + 35 \, a b + 15 \, b^{2}\right)} \cos\left(d x + c\right)^{5} - 20 \, {\left(7 \, a^{2} + 13 \, a b + 6 \, b^{2}\right)} \cos\left(d x + c\right)^{3} - 15 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-\frac{a + b}{a}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(2 \, a^{2} + a b\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a + b}{a}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \sin\left(d x + c\right) + 60 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)}{60 \, {\left(a^{3} d \cos\left(d x + c\right)^{4} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} + a^{3} d\right)} \sin\left(d x + c\right)}, -\frac{2 \, {\left(23 \, a^{2} + 35 \, a b + 15 \, b^{2}\right)} \cos\left(d x + c\right)^{5} - 10 \, {\left(7 \, a^{2} + 13 \, a b + 6 \, b^{2}\right)} \cos\left(d x + c\right)^{3} - 15 \, {\left({\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{\frac{a + b}{a}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a + b}{a}}}{2 \, {\left(a + b\right)} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) + 30 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)}{30 \, {\left(a^{3} d \cos\left(d x + c\right)^{4} - 2 \, a^{3} d \cos\left(d x + c\right)^{2} + a^{3} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[-1/60*(4*(23*a^2 + 35*a*b + 15*b^2)*cos(d*x + c)^5 - 20*(7*a^2 + 13*a*b + 6*b^2)*cos(d*x + c)^3 - 15*((a^2 + 2*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 + 2*a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 + 4*((2*a^2 + a*b)*cos(d*x + c)^3 - (a^2 + a*b)*cos(d*x + c))*sqrt(-(a + b)/a)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c) + 60*(a^2 + 2*a*b + b^2)*cos(d*x + c))/((a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)*sin(d*x + c)), -1/30*(2*(23*a^2 + 35*a*b + 15*b^2)*cos(d*x + c)^5 - 10*(7*a^2 + 13*a*b + 6*b^2)*cos(d*x + c)^3 - 15*((a^2 + 2*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 + 2*a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) + 30*(a^2 + 2*a*b + b^2)*cos(d*x + c))/((a^3*d*cos(d*x + c)^4 - 2*a^3*d*cos(d*x + c)^2 + a^3*d)*sin(d*x + c))]","B",0
456,1,834,0,1.465574," ","integrate(cot(d*x+c)^8/(a+b*sin(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(176 \, a^{3} + 406 \, a^{2} b + 350 \, a b^{2} + 105 \, b^{3}\right)} \cos\left(d x + c\right)^{7} - 28 \, {\left(58 \, a^{3} + 158 \, a^{2} b + 145 \, a b^{2} + 45 \, b^{3}\right)} \cos\left(d x + c\right)^{5} + 140 \, {\left(10 \, a^{3} + 29 \, a^{2} b + 28 \, a b^{2} + 9 \, b^{3}\right)} \cos\left(d x + c\right)^{3} + 105 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-\frac{a + b}{a}} \log\left(\frac{{\left(8 \, a^{2} + 8 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(4 \, a^{2} + 5 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 4 \, {\left({\left(2 \, a^{2} + a b\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + a b\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{a + b}{a}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \sin\left(d x + c\right) - 420 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)}{420 \, {\left(a^{4} d \cos\left(d x + c\right)^{6} - 3 \, a^{4} d \cos\left(d x + c\right)^{4} + 3 \, a^{4} d \cos\left(d x + c\right)^{2} - a^{4} d\right)} \sin\left(d x + c\right)}, \frac{2 \, {\left(176 \, a^{3} + 406 \, a^{2} b + 350 \, a b^{2} + 105 \, b^{3}\right)} \cos\left(d x + c\right)^{7} - 14 \, {\left(58 \, a^{3} + 158 \, a^{2} b + 145 \, a b^{2} + 45 \, b^{3}\right)} \cos\left(d x + c\right)^{5} + 70 \, {\left(10 \, a^{3} + 29 \, a^{2} b + 28 \, a b^{2} + 9 \, b^{3}\right)} \cos\left(d x + c\right)^{3} - 105 \, {\left({\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{a + b}{a}} \arctan\left(\frac{{\left({\left(2 \, a + b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{a + b}{a}}}{2 \, {\left(a + b\right)} \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) \sin\left(d x + c\right) - 210 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)}{210 \, {\left(a^{4} d \cos\left(d x + c\right)^{6} - 3 \, a^{4} d \cos\left(d x + c\right)^{4} + 3 \, a^{4} d \cos\left(d x + c\right)^{2} - a^{4} d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[1/420*(4*(176*a^3 + 406*a^2*b + 350*a*b^2 + 105*b^3)*cos(d*x + c)^7 - 28*(58*a^3 + 158*a^2*b + 145*a*b^2 + 45*b^3)*cos(d*x + c)^5 + 140*(10*a^3 + 29*a^2*b + 28*a*b^2 + 9*b^3)*cos(d*x + c)^3 + 105*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^6 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a*b + b^2)*cos(d*x + c)^2 - 4*((2*a^2 + a*b)*cos(d*x + c)^3 - (a^2 + a*b)*cos(d*x + c))*sqrt(-(a + b)/a)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c) - 420*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))/((a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x + c)), 1/210*(2*(176*a^3 + 406*a^2*b + 350*a*b^2 + 105*b^3)*cos(d*x + c)^7 - 14*(58*a^3 + 158*a^2*b + 145*a*b^2 + 45*b^3)*cos(d*x + c)^5 + 70*(10*a^3 + 29*a^2*b + 28*a*b^2 + 9*b^3)*cos(d*x + c)^3 - 105*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^6 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^2)*sqrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt((a + b)/a)/((a + b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) - 210*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))/((a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x + c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x + c))]","B",0
457,1,47,0,0.646995," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^5,x, algorithm=""fricas"")","-\frac{{\left(3 \, \cos\left(f x + e\right)^{4} + 6 \, \cos\left(f x + e\right)^{2} - 1\right)} \sqrt{a \cos\left(f x + e\right)^{2}}}{3 \, f \cos\left(f x + e\right)^{4}}"," ",0,"-1/3*(3*cos(f*x + e)^4 + 6*cos(f*x + e)^2 - 1)*sqrt(a*cos(f*x + e)^2)/(f*cos(f*x + e)^4)","A",0
458,1,34,0,0.873293," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^3,x, algorithm=""fricas"")","\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(\cos\left(f x + e\right)^{2} + 1\right)}}{f \cos\left(f x + e\right)^{2}}"," ",0,"sqrt(a*cos(f*x + e)^2)*(cos(f*x + e)^2 + 1)/(f*cos(f*x + e)^2)","A",0
459,1,17,0,0.928030," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}}}{f}"," ",0,"-sqrt(a*cos(f*x + e)^2)/f","A",0
460,1,57,0,1.990841," ","integrate(cot(f*x+e)*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(2 \, \cos\left(f x + e\right) - \log\left(-\frac{\cos\left(f x + e\right) + 1}{\cos\left(f x + e\right) - 1}\right)\right)}}{2 \, f \cos\left(f x + e\right)}"," ",0,"1/2*sqrt(a*cos(f*x + e)^2)*(2*cos(f*x + e) - log(-(cos(f*x + e) + 1)/(cos(f*x + e) - 1)))/(f*cos(f*x + e))","A",0
461,1,88,0,0.974743," ","integrate(cot(f*x+e)^3*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(4 \, \cos\left(f x + e\right)^{3} + 3 \, {\left(\cos\left(f x + e\right)^{2} - 1\right)} \log\left(-\frac{\cos\left(f x + e\right) - 1}{\cos\left(f x + e\right) + 1}\right) - 6 \, \cos\left(f x + e\right)\right)}}{4 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)}}"," ",0,"-1/4*sqrt(a*cos(f*x + e)^2)*(4*cos(f*x + e)^3 + 3*(cos(f*x + e)^2 - 1)*log(-(cos(f*x + e) - 1)/(cos(f*x + e) + 1)) - 6*cos(f*x + e))/(f*cos(f*x + e)^3 - f*cos(f*x + e))","A",0
462,1,87,0,1.209629," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^6,x, algorithm=""fricas"")","-\frac{{\left(15 \, \cos\left(f x + e\right)^{4} \log\left(-\frac{\sin\left(f x + e\right) - 1}{\sin\left(f x + e\right) + 1}\right) + 2 \, {\left(8 \, \cos\left(f x + e\right)^{4} + 9 \, \cos\left(f x + e\right)^{2} - 2\right)} \sin\left(f x + e\right)\right)} \sqrt{a \cos\left(f x + e\right)^{2}}}{16 \, f \cos\left(f x + e\right)^{5}}"," ",0,"-1/16*(15*cos(f*x + e)^4*log(-(sin(f*x + e) - 1)/(sin(f*x + e) + 1)) + 2*(8*cos(f*x + e)^4 + 9*cos(f*x + e)^2 - 2)*sin(f*x + e))*sqrt(a*cos(f*x + e)^2)/(f*cos(f*x + e)^5)","A",0
463,1,77,0,0.788197," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^4,x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(3 \, \cos\left(f x + e\right)^{2} \log\left(-\frac{\sin\left(f x + e\right) + 1}{\sin\left(f x + e\right) - 1}\right) - 2 \, {\left(2 \, \cos\left(f x + e\right)^{2} + 1\right)} \sin\left(f x + e\right)\right)}}{4 \, f \cos\left(f x + e\right)^{3}}"," ",0,"-1/4*sqrt(a*cos(f*x + e)^2)*(3*cos(f*x + e)^2*log(-(sin(f*x + e) + 1)/(sin(f*x + e) - 1)) - 2*(2*cos(f*x + e)^2 + 1)*sin(f*x + e))/(f*cos(f*x + e)^3)","A",0
464,1,55,0,0.833768," ","integrate((a-a*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^2,x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(\log\left(-\frac{\sin\left(f x + e\right) - 1}{\sin\left(f x + e\right) + 1}\right) + 2 \, \sin\left(f x + e\right)\right)}}{2 \, f \cos\left(f x + e\right)}"," ",0,"-1/2*sqrt(a*cos(f*x + e)^2)*(log(-(sin(f*x + e) - 1)/(sin(f*x + e) + 1)) + 2*sin(f*x + e))/(f*cos(f*x + e))","A",0
465,1,42,0,1.150867," ","integrate(cot(f*x+e)^2*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(\cos\left(f x + e\right)^{2} - 2\right)}}{f \cos\left(f x + e\right) \sin\left(f x + e\right)}"," ",0,"sqrt(a*cos(f*x + e)^2)*(cos(f*x + e)^2 - 2)/(f*cos(f*x + e)*sin(f*x + e))","A",0
466,1,66,0,0.783243," ","integrate(cot(f*x+e)^4*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, \cos\left(f x + e\right)^{4} - 12 \, \cos\left(f x + e\right)^{2} + 8\right)} \sqrt{a \cos\left(f x + e\right)^{2}}}{3 \, {\left(f \cos\left(f x + e\right)^{3} - f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"-1/3*(3*cos(f*x + e)^4 - 12*cos(f*x + e)^2 + 8)*sqrt(a*cos(f*x + e)^2)/((f*cos(f*x + e)^3 - f*cos(f*x + e))*sin(f*x + e))","A",0
467,1,86,0,1.037269," ","integrate(cot(f*x+e)^6*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\frac{{\left(5 \, \cos\left(f x + e\right)^{6} - 30 \, \cos\left(f x + e\right)^{4} + 40 \, \cos\left(f x + e\right)^{2} - 16\right)} \sqrt{a \cos\left(f x + e\right)^{2}}}{5 \, {\left(f \cos\left(f x + e\right)^{5} - 2 \, f \cos\left(f x + e\right)^{3} + f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"1/5*(5*cos(f*x + e)^6 - 30*cos(f*x + e)^4 + 40*cos(f*x + e)^2 - 16)*sqrt(a*cos(f*x + e)^2)/((f*cos(f*x + e)^5 - 2*f*cos(f*x + e)^3 + f*cos(f*x + e))*sin(f*x + e))","A",0
468,1,50,0,0.929486," ","integrate(tan(f*x+e)^5/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\frac{{\left(15 \, \cos\left(f x + e\right)^{4} - 10 \, \cos\left(f x + e\right)^{2} + 3\right)} \sqrt{a \cos\left(f x + e\right)^{2}}}{15 \, a f \cos\left(f x + e\right)^{6}}"," ",0,"1/15*(15*cos(f*x + e)^4 - 10*cos(f*x + e)^2 + 3)*sqrt(a*cos(f*x + e)^2)/(a*f*cos(f*x + e)^6)","A",0
469,1,40,0,0.898641," ","integrate(tan(f*x+e)^3/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(3 \, \cos\left(f x + e\right)^{2} - 1\right)}}{3 \, a f \cos\left(f x + e\right)^{4}}"," ",0,"-1/3*sqrt(a*cos(f*x + e)^2)*(3*cos(f*x + e)^2 - 1)/(a*f*cos(f*x + e)^4)","A",0
470,1,27,0,0.954702," ","integrate(tan(f*x+e)/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{a \cos\left(f x + e\right)^{2}}}{a f \cos\left(f x + e\right)^{2}}"," ",0,"sqrt(a*cos(f*x + e)^2)/(a*f*cos(f*x + e)^2)","A",0
471,1,84,0,0.753670," ","integrate(cot(f*x+e)/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} \log\left(-\frac{\cos\left(f x + e\right) + 1}{\cos\left(f x + e\right) - 1}\right)}{2 \, a f \cos\left(f x + e\right)}, \frac{\sqrt{-a} \arctan\left(\frac{\sqrt{a \cos\left(f x + e\right)^{2}} \sqrt{-a}}{a}\right)}{a f}\right]"," ",0,"[-1/2*sqrt(a*cos(f*x + e)^2)*log(-(cos(f*x + e) + 1)/(cos(f*x + e) - 1))/(a*f*cos(f*x + e)), sqrt(-a)*arctan(sqrt(a*cos(f*x + e)^2)*sqrt(-a)/a)/(a*f)]","A",0
472,1,79,0,0.666473," ","integrate(cot(f*x+e)^3/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left({\left(\cos\left(f x + e\right)^{2} - 1\right)} \log\left(-\frac{\cos\left(f x + e\right) - 1}{\cos\left(f x + e\right) + 1}\right) - 2 \, \cos\left(f x + e\right)\right)}}{4 \, {\left(a f \cos\left(f x + e\right)^{3} - a f \cos\left(f x + e\right)\right)}}"," ",0,"-1/4*sqrt(a*cos(f*x + e)^2)*((cos(f*x + e)^2 - 1)*log(-(cos(f*x + e) - 1)/(cos(f*x + e) + 1)) - 2*cos(f*x + e))/(a*f*cos(f*x + e)^3 - a*f*cos(f*x + e))","A",0
473,1,80,0,0.505047," ","integrate(tan(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(3 \, \cos\left(f x + e\right)^{4} \log\left(-\frac{\sin\left(f x + e\right) - 1}{\sin\left(f x + e\right) + 1}\right) + 2 \, {\left(5 \, \cos\left(f x + e\right)^{2} - 2\right)} \sin\left(f x + e\right)\right)} \sqrt{a \cos\left(f x + e\right)^{2}}}{16 \, a f \cos\left(f x + e\right)^{5}}"," ",0,"-1/16*(3*cos(f*x + e)^4*log(-(sin(f*x + e) - 1)/(sin(f*x + e) + 1)) + 2*(5*cos(f*x + e)^2 - 2)*sin(f*x + e))*sqrt(a*cos(f*x + e)^2)/(a*f*cos(f*x + e)^5)","A",0
474,1,67,0,0.611829," ","integrate(tan(f*x+e)^2/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(\cos\left(f x + e\right)^{2} \log\left(-\frac{\sin\left(f x + e\right) + 1}{\sin\left(f x + e\right) - 1}\right) - 2 \, \sin\left(f x + e\right)\right)}}{4 \, a f \cos\left(f x + e\right)^{3}}"," ",0,"-1/4*sqrt(a*cos(f*x + e)^2)*(cos(f*x + e)^2*log(-(sin(f*x + e) + 1)/(sin(f*x + e) - 1)) - 2*sin(f*x + e))/(a*f*cos(f*x + e)^3)","A",0
475,1,36,0,0.501297," ","integrate(cot(f*x+e)^2/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}}}{a f \cos\left(f x + e\right) \sin\left(f x + e\right)}"," ",0,"-sqrt(a*cos(f*x + e)^2)/(a*f*cos(f*x + e)*sin(f*x + e))","A",0
476,1,58,0,0.585880," ","integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(3 \, \cos\left(f x + e\right)^{2} - 2\right)}}{3 \, {\left(a f \cos\left(f x + e\right)^{3} - a f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"1/3*sqrt(a*cos(f*x + e)^2)*(3*cos(f*x + e)^2 - 2)/((a*f*cos(f*x + e)^3 - a*f*cos(f*x + e))*sin(f*x + e))","A",0
477,1,79,0,0.663358," ","integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","-\frac{{\left(15 \, \cos\left(f x + e\right)^{4} - 20 \, \cos\left(f x + e\right)^{2} + 8\right)} \sqrt{a \cos\left(f x + e\right)^{2}}}{15 \, {\left(a f \cos\left(f x + e\right)^{5} - 2 \, a f \cos\left(f x + e\right)^{3} + a f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*(15*cos(f*x + e)^4 - 20*cos(f*x + e)^2 + 8)*sqrt(a*cos(f*x + e)^2)/((a*f*cos(f*x + e)^5 - 2*a*f*cos(f*x + e)^3 + a*f*cos(f*x + e))*sin(f*x + e))","A",0
478,1,50,0,0.644270," ","integrate(tan(f*x+e)^5/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\frac{{\left(35 \, \cos\left(f x + e\right)^{4} - 42 \, \cos\left(f x + e\right)^{2} + 15\right)} \sqrt{a \cos\left(f x + e\right)^{2}}}{105 \, a^{2} f \cos\left(f x + e\right)^{8}}"," ",0,"1/105*(35*cos(f*x + e)^4 - 42*cos(f*x + e)^2 + 15)*sqrt(a*cos(f*x + e)^2)/(a^2*f*cos(f*x + e)^8)","A",0
479,1,40,0,0.669945," ","integrate(tan(f*x+e)^3/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(5 \, \cos\left(f x + e\right)^{2} - 3\right)}}{15 \, a^{2} f \cos\left(f x + e\right)^{6}}"," ",0,"-1/15*sqrt(a*cos(f*x + e)^2)*(5*cos(f*x + e)^2 - 3)/(a^2*f*cos(f*x + e)^6)","A",0
480,1,28,0,0.640989," ","integrate(tan(f*x+e)/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{a \cos\left(f x + e\right)^{2}}}{3 \, a^{2} f \cos\left(f x + e\right)^{4}}"," ",0,"1/3*sqrt(a*cos(f*x + e)^2)/(a^2*f*cos(f*x + e)^4)","A",0
481,1,58,0,0.761207," ","integrate(cot(f*x+e)/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(\cos\left(f x + e\right) \log\left(-\frac{\cos\left(f x + e\right) + 1}{\cos\left(f x + e\right) - 1}\right) - 2\right)}}{2 \, a^{2} f \cos\left(f x + e\right)^{2}}"," ",0,"-1/2*sqrt(a*cos(f*x + e)^2)*(cos(f*x + e)*log(-(cos(f*x + e) + 1)/(cos(f*x + e) - 1)) - 2)/(a^2*f*cos(f*x + e)^2)","A",0
482,1,83,0,0.592410," ","integrate(cot(f*x+e)^3/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left({\left(\cos\left(f x + e\right)^{2} - 1\right)} \log\left(-\frac{\cos\left(f x + e\right) + 1}{\cos\left(f x + e\right) - 1}\right) - 2 \, \cos\left(f x + e\right)\right)}}{4 \, {\left(a^{2} f \cos\left(f x + e\right)^{3} - a^{2} f \cos\left(f x + e\right)\right)}}"," ",0,"-1/4*sqrt(a*cos(f*x + e)^2)*((cos(f*x + e)^2 - 1)*log(-(cos(f*x + e) + 1)/(cos(f*x + e) - 1)) - 2*cos(f*x + e))/(a^2*f*cos(f*x + e)^3 - a^2*f*cos(f*x + e))","A",0
483,1,77,0,0.752859," ","integrate(tan(f*x+e)^2/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{{\left(\cos\left(f x + e\right)^{4} \log\left(-\frac{\sin\left(f x + e\right) + 1}{\sin\left(f x + e\right) - 1}\right) + 2 \, {\left(\cos\left(f x + e\right)^{2} - 2\right)} \sin\left(f x + e\right)\right)} \sqrt{a \cos\left(f x + e\right)^{2}}}{16 \, a^{2} f \cos\left(f x + e\right)^{5}}"," ",0,"-1/16*(cos(f*x + e)^4*log(-(sin(f*x + e) + 1)/(sin(f*x + e) - 1)) + 2*(cos(f*x + e)^2 - 2)*sin(f*x + e))*sqrt(a*cos(f*x + e)^2)/(a^2*f*cos(f*x + e)^5)","A",0
484,1,66,0,0.607748," ","integrate(cot(f*x+e)^2/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(\log\left(-\frac{\sin\left(f x + e\right) - 1}{\sin\left(f x + e\right) + 1}\right) \sin\left(f x + e\right) + 2\right)}}{2 \, a^{2} f \cos\left(f x + e\right) \sin\left(f x + e\right)}"," ",0,"-1/2*sqrt(a*cos(f*x + e)^2)*(log(-(sin(f*x + e) - 1)/(sin(f*x + e) + 1))*sin(f*x + e) + 2)/(a^2*f*cos(f*x + e)*sin(f*x + e))","A",0
485,1,50,0,0.635106," ","integrate(cot(f*x+e)^4/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\frac{\sqrt{a \cos\left(f x + e\right)^{2}}}{3 \, {\left(a^{2} f \cos\left(f x + e\right)^{3} - a^{2} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"1/3*sqrt(a*cos(f*x + e)^2)/((a^2*f*cos(f*x + e)^3 - a^2*f*cos(f*x + e))*sin(f*x + e))","A",0
486,1,75,0,0.616784," ","integrate(cot(f*x+e)^6/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","-\frac{\sqrt{a \cos\left(f x + e\right)^{2}} {\left(5 \, \cos\left(f x + e\right)^{2} - 2\right)}}{15 \, {\left(a^{2} f \cos\left(f x + e\right)^{5} - 2 \, a^{2} f \cos\left(f x + e\right)^{3} + a^{2} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"-1/15*sqrt(a*cos(f*x + e)^2)*(5*cos(f*x + e)^2 - 2)/((a^2*f*cos(f*x + e)^5 - 2*a^2*f*cos(f*x + e)^3 + a^2*f*cos(f*x + e))*sin(f*x + e))","A",0
487,1,100,0,0.544137," ","integrate(cot(f*x+e)^8/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\frac{{\left(35 \, \cos\left(f x + e\right)^{4} - 28 \, \cos\left(f x + e\right)^{2} + 8\right)} \sqrt{a \cos\left(f x + e\right)^{2}}}{105 \, {\left(a^{2} f \cos\left(f x + e\right)^{7} - 3 \, a^{2} f \cos\left(f x + e\right)^{5} + 3 \, a^{2} f \cos\left(f x + e\right)^{3} - a^{2} f \cos\left(f x + e\right)\right)} \sin\left(f x + e\right)}"," ",0,"1/105*(35*cos(f*x + e)^4 - 28*cos(f*x + e)^2 + 8)*sqrt(a*cos(f*x + e)^2)/((a^2*f*cos(f*x + e)^7 - 3*a^2*f*cos(f*x + e)^5 + 3*a^2*f*cos(f*x + e)^3 - a^2*f*cos(f*x + e))*sin(f*x + e))","A",0
488,1,354,0,1.685982," ","integrate((a+b*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^5,x, algorithm=""fricas"")","\left[\frac{{\left(8 \, a^{2} + 24 \, a b + 15 \, b^{2}\right)} \sqrt{a + b} \cos\left(f x + e\right)^{4} \log\left(\frac{b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) - 2 \, {\left(8 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(8 \, a^{2} + 17 \, a b + 9 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a^{2} - 4 \, a b - 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{16 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4}}, -\frac{{\left(8 \, a^{2} + 24 \, a b + 15 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) \cos\left(f x + e\right)^{4} + {\left(8 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(f x + e\right)^{4} + {\left(8 \, a^{2} + 17 \, a b + 9 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a^{2} - 4 \, a b - 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{8 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{4}}\right]"," ",0,"[1/16*((8*a^2 + 24*a*b + 15*b^2)*sqrt(a + b)*cos(f*x + e)^4*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) - 2*(8*(a^2 + 2*a*b + b^2)*cos(f*x + e)^4 + (8*a^2 + 17*a*b + 9*b^2)*cos(f*x + e)^2 - 2*a^2 - 4*a*b - 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4), -1/8*((8*a^2 + 24*a*b + 15*b^2)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b))*cos(f*x + e)^4 + (8*(a^2 + 2*a*b + b^2)*cos(f*x + e)^4 + (8*a^2 + 17*a*b + 9*b^2)*cos(f*x + e)^2 - 2*a^2 - 4*a*b - 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^4)]","A",0
489,1,234,0,1.107637," ","integrate((a+b*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^3,x, algorithm=""fricas"")","\left[\frac{{\left(2 \, a + 3 \, b\right)} \sqrt{a + b} \cos\left(f x + e\right)^{2} \log\left(\frac{b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(2 \, {\left(a + b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{4 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{2}}, \frac{{\left(2 \, a + 3 \, b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) \cos\left(f x + e\right)^{2} + {\left(2 \, {\left(a + b\right)} \cos\left(f x + e\right)^{2} + a + b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{2 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{2}}\right]"," ",0,"[1/4*((2*a + 3*b)*sqrt(a + b)*cos(f*x + e)^2*log((b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) + 2*(2*(a + b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a + b)*f*cos(f*x + e)^2), 1/2*((2*a + 3*b)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b))*cos(f*x + e)^2 + (2*(a + b)*cos(f*x + e)^2 + a + b)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a + b)*f*cos(f*x + e)^2)]","A",0
490,1,145,0,1.112160," ","integrate((a+b*sin(f*x+e)^2)^(1/2)*tan(f*x+e),x, algorithm=""fricas"")","\left[\frac{\sqrt{a + b} \log\left(\frac{b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{2 \, f}, -\frac{\sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) + \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{f}\right]"," ",0,"[1/2*(sqrt(a + b)*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) - 2*sqrt(-b*cos(f*x + e)^2 + a + b))/f, -(sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b)) + sqrt(-b*cos(f*x + e)^2 + a + b))/f]","A",0
491,1,135,0,1.330919," ","integrate(cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{2 \, f}, \frac{\sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) + \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{f}\right]"," ",0,"[1/2*(sqrt(a)*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) + 2*sqrt(-b*cos(f*x + e)^2 + a + b))/f, (sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) + sqrt(-b*cos(f*x + e)^2 + a + b))/f]","A",0
492,1,239,0,2.170943," ","integrate(cot(f*x+e)^3*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, a + b\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, {\left(2 \, a \cos\left(f x + e\right)^{2} - 3 \, a\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{4 \, {\left(a f \cos\left(f x + e\right)^{2} - a f\right)}}, -\frac{{\left({\left(2 \, a - b\right)} \cos\left(f x + e\right)^{2} - 2 \, a + b\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) + {\left(2 \, a \cos\left(f x + e\right)^{2} - 3 \, a\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{2 \, {\left(a f \cos\left(f x + e\right)^{2} - a f\right)}}\right]"," ",0,"[-1/4*(((2*a - b)*cos(f*x + e)^2 - 2*a + b)*sqrt(a)*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) + 2*(2*a*cos(f*x + e)^2 - 3*a)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a*f*cos(f*x + e)^2 - a*f), -1/2*(((2*a - b)*cos(f*x + e)^2 - 2*a + b)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) + (2*a*cos(f*x + e)^2 - 3*a)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a*f*cos(f*x + e)^2 - a*f)]","A",0
493,1,415,0,2.651411," ","integrate(cot(f*x+e)^5*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} - 8 \, a b - b^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(8 \, a^{2} \cos\left(f x + e\right)^{4} - {\left(24 \, a^{2} - a b\right)} \cos\left(f x + e\right)^{2} + 14 \, a^{2} - a b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{16 \, {\left(a^{2} f \cos\left(f x + e\right)^{4} - 2 \, a^{2} f \cos\left(f x + e\right)^{2} + a^{2} f\right)}}, \frac{{\left({\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} - 8 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} - 8 \, a b - b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) + {\left(8 \, a^{2} \cos\left(f x + e\right)^{4} - {\left(24 \, a^{2} - a b\right)} \cos\left(f x + e\right)^{2} + 14 \, a^{2} - a b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{8 \, {\left(a^{2} f \cos\left(f x + e\right)^{4} - 2 \, a^{2} f \cos\left(f x + e\right)^{2} + a^{2} f\right)}}\right]"," ",0,"[-1/16*(((8*a^2 - 8*a*b - b^2)*cos(f*x + e)^4 - 2*(8*a^2 - 8*a*b - b^2)*cos(f*x + e)^2 + 8*a^2 - 8*a*b - b^2)*sqrt(a)*log(2*(b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) - 2*(8*a^2*cos(f*x + e)^4 - (24*a^2 - a*b)*cos(f*x + e)^2 + 14*a^2 - a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f), 1/8*(((8*a^2 - 8*a*b - b^2)*cos(f*x + e)^4 - 2*(8*a^2 - 8*a*b - b^2)*cos(f*x + e)^2 + 8*a^2 - 8*a*b - b^2)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) + (8*a^2*cos(f*x + e)^4 - (24*a^2 - a*b)*cos(f*x + e)^2 + 14*a^2 - a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f)]","A",0
494,0,0,0,0.791887," ","integrate((a+b*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^4,x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \tan\left(f x + e\right)^{4}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*tan(f*x + e)^4, x)","F",0
495,0,0,0,0.677831," ","integrate((a+b*sin(f*x+e)^2)^(1/2)*tan(f*x+e)^2,x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \tan\left(f x + e\right)^{2}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*tan(f*x + e)^2, x)","F",0
496,0,0,0,0.717944," ","integrate((a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b), x)","F",0
497,0,0,0,0.703924," ","integrate(cot(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cot\left(f x + e\right)^{2}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*cot(f*x + e)^2, x)","F",0
498,0,0,0,0.856778," ","integrate(cot(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cot\left(f x + e\right)^{4}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*cot(f*x + e)^4, x)","F",0
499,1,385,0,1.528210," ","integrate((a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e)^5,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(8 \, a^{2} + 40 \, a b + 35 \, b^{2}\right)} \sqrt{a + b} \cos\left(f x + e\right)^{4} \log\left(\frac{b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{6} - 16 \, {\left(2 \, a^{2} + 7 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 3 \, {\left(8 \, a^{2} + 21 \, a b + 13 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 6 \, a^{2} + 12 \, a b + 6 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{48 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{4}}, -\frac{3 \, {\left(8 \, a^{2} + 40 \, a b + 35 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) \cos\left(f x + e\right)^{4} - {\left(8 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{6} - 16 \, {\left(2 \, a^{2} + 7 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 3 \, {\left(8 \, a^{2} + 21 \, a b + 13 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 6 \, a^{2} + 12 \, a b + 6 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{24 \, {\left(a + b\right)} f \cos\left(f x + e\right)^{4}}\right]"," ",0,"[1/48*(3*(8*a^2 + 40*a*b + 35*b^2)*sqrt(a + b)*cos(f*x + e)^4*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) + 2*(8*(a*b + b^2)*cos(f*x + e)^6 - 16*(2*a^2 + 7*a*b + 5*b^2)*cos(f*x + e)^4 - 3*(8*a^2 + 21*a*b + 13*b^2)*cos(f*x + e)^2 + 6*a^2 + 12*a*b + 6*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a + b)*f*cos(f*x + e)^4), -1/24*(3*(8*a^2 + 40*a*b + 35*b^2)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b))*cos(f*x + e)^4 - (8*(a*b + b^2)*cos(f*x + e)^6 - 16*(2*a^2 + 7*a*b + 5*b^2)*cos(f*x + e)^4 - 3*(8*a^2 + 21*a*b + 13*b^2)*cos(f*x + e)^2 + 6*a^2 + 12*a*b + 6*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a + b)*f*cos(f*x + e)^4)]","A",0
500,1,265,0,1.352694," ","integrate((a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e)^3,x, algorithm=""fricas"")","\left[\frac{3 \, {\left(2 \, a + 5 \, b\right)} \sqrt{a + b} \cos\left(f x + e\right)^{2} \log\left(\frac{b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) - 2 \, {\left(2 \, b \cos\left(f x + e\right)^{4} - 2 \, {\left(4 \, a + 7 \, b\right)} \cos\left(f x + e\right)^{2} - 3 \, a - 3 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{12 \, f \cos\left(f x + e\right)^{2}}, \frac{3 \, {\left(2 \, a + 5 \, b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) \cos\left(f x + e\right)^{2} - {\left(2 \, b \cos\left(f x + e\right)^{4} - 2 \, {\left(4 \, a + 7 \, b\right)} \cos\left(f x + e\right)^{2} - 3 \, a - 3 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{6 \, f \cos\left(f x + e\right)^{2}}\right]"," ",0,"[1/12*(3*(2*a + 5*b)*sqrt(a + b)*cos(f*x + e)^2*log((b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) - 2*(2*b*cos(f*x + e)^4 - 2*(4*a + 7*b)*cos(f*x + e)^2 - 3*a - 3*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(f*cos(f*x + e)^2), 1/6*(3*(2*a + 5*b)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b))*cos(f*x + e)^2 - (2*b*cos(f*x + e)^4 - 2*(4*a + 7*b)*cos(f*x + e)^2 - 3*a - 3*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(f*cos(f*x + e)^2)]","A",0
501,1,186,0,1.198802," ","integrate((a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a + b\right)}^{\frac{3}{2}} \log\left(\frac{b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(b \cos\left(f x + e\right)^{2} - 4 \, a - 4 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{6 \, f}, -\frac{3 \, {\left(a + b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) - {\left(b \cos\left(f x + e\right)^{2} - 4 \, a - 4 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{3 \, f}\right]"," ",0,"[1/6*(3*(a + b)^(3/2)*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) + 2*(b*cos(f*x + e)^2 - 4*a - 4*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/f, -1/3*(3*(a + b)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b)) - (b*cos(f*x + e)^2 - 4*a - 4*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/f]","A",0
502,1,175,0,1.629481," ","integrate(cot(f*x+e)*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, a^{\frac{3}{2}} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(b \cos\left(f x + e\right)^{2} - 4 \, a - b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{6 \, f}, \frac{3 \, \sqrt{-a} a \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) - {\left(b \cos\left(f x + e\right)^{2} - 4 \, a - b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{3 \, f}\right]"," ",0,"[1/6*(3*a^(3/2)*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) - 2*(b*cos(f*x + e)^2 - 4*a - b)*sqrt(-b*cos(f*x + e)^2 + a + b))/f, 1/3*(3*sqrt(-a)*a*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - (b*cos(f*x + e)^2 - 4*a - b)*sqrt(-b*cos(f*x + e)^2 + a + b))/f]","A",0
503,1,282,0,2.286490," ","integrate(cot(f*x+e)^3*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(2 \, a - 3 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a + 3 \, b\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(2 \, b \cos\left(f x + e\right)^{4} - 2 \, {\left(4 \, a - b\right)} \cos\left(f x + e\right)^{2} + 11 \, a - 4 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{12 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}, -\frac{3 \, {\left({\left(2 \, a - 3 \, b\right)} \cos\left(f x + e\right)^{2} - 2 \, a + 3 \, b\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) - {\left(2 \, b \cos\left(f x + e\right)^{4} - 2 \, {\left(4 \, a - b\right)} \cos\left(f x + e\right)^{2} + 11 \, a - 4 \, b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{6 \, {\left(f \cos\left(f x + e\right)^{2} - f\right)}}\right]"," ",0,"[-1/12*(3*((2*a - 3*b)*cos(f*x + e)^2 - 2*a + 3*b)*sqrt(a)*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) - 2*(2*b*cos(f*x + e)^4 - 2*(4*a - b)*cos(f*x + e)^2 + 11*a - 4*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(f*cos(f*x + e)^2 - f), -1/6*(3*((2*a - 3*b)*cos(f*x + e)^2 - 2*a + 3*b)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - (2*b*cos(f*x + e)^4 - 2*(4*a - b)*cos(f*x + e)^2 + 11*a - 4*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(f*cos(f*x + e)^2 - f)]","A",0
504,1,442,0,3.383843," ","integrate(cot(f*x+e)^5*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(8 \, a b \cos\left(f x + e\right)^{6} - 8 \, {\left(4 \, a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)^{4} + {\left(88 \, a^{2} - 87 \, a b\right)} \cos\left(f x + e\right)^{2} - 50 \, a^{2} + 55 \, a b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{48 \, {\left(a f \cos\left(f x + e\right)^{4} - 2 \, a f \cos\left(f x + e\right)^{2} + a f\right)}}, \frac{3 \, {\left({\left(8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) - {\left(8 \, a b \cos\left(f x + e\right)^{6} - 8 \, {\left(4 \, a^{2} - 3 \, a b\right)} \cos\left(f x + e\right)^{4} + {\left(88 \, a^{2} - 87 \, a b\right)} \cos\left(f x + e\right)^{2} - 50 \, a^{2} + 55 \, a b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{24 \, {\left(a f \cos\left(f x + e\right)^{4} - 2 \, a f \cos\left(f x + e\right)^{2} + a f\right)}}\right]"," ",0,"[1/48*(3*((8*a^2 - 24*a*b + 3*b^2)*cos(f*x + e)^4 - 2*(8*a^2 - 24*a*b + 3*b^2)*cos(f*x + e)^2 + 8*a^2 - 24*a*b + 3*b^2)*sqrt(a)*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) - 2*(8*a*b*cos(f*x + e)^6 - 8*(4*a^2 - 3*a*b)*cos(f*x + e)^4 + (88*a^2 - 87*a*b)*cos(f*x + e)^2 - 50*a^2 + 55*a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a*f*cos(f*x + e)^4 - 2*a*f*cos(f*x + e)^2 + a*f), 1/24*(3*((8*a^2 - 24*a*b + 3*b^2)*cos(f*x + e)^4 - 2*(8*a^2 - 24*a*b + 3*b^2)*cos(f*x + e)^2 + 8*a^2 - 24*a*b + 3*b^2)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - (8*a*b*cos(f*x + e)^6 - 8*(4*a^2 - 3*a*b)*cos(f*x + e)^4 + (88*a^2 - 87*a*b)*cos(f*x + e)^2 - 50*a^2 + 55*a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a*f*cos(f*x + e)^4 - 2*a*f*cos(f*x + e)^2 + a*f)]","A",0
505,0,0,0,0.776323," ","integrate((a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e)^4,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \tan\left(f x + e\right)^{4}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^2 - a - b)*sqrt(-b*cos(f*x + e)^2 + a + b)*tan(f*x + e)^4, x)","F",0
506,0,0,0,0.875510," ","integrate((a+b*sin(f*x+e)^2)^(3/2)*tan(f*x+e)^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \tan\left(f x + e\right)^{2}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^2 - a - b)*sqrt(-b*cos(f*x + e)^2 + a + b)*tan(f*x + e)^2, x)","F",0
507,0,0,0,0.613725," ","integrate((a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{\frac{3}{2}}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^(3/2), x)","F",0
508,0,0,0,0.789416," ","integrate(cot(f*x+e)^2*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cot\left(f x + e\right)^{2}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^2 - a - b)*sqrt(-b*cos(f*x + e)^2 + a + b)*cot(f*x + e)^2, x)","F",0
509,0,0,0,0.996634," ","integrate(cot(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(-{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cot\left(f x + e\right)^{4}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^2 - a - b)*sqrt(-b*cos(f*x + e)^2 + a + b)*cot(f*x + e)^4, x)","F",0
510,1,328,0,1.081194," ","integrate(tan(f*x+e)^5/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \sqrt{a + b} \cos\left(f x + e\right)^{4} \log\left(\frac{b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) - 2 \, {\left({\left(8 \, a^{2} + 13 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a^{2} - 4 \, a b - 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{16 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{4}}, -\frac{{\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) \cos\left(f x + e\right)^{4} + {\left({\left(8 \, a^{2} + 13 \, a b + 5 \, b^{2}\right)} \cos\left(f x + e\right)^{2} - 2 \, a^{2} - 4 \, a b - 2 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{8 \, {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{4}}\right]"," ",0,"[1/16*((8*a^2 + 8*a*b + 3*b^2)*sqrt(a + b)*cos(f*x + e)^4*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) - 2*((8*a^2 + 13*a*b + 5*b^2)*cos(f*x + e)^2 - 2*a^2 - 4*a*b - 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^4), -1/8*((8*a^2 + 8*a*b + 3*b^2)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b))*cos(f*x + e)^4 + ((8*a^2 + 13*a*b + 5*b^2)*cos(f*x + e)^2 - 2*a^2 - 4*a*b - 2*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f*cos(f*x + e)^4)]","A",0
511,1,220,0,0.896285," ","integrate(tan(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(2 \, a + b\right)} \sqrt{a + b} \cos\left(f x + e\right)^{2} \log\left(\frac{b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)}}{4 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2}}, \frac{{\left(2 \, a + b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) \cos\left(f x + e\right)^{2} + \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)}}{2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} f \cos\left(f x + e\right)^{2}}\right]"," ",0,"[1/4*((2*a + b)*sqrt(a + b)*cos(f*x + e)^2*log((b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2), 1/2*((2*a + b)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b))*cos(f*x + e)^2 + sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b))/((a^2 + 2*a*b + b^2)*f*cos(f*x + e)^2)]","A",0
512,1,112,0,0.810086," ","integrate(tan(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right)}{2 \, \sqrt{a + b} f}, -\frac{\sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right)}{{\left(a + b\right)} f}\right]"," ",0,"[1/2*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2)/(sqrt(a + b)*f), -sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b))/((a + b)*f)]","A",0
513,1,100,0,0.795982," ","integrate(cot(f*x+e)/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right)}{2 \, \sqrt{a} f}, \frac{\sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right)}{a f}\right]"," ",0,"[1/2*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1))/(sqrt(a)*f), sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a)/(a*f)]","A",0
514,1,220,0,0.855794," ","integrate(cot(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - b\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a}{4 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)}}, -\frac{{\left({\left(2 \, a + b\right)} \cos\left(f x + e\right)^{2} - 2 \, a - b\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) - \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a}{2 \, {\left(a^{2} f \cos\left(f x + e\right)^{2} - a^{2} f\right)}}\right]"," ",0,"[1/4*(((2*a + b)*cos(f*x + e)^2 - 2*a - b)*sqrt(a)*log(2*(b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*a)/(a^2*f*cos(f*x + e)^2 - a^2*f), -1/2*(((2*a + b)*cos(f*x + e)^2 - 2*a - b)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - sqrt(-b*cos(f*x + e)^2 + a + b)*a)/(a^2*f*cos(f*x + e)^2 - a^2*f)]","A",0
515,1,388,0,0.910084," ","integrate(cot(f*x+e)^5/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left({\left(8 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} - 6 \, a^{2} - 3 \, a b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{16 \, {\left(a^{3} f \cos\left(f x + e\right)^{4} - 2 \, a^{3} f \cos\left(f x + e\right)^{2} + a^{3} f\right)}}, \frac{{\left({\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - 2 \, {\left(8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) - {\left({\left(8 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} - 6 \, a^{2} - 3 \, a b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{8 \, {\left(a^{3} f \cos\left(f x + e\right)^{4} - 2 \, a^{3} f \cos\left(f x + e\right)^{2} + a^{3} f\right)}}\right]"," ",0,"[1/16*(((8*a^2 + 8*a*b + 3*b^2)*cos(f*x + e)^4 - 2*(8*a^2 + 8*a*b + 3*b^2)*cos(f*x + e)^2 + 8*a^2 + 8*a*b + 3*b^2)*sqrt(a)*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) - 2*((8*a^2 + 3*a*b)*cos(f*x + e)^2 - 6*a^2 - 3*a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^3*f*cos(f*x + e)^4 - 2*a^3*f*cos(f*x + e)^2 + a^3*f), 1/8*(((8*a^2 + 8*a*b + 3*b^2)*cos(f*x + e)^4 - 2*(8*a^2 + 8*a*b + 3*b^2)*cos(f*x + e)^2 + 8*a^2 + 8*a*b + 3*b^2)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - ((8*a^2 + 3*a*b)*cos(f*x + e)^2 - 6*a^2 - 3*a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^3*f*cos(f*x + e)^4 - 2*a^3*f*cos(f*x + e)^2 + a^3*f)]","A",0
516,0,0,0,0.686344," ","integrate(tan(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \tan\left(f x + e\right)^{4}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*tan(f*x + e)^4/(b*cos(f*x + e)^2 - a - b), x)","F",0
517,0,0,0,0.634483," ","integrate(tan(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \tan\left(f x + e\right)^{2}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*tan(f*x + e)^2/(b*cos(f*x + e)^2 - a - b), x)","F",0
518,0,0,0,0.566376," ","integrate(1/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)/(b*cos(f*x + e)^2 - a - b), x)","F",0
519,0,0,0,0.677015," ","integrate(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cot\left(f x + e\right)^{2}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*cot(f*x + e)^2/(b*cos(f*x + e)^2 - a - b), x)","F",0
520,0,0,0,0.704972," ","integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cot\left(f x + e\right)^{4}}{b \cos\left(f x + e\right)^{2} - a - b}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*cot(f*x + e)^4/(b*cos(f*x + e)^2 - a - b), x)","F",0
521,1,593,0,1.270376," ","integrate(tan(f*x+e)^5/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(8 \, a^{3} - 9 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4}\right)} \sqrt{a + b} \log\left(\frac{b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) - 2 \, {\left({\left(8 \, a^{3} - 9 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - 2 \, a^{3} - 6 \, a^{2} b - 6 \, a b^{2} - 2 \, b^{3} + {\left(8 \, a^{3} + 19 \, a^{2} b + 14 \, a b^{2} + 3 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{16 \, {\left({\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)^{4}\right)}}, -\frac{{\left({\left(8 \, a^{2} b - 8 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(8 \, a^{3} - 9 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) - {\left({\left(8 \, a^{3} - 9 \, a b^{2} - b^{3}\right)} \cos\left(f x + e\right)^{4} - 2 \, a^{3} - 6 \, a^{2} b - 6 \, a b^{2} - 2 \, b^{3} + {\left(8 \, a^{3} + 19 \, a^{2} b + 14 \, a b^{2} + 3 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{8 \, {\left({\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)^{6} - {\left(a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)^{4}\right)}}\right]"," ",0,"[-1/16*(((8*a^2*b - 8*a*b^2 - b^3)*cos(f*x + e)^6 - (8*a^3 - 9*a*b^2 - b^3)*cos(f*x + e)^4)*sqrt(a + b)*log((b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) - 2*((8*a^3 - 9*a*b^2 - b^3)*cos(f*x + e)^4 - 2*a^3 - 6*a^2*b - 6*a*b^2 - 2*b^3 + (8*a^3 + 19*a^2*b + 14*a*b^2 + 3*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f*cos(f*x + e)^6 - (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*f*cos(f*x + e)^4), -1/8*(((8*a^2*b - 8*a*b^2 - b^3)*cos(f*x + e)^6 - (8*a^3 - 9*a*b^2 - b^3)*cos(f*x + e)^4)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b)) - ((8*a^3 - 9*a*b^2 - b^3)*cos(f*x + e)^4 - 2*a^3 - 6*a^2*b - 6*a*b^2 - 2*b^3 + (8*a^3 + 19*a^2*b + 14*a*b^2 + 3*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f*cos(f*x + e)^6 - (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*f*cos(f*x + e)^4)]","A",0
522,1,445,0,0.970740," ","integrate(tan(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, a^{2} + a b - b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left({\left(2 \, a^{2} + a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{4 \, {\left({\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2}\right)}}, \frac{{\left({\left(2 \, a b - b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, a^{2} + a b - b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) - {\left({\left(2 \, a^{2} + a b - b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{2 \, {\left({\left(a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{4} - {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} f \cos\left(f x + e\right)^{2}\right)}}\right]"," ",0,"[-1/4*(((2*a*b - b^2)*cos(f*x + e)^4 - (2*a^2 + a*b - b^2)*cos(f*x + e)^2)*sqrt(a + b)*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) + 2*((2*a^2 + a*b - b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f*cos(f*x + e)^4 - (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*f*cos(f*x + e)^2), 1/2*(((2*a*b - b^2)*cos(f*x + e)^4 - (2*a^2 + a*b - b^2)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b)) - ((2*a^2 + a*b - b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^3*b + 3*a^2*b^2 + 3*a*b^3 + b^4)*f*cos(f*x + e)^4 - (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*f*cos(f*x + e)^2)]","B",0
523,1,281,0,0.791418," ","integrate(tan(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{a + b} \log\left(\frac{b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)}}{2 \, {\left({\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f\right)}}, -\frac{{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) - \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} {\left(a + b\right)}}{{\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} f}\right]"," ",0,"[1/2*((b*cos(f*x + e)^2 - a - b)*sqrt(a + b)*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b))/((a^2*b + 2*a*b^2 + b^3)*f*cos(f*x + e)^2 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f), -((b*cos(f*x + e)^2 - a - b)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b)) - sqrt(-b*cos(f*x + e)^2 + a + b)*(a + b))/((a^2*b + 2*a*b^2 + b^3)*f*cos(f*x + e)^2 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*f)]","B",0
524,1,225,0,0.712903," ","integrate(cot(f*x+e)/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a}{2 \, {\left(a^{2} b f \cos\left(f x + e\right)^{2} - {\left(a^{3} + a^{2} b\right)} f\right)}}, \frac{{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) - \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} a}{a^{2} b f \cos\left(f x + e\right)^{2} - {\left(a^{3} + a^{2} b\right)} f}\right]"," ",0,"[1/2*((b*cos(f*x + e)^2 - a - b)*sqrt(a)*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*a)/(a^2*b*f*cos(f*x + e)^2 - (a^3 + a^2*b)*f), ((b*cos(f*x + e)^2 - a - b)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - sqrt(-b*cos(f*x + e)^2 + a + b)*a)/(a^2*b*f*cos(f*x + e)^2 - (a^3 + a^2*b)*f)]","B",0
525,1,406,0,0.849858," ","integrate(cot(f*x+e)^3/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(2 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, a^{2} + 7 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, a^{2} + 5 \, a b + 3 \, b^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, {\left({\left(2 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} - 3 \, a^{2} - 3 \, a b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{4 \, {\left(a^{3} b f \cos\left(f x + e\right)^{4} - {\left(a^{4} + 2 \, a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + a^{3} b\right)} f\right)}}, -\frac{{\left({\left(2 \, a b + 3 \, b^{2}\right)} \cos\left(f x + e\right)^{4} - {\left(2 \, a^{2} + 7 \, a b + 6 \, b^{2}\right)} \cos\left(f x + e\right)^{2} + 2 \, a^{2} + 5 \, a b + 3 \, b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) - {\left({\left(2 \, a^{2} + 3 \, a b\right)} \cos\left(f x + e\right)^{2} - 3 \, a^{2} - 3 \, a b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{2 \, {\left(a^{3} b f \cos\left(f x + e\right)^{4} - {\left(a^{4} + 2 \, a^{3} b\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{4} + a^{3} b\right)} f\right)}}\right]"," ",0,"[1/4*(((2*a*b + 3*b^2)*cos(f*x + e)^4 - (2*a^2 + 7*a*b + 6*b^2)*cos(f*x + e)^2 + 2*a^2 + 5*a*b + 3*b^2)*sqrt(a)*log(2*(b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) + 2*((2*a^2 + 3*a*b)*cos(f*x + e)^2 - 3*a^2 - 3*a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^3*b*f*cos(f*x + e)^4 - (a^4 + 2*a^3*b)*f*cos(f*x + e)^2 + (a^4 + a^3*b)*f), -1/2*(((2*a*b + 3*b^2)*cos(f*x + e)^4 - (2*a^2 + 7*a*b + 6*b^2)*cos(f*x + e)^2 + 2*a^2 + 5*a*b + 3*b^2)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - ((2*a^2 + 3*a*b)*cos(f*x + e)^2 - 3*a^2 - 3*a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^3*b*f*cos(f*x + e)^4 - (a^4 + 2*a^3*b)*f*cos(f*x + e)^2 + (a^4 + a^3*b)*f)]","B",0
526,1,652,0,0.908025," ","integrate(cot(f*x+e)^5/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(8 \, a^{2} b + 24 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(8 \, a^{3} + 48 \, a^{2} b + 87 \, a b^{2} + 45 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 8 \, a^{3} - 32 \, a^{2} b - 39 \, a b^{2} - 15 \, b^{3} + {\left(16 \, a^{3} + 72 \, a^{2} b + 102 \, a b^{2} + 45 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left({\left(8 \, a^{3} + 24 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + 14 \, a^{3} + 29 \, a^{2} b + 15 \, a b^{2} - {\left(24 \, a^{3} + 53 \, a^{2} b + 30 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{16 \, {\left(a^{4} b f \cos\left(f x + e\right)^{6} - {\left(a^{5} + 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{4} + {\left(2 \, a^{5} + 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} + a^{4} b\right)} f\right)}}, \frac{{\left({\left(8 \, a^{2} b + 24 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(8 \, a^{3} + 48 \, a^{2} b + 87 \, a b^{2} + 45 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 8 \, a^{3} - 32 \, a^{2} b - 39 \, a b^{2} - 15 \, b^{3} + {\left(16 \, a^{3} + 72 \, a^{2} b + 102 \, a b^{2} + 45 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) - {\left({\left(8 \, a^{3} + 24 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + 14 \, a^{3} + 29 \, a^{2} b + 15 \, a b^{2} - {\left(24 \, a^{3} + 53 \, a^{2} b + 30 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{8 \, {\left(a^{4} b f \cos\left(f x + e\right)^{6} - {\left(a^{5} + 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{4} + {\left(2 \, a^{5} + 3 \, a^{4} b\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{5} + a^{4} b\right)} f\right)}}\right]"," ",0,"[1/16*(((8*a^2*b + 24*a*b^2 + 15*b^3)*cos(f*x + e)^6 - (8*a^3 + 48*a^2*b + 87*a*b^2 + 45*b^3)*cos(f*x + e)^4 - 8*a^3 - 32*a^2*b - 39*a*b^2 - 15*b^3 + (16*a^3 + 72*a^2*b + 102*a*b^2 + 45*b^3)*cos(f*x + e)^2)*sqrt(a)*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) - 2*((8*a^3 + 24*a^2*b + 15*a*b^2)*cos(f*x + e)^4 + 14*a^3 + 29*a^2*b + 15*a*b^2 - (24*a^3 + 53*a^2*b + 30*a*b^2)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^4*b*f*cos(f*x + e)^6 - (a^5 + 3*a^4*b)*f*cos(f*x + e)^4 + (2*a^5 + 3*a^4*b)*f*cos(f*x + e)^2 - (a^5 + a^4*b)*f), 1/8*(((8*a^2*b + 24*a*b^2 + 15*b^3)*cos(f*x + e)^6 - (8*a^3 + 48*a^2*b + 87*a*b^2 + 45*b^3)*cos(f*x + e)^4 - 8*a^3 - 32*a^2*b - 39*a*b^2 - 15*b^3 + (16*a^3 + 72*a^2*b + 102*a*b^2 + 45*b^3)*cos(f*x + e)^2)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - ((8*a^3 + 24*a^2*b + 15*a*b^2)*cos(f*x + e)^4 + 14*a^3 + 29*a^2*b + 15*a*b^2 - (24*a^3 + 53*a^2*b + 30*a*b^2)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^4*b*f*cos(f*x + e)^6 - (a^5 + 3*a^4*b)*f*cos(f*x + e)^4 + (2*a^5 + 3*a^4*b)*f*cos(f*x + e)^2 - (a^5 + a^4*b)*f)]","B",0
527,0,0,0,0.698072," ","integrate(tan(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \tan\left(f x + e\right)^{4}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*tan(f*x + e)^4/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
528,0,0,0,0.747216," ","integrate(tan(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \tan\left(f x + e\right)^{2}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*tan(f*x + e)^2/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
529,0,0,0,0.748618," ","integrate(1/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
530,0,0,0,0.844995," ","integrate(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cot\left(f x + e\right)^{2}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*cot(f*x + e)^2/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
531,0,0,0,0.647823," ","integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cot\left(f x + e\right)^{4}}{b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}}, x\right)"," ",0,"integral(sqrt(-b*cos(f*x + e)^2 + a + b)*cot(f*x + e)^4/(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2), x)","F",0
532,1,995,0,1.197247," ","integrate(tan(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(8 \, a^{2} b^{2} - 24 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(8 \, a^{3} b - 16 \, a^{2} b^{2} - 21 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(8 \, a^{4} - 8 \, a^{3} b - 37 \, a^{2} b^{2} - 18 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right)^{4}\right)} \sqrt{a + b} \log\left(\frac{b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(3 \, {\left(8 \, a^{3} b - 16 \, a^{2} b^{2} - 21 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - 4 \, {\left(8 \, a^{4} - 8 \, a^{3} b - 37 \, a^{2} b^{2} - 18 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 6 \, a^{4} + 24 \, a^{3} b + 36 \, a^{2} b^{2} + 24 \, a b^{3} + 6 \, b^{4} - 3 \, {\left(8 \, a^{4} + 25 \, a^{3} b + 27 \, a^{2} b^{2} + 11 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{48 \, {\left({\left(a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{6} b + 6 \, a^{5} b^{2} + 15 \, a^{4} b^{3} + 20 \, a^{3} b^{4} + 15 \, a^{2} b^{5} + 6 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} + 7 \, a^{6} b + 21 \, a^{5} b^{2} + 35 \, a^{4} b^{3} + 35 \, a^{3} b^{4} + 21 \, a^{2} b^{5} + 7 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{4}\right)}}, -\frac{3 \, {\left({\left(8 \, a^{2} b^{2} - 24 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(8 \, a^{3} b - 16 \, a^{2} b^{2} - 21 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(8 \, a^{4} - 8 \, a^{3} b - 37 \, a^{2} b^{2} - 18 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right)^{4}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) - {\left(3 \, {\left(8 \, a^{3} b - 16 \, a^{2} b^{2} - 21 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right)^{6} - 4 \, {\left(8 \, a^{4} - 8 \, a^{3} b - 37 \, a^{2} b^{2} - 18 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 6 \, a^{4} + 24 \, a^{3} b + 36 \, a^{2} b^{2} + 24 \, a b^{3} + 6 \, b^{4} - 3 \, {\left(8 \, a^{4} + 25 \, a^{3} b + 27 \, a^{2} b^{2} + 11 \, a b^{3} + b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{24 \, {\left({\left(a^{5} b^{2} + 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} + 10 \, a^{2} b^{5} + 5 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{6} b + 6 \, a^{5} b^{2} + 15 \, a^{4} b^{3} + 20 \, a^{3} b^{4} + 15 \, a^{2} b^{5} + 6 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} + 7 \, a^{6} b + 21 \, a^{5} b^{2} + 35 \, a^{4} b^{3} + 35 \, a^{3} b^{4} + 21 \, a^{2} b^{5} + 7 \, a b^{6} + b^{7}\right)} f \cos\left(f x + e\right)^{4}\right)}}\right]"," ",0,"[1/48*(3*((8*a^2*b^2 - 24*a*b^3 + 3*b^4)*cos(f*x + e)^8 - 2*(8*a^3*b - 16*a^2*b^2 - 21*a*b^3 + 3*b^4)*cos(f*x + e)^6 + (8*a^4 - 8*a^3*b - 37*a^2*b^2 - 18*a*b^3 + 3*b^4)*cos(f*x + e)^4)*sqrt(a + b)*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) + 2*(3*(8*a^3*b - 16*a^2*b^2 - 21*a*b^3 + 3*b^4)*cos(f*x + e)^6 - 4*(8*a^4 - 8*a^3*b - 37*a^2*b^2 - 18*a*b^3 + 3*b^4)*cos(f*x + e)^4 + 6*a^4 + 24*a^3*b + 36*a^2*b^2 + 24*a*b^3 + 6*b^4 - 3*(8*a^4 + 25*a^3*b + 27*a^2*b^2 + 11*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f*cos(f*x + e)^8 - 2*(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)*f*cos(f*x + e)^6 + (a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*f*cos(f*x + e)^4), -1/24*(3*((8*a^2*b^2 - 24*a*b^3 + 3*b^4)*cos(f*x + e)^8 - 2*(8*a^3*b - 16*a^2*b^2 - 21*a*b^3 + 3*b^4)*cos(f*x + e)^6 + (8*a^4 - 8*a^3*b - 37*a^2*b^2 - 18*a*b^3 + 3*b^4)*cos(f*x + e)^4)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b)) - (3*(8*a^3*b - 16*a^2*b^2 - 21*a*b^3 + 3*b^4)*cos(f*x + e)^6 - 4*(8*a^4 - 8*a^3*b - 37*a^2*b^2 - 18*a*b^3 + 3*b^4)*cos(f*x + e)^4 + 6*a^4 + 24*a^3*b + 36*a^2*b^2 + 24*a*b^3 + 6*b^4 - 3*(8*a^4 + 25*a^3*b + 27*a^2*b^2 + 11*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^5*b^2 + 5*a^4*b^3 + 10*a^3*b^4 + 10*a^2*b^5 + 5*a*b^6 + b^7)*f*cos(f*x + e)^8 - 2*(a^6*b + 6*a^5*b^2 + 15*a^4*b^3 + 20*a^3*b^4 + 15*a^2*b^5 + 6*a*b^6 + b^7)*f*cos(f*x + e)^6 + (a^7 + 7*a^6*b + 21*a^5*b^2 + 35*a^4*b^3 + 35*a^3*b^4 + 21*a^2*b^5 + 7*a*b^6 + b^7)*f*cos(f*x + e)^4)]","B",0
533,1,769,0,0.837331," ","integrate(tan(f*x+e)^3/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(2 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - 2 \, {\left(2 \, a^{2} b - a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(2 \, a^{3} + a^{2} b - 4 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a + b} \log\left(\frac{b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(3 \, {\left(2 \, a^{2} b - a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 3 \, a^{3} - 9 \, a^{2} b - 9 \, a b^{2} - 3 \, b^{3} - 4 \, {\left(2 \, a^{3} + a^{2} b - 4 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{12 \, {\left({\left(a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6}\right)} f \cos\left(f x + e\right)^{6} - 2 \, {\left(a^{5} b + 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 10 \, a^{2} b^{4} + 5 \, a b^{5} + b^{6}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{6} + 6 \, a^{5} b + 15 \, a^{4} b^{2} + 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} + 6 \, a b^{5} + b^{6}\right)} f \cos\left(f x + e\right)^{2}\right)}}, \frac{3 \, {\left({\left(2 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - 2 \, {\left(2 \, a^{2} b - a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{4} + {\left(2 \, a^{3} + a^{2} b - 4 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) - {\left(3 \, {\left(2 \, a^{2} b - a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 3 \, a^{3} - 9 \, a^{2} b - 9 \, a b^{2} - 3 \, b^{3} - 4 \, {\left(2 \, a^{3} + a^{2} b - 4 \, a b^{2} - 3 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{6 \, {\left({\left(a^{4} b^{2} + 4 \, a^{3} b^{3} + 6 \, a^{2} b^{4} + 4 \, a b^{5} + b^{6}\right)} f \cos\left(f x + e\right)^{6} - 2 \, {\left(a^{5} b + 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 10 \, a^{2} b^{4} + 5 \, a b^{5} + b^{6}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{6} + 6 \, a^{5} b + 15 \, a^{4} b^{2} + 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} + 6 \, a b^{5} + b^{6}\right)} f \cos\left(f x + e\right)^{2}\right)}}\right]"," ",0,"[-1/12*(3*((2*a*b^2 - 3*b^3)*cos(f*x + e)^6 - 2*(2*a^2*b - a*b^2 - 3*b^3)*cos(f*x + e)^4 + (2*a^3 + a^2*b - 4*a*b^2 - 3*b^3)*cos(f*x + e)^2)*sqrt(a + b)*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) + 2*(3*(2*a^2*b - a*b^2 - 3*b^3)*cos(f*x + e)^4 - 3*a^3 - 9*a^2*b - 9*a*b^2 - 3*b^3 - 4*(2*a^3 + a^2*b - 4*a*b^2 - 3*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*f*cos(f*x + e)^6 - 2*(a^5*b + 5*a^4*b^2 + 10*a^3*b^3 + 10*a^2*b^4 + 5*a*b^5 + b^6)*f*cos(f*x + e)^4 + (a^6 + 6*a^5*b + 15*a^4*b^2 + 20*a^3*b^3 + 15*a^2*b^4 + 6*a*b^5 + b^6)*f*cos(f*x + e)^2), 1/6*(3*((2*a*b^2 - 3*b^3)*cos(f*x + e)^6 - 2*(2*a^2*b - a*b^2 - 3*b^3)*cos(f*x + e)^4 + (2*a^3 + a^2*b - 4*a*b^2 - 3*b^3)*cos(f*x + e)^2)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b)) - (3*(2*a^2*b - a*b^2 - 3*b^3)*cos(f*x + e)^4 - 3*a^3 - 9*a^2*b - 9*a*b^2 - 3*b^3 - 4*(2*a^3 + a^2*b - 4*a*b^2 - 3*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^4*b^2 + 4*a^3*b^3 + 6*a^2*b^4 + 4*a*b^5 + b^6)*f*cos(f*x + e)^6 - 2*(a^5*b + 5*a^4*b^2 + 10*a^3*b^3 + 10*a^2*b^4 + 5*a*b^5 + b^6)*f*cos(f*x + e)^4 + (a^6 + 6*a^5*b + 15*a^4*b^2 + 20*a^3*b^3 + 15*a^2*b^4 + 6*a*b^5 + b^6)*f*cos(f*x + e)^2)]","B",0
534,1,521,0,0.692791," ","integrate(tan(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a + b} \log\left(\frac{b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a + b} - 2 \, a - 2 \, b}{\cos\left(f x + e\right)^{2}}\right) + 2 \, {\left(3 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, a^{2} - 8 \, a b - 4 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{6 \, {\left({\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right)} f\right)}}, -\frac{3 \, {\left(b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a - b} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a - b}}{a + b}\right) - {\left(3 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} - 4 \, a^{2} - 8 \, a b - 4 \, b^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{3 \, {\left({\left(a^{3} b^{2} + 3 \, a^{2} b^{3} + 3 \, a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} b + 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} + 4 \, a b^{4} + b^{5}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right)} f\right)}}\right]"," ",0,"[1/6*(3*(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(a + b)*log((b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a + b) - 2*a - 2*b)/cos(f*x + e)^2) + 2*(3*(a*b + b^2)*cos(f*x + e)^2 - 4*a^2 - 8*a*b - 4*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*f*cos(f*x + e)^4 - 2*(a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f*cos(f*x + e)^2 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*f), -1/3*(3*(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-a - b)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a - b)/(a + b)) - (3*(a*b + b^2)*cos(f*x + e)^2 - 4*a^2 - 8*a*b - 4*b^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/((a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + b^5)*f*cos(f*x + e)^4 - 2*(a^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*f*cos(f*x + e)^2 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*f)]","B",0
535,1,382,0,1.120882," ","integrate(cot(f*x+e)/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(3 \, a b \cos\left(f x + e\right)^{2} - 4 \, a^{2} - 3 \, a b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{6 \, {\left(a^{3} b^{2} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f\right)}}, \frac{3 \, {\left(b^{2} \cos\left(f x + e\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(f x + e\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) - {\left(3 \, a b \cos\left(f x + e\right)^{2} - 4 \, a^{2} - 3 \, a b\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{3 \, {\left(a^{3} b^{2} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{4} b + a^{3} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right)} f\right)}}\right]"," ",0,"[1/6*(3*(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) - 2*(3*a*b*cos(f*x + e)^2 - 4*a^2 - 3*a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^3*b^2*f*cos(f*x + e)^4 - 2*(a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^5 + 2*a^4*b + a^3*b^2)*f), 1/3*(3*(b^2*cos(f*x + e)^4 - 2*(a*b + b^2)*cos(f*x + e)^2 + a^2 + 2*a*b + b^2)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - (3*a*b*cos(f*x + e)^2 - 4*a^2 - 3*a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^3*b^2*f*cos(f*x + e)^4 - 2*(a^4*b + a^3*b^2)*f*cos(f*x + e)^2 + (a^5 + 2*a^4*b + a^3*b^2)*f)]","B",0
536,1,666,0,0.777349," ","integrate(cot(f*x+e)^3/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(2 \, a b^{2} + 5 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(4 \, a^{2} b + 16 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 2 \, a^{3} - 9 \, a^{2} b - 12 \, a b^{2} - 5 \, b^{3} + {\left(2 \, a^{3} + 13 \, a^{2} b + 26 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} - 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) + 2 \, {\left(3 \, {\left(2 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + 11 \, a^{3} + 26 \, a^{2} b + 15 \, a b^{2} - 2 \, {\left(4 \, a^{3} + 16 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{12 \, {\left(a^{4} b^{2} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{5} b + 3 \, a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{6} + 4 \, a^{5} b + 3 \, a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right)} f\right)}}, -\frac{3 \, {\left({\left(2 \, a b^{2} + 5 \, b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(4 \, a^{2} b + 16 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(f x + e\right)^{4} - 2 \, a^{3} - 9 \, a^{2} b - 12 \, a b^{2} - 5 \, b^{3} + {\left(2 \, a^{3} + 13 \, a^{2} b + 26 \, a b^{2} + 15 \, b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) - {\left(3 \, {\left(2 \, a^{2} b + 5 \, a b^{2}\right)} \cos\left(f x + e\right)^{4} + 11 \, a^{3} + 26 \, a^{2} b + 15 \, a b^{2} - 2 \, {\left(4 \, a^{3} + 16 \, a^{2} b + 15 \, a b^{2}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{6 \, {\left(a^{4} b^{2} f \cos\left(f x + e\right)^{6} - {\left(2 \, a^{5} b + 3 \, a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{4} + {\left(a^{6} + 4 \, a^{5} b + 3 \, a^{4} b^{2}\right)} f \cos\left(f x + e\right)^{2} - {\left(a^{6} + 2 \, a^{5} b + a^{4} b^{2}\right)} f\right)}}\right]"," ",0,"[1/12*(3*((2*a*b^2 + 5*b^3)*cos(f*x + e)^6 - (4*a^2*b + 16*a*b^2 + 15*b^3)*cos(f*x + e)^4 - 2*a^3 - 9*a^2*b - 12*a*b^2 - 5*b^3 + (2*a^3 + 13*a^2*b + 26*a*b^2 + 15*b^3)*cos(f*x + e)^2)*sqrt(a)*log(2*(b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) + 2*(3*(2*a^2*b + 5*a*b^2)*cos(f*x + e)^4 + 11*a^3 + 26*a^2*b + 15*a*b^2 - 2*(4*a^3 + 16*a^2*b + 15*a*b^2)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^4*b^2*f*cos(f*x + e)^6 - (2*a^5*b + 3*a^4*b^2)*f*cos(f*x + e)^4 + (a^6 + 4*a^5*b + 3*a^4*b^2)*f*cos(f*x + e)^2 - (a^6 + 2*a^5*b + a^4*b^2)*f), -1/6*(3*((2*a*b^2 + 5*b^3)*cos(f*x + e)^6 - (4*a^2*b + 16*a*b^2 + 15*b^3)*cos(f*x + e)^4 - 2*a^3 - 9*a^2*b - 12*a*b^2 - 5*b^3 + (2*a^3 + 13*a^2*b + 26*a*b^2 + 15*b^3)*cos(f*x + e)^2)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - (3*(2*a^2*b + 5*a*b^2)*cos(f*x + e)^4 + 11*a^3 + 26*a^2*b + 15*a*b^2 - 2*(4*a^3 + 16*a^2*b + 15*a*b^2)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^4*b^2*f*cos(f*x + e)^6 - (2*a^5*b + 3*a^4*b^2)*f*cos(f*x + e)^4 + (a^6 + 4*a^5*b + 3*a^4*b^2)*f*cos(f*x + e)^2 - (a^6 + 2*a^5*b + a^4*b^2)*f)]","B",0
537,1,984,0,0.854504," ","integrate(cot(f*x+e)^5/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(8 \, a^{2} b^{2} + 40 \, a b^{3} + 35 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(8 \, a^{3} b + 56 \, a^{2} b^{2} + 115 \, a b^{3} + 70 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(8 \, a^{4} + 88 \, a^{3} b + 323 \, a^{2} b^{2} + 450 \, a b^{3} + 210 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{4} + 56 \, a^{3} b + 123 \, a^{2} b^{2} + 110 \, a b^{3} + 35 \, b^{4} - 2 \, {\left(8 \, a^{4} + 64 \, a^{3} b + 171 \, a^{2} b^{2} + 185 \, a b^{3} + 70 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{a} \log\left(\frac{2 \, {\left(b \cos\left(f x + e\right)^{2} + 2 \, \sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{a} - 2 \, a - b\right)}}{\cos\left(f x + e\right)^{2} - 1}\right) - 2 \, {\left(3 \, {\left(8 \, a^{3} b + 40 \, a^{2} b^{2} + 35 \, a b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(32 \, a^{4} + 232 \, a^{3} b + 500 \, a^{2} b^{2} + 315 \, a b^{3}\right)} \cos\left(f x + e\right)^{4} - 50 \, a^{4} - 205 \, a^{3} b - 260 \, a^{2} b^{2} - 105 \, a b^{3} + {\left(88 \, a^{4} + 413 \, a^{3} b + 640 \, a^{2} b^{2} + 315 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{48 \, {\left(a^{5} b^{2} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} + 6 \, a^{6} b + 6 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f\right)}}, \frac{3 \, {\left({\left(8 \, a^{2} b^{2} + 40 \, a b^{3} + 35 \, b^{4}\right)} \cos\left(f x + e\right)^{8} - 2 \, {\left(8 \, a^{3} b + 56 \, a^{2} b^{2} + 115 \, a b^{3} + 70 \, b^{4}\right)} \cos\left(f x + e\right)^{6} + {\left(8 \, a^{4} + 88 \, a^{3} b + 323 \, a^{2} b^{2} + 450 \, a b^{3} + 210 \, b^{4}\right)} \cos\left(f x + e\right)^{4} + 8 \, a^{4} + 56 \, a^{3} b + 123 \, a^{2} b^{2} + 110 \, a b^{3} + 35 \, b^{4} - 2 \, {\left(8 \, a^{4} + 64 \, a^{3} b + 171 \, a^{2} b^{2} + 185 \, a b^{3} + 70 \, b^{4}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \sqrt{-a}}{a}\right) - {\left(3 \, {\left(8 \, a^{3} b + 40 \, a^{2} b^{2} + 35 \, a b^{3}\right)} \cos\left(f x + e\right)^{6} - {\left(32 \, a^{4} + 232 \, a^{3} b + 500 \, a^{2} b^{2} + 315 \, a b^{3}\right)} \cos\left(f x + e\right)^{4} - 50 \, a^{4} - 205 \, a^{3} b - 260 \, a^{2} b^{2} - 105 \, a b^{3} + {\left(88 \, a^{4} + 413 \, a^{3} b + 640 \, a^{2} b^{2} + 315 \, a b^{3}\right)} \cos\left(f x + e\right)^{2}\right)} \sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{24 \, {\left(a^{5} b^{2} f \cos\left(f x + e\right)^{8} - 2 \, {\left(a^{6} b + 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{6} + {\left(a^{7} + 6 \, a^{6} b + 6 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{4} - 2 \, {\left(a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2}\right)} f \cos\left(f x + e\right)^{2} + {\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} f\right)}}\right]"," ",0,"[1/48*(3*((8*a^2*b^2 + 40*a*b^3 + 35*b^4)*cos(f*x + e)^8 - 2*(8*a^3*b + 56*a^2*b^2 + 115*a*b^3 + 70*b^4)*cos(f*x + e)^6 + (8*a^4 + 88*a^3*b + 323*a^2*b^2 + 450*a*b^3 + 210*b^4)*cos(f*x + e)^4 + 8*a^4 + 56*a^3*b + 123*a^2*b^2 + 110*a*b^3 + 35*b^4 - 2*(8*a^4 + 64*a^3*b + 171*a^2*b^2 + 185*a*b^3 + 70*b^4)*cos(f*x + e)^2)*sqrt(a)*log(2*(b*cos(f*x + e)^2 + 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) - 2*(3*(8*a^3*b + 40*a^2*b^2 + 35*a*b^3)*cos(f*x + e)^6 - (32*a^4 + 232*a^3*b + 500*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^4 - 50*a^4 - 205*a^3*b - 260*a^2*b^2 - 105*a*b^3 + (88*a^4 + 413*a^3*b + 640*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^5*b^2*f*cos(f*x + e)^8 - 2*(a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 + 6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 - 2*(a^7 + 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^2 + (a^7 + 2*a^6*b + a^5*b^2)*f), 1/24*(3*((8*a^2*b^2 + 40*a*b^3 + 35*b^4)*cos(f*x + e)^8 - 2*(8*a^3*b + 56*a^2*b^2 + 115*a*b^3 + 70*b^4)*cos(f*x + e)^6 + (8*a^4 + 88*a^3*b + 323*a^2*b^2 + 450*a*b^3 + 210*b^4)*cos(f*x + e)^4 + 8*a^4 + 56*a^3*b + 123*a^2*b^2 + 110*a*b^3 + 35*b^4 - 2*(8*a^4 + 64*a^3*b + 171*a^2*b^2 + 185*a*b^3 + 70*b^4)*cos(f*x + e)^2)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a) - (3*(8*a^3*b + 40*a^2*b^2 + 35*a*b^3)*cos(f*x + e)^6 - (32*a^4 + 232*a^3*b + 500*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^4 - 50*a^4 - 205*a^3*b - 260*a^2*b^2 - 105*a*b^3 + (88*a^4 + 413*a^3*b + 640*a^2*b^2 + 315*a*b^3)*cos(f*x + e)^2)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^5*b^2*f*cos(f*x + e)^8 - 2*(a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^6 + (a^7 + 6*a^6*b + 6*a^5*b^2)*f*cos(f*x + e)^4 - 2*(a^7 + 3*a^6*b + 2*a^5*b^2)*f*cos(f*x + e)^2 + (a^7 + 2*a^6*b + a^5*b^2)*f)]","B",0
538,0,0,0,0.746829," ","integrate(tan(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \tan\left(f x + e\right)^{4}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*tan(f*x + e)^4/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
539,0,0,0,0.725789," ","integrate(tan(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \tan\left(f x + e\right)^{2}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*tan(f*x + e)^2/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
540,0,0,0,0.593700," ","integrate(1/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
541,0,0,0,0.818156," ","integrate(cot(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cot\left(f x + e\right)^{2}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*cot(f*x + e)^2/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
542,0,0,0,0.690434," ","integrate(cot(f*x+e)^4/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{\sqrt{-b \cos\left(f x + e\right)^{2} + a + b} \cot\left(f x + e\right)^{4}}{b^{3} \cos\left(f x + e\right)^{6} - 3 \, {\left(a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3}\right)} \cos\left(f x + e\right)^{2}}, x\right)"," ",0,"integral(-sqrt(-b*cos(f*x + e)^2 + a + b)*cot(f*x + e)^4/(b^3*cos(f*x + e)^6 - 3*(a*b^2 + b^3)*cos(f*x + e)^4 - a^3 - 3*a^2*b - 3*a*b^2 - b^3 + 3*(a^2*b + 2*a*b^2 + b^3)*cos(f*x + e)^2), x)","F",0
543,0,0,0,1.486233," ","integrate((a+b*sin(f*x+e)^2)^p*(d*tan(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \left(d \tan\left(f x + e\right)\right)^{m}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*(d*tan(f*x + e))^m, x)","F",0
544,0,0,0,0.670883," ","integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c)^3,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \tan\left(d x + c\right)^{3}, x\right)"," ",0,"integral((-b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c)^3, x)","F",0
545,0,0,0,0.633204," ","integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c),x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \tan\left(d x + c\right), x\right)"," ",0,"integral((-b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c), x)","F",0
546,0,0,0,0.607477," ","integrate(cot(d*x+c)*(a+b*sin(d*x+c)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \cot\left(d x + c\right), x\right)"," ",0,"integral((-b*cos(d*x + c)^2 + a + b)^p*cot(d*x + c), x)","F",0
547,0,0,0,0.645926," ","integrate(cot(d*x+c)^3*(a+b*sin(d*x+c)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \cot\left(d x + c\right)^{3}, x\right)"," ",0,"integral((-b*cos(d*x + c)^2 + a + b)^p*cot(d*x + c)^3, x)","F",0
548,0,0,0,0.784829," ","integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c)^4,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \tan\left(d x + c\right)^{4}, x\right)"," ",0,"integral((-b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c)^4, x)","F",0
549,0,0,0,0.669561," ","integrate((a+b*sin(d*x+c)^2)^p*tan(d*x+c)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \tan\left(d x + c\right)^{2}, x\right)"," ",0,"integral((-b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c)^2, x)","F",0
550,0,0,0,0.719763," ","integrate(cot(d*x+c)^2*(a+b*sin(d*x+c)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \cot\left(d x + c\right)^{2}, x\right)"," ",0,"integral((-b*cos(d*x + c)^2 + a + b)^p*cot(d*x + c)^2, x)","F",0
551,0,0,0,0.863345," ","integrate(cot(d*x+c)^4*(a+b*sin(d*x+c)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \cot\left(d x + c\right)^{4}, x\right)"," ",0,"integral((-b*cos(d*x + c)^2 + a + b)^p*cot(d*x + c)^4, x)","F",0
552,1,1764,0,97.680796," ","integrate(cot(x)^3/(a+b*sin(x)^3),x, algorithm=""fricas"")","-\frac{6 \, \sqrt{\frac{1}{3}} {\left(a \cos\left(x\right)^{2} - a\right)} \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} a + 4}{a^{2}}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{3}} \sqrt{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{4} - 4 \, b^{2} \cos\left(x\right)^{2} - 4 \, a b \sin\left(x\right) - 2 \, {\left(a^{2} b \sin\left(x\right) - 2 \, a^{3}\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} + 4 \, a^{2} + 4 \, b^{2}} {\left({\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} a^{3} + 2 \, a^{2}\right)} \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} a + 4}{a^{2}}} + \sqrt{\frac{1}{3}} {\left({\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{5} - 8 \, a^{2} b \sin\left(x\right) + 4 \, a^{3} - 4 \, {\left(a^{3} b \sin\left(x\right) - a^{4}\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}\right)} \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} a + 4}{a^{2}}}}{8 \, b^{2}}\right) - 6 \, \sqrt{\frac{1}{3}} {\left(a \cos\left(x\right)^{2} - a\right)} \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} a + 4}{a^{2}}} \arctan\left(-\frac{2 \, \sqrt{\frac{1}{3}} \sqrt{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{4} - 4 \, b^{2} \cos\left(x\right)^{2} - 4 \, a b \sin\left(x\right) - 2 \, {\left(a^{2} b \sin\left(x\right) - 2 \, a^{3}\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} + 4 \, a^{2} + 4 \, b^{2}} {\left({\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} a^{3} + 2 \, a^{2}\right)} \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} a + 4}{a^{2}}} - \sqrt{\frac{1}{3}} {\left({\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{5} - 8 \, a^{2} b \sin\left(x\right) + 4 \, a^{3} - 4 \, {\left(a^{3} b \sin\left(x\right) - a^{4}\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}\right)} \sqrt{\frac{{\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{2} + 4 \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} a + 4}{a^{2}}}}{8 \, b^{2}}\right) + {\left(a \cos\left(x\right)^{2} - a\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} \log\left(\frac{1}{4} \, {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{4} - b^{2} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + {\left(a^{2} b \sin\left(x\right) + a^{3}\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} + a^{2} + b^{2}\right) - {\left({\left(a \cos\left(x\right)^{2} - a\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} + 6 \, \cos\left(x\right)^{2} - 6\right)} \log\left({\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)}^{2} a^{4} - 4 \, b^{2} \cos\left(x\right)^{2} - 4 \, a b \sin\left(x\right) - 2 \, {\left(a^{2} b \sin\left(x\right) - 2 \, a^{3}\right)} {\left(3 \, {\left(i \, \sqrt{3} + 1\right)} {\left(-\frac{1}{54 \, a^{3}} + \frac{b^{2}}{54 \, a^{5}} + \frac{a^{2} - b^{2}}{54 \, a^{5}}\right)}^{\frac{1}{3}} - \frac{2}{a}\right)} + 4 \, a^{2} + 4 \, b^{2}\right) + 12 \, {\left(\cos\left(x\right)^{2} - 1\right)} \log\left(-\frac{1}{2} \, \sin\left(x\right)\right) - 6}{12 \, {\left(a \cos\left(x\right)^{2} - a\right)}}"," ",0,"-1/12*(6*sqrt(1/3)*(a*cos(x)^2 - a)*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a + 4)/a^2)*arctan(-1/8*(2*sqrt(1/3)*sqrt((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^4 - 4*b^2*cos(x)^2 - 4*a*b*sin(x) - 2*(a^2*b*sin(x) - 2*a^3)*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a) + 4*a^2 + 4*b^2)*((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a^3 + 2*a^2)*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a + 4)/a^2) + sqrt(1/3)*((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^5 - 8*a^2*b*sin(x) + 4*a^3 - 4*(a^3*b*sin(x) - a^4)*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a))*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a + 4)/a^2))/b^2) - 6*sqrt(1/3)*(a*cos(x)^2 - a)*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a + 4)/a^2)*arctan(-1/8*(2*sqrt(1/3)*sqrt((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^4 - 4*b^2*cos(x)^2 - 4*a*b*sin(x) - 2*(a^2*b*sin(x) - 2*a^3)*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a) + 4*a^2 + 4*b^2)*((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a^3 + 2*a^2)*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a + 4)/a^2) - sqrt(1/3)*((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^5 - 8*a^2*b*sin(x) + 4*a^3 - 4*(a^3*b*sin(x) - a^4)*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a))*sqrt(((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^2 + 4*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*a + 4)/a^2))/b^2) + (a*cos(x)^2 - a)*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)*log(1/4*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^4 - b^2*cos(x)^2 + 2*a*b*sin(x) + (a^2*b*sin(x) + a^3)*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a) + a^2 + b^2) - ((a*cos(x)^2 - a)*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a) + 6*cos(x)^2 - 6)*log((3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a)^2*a^4 - 4*b^2*cos(x)^2 - 4*a*b*sin(x) - 2*(a^2*b*sin(x) - 2*a^3)*(3*(I*sqrt(3) + 1)*(-1/54/a^3 + 1/54*b^2/a^5 + 1/54*(a^2 - b^2)/a^5)^(1/3) - 2/a) + 4*a^2 + 4*b^2) + 12*(cos(x)^2 - 1)*log(-1/2*sin(x)) - 6)/(a*cos(x)^2 - a)","C",0
553,-1,0,0,0.000000," ","integrate(cot(x)*(a+b*sin(x)^3)^(1/2),x, algorithm=""fricas"")","\text{Timed out}"," ",0,"Timed out","F(-1)",0
554,-2,0,0,0.000000," ","integrate(cot(x)/(a+b*sin(x)^3)^(1/2),x, algorithm=""fricas"")","\text{Exception raised: TypeError}"," ",0,"Exception raised: TypeError >>  Error detected within library code:   failed of mode Union(SparseUnivariatePolynomial(SimpleAlgebraicExtension(InnerPrimeField(7),SparseUnivariatePolynomial(InnerPrimeField(7)),?^2+3*?+1)),failed) cannot be coerced to mode SparseUnivariatePolynomial(SimpleAlgebraicExtension(InnerPrimeField(7),SparseUnivariatePolynomial(InnerPrimeField(7)),?^2+3*?+1))","F(-2)",0
555,1,195,0,1.344547," ","integrate(cot(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(\frac{8 \, {\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} - 2 \, \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} \sqrt{a} + 2 \, a + b\right)}}{\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1}\right) + 2 \, \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}{4 \, d}, \frac{\sqrt{-a} \arctan\left(\frac{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} \sqrt{-a}}{a}\right) + \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}{2 \, d}\right]"," ",0,"[1/4*(sqrt(a)*log(8*(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 - 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*sqrt(a) + 2*a + b)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)) + 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b))/d, 1/2*(sqrt(-a)*arctan(sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*sqrt(-a)/a) + sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b))/d]","A",0
556,1,361,0,1.143617," ","integrate(tan(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a + b} a \cos\left(d x + c\right)^{2} \log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 4 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} {\left(b \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{a + b} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}}{\cos\left(d x + c\right)^{4}}\right) + 2 \, \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} {\left(a + b\right)}}{4 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} d \cos\left(d x + c\right)^{2}}, -\frac{a \sqrt{-a - b} \arctan\left(\frac{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} {\left(b \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) \cos\left(d x + c\right)^{2} - \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} {\left(a + b\right)}}{2 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} d \cos\left(d x + c\right)^{2}}\right]"," ",0,"[1/4*(sqrt(a + b)*a*cos(d*x + c)^2*log(((a*b + 2*b^2)*cos(d*x + c)^4 - 4*(a*b + b^2)*cos(d*x + c)^2 + 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*(b*cos(d*x + c)^2 - a - b)*sqrt(a + b) + 2*a^2 + 4*a*b + 2*b^2)/cos(d*x + c)^4) + 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*(a + b))/((a^2 + 2*a*b + b^2)*d*cos(d*x + c)^2), -1/2*(a*sqrt(-a - b)*arctan(sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*(b*cos(d*x + c)^2 - a - b)*sqrt(-a - b)/((a*b + b^2)*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*cos(d*x + c)^2 - sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*(a + b))/((a^2 + 2*a*b + b^2)*d*cos(d*x + c)^2)]","B",0
557,1,240,0,1.268207," ","integrate(tan(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{{\left(a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 4 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} - 2 \, \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} {\left(b \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{a + b} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}}{\cos\left(d x + c\right)^{4}}\right)}{4 \, \sqrt{a + b} d}, \frac{\sqrt{-a - b} \arctan\left(\frac{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} {\left(b \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{-a - b}}{{\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a b + b^{2}\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right)}{2 \, {\left(a + b\right)} d}\right]"," ",0,"[1/4*log(((a*b + 2*b^2)*cos(d*x + c)^4 - 4*(a*b + b^2)*cos(d*x + c)^2 - 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*(b*cos(d*x + c)^2 - a - b)*sqrt(a + b) + 2*a^2 + 4*a*b + 2*b^2)/cos(d*x + c)^4)/(sqrt(a + b)*d), 1/2*sqrt(-a - b)*arctan(sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*(b*cos(d*x + c)^2 - a - b)*sqrt(-a - b)/((a*b + b^2)*cos(d*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))/((a + b)*d)]","B",0
558,1,140,0,0.767534," ","integrate(cot(d*x+c)/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{8 \, {\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} - 2 \, \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} \sqrt{a} + 2 \, a + b\right)}}{\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1}\right)}{4 \, \sqrt{a} d}, \frac{\sqrt{-a} \arctan\left(\frac{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} \sqrt{-a}}{a}\right)}{2 \, a d}\right]"," ",0,"[1/4*log(8*(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 - 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*sqrt(a) + 2*a + b)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1))/(sqrt(a)*d), 1/2*sqrt(-a)*arctan(sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*sqrt(-a)/a)/(a*d)]","A",0
559,1,247,0,0.944767," ","integrate(cot(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(\cos\left(d x + c\right)^{2} - 1\right)} \sqrt{a} \log\left(\frac{8 \, {\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + 2 \, \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} \sqrt{a} + 2 \, a + b\right)}}{\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1}\right) + 2 \, \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}{4 \, {\left(a d \cos\left(d x + c\right)^{2} - a d\right)}}, -\frac{{\left(\cos\left(d x + c\right)^{2} - 1\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} \sqrt{-a}}{a}\right) - \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}{2 \, {\left(a d \cos\left(d x + c\right)^{2} - a d\right)}}\right]"," ",0,"[1/4*((cos(d*x + c)^2 - 1)*sqrt(a)*log(8*(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*sqrt(a) + 2*a + b)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)) + 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b))/(a*d*cos(d*x + c)^2 - a*d), -1/2*((cos(d*x + c)^2 - 1)*sqrt(-a)*arctan(sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*sqrt(-a)/a) - sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b))/(a*d*cos(d*x + c)^2 - a*d)]","A",0
560,1,371,0,0.854953," ","integrate(cot(d*x+c)^5/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + 2 \, a - b\right)} \sqrt{a} \log\left(\frac{8 \, {\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + 2 \, \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} \sqrt{a} + 2 \, a + b\right)}}{\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1}\right) + 2 \, \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} {\left(4 \, a \cos\left(d x + c\right)^{2} - 3 \, a\right)}}{8 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)}}, \frac{{\left({\left(2 \, a - b\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(2 \, a - b\right)} \cos\left(d x + c\right)^{2} + 2 \, a - b\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} \sqrt{-a}}{a}\right) - \sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b} {\left(4 \, a \cos\left(d x + c\right)^{2} - 3 \, a\right)}}{4 \, {\left(a^{2} d \cos\left(d x + c\right)^{4} - 2 \, a^{2} d \cos\left(d x + c\right)^{2} + a^{2} d\right)}}\right]"," ",0,"[-1/8*(((2*a - b)*cos(d*x + c)^4 - 2*(2*a - b)*cos(d*x + c)^2 + 2*a - b)*sqrt(a)*log(8*(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*sqrt(a) + 2*a + b)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)) + 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*(4*a*cos(d*x + c)^2 - 3*a))/(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d), 1/4*(((2*a - b)*cos(d*x + c)^4 - 2*(2*a - b)*cos(d*x + c)^2 + 2*a - b)*sqrt(-a)*arctan(sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*sqrt(-a)/a) - sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*(4*a*cos(d*x + c)^2 - 3*a))/(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)]","A",0
561,0,0,0,0.625626," ","integrate(tan(d*x+c)^2/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\tan\left(d x + c\right)^{2}}{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}, x\right)"," ",0,"integral(tan(d*x + c)^2/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b), x)","F",0
562,0,0,0,0.645896," ","integrate(1/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{1}{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}, x\right)"," ",0,"integral(1/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b), x)","F",0
563,0,0,0,0.668824," ","integrate(cot(d*x+c)^2/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\cot\left(d x + c\right)^{2}}{\sqrt{b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b}}, x\right)"," ",0,"integral(cot(d*x + c)^2/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b), x)","F",0
564,0,0,0,3.602562," ","integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \tan\left(d x + c\right)^{m}, x\right)"," ",0,"integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c)^m, x)","F",0
565,0,0,0,0.927314," ","integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^3,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \tan\left(d x + c\right)^{3}, x\right)"," ",0,"integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c)^3, x)","F",0
566,0,0,0,0.682432," ","integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \tan\left(d x + c\right), x\right)"," ",0,"integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c), x)","F",0
567,0,0,0,0.745975," ","integrate(cot(d*x+c)*(a+b*sin(d*x+c)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \cot\left(d x + c\right), x\right)"," ",0,"integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p*cot(d*x + c), x)","F",0
568,0,0,0,0.810934," ","integrate(cot(d*x+c)^3*(a+b*sin(d*x+c)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \cot\left(d x + c\right)^{3}, x\right)"," ",0,"integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p*cot(d*x + c)^3, x)","F",0
569,0,0,0,0.782670," ","integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^4,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \tan\left(d x + c\right)^{4}, x\right)"," ",0,"integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c)^4, x)","F",0
570,0,0,0,0.855743," ","integrate((a+b*sin(d*x+c)^4)^p*tan(d*x+c)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \tan\left(d x + c\right)^{2}, x\right)"," ",0,"integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p*tan(d*x + c)^2, x)","F",0
571,0,0,0,0.736027," ","integrate((a+b*sin(d*x+c)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b\right)}^{p}, x\right)"," ",0,"integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p, x)","F",0
572,0,0,0,0.882641," ","integrate(cot(d*x+c)^2*(a+b*sin(d*x+c)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \cot\left(d x + c\right)^{2}, x\right)"," ",0,"integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p*cot(d*x + c)^2, x)","F",0
573,0,0,0,0.760317," ","integrate(cot(d*x+c)^4*(a+b*sin(d*x+c)^4)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \cos\left(d x + c\right)^{4} - 2 \, b \cos\left(d x + c\right)^{2} + a + b\right)}^{p} \cot\left(d x + c\right)^{4}, x\right)"," ",0,"integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p*cot(d*x + c)^4, x)","F",0
574,0,0,0,0.993677," ","integrate((a+b*sin(d*x+c)^n)^3*tan(d*x+c)^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b^{3} \sin\left(d x + c\right)^{3 \, n} + 3 \, a b^{2} \sin\left(d x + c\right)^{2 \, n} + 3 \, a^{2} b \sin\left(d x + c\right)^{n} + a^{3}\right)} \tan\left(d x + c\right)^{m}, x\right)"," ",0,"integral((b^3*sin(d*x + c)^(3*n) + 3*a*b^2*sin(d*x + c)^(2*n) + 3*a^2*b*sin(d*x + c)^n + a^3)*tan(d*x + c)^m, x)","F",0
575,0,0,0,0.695110," ","integrate((a+b*sin(d*x+c)^n)^2*tan(d*x+c)^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b^{2} \sin\left(d x + c\right)^{2 \, n} + 2 \, a b \sin\left(d x + c\right)^{n} + a^{2}\right)} \tan\left(d x + c\right)^{m}, x\right)"," ",0,"integral((b^2*sin(d*x + c)^(2*n) + 2*a*b*sin(d*x + c)^n + a^2)*tan(d*x + c)^m, x)","F",0
576,0,0,0,0.662607," ","integrate((a+b*sin(d*x+c)^n)*tan(d*x+c)^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right)^{n} + a\right)} \tan\left(d x + c\right)^{m}, x\right)"," ",0,"integral((b*sin(d*x + c)^n + a)*tan(d*x + c)^m, x)","F",0
577,0,0,0,0.901091," ","integrate(tan(d*x+c)^m/(a+b*sin(d*x+c)^n),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\tan\left(d x + c\right)^{m}}{b \sin\left(d x + c\right)^{n} + a}, x\right)"," ",0,"integral(tan(d*x + c)^m/(b*sin(d*x + c)^n + a), x)","F",0
578,0,0,0,0.711410," ","integrate(tan(d*x+c)^m/(a+b*sin(d*x+c)^n)^2,x, algorithm=""fricas"")","{\rm integral}\left(\frac{\tan\left(d x + c\right)^{m}}{b^{2} \sin\left(d x + c\right)^{2 \, n} + 2 \, a b \sin\left(d x + c\right)^{n} + a^{2}}, x\right)"," ",0,"integral(tan(d*x + c)^m/(b^2*sin(d*x + c)^(2*n) + 2*a*b*sin(d*x + c)^n + a^2), x)","F",0
579,1,97,0,0.691090," ","integrate(cot(x)*(a+b*sin(x)^n)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(\frac{b \sin\left(x\right)^{n} - 2 \, \sqrt{b \sin\left(x\right)^{n} + a} \sqrt{a} + 2 \, a}{\sin\left(x\right)^{n}}\right) + 2 \, \sqrt{b \sin\left(x\right)^{n} + a}}{n}, \frac{2 \, {\left(\sqrt{-a} \arctan\left(\frac{\sqrt{b \sin\left(x\right)^{n} + a} \sqrt{-a}}{a}\right) + \sqrt{b \sin\left(x\right)^{n} + a}\right)}}{n}\right]"," ",0,"[(sqrt(a)*log((b*sin(x)^n - 2*sqrt(b*sin(x)^n + a)*sqrt(a) + 2*a)/sin(x)^n) + 2*sqrt(b*sin(x)^n + a))/n, 2*(sqrt(-a)*arctan(sqrt(b*sin(x)^n + a)*sqrt(-a)/a) + sqrt(b*sin(x)^n + a))/n]","A",0
580,1,74,0,0.668661," ","integrate(cot(x)/(a+b*sin(x)^n)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{b \sin\left(x\right)^{n} - 2 \, \sqrt{b \sin\left(x\right)^{n} + a} \sqrt{a} + 2 \, a}{\sin\left(x\right)^{n}}\right)}{\sqrt{a} n}, \frac{2 \, \sqrt{-a} \arctan\left(\frac{\sqrt{b \sin\left(x\right)^{n} + a} \sqrt{-a}}{a}\right)}{a n}\right]"," ",0,"[log((b*sin(x)^n - 2*sqrt(b*sin(x)^n + a)*sqrt(a) + 2*a)/sin(x)^n)/(sqrt(a)*n), 2*sqrt(-a)*arctan(sqrt(b*sin(x)^n + a)*sqrt(-a)/a)/(a*n)]","A",0
581,0,0,0,0.746296," ","integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \tan\left(d x + c\right)^{m}, x\right)"," ",0,"integral((b*sin(d*x + c)^n + a)^p*tan(d*x + c)^m, x)","F",0
582,0,0,0,0.710804," ","integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^3,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \tan\left(d x + c\right)^{3}, x\right)"," ",0,"integral((b*sin(d*x + c)^n + a)^p*tan(d*x + c)^3, x)","F",0
583,0,0,0,0.913199," ","integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \tan\left(d x + c\right), x\right)"," ",0,"integral((b*sin(d*x + c)^n + a)^p*tan(d*x + c), x)","F",0
584,0,0,0,0.862124," ","integrate(cot(d*x+c)*(a+b*sin(d*x+c)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \cot\left(d x + c\right), x\right)"," ",0,"integral((b*sin(d*x + c)^n + a)^p*cot(d*x + c), x)","F",0
585,0,0,0,0.763206," ","integrate(cot(d*x+c)^3*(a+b*sin(d*x+c)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \cot\left(d x + c\right)^{3}, x\right)"," ",0,"integral((b*sin(d*x + c)^n + a)^p*cot(d*x + c)^3, x)","F",0
586,0,0,0,0.925838," ","integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^4,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \tan\left(d x + c\right)^{4}, x\right)"," ",0,"integral((b*sin(d*x + c)^n + a)^p*tan(d*x + c)^4, x)","F",0
587,0,0,0,0.646205," ","integrate((a+b*sin(d*x+c)^n)^p*tan(d*x+c)^2,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \tan\left(d x + c\right)^{2}, x\right)"," ",0,"integral((b*sin(d*x + c)^n + a)^p*tan(d*x + c)^2, x)","F",0
588,0,0,0,0.703525," ","integrate((a+b*sin(d*x+c)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p}, x\right)"," ",0,"integral((b*sin(d*x + c)^n + a)^p, x)","F",0
589,0,0,0,0.550417," ","integrate(cot(d*x+c)^2*(a+b*sin(d*x+c)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \cot\left(d x + c\right)^{2}, x\right)"," ",0,"integral((b*sin(d*x + c)^n + a)^p*cot(d*x + c)^2, x)","F",0
590,0,0,0,0.708032," ","integrate(cot(d*x+c)^4*(a+b*sin(d*x+c)^n)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \sin\left(d x + c\right)^{n} + a\right)}^{p} \cot\left(d x + c\right)^{4}, x\right)"," ",0,"integral((b*sin(d*x + c)^n + a)^p*cot(d*x + c)^4, x)","F",0
591,0,0,0,0.660317," ","integrate((a+b*sin(f*x+e)^2)/(g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(-\frac{{\left(b \cos\left(f x + e\right)^{2} - a - b\right)} \sqrt{g \cos\left(f x + e\right)} \sqrt{d \sin\left(f x + e\right)}}{d g^{3} \cos\left(f x + e\right)^{3} \sin\left(f x + e\right)}, x\right)"," ",0,"integral(-(b*cos(f*x + e)^2 - a - b)*sqrt(g*cos(f*x + e))*sqrt(d*sin(f*x + e))/(d*g^3*cos(f*x + e)^3*sin(f*x + e)), x)","F",0
592,0,0,0,2.311773," ","integrate((c*cos(f*x+e))^m*(d*sin(f*x+e))^n*(a+b*sin(f*x+e)^2)^p,x, algorithm=""fricas"")","{\rm integral}\left({\left(-b \cos\left(f x + e\right)^{2} + a + b\right)}^{p} \left(c \cos\left(f x + e\right)\right)^{m} \left(d \sin\left(f x + e\right)\right)^{n}, x\right)"," ",0,"integral((-b*cos(f*x + e)^2 + a + b)^p*(c*cos(f*x + e))^m*(d*sin(f*x + e))^n, x)","F",0
593,0,0,0,1.038038," ","integrate((a+(c*cos(f*x+e)+b*sin(f*x+e))^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{2 \, b c \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(b^{2} - c^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + a}, x\right)"," ",0,"integral(sqrt(2*b*c*cos(f*x + e)*sin(f*x + e) - (b^2 - c^2)*cos(f*x + e)^2 + b^2 + a), x)","F",0
594,0,0,0,0.857163," ","integrate(1/(a+(c*cos(f*x+e)+b*sin(f*x+e))^2)^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\frac{1}{\sqrt{2 \, b c \cos\left(f x + e\right) \sin\left(f x + e\right) - {\left(b^{2} - c^{2}\right)} \cos\left(f x + e\right)^{2} + b^{2} + a}}, x\right)"," ",0,"integral(1/sqrt(2*b*c*cos(f*x + e)*sin(f*x + e) - (b^2 - c^2)*cos(f*x + e)^2 + b^2 + a), x)","F",0
