Integrand size = 41, antiderivative size = 648 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\frac {\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) x}{2 a^6}+\frac {b \left (20 A b^8+20 a^7 b B-35 a^5 b^3 B+28 a^3 b^5 B-8 a b^7 B-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3 d}+\frac {\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]
1/2*(20*A*b^2-8*B*a*b+a^2*(A+2*C))*x/a^6+1/6*(60*A*b^7+6*a^7*B-65*a^5*b^2* B+68*a^3*b^4*B-24*a*b^6*B+a^4*b^3*(146*A-17*C)-a^2*b^5*(167*A-6*C)-a^6*(24 *A*b-26*C*b))*sin(d*x+c)/a^5/(a^2-b^2)^3/d-1/2*(10*A*b^6-12*a^5*b*B+11*a^3 *b^3*B-4*a*b^5*B-a^6*(A-6*C)+a^4*b^2*(23*A-2*C)-a^2*b^4*(27*A-C))*cos(d*x+ c)*sin(d*x+c)/a^4/(a^2-b^2)^3/d+1/3*(A*b^2-a*(B*b-C*a))*cos(d*x+c)*sin(d*x +c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^3-1/6*(5*A*b^4+7*B*a^3*b-2*B*a*b^3-4*a^ 4*C-a^2*b^2*(10*A+C))*cos(d*x+c)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x +c))^2+1/6*(20*A*b^6-27*a^5*b*B+20*a^3*b^3*B-8*a*b^5*B-a^2*b^4*(53*A-2*C)+ 12*a^6*C+a^4*b^2*(48*A+C))*cos(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*se c(d*x+c))+b*(20*A*b^8+20*a^7*b*B-35*a^5*b^3*B+28*a^3*b^5*B-8*a*b^7*B-a^2*b ^6*(69*A-2*C)-8*a^6*b^2*(5*A-C)+7*a^4*b^4*(12*A-C)-8*a^8*C)*arctanh((a-b)^ (1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^6/(a^2-b^2)^3/d/(a-b)^(1/2)/(a+b)^ (1/2)
Result contains complex when optimal does not.
Time = 10.42 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\frac {\left (a^2 A+20 A b^2-8 a b B+2 a^2 C\right ) (c+d x)}{2 a^6 d}+\frac {b \left (-40 a^6 A b^2+84 a^4 A b^4-69 a^2 A b^6+20 A b^8+20 a^7 b B-35 a^5 b^3 B+28 a^3 b^5 B-8 a b^7 B-8 a^8 C+8 a^6 b^2 C-7 a^4 b^4 C+2 a^2 b^6 C\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 \sqrt {a^2-b^2} \left (-a^2+b^2\right )^3 d}+\frac {(4 A b-a B) \left (-\frac {i \cos (c+d x)}{2 a^5}-\frac {\sin (c+d x)}{2 a^5}\right )}{d}+\frac {(4 A b-a B) \left (\frac {i \cos (c+d x)}{2 a^5}-\frac {\sin (c+d x)}{2 a^5}\right )}{d}+\frac {A b^6 \sin (c+d x)-a b^5 B \sin (c+d x)+a^2 b^4 C \sin (c+d x)}{3 a^5 \left (a^2-b^2\right ) d (b+a \cos (c+d x))^3}+\frac {-18 a^2 A b^5 \sin (c+d x)+13 A b^7 \sin (c+d x)+15 a^3 b^4 B \sin (c+d x)-10 a b^6 B \sin (c+d x)-12 a^4 b^3 C \sin (c+d x)+7 a^2 b^5 C \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {90 a^4 A b^4 \sin (c+d x)-122 a^2 A b^6 \sin (c+d x)+47 A b^8 \sin (c+d x)-60 a^5 b^3 B \sin (c+d x)+71 a^3 b^5 B \sin (c+d x)-26 a b^7 B \sin (c+d x)+36 a^6 b^2 C \sin (c+d x)-32 a^4 b^4 C \sin (c+d x)+11 a^2 b^6 C \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {A \sin (2 (c+d x))}{4 a^4 d} \]
((a^2*A + 20*A*b^2 - 8*a*b*B + 2*a^2*C)*(c + d*x))/(2*a^6*d) + (b*(-40*a^6 *A*b^2 + 84*a^4*A*b^4 - 69*a^2*A*b^6 + 20*A*b^8 + 20*a^7*b*B - 35*a^5*b^3* B + 28*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 8*a^6*b^2*C - 7*a^4*b^4*C + 2*a^2 *b^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^ 2 - b^2]*(-a^2 + b^2)^3*d) + ((4*A*b - a*B)*(((-1/2*I)*Cos[c + d*x])/a^5 - Sin[c + d*x]/(2*a^5)))/d + ((4*A*b - a*B)*(((I/2)*Cos[c + d*x])/a^5 - Sin [c + d*x]/(2*a^5)))/d + (A*b^6*Sin[c + d*x] - a*b^5*B*Sin[c + d*x] + a^2*b ^4*C*Sin[c + d*x])/(3*a^5*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^3) + (-18*a^2 *A*b^5*Sin[c + d*x] + 13*A*b^7*Sin[c + d*x] + 15*a^3*b^4*B*Sin[c + d*x] - 10*a*b^6*B*Sin[c + d*x] - 12*a^4*b^3*C*Sin[c + d*x] + 7*a^2*b^5*C*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) + (90*a^4*A*b^4*Sin[c + d*x] - 122*a^2*A*b^6*Sin[c + d*x] + 47*A*b^8*Sin[c + d*x] - 60*a^5*b^3* B*Sin[c + d*x] + 71*a^3*b^5*B*Sin[c + d*x] - 26*a*b^7*B*Sin[c + d*x] + 36* a^6*b^2*C*Sin[c + d*x] - 32*a^4*b^4*C*Sin[c + d*x] + 11*a^2*b^6*C*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + (A*Sin[2*(c + d*x)])/ (4*a^4*d)
Time = 4.74 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.05, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.463, Rules used = {3042, 4588, 3042, 4588, 3042, 4588, 3042, 4592, 27, 3042, 4592, 27, 3042, 4407, 3042, 4318, 3042, 3138, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\int \frac {\cos ^2(c+d x) \left (-\left ((3 A-2 C) a^2\right )-2 b B a+3 (A b+C b-a B) \sec (c+d x) a+5 A b^2-4 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\int \frac {-\left ((3 A-2 C) a^2\right )-2 b B a+3 (A b+C b-a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a+5 A b^2-4 \left (A b^2-a (b B-a C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (-3 \left (-4 C a^4+7 b B a^3-b^2 (10 A+C) a^2-2 b^3 B a+5 A b^4\right ) \sec ^2(c+d x)+2 a \left (3 B a^3-b (6 A+5 C) a^2+2 b^2 B a+A b^3\right ) \sec (c+d x)+2 \left (3 (A-2 C) a^4+9 b B a^3-b^2 (18 A-C) a^2-4 b^3 B a+10 A b^4\right )\right )}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int \frac {-3 \left (-4 C a^4+7 b B a^3-b^2 (10 A+C) a^2-2 b^3 B a+5 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (3 B a^3-b (6 A+5 C) a^2+2 b^2 B a+A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left (3 (A-2 C) a^4+9 b B a^3-b^2 (18 A-C) a^2-4 b^3 B a+10 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) \left (-2 \left (12 C a^6-27 b B a^5+b^2 (48 A+C) a^4+20 b^3 B a^3-b^4 (53 A-2 C) a^2-8 b^5 B a+20 A b^6\right ) \sec ^2(c+d x)+a \left (-6 B a^5+2 b (9 A+5 C) a^4-7 b^2 B a^3-b^3 (8 A-5 C) a^2-2 b^4 B a+5 A b^5\right ) \sec (c+d x)+6 \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right )\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {-2 \left (12 C a^6-27 b B a^5+b^2 (48 A+C) a^4+20 b^3 B a^3-b^4 (53 A-2 C) a^2-8 b^5 B a+20 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (-6 B a^5+2 b (9 A+5 C) a^4-7 b^2 B a^3-b^3 (8 A-5 C) a^2-2 b^4 B a+5 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+6 \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\int \frac {2 \cos (c+d x) \left (6 B a^7-(24 A b-26 b C) a^6-65 b^2 B a^5+b^3 (146 A-17 C) a^4+68 b^4 B a^3-b^5 (167 A-6 C) a^2-24 b^6 B a+\left (3 (A+2 C) a^6-18 b B a^5+b^2 (27 A+8 C) a^4+7 b^3 B a^3-b^4 (25 A-C) a^2-4 b^5 B a+10 A b^6\right ) \sec (c+d x) a+60 A b^7-3 b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{2 a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\int \frac {\cos (c+d x) \left (6 B a^7-(24 A b-26 b C) a^6-65 b^2 B a^5+b^3 (146 A-17 C) a^4+68 b^4 B a^3-b^5 (167 A-6 C) a^2-24 b^6 B a+\left (3 (A+2 C) a^6-18 b B a^5+b^2 (27 A+8 C) a^4+7 b^3 B a^3-b^4 (25 A-C) a^2-4 b^5 B a+10 A b^6\right ) \sec (c+d x) a+60 A b^7-3 b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\int \frac {6 B a^7-(24 A b-26 b C) a^6-65 b^2 B a^5+b^3 (146 A-17 C) a^4+68 b^4 B a^3-b^5 (167 A-6 C) a^2-24 b^6 B a+\left (3 (A+2 C) a^6-18 b B a^5+b^2 (27 A+8 C) a^4+7 b^3 B a^3-b^4 (25 A-C) a^2-4 b^5 B a+10 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+60 A b^7-3 b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {\sin (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}-\frac {\int -\frac {3 \left (\left (a^2-b^2\right )^3 \left ((A+2 C) a^2-8 b B a+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \int \frac {\left (a^2-b^2\right )^3 \left ((A+2 C) a^2-8 b B a+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a}+\frac {\sin (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \int \frac {\left (a^2-b^2\right )^3 \left ((A+2 C) a^2-8 b B a+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )-12 b B a^5+b^2 (23 A-2 C) a^4+11 b^3 B a^3-b^4 (27 A-C) a^2-4 b^5 B a+10 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {\sin (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \left (\frac {b \left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}+\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)-8 a b B+20 A b^2\right )}{a}\right )}{a}+\frac {\sin (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \left (\frac {b \left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)-8 a b B+20 A b^2\right )}{a}\right )}{a}+\frac {\sin (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \left (\frac {\left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a}+\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)-8 a b B+20 A b^2\right )}{a}\right )}{a}+\frac {\sin (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \left (\frac {\left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a}+\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)-8 a b B+20 A b^2\right )}{a}\right )}{a}+\frac {\sin (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {3 \left (\frac {2 \left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}+\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)-8 a b B+20 A b^2\right )}{a}\right )}{a}+\frac {\sin (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\sin (c+d x) \cos (c+d x) \left (12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \sin (c+d x) \cos (c+d x) \left (-\left (a^6 (A-6 C)\right )-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right )}{a d}-\frac {\frac {\sin (c+d x) \left (6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right )}{a d}+\frac {3 \left (\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)-8 a b B+20 A b^2\right )}{a}+\frac {2 b \left (-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (((5*A*b^4 + 7*a^3*b*B - 2*a*b^3*B - 4*a^4*C - a^2* b^2*(10*A + C))*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (((20*A*b^6 - 27*a^5*b*B + 20*a^3*b^3*B - 8*a*b^5*B - a^2*b^ 4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Cos[c + d*x]*Sin[c + d*x]) /(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])) - ((3*(10*A*b^6 - 12*a^5*b*B + 11* a^3*b^3*B - 4*a*b^5*B - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27 *A - C))*Cos[c + d*x]*Sin[c + d*x])/(a*d) - ((3*(((a^2 - b^2)^3*(20*A*b^2 - 8*a*b*B + a^2*(A + 2*C))*x)/a + (2*b*(20*A*b^8 + 20*a^7*b*B - 35*a^5*b^3 *B + 28*a^3*b^5*B - 8*a*b^7*B - a^2*b^6*(69*A - 2*C) - 8*a^6*b^2*(5*A - C) + 7*a^4*b^4*(12*A - C) - 8*a^8*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/ Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)))/a + ((60*A*b^7 + 6*a^7*B - 6 5*a^5*b^2*B + 68*a^3*b^4*B - 24*a*b^6*B + a^4*b^3*(146*A - 17*C) - a^2*b^5 *(167*A - 6*C) - a^6*(24*A*b - 26*b*C))*Sin[c + d*x])/(a*d))/a)/(a*(a^2 - b^2)))/(2*a*(a^2 - b^2)))/(3*a*(a^2 - b^2))
3.10.29.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc [e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f *x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x ] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[n, 0])
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Time = 1.96 (sec) , antiderivative size = 771, normalized size of antiderivative = 1.19
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}-20 a^{5} b B -5 B \,a^{4} b^{2}+18 a^{3} b^{3} B +2 B \,a^{2} b^{4}-6 a \,b^{5} B +12 a^{6} C +4 a^{5} C b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}-30 a^{5} b B +29 a^{3} b^{3} B -9 a \,b^{5} B +18 a^{6} C -11 a^{4} b^{2} C +3 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}-20 a^{5} b B +5 B \,a^{4} b^{2}+18 a^{3} b^{3} B -2 B \,a^{2} b^{4}-6 a \,b^{5} B +12 a^{6} C -4 a^{5} C b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (40 A \,a^{6} b^{2}-84 a^{4} A \,b^{4}+69 a^{2} A \,b^{6}-20 A \,b^{8}-20 a^{7} b B +35 a^{5} b^{3} B -28 a^{3} b^{5} B +8 a \,b^{7} B +8 a^{8} C -8 a^{6} b^{2} C +7 a^{4} b^{4} C -2 C \,a^{2} b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} A -4 a A b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} a^{2} A -4 a A b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (a^{2} A +20 A \,b^{2}-8 B a b +2 C \,a^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6}}}{d}\) | \(771\) |
| default | \(\frac {\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}-20 a^{5} b B -5 B \,a^{4} b^{2}+18 a^{3} b^{3} B +2 B \,a^{2} b^{4}-6 a \,b^{5} B +12 a^{6} C +4 a^{5} C b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}-30 a^{5} b B +29 a^{3} b^{3} B -9 a \,b^{5} B +18 a^{6} C -11 a^{4} b^{2} C +3 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}-20 a^{5} b B +5 B \,a^{4} b^{2}+18 a^{3} b^{3} B -2 B \,a^{2} b^{4}-6 a \,b^{5} B +12 a^{6} C -4 a^{5} C b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (40 A \,a^{6} b^{2}-84 a^{4} A \,b^{4}+69 a^{2} A \,b^{6}-20 A \,b^{8}-20 a^{7} b B +35 a^{5} b^{3} B -28 a^{3} b^{5} B +8 a \,b^{7} B +8 a^{8} C -8 a^{6} b^{2} C +7 a^{4} b^{4} C -2 C \,a^{2} b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} A -4 a A b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} a^{2} A -4 a A b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (a^{2} A +20 A \,b^{2}-8 B a b +2 C \,a^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6}}}{d}\) | \(771\) |
| risch | \(\text {Expression too large to display}\) | \(3290\) |
1/d*(2*b/a^6*((-1/2*(30*A*a^4*b^2+6*A*a^3*b^3-34*A*a^2*b^4-3*A*a*b^5+12*A* b^6-20*B*a^5*b-5*B*a^4*b^2+18*B*a^3*b^3+2*B*a^2*b^4-6*B*a*b^5+12*C*a^6+4*C *a^5*b-6*C*a^4*b^2-C*a^3*b^3+2*C*a^2*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3*a*b^2+b ^3)*tan(1/2*d*x+1/2*c)^5+2/3*(45*A*a^4*b^2-53*A*a^2*b^4+18*A*b^6-30*B*a^5* b+29*B*a^3*b^3-9*B*a*b^5+18*C*a^6-11*C*a^4*b^2+3*C*a^2*b^4)*a*b/(a^2-2*a*b +b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(30*A*a^4*b^2-6*A*a^3*b^3-3 4*A*a^2*b^4+3*A*a*b^5+12*A*b^6-20*B*a^5*b+5*B*a^4*b^2+18*B*a^3*b^3-2*B*a^2 *b^4-6*B*a*b^5+12*C*a^6-4*C*a^5*b-6*C*a^4*b^2+C*a^3*b^3+2*C*a^2*b^4)*a*b/( a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(tan(1/2*d*x+1/2*c)^2*a -tan(1/2*d*x+1/2*c)^2*b-a-b)^3-1/2*(40*A*a^6*b^2-84*A*a^4*b^4+69*A*a^2*b^6 -20*A*b^8-20*B*a^7*b+35*B*a^5*b^3-28*B*a^3*b^5+8*B*a*b^7+8*C*a^8-8*C*a^6*b ^2+7*C*a^4*b^4-2*C*a^2*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1 /2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))+2/a^6*(((-1/2*a ^2*A-4*a*A*b+B*a^2)*tan(1/2*d*x+1/2*c)^3+(1/2*a^2*A-4*a*A*b+B*a^2)*tan(1/2 *d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^2+1/2*(A*a^2+20*A*b^2-8*B*a*b+2*C*a^ 2)*arctan(tan(1/2*d*x+1/2*c))))
Leaf count of result is larger than twice the leaf count of optimal. 1698 vs. \(2 (624) = 1248\).
Time = 0.68 (sec) , antiderivative size = 3454, normalized size of antiderivative = 5.33 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4, x, algorithm="fricas")
[1/12*(6*((A + 2*C)*a^13 - 8*B*a^12*b + 8*(2*A - C)*a^11*b^2 + 32*B*a^10*b ^3 - 2*(37*A - 6*C)*a^9*b^4 - 48*B*a^8*b^5 + 4*(29*A - 2*C)*a^7*b^6 + 32*B *a^6*b^7 - (79*A - 2*C)*a^5*b^8 - 8*B*a^4*b^9 + 20*A*a^3*b^10)*d*x*cos(d*x + c)^3 + 18*((A + 2*C)*a^12*b - 8*B*a^11*b^2 + 8*(2*A - C)*a^10*b^3 + 32* B*a^9*b^4 - 2*(37*A - 6*C)*a^8*b^5 - 48*B*a^7*b^6 + 4*(29*A - 2*C)*a^6*b^7 + 32*B*a^5*b^8 - (79*A - 2*C)*a^4*b^9 - 8*B*a^3*b^10 + 20*A*a^2*b^11)*d*x *cos(d*x + c)^2 + 18*((A + 2*C)*a^11*b^2 - 8*B*a^10*b^3 + 8*(2*A - C)*a^9* b^4 + 32*B*a^8*b^5 - 2*(37*A - 6*C)*a^7*b^6 - 48*B*a^6*b^7 + 4*(29*A - 2*C )*a^5*b^8 + 32*B*a^4*b^9 - (79*A - 2*C)*a^3*b^10 - 8*B*a^2*b^11 + 20*A*a*b ^12)*d*x*cos(d*x + c) + 6*((A + 2*C)*a^10*b^3 - 8*B*a^9*b^4 + 8*(2*A - C)* a^8*b^5 + 32*B*a^7*b^6 - 2*(37*A - 6*C)*a^6*b^7 - 48*B*a^5*b^8 + 4*(29*A - 2*C)*a^4*b^9 + 32*B*a^3*b^10 - (79*A - 2*C)*a^2*b^11 - 8*B*a*b^12 + 20*A* b^13)*d*x + 3*(8*C*a^8*b^4 - 20*B*a^7*b^5 + 8*(5*A - C)*a^6*b^6 + 35*B*a^5 *b^7 - 7*(12*A - C)*a^4*b^8 - 28*B*a^3*b^9 + (69*A - 2*C)*a^2*b^10 + 8*B*a *b^11 - 20*A*b^12 + (8*C*a^11*b - 20*B*a^10*b^2 + 8*(5*A - C)*a^9*b^3 + 35 *B*a^8*b^4 - 7*(12*A - C)*a^7*b^5 - 28*B*a^6*b^6 + (69*A - 2*C)*a^5*b^7 + 8*B*a^4*b^8 - 20*A*a^3*b^9)*cos(d*x + c)^3 + 3*(8*C*a^10*b^2 - 20*B*a^9*b^ 3 + 8*(5*A - C)*a^8*b^4 + 35*B*a^7*b^5 - 7*(12*A - C)*a^6*b^6 - 28*B*a^5*b ^7 + (69*A - 2*C)*a^4*b^8 + 8*B*a^3*b^9 - 20*A*a^2*b^10)*cos(d*x + c)^2 + 3*(8*C*a^9*b^3 - 20*B*a^8*b^4 + 8*(5*A - C)*a^7*b^5 + 35*B*a^6*b^6 - 7*...
\[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \]
Integral((A + B*sec(c + d*x) + C*sec(c + d*x)**2)*cos(c + d*x)**2/(a + b*s ec(c + d*x))**4, x)
Exception generated. \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4, x, algorithm="maxima")
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 1438 vs. \(2 (624) = 1248\).
Time = 0.41 (sec) , antiderivative size = 1438, normalized size of antiderivative = 2.22 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4, x, algorithm="giac")
-1/6*(6*(8*C*a^8*b - 20*B*a^7*b^2 + 40*A*a^6*b^3 - 8*C*a^6*b^3 + 35*B*a^5* b^4 - 84*A*a^4*b^5 + 7*C*a^4*b^5 - 28*B*a^3*b^6 + 69*A*a^2*b^7 - 2*C*a^2*b ^7 + 8*B*a*b^8 - 20*A*b^9)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2* b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*b^6)*sqrt(-a^2 + b^2)) + 2*( 36*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^5 - 60*B*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 - 60*C*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 + 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^ 5 + 105*B*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^6*b^4*tan(1/2*d*x + 1/2*c )^5 - 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 24*B*a^5*b^5*tan(1/2*d*x + 1/ 2*c)^5 + 45*C*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 - 117*B*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 + 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 + 24*B*a^3*b^7*tan(1/2*d *x + 1/2*c)^5 - 15*C*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^2*b^8*tan(1/2 *d*x + 1/2*c)^5 + 42*B*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^2*b^8*tan(1/ 2*d*x + 1/2*c)^5 - 81*A*a*b^9*tan(1/2*d*x + 1/2*c)^5 - 18*B*a*b^9*tan(1/2* d*x + 1/2*c)^5 + 36*A*b^10*tan(1/2*d*x + 1/2*c)^5 - 72*C*a^8*b^2*tan(1/2*d *x + 1/2*c)^3 + 120*B*a^7*b^3*tan(1/2*d*x + 1/2*c)^3 - 180*A*a^6*b^4*tan(1 /2*d*x + 1/2*c)^3 + 116*C*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 - 236*B*a^5*b^5*t an(1/2*d*x + 1/2*c)^3 + 392*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^3 - 56*C*a^4*b^ 6*tan(1/2*d*x + 1/2*c)^3 + 152*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^3 - 284*A...
Time = 43.03 (sec) , antiderivative size = 21910, normalized size of antiderivative = 33.81 \[ \int \frac {\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)^9*(A*a^8 + 20*A*b^8 - 2*B*a^8 - 59*A*a^2*b^6 + 27*A*a ^3*b^5 + 57*A*a^4*b^4 - 21*A*a^5*b^3 - 11*A*a^6*b^2 + 4*B*a^2*b^6 + 24*B*a ^3*b^5 - 11*B*a^4*b^4 - 26*B*a^5*b^3 + 6*B*a^6*b^2 + 2*C*a^2*b^6 - C*a^3*b ^5 - 6*C*a^4*b^4 + 4*C*a^5*b^3 + 12*C*a^6*b^2 - 10*A*a*b^7 + 7*A*a^7*b - 8 *B*a*b^7 + 2*B*a^7*b))/(a^5*(a + b)^3*(a - b)) - (2*tan(c/2 + (d*x)/2)^3*( 6*A*a^9 - 120*A*b^9 + 6*B*a^9 + 364*A*a^2*b^7 + 71*A*a^3*b^6 - 369*A*a^4*b ^5 - 45*A*a^5*b^4 + 111*A*a^6*b^3 + 3*A*a^7*b^2 + 12*B*a^2*b^7 - 148*B*a^3 *b^6 - 29*B*a^4*b^5 + 159*B*a^5*b^4 + 18*B*a^6*b^3 - 30*B*a^7*b^2 - 12*C*a ^2*b^7 - 3*C*a^3*b^6 + 37*C*a^4*b^5 + 8*C*a^5*b^4 - 60*C*a^6*b^3 - 30*A*a* b^8 - 21*A*a^8*b + 48*B*a*b^8 - 6*B*a^8*b))/(3*a^5*(a + b)^2*(a - b)^3) - (2*tan(c/2 + (d*x)/2)^7*(6*A*a^9 + 120*A*b^9 - 6*B*a^9 - 364*A*a^2*b^7 + 7 1*A*a^3*b^6 + 369*A*a^4*b^5 - 45*A*a^5*b^4 - 111*A*a^6*b^3 + 3*A*a^7*b^2 + 12*B*a^2*b^7 + 148*B*a^3*b^6 - 29*B*a^4*b^5 - 159*B*a^5*b^4 + 18*B*a^6*b^ 3 + 30*B*a^7*b^2 + 12*C*a^2*b^7 - 3*C*a^3*b^6 - 37*C*a^4*b^5 + 8*C*a^5*b^4 + 60*C*a^6*b^3 - 30*A*a*b^8 + 21*A*a^8*b - 48*B*a*b^8 - 6*B*a^8*b))/(3*a^ 5*(a + b)^3*(a - b)^2) + (2*tan(c/2 + (d*x)/2)^5*(9*A*a^10 + 180*A*b^10 - 611*A*a^2*b^8 + 740*A*a^4*b^6 - 324*A*a^6*b^4 + 36*A*a^8*b^2 + 248*B*a^3*b ^7 - 320*B*a^5*b^5 + 132*B*a^7*b^3 + 18*C*a^2*b^8 - 62*C*a^4*b^6 + 110*C*a ^6*b^4 - 36*C*a^8*b^2 - 72*B*a*b^9 - 18*B*a^9*b))/(3*a^5*(a + b)^3*(a - b) ^3) + (tan(c/2 + (d*x)/2)*(A*a^8 + 20*A*b^8 + 2*B*a^8 - 59*A*a^2*b^6 - ...