Integrand size = 33, antiderivative size = 513 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\frac {\left (20 A b^2+a^2 (A+2 C)\right ) x}{2 a^6}+\frac {\left (20 A b^9-a^2 b^7 (69 A-2 C)-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-8 a^8 b C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )^3 d}+\frac {b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac {\left (10 A b^6-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\left (20 A b^6-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]
1/2*(20*A*b^2+a^2*(A+2*C))*x/a^6+1/6*b*(60*A*b^6-a^6*(24*A-26*C)+a^4*b^2*( 146*A-17*C)-a^2*b^4*(167*A-6*C))*sin(d*x+c)/a^5/(a^2-b^2)^3/d-1/2*(10*A*b^ 6-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+C*a^2)*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^4-4*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-a^2*b^4*(53*A-2*C)+1 2*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*sec (d*x+c))+(20*A*b^9-a^2*b^7*(69*A-2*C)-8*a^6*b^3*(5*A-C)+7*a^4*b^5*(12*A-C) -8*a^8*b*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)
Leaf count is larger than twice the leaf count of optimal. \(1452\) vs. \(2(513)=1026\).
Time = 8.67 (sec) , antiderivative size = 1452, normalized size of antiderivative = 2.83 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\frac {b \left (-40 a^6 A b^2+84 a^4 A b^4-69 a^2 A b^6+20 A b^8-8 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 {-72 a^{10} A b (c+d x)-1272 a^8 A b^3 (c+d x)+3288 a^6 A b^5 (c+d x)-1512 a^4 A b^7 (c+d x)-1392 a^2 A b^9 (c+d x)+960 A b^{11} (c+d x)-144 a^{10} b C (c+d x)+336 a^8 b^3 C (c+d x)-144 a^6 b^5 C (c+d x)-144 a^4 b^7 C (c+d x)+96 a^2 b^9 C (c+d x)-36 a^{11} A (c+d x) \cos (c+d x)-756 a^9 A b^2 (c+d x) \cos (c+d x)-396 a^7 A b^4 (c+d x) \cos (c+d x)+6084 a^5 A b^6 (c+d x) \cos (c+d x)-7776 a^3 A b^8 (c+d x) \cos (c+d x)+2880 a A b^{10} (c+d x) \cos (c+d x)-72 a^{11} C (c+d x) \cos (c+d x)-72 a^9 b^2 C (c+d x) \cos (c+d x)+648 a^7 b^4 C (c+d x) \cos (c+d x)-792 a^5 b^6 C (c+d x) \cos (c+d x)+288 a^3 b^8 C (c+d x) \cos (c+d x)-72 a^{10} A b (c+d x) \cos (2 (c+d x))-1224 a^8 A b^3 (c+d x) \cos (2 (c+d x))+4104 a^6 A b^5 (c+d x) \cos (2 (c+d x))-4248 a^4 A b^7 (c+d x) \cos (2 (c+d x))+1440 a^2 A b^9 (c+d x) \cos (2 (c+d x))-144 a^{10} b C (c+d x) \cos (2 (c+d x))+432 a^8 b^3 C (c+d x) \cos (2 (c+d x))-432 a^6 b^5 C (c+d x) \cos (2 (c+d x))+144 a^4 b^7 C (c+d x) \cos (2 (c+d x))-12 a^{11} A (c+d x) \cos (3 (c+d x))-204 a^9 A b^2 (c+d x) \cos (3 (c+d x))+684 a^7 A b^4 (c+d x) \cos (3 (c+d x))-708 a^5 A b^6 (c+d x) \cos (3 (c+d x))+240 a^3 A b^8 (c+d x) \cos (3 (c+d x))-24 a^{11} C (c+d x) \cos (3 (c+d x))+72 a^9 b^2 C (c+d x) \cos (3 (c+d x))-72 a^7 b^4 C (c+d x) \cos (3 (c+d x))+24 a^5 b^6 C (c+d x) \cos (3 (c+d x))-6 a^{11} A \sin (c+d x)+270 a^9 A b^2 \sin (c+d x)-750 a^7 A b^4 \sin (c+d x)-1086 a^5 A b^6 \sin (c+d x)+2232 a^3 A b^8 \sin (c+d x)-960 a A b^{10} \sin (c+d x)-144 a^9 b^2 C \sin (c+d x)-288 a^7 b^4 C \sin (c+d x)+228 a^5 b^6 C \sin (c+d x)-96 a^3 b^8 C \sin (c+d x)+60 a^{10} A b \sin (2 (c+d x))+372 a^8 A b^3 \sin (2 (c+d x))-2772 a^6 A b^5 \sin (2 (c+d x))+3300 a^4 A b^7 \sin (2 (c+d x))-1200 a^2 A b^9 \sin (2 (c+d x))-480 a^8 b^3 C \sin (2 (c+d x))+360 a^6 b^5 C \sin (2 (c+d x))-120 a^4 b^7 C \sin (2 (c+d x))-9 a^{11} A \sin (3 (c+d x))+279 a^9 A b^2 \sin (3 (c+d x))-1143 a^7 A b^4 \sin (3 (c+d x))+1253 a^5 A b^6 \sin (3 (c+d x))-440 a^3 A b^8 \sin (3 (c+d x))-144 a^9 b^2 C \sin (3 (c+d x))+128 a^7 b^4 C \sin (3 (c+d x))-44 a^5 b^6 C \sin (3 (c+d x))+30 a^{10} A b \sin (4 (c+d x))-90 a^8 A b^3 \sin (4 (c+d x))+90 a^6 A b^5 \sin (4 (c+d x))-30 a^4 A b^7 \sin (4 (c+d x))-3 a^{11} A \sin (5 (c+d x))+9 a^9 A b^2 \sin (5 (c+d x))-9 a^7 A b^4 \sin (5 (c+d x))+3 a^5 A b^6 \sin (5 (c+d x))}{96 a^6 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^3} \]
(b*(-40*a^6*A*b^2 + 84*a^4*A*b^4 - 69*a^2*A*b^6 + 20*A*b^8 - 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) - (-72*a^10*A*b*( c + d*x) - 1272*a^8*A*b^3*(c + d*x) + 3288*a^6*A*b^5*(c + d*x) - 1512*a^4* A*b^7*(c + d*x) - 1392*a^2*A*b^9*(c + d*x) + 960*A*b^11*(c + d*x) - 144*a^ 10*b*C*(c + d*x) + 336*a^8*b^3*C*(c + d*x) - 144*a^6*b^5*C*(c + d*x) - 144 *a^4*b^7*C*(c + d*x) + 96*a^2*b^9*C*(c + d*x) - 36*a^11*A*(c + d*x)*Cos[c + d*x] - 756*a^9*A*b^2*(c + d*x)*Cos[c + d*x] - 396*a^7*A*b^4*(c + d*x)*Co s[c + d*x] + 6084*a^5*A*b^6*(c + d*x)*Cos[c + d*x] - 7776*a^3*A*b^8*(c + d *x)*Cos[c + d*x] + 2880*a*A*b^10*(c + d*x)*Cos[c + d*x] - 72*a^11*C*(c + d *x)*Cos[c + d*x] - 72*a^9*b^2*C*(c + d*x)*Cos[c + d*x] + 648*a^7*b^4*C*(c + d*x)*Cos[c + d*x] - 792*a^5*b^6*C*(c + d*x)*Cos[c + d*x] + 288*a^3*b^8*C *(c + d*x)*Cos[c + d*x] - 72*a^10*A*b*(c + d*x)*Cos[2*(c + d*x)] - 1224*a^ 8*A*b^3*(c + d*x)*Cos[2*(c + d*x)] + 4104*a^6*A*b^5*(c + d*x)*Cos[2*(c + d *x)] - 4248*a^4*A*b^7*(c + d*x)*Cos[2*(c + d*x)] + 1440*a^2*A*b^9*(c + d*x )*Cos[2*(c + d*x)] - 144*a^10*b*C*(c + d*x)*Cos[2*(c + d*x)] + 432*a^8*b^3 *C*(c + d*x)*Cos[2*(c + d*x)] - 432*a^6*b^5*C*(c + d*x)*Cos[2*(c + d*x)] + 144*a^4*b^7*C*(c + d*x)*Cos[2*(c + d*x)] - 12*a^11*A*(c + d*x)*Cos[3*(c + d*x)] - 204*a^9*A*b^2*(c + d*x)*Cos[3*(c + d*x)] + 684*a^7*A*b^4*(c + d*x )*Cos[3*(c + d*x)] - 708*a^5*A*b^6*(c + d*x)*Cos[3*(c + d*x)] + 240*a^3...
Time = 3.91 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.576, Rules used = {3042, 4589, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 4589 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\int \frac {\cos ^2(c+d x) \left (-\left ((3 A-2 C) a^2\right )+3 b (A+C) \sec (c+d x) a+5 A b^2-4 \left (C a^2+A b^2\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\int \frac {-\left ((3 A-2 C) a^2\right )+3 b (A+C) \csc \left (c+d x+\frac {\pi }{2}\right ) a+5 A b^2-4 \left (C a^2+A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (-3 \left (-4 C a^4-b^2 (10 A+C) a^2+5 A b^4\right ) \sec ^2(c+d x)+2 a b \left (A b^2-a^2 (6 A+5 C)\right ) \sec (c+d x)+2 \left (3 (A-2 C) a^4-b^2 (18 A-C) a^2+10 A b^4\right )\right )}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int \frac {-3 \left (-4 C a^4-b^2 (10 A+C) a^2+5 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (A b^2-a^2 (6 A+5 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left (3 (A-2 C) a^4-b^2 (18 A-C) a^2+10 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) \left (-2 \left (12 C a^6+b^2 (48 A+C) a^4-b^4 (53 A-2 C) a^2+20 A b^6\right ) \sec ^2(c+d x)+a b \left (2 (9 A+5 C) a^4-b^2 (8 A-5 C) a^2+5 A b^4\right ) \sec (c+d x)+6 \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right )\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {-2 \left (12 C a^6+b^2 (48 A+C) a^4-b^4 (53 A-2 C) a^2+20 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a b \left (2 (9 A+5 C) a^4-b^2 (8 A-5 C) a^2+5 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+6 \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\int \frac {2 \cos (c+d x) \left (60 A b^7-a^2 (167 A-6 C) b^5+a^4 (146 A-17 C) b^3-3 \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \sec ^2(c+d x) b-a^6 (24 A b-26 b C)+a \left (3 (A+2 C) a^6+b^2 (27 A+8 C) a^4-b^4 (25 A-C) a^2+10 A b^6\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{2 a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\int \frac {\cos (c+d x) \left (60 A b^7-a^2 (167 A-6 C) b^5+a^4 (146 A-17 C) b^3-3 \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \sec ^2(c+d x) b-a^6 (24 A b-26 b C)+a \left (3 (A+2 C) a^6+b^2 (27 A+8 C) a^4-b^4 (25 A-C) a^2+10 A b^6\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\int \frac {60 A b^7-a^2 (167 A-6 C) b^5+a^4 (146 A-17 C) b^3-3 \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b-a^6 (24 A b-26 b C)+a \left (3 (A+2 C) a^6+b^2 (27 A+8 C) a^4-b^4 (25 A-C) a^2+10 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \sin (c+d x)}{a d}-\frac {\int -\frac {3 \left (\left (a^2-b^2\right )^3 \left ((A+2 C) a^2+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \int \frac {\left (a^2-b^2\right )^3 \left ((A+2 C) a^2+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \sin (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \int \frac {\left (a^2-b^2\right )^3 \left ((A+2 C) a^2+20 A b^2\right )-a b \left (-\left ((A-6 C) a^6\right )+b^2 (23 A-2 C) a^4-b^4 (27 A-C) a^2+10 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \sin (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \left (\frac {\left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}+\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)+20 A b^2\right )}{a}\right )}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \sin (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \left (\frac {\left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)+20 A b^2\right )}{a}\right )}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \sin (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \left (\frac {\left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a b}+\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)+20 A b^2\right )}{a}\right )}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \sin (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \left (\frac {\left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a b}+\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)+20 A b^2\right )}{a}\right )}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \sin (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {3 \left (\frac {2 \left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a b d}+\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)+20 A b^2\right )}{a}\right )}{a}+\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \sin (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {\left (-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {\left (12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (-\left (a^6 (A-6 C)\right )+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {b \left (-\left (a^6 (24 A-26 C)\right )+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right ) \sin (c+d x)}{a d}+\frac {3 \left (\frac {x \left (a^2-b^2\right )^3 \left (a^2 (A+2 C)+20 A b^2\right )}{a}+\frac {2 \left (-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
((A*b^2 + a^2*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 - 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 - 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 - 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 + a^2*(A + 2*C))*x)/a + (2*(20*A*b^9 - a^2*b^7*(69*A - 2*C) - 8*a^6*b^3*(5*A - C) + 7*a^4*b^5*(12*A - C) - 8*a^8 *b*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]* Sqrt[a + b]*d)))/a + (b*(60*A*b^6 - a^6*(24*A - 26*C) + a^4*b^2*(146*A - 1 7*C) - a^2*b^4*(167*A - 6*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))
3.8.4.3.1 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_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b ^2 + 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] + Simp[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^2*(A + C)*(m + 1) - (A*b^2 + a^2*C)*(m + n + 1) - a*b*(A + C)*(m + 1)*Csc[e + f*x] + (A*b^2 + a^2*C)* (m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, 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 = 1.83 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.21
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} A -4 a A b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} a^{2} A -4 a A b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (a^{2} A +20 A \,b^{2}+2 C \,a^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6}}+\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C +4 a^{5} C b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}+18 a^{6} C -11 a^{4} b^{2} C +3 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C -4 a^{5} C b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (40 A \,a^{6} b^{2}-84 a^{4} A \,b^{4}+69 a^{2} A \,b^{6}-20 A \,b^{8}+8 a^{8} C -8 a^{6} b^{2} C +7 a^{4} b^{4} C -2 C \,a^{2} b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}}{d}\) | \(619\) |
| default | \(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} A -4 a A b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} a^{2} A -4 a A b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (a^{2} A +20 A \,b^{2}+2 C \,a^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6}}+\frac {2 b \left (\frac {-\frac {\left (30 A \,a^{4} b^{2}+6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}-3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C +4 a^{5} C b -6 a^{4} b^{2} C -C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (45 A \,a^{4} b^{2}-53 a^{2} A \,b^{4}+18 A \,b^{6}+18 a^{6} C -11 a^{4} b^{2} C +3 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b^{2}-6 A \,a^{3} b^{3}-34 a^{2} A \,b^{4}+3 A a \,b^{5}+12 A \,b^{6}+12 a^{6} C -4 a^{5} C b -6 a^{4} b^{2} C +C \,a^{3} b^{3}+2 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (40 A \,a^{6} b^{2}-84 a^{4} A \,b^{4}+69 a^{2} A \,b^{6}-20 A \,b^{8}+8 a^{8} C -8 a^{6} b^{2} C +7 a^{4} b^{4} C -2 C \,a^{2} b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}}{d}\) | \(619\) |
| risch | \(\text {Expression too large to display}\) | \(2225\) |
1/d*(2/a^6*(((-1/2*a^2*A-4*a*A*b)*tan(1/2*d*x+1/2*c)^3+(1/2*a^2*A-4*a*A*b) *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+2*C*a^ 2)*arctan(tan(1/2*d*x+1/2*c)))+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+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+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^3-34*A*a^2*b^4+3*A*a*b^5+12*A*b^6+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^2-8 4*A*a^4*b^4+69*A*a^2*b^6-20*A*b^8+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))))
Leaf count of result is larger than twice the leaf count of optimal. 1204 vs. \(2 (490) = 980\).
Time = 0.54 (sec) , antiderivative size = 2465, normalized size of antiderivative = 4.81 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
[1/12*(6*((A + 2*C)*a^13 + 8*(2*A - C)*a^11*b^2 - 2*(37*A - 6*C)*a^9*b^4 + 4*(29*A - 2*C)*a^7*b^6 - (79*A - 2*C)*a^5*b^8 + 20*A*a^3*b^10)*d*x*cos(d* x + c)^3 + 18*((A + 2*C)*a^12*b + 8*(2*A - C)*a^10*b^3 - 2*(37*A - 6*C)*a^ 8*b^5 + 4*(29*A - 2*C)*a^6*b^7 - (79*A - 2*C)*a^4*b^9 + 20*A*a^2*b^11)*d*x *cos(d*x + c)^2 + 18*((A + 2*C)*a^11*b^2 + 8*(2*A - C)*a^9*b^4 - 2*(37*A - 6*C)*a^7*b^6 + 4*(29*A - 2*C)*a^5*b^8 - (79*A - 2*C)*a^3*b^10 + 20*A*a*b^ 12)*d*x*cos(d*x + c) + 6*((A + 2*C)*a^10*b^3 + 8*(2*A - C)*a^8*b^5 - 2*(37 *A - 6*C)*a^6*b^7 + 4*(29*A - 2*C)*a^4*b^9 - (79*A - 2*C)*a^2*b^11 + 20*A* b^13)*d*x + 3*(8*C*a^8*b^4 + 8*(5*A - C)*a^6*b^6 - 7*(12*A - C)*a^4*b^8 + (69*A - 2*C)*a^2*b^10 - 20*A*b^12 + (8*C*a^11*b + 8*(5*A - C)*a^9*b^3 - 7* (12*A - C)*a^7*b^5 + (69*A - 2*C)*a^5*b^7 - 20*A*a^3*b^9)*cos(d*x + c)^3 + 3*(8*C*a^10*b^2 + 8*(5*A - C)*a^8*b^4 - 7*(12*A - C)*a^6*b^6 + (69*A - 2* C)*a^4*b^8 - 20*A*a^2*b^10)*cos(d*x + c)^2 + 3*(8*C*a^9*b^3 + 8*(5*A - C)* a^7*b^5 - 7*(12*A - C)*a^5*b^7 + (69*A - 2*C)*a^3*b^9 - 20*A*a*b^11)*cos(d *x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2) /(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - 2*(2*(12*A - 13*C)*a^9 *b^4 - (170*A - 43*C)*a^7*b^6 + (313*A - 23*C)*a^5*b^8 - (227*A - 6*C)*a^3 *b^10 + 60*A*a*b^12 - 3*(A*a^13 - 4*A*a^11*b^2 + 6*A*a^9*b^4 - 4*A*a^7*b^6 + A*a^5*b^8)*cos(d*x + c)^4 + 15*(A*a^12*b - 4*A*a^10*b^3 + 6*A*a^8*b^...
\[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]
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. 1031 vs. \(2 (490) = 980\).
Time = 0.40 (sec) , antiderivative size = 1031, normalized size of antiderivative = 2.01 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
-1/6*(6*(8*C*a^8*b + 40*A*a^6*b^3 - 8*C*a^6*b^3 - 84*A*a^4*b^5 + 7*C*a^4*b ^5 + 69*A*a^2*b^7 - 2*C*a^2*b^7 - 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*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 - 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 + 45*C*a^5*b^5*ta n(1/2*d*x + 1/2*c)^5 - 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 - 6*C*a^4*b^6*t an(1/2*d*x + 1/2*c)^5 + 213*A*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 + 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 + 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 - 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 + 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 - 284* A*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 + 12*C*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 + 7 2*A*b^10*tan(1/2*d*x + 1/2*c)^3 + 36*C*a^8*b^2*tan(1/2*d*x + 1/2*c) + 60*C *a^7*b^3*tan(1/2*d*x + 1/2*c) + 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c) - 6*C*a^ 6*b^4*tan(1/2*d*x + 1/2*c) + 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c) - 45*C*a^5 *b^5*tan(1/2*d*x + 1/2*c) - 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c) - 6*C*a^4*b^ 6*tan(1/2*d*x + 1/2*c) - 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c) + 15*C*a^3*...
Time = 37.29 (sec) , antiderivative size = 14266, normalized size of antiderivative = 27.81 \[ \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)*(A*a^8 + 20*A*b^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 + 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))/(a^5*(a + b) *(a - b)^3) + (tan(c/2 + (d*x)/2)^9*(A*a^8 + 20*A*b^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 + 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 ))/(a^5*(a + b)^3*(a - b)) + (2*tan(c/2 + (d*x)/2)^3*(120*A*b^9 - 6*A*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*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))/(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 - 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*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))/(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 + 18*C*a^2*b^8 - 62*C*a^4*b^6 + 110*C*a^6*b^4 - 36*C*a^8*b^2))/(3 *a^5*(a + b)^3*(a - b)^3))/(d*(tan(c/2 + (d*x)/2)^2*(9*a*b^2 + 3*a^2*b - a ^3 + 5*b^3) + tan(c/2 + (d*x)/2)^4*(6*a*b^2 - 6*a^2*b - 2*a^3 + 10*b^3) - tan(c/2 + (d*x)/2)^6*(6*a*b^2 + 6*a^2*b - 2*a^3 - 10*b^3) + 3*a*b^2 + 3*a^ 2*b + a^3 + b^3 - tan(c/2 + (d*x)/2)^10*(3*a*b^2 - 3*a^2*b + a^3 - b^3)...