\(\int \frac {(A+C \cos ^2(c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx\) [591]

Optimal result

Integrand size = 33, antiderivative size = 376 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {4 A b \text {arctanh}(\sin (c+d x))}{a^5 d}+\frac {\left (68 a^2 A b^4-24 A b^6+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2+a^2 C\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\left (4 A b^4-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \] Output:

-(35*a^4*A*b^4-28*a^2*A*b^6+8*A*b^8-2*a^8*C-a^6*b^2*(20*A+3*C))*arctan((a- 
b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(7/2)/(a+b)^(7/2)/d-4*A 
*b*arctanh(sin(d*x+c))/a^5/d+1/6*(68*a^2*A*b^4-24*A*b^6+a^6*(6*A-11*C)-a^4 
*b^2*(65*A+4*C))*tan(d*x+c)/a^4/(a^2-b^2)^3/d+1/3*(A*b^2+C*a^2)*tan(d*x+c) 
/a/(a^2-b^2)/d/(a+b*cos(d*x+c))^3-1/6*(4*A*b^4-3*a^4*C-a^2*b^2*(9*A+2*C))* 
tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^2-1/2*(11*a^2*A*b^4-4*A*b^6- 
2*a^6*C-3*a^4*b^2*(4*A+C))*tan(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*cos(d*x+c))
 

Mathematica [A] (warning: unable to verify)

Time = 5.75 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.37 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\frac {\cos (c+d x) \left (C+A \sec ^2(c+d x)\right ) \left (\frac {24 \left (-35 a^4 A b^4+28 a^2 A b^6-8 A b^8+2 a^8 C+a^6 b^2 (20 A+3 C)\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right ) \cos (c+d x)}{\left (-a^2+b^2\right )^{7/2}}+96 A b \cos (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-96 A b \cos (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a \left (24 a^9 A-36 a^7 A b^2-246 a^5 A b^4+318 a^3 A b^6-120 a A b^8-54 a^7 b^2 C-6 a^5 b^4 C-b \left (-28 a^2 A b^6+72 A b^8-5 a^4 b^4 (61 A-4 C)-72 a^8 (A-C)+a^6 b^2 (438 A+13 C)\right ) \cos (c+d x)+6 a b^2 \left (57 a^2 A b^4-20 A b^6+a^6 (6 A-9 C)-a^4 b^2 (53 A+C)\right ) \cos (2 (c+d x))+6 a^6 A b^3 \cos (3 (c+d x))-65 a^4 A b^5 \cos (3 (c+d x))+68 a^2 A b^7 \cos (3 (c+d x))-24 A b^9 \cos (3 (c+d x))-11 a^6 b^3 C \cos (3 (c+d x))-4 a^4 b^5 C \cos (3 (c+d x))\right ) \sin (c+d x)}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}\right )}{12 a^5 d (2 A+C+C \cos (2 (c+d x)))} \] Input:

Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x 
]
 

Output:

(Cos[c + d*x]*(C + A*Sec[c + d*x]^2)*((24*(-35*a^4*A*b^4 + 28*a^2*A*b^6 - 
8*A*b^8 + 2*a^8*C + a^6*b^2*(20*A + 3*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2 
])/Sqrt[-a^2 + b^2]]*Cos[c + d*x])/(-a^2 + b^2)^(7/2) + 96*A*b*Cos[c + d*x 
]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 96*A*b*Cos[c + d*x]*Log[Cos[( 
c + d*x)/2] + Sin[(c + d*x)/2]] + (a*(24*a^9*A - 36*a^7*A*b^2 - 246*a^5*A* 
b^4 + 318*a^3*A*b^6 - 120*a*A*b^8 - 54*a^7*b^2*C - 6*a^5*b^4*C - b*(-28*a^ 
2*A*b^6 + 72*A*b^8 - 5*a^4*b^4*(61*A - 4*C) - 72*a^8*(A - C) + a^6*b^2*(43 
8*A + 13*C))*Cos[c + d*x] + 6*a*b^2*(57*a^2*A*b^4 - 20*A*b^6 + a^6*(6*A - 
9*C) - a^4*b^2*(53*A + C))*Cos[2*(c + d*x)] + 6*a^6*A*b^3*Cos[3*(c + d*x)] 
 - 65*a^4*A*b^5*Cos[3*(c + d*x)] + 68*a^2*A*b^7*Cos[3*(c + d*x)] - 24*A*b^ 
9*Cos[3*(c + d*x)] - 11*a^6*b^3*C*Cos[3*(c + d*x)] - 4*a^4*b^5*C*Cos[3*(c 
+ d*x)])*Sin[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3)))/(12*a^5*d* 
(2*A + C + C*Cos[2*(c + d*x)]))
 

Rubi [A] (verified)

Time = 2.82 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.15, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 3535, 25, 3042, 3534, 3042, 3534, 25, 3042, 3534, 27, 3042, 3480, 3042, 3138, 218, 4257}

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.

\(\displaystyle \int \frac {\sec ^2(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\)

\(\Big \downarrow \) 3535

\(\displaystyle \frac {\int -\frac {\left (-\left ((3 A-C) a^2\right )+3 b (A+C) \cos (c+d x) a+4 A b^2-3 \left (C a^2+A b^2\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}+\frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\int \frac {\left (-\left ((3 A-C) a^2\right )+3 b (A+C) \cos (c+d x) a+4 A b^2-3 \left (C a^2+A b^2\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\int \frac {-\left ((3 A-C) a^2\right )+3 b (A+C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+4 A b^2-3 \left (C a^2+A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\int \frac {\left (-\left ((6 A-5 C) a^4\right )+23 A b^2 a^2-2 b \left (A b^2-a^2 (6 A+5 C)\right ) \cos (c+d x) a-12 A b^4+2 \left (-3 C a^4-b^2 (9 A+2 C) a^2+4 A b^4\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\int \frac {-\left ((6 A-5 C) a^4\right )+23 A b^2 a^2-2 b \left (A b^2-a^2 (6 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-12 A b^4+2 \left (-3 C a^4-b^2 (9 A+2 C) a^2+4 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {\int -\frac {\left ((6 A-11 C) a^6-b^2 (65 A+4 C) a^4+68 A b^4 a^2-b \left ((18 A+11 C) a^4-b^2 (7 A-4 C) a^2+4 A b^4\right ) \cos (c+d x) a-24 A b^6-3 \left (-2 C a^6-3 b^2 (4 A+C) a^4+11 A b^4 a^2-4 A b^6\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {\left ((6 A-11 C) a^6-b^2 (65 A+4 C) a^4+68 A b^4 a^2-b \left ((18 A+11 C) a^4-b^2 (7 A-4 C) a^2+4 A b^4\right ) \cos (c+d x) a-24 A b^6-3 \left (-2 C a^6-3 b^2 (4 A+C) a^4+11 A b^4 a^2-4 A b^6\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {(6 A-11 C) a^6-b^2 (65 A+4 C) a^4+68 A b^4 a^2-b \left ((18 A+11 C) a^4-b^2 (7 A-4 C) a^2+4 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-24 A b^6-3 \left (-2 C a^6-3 b^2 (4 A+C) a^4+11 A b^4 a^2-4 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\int -\frac {3 \left (8 A b \left (a^2-b^2\right )^3+a \left (-2 C a^6-3 b^2 (4 A+C) a^4+11 A b^4 a^2-4 A b^6\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \tan (c+d x)}{a d}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \int \frac {\left (8 A b \left (a^2-b^2\right )^3+a \left (-2 C a^6-3 b^2 (4 A+C) a^4+11 A b^4 a^2-4 A b^6\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \int \frac {8 A b \left (a^2-b^2\right )^3+a \left (-2 C a^6-3 b^2 (4 A+C) a^4+11 A b^4 a^2-4 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3480

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {8 A b \left (a^2-b^2\right )^3 \int \sec (c+d x)dx}{a}+\frac {\left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {8 A b \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {\left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {8 A b \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {2 \left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {8 A b \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {2 \left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\left (a^2 C+A b^2\right ) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\left (-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\frac {3 \left (-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\left (a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {8 A b \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{a d}+\frac {2 \left (-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\)

Input:

Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]
 

Output:

((A*b^2 + a^2*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) 
- (((4*A*b^4 - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Tan[c + d*x])/(2*a*(a^2 - b^ 
2)*d*(a + b*Cos[c + d*x])^2) + ((3*(11*a^2*A*b^4 - 4*A*b^6 - 2*a^6*C - 3*a 
^4*b^2*(4*A + C))*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) - ( 
(-3*((2*(35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 - 2*a^8*C - a^6*b^2*(20*A + 
 3*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]* 
Sqrt[a + b]*d) + (8*A*b*(a^2 - b^2)^3*ArcTanh[Sin[c + d*x]])/(a*d)))/a + ( 
(68*a^2*A*b^4 - 24*A*b^6 + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Tan[c 
+ d*x])/(a*d))/(a*(a^2 - b^2)))/(2*a*(a^2 - b^2)))/(3*a*(a^2 - b^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3138

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]
 

rule 3480

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b 
- a*B)/(b*c - a*d)   Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ 
(b*c - a*d)   Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f 
, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3534

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

rule 3535

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S 
in[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m 
+ 1)*(b*c - a*d)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin 
[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n 
+ 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d 
*(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 
- d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) || 
 !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 
 0])))
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 1.84 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.40

method	result	size
derivativedivides	\(\frac {-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4 A b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{5}}+\frac {\frac {2 \left (-\frac {\left (20 A \,a^{4} b^{2}+5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}-2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C +3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (30 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+9 A \,b^{6}+9 a^{6} C +a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b^{2}-5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}+2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C -3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (20 A \,a^{6} b^{2}-35 a^{4} A \,b^{4}+28 a^{2} A \,b^{6}-8 A \,b^{8}+2 a^{8} C +3 a^{6} b^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{5}}-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {4 A b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{5}}}{d}\)	\(528\)
default	\(\frac {-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4 A b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{5}}+\frac {\frac {2 \left (-\frac {\left (20 A \,a^{4} b^{2}+5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}-2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C +3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (30 A \,a^{4} b^{2}-29 a^{2} A \,b^{4}+9 A \,b^{6}+9 a^{6} C +a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b^{2}-5 A \,a^{3} b^{3}-18 a^{2} A \,b^{4}+2 A a \,b^{5}+6 A \,b^{6}+6 a^{6} C -3 C \,a^{5} b +2 a^{4} b^{2} C \right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (20 A \,a^{6} b^{2}-35 a^{4} A \,b^{4}+28 a^{2} A \,b^{6}-8 A \,b^{8}+2 a^{8} C +3 a^{6} b^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{5}}-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {4 A b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{5}}}{d}\)	\(528\)
risch	\(\text {Expression too large to display}\)	\(2029\)

Input:

int((A+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x,method=_RETURNVER 
BOSE)
 

Output:

1/d*(-A/a^4/(tan(1/2*d*x+1/2*c)-1)+4*A*b/a^5*ln(tan(1/2*d*x+1/2*c)-1)+2/a^ 
5*((-1/2*(20*A*a^4*b^2+5*A*a^3*b^3-18*A*a^2*b^4-2*A*a*b^5+6*A*b^6+6*C*a^6+ 
3*C*a^5*b+2*C*a^4*b^2)*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*(30*A*a^4*b^2-29*A*a^2*b^4+9*A*b^6+9*C*a^6+C*a^4*b^2)*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*(20*A*a^4*b^2-5*A*a^3*b 
^3-18*A*a^2*b^4+2*A*a*b^5+6*A*b^6+6*C*a^6-3*C*a^5*b+2*C*a^4*b^2)*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*(20*A*a^6*b^2-35*A*a^4*b^4+28*A*a^2*b^6-8*A 
*b^8+2*C*a^8+3*C*a^6*b^2)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2 
)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))-A/a^4/(tan(1/2*d*x 
+1/2*c)+1)-4*A*b/a^5*ln(tan(1/2*d*x+1/2*c)+1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1170 vs. \(2 (356) = 712\).

Time = 18.57 (sec) , antiderivative size = 2410, normalized size of antiderivative = 6.41 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x, algorithm= 
"fricas")
 

Output:

[-1/12*(3*((2*C*a^8*b^3 + (20*A + 3*C)*a^6*b^5 - 35*A*a^4*b^7 + 28*A*a^2*b 
^9 - 8*A*b^11)*cos(d*x + c)^4 + 3*(2*C*a^9*b^2 + (20*A + 3*C)*a^7*b^4 - 35 
*A*a^5*b^6 + 28*A*a^3*b^8 - 8*A*a*b^10)*cos(d*x + c)^3 + 3*(2*C*a^10*b + ( 
20*A + 3*C)*a^8*b^3 - 35*A*a^6*b^5 + 28*A*a^4*b^7 - 8*A*a^2*b^9)*cos(d*x + 
 c)^2 + (2*C*a^11 + (20*A + 3*C)*a^9*b^2 - 35*A*a^7*b^4 + 28*A*a^5*b^6 - 8 
*A*a^3*b^8)*cos(d*x + c))*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^ 
2 - b^2)*cos(d*x + c)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)*sin(d*x 
+ c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) + 24* 
((A*a^8*b^4 - 4*A*a^6*b^6 + 6*A*a^4*b^8 - 4*A*a^2*b^10 + A*b^12)*cos(d*x + 
 c)^4 + 3*(A*a^9*b^3 - 4*A*a^7*b^5 + 6*A*a^5*b^7 - 4*A*a^3*b^9 + A*a*b^11) 
*cos(d*x + c)^3 + 3*(A*a^10*b^2 - 4*A*a^8*b^4 + 6*A*a^6*b^6 - 4*A*a^4*b^8 
+ A*a^2*b^10)*cos(d*x + c)^2 + (A*a^11*b - 4*A*a^9*b^3 + 6*A*a^7*b^5 - 4*A 
*a^5*b^7 + A*a^3*b^9)*cos(d*x + c))*log(sin(d*x + c) + 1) - 24*((A*a^8*b^4 
 - 4*A*a^6*b^6 + 6*A*a^4*b^8 - 4*A*a^2*b^10 + A*b^12)*cos(d*x + c)^4 + 3*( 
A*a^9*b^3 - 4*A*a^7*b^5 + 6*A*a^5*b^7 - 4*A*a^3*b^9 + A*a*b^11)*cos(d*x + 
c)^3 + 3*(A*a^10*b^2 - 4*A*a^8*b^4 + 6*A*a^6*b^6 - 4*A*a^4*b^8 + A*a^2*b^1 
0)*cos(d*x + c)^2 + (A*a^11*b - 4*A*a^9*b^3 + 6*A*a^7*b^5 - 4*A*a^5*b^7 + 
A*a^3*b^9)*cos(d*x + c))*log(-sin(d*x + c) + 1) - 2*(6*A*a^12 - 24*A*a^10* 
b^2 + 36*A*a^8*b^4 - 24*A*a^6*b^6 + 6*A*a^4*b^8 + ((6*A - 11*C)*a^9*b^3 - 
(71*A - 7*C)*a^7*b^5 + (133*A + 4*C)*a^5*b^7 - 92*A*a^3*b^9 + 24*A*a*b^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Timed out} \] Input:

integrate((A+C*cos(d*x+c)**2)*sec(d*x+c)**2/(a+b*cos(d*x+c))**4,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x, algorithm= 
"maxima")
 

Output:

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*b^2-4*a^2>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 871 vs. \(2 (356) = 712\).

Time = 0.23 (sec) , antiderivative size = 871, normalized size of antiderivative = 2.32 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x, algorithm= 
"giac")
 

Output:

-1/3*(3*(2*C*a^8 + 20*A*a^6*b^2 + 3*C*a^6*b^2 - 35*A*a^4*b^4 + 28*A*a^2*b^ 
6 - 8*A*b^8)*(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^11 
- 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(a^2 - b^2)) + 12*A*b*log(abs(tan(1 
/2*d*x + 1/2*c) + 1))/a^5 - 12*A*b*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^5 
+ (18*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 - 27*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 
 + 60*A*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^ 
5 - 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*C*a^5*b^4*tan(1/2*d*x + 1/2*c 
)^5 - 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^4*b^5*tan(1/2*d*x + 1/2* 
c)^5 + 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 24*A*a^2*b^7*tan(1/2*d*x + 1 
/2*c)^5 - 42*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 + 18*A*b^9*tan(1/2*d*x + 1/2*c 
)^5 + 36*C*a^8*b*tan(1/2*d*x + 1/2*c)^3 + 120*A*a^6*b^3*tan(1/2*d*x + 1/2* 
c)^3 - 32*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 - 236*A*a^4*b^5*tan(1/2*d*x + 1 
/2*c)^3 - 4*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 + 152*A*a^2*b^7*tan(1/2*d*x + 
 1/2*c)^3 - 36*A*b^9*tan(1/2*d*x + 1/2*c)^3 + 18*C*a^8*b*tan(1/2*d*x + 1/2 
*c) + 27*C*a^7*b^2*tan(1/2*d*x + 1/2*c) + 60*A*a^6*b^3*tan(1/2*d*x + 1/2*c 
) + 6*C*a^6*b^3*tan(1/2*d*x + 1/2*c) + 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c) 
+ 3*C*a^5*b^4*tan(1/2*d*x + 1/2*c) - 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6 
*C*a^4*b^5*tan(1/2*d*x + 1/2*c) - 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c) - 24* 
A*a^2*b^7*tan(1/2*d*x + 1/2*c) + 42*A*a*b^8*tan(1/2*d*x + 1/2*c) + 18*A...
 

Mupad [B] (verification not implemented)

Time = 8.12 (sec) , antiderivative size = 10078, normalized size of antiderivative = 26.80 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

int((A + C*cos(c + d*x)^2)/(cos(c + d*x)^2*(a + b*cos(c + d*x))^4),x)
 

Output:

((tan(c/2 + (d*x)/2)^3*(18*A*a^8 + 72*A*b^8 - 236*A*a^2*b^6 + 47*A*a^3*b^5 
 + 273*A*a^4*b^4 - 60*A*a^5*b^3 - 72*A*a^6*b^2 + 10*C*a^4*b^4 - 7*C*a^5*b^ 
3 + 45*C*a^6*b^2 - 12*A*a*b^7 - 18*C*a^7*b))/(3*a^4*(a + b)^2*(a - b)^3) + 
 (tan(c/2 + (d*x)/2)^5*(18*A*a^8 + 72*A*b^8 - 236*A*a^2*b^6 - 47*A*a^3*b^5 
 + 273*A*a^4*b^4 + 60*A*a^5*b^3 - 72*A*a^6*b^2 + 10*C*a^4*b^4 + 7*C*a^5*b^ 
3 + 45*C*a^6*b^2 + 12*A*a*b^7 + 18*C*a^7*b))/(3*a^4*(a + b)^3*(a - b)^2) - 
 (tan(c/2 + (d*x)/2)*(8*A*b^7 - 2*A*a^7 - 24*A*a^2*b^5 - 11*A*a^3*b^4 + 26 
*A*a^4*b^3 + 6*A*a^5*b^2 + 2*C*a^4*b^3 - 3*C*a^5*b^2 + 4*A*a*b^6 - 2*A*a^6 
*b + 6*C*a^6*b))/(a^4*(a + b)*(a - b)^3) + (tan(c/2 + (d*x)/2)^7*(2*A*a^7 
+ 8*A*b^7 - 24*A*a^2*b^5 + 11*A*a^3*b^4 + 26*A*a^4*b^3 - 6*A*a^5*b^2 + 2*C 
*a^4*b^3 + 3*C*a^5*b^2 - 4*A*a*b^6 - 2*A*a^6*b + 6*C*a^6*b))/(a^4*(a + b)^ 
3*(a - b)))/(d*(3*a*b^2 + 3*a^2*b - tan(c/2 + (d*x)/2)^4*(6*a^2*b - 6*b^3) 
 - tan(c/2 + (d*x)/2)^2*(6*a*b^2 - 2*a^3 + 4*b^3) - tan(c/2 + (d*x)/2)^6*( 
2*a^3 - 6*a*b^2 + 4*b^3) + a^3 + b^3 - tan(c/2 + (d*x)/2)^8*(3*a*b^2 - 3*a 
^2*b + a^3 - b^3))) + (A*b*atan(((A*b*((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 
 + 4*C^2*a^16 - 128*A^2*a*b^15 - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 192 
0*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 
+ 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^ 
5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 + 9*C^2*a^12*b^4 
+ 12*C^2*a^14*b^2 - 48*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 98*A*C*a^10*b^6...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 4678, normalized size of antiderivative = 12.44 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:

int((A+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x)
 

Output:

(36*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( 
a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**8*b**2*c + 360*sqrt(a**2 - b 
**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos 
(c + d*x)*sin(c + d*x)**2*a**7*b**4 + 54*sqrt(a**2 - b**2)*atan((tan((c + 
d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d* 
x)**2*a**6*b**4*c - 630*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan(( 
c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**5*b**6 + 
 504*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt 
(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**3*b**8 - 144*sqrt(a**2 - b* 
*2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos( 
c + d*x)*sin(c + d*x)**2*a*b**10 - 12*sqrt(a**2 - b**2)*atan((tan((c + d*x 
)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**10*c - 120 
*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a** 
2 - b**2))*cos(c + d*x)*a**9*b**2 - 54*sqrt(a**2 - b**2)*atan((tan((c + d* 
x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**8*b**2*c 
- 150*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqr 
t(a**2 - b**2))*cos(c + d*x)*a**7*b**4 - 54*sqrt(a**2 - b**2)*atan((tan((c 
 + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**6*b* 
*4*c + 462*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b 
)/sqrt(a**2 - b**2))*cos(c + d*x)*a**5*b**6 - 456*sqrt(a**2 - b**2)*ata...