\(\int \frac {(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx\) [1007]

Optimal result

Integrand size = 41, antiderivative size = 480 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=-\frac {\left (35 a^4 A b^4-28 a^2 A b^6+8 A b^8+8 a^7 b B-8 a^5 b^3 B+7 a^3 b^5 B-2 a b^7 B-2 a^8 C-a^6 b^2 (20 A+3 C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {(4 A b-a B) \text {arctanh}(\sin (c+d x))}{a^5 d}+\frac {\left (68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B+a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac {\left (4 A b^4+6 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (9 A+2 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {\left (11 a^2 A b^4-4 A b^6+6 a^5 b B-2 a^3 b^3 B+a b^5 B-2 a^6 C-3 a^4 b^2 (4 A+C)\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \] Output:

-(35*a^4*A*b^4-28*a^2*A*b^6+8*A*b^8+8*a^7*b*B-8*a^5*b^3*B+7*a^3*b^5*B-2*a* 
b^7*B-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-B*a)*arctanh(sin(d*x+c))/a 
^5/d+1/6*(68*a^2*A*b^4-24*A*b^6+26*B*a^5*b-17*B*a^3*b^3+6*B*a*b^5+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-a*(B*b- 
C*a))*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c))^3-1/6*(4*A*b^4+6*B*a^3*b-B 
*a*b^3-3*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+6*B*a^5*b-2*B*a^3*b^3+B*a*b^5-2*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 = 7.47 (sec) , antiderivative size = 709, normalized size of antiderivative = 1.48 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\frac {\cos (c+d x) \left (C+B \sec (c+d x)+A \sec ^2(c+d x)\right ) \left (\frac {48 \left (-35 a^4 A b^4+28 a^2 A b^6-8 A b^8-8 a^7 b B+8 a^5 b^3 B-7 a^3 b^5 B+2 a b^7 B+2 a^8 C+a^6 b^2 (20 A+3 C)\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right ) \cos (c+d x)}{\left (-a^2+b^2\right )^{7/2}}+48 (4 A b-a B) \cos (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 (-4 A b+a B) \cos (c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \left (24 a^9 A-36 a^7 A b^2-246 a^5 A b^4+318 a^3 A b^6-120 a A b^8+120 a^6 b^3 B-90 a^4 b^5 B+30 a^2 b^7 B-54 a^7 b^2 C-6 a^5 b^4 C-b \left (-28 a^2 A b^6+72 A b^8-144 a^7 b B+50 a^5 b^3 B+7 a^3 b^5 B-18 a b^7 B-5 a^4 b^4 (61 A-4 C)-72 a^8 (A-C)+a^6 b^2 (438 A+13 C)\right ) \cos (c+d x)+6 a b^2 \left (57 a^2 A b^4-20 A b^6+20 a^5 b B-15 a^3 b^3 B+5 a b^5 B+a^6 (6 A-9 C)-a^4 b^2 (53 A+C)\right ) \cos (2 (c+d x))+6 a^6 A b^3 \cos (3 (c+d x))-65 a^4 A b^5 \cos (3 (c+d x))+68 a^2 A b^7 \cos (3 (c+d x))-24 A b^9 \cos (3 (c+d x))+26 a^5 b^4 B \cos (3 (c+d x))-17 a^3 b^6 B \cos (3 (c+d x))+6 a b^8 B \cos (3 (c+d x))-11 a^6 b^3 C \cos (3 (c+d x))-4 a^4 b^5 C \cos (3 (c+d x))\right ) \sin (c+d x)}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}\right )}{24 a^5 d (2 A+C+2 B \cos (c+d x)+C \cos (2 (c+d x)))} \] Input:

Integrate[((A + B*Cos[c + d*x] + 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 + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((48*(-35*a^4*A*b^4 
+ 28*a^2*A*b^6 - 8*A*b^8 - 8*a^7*b*B + 8*a^5*b^3*B - 7*a^3*b^5*B + 2*a*b^7 
*B + 2*a^8*C + a^6*b^2*(20*A + 3*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sq 
rt[-a^2 + b^2]]*Cos[c + d*x])/(-a^2 + b^2)^(7/2) + 48*(4*A*b - a*B)*Cos[c 
+ d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*(-4*A*b + a*B)*Cos[c 
+ d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (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 + 120*a^6*b^3*B - 90*a 
^4*b^5*B + 30*a^2*b^7*B - 54*a^7*b^2*C - 6*a^5*b^4*C - b*(-28*a^2*A*b^6 + 
72*A*b^8 - 144*a^7*b*B + 50*a^5*b^3*B + 7*a^3*b^5*B - 18*a*b^7*B - 5*a^4*b 
^4*(61*A - 4*C) - 72*a^8*(A - C) + a^6*b^2*(438*A + 13*C))*Cos[c + d*x] + 
6*a*b^2*(57*a^2*A*b^4 - 20*A*b^6 + 20*a^5*b*B - 15*a^3*b^3*B + 5*a*b^5*B + 
 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)] + 26*a^5*b^4*B*Cos[3*(c + d*x)] - 17*a^3*b^ 
6*B*Cos[3*(c + d*x)] + 6*a*b^8*B*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)))/(24*a^5*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + 
d*x)]))
 

Rubi [A] (verified)

Time = 3.60 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.12, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.415, Rules used = {3042, 3534, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\int \frac {\left (-\left ((6 A-5 C) a^4\right )-8 b B a^3+23 A b^2 a^2+3 b^3 B a-2 \left (3 B a^3-b (6 A+5 C) a^2+2 b^2 B a+A b^3\right ) \cos (c+d x) a-12 A b^4+2 \left (-3 C a^4+6 b B a^3-b^2 (9 A+2 C) a^2-b^3 B a+4 A b^4\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\int \frac {-\left ((6 A-5 C) a^4\right )-8 b B a^3+23 A b^2 a^2+3 b^3 B a-2 \left (3 B a^3-b (6 A+5 C) a^2+2 b^2 B a+A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-12 A b^4+2 \left (-3 C a^4+6 b B a^3-b^2 (9 A+2 C) a^2-b^3 B a+4 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {\int -\frac {\left ((6 A-11 C) a^6+26 b B a^5-b^2 (65 A+4 C) a^4-17 b^3 B a^3+68 A b^4 a^2+6 b^5 B a-\left (-6 B a^5+b (18 A+11 C) a^4-8 b^2 B a^3-b^3 (7 A-4 C) a^2-b^4 B a+4 A b^5\right ) \cos (c+d x) a-24 A b^6-3 \left (-2 C a^6+6 b B a^5-3 b^2 (4 A+C) a^4-2 b^3 B a^3+11 A b^4 a^2+b^5 B a-4 A b^6\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {3 \tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {\left ((6 A-11 C) a^6+26 b B a^5-b^2 (65 A+4 C) a^4-17 b^3 B a^3+68 A b^4 a^2+6 b^5 B a-\left (-6 B a^5+b (18 A+11 C) a^4-8 b^2 B a^3-b^3 (7 A-4 C) a^2-b^4 B a+4 A b^5\right ) \cos (c+d x) a-24 A b^6-3 \left (-2 C a^6+6 b B a^5-3 b^2 (4 A+C) a^4-2 b^3 B a^3+11 A b^4 a^2+b^5 B a-4 A b^6\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int \frac {(6 A-11 C) a^6+26 b B a^5-b^2 (65 A+4 C) a^4-17 b^3 B a^3+68 A b^4 a^2+6 b^5 B a-\left (-6 B a^5+b (18 A+11 C) a^4-8 b^2 B a^3-b^3 (7 A-4 C) a^2-b^4 B a+4 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-24 A b^6-3 \left (-2 C a^6+6 b B a^5-3 b^2 (4 A+C) a^4-2 b^3 B a^3+11 A b^4 a^2+b^5 B a-4 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\int -\frac {3 \left (2 (4 A b-a B) \left (a^2-b^2\right )^3+a \left (-2 C a^6+6 b B a^5-3 b^2 (4 A+C) a^4-2 b^3 B a^3+11 A b^4 a^2+b^5 B a-4 A b^6\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {\tan (c+d x) \left (a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right )}{a d}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\tan (c+d x) \left (a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right )}{a d}-\frac {3 \int \frac {\left (2 (4 A b-a B) \left (a^2-b^2\right )^3+a \left (-2 C a^6+6 b B a^5-3 b^2 (4 A+C) a^4-2 b^3 B a^3+11 A b^4 a^2+b^5 B a-4 A b^6\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\tan (c+d x) \left (a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right )}{a d}-\frac {3 \int \frac {2 (4 A b-a B) \left (a^2-b^2\right )^3+a \left (-2 C a^6+6 b B a^5-3 b^2 (4 A+C) a^4-2 b^3 B a^3+11 A b^4 a^2+b^5 B a-4 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3480

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\tan (c+d x) \left (a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right )}{a d}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 (4 A b-a B) \int \sec (c+d x)dx}{a}+\frac {\left (-2 a^8 C+8 a^7 b B-a^6 b^2 (20 A+3 C)-8 a^5 b^3 B+35 a^4 A b^4+7 a^3 b^5 B-28 a^2 A b^6-2 a b^7 B+8 A b^8\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\tan (c+d x) \left (a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right )}{a d}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 (4 A b-a B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {\left (-2 a^8 C+8 a^7 b B-a^6 b^2 (20 A+3 C)-8 a^5 b^3 B+35 a^4 A b^4+7 a^3 b^5 B-28 a^2 A b^6-2 a b^7 B+8 A b^8\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\tan (c+d x) \left (a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right )}{a d}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 (4 A b-a B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {2 \left (-2 a^8 C+8 a^7 b B-a^6 b^2 (20 A+3 C)-8 a^5 b^3 B+35 a^4 A b^4+7 a^3 b^5 B-28 a^2 A b^6-2 a b^7 B+8 A b^8\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\frac {3 \tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\tan (c+d x) \left (a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right )}{a d}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 (4 A b-a B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}+\frac {2 \left (-2 a^8 C+8 a^7 b B-a^6 b^2 (20 A+3 C)-8 a^5 b^3 B+35 a^4 A b^4+7 a^3 b^5 B-28 a^2 A b^6-2 a b^7 B+8 A b^8\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}-\frac {\frac {\tan (c+d x) \left (-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\frac {3 \tan (c+d x) \left (-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\tan (c+d x) \left (a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right )}{a d}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 (4 A b-a B) \text {arctanh}(\sin (c+d x))}{a d}+\frac {2 \left (-2 a^8 C+8 a^7 b B-a^6 b^2 (20 A+3 C)-8 a^5 b^3 B+35 a^4 A b^4+7 a^3 b^5 B-28 a^2 A b^6-2 a b^7 B+8 A b^8\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\)

Input:

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

Output:

((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d 
*x])^3) - (((4*A*b^4 + 6*a^3*b*B - a*b^3*B - 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 + 6*a^5*b*B - 2*a^3*b^3*B + a*b^5*B - 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 + 8*a^7*b*B - 8*a^5*b^3*B + 7*a^3*b 
^5*B - 2*a*b^7*B - 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) + (2*(a^2 - b^2 
)^3*(4*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a*d)))/a + ((68*a^2*A*b^4 - 24*A 
*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B + 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 4257

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

Maple [A] (verified)

Time = 3.57 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.41

method	result	size
derivativedivides	\(\frac {-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-4 A b +B a \right ) \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}-12 B \,a^{5} b -4 B \,a^{4} b^{2}+6 B \,a^{3} b^{3}+B \,a^{2} b^{4}-2 B a \,b^{5}+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}-18 B \,a^{5} b +11 B \,a^{3} b^{3}-3 B a \,b^{5}+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}-12 B \,a^{5} b +4 B \,a^{4} b^{2}+6 B \,a^{3} b^{3}-B \,a^{2} b^{4}-2 B a \,b^{5}+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}-8 a^{7} b B +8 a^{5} b^{3} B -7 a^{3} b^{5} B +2 a \,b^{7} B +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 {\left (4 A b -B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{5}}}{d}\)	\(675\)
default	\(\frac {-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-4 A b +B a \right ) \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}-12 B \,a^{5} b -4 B \,a^{4} b^{2}+6 B \,a^{3} b^{3}+B \,a^{2} b^{4}-2 B a \,b^{5}+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}-18 B \,a^{5} b +11 B \,a^{3} b^{3}-3 B a \,b^{5}+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}-12 B \,a^{5} b +4 B \,a^{4} b^{2}+6 B \,a^{3} b^{3}-B \,a^{2} b^{4}-2 B a \,b^{5}+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}-8 a^{7} b B +8 a^{5} b^{3} B -7 a^{3} b^{5} B +2 a \,b^{7} B +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 {\left (4 A b -B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{5}}}{d}\)	\(675\)
risch	\(\text {Expression too large to display}\)	\(3248\)

Input:

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x,meth 
od=_RETURNVERBOSE)
 

Output:

1/d*(-A/a^4/(tan(1/2*d*x+1/2*c)+1)+1/a^5*(-4*A*b+B*a)*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 
-12*B*a^5*b-4*B*a^4*b^2+6*B*a^3*b^3+B*a^2*b^4-2*B*a*b^5+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^6-18*B*a^5*b+11*B*a^3*b^3-3*B*a*b^5+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^6-12*B*a^5*b+4*B*a 
^4*b^2+6*B*a^3*b^3-B*a^2*b^4-2*B*a*b^5+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^8-8*B*a^7*b+8*B*a^5*b^3-7*B*a^3*b^5+2*B*a*b^7+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-B*a)/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. 1764 vs. \(2 (457) = 914\).

Time = 162.44 (sec) , antiderivative size = 3598, normalized size of antiderivative = 7.50 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

integrate((A+B*cos(d*x+c)+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. 1255 vs. \(2 (457) = 914\).

Time = 0.22 (sec) , antiderivative size = 1255, normalized size of antiderivative = 2.61 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*cos(d*x+c)+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 - 8*B*a^7*b + 20*A*a^6*b^2 + 3*C*a^6*b^2 + 8*B*a^5*b^3 - 
35*A*a^4*b^4 - 7*B*a^3*b^5 + 28*A*a^2*b^6 + 2*B*a*b^7 - 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)) + (18*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 - 36*B* 
a^7*b^2*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 + 60*B*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 + 
 6*B*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 - 45*B*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 + 6*B*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 + 15*B*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 - 6*B*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 - 72*B*a^7*b^2*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 + 116*B*a^5*b^4*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 - 56*B*a^3*b^6*tan(1/2*d*x + 1/ 
2*c)^3 + 152*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 + 12*B*a*b^8*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/...
 

Mupad [B] (verification not implemented)

Time = 9.18 (sec) , antiderivative size = 15980, normalized size of antiderivative = 33.29 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

int((A + B*cos(c + d*x) + 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)^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 + B*a^2*b^5 + 6*B*a^3*b^4 - 4*B*a^4*b^3 - 12*B* 
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 - 2*B*a*b^6 + 
6*C*a^6*b))/(a^4*(a + b)^3*(a - b)) + (tan(c/2 + (d*x)/2)*(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 + B*a^2*b^5 
- 6*B*a^3*b^4 - 4*B*a^4*b^3 + 12*B*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 + 2*B*a*b^6 - 6*C*a^6*b))/(a^4*(a + b)*(a - b)^3) + ( 
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 + 3*B*a^2*b^6 + 59*B*a^3*b^5 
- 14*B*a^4*b^4 - 96*B*a^5*b^3 + 36*B*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*B*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 - 4 
7*A*a^3*b^5 + 273*A*a^4*b^4 + 60*A*a^5*b^3 - 72*A*a^6*b^2 - 3*B*a^2*b^6 + 
59*B*a^3*b^5 + 14*B*a^4*b^4 - 96*B*a^5*b^3 - 36*B*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*B*a*b^7 + 18*C*a^7*b))/(3*a^ 
4*(a + b)^3*(a - b)^2))/(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))) + (atan((((4*A*b - B*a)*(((4*A*b - B*a)*(( 
8*(4*B*a^24 + 4*C*a^24 + 16*A*a^10*b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 4678, normalized size of antiderivative = 9.75 \[ \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:

int((A+B*cos(d*x+c)+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 + 216*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 - 486*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 + 
 378*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 - 108*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 - 72* 
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 - 
 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**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 + 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)*a**5*b**6 - 342*sqrt(a**2 - b**2)*atan(...