3.4.42 \(\int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx\) [342]

3.4.42.1 Optimal result

Integrand size = 29, antiderivative size = 411 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=-\frac {(4 A b-a B) x}{a^5}+\frac {b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]

output

-(4*A*b-B*a)*x/a^5+b*(20*A*a^6*b-35*A*a^4*b^3+28*A*a^2*b^5-8*A*b^7-8*B*a^7 
+8*B*a^5*b^2-7*B*a^3*b^4+2*B*a*b^6)*arctanh((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+1/6*(6*A*a^6-65*A*a^4*b^2+68*A 
*a^2*b^4-24*A*b^6+26*B*a^5*b-17*B*a^3*b^3+6*B*a*b^5)*sin(d*x+c)/a^4/(a^2-b 
^2)^3/d+1/3*b*(A*b-B*a)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^3+1/6*b* 
(9*A*a^2*b-4*A*b^3-6*B*a^3+B*a*b^2)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec( 
d*x+c))^2+1/2*b*(12*A*a^4*b-11*A*a^2*b^3+4*A*b^5-6*B*a^5+2*B*a^3*b^2-B*a*b 
^4)*sin(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))
 

3.4.42.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1205\) vs. \(2(411)=822\).

Time = 6.64 (sec) , antiderivative size = 1205, normalized size of antiderivative = 2.93 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\frac {(b+a \cos (c+d x)) \sec ^3(c+d x) (A+B \sec (c+d x)) \left (\frac {24 b \left (-20 a^6 A b+35 a^4 A b^3-28 a^2 A b^5+8 A b^7+8 a^7 B-8 a^5 b^2 B+7 a^3 b^4 B-2 a b^6 B\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^3}{\left (a^2-b^2\right )^{7/2}}+\frac {-144 a^8 A b^2 c+336 a^6 A b^4 c-144 a^4 A b^6 c-144 a^2 A b^8 c+96 A b^{10} c+36 a^9 b B c-84 a^7 b^3 B c+36 a^5 b^5 B c+36 a^3 b^7 B c-24 a b^9 B c-144 a^8 A b^2 d x+336 a^6 A b^4 d x-144 a^4 A b^6 d x-144 a^2 A b^8 d x+96 A b^{10} d x+36 a^9 b B d x-84 a^7 b^3 B d x+36 a^5 b^5 B d x+36 a^3 b^7 B d x-24 a b^9 B d x+18 a \left (a^2-b^2\right )^3 \left (a^2+4 b^2\right ) (-4 A b+a B) (c+d x) \cos (c+d x)+36 a^2 b \left (a^2-b^2\right )^3 (-4 A b+a B) (c+d x) \cos (2 (c+d x))-24 a^9 A b c \cos (3 (c+d x))+72 a^7 A b^3 c \cos (3 (c+d x))-72 a^5 A b^5 c \cos (3 (c+d x))+24 a^3 A b^7 c \cos (3 (c+d x))+6 a^{10} B c \cos (3 (c+d x))-18 a^8 b^2 B c \cos (3 (c+d x))+18 a^6 b^4 B c \cos (3 (c+d x))-6 a^4 b^6 B c \cos (3 (c+d x))-24 a^9 A b d x \cos (3 (c+d x))+72 a^7 A b^3 d x \cos (3 (c+d x))-72 a^5 A b^5 d x \cos (3 (c+d x))+24 a^3 A b^7 d x \cos (3 (c+d x))+6 a^{10} B d x \cos (3 (c+d x))-18 a^8 b^2 B d x \cos (3 (c+d x))+18 a^6 b^4 B d x \cos (3 (c+d x))-6 a^4 b^6 B d x \cos (3 (c+d x))+18 a^9 A b \sin (c+d x)-90 a^7 A b^3 \sin (c+d x)-135 a^5 A b^5 \sin (c+d x)+228 a^3 A b^7 \sin (c+d x)-96 a A b^9 \sin (c+d x)+36 a^8 b^2 B \sin (c+d x)+72 a^6 b^4 B \sin (c+d x)-57 a^4 b^6 B \sin (c+d x)+24 a^2 b^8 B \sin (c+d x)+6 a^{10} A \sin (2 (c+d x))+18 a^8 A b^2 \sin (2 (c+d x))-300 a^6 A b^4 \sin (2 (c+d x))+336 a^4 A b^6 \sin (2 (c+d x))-120 a^2 A b^8 \sin (2 (c+d x))+120 a^7 b^3 B \sin (2 (c+d x))-90 a^5 b^5 B \sin (2 (c+d x))+30 a^3 b^7 B \sin (2 (c+d x))+18 a^9 A b \sin (3 (c+d x))-114 a^7 A b^3 \sin (3 (c+d x))+125 a^5 A b^5 \sin (3 (c+d x))-44 a^3 A b^7 \sin (3 (c+d x))+36 a^8 b^2 B \sin (3 (c+d x))-32 a^6 b^4 B \sin (3 (c+d x))+11 a^4 b^6 B \sin (3 (c+d x))+3 a^{10} A \sin (4 (c+d x))-9 a^8 A b^2 \sin (4 (c+d x))+9 a^6 A b^4 \sin (4 (c+d x))-3 a^4 A b^6 \sin (4 (c+d x))}{\left (a^2-b^2\right )^3}\right )}{24 a^5 d (B+A \cos (c+d x)) (a+b \sec (c+d x))^4} \]

input

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

output

((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x])*((24*b*(-20*a^6* 
A*b + 35*a^4*A*b^3 - 28*a^2*A*b^5 + 8*A*b^7 + 8*a^7*B - 8*a^5*b^2*B + 7*a^ 
3*b^4*B - 2*a*b^6*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]* 
(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (-144*a^8*A*b^2*c + 336*a^6*A* 
b^4*c - 144*a^4*A*b^6*c - 144*a^2*A*b^8*c + 96*A*b^10*c + 36*a^9*b*B*c - 8 
4*a^7*b^3*B*c + 36*a^5*b^5*B*c + 36*a^3*b^7*B*c - 24*a*b^9*B*c - 144*a^8*A 
*b^2*d*x + 336*a^6*A*b^4*d*x - 144*a^4*A*b^6*d*x - 144*a^2*A*b^8*d*x + 96* 
A*b^10*d*x + 36*a^9*b*B*d*x - 84*a^7*b^3*B*d*x + 36*a^5*b^5*B*d*x + 36*a^3 
*b^7*B*d*x - 24*a*b^9*B*d*x + 18*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(-4*A*b + a 
*B)*(c + d*x)*Cos[c + d*x] + 36*a^2*b*(a^2 - b^2)^3*(-4*A*b + a*B)*(c + d* 
x)*Cos[2*(c + d*x)] - 24*a^9*A*b*c*Cos[3*(c + d*x)] + 72*a^7*A*b^3*c*Cos[3 
*(c + d*x)] - 72*a^5*A*b^5*c*Cos[3*(c + d*x)] + 24*a^3*A*b^7*c*Cos[3*(c + 
d*x)] + 6*a^10*B*c*Cos[3*(c + d*x)] - 18*a^8*b^2*B*c*Cos[3*(c + d*x)] + 18 
*a^6*b^4*B*c*Cos[3*(c + d*x)] - 6*a^4*b^6*B*c*Cos[3*(c + d*x)] - 24*a^9*A* 
b*d*x*Cos[3*(c + d*x)] + 72*a^7*A*b^3*d*x*Cos[3*(c + d*x)] - 72*a^5*A*b^5* 
d*x*Cos[3*(c + d*x)] + 24*a^3*A*b^7*d*x*Cos[3*(c + d*x)] + 6*a^10*B*d*x*Co 
s[3*(c + d*x)] - 18*a^8*b^2*B*d*x*Cos[3*(c + d*x)] + 18*a^6*b^4*B*d*x*Cos[ 
3*(c + d*x)] - 6*a^4*b^6*B*d*x*Cos[3*(c + d*x)] + 18*a^9*A*b*Sin[c + d*x] 
- 90*a^7*A*b^3*Sin[c + d*x] - 135*a^5*A*b^5*Sin[c + d*x] + 228*a^3*A*b^7*S 
in[c + d*x] - 96*a*A*b^9*Sin[c + d*x] + 36*a^8*b^2*B*Sin[c + d*x] + 72*...
 

3.4.42.3 Rubi [A] (verified)

Time = 3.04 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.14, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {3042, 4518, 25, 3042, 4588, 25, 3042, 4588, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4518

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4588

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4588

\(\displaystyle \frac {\frac {\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int -\frac {\cos (c+d x) \left (6 A a^6+26 b B a^5-65 A b^2 a^4-17 b^3 B a^3+68 A b^4 a^2+6 b^5 B a-\left (-6 B a^5+18 A b a^4-8 b^2 B a^3-7 A b^3 a^2-b^4 B a+4 A b^5\right ) \sec (c+d x) a-24 A b^6+3 b \left (-6 B a^5+12 A b a^4+2 b^2 B a^3-11 A b^3 a^2-b^4 B a+4 A b^5\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {\cos (c+d x) \left (6 A a^6+26 b B a^5-65 A b^2 a^4-17 b^3 B a^3+68 A b^4 a^2+6 b^5 B a-\left (-6 B a^5+18 A b a^4-8 b^2 B a^3-7 A b^3 a^2-b^4 B a+4 A b^5\right ) \sec (c+d x) a-24 A b^6+3 b \left (-6 B a^5+12 A b a^4+2 b^2 B a^3-11 A b^3 a^2-b^4 B a+4 A b^5\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {6 A a^6+26 b B a^5-65 A b^2 a^4-17 b^3 B a^3+68 A b^4 a^2+6 b^5 B a-\left (-6 B a^5+18 A b a^4-8 b^2 B a^3-7 A b^3 a^2-b^4 B a+4 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a-24 A b^6+3 b \left (-6 B a^5+12 A b a^4+2 b^2 B a^3-11 A b^3 a^2-b^4 B a+4 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 4592

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {\int \frac {3 \left (2 \left (a^2-b^2\right )^3 (4 A b-a B)-a b \left (-6 B a^5+12 A b a^4+2 b^2 B a^3-11 A b^3 a^2-b^4 B a+4 A b^5\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \int \frac {2 \left (a^2-b^2\right )^3 (4 A b-a B)-a b \left (-6 B a^5+12 A b a^4+2 b^2 B a^3-11 A b^3 a^2-b^4 B a+4 A b^5\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \int \frac {2 \left (a^2-b^2\right )^3 (4 A b-a B)-a b \left (-6 B a^5+12 A b a^4+2 b^2 B a^3-11 A b^3 a^2-b^4 B a+4 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 4407

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3 (4 A b-a B)}{a}-\frac {b \left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3 (4 A b-a B)}{a}-\frac {b \left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 4318

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3 (4 A b-a B)}{a}-\frac {\left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3 (4 A b-a B)}{a}-\frac {\left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3 (4 A b-a B)}{a}-\frac {2 \left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {2 x \left (a^2-b^2\right )^3 (4 A b-a B)}{a}-\frac {2 b \left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\)

input

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

output

(b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + 
((b*(9*a^2*A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*Sin[c + d*x])/(2*a*(a^2 - b^ 
2)*d*(a + b*Sec[c + d*x])^2) + ((3*b*(12*a^4*A*b - 11*a^2*A*b^3 + 4*A*b^5 
- 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*S 
ec[c + d*x])) + ((-3*((2*(a^2 - b^2)^3*(4*A*b - a*B)*x)/a - (2*b*(20*a^6*A 
*b - 35*a^4*A*b^3 + 28*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 8*a^5*b^2*B - 7*a^3 
*b^4*B + 2*a*b^6*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/( 
a*Sqrt[a - b]*Sqrt[a + b]*d)))/a + ((6*a^6*A - 65*a^4*A*b^2 + 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)*Sin[c + d*x])/(a*d))/ 
(a*(a^2 - b^2)))/(2*a*(a^2 - b^2)))/(3*a*(a^2 - b^2))
 

3.4.42.3.1 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 221

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]
 

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 4318

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]
 

rule 4407

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]
 

rule 4518

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( 
a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[b*(A*b 
- a*B)*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*(a^2*(m + 1) - b^2*(m + n + 1)) 
 + a*b*B*n - a*(A*b - a*B)*(m + 1)*Csc[e + f*x] + b*(A*b - a*B)*(m + n + 2) 
*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A* 
b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] && IL 
tQ[n, 0])
 

rule 4588

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])
 

rule 4592

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]
 

3.4.42.4 Maple [A] (verified)

Time = 2.04 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.36

method	result	size
derivativedivides	\(\frac {-\frac {2 b \left (\frac {-\frac {\left (20 A \,a^{4} b +5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}-2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}-4 B \,a^{4} b +6 B \,a^{3} b^{2}+B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b -29 A \,a^{2} b^{3}+9 A \,b^{5}-18 B \,a^{5}+11 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b -5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}+2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}+4 B \,a^{4} b +6 B \,a^{3} b^{2}-B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b -35 A \,a^{4} b^{3}+28 A \,a^{2} b^{5}-8 A \,b^{7}-8 B \,a^{7}+8 B \,a^{5} b^{2}-7 B \,a^{3} b^{4}+2 B a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\left (4 A b -B a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}}{d}\)	\(558\)
default	\(\frac {-\frac {2 b \left (\frac {-\frac {\left (20 A \,a^{4} b +5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}-2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}-4 B \,a^{4} b +6 B \,a^{3} b^{2}+B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b -29 A \,a^{2} b^{3}+9 A \,b^{5}-18 B \,a^{5}+11 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b -5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}+2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}+4 B \,a^{4} b +6 B \,a^{3} b^{2}-B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b -35 A \,a^{4} b^{3}+28 A \,a^{2} b^{5}-8 A \,b^{7}-8 B \,a^{7}+8 B \,a^{5} b^{2}-7 B \,a^{3} b^{4}+2 B a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\left (4 A b -B a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}}{d}\)	\(558\)
risch	\(\text {Expression too large to display}\)	\(2154\)

input

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

output

1/d*(-2*b/a^5*((-1/2*(20*A*a^4*b+5*A*a^3*b^2-18*A*a^2*b^3-2*A*a*b^4+6*A*b^ 
5-12*B*a^5-4*B*a^4*b+6*B*a^3*b^2+B*a^2*b^3-2*B*a*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*(30*A*a^4*b-29*A*a^2*b^3+9*A*b^5- 
18*B*a^5+11*B*a^3*b^2-3*B*a*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*(20*A*a^4*b-5*A*a^3*b^2-18*A*a^2*b^3+2*A*a*b^4+6*A*b^5 
-12*B*a^5+4*B*a^4*b+6*B*a^3*b^2-B*a^2*b^3-2*B*a*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*(20*A*a^6*b-35*A*a^4*b^3+28*A*a^2*b^5-8*A*b^7-8*B*a^7+8 
*B*a^5*b^2-7*B*a^3*b^4+2*B*a*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+ 
b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2)))-2/a^5*(-A 
*a*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+(4*A*b-B*a)*arctan(tan(1/2* 
d*x+1/2*c))))
 

3.4.42.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1251 vs. \(2 (393) = 786\).

Time = 0.53 (sec) , antiderivative size = 2560, normalized size of antiderivative = 6.23 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]

input

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

output

[1/12*(12*(B*a^12 - 4*A*a^11*b - 4*B*a^10*b^2 + 16*A*a^9*b^3 + 6*B*a^8*b^4 
 - 24*A*a^7*b^5 - 4*B*a^6*b^6 + 16*A*a^5*b^7 + B*a^4*b^8 - 4*A*a^3*b^9)*d* 
x*cos(d*x + c)^3 + 36*(B*a^11*b - 4*A*a^10*b^2 - 4*B*a^9*b^3 + 16*A*a^8*b^ 
4 + 6*B*a^7*b^5 - 24*A*a^6*b^6 - 4*B*a^5*b^7 + 16*A*a^4*b^8 + B*a^3*b^9 - 
4*A*a^2*b^10)*d*x*cos(d*x + c)^2 + 36*(B*a^10*b^2 - 4*A*a^9*b^3 - 4*B*a^8* 
b^4 + 16*A*a^7*b^5 + 6*B*a^6*b^6 - 24*A*a^5*b^7 - 4*B*a^4*b^8 + 16*A*a^3*b 
^9 + B*a^2*b^10 - 4*A*a*b^11)*d*x*cos(d*x + c) + 12*(B*a^9*b^3 - 4*A*a^8*b 
^4 - 4*B*a^7*b^5 + 16*A*a^6*b^6 + 6*B*a^5*b^7 - 24*A*a^4*b^8 - 4*B*a^3*b^9 
 + 16*A*a^2*b^10 + B*a*b^11 - 4*A*b^12)*d*x - 3*(8*B*a^7*b^4 - 20*A*a^6*b^ 
5 - 8*B*a^5*b^6 + 35*A*a^4*b^7 + 7*B*a^3*b^8 - 28*A*a^2*b^9 - 2*B*a*b^10 + 
 8*A*b^11 + (8*B*a^10*b - 20*A*a^9*b^2 - 8*B*a^8*b^3 + 35*A*a^7*b^4 + 7*B* 
a^6*b^5 - 28*A*a^5*b^6 - 2*B*a^4*b^7 + 8*A*a^3*b^8)*cos(d*x + c)^3 + 3*(8* 
B*a^9*b^2 - 20*A*a^8*b^3 - 8*B*a^7*b^4 + 35*A*a^6*b^5 + 7*B*a^5*b^6 - 28*A 
*a^4*b^7 - 2*B*a^3*b^8 + 8*A*a^2*b^9)*cos(d*x + c)^2 + 3*(8*B*a^8*b^3 - 20 
*A*a^7*b^4 - 8*B*a^6*b^5 + 35*A*a^5*b^6 + 7*B*a^4*b^7 - 28*A*a^3*b^8 - 2*B 
*a^2*b^9 + 8*A*a*b^10)*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*(6*A*a^9*b^3 + 26*B*a^8*b^4 - 71*A*a^7*b^5 - 43*B*a^6*b^6 + 133*A* 
a^5*b^7 + 23*B*a^4*b^8 - 92*A*a^3*b^9 - 6*B*a^2*b^10 + 24*A*a*b^11 + 6*...
 

3.4.42.6 Sympy [F]

\[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \]

input

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

output

Integral((A + B*sec(c + d*x))*cos(c + d*x)/(a + b*sec(c + d*x))**4, x)
 

3.4.42.7 Maxima [F(-2)]

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

input

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

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

3.4.42.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 966 vs. \(2 (393) = 786\).

Time = 0.40 (sec) , antiderivative size = 966, normalized size of antiderivative = 2.35 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]

input

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

output

-1/3*(3*(8*B*a^7*b - 20*A*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)) + (36*B*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 + 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 + 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 - 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 - 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 + 116*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 - 236*A*a^4*b^5*ta 
n(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 + 36*B*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) + 60*B*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) - 6*B*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) - 45*B*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) - 6*B*a^3*b^6*tan(1/2*d*x + 1/2*c) + 24*A*a^2*b^7*tan(1/2*...
 

3.4.42.9 Mupad [B] (verification not implemented)

Time = 27.60 (sec) , antiderivative size = 7863, normalized size of antiderivative = 19.13 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]

input

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

output

(log(tan(c/2 + (d*x)/2) - 1i)*(4*A*b - B*a)*1i)/(a^5*d) - ((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 - 4*A*a*b^ 
6 + 2*A*a^6*b + 2*B*a*b^6))/((a + b)*(3*a^6*b - a^7 + a^4*b^3 - 3*a^5*b^2) 
) - (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 - 12*A*a*b^7 - 18*B*a*b^ 
7))/(3*(a + b)^2*(3*a^6*b - a^7 + a^4*b^3 - 3*a^5*b^2)) + (tan(c/2 + (d*x) 
/2)^7*(24*A*a^2*b^5 - 8*A*b^7 - 2*A*a^7 - 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 + 4*A*a*b 
^6 + 2*A*a^6*b + 2*B*a*b^6))/((a^4*b - a^5)*(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 - 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 + 12*A*a*b^7 - 18*B*a*b^7))/(3*(a^4*b - a^5)* 
(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))) - (log(tan(c/2 + (d*x)/2) + 1i)*(A*b*4i - B*a* 
1i))/(a^5*d) - (b*atan(((b*((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^ 
16 - 128*A^2*a*b^15 - 8*B^2*a^15*b - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^1...