3.4.27 \(\int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx\) [327]

3.4.27.1 Optimal result

Integrand size = 31, antiderivative size = 346 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=-\frac {\left (2 a^2 A b+8 A b^3-a^3 B-6 a b^2 B\right ) x}{2 a^5}+\frac {2 b^3 \left (5 a^2 A b-4 A b^3-4 a^3 B+3 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {\left (2 a^4 A+7 a^2 A b^2-12 A b^4-6 a^3 b B+9 a b^3 B\right ) \sin (c+d x)}{3 a^4 \left (a^2-b^2\right ) d}-\frac {\left (2 a^2 A b-4 A b^3-a^3 B+3 a b^2 B\right ) \cos (c+d x) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 A-4 A b^2+3 a b B\right ) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d}+\frac {b (A b-a B) \cos ^2(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]

output

-1/2*(2*A*a^2*b+8*A*b^3-B*a^3-6*B*a*b^2)*x/a^5+2*b^3*(5*A*a^2*b-4*A*b^3-4* 
B*a^3+3*B*a*b^2)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/( 
a-b)^(3/2)/(a+b)^(3/2)/d+1/3*(2*A*a^4+7*A*a^2*b^2-12*A*b^4-6*B*a^3*b+9*B*a 
*b^3)*sin(d*x+c)/a^4/(a^2-b^2)/d-1/2*(2*A*a^2*b-4*A*b^3-B*a^3+3*B*a*b^2)*c 
os(d*x+c)*sin(d*x+c)/a^3/(a^2-b^2)/d+1/3*(A*a^2-4*A*b^2+3*B*a*b)*cos(d*x+c 
)^2*sin(d*x+c)/a^2/(a^2-b^2)/d+b*(A*b-B*a)*cos(d*x+c)^2*sin(d*x+c)/a/(a^2- 
b^2)/d/(a+b*sec(d*x+c))
 

3.4.27.2 Mathematica [A] (verified)

Time = 2.18 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.65 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\frac {6 \left (-2 a^2 A b-8 A b^3+a^3 B+6 a b^2 B\right ) (c+d x)+\frac {24 b^3 \left (-5 a^2 A b+4 A b^3+4 a^3 B-3 a b^2 B\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+3 a \left (3 a^2 A+12 A b^2-8 a b B\right ) \sin (c+d x)+\frac {12 a b^4 (-A b+a B) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))}+3 a^2 (-2 A b+a B) \sin (2 (c+d x))+a^3 A \sin (3 (c+d x))}{12 a^5 d} \]

input

Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]
 

output

(6*(-2*a^2*A*b - 8*A*b^3 + a^3*B + 6*a*b^2*B)*(c + d*x) + (24*b^3*(-5*a^2* 
A*b + 4*A*b^3 + 4*a^3*B - 3*a*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/S 
qrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 3*a*(3*a^2*A + 12*A*b^2 - 8*a*b*B)*Si 
n[c + d*x] + (12*a*b^4*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(b + 
a*Cos[c + d*x])) + 3*a^2*(-2*A*b + a*B)*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c 
+ d*x)])/(12*a^5*d)
 

3.4.27.3 Rubi [A] (verified)

Time = 2.49 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {3042, 4518, 25, 3042, 4592, 3042, 4592, 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.

\(\displaystyle \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4518

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4592

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4592

\(\displaystyle \frac {\frac {\left (a^2 A+3 a b B-4 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 \left (a^3 (-B)+2 a^2 A b+3 a b^2 B-4 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (-3 b \left (-B a^3+2 A b a^2+3 b^2 B a-4 A b^3\right ) \sec ^2(c+d x)-a \left (-3 B a^3+2 A b a^2-3 b^2 B a+4 A b^3\right ) \sec (c+d x)+2 \left (2 A a^4-6 b B a^3+7 A b^2 a^2+9 b^3 B a-12 A b^4\right )\right )}{a+b \sec (c+d x)}dx}{2 a}}{3 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\left (a^2 A+3 a b B-4 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 \left (a^3 (-B)+2 a^2 A b+3 a b^2 B-4 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\int \frac {-3 b \left (-B a^3+2 A b a^2+3 b^2 B a-4 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a \left (-3 B a^3+2 A b a^2-3 b^2 B a+4 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left (2 A a^4-6 b B a^3+7 A b^2 a^2+9 b^3 B a-12 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a}}{3 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\)

\(\Big \downarrow \) 4592

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\left (a^2 A+3 a b B-4 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 \left (a^3 (-B)+2 a^2 A b+3 a b^2 B-4 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (2 a^4 A-6 a^3 b B+7 a^2 A b^2+9 a b^3 B-12 A b^4\right ) \sin (c+d x)}{a d}-\frac {3 \int \frac {\left (a^2-b^2\right ) \left (-B a^3+2 A b a^2-6 b^2 B a+8 A b^3\right )+a b \left (-B a^3+2 A b a^2+3 b^2 B a-4 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\)

\(\Big \downarrow \) 4407

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4318

\(\displaystyle \frac {\frac {\left (a^2 A+3 a b B-4 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 \left (a^3 (-B)+2 a^2 A b+3 a b^2 B-4 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (2 a^4 A-6 a^3 b B+7 a^2 A b^2+9 a b^3 B-12 A b^4\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {x \left (a^2-b^2\right ) \left (a^3 (-B)+2 a^2 A b-6 a b^2 B+8 A b^3\right )}{a}-\frac {2 b^2 \left (-4 a^3 B+5 a^2 A b+3 a b^2 B-4 A b^3\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a}\right )}{a}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\left (a^2 A+3 a b B-4 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 \left (a^3 (-B)+2 a^2 A b+3 a b^2 B-4 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (2 a^4 A-6 a^3 b B+7 a^2 A b^2+9 a b^3 B-12 A b^4\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {x \left (a^2-b^2\right ) \left (a^3 (-B)+2 a^2 A b-6 a b^2 B+8 A b^3\right )}{a}-\frac {2 b^2 \left (-4 a^3 B+5 a^2 A b+3 a b^2 B-4 A b^3\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a}\right )}{a}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\frac {\left (a^2 A+3 a b B-4 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 \left (a^3 (-B)+2 a^2 A b+3 a b^2 B-4 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (2 a^4 A-6 a^3 b B+7 a^2 A b^2+9 a b^3 B-12 A b^4\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {x \left (a^2-b^2\right ) \left (a^3 (-B)+2 a^2 A b-6 a b^2 B+8 A b^3\right )}{a}-\frac {4 b^2 \left (-4 a^3 B+5 a^2 A b+3 a b^2 B-4 A b^3\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {\frac {\left (a^2 A+3 a b B-4 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {\frac {3 \left (a^3 (-B)+2 a^2 A b+3 a b^2 B-4 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (2 a^4 A-6 a^3 b B+7 a^2 A b^2+9 a b^3 B-12 A b^4\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {x \left (a^2-b^2\right ) \left (a^3 (-B)+2 a^2 A b-6 a b^2 B+8 A b^3\right )}{a}-\frac {4 b^3 \left (-4 a^3 B+5 a^2 A b+3 a b^2 B-4 A b^3\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{2 a}}{3 a}}{a \left (a^2-b^2\right )}\)

input

Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]
 

output

(b*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c 
+ d*x])) + (((a^2*A - 4*A*b^2 + 3*a*b*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a 
*d) - ((3*(2*a^2*A*b - 4*A*b^3 - a^3*B + 3*a*b^2*B)*Cos[c + d*x]*Sin[c + d 
*x])/(2*a*d) - ((-3*(((a^2 - b^2)*(2*a^2*A*b + 8*A*b^3 - a^3*B - 6*a*b^2*B 
)*x)/a - (4*b^3*(5*a^2*A*b - 4*A*b^3 - 4*a^3*B + 3*a*b^2*B)*ArcTanh[(Sqrt[ 
a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)))/a + 
 (2*(2*a^4*A + 7*a^2*A*b^2 - 12*A*b^4 - 6*a^3*b*B + 9*a*b^3*B)*Sin[c + d*x 
])/(a*d))/(2*a))/(3*a))/(a*(a^2 - b^2))
 

3.4.27.3.1 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 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]
 

3.4.27.4 Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {-\frac {2 b^{3} \left (-\frac {a b \left (A b -B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (5 A \,a^{2} b -4 A \,b^{3}-4 B \,a^{3}+3 B a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {2 \left (\frac {\left (-a^{3} A -A \,a^{2} b -3 A a \,b^{2}+\frac {1}{2} B \,a^{3}+2 B \,a^{2} b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {2}{3} a^{3} A -6 A a \,b^{2}+4 B \,a^{2} b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{3} A -3 A a \,b^{2}+2 B \,a^{2} b +A \,a^{2} b -\frac {1}{2} B \,a^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {\left (2 A \,a^{2} b +8 A \,b^{3}-B \,a^{3}-6 B a \,b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{5}}}{d}\)	\(346\)
default	\(\frac {-\frac {2 b^{3} \left (-\frac {a b \left (A b -B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (5 A \,a^{2} b -4 A \,b^{3}-4 B \,a^{3}+3 B a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {2 \left (\frac {\left (-a^{3} A -A \,a^{2} b -3 A a \,b^{2}+\frac {1}{2} B \,a^{3}+2 B \,a^{2} b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {2}{3} a^{3} A -6 A a \,b^{2}+4 B \,a^{2} b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{3} A -3 A a \,b^{2}+2 B \,a^{2} b +A \,a^{2} b -\frac {1}{2} B \,a^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {\left (2 A \,a^{2} b +8 A \,b^{3}-B \,a^{3}-6 B a \,b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{5}}}{d}\)	\(346\)
risch	\(-\frac {x A b}{a^{3}}-\frac {4 x A \,b^{3}}{a^{5}}+\frac {x B}{2 a^{2}}+\frac {3 x B \,b^{2}}{a^{4}}+\frac {2 i b^{4} \left (-A b +B a \right ) \left (b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{a^{5} \left (a^{2}-b^{2}\right ) d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} B b}{a^{3} d}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )} A b}{4 a^{3} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} A b}{4 a^{3} d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B}{8 a^{2} d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A \,b^{2}}{2 a^{4} d}-\frac {3 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{2} d}+\frac {i {\mathrm e}^{i \left (d x +c \right )} B b}{a^{3} d}+\frac {3 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{2} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B}{8 a^{2} d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{2}}{2 a^{4} d}+\frac {5 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}-\frac {4 b^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{5}}-\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}+\frac {3 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{4}}-\frac {5 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}+\frac {4 b^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{5}}+\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}-\frac {3 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{4}}+\frac {A \sin \left (3 d x +3 c \right )}{12 a^{2} d}\)	\(1030\)

input

int(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^2,x,method=_RETURNVERBO 
SE)
 

output

1/d*(-2*b^3/a^5*(-a*b*(A*b-B*a)/(a^2-b^2)*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)-(5*A*a^2*b-4*A*b^3-4*B*a^3+3*B*a*b^ 
2)/(a-b)/(a+b)/((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^3*A-A*a^2*b-3*A*a*b^2+1/2*B*a^3+2*B*a^2*b)*tan 
(1/2*d*x+1/2*c)^5+(-2/3*a^3*A-6*A*a*b^2+4*B*a^2*b)*tan(1/2*d*x+1/2*c)^3+(- 
a^3*A-3*A*a*b^2+2*B*a^2*b+A*a^2*b-1/2*B*a^3)*tan(1/2*d*x+1/2*c))/(1+tan(1/ 
2*d*x+1/2*c)^2)^3+1/2*(2*A*a^2*b+8*A*b^3-B*a^3-6*B*a*b^2)*arctan(tan(1/2*d 
*x+1/2*c))))
 

3.4.27.5 Fricas [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 1167, normalized size of antiderivative = 3.37 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]

input

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^2,x, algorithm="f 
ricas")
 

output

[1/6*(3*(B*a^8 - 2*A*a^7*b + 4*B*a^6*b^2 - 4*A*a^5*b^3 - 11*B*a^4*b^4 + 14 
*A*a^3*b^5 + 6*B*a^2*b^6 - 8*A*a*b^7)*d*x*cos(d*x + c) + 3*(B*a^7*b - 2*A* 
a^6*b^2 + 4*B*a^5*b^3 - 4*A*a^4*b^4 - 11*B*a^3*b^5 + 14*A*a^2*b^6 + 6*B*a* 
b^7 - 8*A*b^8)*d*x + 3*(4*B*a^3*b^4 - 5*A*a^2*b^5 - 3*B*a*b^6 + 4*A*b^7 + 
(4*B*a^4*b^3 - 5*A*a^3*b^4 - 3*B*a^2*b^5 + 4*A*a*b^6)*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)) + (4*A*a^7*b - 12*B*a^6*b^2 + 10*A*a^5* 
b^3 + 30*B*a^4*b^4 - 38*A*a^3*b^5 - 18*B*a^2*b^6 + 24*A*a*b^7 + 2*(A*a^8 - 
 2*A*a^6*b^2 + A*a^4*b^4)*cos(d*x + c)^3 + (3*B*a^8 - 4*A*a^7*b - 6*B*a^6* 
b^2 + 8*A*a^5*b^3 + 3*B*a^4*b^4 - 4*A*a^3*b^5)*cos(d*x + c)^2 + (4*A*a^8 - 
 9*B*a^7*b + 4*A*a^6*b^2 + 18*B*a^5*b^3 - 20*A*a^4*b^4 - 9*B*a^3*b^5 + 12* 
A*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^10 - 2*a^8*b^2 + a^6*b^4)*d*cos 
(d*x + c) + (a^9*b - 2*a^7*b^3 + a^5*b^5)*d), 1/6*(3*(B*a^8 - 2*A*a^7*b + 
4*B*a^6*b^2 - 4*A*a^5*b^3 - 11*B*a^4*b^4 + 14*A*a^3*b^5 + 6*B*a^2*b^6 - 8* 
A*a*b^7)*d*x*cos(d*x + c) + 3*(B*a^7*b - 2*A*a^6*b^2 + 4*B*a^5*b^3 - 4*A*a 
^4*b^4 - 11*B*a^3*b^5 + 14*A*a^2*b^6 + 6*B*a*b^7 - 8*A*b^8)*d*x - 6*(4*B*a 
^3*b^4 - 5*A*a^2*b^5 - 3*B*a*b^6 + 4*A*b^7 + (4*B*a^4*b^3 - 5*A*a^3*b^4 - 
3*B*a^2*b^5 + 4*A*a*b^6)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 
+ b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + (4*A*a^7*b - ...
 

3.4.27.6 Sympy [F]

\[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]

input

integrate(cos(d*x+c)**3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))**2,x)
 

output

Integral((A + B*sec(c + d*x))*cos(c + d*x)**3/(a + b*sec(c + d*x))**2, x)
 

3.4.27.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]

input

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^2,x, algorithm="m 
axima")
 

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
 

3.4.27.8 Giac [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.37 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=-\frac {\frac {12 \, {\left (4 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4} - 3 \, B a b^{5} + 4 \, A b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{7} - a^{5} b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {12 \, {\left (B a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}} - \frac {3 \, {\left (B a^{3} - 2 \, A a^{2} b + 6 \, B a b^{2} - 8 \, A b^{3}\right )} {\left (d x + c\right )}}{a^{5}} - \frac {2 \, {\left (6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{4}}}{6 \, d} \]

input

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^2,x, algorithm="g 
iac")
 

output

-1/6*(12*(4*B*a^3*b^3 - 5*A*a^2*b^4 - 3*B*a*b^5 + 4*A*b^6)*(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*t 
an(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^7 - a^5*b^2)*sqrt(-a^2 + b^2)) 
 + 12*(B*a*b^4*tan(1/2*d*x + 1/2*c) - A*b^5*tan(1/2*d*x + 1/2*c))/((a^6 - 
a^4*b^2)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)) - 
3*(B*a^3 - 2*A*a^2*b + 6*B*a*b^2 - 8*A*b^3)*(d*x + c)/a^5 - 2*(6*A*a^2*tan 
(1/2*d*x + 1/2*c)^5 - 3*B*a^2*tan(1/2*d*x + 1/2*c)^5 + 6*A*a*b*tan(1/2*d*x 
 + 1/2*c)^5 - 12*B*a*b*tan(1/2*d*x + 1/2*c)^5 + 18*A*b^2*tan(1/2*d*x + 1/2 
*c)^5 + 4*A*a^2*tan(1/2*d*x + 1/2*c)^3 - 24*B*a*b*tan(1/2*d*x + 1/2*c)^3 + 
 36*A*b^2*tan(1/2*d*x + 1/2*c)^3 + 6*A*a^2*tan(1/2*d*x + 1/2*c) + 3*B*a^2* 
tan(1/2*d*x + 1/2*c) - 6*A*a*b*tan(1/2*d*x + 1/2*c) - 12*B*a*b*tan(1/2*d*x 
 + 1/2*c) + 18*A*b^2*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3 
*a^4))/d
 

3.4.27.9 Mupad [B] (verification not implemented)

Time = 25.59 (sec) , antiderivative size = 7763, normalized size of antiderivative = 22.44 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]

input

int((cos(c + d*x)^3*(A + B/cos(c + d*x)))/(a + b/cos(c + d*x))^2,x)
 

output

- ((tan(c/2 + (d*x)/2)^7*(2*A*a^5 + 8*A*b^5 - B*a^5 - 6*A*a^2*b^3 + 2*A*a^ 
3*b^2 + 3*B*a^2*b^3 + 5*B*a^3*b^2 - 4*A*a*b^4 - 6*B*a*b^4 - 3*B*a^4*b))/(a 
^4*(a + b)*(a - b)) - (tan(c/2 + (d*x)/2)*(2*A*a^5 - 8*A*b^5 + B*a^5 + 6*A 
*a^2*b^3 + 2*A*a^3*b^2 + 3*B*a^2*b^3 - 5*B*a^3*b^2 - 4*A*a*b^4 + 6*B*a*b^4 
 - 3*B*a^4*b))/(a^4*(a + b)*(a - b)) + (tan(c/2 + (d*x)/2)^3*(2*A*a^5 + 72 
*A*b^5 + 3*B*a^5 - 38*A*a^2*b^3 - 14*A*a^3*b^2 - 9*B*a^2*b^3 + 33*B*a^3*b^ 
2 + 12*A*a*b^4 - 16*A*a^4*b - 54*B*a*b^4 + 9*B*a^4*b))/(3*a^4*(a + b)*(a - 
 b)) - (tan(c/2 + (d*x)/2)^5*(2*A*a^5 - 72*A*b^5 - 3*B*a^5 + 38*A*a^2*b^3 
- 14*A*a^3*b^2 - 9*B*a^2*b^3 - 33*B*a^3*b^2 + 12*A*a*b^4 + 16*A*a^4*b + 54 
*B*a*b^4 + 9*B*a^4*b))/(3*a^4*(a + b)*(a - b)))/(d*(a + b - tan(c/2 + (d*x 
)/2)^8*(a - b) + tan(c/2 + (d*x)/2)^2*(2*a + 4*b) - tan(c/2 + (d*x)/2)^6*( 
2*a - 4*b) + 6*b*tan(c/2 + (d*x)/2)^4)) - (atan(((((((8*(2*B*a^18 + 16*A*a 
^10*b^8 - 8*A*a^11*b^7 - 36*A*a^12*b^6 + 16*A*a^13*b^5 + 20*A*a^14*b^4 - 4 
*A*a^15*b^3 - 12*B*a^11*b^7 + 6*B*a^12*b^6 + 28*B*a^13*b^5 - 14*B*a^14*b^4 
 - 16*B*a^15*b^3 + 6*B*a^16*b^2 - 4*A*a^17*b))/(a^14*b + a^15 - a^12*b^3 - 
 a^13*b^2) - (8*tan(c/2 + (d*x)/2)*(A*b^3*4i - (B*a^3*1i)/2 + A*a^2*b*1i - 
 B*a*b^2*3i)*(8*a^15*b - 8*a^10*b^6 + 8*a^11*b^5 + 16*a^12*b^4 - 16*a^13*b 
^3 - 8*a^14*b^2))/(a^5*(a^10*b + a^11 - a^8*b^3 - a^9*b^2)))*(A*b^3*4i - ( 
B*a^3*1i)/2 + A*a^2*b*1i - B*a*b^2*3i))/a^5 + (8*tan(c/2 + (d*x)/2)*(128*A 
^2*b^12 + B^2*a^12 - 128*A^2*a*b^11 - 2*B^2*a^11*b - 192*A^2*a^2*b^10 +...