\(\int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx\) [117]

Optimal result

Integrand size = 31, antiderivative size = 256 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {(44 A-21 B) x}{2 a^4}+\frac {8 (227 A-108 B) \sin (c+d x)}{35 a^4 d}-\frac {(44 A-21 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(178 A-87 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(44 A-21 B) \cos ^2(c+d x) \sin (c+d x)}{3 a^4 d (1+\sec (c+d x))}-\frac {(A-B) \cos ^2(c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(16 A-9 B) \cos ^2(c+d x) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {8 (227 A-108 B) \sin ^3(c+d x)}{105 a^4 d} \] Output:

-1/2*(44*A-21*B)*x/a^4+8/35*(227*A-108*B)*sin(d*x+c)/a^4/d-1/2*(44*A-21*B) 
*cos(d*x+c)*sin(d*x+c)/a^4/d-1/105*(178*A-87*B)*cos(d*x+c)^2*sin(d*x+c)/a^ 
4/d/(1+sec(d*x+c))^2-1/3*(44*A-21*B)*cos(d*x+c)^2*sin(d*x+c)/a^4/d/(1+sec( 
d*x+c))-1/7*(A-B)*cos(d*x+c)^2*sin(d*x+c)/d/(a+a*sec(d*x+c))^4-1/35*(16*A- 
9*B)*cos(d*x+c)^2*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^3-8/105*(227*A-108*B)*si 
n(d*x+c)^3/a^4/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(611\) vs. \(2(256)=512\).

Time = 5.84 (sec) , antiderivative size = 611, normalized size of antiderivative = 2.39 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (-14700 (44 A-21 B) d x \cos \left (\frac {d x}{2}\right )-14700 (44 A-21 B) d x \cos \left (c+\frac {d x}{2}\right )-388080 A d x \cos \left (c+\frac {3 d x}{2}\right )+185220 B d x \cos \left (c+\frac {3 d x}{2}\right )-388080 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+185220 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-129360 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+61740 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-129360 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+61740 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-18480 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+8820 B d x \cos \left (3 c+\frac {7 d x}{2}\right )-18480 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+8820 B d x \cos \left (4 c+\frac {7 d x}{2}\right )+1010660 A \sin \left (\frac {d x}{2}\right )-539490 B \sin \left (\frac {d x}{2}\right )-687260 A \sin \left (c+\frac {d x}{2}\right )+386190 B \sin \left (c+\frac {d x}{2}\right )+814107 A \sin \left (c+\frac {3 d x}{2}\right )-422478 B \sin \left (c+\frac {3 d x}{2}\right )-204645 A \sin \left (2 c+\frac {3 d x}{2}\right )+132930 B \sin \left (2 c+\frac {3 d x}{2}\right )+357609 A \sin \left (2 c+\frac {5 d x}{2}\right )-181461 B \sin \left (2 c+\frac {5 d x}{2}\right )+18025 A \sin \left (3 c+\frac {5 d x}{2}\right )+3675 B \sin \left (3 c+\frac {5 d x}{2}\right )+72522 A \sin \left (3 c+\frac {7 d x}{2}\right )-36003 B \sin \left (3 c+\frac {7 d x}{2}\right )+24010 A \sin \left (4 c+\frac {7 d x}{2}\right )-9555 B \sin \left (4 c+\frac {7 d x}{2}\right )+2310 A \sin \left (4 c+\frac {9 d x}{2}\right )-945 B \sin \left (4 c+\frac {9 d x}{2}\right )+2310 A \sin \left (5 c+\frac {9 d x}{2}\right )-945 B \sin \left (5 c+\frac {9 d x}{2}\right )-175 A \sin \left (5 c+\frac {11 d x}{2}\right )+105 B \sin \left (5 c+\frac {11 d x}{2}\right )-175 A \sin \left (6 c+\frac {11 d x}{2}\right )+105 B \sin \left (6 c+\frac {11 d x}{2}\right )+35 A \sin \left (6 c+\frac {13 d x}{2}\right )+35 A \sin \left (7 c+\frac {13 d x}{2}\right )\right )}{6720 a^4 d (1+\cos (c+d x))^4} \] Input:

Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]
 

Output:

(Cos[(c + d*x)/2]*Sec[c/2]*(-14700*(44*A - 21*B)*d*x*Cos[(d*x)/2] - 14700* 
(44*A - 21*B)*d*x*Cos[c + (d*x)/2] - 388080*A*d*x*Cos[c + (3*d*x)/2] + 185 
220*B*d*x*Cos[c + (3*d*x)/2] - 388080*A*d*x*Cos[2*c + (3*d*x)/2] + 185220* 
B*d*x*Cos[2*c + (3*d*x)/2] - 129360*A*d*x*Cos[2*c + (5*d*x)/2] + 61740*B*d 
*x*Cos[2*c + (5*d*x)/2] - 129360*A*d*x*Cos[3*c + (5*d*x)/2] + 61740*B*d*x* 
Cos[3*c + (5*d*x)/2] - 18480*A*d*x*Cos[3*c + (7*d*x)/2] + 8820*B*d*x*Cos[3 
*c + (7*d*x)/2] - 18480*A*d*x*Cos[4*c + (7*d*x)/2] + 8820*B*d*x*Cos[4*c + 
(7*d*x)/2] + 1010660*A*Sin[(d*x)/2] - 539490*B*Sin[(d*x)/2] - 687260*A*Sin 
[c + (d*x)/2] + 386190*B*Sin[c + (d*x)/2] + 814107*A*Sin[c + (3*d*x)/2] - 
422478*B*Sin[c + (3*d*x)/2] - 204645*A*Sin[2*c + (3*d*x)/2] + 132930*B*Sin 
[2*c + (3*d*x)/2] + 357609*A*Sin[2*c + (5*d*x)/2] - 181461*B*Sin[2*c + (5* 
d*x)/2] + 18025*A*Sin[3*c + (5*d*x)/2] + 3675*B*Sin[3*c + (5*d*x)/2] + 725 
22*A*Sin[3*c + (7*d*x)/2] - 36003*B*Sin[3*c + (7*d*x)/2] + 24010*A*Sin[4*c 
 + (7*d*x)/2] - 9555*B*Sin[4*c + (7*d*x)/2] + 2310*A*Sin[4*c + (9*d*x)/2] 
- 945*B*Sin[4*c + (9*d*x)/2] + 2310*A*Sin[5*c + (9*d*x)/2] - 945*B*Sin[5*c 
 + (9*d*x)/2] - 175*A*Sin[5*c + (11*d*x)/2] + 105*B*Sin[5*c + (11*d*x)/2] 
- 175*A*Sin[6*c + (11*d*x)/2] + 105*B*Sin[6*c + (11*d*x)/2] + 35*A*Sin[6*c 
 + (13*d*x)/2] + 35*A*Sin[7*c + (13*d*x)/2]))/(6720*a^4*d*(1 + Cos[c + d*x 
])^4)
 

Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.516, Rules used = {3042, 4508, 3042, 4508, 3042, 4508, 3042, 4508, 27, 3042, 4274, 3042, 3113, 2009, 3115, 24}

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.

\(\displaystyle \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a \sec (c+d x)+a)^4} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4508

\(\displaystyle \frac {\int \frac {\cos ^3(c+d x) (a (10 A-3 B)-6 a (A-B) \sec (c+d x))}{(\sec (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a (10 A-3 B)-6 a (A-B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 4508

\(\displaystyle \frac {\frac {\int \frac {\cos ^3(c+d x) \left (14 a^2 (7 A-3 B)-5 a^2 (16 A-9 B) \sec (c+d x)\right )}{(\sec (c+d x) a+a)^2}dx}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {14 a^2 (7 A-3 B)-5 a^2 (16 A-9 B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 4508

\(\displaystyle \frac {\frac {\frac {\int \frac {\cos ^3(c+d x) \left (9 a^3 (92 A-43 B)-4 a^3 (178 A-87 B) \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {9 a^3 (92 A-43 B)-4 a^3 (178 A-87 B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 4508

\(\displaystyle \frac {\frac {\frac {\frac {\int 3 \cos ^3(c+d x) \left (8 a^4 (227 A-108 B)-35 a^4 (44 A-21 B) \sec (c+d x)\right )dx}{a^2}-\frac {35 a^3 (44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {3 \int \cos ^3(c+d x) \left (8 a^4 (227 A-108 B)-35 a^4 (44 A-21 B) \sec (c+d x)\right )dx}{a^2}-\frac {35 a^3 (44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {3 \int \frac {8 a^4 (227 A-108 B)-35 a^4 (44 A-21 B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx}{a^2}-\frac {35 a^3 (44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 4274

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (8 a^4 (227 A-108 B) \int \cos ^3(c+d x)dx-35 a^4 (44 A-21 B) \int \cos ^2(c+d x)dx\right )}{a^2}-\frac {35 a^3 (44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (8 a^4 (227 A-108 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx-35 a^4 (44 A-21 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )}{a^2}-\frac {35 a^3 (44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 3113

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (-\frac {8 a^4 (227 A-108 B) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}-35 a^4 (44 A-21 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )}{a^2}-\frac {35 a^3 (44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (-35 a^4 (44 A-21 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {8 a^4 (227 A-108 B) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}-\frac {35 a^3 (44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (-35 a^4 (44 A-21 B) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {8 a^4 (227 A-108 B) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )}{a^2}-\frac {35 a^3 (44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\frac {\frac {\frac {3 \left (-\frac {8 a^4 (227 A-108 B) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}-35 a^4 (44 A-21 B) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2}-\frac {35 a^3 (44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}-\frac {a (16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}-\frac {(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}\)

Input:

Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]
 

Output:

-1/7*((A - B)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^4) + (- 
1/5*(a*(16*A - 9*B)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^3 
) + (-1/3*((178*A - 87*B)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(1 + Sec[c + d*x 
])^2) + ((-35*a^3*(44*A - 21*B)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec 
[c + d*x])) + (3*(-35*a^4*(44*A - 21*B)*(x/2 + (Cos[c + d*x]*Sin[c + d*x]) 
/(2*d)) - (8*a^4*(227*A - 108*B)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d))/a 
^2)/(3*a^2))/(5*a^2))/(7*a^2)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 4274

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[a   Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d   In 
t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
 

rule 4508

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

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.55

method	result	size
parallelrisch	\(\frac {2170 \left (\left (\frac {238253 A}{2170}-\frac {56256 B}{1085}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {25218 A}{1085}-\frac {23619 B}{2170}\right ) \cos \left (3 d x +3 c \right )+\left (A -\frac {12 B}{31}\right ) \cos \left (4 d x +4 c \right )+\left (-\frac {2 A}{31}+\frac {3 B}{62}\right ) \cos \left (5 d x +5 c \right )+\frac {A \cos \left (6 d x +6 c \right )}{62}+\left (\frac {271492 A}{1085}-\frac {128643 B}{1085}\right ) \cos \left (d x +c \right )+\frac {176171 A}{1085}-\frac {83484 B}{1085}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-591360 \left (A -\frac {21 B}{44}\right ) d x}{26880 a^{4} d}\)	\(140\)
derivativedivides	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\frac {59 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B +209 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -\frac {16 \left (\left (-13 A +\frac {9 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {62 A}{3}+8 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-9 A +\frac {7 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-8 \left (44 A -21 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(210\)
default	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\frac {59 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B +209 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -\frac {16 \left (\left (-13 A +\frac {9 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {62 A}{3}+8 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-9 A +\frac {7 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-8 \left (44 A -21 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(210\)
norman	\(\frac {-\frac {\left (44 A -21 B \right ) x}{2 a}-\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{56 a d}+\frac {\left (31 A -24 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{140 a d}-\frac {3 \left (44 A -21 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}-\frac {3 \left (44 A -21 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}-\frac {\left (44 A -21 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a}+\frac {7 \left (67 A -32 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 a d}+\frac {\left (353 A -167 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {3 \left (1297 A -613 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{40 a d}+\frac {\left (1369 A -676 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{70 a d}-\frac {\left (1417 A -843 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{840 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} a^{3}}\)	\(284\)
risch	\(-\frac {22 A x}{a^{4}}+\frac {21 x B}{2 a^{4}}-\frac {i A \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 a^{4} d}+\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{2 a^{4} d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B}{8 a^{4} d}-\frac {43 i A \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B}{a^{4} d}+\frac {43 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{4} d}-\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B}{a^{4} d}-\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{2 a^{4} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B}{8 a^{4} d}+\frac {i A \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 a^{4} d}+\frac {2 i \left (3675 A \,{\mathrm e}^{6 i \left (d x +c \right )}-2100 B \,{\mathrm e}^{6 i \left (d x +c \right )}+19845 A \,{\mathrm e}^{5 i \left (d x +c \right )}-11025 B \,{\mathrm e}^{5 i \left (d x +c \right )}+46550 A \,{\mathrm e}^{4 i \left (d x +c \right )}-25515 B \,{\mathrm e}^{4 i \left (d x +c \right )}+59570 A \,{\mathrm e}^{3 i \left (d x +c \right )}-32340 B \,{\mathrm e}^{3 i \left (d x +c \right )}+43827 A \,{\mathrm e}^{2 i \left (d x +c \right )}-23688 B \,{\mathrm e}^{2 i \left (d x +c \right )}+17549 \,{\mathrm e}^{i \left (d x +c \right )} A -9471 B \,{\mathrm e}^{i \left (d x +c \right )}+3032 A -1653 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\)	\(379\)

Input:

int(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x,method=_RETURNVERBO 
SE)
 

Output:

1/26880*(2170*((238253/2170*A-56256/1085*B)*cos(2*d*x+2*c)+(25218/1085*A-2 
3619/2170*B)*cos(3*d*x+3*c)+(A-12/31*B)*cos(4*d*x+4*c)+(-2/31*A+3/62*B)*co 
s(5*d*x+5*c)+1/62*A*cos(6*d*x+6*c)+(271492/1085*A-128643/1085*B)*cos(d*x+c 
)+176171/1085*A-83484/1085*B)*tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^6-5913 
60*(A-21/44*B)*d*x)/a^4/d
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {105 \, {\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (44 \, A - 21 \, B\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (44 \, A - 21 \, B\right )} d x - {\left (70 \, A \cos \left (d x + c\right )^{6} - 35 \, {\left (4 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{5} + 140 \, {\left (7 \, A - 3 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (3196 \, A - 1509 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (7184 \, A - 3411 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (24436 \, A - 11619 \, B\right )} \cos \left (d x + c\right ) + 7264 \, A - 3456 \, B\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \] Input:

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="f 
ricas")
 

Output:

-1/210*(105*(44*A - 21*B)*d*x*cos(d*x + c)^4 + 420*(44*A - 21*B)*d*x*cos(d 
*x + c)^3 + 630*(44*A - 21*B)*d*x*cos(d*x + c)^2 + 420*(44*A - 21*B)*d*x*c 
os(d*x + c) + 105*(44*A - 21*B)*d*x - (70*A*cos(d*x + c)^6 - 35*(4*A - 3*B 
)*cos(d*x + c)^5 + 140*(7*A - 3*B)*cos(d*x + c)^4 + 4*(3196*A - 1509*B)*co 
s(d*x + c)^3 + 4*(7184*A - 3411*B)*cos(d*x + c)^2 + (24436*A - 11619*B)*co 
s(d*x + c) + 7264*A - 3456*B)*sin(d*x + c))/(a^4*d*cos(d*x + c)^4 + 4*a^4* 
d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c) + a^4*d)
 

Sympy [F]

\[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \] Input:

integrate(cos(d*x+c)**3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))**4,x)
 

Output:

(Integral(A*cos(c + d*x)**3/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c 
 + d*x)**2 + 4*sec(c + d*x) + 1), x) + Integral(B*cos(c + d*x)**3*sec(c + 
d*x)/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + 
d*x) + 1), x))/a**4
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.77 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {A {\left (\frac {560 \, {\left (\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {62 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {39 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{4} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {21945 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2065 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {231 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {36960 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - 3 \, B {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {5880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \] Input:

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="m 
axima")
 

Output:

1/840*(A*(560*(27*sin(d*x + c)/(cos(d*x + c) + 1) + 62*sin(d*x + c)^3/(cos 
(d*x + c) + 1)^3 + 39*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a^4 + 3*a^4*si 
n(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^4*sin(d*x + c)^4/(cos(d*x + c) + 1 
)^4 + a^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (21945*sin(d*x + c)/(cos( 
d*x + c) + 1) - 2065*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 231*sin(d*x + c 
)^5/(cos(d*x + c) + 1)^5 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 3 
6960*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4) - 3*B*(280*(7*sin(d*x + 
c)/(cos(d*x + c) + 1) + 9*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^4 + 2*a^ 
4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^4*sin(d*x + c)^4/(cos(d*x + c) + 
 1)^4) + (3885*sin(d*x + c)/(cos(d*x + c) + 1) - 455*sin(d*x + c)^3/(cos(d 
*x + c) + 1)^3 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin(d*x + c)^7 
/(cos(d*x + c) + 1)^7)/a^4 - 5880*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/ 
a^4))/d
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {\frac {420 \, {\left (d x + c\right )} {\left (44 \, A - 21 \, B\right )}}{a^{4}} - \frac {280 \, {\left (78 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 124 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 54 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 231 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 189 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2065 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1365 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21945 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11655 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \] Input:

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="g 
iac")
 

Output:

-1/840*(420*(d*x + c)*(44*A - 21*B)/a^4 - 280*(78*A*tan(1/2*d*x + 1/2*c)^5 
 - 27*B*tan(1/2*d*x + 1/2*c)^5 + 124*A*tan(1/2*d*x + 1/2*c)^3 - 48*B*tan(1 
/2*d*x + 1/2*c)^3 + 54*A*tan(1/2*d*x + 1/2*c) - 21*B*tan(1/2*d*x + 1/2*c)) 
/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*a^4) + (15*A*a^24*tan(1/2*d*x + 1/2*c)^7 
- 15*B*a^24*tan(1/2*d*x + 1/2*c)^7 - 231*A*a^24*tan(1/2*d*x + 1/2*c)^5 + 1 
89*B*a^24*tan(1/2*d*x + 1/2*c)^5 + 2065*A*a^24*tan(1/2*d*x + 1/2*c)^3 - 13 
65*B*a^24*tan(1/2*d*x + 1/2*c)^3 - 21945*A*a^24*tan(1/2*d*x + 1/2*c) + 116 
55*B*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d
 

Mupad [B] (verification not implemented)

Time = 11.10 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.17 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\left (26\,A-9\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {124\,A}{3}-16\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (18\,A-7\,B\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}-\frac {x\,\left (44\,A-21\,B\right )}{2\,a^4}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,\left (A-B\right )}{12\,a^4}+\frac {7\,A-5\,B}{6\,a^4}+\frac {21\,A-9\,B}{24\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {A-B}{10\,a^4}+\frac {7\,A-5\,B}{40\,a^4}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A-B\right )}{2\,a^4}+\frac {5\,\left (7\,A-5\,B\right )}{4\,a^4}+\frac {21\,A-9\,B}{2\,a^4}+\frac {35\,A-5\,B}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4\,d} \] Input:

int((cos(c + d*x)^3*(A + B/cos(c + d*x)))/(a + a/cos(c + d*x))^4,x)
 

Output:

(tan(c/2 + (d*x)/2)^5*(26*A - 9*B) + tan(c/2 + (d*x)/2)^3*((124*A)/3 - 16* 
B) + tan(c/2 + (d*x)/2)*(18*A - 7*B))/(d*(3*a^4*tan(c/2 + (d*x)/2)^2 + 3*a 
^4*tan(c/2 + (d*x)/2)^4 + a^4*tan(c/2 + (d*x)/2)^6 + a^4)) - (x*(44*A - 21 
*B))/(2*a^4) - (tan(c/2 + (d*x)/2)^3*((5*(A - B))/(12*a^4) + (7*A - 5*B)/( 
6*a^4) + (21*A - 9*B)/(24*a^4)))/d + (tan(c/2 + (d*x)/2)^5*((A - B)/(10*a^ 
4) + (7*A - 5*B)/(40*a^4)))/d + (tan(c/2 + (d*x)/2)*((5*(A - B))/(2*a^4) + 
 (5*(7*A - 5*B))/(4*a^4) + (21*A - 9*B)/(2*a^4) + (35*A - 5*B)/(8*a^4)))/d 
 - (tan(c/2 + (d*x)/2)^7*(A - B))/(56*a^4*d)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.36 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {-70 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a +1330 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a -525 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b -4620 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a d x +2205 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b d x -18402 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a +8658 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b +18480 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a d x -8820 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b d x -30 \cos \left (d x +c \right ) a +30 \cos \left (d x +c \right ) b +210 \sin \left (d x +c \right )^{6} a -105 \sin \left (d x +c \right )^{6} b +11174 \sin \left (d x +c \right )^{4} a -5301 \sin \left (d x +c \right )^{4} b -13860 \sin \left (d x +c \right )^{3} a d x +6615 \sin \left (d x +c \right )^{3} b d x -18678 \sin \left (d x +c \right )^{2} a +8892 \sin \left (d x +c \right )^{2} b +18480 \sin \left (d x +c \right ) a d x -8820 \sin \left (d x +c \right ) b d x +30 a -30 b}{210 \sin \left (d x +c \right ) a^{4} d \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-4 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )^{2}-4\right )} \] Input:

int(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x)
 

Output:

( - 70*cos(c + d*x)*sin(c + d*x)**6*a + 1330*cos(c + d*x)*sin(c + d*x)**4* 
a - 525*cos(c + d*x)*sin(c + d*x)**4*b - 4620*cos(c + d*x)*sin(c + d*x)**3 
*a*d*x + 2205*cos(c + d*x)*sin(c + d*x)**3*b*d*x - 18402*cos(c + d*x)*sin( 
c + d*x)**2*a + 8658*cos(c + d*x)*sin(c + d*x)**2*b + 18480*cos(c + d*x)*s 
in(c + d*x)*a*d*x - 8820*cos(c + d*x)*sin(c + d*x)*b*d*x - 30*cos(c + d*x) 
*a + 30*cos(c + d*x)*b + 210*sin(c + d*x)**6*a - 105*sin(c + d*x)**6*b + 1 
1174*sin(c + d*x)**4*a - 5301*sin(c + d*x)**4*b - 13860*sin(c + d*x)**3*a* 
d*x + 6615*sin(c + d*x)**3*b*d*x - 18678*sin(c + d*x)**2*a + 8892*sin(c + 
d*x)**2*b + 18480*sin(c + d*x)*a*d*x - 8820*sin(c + d*x)*b*d*x + 30*a - 30 
*b)/(210*sin(c + d*x)*a**4*d*(cos(c + d*x)*sin(c + d*x)**2 - 4*cos(c + d*x 
) + 3*sin(c + d*x)**2 - 4))