\(\int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx\) [272]

Optimal result

Integrand size = 31, antiderivative size = 402 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=-\frac {b^2 \left (20 a^4 A b-29 a^2 A b^3+12 A b^5-12 a^5 B+15 a^3 b^2 B-6 a b^4 B\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {\left (a^2 A+12 A b^2-6 a b B\right ) \text {arctanh}(\sin (c+d x))}{2 a^5 d}-\frac {\left (6 a^4 A b-21 a^2 A b^3+12 A b^5-2 a^5 B+11 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^4 A-10 a^2 A b^2+6 A b^4+6 a^3 b B-3 a b^3 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {b \left (7 a^2 A b-4 A b^3-5 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \] Output:

-b^2*(20*A*a^4*b-29*A*a^2*b^3+12*A*b^5-12*B*a^5+15*B*a^3*b^2-6*B*a*b^4)*ar 
ctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(5/2)/(a+b)^(5/ 
2)/d+1/2*(A*a^2+12*A*b^2-6*B*a*b)*arctanh(sin(d*x+c))/a^5/d-1/2*(6*A*a^4*b 
-21*A*a^2*b^3+12*A*b^5-2*B*a^5+11*B*a^3*b^2-6*B*a*b^4)*tan(d*x+c)/a^4/(a^2 
-b^2)^2/d+1/2*(A*a^4-10*A*a^2*b^2+6*A*b^4+6*B*a^3*b-3*B*a*b^3)*sec(d*x+c)* 
tan(d*x+c)/a^3/(a^2-b^2)^2/d+1/2*b*(A*b-B*a)*sec(d*x+c)*tan(d*x+c)/a/(a^2- 
b^2)/d/(a+b*cos(d*x+c))^2+1/2*b*(7*A*a^2*b-4*A*b^3-5*B*a^3+2*B*a*b^2)*sec( 
d*x+c)*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))
 

Mathematica [A] (verified)

Time = 5.14 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.26 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {16 b^2 \left (20 a^4 A b-29 a^2 A b^3+12 A b^5-12 a^5 B+15 a^3 b^2 B-6 a b^4 B\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}-8 \left (a^2 A+12 A b^2-6 a b B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 \left (a^2 A+12 A b^2-6 a b B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \left (4 a^7 A-30 a^5 A b^2+68 a^3 A b^4-36 a A b^6+8 a^6 b B-32 a^4 b^3 B+18 a^2 b^5 B+\left (-16 a^6 A b+14 a^4 A b^3+47 a^2 A b^5-36 A b^7+8 a^7 B-10 a^5 b^2 B-25 a^3 b^4 B+18 a b^6 B\right ) \cos (c+d x)+2 a b \left (-11 a^4 A b+32 a^2 A b^3-18 A b^5+4 a^5 B-16 a^3 b^2 B+9 a b^4 B\right ) \cos (2 (c+d x))-6 a^4 A b^3 \cos (3 (c+d x))+21 a^2 A b^5 \cos (3 (c+d x))-12 A b^7 \cos (3 (c+d x))+2 a^5 b^2 B \cos (3 (c+d x))-11 a^3 b^4 B \cos (3 (c+d x))+6 a b^6 B \cos (3 (c+d x))\right ) \sec (c+d x) \tan (c+d x)}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}}{16 a^5 d} \] Input:

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

Output:

((16*b^2*(20*a^4*A*b - 29*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 15*a^3*b^2*B - 
 6*a*b^4*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + 
b^2)^(5/2) - 8*(a^2*A + 12*A*b^2 - 6*a*b*B)*Log[Cos[(c + d*x)/2] - Sin[(c 
+ d*x)/2]] + 8*(a^2*A + 12*A*b^2 - 6*a*b*B)*Log[Cos[(c + d*x)/2] + Sin[(c 
+ d*x)/2]] + (2*a*(4*a^7*A - 30*a^5*A*b^2 + 68*a^3*A*b^4 - 36*a*A*b^6 + 8* 
a^6*b*B - 32*a^4*b^3*B + 18*a^2*b^5*B + (-16*a^6*A*b + 14*a^4*A*b^3 + 47*a 
^2*A*b^5 - 36*A*b^7 + 8*a^7*B - 10*a^5*b^2*B - 25*a^3*b^4*B + 18*a*b^6*B)* 
Cos[c + d*x] + 2*a*b*(-11*a^4*A*b + 32*a^2*A*b^3 - 18*A*b^5 + 4*a^5*B - 16 
*a^3*b^2*B + 9*a*b^4*B)*Cos[2*(c + d*x)] - 6*a^4*A*b^3*Cos[3*(c + d*x)] + 
21*a^2*A*b^5*Cos[3*(c + d*x)] - 12*A*b^7*Cos[3*(c + d*x)] + 2*a^5*b^2*B*Co 
s[3*(c + d*x)] - 11*a^3*b^4*B*Cos[3*(c + d*x)] + 6*a*b^6*B*Cos[3*(c + d*x) 
])*Sec[c + d*x]*Tan[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(16* 
a^5*d)
 

Rubi [A] (verified)

Time = 2.77 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.06, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.516, Rules used = {3042, 3479, 3042, 3534, 3042, 3534, 27, 3042, 3534, 25, 3042, 3480, 3042, 3138, 218, 4257}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3479

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {\int \frac {\left (2 b \left (-5 B a^3+7 A b a^2+2 b^2 B a-4 A b^3\right ) \cos ^2(c+d x)-a \left (-2 B a^3+4 A b a^2-b^2 B a-A b^3\right ) \cos (c+d x)+2 \left (A a^4+6 b B a^3-10 A b^2 a^2-3 b^3 B a+6 A b^4\right )\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {2 b \left (-5 B a^3+7 A b a^2+2 b^2 B a-4 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-a \left (-2 B a^3+4 A b a^2-b^2 B a-A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (A a^4+6 b B a^3-10 A b^2 a^2-3 b^3 B a+6 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {\frac {\int -\frac {2 \left (-2 B a^5+6 A b a^4+11 b^2 B a^3-21 A b^3 a^2-6 b^4 B a-\left (A a^4-4 b B a^3+4 A b^2 a^2+b^3 B a-2 A b^4\right ) \cos (c+d x) a+12 A b^5-b \left (A a^4+6 b B a^3-10 A b^2 a^2-3 b^3 B a+6 A b^4\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{2 a}+\frac {\left (a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\left (a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\int \frac {\left (-2 B a^5+6 A b a^4+11 b^2 B a^3-21 A b^3 a^2-6 b^4 B a-\left (A a^4-4 b B a^3+4 A b^2 a^2+b^3 B a-2 A b^4\right ) \cos (c+d x) a+12 A b^5-b \left (A a^4+6 b B a^3-10 A b^2 a^2-3 b^3 B a+6 A b^4\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\int \frac {-2 B a^5+6 A b a^4+11 b^2 B a^3-21 A b^3 a^2-6 b^4 B a-\left (A a^4-4 b B a^3+4 A b^2 a^2+b^3 B a-2 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+12 A b^5-b \left (A a^4+6 b B a^3-10 A b^2 a^2-3 b^3 B a+6 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {\frac {\left (a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\int -\frac {\left (\left (A a^2-6 b B a+12 A b^2\right ) \left (a^2-b^2\right )^2+a b \left (A a^4+6 b B a^3-10 A b^2 a^2-3 b^3 B a+6 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {\left (-2 a^5 B+6 a^4 A b+11 a^3 b^2 B-21 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \tan (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\left (a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-2 a^5 B+6 a^4 A b+11 a^3 b^2 B-21 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \tan (c+d x)}{a d}-\frac {\int \frac {\left (\left (A a^2-6 b B a+12 A b^2\right ) \left (a^2-b^2\right )^2+a b \left (A a^4+6 b B a^3-10 A b^2 a^2-3 b^3 B a+6 A b^4\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-2 a^5 B+6 a^4 A b+11 a^3 b^2 B-21 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \tan (c+d x)}{a d}-\frac {\int \frac {\left (A a^2-6 b B a+12 A b^2\right ) \left (a^2-b^2\right )^2+a b \left (A a^4+6 b B a^3-10 A b^2 a^2-3 b^3 B a+6 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3480

\(\displaystyle \frac {\frac {\frac {\left (a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-2 a^5 B+6 a^4 A b+11 a^3 b^2 B-21 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \tan (c+d x)}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 A-6 a b B+12 A b^2\right ) \int \sec (c+d x)dx}{a}-\frac {b^2 \left (-12 a^5 B+20 a^4 A b+15 a^3 b^2 B-29 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a}}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-2 a^5 B+6 a^4 A b+11 a^3 b^2 B-21 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \tan (c+d x)}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 A-6 a b B+12 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b^2 \left (-12 a^5 B+20 a^4 A b+15 a^3 b^2 B-29 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\frac {\frac {\left (a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-2 a^5 B+6 a^4 A b+11 a^3 b^2 B-21 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \tan (c+d x)}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 A-6 a b B+12 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b^2 \left (-12 a^5 B+20 a^4 A b+15 a^3 b^2 B-29 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\left (a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-2 a^5 B+6 a^4 A b+11 a^3 b^2 B-21 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \tan (c+d x)}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 A-6 a b B+12 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b^2 \left (-12 a^5 B+20 a^4 A b+15 a^3 b^2 B-29 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\frac {b \left (-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\frac {\left (a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+6 A b^4\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-2 a^5 B+6 a^4 A b+11 a^3 b^2 B-21 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \tan (c+d x)}{a d}-\frac {\frac {\left (a^2-b^2\right )^2 \left (a^2 A-6 a b B+12 A b^2\right ) \text {arctanh}(\sin (c+d x))}{a d}-\frac {2 b^2 \left (-12 a^5 B+20 a^4 A b+15 a^3 b^2 B-29 a^2 A b^3-6 a b^4 B+12 A b^5\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}\)

Input:

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

Output:

(b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[c 
+ d*x])^2) + ((b*(7*a^2*A*b - 4*A*b^3 - 5*a^3*B + 2*a*b^2*B)*Sec[c + d*x]* 
Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) + (((a^4*A - 10*a^2*A 
*b^2 + 6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B)*Sec[c + d*x]*Tan[c + d*x])/(a*d) - 
 (-(((-2*b^2*(20*a^4*A*b - 29*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 15*a^3*b^2 
*B - 6*a*b^4*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqr 
t[a - b]*Sqrt[a + b]*d) + ((a^2 - b^2)^2*(a^2*A + 12*A*b^2 - 6*a*b*B)*ArcT 
anh[Sin[c + d*x]])/(a*d))/a) + ((6*a^4*A*b - 21*a^2*A*b^3 + 12*A*b^5 - 2*a 
^5*B + 11*a^3*b^2*B - 6*a*b^4*B)*Tan[c + d*x])/(a*d))/a)/(a*(a^2 - b^2)))/ 
(2*a*(a^2 - b^2))
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 218

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

rule 3042

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

rule 3138

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ 
e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + b + 
(a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] 
 && NeQ[a^2 - b^2, 0]
 

rule 3479

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

rule 3480

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

rule 3534

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

rule 4257

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

Maple [A] (verified)

Time = 3.55 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.14

method	result	size
derivativedivides	\(\frac {-\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-A a -6 A b +2 B a}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a^{2} A +12 A \,b^{2}-6 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{5}}-\frac {2 b^{2} \left (\frac {-\frac {\left (10 A \,a^{2} b +A a \,b^{2}-6 A \,b^{3}-8 a^{3} B -B \,a^{2} b +4 B a \,b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (10 A \,a^{2} b -A a \,b^{2}-6 A \,b^{3}-8 a^{3} B +B \,a^{2} b +4 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (20 A \,a^{4} b -29 A \,a^{2} b^{3}+12 A \,b^{5}-12 B \,a^{5}+15 B \,a^{3} b^{2}-6 B a \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}+\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-A a -6 A b +2 B a}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2} A -12 A \,b^{2}+6 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{5}}}{d}\)	\(460\)
default	\(\frac {-\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-A a -6 A b +2 B a}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a^{2} A +12 A \,b^{2}-6 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{5}}-\frac {2 b^{2} \left (\frac {-\frac {\left (10 A \,a^{2} b +A a \,b^{2}-6 A \,b^{3}-8 a^{3} B -B \,a^{2} b +4 B a \,b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (10 A \,a^{2} b -A a \,b^{2}-6 A \,b^{3}-8 a^{3} B +B \,a^{2} b +4 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (20 A \,a^{4} b -29 A \,a^{2} b^{3}+12 A \,b^{5}-12 B \,a^{5}+15 B \,a^{3} b^{2}-6 B a \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}+\frac {A}{2 a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-A a -6 A b +2 B a}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2} A -12 A \,b^{2}+6 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{5}}}{d}\)	\(460\)
risch	\(\text {Expression too large to display}\)	\(2118\)

Input:

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

Output:

1/d*(-1/2*A/a^3/(tan(1/2*d*x+1/2*c)+1)^2-1/2*(-A*a-6*A*b+2*B*a)/a^4/(tan(1 
/2*d*x+1/2*c)+1)+1/2*(A*a^2+12*A*b^2-6*B*a*b)/a^5*ln(tan(1/2*d*x+1/2*c)+1) 
-2*b^2/a^5*((-1/2*(10*A*a^2*b+A*a*b^2-6*A*b^3-8*B*a^3-B*a^2*b+4*B*a*b^2)*a 
*b/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*b*a*(10*A*a^2*b-A*a*b^2- 
6*A*b^3-8*B*a^3+B*a^2*b+4*B*a*b^2)/(a+b)/(a-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)^2+1/2*(20*A*a^4*b-29*A*a^2* 
b^3+12*A*b^5-12*B*a^5+15*B*a^3*b^2-6*B*a*b^4)/(a^4-2*a^2*b^2+b^4)/((a-b)*( 
a+b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2)))+1/2*A/a^ 
3/(tan(1/2*d*x+1/2*c)-1)^2-1/2*(-A*a-6*A*b+2*B*a)/a^4/(tan(1/2*d*x+1/2*c)- 
1)+1/2/a^5*(-A*a^2-12*A*b^2+6*B*a*b)*ln(tan(1/2*d*x+1/2*c)-1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1174 vs. \(2 (382) = 764\).

Time = 37.32 (sec) , antiderivative size = 2416, normalized size of antiderivative = 6.01 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(((12*B*a^5*b^4 - 20*A*a^4*b^5 - 15*B*a^3*b^6 + 29*A*a^2*b^7 + 6*B*a* 
b^8 - 12*A*b^9)*cos(d*x + c)^4 + 2*(12*B*a^6*b^3 - 20*A*a^5*b^4 - 15*B*a^4 
*b^5 + 29*A*a^3*b^6 + 6*B*a^2*b^7 - 12*A*a*b^8)*cos(d*x + c)^3 + (12*B*a^7 
*b^2 - 20*A*a^6*b^3 - 15*B*a^5*b^4 + 29*A*a^4*b^5 + 6*B*a^3*b^6 - 12*A*a^2 
*b^7)*cos(d*x + c)^2)*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - 
b^2)*cos(d*x + c)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)*sin(d*x + c) 
 - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) + ((A*a^8 
*b^2 - 6*B*a^7*b^3 + 9*A*a^6*b^4 + 18*B*a^5*b^5 - 33*A*a^4*b^6 - 18*B*a^3* 
b^7 + 35*A*a^2*b^8 + 6*B*a*b^9 - 12*A*b^10)*cos(d*x + c)^4 + 2*(A*a^9*b - 
6*B*a^8*b^2 + 9*A*a^7*b^3 + 18*B*a^6*b^4 - 33*A*a^5*b^5 - 18*B*a^4*b^6 + 3 
5*A*a^3*b^7 + 6*B*a^2*b^8 - 12*A*a*b^9)*cos(d*x + c)^3 + (A*a^10 - 6*B*a^9 
*b + 9*A*a^8*b^2 + 18*B*a^7*b^3 - 33*A*a^6*b^4 - 18*B*a^5*b^5 + 35*A*a^4*b 
^6 + 6*B*a^3*b^7 - 12*A*a^2*b^8)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - ( 
(A*a^8*b^2 - 6*B*a^7*b^3 + 9*A*a^6*b^4 + 18*B*a^5*b^5 - 33*A*a^4*b^6 - 18* 
B*a^3*b^7 + 35*A*a^2*b^8 + 6*B*a*b^9 - 12*A*b^10)*cos(d*x + c)^4 + 2*(A*a^ 
9*b - 6*B*a^8*b^2 + 9*A*a^7*b^3 + 18*B*a^6*b^4 - 33*A*a^5*b^5 - 18*B*a^4*b 
^6 + 35*A*a^3*b^7 + 6*B*a^2*b^8 - 12*A*a*b^9)*cos(d*x + c)^3 + (A*a^10 - 6 
*B*a^9*b + 9*A*a^8*b^2 + 18*B*a^7*b^3 - 33*A*a^6*b^4 - 18*B*a^5*b^5 + 35*A 
*a^4*b^6 + 6*B*a^3*b^7 - 12*A*a^2*b^8)*cos(d*x + c)^2)*log(-sin(d*x + c) + 
 1) + 2*(A*a^10 - 3*A*a^8*b^2 + 3*A*a^6*b^4 - A*a^4*b^6 + (2*B*a^8*b^2 ...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((A+B*cos(d*x+c))*sec(d*x+c)^3/(a+b*cos(d*x+c))^3,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*b^2-4*a^2>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1395 vs. \(2 (382) = 764\).

Time = 0.23 (sec) , antiderivative size = 1395, normalized size of antiderivative = 3.47 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(2*(12*B*a^5*b^2 - 20*A*a^4*b^3 - 15*B*a^3*b^4 + 29*A*a^2*b^5 + 6*B*a 
*b^6 - 12*A*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arcta 
n(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a 
^9 - 2*a^7*b^2 + a^5*b^4)*sqrt(a^2 - b^2)) - 2*(A*a^7*tan(1/2*d*x + 1/2*c) 
^7 - 2*B*a^7*tan(1/2*d*x + 1/2*c)^7 + 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^7 + 4 
*B*a^6*b*tan(1/2*d*x + 1/2*c)^7 - 13*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 + 2* 
B*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 2*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 16 
*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 + 
9*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 17*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 
 9*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 18*A*a*b^6*tan(1/2*d*x + 1/2*c)^7 - 
6*B*a*b^6*tan(1/2*d*x + 1/2*c)^7 + 12*A*b^7*tan(1/2*d*x + 1/2*c)^7 + 3*A*a 
^7*tan(1/2*d*x + 1/2*c)^5 - 2*B*a^7*tan(1/2*d*x + 1/2*c)^5 + 4*A*a^6*b*tan 
(1/2*d*x + 1/2*c)^5 - 4*B*a^6*b*tan(1/2*d*x + 1/2*c)^5 + 5*A*a^5*b^2*tan(1 
/2*d*x + 1/2*c)^5 + 10*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 - 26*A*a^4*b^3*tan 
(1/2*d*x + 1/2*c)^5 + 16*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 29*A*a^3*b^4*t 
an(1/2*d*x + 1/2*c)^5 - 35*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 67*A*a^2*b^5 
*tan(1/2*d*x + 1/2*c)^5 - 9*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 18*A*a*b^6* 
tan(1/2*d*x + 1/2*c)^5 + 18*B*a*b^6*tan(1/2*d*x + 1/2*c)^5 - 36*A*b^7*tan( 
1/2*d*x + 1/2*c)^5 + 3*A*a^7*tan(1/2*d*x + 1/2*c)^3 + 2*B*a^7*tan(1/2*d*x 
+ 1/2*c)^3 - 4*A*a^6*b*tan(1/2*d*x + 1/2*c)^3 - 4*B*a^6*b*tan(1/2*d*x +...
 

Mupad [B] (verification not implemented)

Time = 33.35 (sec) , antiderivative size = 10547, normalized size of antiderivative = 26.24 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((tan(c/2 + (d*x)/2)^3*(3*A*a^7 + 36*A*b^7 + 2*B*a^7 - 67*A*a^2*b^5 - 29*A 
*a^3*b^4 + 26*A*a^4*b^3 + 5*A*a^5*b^2 - 9*B*a^2*b^5 + 35*B*a^3*b^4 + 16*B* 
a^4*b^3 - 10*B*a^5*b^2 + 18*A*a*b^6 - 4*A*a^6*b - 18*B*a*b^6 - 4*B*a^6*b)) 
/((a + b)^2*(a^6 - 2*a^5*b + a^4*b^2)) + (tan(c/2 + (d*x)/2)^5*(3*A*a^7 - 
36*A*b^7 - 2*B*a^7 + 67*A*a^2*b^5 - 29*A*a^3*b^4 - 26*A*a^4*b^3 + 5*A*a^5* 
b^2 - 9*B*a^2*b^5 - 35*B*a^3*b^4 + 16*B*a^4*b^3 + 10*B*a^5*b^2 + 18*A*a*b^ 
6 + 4*A*a^6*b + 18*B*a*b^6 - 4*B*a^6*b))/((a + b)^2*(a^6 - 2*a^5*b + a^4*b 
^2)) - (tan(c/2 + (d*x)/2)^7*(A*a^6 - 12*A*b^6 - 2*B*a^6 + 23*A*a^2*b^4 - 
10*A*a^3*b^3 - 8*A*a^4*b^2 - 3*B*a^2*b^4 - 12*B*a^3*b^3 + 4*B*a^4*b^2 + 6* 
A*a*b^5 + 5*A*a^5*b + 6*B*a*b^5 + 2*B*a^5*b))/((a^4*b - a^5)*(a + b)^2) + 
(tan(c/2 + (d*x)/2)*(A*a^6 - 12*A*b^6 + 2*B*a^6 + 23*A*a^2*b^4 + 10*A*a^3* 
b^3 - 8*A*a^4*b^2 + 3*B*a^2*b^4 - 12*B*a^3*b^3 - 4*B*a^4*b^2 - 6*A*a*b^5 - 
 5*A*a^5*b + 6*B*a*b^5 + 2*B*a^5*b))/((a + b)*(a^6 - 2*a^5*b + a^4*b^2)))/ 
(d*(2*a*b - tan(c/2 + (d*x)/2)^4*(2*a^2 - 6*b^2) - tan(c/2 + (d*x)/2)^2*(4 
*a*b + 4*b^2) + tan(c/2 + (d*x)/2)^6*(4*a*b - 4*b^2) + tan(c/2 + (d*x)/2)^ 
8*(a^2 - 2*a*b + b^2) + a^2 + b^2)) - (atan(((((8*tan(c/2 + (d*x)/2)*(A^2* 
a^14 + 288*A^2*b^14 - 288*A^2*a*b^13 - 2*A^2*a^13*b - 1104*A^2*a^2*b^12 + 
1104*A^2*a^3*b^11 + 1538*A^2*a^4*b^10 - 1538*A^2*a^5*b^9 - 827*A^2*a^6*b^8 
 + 872*A^2*a^7*b^7 + 18*A^2*a^8*b^6 - 108*A^2*a^9*b^5 + 74*A^2*a^10*b^4 - 
40*A^2*a^11*b^3 + 21*A^2*a^12*b^2 + 72*B^2*a^2*b^12 - 72*B^2*a^3*b^11 -...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1637, normalized size of antiderivative = 4.07 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 16*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sq 
rt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**2*b**4 + 12*sqrt(a**2 - b 
**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos 
(c + d*x)*sin(c + d*x)**2*b**6 + 16*sqrt(a**2 - b**2)*atan((tan((c + d*x)/ 
2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**2*b**4 - 12* 
sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 
 - b**2))*cos(c + d*x)*b**6 - 16*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)* 
a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a**3*b**3 + 12* 
sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 
 - b**2))*sin(c + d*x)**2*a*b**5 + 16*sqrt(a**2 - b**2)*atan((tan((c + d*x 
)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*a**3*b**3 - 12*sqrt(a**2 - 
 b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*a 
*b**5 - cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**6*b - 4* 
cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**4*b**3 + 11*cos( 
c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**2*b**5 - 6*cos(c + d 
*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*b**7 + cos(c + d*x)*log(tan( 
(c + d*x)/2) - 1)*a**6*b + 4*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**4*b 
**3 - 11*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**2*b**5 + 6*cos(c + d*x) 
*log(tan((c + d*x)/2) - 1)*b**7 + cos(c + d*x)*log(tan((c + d*x)/2) + 1)*s 
in(c + d*x)**2*a**6*b + 4*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c ...