\(\int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx\) [265]

Optimal result

Integrand size = 31, antiderivative size = 398 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx=-\frac {\left (6 a A b-12 a^2 B-b^2 B\right ) x}{2 b^5}+\frac {a^2 \left (6 a^4 A b-15 a^2 A b^3+12 A b^5-12 a^5 B+29 a^3 b^2 B-20 a b^4 B\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}+\frac {\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac {a (A b-a B) \cos ^3(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \] Output:

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

Mathematica [A] (verified)

Time = 6.61 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.84 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {16 a^2 \left (-6 a^4 A b+15 a^2 A b^3-12 A b^5+12 a^5 B-29 a^3 b^2 B+20 a b^4 B\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+\frac {-48 a^7 A b c+72 a^5 A b^3 c-24 a A b^7 c+96 a^8 B c-136 a^6 b^2 B c-12 a^4 b^4 B c+48 a^2 b^6 B c+4 b^8 B c-48 a^7 A b d x+72 a^5 A b^3 d x-24 a A b^7 d x+96 a^8 B d x-136 a^6 b^2 B d x-12 a^4 b^4 B d x+48 a^2 b^6 B d x+4 b^8 B d x+16 a b \left (a^2-b^2\right )^2 \left (-6 a A b+12 a^2 B+b^2 B\right ) (c+d x) \cos (c+d x)+4 \left (-a^2 b+b^3\right )^2 \left (-6 a A b+12 a^2 B+b^2 B\right ) (c+d x) \cos (2 (c+d x))+48 a^6 A b^2 \sin (c+d x)-84 a^4 A b^4 \sin (c+d x)+8 a^2 A b^6 \sin (c+d x)+4 A b^8 \sin (c+d x)-96 a^7 b B \sin (c+d x)+160 a^5 b^3 B \sin (c+d x)-32 a^3 b^5 B \sin (c+d x)-8 a b^7 B \sin (c+d x)+36 a^5 A b^3 \sin (2 (c+d x))-64 a^3 A b^5 \sin (2 (c+d x))+16 a A b^7 \sin (2 (c+d x))-72 a^6 b^2 B \sin (2 (c+d x))+130 a^4 b^4 B \sin (2 (c+d x))-48 a^2 b^6 B \sin (2 (c+d x))+2 b^8 B \sin (2 (c+d x))+4 a^4 A b^4 \sin (3 (c+d x))-8 a^2 A b^6 \sin (3 (c+d x))+4 A b^8 \sin (3 (c+d x))-8 a^5 b^3 B \sin (3 (c+d x))+16 a^3 b^5 B \sin (3 (c+d x))-8 a b^7 B \sin (3 (c+d x))+a^4 b^4 B \sin (4 (c+d x))-2 a^2 b^6 B \sin (4 (c+d x))+b^8 B \sin (4 (c+d x))}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}}{16 b^5 d} \] Input:

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

Output:

((16*a^2*(-6*a^4*A*b + 15*a^2*A*b^3 - 12*A*b^5 + 12*a^5*B - 29*a^3*b^2*B + 
 20*a*b^4*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + 
 b^2)^(5/2) + (-48*a^7*A*b*c + 72*a^5*A*b^3*c - 24*a*A*b^7*c + 96*a^8*B*c 
- 136*a^6*b^2*B*c - 12*a^4*b^4*B*c + 48*a^2*b^6*B*c + 4*b^8*B*c - 48*a^7*A 
*b*d*x + 72*a^5*A*b^3*d*x - 24*a*A*b^7*d*x + 96*a^8*B*d*x - 136*a^6*b^2*B* 
d*x - 12*a^4*b^4*B*d*x + 48*a^2*b^6*B*d*x + 4*b^8*B*d*x + 16*a*b*(a^2 - b^ 
2)^2*(-6*a*A*b + 12*a^2*B + b^2*B)*(c + d*x)*Cos[c + d*x] + 4*(-(a^2*b) + 
b^3)^2*(-6*a*A*b + 12*a^2*B + b^2*B)*(c + d*x)*Cos[2*(c + d*x)] + 48*a^6*A 
*b^2*Sin[c + d*x] - 84*a^4*A*b^4*Sin[c + d*x] + 8*a^2*A*b^6*Sin[c + d*x] + 
 4*A*b^8*Sin[c + d*x] - 96*a^7*b*B*Sin[c + d*x] + 160*a^5*b^3*B*Sin[c + d* 
x] - 32*a^3*b^5*B*Sin[c + d*x] - 8*a*b^7*B*Sin[c + d*x] + 36*a^5*A*b^3*Sin 
[2*(c + d*x)] - 64*a^3*A*b^5*Sin[2*(c + d*x)] + 16*a*A*b^7*Sin[2*(c + d*x) 
] - 72*a^6*b^2*B*Sin[2*(c + d*x)] + 130*a^4*b^4*B*Sin[2*(c + d*x)] - 48*a^ 
2*b^6*B*Sin[2*(c + d*x)] + 2*b^8*B*Sin[2*(c + d*x)] + 4*a^4*A*b^4*Sin[3*(c 
 + d*x)] - 8*a^2*A*b^6*Sin[3*(c + d*x)] + 4*A*b^8*Sin[3*(c + d*x)] - 8*a^5 
*b^3*B*Sin[3*(c + d*x)] + 16*a^3*b^5*B*Sin[3*(c + d*x)] - 8*a*b^7*B*Sin[3* 
(c + d*x)] + a^4*b^4*B*Sin[4*(c + d*x)] - 2*a^2*b^6*B*Sin[4*(c + d*x)] + b 
^8*B*Sin[4*(c + d*x)])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(16*b^5*d)
 

Rubi [A] (verified)

Time = 2.15 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.516, Rules used = {3042, 3468, 25, 3042, 3526, 25, 3042, 3528, 27, 3042, 3502, 3042, 3214, 3042, 3138, 218}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3468

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3214

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3138

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(a*(A*b - a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[ 
c + d*x])^2) + ((a*(2*a^2*A*b - 5*A*b^3 - 4*a^3*B + 7*a*b^2*B)*Cos[c + d*x 
]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) + (-(((3*a^3*A*b 
- 6*a*A*b^3 - 6*a^4*B + 10*a^2*b^2*B - b^4*B)*Cos[c + d*x]*Sin[c + d*x])/( 
b*d)) - ((((a^2 - b^2)^2*(6*a*A*b - 12*a^2*B - b^2*B)*x)/b - (2*a^2*(6*a^4 
*A*b - 15*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 29*a^3*b^2*B - 20*a*b^4*B)*Arc 
Tan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b 
]*d))/b - ((6*a^4*A*b - 11*a^2*A*b^3 + 2*A*b^5 - 12*a^5*B + 21*a^3*b^2*B - 
 6*a*b^4*B)*Sin[c + d*x])/(b*d))/b)/(b*(a^2 - b^2)))/(2*b*(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 3214

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

rule 3468

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[(-(b*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c 
 + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 
1)*(c^2 - d^2))   Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n 
+ 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a 
*B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) 
 - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - 
a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, 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] && GtQ[m, 1] && LtQ[n, -1]
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

rule 3526

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[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - 
d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin[e + f*x])^(m - 
 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* 
d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 
1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x 
] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f 
*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d 
, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
 

rule 3528

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[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x 
])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*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] && GtQ[ 
m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
 

Maple [A] (verified)

Time = 2.72 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.01

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {\left (-A \,b^{2}+3 B a b +\frac {1}{2} B \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-A \,b^{2}+3 B a b -\frac {1}{2} B \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (6 A a b -12 a^{2} B -B \,b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}+\frac {2 a^{2} \left (\frac {\frac {\left (4 A \,a^{2} b -A a \,b^{2}-8 A \,b^{3}-6 a^{3} B +B \,a^{2} b +10 B a \,b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (4 A \,a^{2} b +A a \,b^{2}-8 A \,b^{3}-6 a^{3} B -B \,a^{2} b +10 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (6 A \,a^{4} b -15 A \,a^{2} b^{3}+12 A \,b^{5}-12 B \,a^{5}+29 B \,a^{3} b^{2}-20 B a \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\)	\(402\)
default	\(\frac {-\frac {2 \left (\frac {\left (-A \,b^{2}+3 B a b +\frac {1}{2} B \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-A \,b^{2}+3 B a b -\frac {1}{2} B \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (6 A a b -12 a^{2} B -B \,b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}+\frac {2 a^{2} \left (\frac {\frac {\left (4 A \,a^{2} b -A a \,b^{2}-8 A \,b^{3}-6 a^{3} B +B \,a^{2} b +10 B a \,b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (4 A \,a^{2} b +A a \,b^{2}-8 A \,b^{3}-6 a^{3} B -B \,a^{2} b +10 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (6 A \,a^{4} b -15 A \,a^{2} b^{3}+12 A \,b^{5}-12 B \,a^{5}+29 B \,a^{3} b^{2}-20 B a \,b^{4}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{5}}}{d}\)	\(402\)
risch	\(\text {Expression too large to display}\)	\(1509\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 871 vs. \(2 (378) = 756\).

Time = 0.23 (sec) , antiderivative size = 1812, normalized size of antiderivative = 4.55 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(2*(12*B*a^8*b^2 - 6*A*a^7*b^3 - 35*B*a^6*b^4 + 18*A*a^5*b^5 + 33*B*a 
^4*b^6 - 18*A*a^3*b^7 - 9*B*a^2*b^8 + 6*A*a*b^9 - B*b^10)*d*x*cos(d*x + c) 
^2 + 4*(12*B*a^9*b - 6*A*a^8*b^2 - 35*B*a^7*b^3 + 18*A*a^6*b^4 + 33*B*a^5* 
b^5 - 18*A*a^4*b^6 - 9*B*a^3*b^7 + 6*A*a^2*b^8 - B*a*b^9)*d*x*cos(d*x + c) 
 + 2*(12*B*a^10 - 6*A*a^9*b - 35*B*a^8*b^2 + 18*A*a^7*b^3 + 33*B*a^6*b^4 - 
 18*A*a^5*b^5 - 9*B*a^4*b^6 + 6*A*a^3*b^7 - B*a^2*b^8)*d*x + (12*B*a^9 - 6 
*A*a^8*b - 29*B*a^7*b^2 + 15*A*a^6*b^3 + 20*B*a^5*b^4 - 12*A*a^4*b^5 + (12 
*B*a^7*b^2 - 6*A*a^6*b^3 - 29*B*a^5*b^4 + 15*A*a^4*b^5 + 20*B*a^3*b^6 - 12 
*A*a^2*b^7)*cos(d*x + c)^2 + 2*(12*B*a^8*b - 6*A*a^7*b^2 - 29*B*a^6*b^3 + 
15*A*a^5*b^4 + 20*B*a^4*b^5 - 12*A*a^3*b^6)*cos(d*x + c))*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)) - 2*(12*B*a^9*b - 6*A*a^8*b^2 - 33*B*a^7*b^3 + 
17*A*a^6*b^4 + 27*B*a^5*b^5 - 13*A*a^4*b^6 - 6*B*a^3*b^7 + 2*A*a^2*b^8 - ( 
B*a^6*b^4 - 3*B*a^4*b^6 + 3*B*a^2*b^8 - B*b^10)*cos(d*x + c)^3 + 2*(2*B*a^ 
7*b^3 - A*a^6*b^4 - 6*B*a^5*b^5 + 3*A*a^4*b^6 + 6*B*a^3*b^7 - 3*A*a^2*b^8 
- 2*B*a*b^9 + A*b^10)*cos(d*x + c)^2 + (18*B*a^8*b^2 - 9*A*a^7*b^3 - 50*B* 
a^6*b^4 + 25*A*a^5*b^5 + 43*B*a^4*b^6 - 20*A*a^3*b^7 - 11*B*a^2*b^8 + 4*A* 
a*b^9)*cos(d*x + c))*sin(d*x + c))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^ 
13)*d*cos(d*x + c)^2 + 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(cos(d*x+c)^4*(A+B*cos(d*x+c))/(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. 2712 vs. \(2 (378) = 756\).

Time = 0.45 (sec) , antiderivative size = 2712, normalized size of antiderivative = 6.81 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*((3*(2*a^5*b - a^4*b^2 - 4*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*sqrt(a^2 - b 
^2)*A*abs(a^4*b^5 - 2*a^2*b^7 + b^9)*abs(-a + b) - (12*a^6 - 6*a^5*b - 23* 
a^4*b^2 + 10*a^3*b^3 + 10*a^2*b^4 - a*b^5 + b^6)*sqrt(a^2 - b^2)*B*abs(a^4 
*b^5 - 2*a^2*b^7 + b^9)*abs(-a + b) + 3*(4*a^10*b^5 - 2*a^9*b^6 - 17*a^8*b 
^7 + 8*a^7*b^8 + 28*a^6*b^9 - 12*a^5*b^10 - 21*a^4*b^11 + 8*a^3*b^12 + 6*a 
^2*b^13 - 2*a*b^14)*sqrt(a^2 - b^2)*A*abs(-a + b) - (24*a^11*b^4 - 12*a^10 
*b^5 - 100*a^9*b^6 + 47*a^8*b^7 + 158*a^7*b^8 - 68*a^6*b^9 - 111*a^5*b^10 
+ 42*a^4*b^11 + 28*a^3*b^12 - 8*a^2*b^13 + a*b^14 - b^15)*sqrt(a^2 - b^2)* 
B*abs(-a + b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*tan(1/2*d*x + 
1/2*c)/sqrt((4*a^5*b^4 - 8*a^3*b^6 + 4*a*b^8 + sqrt(-16*(a^5*b^4 + a^4*b^5 
 - 2*a^3*b^6 - 2*a^2*b^7 + a*b^8 + b^9)*(a^5*b^4 - a^4*b^5 - 2*a^3*b^6 + 2 
*a^2*b^7 + a*b^8 - b^9) + 16*(a^5*b^4 - 2*a^3*b^6 + a*b^8)^2))/(a^5*b^4 - 
a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b^9))))/((a^4*b^5 - 2*a^2*b^7 + 
b^9)^2*(a^2 - 2*a*b + b^2) + (a^7*b^4 - 2*a^6*b^5 - a^5*b^6 + 4*a^4*b^7 - 
a^3*b^8 - 2*a^2*b^9 + a*b^10)*abs(a^4*b^5 - 2*a^2*b^7 + b^9)) + (24*B*a^11 
*b^4 - 12*A*a^10*b^5 - 12*B*a^10*b^5 + 6*A*a^9*b^6 - 100*B*a^9*b^6 + 51*A* 
a^8*b^7 + 47*B*a^8*b^7 - 24*A*a^7*b^8 + 158*B*a^7*b^8 - 84*A*a^6*b^9 - 68* 
B*a^6*b^9 + 36*A*a^5*b^10 - 111*B*a^5*b^10 + 63*A*a^4*b^11 + 42*B*a^4*b^11 
 - 24*A*a^3*b^12 + 28*B*a^3*b^12 - 18*A*a^2*b^13 - 8*B*a^2*b^13 + 6*A*a*b^ 
14 + B*a*b^14 - B*b^15 - 12*B*a^6*abs(a^4*b^5 - 2*a^2*b^7 + b^9) + 6*A*...
 

Mupad [B] (verification not implemented)

Time = 34.68 (sec) , antiderivative size = 10598, normalized size of antiderivative = 26.63 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((tan(c/2 + (d*x)/2)^5*(3*B*b^7 - 36*B*a^7 - 2*A*b^7 + 10*A*a^2*b^5 + 16*A 
*a^3*b^4 - 35*A*a^4*b^3 - 9*A*a^5*b^2 + 5*B*a^2*b^5 - 26*B*a^3*b^4 - 29*B* 
a^4*b^3 + 67*B*a^5*b^2 - 4*A*a*b^6 + 18*A*a^6*b + 4*B*a*b^6 + 18*B*a^6*b)) 
/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b^4)) - (tan(c/2 + (d*x)/2)^3*(2*A*b^7 + 
36*B*a^7 + 3*B*b^7 - 10*A*a^2*b^5 + 16*A*a^3*b^4 + 35*A*a^4*b^3 - 9*A*a^5* 
b^2 + 5*B*a^2*b^5 + 26*B*a^3*b^4 - 29*B*a^4*b^3 - 67*B*a^5*b^2 - 4*A*a*b^6 
 - 18*A*a^6*b - 4*B*a*b^6 + 18*B*a^6*b))/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b 
^4)) + (tan(c/2 + (d*x)/2)^7*(B*b^6 - 12*B*a^6 - 2*A*b^6 + 4*A*a^2*b^4 - 1 
2*A*a^3*b^3 - 3*A*a^4*b^2 - 8*B*a^2*b^4 - 10*B*a^3*b^3 + 23*B*a^4*b^2 + 2* 
A*a*b^5 + 6*A*a^5*b + 5*B*a*b^5 + 6*B*a^5*b))/((a*b^4 - b^5)*(a + b)^2) + 
(tan(c/2 + (d*x)/2)*(2*A*b^6 - 12*B*a^6 + B*b^6 - 4*A*a^2*b^4 - 12*A*a^3*b 
^3 + 3*A*a^4*b^2 - 8*B*a^2*b^4 + 10*B*a^3*b^3 + 23*B*a^4*b^2 + 2*A*a*b^5 + 
 6*A*a^5*b - 5*B*a*b^5 - 6*B*a^5*b))/((a + b)*(b^6 - 2*a*b^5 + a^2*b^4)))/ 
(d*(2*a*b + tan(c/2 + (d*x)/2)^4*(6*a^2 - 2*b^2) + tan(c/2 + (d*x)/2)^2*(4 
*a*b + 4*a^2) - tan(c/2 + (d*x)/2)^6*(4*a*b - 4*a^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)*(288* 
B^2*a^14 + B^2*b^14 - 2*B^2*a*b^13 - 288*B^2*a^13*b + 36*A^2*a^2*b^12 - 72 
*A^2*a^3*b^11 + 36*A^2*a^4*b^10 + 288*A^2*a^5*b^9 - 288*A^2*a^6*b^8 - 432* 
A^2*a^7*b^7 + 441*A^2*a^8*b^6 + 288*A^2*a^9*b^5 - 288*A^2*a^10*b^4 - 72*A^ 
2*a^11*b^3 + 72*A^2*a^12*b^2 + 21*B^2*a^2*b^12 - 40*B^2*a^3*b^11 + 74*B...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 623, normalized size of antiderivative = 1.57 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx=\frac {-12 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {a^{2}-b^{2}}}\right ) \cos \left (d x +c \right ) a^{5} b +16 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {a^{2}-b^{2}}}\right ) \cos \left (d x +c \right ) a^{3} b^{3}+6 \cos \left (d x +c \right ) a^{6} b d x -11 \cos \left (d x +c \right ) a^{4} b^{3} d x +4 \cos \left (d x +c \right ) a^{2} b^{5} d x +6 a^{7} c +\cos \left (d x +c \right ) b^{7} c +\cos \left (d x +c \right ) b^{7} d x -\sin \left (d x +c \right )^{3} a^{4} b^{3}+a \,b^{6} c +a \,b^{6} d x +16 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {a^{2}-b^{2}}}\right ) a^{4} b^{2}-3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{5} b^{2}+6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b^{4}-3 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{6}-\sin \left (d x +c \right )^{3} b^{7}+\sin \left (d x +c \right ) b^{7}+6 \cos \left (d x +c \right ) a^{6} b c -11 \cos \left (d x +c \right ) a^{4} b^{3} c +4 \cos \left (d x +c \right ) a^{2} b^{5} c -11 a^{5} b^{2} d x +4 a^{3} b^{4} d x -12 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {a^{2}-b^{2}}}\right ) a^{6}+2 \sin \left (d x +c \right )^{3} a^{2} b^{5}-6 \sin \left (d x +c \right ) a^{6} b +11 \sin \left (d x +c \right ) a^{4} b^{3}-6 \sin \left (d x +c \right ) a^{2} b^{5}+6 a^{7} d x -11 a^{5} b^{2} c +4 a^{3} b^{4} c}{2 b^{4} d \left (\cos \left (d x +c \right ) a^{4} b -2 \cos \left (d x +c \right ) a^{2} b^{3}+\cos \left (d x +c \right ) b^{5}+a^{5}-2 a^{3} b^{2}+a \,b^{4}\right )} \] Input:

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

Output:

( - 12*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)*a**5*b + 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**3*b**3 
 - 12*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqr 
t(a**2 - b**2))*a**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))*a**4*b**2 - 3*cos(c + d*x)*sin(c + d*x 
)*a**5*b**2 + 6*cos(c + d*x)*sin(c + d*x)*a**3*b**4 - 3*cos(c + d*x)*sin(c 
 + d*x)*a*b**6 + 6*cos(c + d*x)*a**6*b*c + 6*cos(c + d*x)*a**6*b*d*x - 11* 
cos(c + d*x)*a**4*b**3*c - 11*cos(c + d*x)*a**4*b**3*d*x + 4*cos(c + d*x)* 
a**2*b**5*c + 4*cos(c + d*x)*a**2*b**5*d*x + cos(c + d*x)*b**7*c + cos(c + 
 d*x)*b**7*d*x - sin(c + d*x)**3*a**4*b**3 + 2*sin(c + d*x)**3*a**2*b**5 - 
 sin(c + d*x)**3*b**7 - 6*sin(c + d*x)*a**6*b + 11*sin(c + d*x)*a**4*b**3 
- 6*sin(c + d*x)*a**2*b**5 + sin(c + d*x)*b**7 + 6*a**7*c + 6*a**7*d*x - 1 
1*a**5*b**2*c - 11*a**5*b**2*d*x + 4*a**3*b**4*c + 4*a**3*b**4*d*x + a*b** 
6*c + a*b**6*d*x)/(2*b**4*d*(cos(c + d*x)*a**4*b - 2*cos(c + d*x)*a**2*b** 
3 + cos(c + d*x)*b**5 + a**5 - 2*a**3*b**2 + a*b**4))