\(\int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx\) [274]

Optimal result

Integrand size = 31, antiderivative size = 301 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx=\frac {B x}{b^4}-\frac {\left (3 a^2 A b^5+2 A b^7+2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B-8 a b^6 B\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^4 (a+b)^{7/2} d}+\frac {a (A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {a^2 \left (5 A b^3+3 a^3 B-8 a b^2 B\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}-\frac {a \left (a^2 A b^3-16 A b^5+9 a^5 B-28 a^3 b^2 B+34 a b^4 B\right ) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \] Output:

B*x/b^4-(3*A*a^2*b^5+2*A*b^7+2*B*a^7-7*B*a^5*b^2+8*B*a^3*b^4-8*B*a*b^6)*ar 
ctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^4/(a+b)^(7/ 
2)/d+1/3*a*(A*b-B*a)*cos(d*x+c)^2*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c) 
)^3+1/6*a^2*(5*A*b^3+3*B*a^3-8*B*a*b^2)*sin(d*x+c)/b^3/(a^2-b^2)^2/d/(a+b* 
cos(d*x+c))^2-1/6*a*(A*a^2*b^3-16*A*b^5+9*B*a^5-28*B*a^3*b^2+34*B*a*b^4)*s 
in(d*x+c)/b^3/(a^2-b^2)^3/d/(a+b*cos(d*x+c))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(717\) vs. \(2(301)=602\).

Time = 6.40 (sec) , antiderivative size = 717, normalized size of antiderivative = 2.38 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx=\frac {-\frac {24 \left (3 a^2 A b^5+2 A b^7+2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B-8 a b^6 B\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}+\frac {24 a^9 B c-36 a^7 b^2 B c-36 a^5 b^4 B c+84 a^3 b^6 B c-36 a b^8 B c+24 a^9 B d x-36 a^7 b^2 B d x-36 a^5 b^4 B d x+84 a^3 b^6 B d x-36 a b^8 B d x+18 b \left (a^2-b^2\right )^3 \left (4 a^2+b^2\right ) B (c+d x) \cos (c+d x)+36 a b^2 \left (a^2-b^2\right )^3 B (c+d x) \cos (2 (c+d x))+6 a^6 b^3 B c \cos (3 (c+d x))-18 a^4 b^5 B c \cos (3 (c+d x))+18 a^2 b^7 B c \cos (3 (c+d x))-6 b^9 B c \cos (3 (c+d x))+6 a^6 b^3 B d x \cos (3 (c+d x))-18 a^4 b^5 B d x \cos (3 (c+d x))+18 a^2 b^7 B d x \cos (3 (c+d x))-6 b^9 B d x \cos (3 (c+d x))+18 a^5 A b^4 \sin (c+d x)+39 a^3 A b^6 \sin (c+d x)+18 a A b^8 \sin (c+d x)-24 a^8 b B \sin (c+d x)+57 a^6 b^3 B \sin (c+d x)-72 a^4 b^5 B \sin (c+d x)-36 a^2 b^7 B \sin (c+d x)+6 a^4 A b^5 \sin (2 (c+d x))+54 a^2 A b^7 \sin (2 (c+d x))-30 a^7 b^2 B \sin (2 (c+d x))+90 a^5 b^4 B \sin (2 (c+d x))-120 a^3 b^6 B \sin (2 (c+d x))+2 a^5 A b^4 \sin (3 (c+d x))-5 a^3 A b^6 \sin (3 (c+d x))+18 a A b^8 \sin (3 (c+d x))-11 a^6 b^3 B \sin (3 (c+d x))+32 a^4 b^5 B \sin (3 (c+d x))-36 a^2 b^7 B \sin (3 (c+d x))}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}}{24 b^4 d} \] Input:

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

Output:

((-24*(3*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^3*b^4*B - 8*a*b 
^6*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^( 
7/2) + (24*a^9*B*c - 36*a^7*b^2*B*c - 36*a^5*b^4*B*c + 84*a^3*b^6*B*c - 36 
*a*b^8*B*c + 24*a^9*B*d*x - 36*a^7*b^2*B*d*x - 36*a^5*b^4*B*d*x + 84*a^3*b 
^6*B*d*x - 36*a*b^8*B*d*x + 18*b*(a^2 - b^2)^3*(4*a^2 + b^2)*B*(c + d*x)*C 
os[c + d*x] + 36*a*b^2*(a^2 - b^2)^3*B*(c + d*x)*Cos[2*(c + d*x)] + 6*a^6* 
b^3*B*c*Cos[3*(c + d*x)] - 18*a^4*b^5*B*c*Cos[3*(c + d*x)] + 18*a^2*b^7*B* 
c*Cos[3*(c + d*x)] - 6*b^9*B*c*Cos[3*(c + d*x)] + 6*a^6*b^3*B*d*x*Cos[3*(c 
 + d*x)] - 18*a^4*b^5*B*d*x*Cos[3*(c + d*x)] + 18*a^2*b^7*B*d*x*Cos[3*(c + 
 d*x)] - 6*b^9*B*d*x*Cos[3*(c + d*x)] + 18*a^5*A*b^4*Sin[c + d*x] + 39*a^3 
*A*b^6*Sin[c + d*x] + 18*a*A*b^8*Sin[c + d*x] - 24*a^8*b*B*Sin[c + d*x] + 
57*a^6*b^3*B*Sin[c + d*x] - 72*a^4*b^5*B*Sin[c + d*x] - 36*a^2*b^7*B*Sin[c 
 + d*x] + 6*a^4*A*b^5*Sin[2*(c + d*x)] + 54*a^2*A*b^7*Sin[2*(c + d*x)] - 3 
0*a^7*b^2*B*Sin[2*(c + d*x)] + 90*a^5*b^4*B*Sin[2*(c + d*x)] - 120*a^3*b^6 
*B*Sin[2*(c + d*x)] + 2*a^5*A*b^4*Sin[3*(c + d*x)] - 5*a^3*A*b^6*Sin[3*(c 
+ d*x)] + 18*a*A*b^8*Sin[3*(c + d*x)] - 11*a^6*b^3*B*Sin[3*(c + d*x)] + 32 
*a^4*b^5*B*Sin[3*(c + d*x)] - 36*a^2*b^7*B*Sin[3*(c + d*x)])/((a^2 - b^2)^ 
3*(a + b*Cos[c + d*x])^3))/(24*b^4*d)
 

Rubi [A] (verified)

Time = 1.50 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.18, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {3042, 3468, 25, 3042, 3510, 25, 3042, 3500, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3468

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3510

\(\displaystyle \frac {\frac {\int -\frac {-6 b \left (a^2-b^2\right )^2 B \cos ^2(c+d x)+\left (3 B a^5-10 b^2 B a^3+A b^3 a^2+12 b^4 B a-6 A b^5\right ) \cos (c+d x)+2 a b \left (3 B a^3-8 b^2 B a+5 A b^3\right )}{(a+b \cos (c+d x))^2}dx}{2 b^2 \left (a^2-b^2\right )}+\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {-6 b \left (a^2-b^2\right )^2 B \cos ^2(c+d x)+\left (3 B a^5-10 b^2 B a^3+A b^3 a^2+12 b^4 B a-6 A b^5\right ) \cos (c+d x)+2 a b \left (3 B a^3-8 b^2 B a+5 A b^3\right )}{(a+b \cos (c+d x))^2}dx}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int \frac {-6 b \left (a^2-b^2\right )^2 B \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (3 B a^5-10 b^2 B a^3+A b^3 a^2+12 b^4 B a-6 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a b \left (3 B a^3-8 b^2 B a+5 A b^3\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3500

\(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int -\frac {3 \left (b \left (2 A b^6-6 a B b^5+3 a^2 A b^4+2 a^3 B b^3-a^5 B b\right )-2 b \left (a^2-b^2\right )^3 B \cos (c+d x)\right )}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {3 \int \frac {b \left (2 A b^6-6 a B b^5+3 a^2 A b^4+2 a^3 B b^3-a^5 B b\right )-2 b \left (a^2-b^2\right )^3 B \cos (c+d x)}{a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {3 \int \frac {b \left (2 A b^6-6 a B b^5+3 a^2 A b^4+2 a^3 B b^3-a^5 B b\right )-2 b \left (a^2-b^2\right )^3 B \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3214

\(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {3 \left (\left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \int \frac {1}{a+b \cos (c+d x)}dx-2 B x \left (a^2-b^2\right )^3\right )}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {3 \left (\left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-2 B x \left (a^2-b^2\right )^3\right )}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {3 \left (\frac {2 \left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{d}-2 B x \left (a^2-b^2\right )^3\right )}{b \left (a^2-b^2\right )}+\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}+\frac {a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {\frac {a^2 \left (3 a^3 B-8 a b^2 B+5 A b^3\right ) \sin (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {a \left (9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {3 \left (\frac {2 \left (2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}-2 B x \left (a^2-b^2\right )^3\right )}{b \left (a^2-b^2\right )}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\)

Input:

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

Output:

(a*(A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[ 
c + d*x])^3) + ((a^2*(5*A*b^3 + 3*a^3*B - 8*a*b^2*B)*Sin[c + d*x])/(2*b^2* 
(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - ((3*(-2*(a^2 - b^2)^3*B*x + (2*(3* 
a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^3*b^4*B - 8*a*b^6*B)*Arc 
Tan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]* 
d)))/(b*(a^2 - b^2)) + (a*(a^2*A*b^3 - 16*A*b^5 + 9*a^5*B - 28*a^3*b^2*B + 
 34*a*b^4*B)*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x])))/(2*b^2*(a 
^2 - b^2)))/(3*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 3500

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(b*f*(m + 1)* 
(a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x 
])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A 
*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, 
B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
 

rule 3510

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f 
_.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ 
e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S 
imp[1/(b^2*(m + 1)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( 
m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m 
 + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) 
))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F 
reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 
 0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 2.14 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.52

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {-\frac {\left (2 A \,a^{2} b^{3}+3 A a \,b^{4}+6 A \,b^{5}-2 B \,a^{5}+B \,a^{4} b +6 B \,a^{3} b^{2}-4 B \,a^{2} b^{3}-12 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (A \,a^{2} b^{3}+9 A \,b^{5}-3 B \,a^{5}+11 B \,a^{3} b^{2}-18 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 A \,a^{2} b^{3}-3 A a \,b^{4}+6 A \,b^{5}-2 B \,a^{5}-B \,a^{4} b +6 B \,a^{3} b^{2}+4 B \,a^{2} b^{3}-12 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (3 A \,a^{2} b^{5}+2 A \,b^{7}+2 B \,a^{7}-7 B \,a^{5} b^{2}+8 B \,a^{3} b^{4}-8 B a \,b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 b^{4} a^{2}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4}}+\frac {2 B \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}}{d}\)	\(459\)
default	\(\frac {-\frac {2 \left (\frac {-\frac {\left (2 A \,a^{2} b^{3}+3 A a \,b^{4}+6 A \,b^{5}-2 B \,a^{5}+B \,a^{4} b +6 B \,a^{3} b^{2}-4 B \,a^{2} b^{3}-12 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (A \,a^{2} b^{3}+9 A \,b^{5}-3 B \,a^{5}+11 B \,a^{3} b^{2}-18 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 A \,a^{2} b^{3}-3 A a \,b^{4}+6 A \,b^{5}-2 B \,a^{5}-B \,a^{4} b +6 B \,a^{3} b^{2}+4 B \,a^{2} b^{3}-12 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (3 A \,a^{2} b^{5}+2 A \,b^{7}+2 B \,a^{7}-7 B \,a^{5} b^{2}+8 B \,a^{3} b^{4}-8 B a \,b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 b^{4} a^{2}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4}}+\frac {2 B \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}}{d}\)	\(459\)
risch	\(\text {Expression too large to display}\)	\(1763\)

Input:

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

Output:

1/d*(-2/b^4*((-1/2*(2*A*a^2*b^3+3*A*a*b^4+6*A*b^5-2*B*a^5+B*a^4*b+6*B*a^3* 
b^2-4*B*a^2*b^3-12*B*a*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d* 
x+1/2*c)^5-2/3*(A*a^2*b^3+9*A*b^5-3*B*a^5+11*B*a^3*b^2-18*B*a*b^4)*a*b/(a^ 
2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(2*A*a^2*b^3-3*A*a*b 
^4+6*A*b^5-2*B*a^5-B*a^4*b+6*B*a^3*b^2+4*B*a^2*b^3-12*B*a*b^4)*a*b/(a+b)/( 
a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(tan(1/2*d*x+1/2*c)^2*a-tan(1 
/2*d*x+1/2*c)^2*b+a+b)^3+1/2*(3*A*a^2*b^5+2*A*b^7+2*B*a^7-7*B*a^5*b^2+8*B* 
a^3*b^4-8*B*a*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arcta 
n((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2)))+2*B/b^4*arctan(tan(1/2*d* 
x+1/2*c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 894 vs. \(2 (286) = 572\).

Time = 0.23 (sec) , antiderivative size = 1857, normalized size of antiderivative = 6.17 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/12*(12*(B*a^8*b^3 - 4*B*a^6*b^5 + 6*B*a^4*b^7 - 4*B*a^2*b^9 + B*b^11)*d 
*x*cos(d*x + c)^3 + 36*(B*a^9*b^2 - 4*B*a^7*b^4 + 6*B*a^5*b^6 - 4*B*a^3*b^ 
8 + B*a*b^10)*d*x*cos(d*x + c)^2 + 36*(B*a^10*b - 4*B*a^8*b^3 + 6*B*a^6*b^ 
5 - 4*B*a^4*b^7 + B*a^2*b^9)*d*x*cos(d*x + c) + 12*(B*a^11 - 4*B*a^9*b^2 + 
 6*B*a^7*b^4 - 4*B*a^5*b^6 + B*a^3*b^8)*d*x + 3*(2*B*a^10 - 7*B*a^8*b^2 + 
8*B*a^6*b^4 + 3*A*a^5*b^5 - 8*B*a^4*b^6 + 2*A*a^3*b^7 + (2*B*a^7*b^3 - 7*B 
*a^5*b^5 + 8*B*a^3*b^7 + 3*A*a^2*b^8 - 8*B*a*b^9 + 2*A*b^10)*cos(d*x + c)^ 
3 + 3*(2*B*a^8*b^2 - 7*B*a^6*b^4 + 8*B*a^4*b^6 + 3*A*a^3*b^7 - 8*B*a^2*b^8 
 + 2*A*a*b^9)*cos(d*x + c)^2 + 3*(2*B*a^9*b - 7*B*a^7*b^3 + 8*B*a^5*b^5 + 
3*A*a^4*b^6 - 8*B*a^3*b^7 + 2*A*a^2*b^8)*cos(d*x + c))*sqrt(-a^2 + b^2)*lo 
g((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*(6*B*a^10*b - 23*B*a^8*b^3 - 4*A*a^7*b^4 + 43* 
B*a^6*b^5 - 7*A*a^5*b^6 - 26*B*a^4*b^7 + 11*A*a^3*b^8 + (11*B*a^8*b^3 - 2* 
A*a^7*b^4 - 43*B*a^6*b^5 + 7*A*a^5*b^6 + 68*B*a^4*b^7 - 23*A*a^3*b^8 - 36* 
B*a^2*b^9 + 18*A*a*b^10)*cos(d*x + c)^2 + 3*(5*B*a^9*b^2 - 20*B*a^7*b^4 - 
A*a^6*b^5 + 35*B*a^5*b^6 - 8*A*a^4*b^7 - 20*B*a^3*b^8 + 9*A*a^2*b^9)*cos(d 
*x + c))*sin(d*x + c))/((a^8*b^7 - 4*a^6*b^9 + 6*a^4*b^11 - 4*a^2*b^13 + b 
^15)*d*cos(d*x + c)^3 + 3*(a^9*b^6 - 4*a^7*b^8 + 6*a^5*b^10 - 4*a^3*b^12 + 
 a*b^14)*d*cos(d*x + c)^2 + 3*(a^10*b^5 - 4*a^8*b^7 + 6*a^6*b^9 - 4*a^4...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(cos(d*x+c)^3*(A+B*cos(d*x+c))/(a+b*cos(d*x+c))^4,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. 813 vs. \(2 (286) = 572\).

Time = 0.22 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.70 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*(3*(2*B*a^7 - 7*B*a^5*b^2 + 8*B*a^3*b^4 + 3*A*a^2*b^5 - 8*B*a*b^6 + 2* 
A*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan( 
1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/((a^6*b^4 - 3 
*a^4*b^6 + 3*a^2*b^8 - b^10)*sqrt(a^2 - b^2)) + 3*(d*x + c)*B/b^4 - (6*B*a 
^8*tan(1/2*d*x + 1/2*c)^5 - 15*B*a^7*b*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^6*b^ 
2*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 45*B*a^5*b 
^3*tan(1/2*d*x + 1/2*c)^5 + 3*A*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^4*b 
^4*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 60*B*a^3* 
b^5*tan(1/2*d*x + 1/2*c)^5 + 27*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 + 36*B*a^ 
2*b^6*tan(1/2*d*x + 1/2*c)^5 - 18*A*a*b^7*tan(1/2*d*x + 1/2*c)^5 + 12*B*a^ 
8*tan(1/2*d*x + 1/2*c)^3 - 56*B*a^6*b^2*tan(1/2*d*x + 1/2*c)^3 - 4*A*a^5*b 
^3*tan(1/2*d*x + 1/2*c)^3 + 116*B*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 - 32*A*a^ 
3*b^5*tan(1/2*d*x + 1/2*c)^3 - 72*B*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 + 36*A* 
a*b^7*tan(1/2*d*x + 1/2*c)^3 + 6*B*a^8*tan(1/2*d*x + 1/2*c) + 15*B*a^7*b*t 
an(1/2*d*x + 1/2*c) - 6*B*a^6*b^2*tan(1/2*d*x + 1/2*c) - 6*A*a^5*b^3*tan(1 
/2*d*x + 1/2*c) - 45*B*a^5*b^3*tan(1/2*d*x + 1/2*c) - 3*A*a^4*b^4*tan(1/2* 
d*x + 1/2*c) - 6*B*a^4*b^4*tan(1/2*d*x + 1/2*c) - 6*A*a^3*b^5*tan(1/2*d*x 
+ 1/2*c) + 60*B*a^3*b^5*tan(1/2*d*x + 1/2*c) - 27*A*a^2*b^6*tan(1/2*d*x + 
1/2*c) + 36*B*a^2*b^6*tan(1/2*d*x + 1/2*c) - 18*A*a*b^7*tan(1/2*d*x + 1/2* 
c))/((a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*(a*tan(1/2*d*x + 1/2*c)^2 ...
 

Mupad [B] (verification not implemented)

Time = 32.60 (sec) , antiderivative size = 9733, normalized size of antiderivative = 32.34 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

((tan(c/2 + (d*x)/2)^5*(3*A*a^2*b^4 - 2*B*a^6 + 2*A*a^3*b^3 - 12*B*a^2*b^4 
 - 4*B*a^3*b^3 + 6*B*a^4*b^2 + 6*A*a*b^5 + B*a^5*b))/((a*b^3 - b^4)*(a + b 
)^3) - (tan(c/2 + (d*x)/2)*(2*B*a^6 + 3*A*a^2*b^4 - 2*A*a^3*b^3 + 12*B*a^2 
*b^4 - 4*B*a^3*b^3 - 6*B*a^4*b^2 - 6*A*a*b^5 + B*a^5*b))/((a + b)*(3*a*b^5 
 - b^6 - 3*a^2*b^4 + a^3*b^3)) + (4*tan(c/2 + (d*x)/2)^3*(A*a^3*b^3 - 3*B* 
a^6 - 18*B*a^2*b^4 + 11*B*a^4*b^2 + 9*A*a*b^5))/(3*(a + b)^2*(b^5 - 2*a*b^ 
4 + a^2*b^3)))/(d*(3*a*b^2 - tan(c/2 + (d*x)/2)^4*(3*a*b^2 + 3*a^2*b - 3*a 
^3 - 3*b^3) - tan(c/2 + (d*x)/2)^2*(3*a*b^2 - 3*a^2*b - 3*a^3 + 3*b^3) + 3 
*a^2*b + a^3 + b^3 + tan(c/2 + (d*x)/2)^6*(3*a*b^2 - 3*a^2*b + a^3 - b^3)) 
) + (2*B*atan(((B*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 8*B^2*a^14 + 4*B^2* 
b^14 - 8*B^2*a*b^13 - 8*B^2*a^13*b + 12*A^2*a^2*b^12 + 9*A^2*a^4*b^10 + 44 
*B^2*a^2*b^12 + 48*B^2*a^3*b^11 - 92*B^2*a^4*b^10 - 120*B^2*a^5*b^9 + 156* 
B^2*a^6*b^8 + 160*B^2*a^7*b^7 - 164*B^2*a^8*b^6 - 120*B^2*a^9*b^5 + 117*B^ 
2*a^10*b^4 + 48*B^2*a^11*b^3 - 48*B^2*a^12*b^2 - 32*A*B*a*b^13 - 16*A*B*a^ 
3*b^11 + 20*A*B*a^5*b^9 - 34*A*B*a^7*b^7 + 12*A*B*a^9*b^5))/(a*b^16 + b^17 
 - 5*a^2*b^15 - 5*a^3*b^14 + 10*a^4*b^13 + 10*a^5*b^12 - 10*a^6*b^11 - 10* 
a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - a^10*b^7 - a^11*b^6) + (B*((8*(4*A*b^21 
 + 4*B*b^21 - 6*A*a^2*b^19 + 6*A*a^3*b^18 - 6*A*a^4*b^17 + 6*A*a^5*b^16 + 
14*A*a^6*b^15 - 14*A*a^7*b^14 - 6*A*a^8*b^13 + 6*A*a^9*b^12 - 12*B*a^2*b^1 
9 + 64*B*a^3*b^18 + 20*B*a^4*b^17 - 110*B*a^5*b^16 - 30*B*a^6*b^15 + 11...
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1004, normalized size of antiderivative = 3.34 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 8*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqr 
t(a**2 - b**2))*cos(c + d*x)*a**6*b + 20*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**4*b**3 
- 24*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**5 + 4*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**5*b 
**2 - 10*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**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))*sin(c + d*x)** 
2*a*b**6 - 4*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2) 
*b)/sqrt(a**2 - b**2))*a**7 + 6*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a 
 - tan((c + d*x)/2)*b)/sqrt(a**2 - b**2))*a**5*b**2 - 2*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** 
4 - 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))*a*b**6 - 3*cos(c + d*x)*sin(c + d*x)*a**6*b**2 + 9*cos(c 
+ d*x)*sin(c + d*x)*a**4*b**4 - 6*cos(c + d*x)*sin(c + d*x)*a**2*b**6 + 4* 
cos(c + d*x)*a**7*b*d*x - 12*cos(c + d*x)*a**5*b**3*d*x + 12*cos(c + d*x)* 
a**3*b**5*d*x - 4*cos(c + d*x)*a*b**7*d*x - 2*sin(c + d*x)**2*a**6*b**2*d* 
x + 6*sin(c + d*x)**2*a**4*b**4*d*x - 6*sin(c + d*x)**2*a**2*b**6*d*x + 2* 
sin(c + d*x)**2*b**8*d*x - 2*sin(c + d*x)*a**7*b + 7*sin(c + d*x)*a**5*...