\(\int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx\) [279]

Optimal result

Integrand size = 31, antiderivative size = 420 \[ \int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\frac {b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {(4 A b-a B) \text {arctanh}(\sin (c+d x))}{a^5 d}+\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \tan (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \tan (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \] Output:

b*(20*A*a^6*b-35*A*a^4*b^3+28*A*a^2*b^5-8*A*b^7-8*B*a^7+8*B*a^5*b^2-7*B*a^ 
3*b^4+2*B*a*b^6)*arctan((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a 
-b)^(7/2)/(a+b)^(7/2)/d-(4*A*b-B*a)*arctanh(sin(d*x+c))/a^5/d+1/6*(6*A*a^6 
-65*A*a^4*b^2+68*A*a^2*b^4-24*A*b^6+26*B*a^5*b-17*B*a^3*b^3+6*B*a*b^5)*tan 
(d*x+c)/a^4/(a^2-b^2)^3/d+1/3*b*(A*b-B*a)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*co 
s(d*x+c))^3+1/6*b*(9*A*a^2*b-4*A*b^3-6*B*a^3+B*a*b^2)*tan(d*x+c)/a^2/(a^2- 
b^2)^2/d/(a+b*cos(d*x+c))^2+1/2*b*(12*A*a^4*b-11*A*a^2*b^3+4*A*b^5-6*B*a^5 
+2*B*a^3*b^2-B*a*b^4)*tan(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*cos(d*x+c))
 

Mathematica [A] (verified)

Time = 4.94 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.31 \[ \int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\frac {-\frac {48 b \left (-20 a^6 A b+35 a^4 A b^3-28 a^2 A b^5+8 A b^7+8 a^7 B-8 a^5 b^2 B+7 a^3 b^4 B-2 a b^6 B\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}+48 (4 A b-a B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 (-4 A b+a B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \left (24 a^9 A-36 a^7 A b^2-246 a^5 A b^4+318 a^3 A b^6-120 a A b^8+120 a^6 b^3 B-90 a^4 b^5 B+30 a^2 b^7 B+b \left (72 a^8 A-438 a^6 A b^2+305 a^4 A b^4+28 a^2 A b^6-72 A b^8+144 a^7 b B-50 a^5 b^3 B-7 a^3 b^5 B+18 a b^7 B\right ) \cos (c+d x)+6 a b^2 \left (6 a^6 A-53 a^4 A b^2+57 a^2 A b^4-20 A b^6+20 a^5 b B-15 a^3 b^3 B+5 a b^5 B\right ) \cos (2 (c+d x))+6 a^6 A b^3 \cos (3 (c+d x))-65 a^4 A b^5 \cos (3 (c+d x))+68 a^2 A b^7 \cos (3 (c+d x))-24 A b^9 \cos (3 (c+d x))+26 a^5 b^4 B \cos (3 (c+d x))-17 a^3 b^6 B \cos (3 (c+d x))+6 a b^8 B \cos (3 (c+d x))\right ) \tan (c+d x)}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}}{48 a^5 d} \] Input:

Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]
 

Output:

((-48*b*(-20*a^6*A*b + 35*a^4*A*b^3 - 28*a^2*A*b^5 + 8*A*b^7 + 8*a^7*B - 8 
*a^5*b^2*B + 7*a^3*b^4*B - 2*a*b^6*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/S 
qrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + 48*(4*A*b - a*B)*Log[Cos[(c + d*x)/ 
2] - Sin[(c + d*x)/2]] + 48*(-4*A*b + a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + 
 d*x)/2]] + (2*a*(24*a^9*A - 36*a^7*A*b^2 - 246*a^5*A*b^4 + 318*a^3*A*b^6 
- 120*a*A*b^8 + 120*a^6*b^3*B - 90*a^4*b^5*B + 30*a^2*b^7*B + b*(72*a^8*A 
- 438*a^6*A*b^2 + 305*a^4*A*b^4 + 28*a^2*A*b^6 - 72*A*b^8 + 144*a^7*b*B - 
50*a^5*b^3*B - 7*a^3*b^5*B + 18*a*b^7*B)*Cos[c + d*x] + 6*a*b^2*(6*a^6*A - 
 53*a^4*A*b^2 + 57*a^2*A*b^4 - 20*A*b^6 + 20*a^5*b*B - 15*a^3*b^3*B + 5*a* 
b^5*B)*Cos[2*(c + d*x)] + 6*a^6*A*b^3*Cos[3*(c + d*x)] - 65*a^4*A*b^5*Cos[ 
3*(c + d*x)] + 68*a^2*A*b^7*Cos[3*(c + d*x)] - 24*A*b^9*Cos[3*(c + d*x)] + 
 26*a^5*b^4*B*Cos[3*(c + d*x)] - 17*a^3*b^6*B*Cos[3*(c + d*x)] + 6*a*b^8*B 
*Cos[3*(c + d*x)])*Tan[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3))/( 
48*a^5*d)
 

Rubi [A] (verified)

Time = 3.11 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.13, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {3042, 3479, 3042, 3534, 3042, 3534, 3042, 3534, 27, 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.

\(\displaystyle \int \frac {\sec ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3479

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {\int \frac {\left (6 A a^4+8 b B a^3-23 A b^2 a^2-3 b^3 B a-2 \left (-3 B a^3+6 A b a^2-2 b^2 B a-A b^3\right ) \cos (c+d x) a+12 A b^4+2 b \left (-6 B a^3+9 A b a^2+b^2 B a-4 A b^3\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {6 A a^4+8 b B a^3-23 A b^2 a^2-3 b^3 B a-2 \left (-3 B a^3+6 A b a^2-2 b^2 B a-A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+12 A b^4+2 b \left (-6 B a^3+9 A b a^2+b^2 B a-4 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (6 A a^6+26 b B a^5-65 A b^2 a^4-17 b^3 B a^3+68 A b^4 a^2+6 b^5 B a-\left (-6 B a^5+18 A b a^4-8 b^2 B a^3-7 A b^3 a^2-b^4 B a+4 A b^5\right ) \cos (c+d x) a-24 A b^6+3 b \left (-6 B a^5+12 A b a^4+2 b^2 B a^3-11 A b^3 a^2-b^4 B a+4 A b^5\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {6 A a^6+26 b B a^5-65 A b^2 a^4-17 b^3 B a^3+68 A b^4 a^2+6 b^5 B a-\left (-6 B a^5+18 A b a^4-8 b^2 B a^3-7 A b^3 a^2-b^4 B a+4 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-24 A b^6+3 b \left (-6 B a^5+12 A b a^4+2 b^2 B a^3-11 A b^3 a^2-b^4 B a+4 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {\frac {\frac {\int -\frac {3 \left (2 \left (a^2-b^2\right )^3 (4 A b-a B)-a b \left (-6 B a^5+12 A b a^4+2 b^2 B a^3-11 A b^3 a^2-b^4 B a+4 A b^5\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \tan (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \int \frac {\left (2 \left (a^2-b^2\right )^3 (4 A b-a B)-a b \left (-6 B a^5+12 A b a^4+2 b^2 B a^3-11 A b^3 a^2-b^4 B a+4 A b^5\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \int \frac {2 \left (a^2-b^2\right )^3 (4 A b-a B)-a b \left (-6 B a^5+12 A b a^4+2 b^2 B a^3-11 A b^3 a^2-b^4 B a+4 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3480

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 (4 A b-a B) \int \sec (c+d x)dx}{a}-\frac {b \left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 (4 A b-a B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b \left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 (4 A b-a B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b \left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 (4 A b-a B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b \left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a \left (a^2-b^2\right )}+\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\frac {3 b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {2 \left (a^2-b^2\right )^3 (4 A b-a B) \text {arctanh}(\sin (c+d x))}{a d}-\frac {2 b \left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\)

Input:

Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]
 

Output:

(b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + 
((b*(9*a^2*A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*Tan[c + d*x])/(2*a*(a^2 - b^ 
2)*d*(a + b*Cos[c + d*x])^2) + ((3*b*(12*a^4*A*b - 11*a^2*A*b^3 + 4*A*b^5 
- 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*C 
os[c + d*x])) + ((-3*((-2*b*(20*a^6*A*b - 35*a^4*A*b^3 + 28*a^2*A*b^5 - 8* 
A*b^7 - 8*a^7*B + 8*a^5*b^2*B - 7*a^3*b^4*B + 2*a*b^6*B)*ArcTan[(Sqrt[a - 
b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d) + (2*(a^2 
 - b^2)^3*(4*A*b - a*B)*ArcTanh[Sin[c + d*x]])/(a*d)))/a + ((6*a^6*A - 65* 
a^4*A*b^2 + 68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5* 
B)*Tan[c + d*x])/(a*d))/(a*(a^2 - b^2)))/(2*a*(a^2 - b^2)))/(3*a*(a^2 - b^ 
2))
 

Defintions of rubi rules used

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 = 3.99 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.40

method	result	size
derivativedivides	\(\frac {\frac {2 b \left (\frac {-\frac {\left (20 A \,a^{4} b +5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}-2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}-4 B \,a^{4} b +6 B \,a^{3} b^{2}+B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (30 A \,a^{4} b -29 A \,a^{2} b^{3}+9 A \,b^{5}-18 B \,a^{5}+11 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b -5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}+2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}+4 B \,a^{4} b +6 B \,a^{3} b^{2}-B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (20 A \,a^{6} b -35 A \,a^{4} b^{3}+28 A \,a^{2} b^{5}-8 A \,b^{7}-8 B \,a^{7}+8 B \,a^{5} b^{2}-7 B \,a^{3} b^{4}+2 B a \,b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 b^{4} a^{2}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-4 A b +B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{5}}-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (4 A b -B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{5}}}{d}\)	\(587\)
default	\(\frac {\frac {2 b \left (\frac {-\frac {\left (20 A \,a^{4} b +5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}-2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}-4 B \,a^{4} b +6 B \,a^{3} b^{2}+B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (30 A \,a^{4} b -29 A \,a^{2} b^{3}+9 A \,b^{5}-18 B \,a^{5}+11 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b -5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}+2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}+4 B \,a^{4} b +6 B \,a^{3} b^{2}-B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (20 A \,a^{6} b -35 A \,a^{4} b^{3}+28 A \,a^{2} b^{5}-8 A \,b^{7}-8 B \,a^{7}+8 B \,a^{5} b^{2}-7 B \,a^{3} b^{4}+2 B a \,b^{6}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 b^{4} a^{2}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-4 A b +B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{5}}-\frac {A}{a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (4 A b -B a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{5}}}{d}\)	\(587\)
risch	\(\text {Expression too large to display}\)	\(2561\)

Input:

int((A+B*cos(d*x+c))*sec(d*x+c)^2/(a+cos(d*x+c)*b)^4,x,method=_RETURNVERBO 
SE)
 

Output:

1/d*(2*b/a^5*((-1/2*(20*A*a^4*b+5*A*a^3*b^2-18*A*a^2*b^3-2*A*a*b^4+6*A*b^5 
-12*B*a^5-4*B*a^4*b+6*B*a^3*b^2+B*a^2*b^3-2*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*(30*A*a^4*b-29*A*a^2*b^3+9*A*b^5-1 
8*B*a^5+11*B*a^3*b^2-3*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*(20*A*a^4*b-5*A*a^3*b^2-18*A*a^2*b^3+2*A*a*b^4+6*A*b^5- 
12*B*a^5+4*B*a^4*b+6*B*a^3*b^2-B*a^2*b^3-2*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*(20*A*a^6*b-35*A*a^4*b^3+28*A*a^2*b^5-8*A*b^7-8*B*a^7+8* 
B*a^5*b^2-7*B*a^3*b^4+2*B*a*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b 
))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2)))-A/a^4/(tan( 
1/2*d*x+1/2*c)+1)+1/a^5*(-4*A*b+B*a)*ln(tan(1/2*d*x+1/2*c)+1)-A/a^4/(tan(1 
/2*d*x+1/2*c)-1)+(4*A*b-B*a)/a^5*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. 1662 vs. \(2 (402) = 804\).

Time = 66.78 (sec) , antiderivative size = 3393, normalized size of antiderivative = 8.08 \[ \int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*cos(d*x+c))*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x, algorithm="f 
ricas")
 

Output:

Too large to include
 

Sympy [F]

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

integrate((A+B*cos(d*x+c))*sec(d*x+c)**2/(a+b*cos(d*x+c))**4,x)
 

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((A+B*cos(d*x+c))*sec(d*x+c)^2/(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. 996 vs. \(2 (402) = 804\).

Time = 0.32 (sec) , antiderivative size = 996, normalized size of antiderivative = 2.37 \[ \int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*cos(d*x+c))*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x, algorithm="g 
iac")
 

Output:

1/3*(3*(8*B*a^7*b - 20*A*a^6*b^2 - 8*B*a^5*b^3 + 35*A*a^4*b^4 + 7*B*a^3*b^ 
5 - 28*A*a^2*b^6 - 2*B*a*b^7 + 8*A*b^8)*(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^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(a^2 - b 
^2)) + (36*B*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 - 60*A*a^6*b^3*tan(1/2*d*x + 1 
/2*c)^5 - 60*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 105*A*a^5*b^4*tan(1/2*d*x 
+ 1/2*c)^5 - 6*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 + 24*A*a^4*b^5*tan(1/2*d*x 
 + 1/2*c)^5 + 45*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 - 117*A*a^3*b^6*tan(1/2* 
d*x + 1/2*c)^5 - 6*B*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 24*A*a^2*b^7*tan(1/2 
*d*x + 1/2*c)^5 - 15*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 + 42*A*a*b^8*tan(1/2 
*d*x + 1/2*c)^5 + 6*B*a*b^8*tan(1/2*d*x + 1/2*c)^5 - 18*A*b^9*tan(1/2*d*x 
+ 1/2*c)^5 + 72*B*a^7*b^2*tan(1/2*d*x + 1/2*c)^3 - 120*A*a^6*b^3*tan(1/2*d 
*x + 1/2*c)^3 - 116*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 + 236*A*a^4*b^5*tan(1 
/2*d*x + 1/2*c)^3 + 56*B*a^3*b^6*tan(1/2*d*x + 1/2*c)^3 - 152*A*a^2*b^7*ta 
n(1/2*d*x + 1/2*c)^3 - 12*B*a*b^8*tan(1/2*d*x + 1/2*c)^3 + 36*A*b^9*tan(1/ 
2*d*x + 1/2*c)^3 + 36*B*a^7*b^2*tan(1/2*d*x + 1/2*c) - 60*A*a^6*b^3*tan(1/ 
2*d*x + 1/2*c) + 60*B*a^6*b^3*tan(1/2*d*x + 1/2*c) - 105*A*a^5*b^4*tan(1/2 
*d*x + 1/2*c) - 6*B*a^5*b^4*tan(1/2*d*x + 1/2*c) + 24*A*a^4*b^5*tan(1/2*d* 
x + 1/2*c) - 45*B*a^4*b^5*tan(1/2*d*x + 1/2*c) + 117*A*a^3*b^6*tan(1/2*d*x 
 + 1/2*c) - 6*B*a^3*b^6*tan(1/2*d*x + 1/2*c) + 24*A*a^2*b^7*tan(1/2*d*x...
 

Mupad [B] (verification not implemented)

Time = 41.46 (sec) , antiderivative size = 13119, normalized size of antiderivative = 31.24 \[ \int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \] Input:

int((A + B*cos(c + d*x))/(cos(c + d*x)^2*(a + b*cos(c + d*x))^4),x)
 

Output:

((tan(c/2 + (d*x)/2)^3*(18*A*a^8 + 72*A*b^8 - 236*A*a^2*b^6 + 47*A*a^3*b^5 
 + 273*A*a^4*b^4 - 60*A*a^5*b^3 - 72*A*a^6*b^2 + 3*B*a^2*b^6 + 59*B*a^3*b^ 
5 - 14*B*a^4*b^4 - 96*B*a^5*b^3 + 36*B*a^6*b^2 - 12*A*a*b^7 - 18*B*a*b^7)) 
/(3*a^4*(a + b)^2*(a - b)^3) - (tan(c/2 + (d*x)/2)^7*(24*A*a^2*b^5 - 8*A*b 
^7 - 2*A*a^7 - 11*A*a^3*b^4 - 26*A*a^4*b^3 + 6*A*a^5*b^2 - B*a^2*b^5 - 6*B 
*a^3*b^4 + 4*B*a^4*b^3 + 12*B*a^5*b^2 + 4*A*a*b^6 + 2*A*a^6*b + 2*B*a*b^6) 
)/(a^4*(a + b)^3*(a - b)) + (tan(c/2 + (d*x)/2)^5*(18*A*a^8 + 72*A*b^8 - 2 
36*A*a^2*b^6 - 47*A*a^3*b^5 + 273*A*a^4*b^4 + 60*A*a^5*b^3 - 72*A*a^6*b^2 
- 3*B*a^2*b^6 + 59*B*a^3*b^5 + 14*B*a^4*b^4 - 96*B*a^5*b^3 - 36*B*a^6*b^2 
+ 12*A*a*b^7 - 18*B*a*b^7))/(3*a^4*(a + b)^3*(a - b)^2) + (tan(c/2 + (d*x) 
/2)*(2*A*a^7 - 8*A*b^7 + 24*A*a^2*b^5 + 11*A*a^3*b^4 - 26*A*a^4*b^3 - 6*A* 
a^5*b^2 + B*a^2*b^5 - 6*B*a^3*b^4 - 4*B*a^4*b^3 + 12*B*a^5*b^2 - 4*A*a*b^6 
 + 2*A*a^6*b + 2*B*a*b^6))/(a^4*(a + b)*(a - b)^3))/(d*(3*a*b^2 + 3*a^2*b 
- tan(c/2 + (d*x)/2)^4*(6*a^2*b - 6*b^3) - tan(c/2 + (d*x)/2)^2*(6*a*b^2 - 
 2*a^3 + 4*b^3) - tan(c/2 + (d*x)/2)^6*(2*a^3 - 6*a*b^2 + 4*b^3) + a^3 + b 
^3 - tan(c/2 + (d*x)/2)^8*(3*a*b^2 - 3*a^2*b + a^3 - b^3))) + (atan((((((4 
*A*b - B*a)*((8*(4*B*a^24 + 16*A*a^10*b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^ 
12 + 50*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 
174*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72*A*a 
^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13 + 2*B*a^12*b^12 + 26*B*a^13*b^1...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 2086, normalized size of antiderivative = 4.97 \[ \int \frac {(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx =\text {Too large to display} \] Input:

int((A+B*cos(d*x+c))*sec(d*x+c)^2/(a+b*cos(d*x+c))^4,x)
 

Output:

(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)*sin(c + d*x)**2*a**4*b**4 - 30*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*a**2*b**6 + 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**8 - 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**6*b**2 + 6*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**4 + 18*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**6 - 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**8 + 48*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**3 - 60*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**5 + 24*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**7 - 48*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* 
*3 + 60*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/s 
qrt(a**2 - b**2))*a**3*b**5 - 24*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**7 + 6*cos(c + d*x)*log(...