\(\int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx\) [341]

Optimal result

Integrand size = 23, antiderivative size = 292 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {A x}{a^4}-\frac {\left (8 a^6 A b-8 a^4 A b^3+7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {b (A b-a B) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b \left (8 a^2 A b-3 A b^3-5 a^3 B\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b \left (26 a^4 A b-17 a^2 A b^3+6 A b^5-11 a^5 B-4 a^3 b^2 B\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \] Output:

A*x/a^4-(8*A*a^6*b-8*A*a^4*b^3+7*A*a^2*b^5-2*A*b^7-2*B*a^7-3*B*a^5*b^2)*ar 
ctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^4/(a-b)^(7/2)/(a+b)^(7 
/2)/d+1/3*b*(A*b-B*a)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^3+1/6*b*(8 
*A*a^2*b-3*A*b^3-5*B*a^3)*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2+ 
1/6*b*(26*A*a^4*b-17*A*a^2*b^3+6*A*b^5-11*B*a^5-4*B*a^3*b^2)*tan(d*x+c)/a^ 
3/(a^2-b^2)^3/d/(a+b*sec(d*x+c))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(769\) vs. \(2(292)=584\).

Time = 3.72 (sec) , antiderivative size = 769, normalized size of antiderivative = 2.63 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {(b+a \cos (c+d x)) \sec ^3(c+d x) (A+B \sec (c+d x)) \left (-\frac {24 \left (-8 a^6 A b+8 a^4 A b^3-7 a^2 A b^5+2 A b^7+2 a^7 B+3 a^5 b^2 B\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^3}{\left (a^2-b^2\right )^{7/2}}+\frac {36 a^8 A b c-84 a^6 A b^3 c+36 a^4 A b^5 c+36 a^2 A b^7 c-24 A b^9 c+36 a^8 A b d x-84 a^6 A b^3 d x+36 a^4 A b^5 d x+36 a^2 A b^7 d x-24 A b^9 d x+18 a A \left (a^2-b^2\right )^3 \left (a^2+4 b^2\right ) (c+d x) \cos (c+d x)+36 a^2 A b \left (a^2-b^2\right )^3 (c+d x) \cos (2 (c+d x))+6 a^9 A c \cos (3 (c+d x))-18 a^7 A b^2 c \cos (3 (c+d x))+18 a^5 A b^4 c \cos (3 (c+d x))-6 a^3 A b^6 c \cos (3 (c+d x))+6 a^9 A d x \cos (3 (c+d x))-18 a^7 A b^2 d x \cos (3 (c+d x))+18 a^5 A b^4 d x \cos (3 (c+d x))-6 a^3 A b^6 d x \cos (3 (c+d x))+36 a^7 A b^2 \sin (c+d x)+72 a^5 A b^4 \sin (c+d x)-57 a^3 A b^6 \sin (c+d x)+24 a A b^8 \sin (c+d x)-18 a^8 b B \sin (c+d x)-39 a^6 b^3 B \sin (c+d x)-18 a^4 b^5 B \sin (c+d x)+120 a^6 A b^3 \sin (2 (c+d x))-90 a^4 A b^5 \sin (2 (c+d x))+30 a^2 A b^7 \sin (2 (c+d x))-54 a^7 b^2 B \sin (2 (c+d x))-6 a^5 b^4 B \sin (2 (c+d x))+36 a^7 A b^2 \sin (3 (c+d x))-32 a^5 A b^4 \sin (3 (c+d x))+11 a^3 A b^6 \sin (3 (c+d x))-18 a^8 b B \sin (3 (c+d x))+5 a^6 b^3 B \sin (3 (c+d x))-2 a^4 b^5 B \sin (3 (c+d x))}{\left (a^2-b^2\right )^3}\right )}{24 a^4 d (B+A \cos (c+d x)) (a+b \sec (c+d x))^4} \] Input:

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

Output:

((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x])*((-24*(-8*a^6*A* 
b + 8*a^4*A*b^3 - 7*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B + 3*a^5*b^2*B)*ArcTanh[( 
(-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3)/(a^2 - 
 b^2)^(7/2) + (36*a^8*A*b*c - 84*a^6*A*b^3*c + 36*a^4*A*b^5*c + 36*a^2*A*b 
^7*c - 24*A*b^9*c + 36*a^8*A*b*d*x - 84*a^6*A*b^3*d*x + 36*a^4*A*b^5*d*x + 
 36*a^2*A*b^7*d*x - 24*A*b^9*d*x + 18*a*A*(a^2 - b^2)^3*(a^2 + 4*b^2)*(c + 
 d*x)*Cos[c + d*x] + 36*a^2*A*b*(a^2 - b^2)^3*(c + d*x)*Cos[2*(c + d*x)] + 
 6*a^9*A*c*Cos[3*(c + d*x)] - 18*a^7*A*b^2*c*Cos[3*(c + d*x)] + 18*a^5*A*b 
^4*c*Cos[3*(c + d*x)] - 6*a^3*A*b^6*c*Cos[3*(c + d*x)] + 6*a^9*A*d*x*Cos[3 
*(c + d*x)] - 18*a^7*A*b^2*d*x*Cos[3*(c + d*x)] + 18*a^5*A*b^4*d*x*Cos[3*( 
c + d*x)] - 6*a^3*A*b^6*d*x*Cos[3*(c + d*x)] + 36*a^7*A*b^2*Sin[c + d*x] + 
 72*a^5*A*b^4*Sin[c + d*x] - 57*a^3*A*b^6*Sin[c + d*x] + 24*a*A*b^8*Sin[c 
+ d*x] - 18*a^8*b*B*Sin[c + d*x] - 39*a^6*b^3*B*Sin[c + d*x] - 18*a^4*b^5* 
B*Sin[c + d*x] + 120*a^6*A*b^3*Sin[2*(c + d*x)] - 90*a^4*A*b^5*Sin[2*(c + 
d*x)] + 30*a^2*A*b^7*Sin[2*(c + d*x)] - 54*a^7*b^2*B*Sin[2*(c + d*x)] - 6* 
a^5*b^4*B*Sin[2*(c + d*x)] + 36*a^7*A*b^2*Sin[3*(c + d*x)] - 32*a^5*A*b^4* 
Sin[3*(c + d*x)] + 11*a^3*A*b^6*Sin[3*(c + d*x)] - 18*a^8*b*B*Sin[3*(c + d 
*x)] + 5*a^6*b^3*B*Sin[3*(c + d*x)] - 2*a^4*b^5*B*Sin[3*(c + d*x)])/(a^2 - 
 b^2)^3))/(24*a^4*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^4)
 

Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.22, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {3042, 4411, 25, 3042, 4548, 25, 3042, 4548, 27, 3042, 4407, 3042, 4318, 3042, 3138, 221}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4411

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4548

\(\displaystyle \frac {\frac {b \left (-5 a^3 B+8 a^2 A b-3 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int -\frac {6 A \left (a^2-b^2\right )^2+b \left (-5 B a^3+8 A b a^2-3 A b^3\right ) \sec ^2(c+d x)-2 a \left (-3 B a^3+6 A b a^2-2 b^2 B a-A b^3\right ) \sec (c+d x)}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {6 A \left (a^2-b^2\right )^2+b \left (-5 B a^3+8 A b a^2-3 A b^3\right ) \sec ^2(c+d x)-2 a \left (-3 B a^3+6 A b a^2-2 b^2 B a-A b^3\right ) \sec (c+d x)}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+8 a^2 A b-3 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {6 A \left (a^2-b^2\right )^2+b \left (-5 B a^3+8 A b a^2-3 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a \left (-3 B a^3+6 A b a^2-2 b^2 B a-A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+8 a^2 A b-3 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 4548

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 \int \frac {2 A \left (a^2-b^2\right )^3-a \left (-2 B a^5+6 A b a^4-3 b^2 B a^3-2 A b^3 a^2+A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a \left (a^2-b^2\right )}+\frac {b \left (-11 a^5 B+26 a^4 A b-4 a^3 b^2 B-17 a^2 A b^3+6 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+8 a^2 A b-3 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 4407

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A x \left (a^2-b^2\right )^3}{a}-\frac {\left (-2 a^7 B+8 a^6 A b-3 a^5 b^2 B-8 a^4 A b^3+7 a^2 A b^5-2 A b^7\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a \left (a^2-b^2\right )}+\frac {b \left (-11 a^5 B+26 a^4 A b-4 a^3 b^2 B-17 a^2 A b^3+6 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+8 a^2 A b-3 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 4318

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A x \left (a^2-b^2\right )^3}{a}-\frac {\left (-2 a^7 B+8 a^6 A b-3 a^5 b^2 B-8 a^4 A b^3+7 a^2 A b^5-2 A b^7\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a b}\right )}{a \left (a^2-b^2\right )}+\frac {b \left (-11 a^5 B+26 a^4 A b-4 a^3 b^2 B-17 a^2 A b^3+6 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+8 a^2 A b-3 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A x \left (a^2-b^2\right )^3}{a}-\frac {2 \left (-2 a^7 B+8 a^6 A b-3 a^5 b^2 B-8 a^4 A b^3+7 a^2 A b^5-2 A b^7\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a b d}\right )}{a \left (a^2-b^2\right )}+\frac {b \left (-11 a^5 B+26 a^4 A b-4 a^3 b^2 B-17 a^2 A b^3+6 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-5 a^3 B+8 a^2 A b-3 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {b (A b-a B) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {\frac {b \left (-5 a^3 B+8 a^2 A b-3 A b^3\right ) \tan (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\frac {b \left (-11 a^5 B+26 a^4 A b-4 a^3 b^2 B-17 a^2 A b^3+6 A b^5\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {3 \left (\frac {2 A x \left (a^2-b^2\right )^3}{a}-\frac {2 \left (-2 a^7 B+8 a^6 A b-3 a^5 b^2 B-8 a^4 A b^3+7 a^2 A b^5-2 A b^7\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\)

Input:

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

Output:

(b*(A*b - a*B)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + 
((b*(8*a^2*A*b - 3*A*b^3 - 5*a^3*B)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + 
b*Sec[c + d*x])^2) + ((3*((2*A*(a^2 - b^2)^3*x)/a - (2*(8*a^6*A*b - 8*a^4* 
A*b^3 + 7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B)*ArcTanh[(Sqrt[a - b 
]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)))/(a*(a^2 
- b^2)) + (b*(26*a^4*A*b - 17*a^2*A*b^3 + 6*A*b^5 - 11*a^5*B - 4*a^3*b^2*B 
)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])))/(2*a*(a^2 - b^2))) 
/(3*a*(a^2 - b^2))
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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 4318

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo 
l] :> Simp[1/b   Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, 
f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4407

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

rule 4411

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d 
_.) + (c_)), x_Symbol] :> Simp[b*(b*c - a*d)*Cot[e + f*x]*((a + b*Csc[e + f 
*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2) 
)   Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - 
 a*d)*(m + 1))*Csc[e + f*x] + b*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && N 
eQ[a^2 - b^2, 0] && IntegerQ[2*m]
 

rule 4548

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

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.59

method	result	size
derivativedivides	\(\frac {\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {\frac {2 \left (-\frac {\left (12 A \,a^{4} b +4 A \,a^{3} b^{2}-6 A \,a^{2} b^{3}-A a \,b^{4}+2 A \,b^{5}-6 B \,a^{5}-3 B \,a^{4} b -2 B \,a^{3} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (18 A \,a^{4} b -11 A \,a^{2} b^{3}+3 A \,b^{5}-9 B \,a^{5}-B \,a^{3} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (12 A \,a^{4} b -4 A \,a^{3} b^{2}-6 A \,a^{2} b^{3}+A a \,b^{4}+2 A \,b^{5}-6 B \,a^{5}+3 B \,a^{4} b -2 B \,a^{3} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (8 A \,a^{6} b -8 A \,a^{4} b^{3}+7 A \,a^{2} b^{5}-2 A \,b^{7}-2 B \,a^{7}-3 B \,a^{5} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{4}}}{d}\)	\(464\)
default	\(\frac {\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {\frac {2 \left (-\frac {\left (12 A \,a^{4} b +4 A \,a^{3} b^{2}-6 A \,a^{2} b^{3}-A a \,b^{4}+2 A \,b^{5}-6 B \,a^{5}-3 B \,a^{4} b -2 B \,a^{3} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (18 A \,a^{4} b -11 A \,a^{2} b^{3}+3 A \,b^{5}-9 B \,a^{5}-B \,a^{3} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (12 A \,a^{4} b -4 A \,a^{3} b^{2}-6 A \,a^{2} b^{3}+A a \,b^{4}+2 A \,b^{5}-6 B \,a^{5}+3 B \,a^{4} b -2 B \,a^{3} b^{2}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (8 A \,a^{6} b -8 A \,a^{4} b^{3}+7 A \,a^{2} b^{5}-2 A \,b^{7}-2 B \,a^{7}-3 B \,a^{5} b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{4}}}{d}\)	\(464\)
risch	\(\text {Expression too large to display}\)	\(1763\)

Input:

int((A+B*sec(d*x+c))/(a+b*sec(d*x+c))^4,x,method=_RETURNVERBOSE)
 

Output:

1/d*(2*A/a^4*arctan(tan(1/2*d*x+1/2*c))+2/a^4*((-1/2*(12*A*a^4*b+4*A*a^3*b 
^2-6*A*a^2*b^3-A*a*b^4+2*A*b^5-6*B*a^5-3*B*a^4*b-2*B*a^3*b^2)*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*(18*A*a^4*b-11*A*a^2*b^3+ 
3*A*b^5-9*B*a^5-B*a^3*b^2)*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*(12*A*a^4*b-4*A*a^3*b^2-6*A*a^2*b^3+A*a*b^4+2*A*b^5-6*B*a^5+ 
3*B*a^4*b-2*B*a^3*b^2)*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*(8*A*a^6*b- 
8*A*a^4*b^3+7*A*a^2*b^5-2*A*b^7-2*B*a^7-3*B*a^5*b^2)/(a^6-3*a^4*b^2+3*a^2* 
b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((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. 905 vs. \(2 (276) = 552\).

Time = 0.22 (sec) , antiderivative size = 1867, normalized size of antiderivative = 6.39 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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*a^2-4*b^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. 814 vs. \(2 (276) = 552\).

Time = 0.27 (sec) , antiderivative size = 814, normalized size of antiderivative = 2.79 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 23.29 (sec) , antiderivative size = 9721, normalized size of antiderivative = 33.29 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

((tan(c/2 + (d*x)/2)^5*(6*A*a^2*b^4 - 2*A*b^6 - 4*A*a^3*b^3 - 12*A*a^4*b^2 
 + 2*B*a^3*b^3 + 3*B*a^4*b^2 + A*a*b^5 + 6*B*a^5*b))/((a^3*b - a^4)*(a + b 
)^3) - (tan(c/2 + (d*x)/2)*(2*A*b^6 - 6*A*a^2*b^4 - 4*A*a^3*b^3 + 12*A*a^4 
*b^2 - 2*B*a^3*b^3 + 3*B*a^4*b^2 + A*a*b^5 - 6*B*a^5*b))/((a + b)*(3*a^5*b 
 - a^6 + a^3*b^3 - 3*a^4*b^2)) + (4*tan(c/2 + (d*x)/2)^3*(11*A*a^2*b^4 - 3 
*A*b^6 - 18*A*a^4*b^2 + B*a^3*b^3 + 9*B*a^5*b))/(3*(a + b)^2*(a^5 - 2*a^4* 
b + a^3*b^2)))/(d*(tan(c/2 + (d*x)/2)^2*(3*a*b^2 - 3*a^2*b - 3*a^3 + 3*b^3 
) - tan(c/2 + (d*x)/2)^4*(3*a*b^2 + 3*a^2*b - 3*a^3 - 3*b^3) + 3*a*b^2 + 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*A*atan(-((A*((8*tan(c/2 + (d*x)/2)*(4*A^2*a^14 + 8*A^2*b^14 + 4*B^2 
*a^14 - 8*A^2*a*b^13 - 8*A^2*a^13*b - 48*A^2*a^2*b^12 + 48*A^2*a^3*b^11 + 
117*A^2*a^4*b^10 - 120*A^2*a^5*b^9 - 164*A^2*a^6*b^8 + 160*A^2*a^7*b^7 + 1 
56*A^2*a^8*b^6 - 120*A^2*a^9*b^5 - 92*A^2*a^10*b^4 + 48*A^2*a^11*b^3 + 44* 
A^2*a^12*b^2 + 9*B^2*a^10*b^4 + 12*B^2*a^12*b^2 - 32*A*B*a^13*b + 12*A*B*a 
^5*b^9 - 34*A*B*a^7*b^7 + 20*A*B*a^9*b^5 - 16*A*B*a^11*b^3))/(a^16*b + a^1 
7 - a^6*b^11 - a^7*b^10 + 5*a^8*b^9 + 5*a^9*b^8 - 10*a^10*b^7 - 10*a^11*b^ 
6 + 10*a^12*b^5 + 10*a^13*b^4 - 5*a^14*b^3 - 5*a^15*b^2) + (A*((8*(4*A*a^2 
1 + 4*B*a^21 - 4*A*a^8*b^13 + 2*A*a^9*b^12 + 26*A*a^10*b^11 - 14*A*a^11*b^ 
10 - 70*A*a^12*b^9 + 30*A*a^13*b^8 + 110*A*a^14*b^7 - 30*A*a^15*b^6 - 110* 
A*a^16*b^5 + 20*A*a^17*b^4 + 64*A*a^18*b^3 - 12*A*a^19*b^2 + 6*B*a^12*b...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1004, normalized size of antiderivative = 3.44 \[ \int \frac {A+B \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx =\text {Too large to display} \] Input:

int((A+B*sec(d*x+c))/(a+b*sec(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)*a**5*b**2 + 20*sqrt( - a**2 + b**2)*at 
an((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + 
 d*x)*a**3*b**4 - 8*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*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 
))*sin(c + d*x)**2*a**6*b - 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**4*b**3 + 
 4*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqr 
t( - a**2 + b**2))*sin(c + d*x)**2*a**2*b**5 - 12*sqrt( - a**2 + b**2)*ata 
n((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*a**6*b - 
 2*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**4*b**3 + 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**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))*b**7 + 6*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 + 3*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**8*d*x + 6*sin(c + d*x)* 
*2*a**6*b**2*d*x - 6*sin(c + d*x)**2*a**4*b**4*d*x + 2*sin(c + d*x)**2*...