\(\int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx\) [927]

Optimal result

Integrand size = 33, antiderivative size = 336 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\frac {A x}{a^4}-\frac {\left (7 a^2 A b^5-2 A b^7-2 a^7 B-3 a^5 b^2 B-a^4 b^3 (8 A-C)+4 a^6 b (2 A+C)\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\left (3 A b^4+5 a^3 b B-2 a^4 C-a^2 b^2 (8 A+3 C)\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (17 a^2 A b^4-6 A b^6+11 a^5 b B+4 a^3 b^3 B-2 a^6 C-13 a^4 b^2 (2 A+C)\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \] Output:

A*x/a^4-(7*a^2*A*b^5-2*A*b^7-2*a^7*B-3*a^5*b^2*B-a^4*b^3*(8*A-C)+4*a^6*b*( 
2*A+C))*arctanh((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*(A*b^2-a*(B*b-C*a))*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*sec( 
d*x+c))^3-1/6*(3*A*b^4+5*B*a^3*b-2*a^4*C-a^2*b^2*(8*A+3*C))*tan(d*x+c)/a^2 
/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2-1/6*(17*a^2*A*b^4-6*A*b^6+11*a^5*b*B+4*a 
^3*b^3*B-2*a^6*C-13*a^4*b^2*(2*A+C))*tan(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*sec 
(d*x+c))
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 9.25 (sec) , antiderivative size = 1230, normalized size of antiderivative = 3.66 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*A*x*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c 
 + d*x]^2))/(a^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b* 
Sec[c + d*x])^4) + ((-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 - 4*a^6*b*C - a^4*b^3*C)*(b + a*Cos[c + d*x])^4*Sec[c 
+ d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((2*I)*ArcTan[Sec[(d*x)/ 
2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqr 
t[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[ 
c + (d*x)/2])]*Cos[c])/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) 
 + (2*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2 
*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b* 
Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[ 
Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^3*(A + 2*C + 2*B*Cos[c + d*x] + A* 
Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) - (2*(b + a*Cos[c + d*x])*Sec[c] 
*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*b^5*Sin[c] - a* 
b^4*B*Sin[c] + a^2*b^3*C*Sin[c] - a*A*b^4*Sin[d*x] + a^2*b^3*B*Sin[d*x] - 
a^3*b^2*C*Sin[d*x]))/(3*a^4*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A* 
Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c] 
*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(14*a^2*A*b^4*Sin[ 
c] - 9*A*b^6*Sin[c] - 11*a^3*b^3*B*Sin[c] + 6*a*b^5*B*Sin[c] + 8*a^4*b^2*C 
*Sin[c] - 3*a^2*b^4*C*Sin[c] - 12*a^3*A*b^3*Sin[d*x] + 7*a*A*b^5*Sin[d*...
 

Rubi [A] (verified)

Time = 1.92 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.19, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {3042, 4548, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4548

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4548

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4548

\(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \left (2 A \left (a^2-b^2\right )^3-a \left (-2 B a^5+(6 A b+4 C b) a^4-3 b^2 B a^3-b^3 (2 A-C) a^2+A b^5\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \frac {2 A \left (a^2-b^2\right )^3-a \left (-2 B a^5+(6 A b+4 C b) a^4-3 b^2 B a^3-b^3 (2 A-C) a^2+A b^5\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 \int \frac {2 A \left (a^2-b^2\right )^3-a \left (-2 B a^5+(6 A b+4 C b) a^4-3 b^2 B a^3-b^3 (2 A-C) a^2+A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 4407

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A x \left (a^2-b^2\right )^3}{a}-\frac {\left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}\right )}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A x \left (a^2-b^2\right )^3}{a}-\frac {\left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 4318

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A x \left (a^2-b^2\right )^3}{a}-\frac {\left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a b}\right )}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A x \left (a^2-b^2\right )^3}{a}-\frac {\left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a b}\right )}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {2 A x \left (a^2-b^2\right )^3}{a}-\frac {2 \left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a b d}\right )}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {\frac {\frac {3 \left (\frac {2 A x \left (a^2-b^2\right )^3}{a}-\frac {2 \left (-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}\)

Input:

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

Output:

((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d 
*x])^3) + (-1/2*((3*A*b^4 + 5*a^3*b*B - 2*a^4*C - a^2*b^2*(8*A + 3*C))*Tan 
[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*((2*A*(a^2 - b^2 
)^3*x)/a - (2*(7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B - a^4*b^3*(8* 
A - C) + 4*a^6*b*(2*A + C))*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)) - ((17*a^2*A*b^4 - 
6*A*b^6 + 11*a^5*b*B + 4*a^3*b^3*B - 2*a^6*C - 13*a^4*b^2*(2*A + C))*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 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 = 0.79 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.68

method	result	size
derivativedivides	\(\frac {\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {\frac {2 \left (-\frac {\left (12 A \,a^{4} b^{2}+4 A \,a^{3} b^{3}-6 a^{2} A \,b^{4}-A \,b^{5} a +2 A \,b^{6}-6 a^{5} b B -3 B \,a^{4} b^{2}-2 a^{3} b^{3} B +2 a^{6} C +2 a^{5} C b +6 a^{4} b^{2} C +C \,a^{3} b^{3}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (18 A \,a^{4} b^{2}-11 a^{2} A \,b^{4}+3 A \,b^{6}-9 a^{5} b B -a^{3} b^{3} B +3 a^{6} C +7 a^{4} b^{2} C \right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (12 A \,a^{4} b^{2}-4 A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+A \,b^{5} a +2 A \,b^{6}-6 a^{5} b B +3 B \,a^{4} b^{2}-2 a^{3} b^{3} B +2 a^{6} C -2 a^{5} C b +6 a^{4} b^{2} C -C \,a^{3} b^{3}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (8 A \,a^{6} b -8 A \,a^{4} b^{3}+7 a^{2} A \,b^{5}-2 A \,b^{7}-2 a^{7} B -3 a^{5} b^{2} B +4 C \,a^{6} b +C \,a^{4} b^{3}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{4}}}{d}\)	\(565\)
default	\(\frac {\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {\frac {2 \left (-\frac {\left (12 A \,a^{4} b^{2}+4 A \,a^{3} b^{3}-6 a^{2} A \,b^{4}-A \,b^{5} a +2 A \,b^{6}-6 a^{5} b B -3 B \,a^{4} b^{2}-2 a^{3} b^{3} B +2 a^{6} C +2 a^{5} C b +6 a^{4} b^{2} C +C \,a^{3} b^{3}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (18 A \,a^{4} b^{2}-11 a^{2} A \,b^{4}+3 A \,b^{6}-9 a^{5} b B -a^{3} b^{3} B +3 a^{6} C +7 a^{4} b^{2} C \right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (12 A \,a^{4} b^{2}-4 A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+A \,b^{5} a +2 A \,b^{6}-6 a^{5} b B +3 B \,a^{4} b^{2}-2 a^{3} b^{3} B +2 a^{6} C -2 a^{5} C b +6 a^{4} b^{2} C -C \,a^{3} b^{3}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (8 A \,a^{6} b -8 A \,a^{4} b^{3}+7 a^{2} A \,b^{5}-2 A \,b^{7}-2 a^{7} B -3 a^{5} b^{2} B +4 C \,a^{6} b +C \,a^{4} b^{3}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{4}}}{d}\)	\(565\)
risch	\(\text {Expression too large to display}\)	\(2437\)

Input:

int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x,method=_RETURNVER 
BOSE)
 

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^2+4*A*a^3 
*b^3-6*A*a^2*b^4-A*a*b^5+2*A*b^6-6*B*a^5*b-3*B*a^4*b^2-2*B*a^3*b^3+2*C*a^6 
+2*C*a^5*b+6*C*a^4*b^2+C*a^3*b^3)*a/(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^2-11*A*a^2*b^4+3*A*b^6-9*B*a^5*b-B*a^3*b^3+ 
3*C*a^6+7*C*a^4*b^2)*a/(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^2-4*A*a^3*b^3-6*A*a^2*b^4+A*a*b^5+2*A*b^6-6*B*a^5*b+3*B* 
a^4*b^2-2*B*a^3*b^3+2*C*a^6-2*C*a^5*b+6*C*a^4*b^2-C*a^3*b^3)*a/(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+4*C*a^6*b+C*a^4*b^3)/(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. 999 vs. \(2 (317) = 634\).

Time = 0.23 (sec) , antiderivative size = 2056, normalized size of antiderivative = 6.12 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(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 - 4*(2*A + C) 
*a^6*b^4 + 3*B*a^5*b^5 + (8*A - C)*a^4*b^6 - 7*A*a^2*b^8 + 2*A*b^10 + (2*B 
*a^10 - 4*(2*A + C)*a^9*b + 3*B*a^8*b^2 + (8*A - C)*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 - 4*(2*A + C)*a^8*b^2 + 3*B*a 
^7*b^3 + (8*A - C)*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 - 4*(2*A + C)*a^7*b^3 + 3*B*a^6*b^4 + (8*A - C)*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*(2*C*a^9*b^2 - 11*B*a^8*b^3 + (26*A + 11*C)*a^7*b^4 + 7*B*a^6*b^5 
- (43*A + 13*C)*a^5*b^6 + 4*B*a^4*b^7 + 23*A*a^3*b^8 - 6*A*a*b^10 + (6*C*a 
^11 - 18*B*a^10*b + 4*(9*A + C)*a^9*b^2 + 23*B*a^8*b^3 - (68*A + 11*C)*a^7 
*b^4 - 7*B*a^6*b^5 + (43*A + C)*a^5*b^6 + 2*B*a^4*b^7 - 11*A*a^3*b^8)*cos( 
d*x + c)^2 + 3*(2*C*a^10*b - 9*B*a^9*b^2 + (20*A + 7*C)*a^8*b^3 + 8*B*a^7* 
b^4 - 5*(7*A + 2*C)*a^6*b^5 + B*a^5*b^6 + (20*A + C)*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*...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(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. 1106 vs. \(2 (317) = 634\).

Time = 0.35 (sec) , antiderivative size = 1106, normalized size of antiderivative = 3.29 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*(3*(2*B*a^7 - 8*A*a^6*b - 4*C*a^6*b + 3*B*a^5*b^2 + 8*A*a^4*b^3 - C*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 - (6*C*a^8*tan(1/2*d*x + 1/2*c)^5 - 18*B*a^7*b*tan(1/2*d 
*x + 1/2*c)^5 - 6*C*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 + 12*C*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 - 27*C*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 + 12*C*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 + 3*C*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 - 12*C*a^8*tan(1/2*d*x + 1/2*c)^3 + 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 - 16*C*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 + 28*C*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 + 12*A*b^8*tan(1/2*d 
*x + 1/2*c)^3 + 6*C*a^8*tan(1/2*d*x + 1/2*c) - 18*B*a^7*b*tan(1/2*d*x + 1/ 
2*c) + 6*C*a^7*b*tan(1/2*d*x + 1/2*c) + 36*A*a^6*b^2*tan(1/2*d*x + 1/2*...
 

Mupad [B] (verification not implemented)

Time = 26.63 (sec) , antiderivative size = 11934, normalized size of antiderivative = 35.52 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

(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 + 156*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 + C^2*a^8*b^6 + 8*C^2*a^10*b^4 
+ 16*C^2*a^12*b^2 - 32*A*B*a^13*b - 16*B*C*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 - 4*A*C*a^4*b^10 - 2*A*C*a^6* 
b^8 + 40*A*C*a^8*b^6 - 48*A*C*a^10*b^4 + 64*A*C*a^12*b^2 - 6*B*C*a^9*b^5 - 
 28*B*C*a^11*b^3))/(a^16*b + a^17 - 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^21 + 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^9 - 6*B*a^13*b^8 - 14*B*a^14*b^7 + 14*B*a^15 
*b^6 + 6*B*a^16*b^5 - 6*B*a^17*b^4 + 6*B*a^18*b^3 - 6*B*a^19*b^2 - 2*C*a^1 
1*b^10 + 2*C*a^12*b^9 - 2*C*a^13*b^8 + 2*C*a^14*b^7 + 18*C*a^15*b^6 - 18*C 
*a^16*b^5 - 22*C*a^17*b^4 + 22*C*a^18*b^3 + 8*C*a^19*b^2 - 16*A*a^20*b - 4 
*B*a^20*b - 8*C*a^20*b))/(a^19*b + a^20 - a^9*b^11 - a^10*b^10 + 5*a^11*b^ 
9 + 5*a^12*b^8 - 10*a^13*b^7 - 10*a^14*b^6 + 10*a^15*b^5 + 10*a^16*b^4 - 5 
*a^17*b^3 - 5*a^18*b^2) - (A*tan(c/2 + (d*x)/2)*(8*a^21*b - 8*a^8*b^14 ...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 2797, normalized size of antiderivative = 8.32 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 36*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**9*b - 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**8*b*c + 66*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**7*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))*cos(c + d*x)*sin(c + d*x 
)**2*a**6*b**3*c - 42*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**5*b* 
*5 + 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*a**3*b**7 + 36*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**9*b + 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**8*b*c + 
 42*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**7*b**3 + 78*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**3*c - 156*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**5 + 18*sqrt( - a 
**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2...