\(\int \frac {\sec ^3(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^4} \, dx\) [924]

Optimal result

Integrand size = 41, antiderivative size = 358 \[ \int \frac {\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\frac {C \text {arctanh}(\sin (c+d x))}{b^4 d}-\frac {\left (3 a^2 b^5 B+2 b^7 B-a^3 b^4 (A-8 C)+2 a^7 C-7 a^5 b^2 C-4 a b^6 (A+2 C)\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^4 (a+b)^{7/2} d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {a \left (2 A b^4-5 a b^3 B-3 a^4 C+a^2 b^2 (3 A+8 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}-\frac {\left (4 A b^6+a^3 b^3 B-16 a b^5 B+9 a^6 C+2 a^2 b^4 (7 A+17 C)-a^4 b^2 (3 A+28 C)\right ) \tan (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \] Output:

C*arctanh(sin(d*x+c))/b^4/d-(3*a^2*b^5*B+2*b^7*B-a^3*b^4*(A-8*C)+2*a^7*C-7 
*a^5*b^2*C-4*a*b^6*(A+2*C))*arctanh((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*b^2-a*(B*b-C*a))*sec(d*x+c)^2*t 
an(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(d*x+c))^3-1/6*a*(2*A*b^4-5*B*a*b^3-3*a^4* 
C+a^2*b^2*(3*A+8*C))*tan(d*x+c)/b^3/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2-1/6*( 
4*A*b^6+a^3*b^3*B-16*a*b^5*B+9*a^6*C+2*a^2*b^4*(7*A+17*C)-a^4*b^2*(3*A+28* 
C))*tan(d*x+c)/b^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 = 8.20 (sec) , antiderivative size = 1302, normalized size of antiderivative = 3.64 \[ \int \frac {\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx =\text {Too large to display} \] Input:

Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b* 
Sec[c + d*x])^4,x]
 

Output:

(-2*C*(b + a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]* 
Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^4*d*(A + 2*C + 
2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*C*(b + 
 a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d* 
x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^4*d*(A + 2*C + 2*B*Cos[c 
+ d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((-(a^3*A*b^4) - 4* 
a*A*b^6 + 3*a^2*b^5*B + 2*b^7*B + 2*a^7*C - 7*a^5*b^2*C + 8*a^3*b^4*C - 8* 
a*b^6*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[Co 
s[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])]*Cos[c])/(b^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])/(b^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^3*Sin[c] - a*b^2*B*Sin[c] + a^2*b*C*Sin[c] - a*A 
*b^2*Sin[d*x] + a^2*b*B*Sin[d*x] - a^3*C*Sin[d*x]))/(3*a*b*(-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])^...
 

Rubi [A] (verified)

Time = 2.48 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.16, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.390, Rules used = {3042, 4586, 3042, 4578, 25, 3042, 4568, 27, 3042, 4486, 3042, 4257, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4586

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4578

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4568

\(\displaystyle -\frac {\frac {a \tan (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {\int \frac {3 \sec (c+d x) \left (2 B b^7-2 a (2 A+3 C) b^6+3 a^2 B b^5-a^3 (A-2 C) b^4-a^5 C b^2-2 \left (a^2-b^2\right )^3 C \sec (c+d x) b\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {a \tan (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {3 \int \frac {\sec (c+d x) \left (2 B b^7-2 a (2 A+3 C) b^6+3 a^2 B b^5-a^3 (A-2 C) b^4-a^5 C b^2-2 \left (a^2-b^2\right )^3 C \sec (c+d x) b\right )}{a+b \sec (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {a \tan (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {3 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (2 B b^7-2 a (2 A+3 C) b^6+3 a^2 B b^5-a^3 (A-2 C) b^4-a^5 C b^2-2 \left (a^2-b^2\right )^3 C \csc \left (c+d x+\frac {\pi }{2}\right ) b\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 4486

\(\displaystyle -\frac {\frac {a \tan (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {3 \left (\left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx-2 C \left (a^2-b^2\right )^3 \int \sec (c+d x)dx\right )}{b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {a \tan (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {3 \left (\left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx-2 C \left (a^2-b^2\right )^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx\right )}{b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 4257

\(\displaystyle -\frac {\frac {a \tan (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {3 \left (\left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 C \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}\right )}{b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 4318

\(\displaystyle -\frac {\frac {a \tan (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {3 \left (\frac {\left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{b}-\frac {2 C \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}\right )}{b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {a \tan (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {3 \left (\frac {\left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{b}-\frac {2 C \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}\right )}{b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 3138

\(\displaystyle -\frac {\frac {a \tan (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {3 \left (\frac {2 \left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}-\frac {2 C \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}\right )}{b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\tan (c+d x) \sec ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\frac {a \tan (c+d x) \left (-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {-\frac {3 \left (\frac {2 \left (2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}-\frac {2 C \left (a^2-b^2\right )^3 \text {arctanh}(\sin (c+d x))}{d}\right )}{b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right )}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b^2 \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\)

Input:

Int[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c 
+ d*x])^4,x]
 

Output:

-1/3*((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)* 
d*(a + b*Sec[c + d*x])^3) - ((a*(2*A*b^4 - 5*a*b^3*B - 3*a^4*C + a^2*b^2*( 
3*A + 8*C))*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ( 
(-3*((-2*(a^2 - b^2)^3*C*ArcTanh[Sin[c + d*x]])/d + (2*(3*a^2*b^5*B + 2*b^ 
7*B - a^3*b^4*(A - 8*C) + 2*a^7*C - 7*a^5*b^2*C - 4*a*b^6*(A + 2*C))*ArcTa 
nh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d 
)))/(b*(a^2 - b^2)) - ((4*A*b^6 + a^3*b^3*B - 16*a*b^5*B + 9*a^6*C + 2*a^2 
*b^4*(7*A + 17*C) - a^4*b^2*(3*A + 28*C))*Tan[c + d*x])/((a^2 - b^2)*d*(a 
+ b*Sec[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 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 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

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 4486

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

rule 4568

Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e 
_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S 
ymbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cot[e + f*x]*((a + b*Csc[e + f*x] 
)^(m + 1)/(b*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) 
 Int[Csc[e + f*x]*(a + b*Csc[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))*Csc[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 4578

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[ 
(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x 
_Symbol] :> Simp[a*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x 
])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - Simp[1/(b^2*(m + 1)*(a^2 - b^ 
2))   Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[b*(m + 1)*((-a)*(b 
*B - a*C) + A*b^2) + (b*B*(a^2 + b^2*(m + 1)) - a*(A*b^2*(m + 2) + C*(a^2 + 
 b^2*(m + 1))))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^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 
]
 

rule 4586

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

Maple [A] (verified)

Time = 1.47 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.63

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {-\frac {\left (A \,a^{3} b^{3}+6 a^{2} A \,b^{4}+2 A \,b^{5} a +2 A \,b^{6}-2 a^{3} b^{3} B -3 B \,a^{2} b^{4}-6 a \,b^{5} B +2 a^{6} C -a^{5} C b -6 a^{4} b^{2} C +4 C \,a^{3} b^{3}+12 C \,a^{2} b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (7 a^{2} A \,b^{4}+3 A \,b^{6}-a^{3} b^{3} B -9 a \,b^{5} B +3 a^{6} C -11 a^{4} b^{2} C +18 C \,a^{2} b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+2 A \,b^{5} a -2 A \,b^{6}+2 a^{3} b^{3} B -3 B \,a^{2} b^{4}+6 a \,b^{5} B -2 a^{6} C -a^{5} C b +6 a^{4} b^{2} C +4 C \,a^{3} b^{3}-12 C \,a^{2} b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (A \,a^{3} b^{4}+4 A a \,b^{6}-3 a^{2} b^{5} B -2 b^{7} B -2 a^{7} C +7 a^{5} b^{2} C -8 C \,a^{3} b^{4}+8 C a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}}{d}\)	\(584\)
default	\(\frac {-\frac {2 \left (\frac {-\frac {\left (A \,a^{3} b^{3}+6 a^{2} A \,b^{4}+2 A \,b^{5} a +2 A \,b^{6}-2 a^{3} b^{3} B -3 B \,a^{2} b^{4}-6 a \,b^{5} B +2 a^{6} C -a^{5} C b -6 a^{4} b^{2} C +4 C \,a^{3} b^{3}+12 C \,a^{2} b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (7 a^{2} A \,b^{4}+3 A \,b^{6}-a^{3} b^{3} B -9 a \,b^{5} B +3 a^{6} C -11 a^{4} b^{2} C +18 C \,a^{2} b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (A \,a^{3} b^{3}-6 a^{2} A \,b^{4}+2 A \,b^{5} a -2 A \,b^{6}+2 a^{3} b^{3} B -3 B \,a^{2} b^{4}+6 a \,b^{5} B -2 a^{6} C -a^{5} C b +6 a^{4} b^{2} C +4 C \,a^{3} b^{3}-12 C \,a^{2} b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (A \,a^{3} b^{4}+4 A a \,b^{6}-3 a^{2} b^{5} B -2 b^{7} B -2 a^{7} C +7 a^{5} b^{2} C -8 C \,a^{3} b^{4}+8 C a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}+\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}-\frac {C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}}{d}\)	\(584\)
risch	\(\text {Expression too large to display}\)	\(2287\)

Input:

int(sec(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x,meth 
od=_RETURNVERBOSE)
 

Output:

1/d*(-2/b^4*((-1/2*(A*a^3*b^3+6*A*a^2*b^4+2*A*a*b^5+2*A*b^6-2*B*a^3*b^3-3* 
B*a^2*b^4-6*B*a*b^5+2*C*a^6-C*a^5*b-6*C*a^4*b^2+4*C*a^3*b^3+12*C*a^2*b^4)* 
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*(7*A*a^2*b^4+3* 
A*b^6-B*a^3*b^3-9*B*a*b^5+3*C*a^6-11*C*a^4*b^2+18*C*a^2*b^4)*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*(A*a^3*b^3-6*A*a^2*b^4+2*A*a 
*b^5-2*A*b^6+2*B*a^3*b^3-3*B*a^2*b^4+6*B*a*b^5-2*C*a^6-C*a^5*b+6*C*a^4*b^2 
+4*C*a^3*b^3-12*C*a^2*b^4)*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*(A*a^3*b^ 
4+4*A*a*b^6-3*B*a^2*b^5-2*B*b^7-2*C*a^7+7*C*a^5*b^2-8*C*a^3*b^4+8*C*a*b^6) 
/(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)))+C/b^4*ln(tan(1/2*d*x+1/2*c)+1)-C/b^4*ln(ta 
n(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. 1204 vs. \(2 (343) = 686\).

Time = 46.73 (sec) , antiderivative size = 2466, normalized size of antiderivative = 6.89 \[ \int \frac {\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^3*(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*(3*(2*C*a^7*b^3 - 7*C*a^5*b^5 - (A - 8*C)*a^3*b^7 + 3*B*a^2*b^8 - 4 
*(A + 2*C)*a*b^9 + 2*B*b^10 + (2*C*a^10 - 7*C*a^8*b^2 - (A - 8*C)*a^6*b^4 
+ 3*B*a^5*b^5 - 4*(A + 2*C)*a^4*b^6 + 2*B*a^3*b^7)*cos(d*x + c)^3 + 3*(2*C 
*a^9*b - 7*C*a^7*b^3 - (A - 8*C)*a^5*b^5 + 3*B*a^4*b^6 - 4*(A + 2*C)*a^3*b 
^7 + 2*B*a^2*b^8)*cos(d*x + c)^2 + 3*(2*C*a^8*b^2 - 7*C*a^6*b^4 - (A - 8*C 
)*a^4*b^6 + 3*B*a^3*b^7 - 4*(A + 2*C)*a^2*b^8 + 2*B*a*b^9)*cos(d*x + c))*s 
qrt(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)) - 6*(C*a^8*b^3 - 4*C*a^6*b^5 + 6*C 
*a^4*b^7 - 4*C*a^2*b^9 + C*b^11 + (C*a^11 - 4*C*a^9*b^2 + 6*C*a^7*b^4 - 4* 
C*a^5*b^6 + C*a^3*b^8)*cos(d*x + c)^3 + 3*(C*a^10*b - 4*C*a^8*b^3 + 6*C*a^ 
6*b^5 - 4*C*a^4*b^7 + C*a^2*b^9)*cos(d*x + c)^2 + 3*(C*a^9*b^2 - 4*C*a^7*b 
^4 + 6*C*a^5*b^6 - 4*C*a^3*b^8 + C*a*b^10)*cos(d*x + c))*log(sin(d*x + c) 
+ 1) + 6*(C*a^8*b^3 - 4*C*a^6*b^5 + 6*C*a^4*b^7 - 4*C*a^2*b^9 + C*b^11 + ( 
C*a^11 - 4*C*a^9*b^2 + 6*C*a^7*b^4 - 4*C*a^5*b^6 + C*a^3*b^8)*cos(d*x + c) 
^3 + 3*(C*a^10*b - 4*C*a^8*b^3 + 6*C*a^6*b^5 - 4*C*a^4*b^7 + C*a^2*b^9)*co 
s(d*x + c)^2 + 3*(C*a^9*b^2 - 4*C*a^7*b^4 + 6*C*a^5*b^6 - 4*C*a^3*b^8 + C* 
a*b^10)*cos(d*x + c))*log(-sin(d*x + c) + 1) + 2*(11*C*a^8*b^3 - 2*B*a^7*b 
^4 - (A + 43*C)*a^6*b^5 + 7*B*a^5*b^6 + (11*A + 68*C)*a^4*b^7 - 23*B*a^3*b 
^8 - 4*(A + 9*C)*a^2*b^9 + 18*B*a*b^10 - 6*A*b^11 + (6*C*a^10*b - 23*C*...
 

Sympy [F]

\[ \int \frac {\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \] Input:

integrate(sec(d*x+c)**3*(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)*sec(c + d*x)**3/(a + b*s 
ec(c + d*x))**4, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(sec(d*x+c)^3*(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. 1135 vs. \(2 (343) = 686\).

Time = 0.34 (sec) , antiderivative size = 1135, normalized size of antiderivative = 3.17 \[ \int \frac {\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^3*(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*C*a^7 - 7*C*a^5*b^2 - A*a^3*b^4 + 8*C*a^3*b^4 + 3*B*a^2*b^5 - 4 
*A*a*b^6 - 8*C*a*b^6 + 2*B*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*C*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^4 + 3*C*log(abs(tan(1/2*d*x + 1/ 
2*c) - 1))/b^4 - (6*C*a^8*tan(1/2*d*x + 1/2*c)^5 - 15*C*a^7*b*tan(1/2*d*x 
+ 1/2*c)^5 - 6*C*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 + 3*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 + 45*C*a^5*b^3*tan(1/2*d*x 
 + 1/2*c)^5 + 12*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 - 6*C*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 27*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 - 60*C*a^3*b^5*tan(1/2* 
d*x + 1/2*c)^5 + 12*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 + 27*B*a^2*b^6*tan(1/ 
2*d*x + 1/2*c)^5 + 36*C*a^2*b^6*tan(1/2*d*x + 1/2*c)^5 - 6*A*a*b^7*tan(1/2 
*d*x + 1/2*c)^5 - 18*B*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 + 56*C*a^6*b^2*tan(1/2*d*x + 
1/2*c)^3 + 4*B*a^5*b^3*tan(1/2*d*x + 1/2*c)^3 - 28*A*a^4*b^4*tan(1/2*d*x + 
 1/2*c)^3 - 116*C*a^4*b^4*tan(1/2*d*x + 1/2*c)^3 + 32*B*a^3*b^5*tan(1/2*d* 
x + 1/2*c)^3 + 16*A*a^2*b^6*tan(1/2*d*x + 1/2*c)^3 + 72*C*a^2*b^6*tan(1/2* 
d*x + 1/2*c)^3 - 36*B*a*b^7*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) + 15*C*a^7*b*tan(1/2*d*x + 1/...
 

Mupad [B] (verification not implemented)

Time = 26.27 (sec) , antiderivative size = 11926, normalized size of antiderivative = 33.31 \[ \int \frac {\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:

int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/(cos(c + d*x)^3*(a + b/cos(c + 
 d*x))^4),x)
 

Output:

- ((tan(c/2 + (d*x)/2)*(2*A*b^6 + 2*C*a^6 + 6*A*a^2*b^4 - A*a^3*b^3 + 3*B* 
a^2*b^4 - 2*B*a^3*b^3 + 12*C*a^2*b^4 - 4*C*a^3*b^3 - 6*C*a^4*b^2 - 2*A*a*b 
^5 - 6*B*a*b^5 + C*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*(3*A*b^6 + 3*C*a^6 + 7*A*a^2*b^4 - B*a^3*b^3 + 
18*C*a^2*b^4 - 11*C*a^4*b^2 - 9*B*a*b^5))/(3*(a + b)^2*(b^5 - 2*a*b^4 + a^ 
2*b^3)) + (tan(c/2 + (d*x)/2)^5*(2*A*b^6 + 2*C*a^6 + 6*A*a^2*b^4 + A*a^3*b 
^3 - 3*B*a^2*b^4 - 2*B*a^3*b^3 + 12*C*a^2*b^4 + 4*C*a^3*b^3 - 6*C*a^4*b^2 
+ 2*A*a*b^5 - 6*B*a*b^5 - C*a^5*b))/((a*b^3 - b^4)*(a + b)^3))/(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))) - (C*atan(((C*((8*tan(c/ 
2 + (d*x)/2)*(4*B^2*b^14 + 8*C^2*a^14 + 4*C^2*b^14 - 8*C^2*a*b^13 - 8*C^2* 
a^13*b + 16*A^2*a^2*b^12 + 8*A^2*a^4*b^10 + A^2*a^6*b^8 + 12*B^2*a^2*b^12 
+ 9*B^2*a^4*b^10 + 44*C^2*a^2*b^12 + 48*C^2*a^3*b^11 - 92*C^2*a^4*b^10 - 1 
20*C^2*a^5*b^9 + 156*C^2*a^6*b^8 + 160*C^2*a^7*b^7 - 164*C^2*a^8*b^6 - 120 
*C^2*a^9*b^5 + 117*C^2*a^10*b^4 + 48*C^2*a^11*b^3 - 48*C^2*a^12*b^2 - 16*A 
*B*a*b^13 - 32*B*C*a*b^13 - 28*A*B*a^3*b^11 - 6*A*B*a^5*b^9 + 64*A*C*a^2*b 
^12 - 48*A*C*a^4*b^10 + 40*A*C*a^6*b^8 - 2*A*C*a^8*b^6 - 4*A*C*a^10*b^4 - 
16*B*C*a^3*b^11 + 20*B*C*a^5*b^9 - 34*B*C*a^7*b^7 + 12*B*C*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^...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 3878, normalized size of antiderivative = 10.83 \[ \int \frac {\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx =\text {Too large to display} \] Input:

int(sec(d*x+c)^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x)
 

Output:

( - 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**10*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**8*b**2*c + 6*sqrt( - a**2 + b**2)*a 
tan((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**4 - 48*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**4*c + 6*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - t 
an((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**5 
*b**6 + 48*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**6*c - 12*s 
qrt( - 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**8 + 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)*a**10*c - 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**8*b**2*c - 6*sqr 
t( - 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**7*b**4 - 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**4*c - 24*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d...