\(\int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [878]

Optimal result

Integrand size = 41, antiderivative size = 381 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {\left (6 a^4 b B-104 a^2 b^3 B-32 b^5 B-2 a^5 C-24 a b^4 (5 A+4 C)-a^3 b^2 (30 A+17 C)\right ) \tan (c+d x)}{60 b^2 d}-\frac {\left (12 a^3 b B-142 a b^3 B-4 a^4 C-12 a^2 b^2 (5 A+3 C)-15 b^4 (6 A+5 C)\right ) \sec (c+d x) \tan (c+d x)}{240 b d}-\frac {\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac {\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac {(3 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{15 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d} \] Output:

1/16*(8*B*a^3+18*B*a*b^2+6*a^2*b*(4*A+3*C)+b^3*(6*A+5*C))*arctanh(sin(d*x+ 
c))/d-1/60*(6*B*a^4*b-104*B*a^2*b^3-32*B*b^5-2*a^5*C-24*a*b^4*(5*A+4*C)-a^ 
3*b^2*(30*A+17*C))*tan(d*x+c)/b^2/d-1/240*(12*B*a^3*b-142*B*a*b^3-4*a^4*C- 
12*a^2*b^2*(5*A+3*C)-15*b^4*(6*A+5*C))*sec(d*x+c)*tan(d*x+c)/b/d-1/120*(6* 
B*a^2*b-32*B*b^3-2*a^3*C-3*a*b^2*(10*A+7*C))*(a+b*sec(d*x+c))^2*tan(d*x+c) 
/b^2/d+1/120*(30*A*b^2-6*B*a*b+2*C*a^2+25*C*b^2)*(a+b*sec(d*x+c))^3*tan(d* 
x+c)/b^2/d+1/15*(3*B*b-C*a)*(a+b*sec(d*x+c))^4*tan(d*x+c)/b^2/d+1/6*C*sec( 
d*x+c)*(a+b*sec(d*x+c))^4*tan(d*x+c)/b/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 11.16 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.86 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {C \sec ^2(c+d x) (a+b \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {1}{6} \left (\frac {3 (2 b B+a C) \sec ^2(c+d x) (a+b \sec (c+d x))^2 \tan (c+d x)}{5 d}+\frac {1}{5} \left (\frac {b \left (30 A b^2+42 a b B+6 a^2 C+25 b^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \left (\frac {\left (24 a^2 (15 a A+6 b B+8 a C)+48 \left (12 a^2 b B+4 b^3 B+a^3 C+3 a b^2 (5 A+4 C)\right )\right ) \tan (c+d x)}{3 d}+\frac {8 \left (12 a^2 b B+4 b^3 B+a^3 C+3 a b^2 (5 A+4 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{d}+15 \left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\sec (c+d x) \tan (c+d x)}{2 d}\right )\right )\right )\right ) \] Input:

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

Output:

(C*Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(6*d) + ((3*(2*b*B 
+ a*C)*Sec[c + d*x]^2*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(5*d) + ((b*(30 
*A*b^2 + 42*a*b*B + 6*a^2*C + 25*b^2*C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) 
 + (((24*a^2*(15*a*A + 6*b*B + 8*a*C) + 48*(12*a^2*b*B + 4*b^3*B + a^3*C + 
 3*a*b^2*(5*A + 4*C)))*Tan[c + d*x])/(3*d) + (8*(12*a^2*b*B + 4*b^3*B + a^ 
3*C + 3*a*b^2*(5*A + 4*C))*Sec[c + d*x]^2*Tan[c + d*x])/d + 15*(8*a^3*B + 
18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*(ArcTanh[Sin[c + d*x]] 
/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/4)/5)/6
 

Rubi [A] (verified)

Time = 2.22 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.415, Rules used = {3042, 4580, 3042, 4570, 3042, 4490, 27, 3042, 4490, 3042, 4485, 3042, 4274, 3042, 4254, 24, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4580

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4570

\(\displaystyle \frac {\frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (3 b (8 b B-a C)+\left (2 C a^2-6 b B a+30 A b^2+25 b^2 C\right ) \sec (c+d x)\right )dx}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (3 b (8 b B-a C)+\left (2 C a^2-6 b B a+30 A b^2+25 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 4490

\(\displaystyle \frac {\frac {\frac {1}{4} \int 3 \sec (c+d x) (a+b \sec (c+d x))^2 \left (b \left (-2 C a^2+26 b B a+30 A b^2+25 b^2 C\right )-\left (-2 C a^3+6 b B a^2-3 b^2 (10 A+7 C) a-32 b^3 B\right ) \sec (c+d x)\right )dx+\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3}{4} \int \sec (c+d x) (a+b \sec (c+d x))^2 \left (b \left (-2 C a^2+26 b B a+30 A b^2+25 b^2 C\right )-\left (-2 C a^3+6 b B a^2-3 b^2 (10 A+7 C) a-32 b^3 B\right ) \sec (c+d x)\right )dx+\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b \left (-2 C a^2+26 b B a+30 A b^2+25 b^2 C\right )+\left (2 C a^3-6 b B a^2+3 b^2 (10 A+7 C) a+32 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 4490

\(\displaystyle \frac {\frac {\frac {3}{4} \left (\frac {1}{3} \int \sec (c+d x) (a+b \sec (c+d x)) \left (b \left (-2 C a^3+66 b B a^2+3 b^2 (50 A+39 C) a+64 b^3 B\right )-\left (-4 C a^4+12 b B a^3-12 b^2 (5 A+3 C) a^2-142 b^3 B a-15 b^4 (6 A+5 C)\right ) \sec (c+d x)\right )dx-\frac {\tan (c+d x) \left (-2 a^3 C+6 a^2 b B-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3}{4} \left (\frac {1}{3} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left (-2 C a^3+66 b B a^2+3 b^2 (50 A+39 C) a+64 b^3 B\right )+\left (4 C a^4-12 b B a^3+12 b^2 (5 A+3 C) a^2+142 b^3 B a+15 b^4 (6 A+5 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {\tan (c+d x) \left (-2 a^3 C+6 a^2 b B-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 4485

\(\displaystyle \frac {\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \sec (c+d x) \left (15 b^2 \left (8 B a^3+6 b (4 A+3 C) a^2+18 b^2 B a+b^3 (6 A+5 C)\right )-4 \left (-2 C a^5+6 b B a^4-b^2 (30 A+17 C) a^3-104 b^3 B a^2-24 b^4 (5 A+4 C) a-32 b^5 B\right ) \sec (c+d x)\right )dx-\frac {b \tan (c+d x) \sec (c+d x) \left (-4 a^4 C+12 a^3 b B-12 a^2 b^2 (5 A+3 C)-142 a b^3 B-15 b^4 (6 A+5 C)\right )}{2 d}\right )-\frac {\tan (c+d x) \left (-2 a^3 C+6 a^2 b B-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (15 b^2 \left (8 B a^3+6 b (4 A+3 C) a^2+18 b^2 B a+b^3 (6 A+5 C)\right )-4 \left (-2 C a^5+6 b B a^4-b^2 (30 A+17 C) a^3-104 b^3 B a^2-24 b^4 (5 A+4 C) a-32 b^5 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {b \tan (c+d x) \sec (c+d x) \left (-4 a^4 C+12 a^3 b B-12 a^2 b^2 (5 A+3 C)-142 a b^3 B-15 b^4 (6 A+5 C)\right )}{2 d}\right )-\frac {\tan (c+d x) \left (-2 a^3 C+6 a^2 b B-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 4274

\(\displaystyle \frac {\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b^2 \left (8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right ) \int \sec (c+d x)dx-4 \left (-2 a^5 C+6 a^4 b B-a^3 b^2 (30 A+17 C)-104 a^2 b^3 B-24 a b^4 (5 A+4 C)-32 b^5 B\right ) \int \sec ^2(c+d x)dx\right )-\frac {b \tan (c+d x) \sec (c+d x) \left (-4 a^4 C+12 a^3 b B-12 a^2 b^2 (5 A+3 C)-142 a b^3 B-15 b^4 (6 A+5 C)\right )}{2 d}\right )-\frac {\tan (c+d x) \left (-2 a^3 C+6 a^2 b B-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b^2 \left (8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-4 \left (-2 a^5 C+6 a^4 b B-a^3 b^2 (30 A+17 C)-104 a^2 b^3 B-24 a b^4 (5 A+4 C)-32 b^5 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx\right )-\frac {b \tan (c+d x) \sec (c+d x) \left (-4 a^4 C+12 a^3 b B-12 a^2 b^2 (5 A+3 C)-142 a b^3 B-15 b^4 (6 A+5 C)\right )}{2 d}\right )-\frac {\tan (c+d x) \left (-2 a^3 C+6 a^2 b B-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 4254

\(\displaystyle \frac {\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b^2 \left (8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {4 \left (-2 a^5 C+6 a^4 b B-a^3 b^2 (30 A+17 C)-104 a^2 b^3 B-24 a b^4 (5 A+4 C)-32 b^5 B\right ) \int 1d(-\tan (c+d x))}{d}\right )-\frac {b \tan (c+d x) \sec (c+d x) \left (-4 a^4 C+12 a^3 b B-12 a^2 b^2 (5 A+3 C)-142 a b^3 B-15 b^4 (6 A+5 C)\right )}{2 d}\right )-\frac {\tan (c+d x) \left (-2 a^3 C+6 a^2 b B-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\frac {\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b^2 \left (8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {4 \tan (c+d x) \left (-2 a^5 C+6 a^4 b B-a^3 b^2 (30 A+17 C)-104 a^2 b^3 B-24 a b^4 (5 A+4 C)-32 b^5 B\right )}{d}\right )-\frac {b \tan (c+d x) \sec (c+d x) \left (-4 a^4 C+12 a^3 b B-12 a^2 b^2 (5 A+3 C)-142 a b^3 B-15 b^4 (6 A+5 C)\right )}{2 d}\right )-\frac {\tan (c+d x) \left (-2 a^3 C+6 a^2 b B-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\frac {\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{4 d}+\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {15 b^2 \left (8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{d}-\frac {4 \tan (c+d x) \left (-2 a^5 C+6 a^4 b B-a^3 b^2 (30 A+17 C)-104 a^2 b^3 B-24 a b^4 (5 A+4 C)-32 b^5 B\right )}{d}\right )-\frac {b \tan (c+d x) \sec (c+d x) \left (-4 a^4 C+12 a^3 b B-12 a^2 b^2 (5 A+3 C)-142 a b^3 B-15 b^4 (6 A+5 C)\right )}{2 d}\right )-\frac {\tan (c+d x) \left (-2 a^3 C+6 a^2 b B-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{3 d}\right )}{5 b}+\frac {2 (3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\)

Input:

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

Output:

(C*Sec[c + d*x]*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(6*b*d) + ((2*(3*b*B 
- a*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d) + (((30*A*b^2 - 6*a*b* 
B + 2*a^2*C + 25*b^2*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + (3*(- 
1/3*((6*a^2*b*B - 32*b^3*B - 2*a^3*C - 3*a*b^2*(10*A + 7*C))*(a + b*Sec[c 
+ d*x])^2*Tan[c + d*x])/d + (-1/2*(b*(12*a^3*b*B - 142*a*b^3*B - 4*a^4*C - 
 12*a^2*b^2*(5*A + 3*C) - 15*b^4*(6*A + 5*C))*Sec[c + d*x]*Tan[c + d*x])/d 
 + ((15*b^2*(8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C)) 
*ArcTanh[Sin[c + d*x]])/d - (4*(6*a^4*b*B - 104*a^2*b^3*B - 32*b^5*B - 2*a 
^5*C - 24*a*b^4*(5*A + 4*C) - a^3*b^2*(30*A + 17*C))*Tan[c + d*x])/d)/2)/3 
))/4)/(5*b))/(6*b)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4254

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, 
 d}, x] && IGtQ[n/2, 0]
 

rule 4257

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

rule 4274

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[a   Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d   In 
t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
 

rule 4485

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

rule 4490

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

rule 4570

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[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2) 
)), x] + Simp[1/(b*(m + 2))   Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[ 
b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; 
 FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]
 

rule 4580

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[(-C)*Csc[e + f*x]*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 
1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3))   Int[Csc[e + f*x]*(a + b*Csc[e 
+ f*x])^m*Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B* 
(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] & 
& NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]
 

Maple [A] (verified)

Time = 11.05 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.74

method	result	size
parts	\(\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (A \,b^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (3 a A \,b^{2}+3 B \,a^{2} b +a^{3} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {a^{3} A \tan \left (d x +c \right )}{d}\)	\(283\)
derivativedivides	\(\frac {a^{3} A \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{2} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{2} b C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a A \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B a \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a \,b^{2} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+A \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(444\)
default	\(\frac {a^{3} A \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 B \,a^{2} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{2} b C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a A \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B a \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a \,b^{2} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+A \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(444\)
parallelrisch	\(\frac {-2160 \left (\frac {\cos \left (6 d x +6 c \right )}{6}+\cos \left (4 d x +4 c \right )+\frac {5 \cos \left (2 d x +2 c \right )}{2}+\frac {5}{3}\right ) \left (\left (\frac {A}{4}+\frac {5 C}{24}\right ) b^{3}+\frac {3 B a \,b^{2}}{4}+a^{2} \left (A +\frac {3 C}{4}\right ) b +\frac {B \,a^{3}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2160 \left (\frac {\cos \left (6 d x +6 c \right )}{6}+\cos \left (4 d x +4 c \right )+\frac {5 \cos \left (2 d x +2 c \right )}{2}+\frac {5}{3}\right ) \left (\left (\frac {A}{4}+\frac {5 C}{24}\right ) b^{3}+\frac {3 B a \,b^{2}}{4}+a^{2} \left (A +\frac {3 C}{4}\right ) b +\frac {B \,a^{3}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (1920 B \,b^{3}+4320 a \left (A +\frac {4 C}{3}\right ) b^{2}+4320 B \,a^{2} b +1200 a^{3} \left (A +\frac {6 C}{5}\right )\right ) \sin \left (2 d x +2 c \right )+\left (\left (1020 A +850 C \right ) b^{3}+3060 B a \,b^{2}+2160 a^{2} \left (A +\frac {17 C}{12}\right ) b +720 B \,a^{3}\right ) \sin \left (3 d x +3 c \right )+\left (768 B \,b^{3}+2880 \left (A +\frac {4 C}{5}\right ) a \,b^{2}+2880 B \,a^{2} b +960 a^{3} \left (A +C \right )\right ) \sin \left (4 d x +4 c \right )+\left (\left (180 A +150 C \right ) b^{3}+540 B a \,b^{2}+720 a^{2} \left (A +\frac {3 C}{4}\right ) b +240 B \,a^{3}\right ) \sin \left (5 d x +5 c \right )+\left (128 B \,b^{3}+480 \left (A +\frac {4 C}{5}\right ) a \,b^{2}+480 B \,a^{2} b +240 a^{3} \left (A +\frac {2 C}{3}\right )\right ) \sin \left (6 d x +6 c \right )+1440 \sin \left (d x +c \right ) \left (\left (\frac {7 A}{12}+\frac {11 C}{8}\right ) b^{3}+\frac {7 B a \,b^{2}}{4}+a^{2} \left (A +\frac {7 C}{4}\right ) b +\frac {B \,a^{3}}{3}\right )}{240 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\)	\(474\)
norman	\(\frac {-\frac {\left (16 a^{3} A -24 A \,a^{2} b +48 a A \,b^{2}-10 A \,b^{3}-8 B \,a^{3}+48 B \,a^{2} b -30 B a \,b^{2}+16 B \,b^{3}+16 a^{3} C -30 a^{2} b C +48 C a \,b^{2}-11 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (16 a^{3} A +24 A \,a^{2} b +48 a A \,b^{2}+10 A \,b^{3}+8 B \,a^{3}+48 B \,a^{2} b +30 B a \,b^{2}+16 B \,b^{3}+16 a^{3} C +30 a^{2} b C +48 C a \,b^{2}+11 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (240 a^{3} A -216 A \,a^{2} b +528 a A \,b^{2}-42 A \,b^{3}-72 B \,a^{3}+528 B \,a^{2} b -126 B a \,b^{2}+112 B \,b^{3}+176 a^{3} C -126 a^{2} b C +336 C a \,b^{2}+5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {\left (240 a^{3} A +216 A \,a^{2} b +528 a A \,b^{2}+42 A \,b^{3}+72 B \,a^{3}+528 B \,a^{2} b +126 B a \,b^{2}+112 B \,b^{3}+176 a^{3} C +126 a^{2} b C +336 C a \,b^{2}-5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}-\frac {\left (400 a^{3} A -120 A \,a^{2} b +720 a A \,b^{2}-10 A \,b^{3}-40 B \,a^{3}+720 B \,a^{2} b -30 B a \,b^{2}+208 B \,b^{3}+240 a^{3} C -30 a^{2} b C +624 C a \,b^{2}-75 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (400 a^{3} A +120 A \,a^{2} b +720 a A \,b^{2}+10 A \,b^{3}+40 B \,a^{3}+720 B \,a^{2} b +30 B a \,b^{2}+208 B \,b^{3}+240 a^{3} C +30 a^{2} b C +624 C a \,b^{2}+75 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {\left (24 A \,a^{2} b +6 A \,b^{3}+8 B \,a^{3}+18 B a \,b^{2}+18 a^{2} b C +5 C \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {\left (24 A \,a^{2} b +6 A \,b^{3}+8 B \,a^{3}+18 B a \,b^{2}+18 a^{2} b C +5 C \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\)	\(701\)
risch	\(\text {Expression too large to display}\)	\(1245\)

Input:

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

Output:

(3*A*a^2*b+B*a^3)/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c 
)))-(B*b^3+3*C*a*b^2)/d*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x 
+c)+(A*b^3+3*B*a*b^2+3*C*a^2*b)/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan 
(d*x+c)+3/8*ln(sec(d*x+c)+tan(d*x+c)))-(3*A*a*b^2+3*B*a^2*b+C*a^3)/d*(-2/3 
-1/3*sec(d*x+c)^2)*tan(d*x+c)+C*b^3/d*(-(-1/6*sec(d*x+c)^5-5/24*sec(d*x+c) 
^3-5/16*sec(d*x+c))*tan(d*x+c)+5/16*ln(sec(d*x+c)+tan(d*x+c)))+a^3*A/d*tan 
(d*x+c)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.90 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (8 \, B a^{3} + 6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, B a^{3} + 6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (5 \, {\left (3 \, A + 2 \, C\right )} a^{3} + 30 \, B a^{2} b + 6 \, {\left (5 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )^{5} + 15 \, {\left (8 \, B a^{3} + 6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} + 40 \, C b^{3} + 16 \, {\left (5 \, C a^{3} + 15 \, B a^{2} b + 3 \, {\left (5 \, A + 4 \, C\right )} a b^{2} + 4 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (18 \, C a^{2} b + 18 \, B a b^{2} + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \] Input:

integrate(sec(d*x+c)^2*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), 
x, algorithm="fricas")
 

Output:

1/480*(15*(8*B*a^3 + 6*(4*A + 3*C)*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3)*c 
os(d*x + c)^6*log(sin(d*x + c) + 1) - 15*(8*B*a^3 + 6*(4*A + 3*C)*a^2*b + 
18*B*a*b^2 + (6*A + 5*C)*b^3)*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(1 
6*(5*(3*A + 2*C)*a^3 + 30*B*a^2*b + 6*(5*A + 4*C)*a*b^2 + 8*B*b^3)*cos(d*x 
 + c)^5 + 15*(8*B*a^3 + 6*(4*A + 3*C)*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b^3 
)*cos(d*x + c)^4 + 40*C*b^3 + 16*(5*C*a^3 + 15*B*a^2*b + 3*(5*A + 4*C)*a*b 
^2 + 4*B*b^3)*cos(d*x + c)^3 + 10*(18*C*a^2*b + 18*B*a*b^2 + (6*A + 5*C)*b 
^3)*cos(d*x + c)^2 + 48*(3*C*a*b^2 + B*b^3)*cos(d*x + c))*sin(d*x + c))/(d 
*cos(d*x + c)^6)
 

Sympy [F]

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

integrate(sec(d*x+c)**2*(a+b*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)** 
2),x)
 

Output:

Integral((a + b*sec(c + d*x))**3*(A + B*sec(c + d*x) + C*sec(c + d*x)**2)* 
sec(c + d*x)**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.48 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

integrate(sec(d*x+c)^2*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), 
x, algorithm="maxima")
 

Output:

1/480*(160*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^3 + 480*(tan(d*x + c)^3 + 
 3*tan(d*x + c))*B*a^2*b + 480*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a*b^2 + 
 96*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a*b^2 + 32* 
(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*b^3 - 5*C*b^3*( 
2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^ 
6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 
15*log(sin(d*x + c) - 1)) - 90*C*a^2*b*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + 
c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3* 
log(sin(d*x + c) - 1)) - 90*B*a*b^2*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c)) 
/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log 
(sin(d*x + c) - 1)) - 30*A*b^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin 
(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin( 
d*x + c) - 1)) - 120*B*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin( 
d*x + c) + 1) + log(sin(d*x + c) - 1)) - 360*A*a^2*b*(2*sin(d*x + c)/(sin( 
d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 480*A*a 
^3*tan(d*x + c))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1370 vs. \(2 (365) = 730\).

Time = 0.32 (sec) , antiderivative size = 1370, normalized size of antiderivative = 3.60 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^2*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), 
x, algorithm="giac")
 

Output:

1/240*(15*(8*B*a^3 + 24*A*a^2*b + 18*C*a^2*b + 18*B*a*b^2 + 6*A*b^3 + 5*C* 
b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(8*B*a^3 + 24*A*a^2*b + 18*C* 
a^2*b + 18*B*a*b^2 + 6*A*b^3 + 5*C*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) 
 - 2*(240*A*a^3*tan(1/2*d*x + 1/2*c)^11 - 120*B*a^3*tan(1/2*d*x + 1/2*c)^1 
1 + 240*C*a^3*tan(1/2*d*x + 1/2*c)^11 - 360*A*a^2*b*tan(1/2*d*x + 1/2*c)^1 
1 + 720*B*a^2*b*tan(1/2*d*x + 1/2*c)^11 - 450*C*a^2*b*tan(1/2*d*x + 1/2*c) 
^11 + 720*A*a*b^2*tan(1/2*d*x + 1/2*c)^11 - 450*B*a*b^2*tan(1/2*d*x + 1/2* 
c)^11 + 720*C*a*b^2*tan(1/2*d*x + 1/2*c)^11 - 150*A*b^3*tan(1/2*d*x + 1/2* 
c)^11 + 240*B*b^3*tan(1/2*d*x + 1/2*c)^11 - 165*C*b^3*tan(1/2*d*x + 1/2*c) 
^11 - 1200*A*a^3*tan(1/2*d*x + 1/2*c)^9 + 360*B*a^3*tan(1/2*d*x + 1/2*c)^9 
 - 880*C*a^3*tan(1/2*d*x + 1/2*c)^9 + 1080*A*a^2*b*tan(1/2*d*x + 1/2*c)^9 
- 2640*B*a^2*b*tan(1/2*d*x + 1/2*c)^9 + 630*C*a^2*b*tan(1/2*d*x + 1/2*c)^9 
 - 2640*A*a*b^2*tan(1/2*d*x + 1/2*c)^9 + 630*B*a*b^2*tan(1/2*d*x + 1/2*c)^ 
9 - 1680*C*a*b^2*tan(1/2*d*x + 1/2*c)^9 + 210*A*b^3*tan(1/2*d*x + 1/2*c)^9 
 - 560*B*b^3*tan(1/2*d*x + 1/2*c)^9 - 25*C*b^3*tan(1/2*d*x + 1/2*c)^9 + 24 
00*A*a^3*tan(1/2*d*x + 1/2*c)^7 - 240*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 1440* 
C*a^3*tan(1/2*d*x + 1/2*c)^7 - 720*A*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 4320*B 
*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 180*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 4320* 
A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 180*B*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 3744 
*C*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 60*A*b^3*tan(1/2*d*x + 1/2*c)^7 + 124...
 

Mupad [B] (verification not implemented)

Time = 15.04 (sec) , antiderivative size = 769, normalized size of antiderivative = 2.02 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(atanh((4*tan(c/2 + (d*x)/2)*((3*A*b^3)/8 + (B*a^3)/2 + (5*C*b^3)/16 + (3* 
A*a^2*b)/2 + (9*B*a*b^2)/8 + (9*C*a^2*b)/8))/((3*A*b^3)/2 + 2*B*a^3 + (5*C 
*b^3)/4 + 6*A*a^2*b + (9*B*a*b^2)/2 + (9*C*a^2*b)/2))*((3*A*b^3)/4 + B*a^3 
 + (5*C*b^3)/8 + 3*A*a^2*b + (9*B*a*b^2)/4 + (9*C*a^2*b)/4))/d + (tan(c/2 
+ (d*x)/2)*(2*A*a^3 + (5*A*b^3)/4 + B*a^3 + 2*B*b^3 + 2*C*a^3 + (11*C*b^3) 
/8 + 6*A*a*b^2 + 3*A*a^2*b + (15*B*a*b^2)/4 + 6*B*a^2*b + 6*C*a*b^2 + (15* 
C*a^2*b)/4) - tan(c/2 + (d*x)/2)^11*(2*A*a^3 - (5*A*b^3)/4 - B*a^3 + 2*B*b 
^3 + 2*C*a^3 - (11*C*b^3)/8 + 6*A*a*b^2 - 3*A*a^2*b - (15*B*a*b^2)/4 + 6*B 
*a^2*b + 6*C*a*b^2 - (15*C*a^2*b)/4) - tan(c/2 + (d*x)/2)^3*(10*A*a^3 + (7 
*A*b^3)/4 + 3*B*a^3 + (14*B*b^3)/3 + (22*C*a^3)/3 - (5*C*b^3)/24 + 22*A*a* 
b^2 + 9*A*a^2*b + (21*B*a*b^2)/4 + 22*B*a^2*b + 14*C*a*b^2 + (21*C*a^2*b)/ 
4) + tan(c/2 + (d*x)/2)^9*(10*A*a^3 - (7*A*b^3)/4 - 3*B*a^3 + (14*B*b^3)/3 
 + (22*C*a^3)/3 + (5*C*b^3)/24 + 22*A*a*b^2 - 9*A*a^2*b - (21*B*a*b^2)/4 + 
 22*B*a^2*b + 14*C*a*b^2 - (21*C*a^2*b)/4) + tan(c/2 + (d*x)/2)^5*(20*A*a^ 
3 + (A*b^3)/2 + 2*B*a^3 + (52*B*b^3)/5 + 12*C*a^3 + (15*C*b^3)/4 + 36*A*a* 
b^2 + 6*A*a^2*b + (3*B*a*b^2)/2 + 36*B*a^2*b + (156*C*a*b^2)/5 + (3*C*a^2* 
b)/2) - tan(c/2 + (d*x)/2)^7*(20*A*a^3 - (A*b^3)/2 - 2*B*a^3 + (52*B*b^3)/ 
5 + 12*C*a^3 - (15*C*b^3)/4 + 36*A*a*b^2 - 6*A*a^2*b - (3*B*a*b^2)/2 + 36* 
B*a^2*b + (156*C*a*b^2)/5 - (3*C*a^2*b)/2))/(d*(15*tan(c/2 + (d*x)/2)^4 - 
6*tan(c/2 + (d*x)/2)^2 - 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2...
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1270, normalized size of antiderivative = 3.33 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

( - 240*cos(c + d*x)*sin(c + d*x)**5*a**4 - 160*cos(c + d*x)*sin(c + d*x)* 
*5*a**3*c - 960*cos(c + d*x)*sin(c + d*x)**5*a**2*b**2 - 384*cos(c + d*x)* 
sin(c + d*x)**5*a*b**2*c - 128*cos(c + d*x)*sin(c + d*x)**5*b**4 + 480*cos 
(c + d*x)*sin(c + d*x)**3*a**4 + 400*cos(c + d*x)*sin(c + d*x)**3*a**3*c + 
 2400*cos(c + d*x)*sin(c + d*x)**3*a**2*b**2 + 960*cos(c + d*x)*sin(c + d* 
x)**3*a*b**2*c + 320*cos(c + d*x)*sin(c + d*x)**3*b**4 - 240*cos(c + d*x)* 
sin(c + d*x)*a**4 - 240*cos(c + d*x)*sin(c + d*x)*a**3*c - 1440*cos(c + d* 
x)*sin(c + d*x)*a**2*b**2 - 720*cos(c + d*x)*sin(c + d*x)*a*b**2*c - 240*c 
os(c + d*x)*sin(c + d*x)*b**4 - 480*log(tan((c + d*x)/2) - 1)*sin(c + d*x) 
**6*a**3*b - 270*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**2*b*c - 360* 
log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a*b**3 - 75*log(tan((c + d*x)/2) 
 - 1)*sin(c + d*x)**6*b**3*c + 1440*log(tan((c + d*x)/2) - 1)*sin(c + d*x) 
**4*a**3*b + 810*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**2*b*c + 1080 
*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a*b**3 + 225*log(tan((c + d*x)/ 
2) - 1)*sin(c + d*x)**4*b**3*c - 1440*log(tan((c + d*x)/2) - 1)*sin(c + d* 
x)**2*a**3*b - 810*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**2*b*c - 10 
80*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a*b**3 - 225*log(tan((c + d*x 
)/2) - 1)*sin(c + d*x)**2*b**3*c + 480*log(tan((c + d*x)/2) - 1)*a**3*b + 
270*log(tan((c + d*x)/2) - 1)*a**2*b*c + 360*log(tan((c + d*x)/2) - 1)*a*b 
**3 + 75*log(tan((c + d*x)/2) - 1)*b**3*c + 480*log(tan((c + d*x)/2) + ...