\(\int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx\) [302]

Optimal result

Integrand size = 31, antiderivative size = 334 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {\left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (24 a^4 A b+224 a^2 A b^3+32 A b^5-4 a^5 B+121 a^3 b^2 B+128 a b^4 B\right ) \tan (c+d x)}{60 b d}+\frac {\left (48 a^3 A b+232 a A b^3-8 a^4 B+178 a^2 b^2 B+75 b^4 B\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {\left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac {\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac {(6 A b-a B) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac {B (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d} \] Output:

1/16*(32*A*a^3*b+24*A*a*b^3+8*B*a^4+36*B*a^2*b^2+5*B*b^4)*arctanh(sin(d*x+ 
c))/d+1/60*(24*A*a^4*b+224*A*a^2*b^3+32*A*b^5-4*B*a^5+121*B*a^3*b^2+128*B* 
a*b^4)*tan(d*x+c)/b/d+1/240*(48*A*a^3*b+232*A*a*b^3-8*B*a^4+178*B*a^2*b^2+ 
75*B*b^4)*sec(d*x+c)*tan(d*x+c)/d+1/120*(24*A*a^2*b+32*A*b^3-4*B*a^3+53*B* 
a*b^2)*(a+b*sec(d*x+c))^2*tan(d*x+c)/b/d+1/120*(24*A*a*b-4*B*a^2+25*B*b^2) 
*(a+b*sec(d*x+c))^3*tan(d*x+c)/b/d+1/30*(6*A*b-B*a)*(a+b*sec(d*x+c))^4*tan 
(d*x+c)/b/d+1/6*B*(a+b*sec(d*x+c))^5*tan(d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 1.89 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.73 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {15 \left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (240 \left (a^4 A+6 a^2 A b^2+A b^4+4 a^3 b B+4 a b^3 B\right )+15 \left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right ) \sec (c+d x)+10 b^2 \left (24 a A b+36 a^2 B+5 b^2 B\right ) \sec ^3(c+d x)+40 b^4 B \sec ^5(c+d x)+160 b \left (3 a^2 A b+A b^3+2 a^3 B+4 a b^2 B\right ) \tan ^2(c+d x)+48 b^3 (A b+4 a B) \tan ^4(c+d x)\right )}{240 d} \] Input:

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

Output:

(15*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*ArcTanh[S 
in[c + d*x]] + Tan[c + d*x]*(240*(a^4*A + 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B 
+ 4*a*b^3*B) + 15*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^ 
4*B)*Sec[c + d*x] + 10*b^2*(24*a*A*b + 36*a^2*B + 5*b^2*B)*Sec[c + d*x]^3 
+ 40*b^4*B*Sec[c + d*x]^5 + 160*b*(3*a^2*A*b + A*b^3 + 2*a^3*B + 4*a*b^2*B 
)*Tan[c + d*x]^2 + 48*b^3*(A*b + 4*a*B)*Tan[c + d*x]^4))/(240*d)
 

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {3042, 4498, 3042, 4490, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4498

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4490

\(\displaystyle \frac {\frac {1}{5} \int \sec (c+d x) (a+b \sec (c+d x))^3 \left (3 b (8 A b+7 a B)+\left (-4 B a^2+24 A b a+25 b^2 B\right ) \sec (c+d x)\right )dx+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (3 b (8 A b+7 a B)+\left (-4 B a^2+24 A b a+25 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4490

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int 3 \sec (c+d x) (a+b \sec (c+d x))^2 \left (b \left (24 B a^2+56 A b a+25 b^2 B\right )+\left (-4 B a^3+24 A b a^2+53 b^2 B a+32 A b^3\right ) \sec (c+d x)\right )dx+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int \sec (c+d x) (a+b \sec (c+d x))^2 \left (b \left (24 B a^2+56 A b a+25 b^2 B\right )+\left (-4 B a^3+24 A b a^2+53 b^2 B a+32 A b^3\right ) \sec (c+d x)\right )dx+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b \left (24 B a^2+56 A b a+25 b^2 B\right )+\left (-4 B a^3+24 A b a^2+53 b^2 B a+32 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4490

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \sec (c+d x) (a+b \sec (c+d x)) \left (b \left (64 B a^3+216 A b a^2+181 b^2 B a+64 A b^3\right )+\left (-8 B a^4+48 A b a^3+178 b^2 B a^2+232 A b^3 a+75 b^4 B\right ) \sec (c+d x)\right )dx+\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left (64 B a^3+216 A b a^2+181 b^2 B a+64 A b^3\right )+\left (-8 B a^4+48 A b a^3+178 b^2 B a^2+232 A b^3 a+75 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4485

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \sec (c+d x) \left (15 b \left (8 B a^4+32 A b a^3+36 b^2 B a^2+24 A b^3 a+5 b^4 B\right )+4 \left (-4 B a^5+24 A b a^4+121 b^2 B a^3+224 A b^3 a^2+128 b^4 B a+32 A b^5\right ) \sec (c+d x)\right )dx+\frac {b \left (-8 a^4 B+48 a^3 A b+178 a^2 b^2 B+232 a A b^3+75 b^4 B\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (15 b \left (8 B a^4+32 A b a^3+36 b^2 B a^2+24 A b^3 a+5 b^4 B\right )+4 \left (-4 B a^5+24 A b a^4+121 b^2 B a^3+224 A b^3 a^2+128 b^4 B a+32 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {b \left (-8 a^4 B+48 a^3 A b+178 a^2 b^2 B+232 a A b^3+75 b^4 B\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4274

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b \left (8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right ) \int \sec (c+d x)dx+4 \left (-4 a^5 B+24 a^4 A b+121 a^3 b^2 B+224 a^2 A b^3+128 a b^4 B+32 A b^5\right ) \int \sec ^2(c+d x)dx\right )+\frac {b \left (-8 a^4 B+48 a^3 A b+178 a^2 b^2 B+232 a A b^3+75 b^4 B\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b \left (8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+4 \left (-4 a^5 B+24 a^4 A b+121 a^3 b^2 B+224 a^2 A b^3+128 a b^4 B+32 A b^5\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {b \left (-8 a^4 B+48 a^3 A b+178 a^2 b^2 B+232 a A b^3+75 b^4 B\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4254

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b \left (8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {4 \left (-4 a^5 B+24 a^4 A b+121 a^3 b^2 B+224 a^2 A b^3+128 a b^4 B+32 A b^5\right ) \int 1d(-\tan (c+d x))}{d}\right )+\frac {b \left (-8 a^4 B+48 a^3 A b+178 a^2 b^2 B+232 a A b^3+75 b^4 B\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b \left (8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {4 \left (-4 a^5 B+24 a^4 A b+121 a^3 b^2 B+224 a^2 A b^3+128 a b^4 B+32 A b^5\right ) \tan (c+d x)}{d}\right )+\frac {b \left (-8 a^4 B+48 a^3 A b+178 a^2 b^2 B+232 a A b^3+75 b^4 B\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )+\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\frac {1}{5} \left (\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac {3}{4} \left (\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {b \left (-8 a^4 B+48 a^3 A b+178 a^2 b^2 B+232 a A b^3+75 b^4 B\right ) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {1}{2} \left (\frac {15 b \left (8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {4 \left (-4 a^5 B+24 a^4 A b+121 a^3 b^2 B+224 a^2 A b^3+128 a b^4 B+32 A b^5\right ) \tan (c+d x)}{d}\right )\right )\right )\right )+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}}{6 b}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}\)

Input:

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

Output:

(B*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(6*b*d) + (((6*A*b - a*B)*(a + b*S 
ec[c + d*x])^4*Tan[c + d*x])/(5*d) + (((24*a*A*b - 4*a^2*B + 25*b^2*B)*(a 
+ b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + (3*(((24*a^2*A*b + 32*A*b^3 - 4* 
a^3*B + 53*a*b^2*B)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + ((b*(48*a 
^3*A*b + 232*a*A*b^3 - 8*a^4*B + 178*a^2*b^2*B + 75*b^4*B)*Sec[c + d*x]*Ta 
n[c + d*x])/(2*d) + ((15*b*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2 
*B + 5*b^4*B)*ArcTanh[Sin[c + d*x]])/d + (4*(24*a^4*A*b + 224*a^2*A*b^3 + 
32*A*b^5 - 4*a^5*B + 121*a^3*b^2*B + 128*a*b^4*B)*Tan[c + d*x])/d)/2)/3))/ 
4)/5)/(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 4498

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

Maple [A] (verified)

Time = 4.61 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.83

method	result	size
parts	\(-\frac {\left (A \,b^{4}+4 a \,b^{3} B \right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 A \,a^{2} b^{2}+4 a^{3} b B \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {a^{4} A \tan \left (d x +c \right )}{d}\)	\(277\)
derivativedivides	\(\frac {a^{4} A \tan \left (d x +c \right )+B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 a^{3} b B \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-6 A \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 B \,a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 A a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a \,b^{3} B \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-A \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(375\)
default	\(\frac {a^{4} A \tan \left (d x +c \right )+B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 a^{3} b B \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-6 A \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 B \,a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 A a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a \,b^{3} B \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-A \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(375\)
parallelrisch	\(\frac {-480 \left (A \,a^{3} b +\frac {3}{4} A a \,b^{3}+\frac {1}{4} B \,a^{4}+\frac {9}{8} B \,a^{2} b^{2}+\frac {5}{32} B \,b^{4}\right ) \left (10+\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+480 \left (A \,a^{3} b +\frac {3}{4} A a \,b^{3}+\frac {1}{4} B \,a^{4}+\frac {9}{8} B \,a^{2} b^{2}+\frac {5}{32} B \,b^{4}\right ) \left (10+\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (1200 a^{4} A +8640 A \,a^{2} b^{2}+1920 A \,b^{4}+5760 a^{3} b B +7680 a \,b^{3} B \right ) \sin \left (2 d x +2 c \right )+\left (2880 A \,a^{3} b +4080 A a \,b^{3}+720 B \,a^{4}+6120 B \,a^{2} b^{2}+850 B \,b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (960 a^{4} A +5760 A \,a^{2} b^{2}+768 A \,b^{4}+3840 a^{3} b B +3072 a \,b^{3} B \right ) \sin \left (4 d x +4 c \right )+\left (960 A \,a^{3} b +720 A a \,b^{3}+240 B \,a^{4}+1080 B \,a^{2} b^{2}+150 B \,b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (240 a^{4} A +960 A \,a^{2} b^{2}+128 A \,b^{4}+640 a^{3} b B +512 a \,b^{3} B \right ) \sin \left (6 d x +6 c \right )+1920 \sin \left (d x +c \right ) \left (A \,a^{3} b +\frac {7}{4} A a \,b^{3}+\frac {1}{4} B \,a^{4}+\frac {21}{8} B \,a^{2} b^{2}+\frac {33}{32} B \,b^{4}\right )}{240 d \left (10+\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )\right )}\)	\(479\)
norman	\(\frac {-\frac {\left (16 a^{4} A -32 A \,a^{3} b +96 A \,a^{2} b^{2}-40 A a \,b^{3}+16 A \,b^{4}-8 B \,a^{4}+64 a^{3} b B -60 B \,a^{2} b^{2}+64 a \,b^{3} B -11 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (16 a^{4} A +32 A \,a^{3} b +96 A \,a^{2} b^{2}+40 A a \,b^{3}+16 A \,b^{4}+8 B \,a^{4}+64 a^{3} b B +60 B \,a^{2} b^{2}+64 a \,b^{3} B +11 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (240 a^{4} A -288 A \,a^{3} b +1056 A \,a^{2} b^{2}-168 A a \,b^{3}+112 A \,b^{4}-72 B \,a^{4}+704 a^{3} b B -252 B \,a^{2} b^{2}+448 a \,b^{3} B +5 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {\left (240 a^{4} A +288 A \,a^{3} b +1056 A \,a^{2} b^{2}+168 A a \,b^{3}+112 A \,b^{4}+72 B \,a^{4}+704 a^{3} b B +252 B \,a^{2} b^{2}+448 a \,b^{3} B -5 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}-\frac {\left (400 a^{4} A -160 A \,a^{3} b +1440 A \,a^{2} b^{2}-40 A a \,b^{3}+208 A \,b^{4}-40 B \,a^{4}+960 a^{3} b B -60 B \,a^{2} b^{2}+832 a \,b^{3} B -75 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (400 a^{4} A +160 A \,a^{3} b +1440 A \,a^{2} b^{2}+40 A a \,b^{3}+208 A \,b^{4}+40 B \,a^{4}+960 a^{3} b B +60 B \,a^{2} b^{2}+832 a \,b^{3} B +75 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {\left (32 A \,a^{3} b +24 A a \,b^{3}+8 B \,a^{4}+36 B \,a^{2} b^{2}+5 B \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {\left (32 A \,a^{3} b +24 A a \,b^{3}+8 B \,a^{4}+36 B \,a^{2} b^{2}+5 B \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\)	\(645\)
risch	\(\text {Expression too large to display}\)	\(1071\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.98 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (15 \, A a^{4} + 40 \, B a^{3} b + 60 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{5} + 40 \, B b^{4} + 15 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{4} + 32 \, {\left (10 \, B a^{3} b + 15 \, A a^{2} b^{2} + 8 \, B a b^{3} + 2 \, A b^{4}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \] Input:

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

Output:

1/480*(15*(8*B*a^4 + 32*A*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 5*B*b^4)*cos 
(d*x + c)^6*log(sin(d*x + c) + 1) - 15*(8*B*a^4 + 32*A*a^3*b + 36*B*a^2*b^ 
2 + 24*A*a*b^3 + 5*B*b^4)*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(16*(1 
5*A*a^4 + 40*B*a^3*b + 60*A*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4)*cos(d*x + c)^5 
 + 40*B*b^4 + 15*(8*B*a^4 + 32*A*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 5*B*b 
^4)*cos(d*x + c)^4 + 32*(10*B*a^3*b + 15*A*a^2*b^2 + 8*B*a*b^3 + 2*A*b^4)* 
cos(d*x + c)^3 + 10*(36*B*a^2*b^2 + 24*A*a*b^3 + 5*B*b^4)*cos(d*x + c)^2 + 
 48*(4*B*a*b^3 + A*b^4)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.42 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} b + 960 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} b^{2} + 128 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a b^{3} + 32 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A b^{4} - 5 \, B b^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, B a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, A a b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, B a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 480 \, A a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, A a^{4} \tan \left (d x + c\right )}{480 \, d} \] Input:

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

Output:

1/480*(640*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^3*b + 960*(tan(d*x + c)^3 
 + 3*tan(d*x + c))*A*a^2*b^2 + 128*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 
 15*tan(d*x + c))*B*a*b^3 + 32*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15* 
tan(d*x + c))*A*b^4 - 5*B*b^4*(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)) - 180*B*a^2*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)) - 120*A*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^4*(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*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^4*tan(d*x + c))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1186 vs. \(2 (319) = 638\).

Time = 0.22 (sec) , antiderivative size = 1186, normalized size of antiderivative = 3.55 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Too large to display} \] Input:

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

Output:

1/240*(15*(8*B*a^4 + 32*A*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 5*B*b^4)*log 
(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(8*B*a^4 + 32*A*a^3*b + 36*B*a^2*b^2 
+ 24*A*a*b^3 + 5*B*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(240*A*a^4* 
tan(1/2*d*x + 1/2*c)^11 - 120*B*a^4*tan(1/2*d*x + 1/2*c)^11 - 480*A*a^3*b* 
tan(1/2*d*x + 1/2*c)^11 + 960*B*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 1440*A*a^2 
*b^2*tan(1/2*d*x + 1/2*c)^11 - 900*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 600 
*A*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 960*B*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 2 
40*A*b^4*tan(1/2*d*x + 1/2*c)^11 - 165*B*b^4*tan(1/2*d*x + 1/2*c)^11 - 120 
0*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 360*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 1440*A 
*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 3520*B*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 5280 
*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 1260*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 
+ 840*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 2240*B*a*b^3*tan(1/2*d*x + 1/2*c)^9 
 - 560*A*b^4*tan(1/2*d*x + 1/2*c)^9 - 25*B*b^4*tan(1/2*d*x + 1/2*c)^9 + 24 
00*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 240*B*a^4*tan(1/2*d*x + 1/2*c)^7 - 960*A 
*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 5760*B*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 8640 
*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 360*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 
 240*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 4992*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 
+ 1248*A*b^4*tan(1/2*d*x + 1/2*c)^7 - 450*B*b^4*tan(1/2*d*x + 1/2*c)^7 - 2 
400*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 240*B*a^4*tan(1/2*d*x + 1/2*c)^5 - 960* 
A*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 5760*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 - ...
 

Mupad [B] (verification not implemented)

Time = 14.64 (sec) , antiderivative size = 709, normalized size of antiderivative = 2.12 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx =\text {Too large to display} \] Input:

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

Output:

(atanh((4*tan(c/2 + (d*x)/2)*((B*a^4)/2 + (5*B*b^4)/16 + (9*B*a^2*b^2)/4 + 
 (3*A*a*b^3)/2 + 2*A*a^3*b))/(2*B*a^4 + (5*B*b^4)/4 + 9*B*a^2*b^2 + 6*A*a* 
b^3 + 8*A*a^3*b))*(B*a^4 + (5*B*b^4)/8 + (9*B*a^2*b^2)/2 + 3*A*a*b^3 + 4*A 
*a^3*b))/d + (tan(c/2 + (d*x)/2)*(2*A*a^4 + 2*A*b^4 + B*a^4 + (11*B*b^4)/8 
 + 12*A*a^2*b^2 + (15*B*a^2*b^2)/2 + 5*A*a*b^3 + 4*A*a^3*b + 8*B*a*b^3 + 8 
*B*a^3*b) - tan(c/2 + (d*x)/2)^11*(2*A*a^4 + 2*A*b^4 - B*a^4 - (11*B*b^4)/ 
8 + 12*A*a^2*b^2 - (15*B*a^2*b^2)/2 - 5*A*a*b^3 - 4*A*a^3*b + 8*B*a*b^3 + 
8*B*a^3*b) - tan(c/2 + (d*x)/2)^3*(10*A*a^4 + (14*A*b^4)/3 + 3*B*a^4 - (5* 
B*b^4)/24 + 44*A*a^2*b^2 + (21*B*a^2*b^2)/2 + 7*A*a*b^3 + 12*A*a^3*b + (56 
*B*a*b^3)/3 + (88*B*a^3*b)/3) + tan(c/2 + (d*x)/2)^9*(10*A*a^4 + (14*A*b^4 
)/3 - 3*B*a^4 + (5*B*b^4)/24 + 44*A*a^2*b^2 - (21*B*a^2*b^2)/2 - 7*A*a*b^3 
 - 12*A*a^3*b + (56*B*a*b^3)/3 + (88*B*a^3*b)/3) + tan(c/2 + (d*x)/2)^5*(2 
0*A*a^4 + (52*A*b^4)/5 + 2*B*a^4 + (15*B*b^4)/4 + 72*A*a^2*b^2 + 3*B*a^2*b 
^2 + 2*A*a*b^3 + 8*A*a^3*b + (208*B*a*b^3)/5 + 48*B*a^3*b) - tan(c/2 + (d* 
x)/2)^7*(20*A*a^4 + (52*A*b^4)/5 - 2*B*a^4 - (15*B*b^4)/4 + 72*A*a^2*b^2 - 
 3*B*a^2*b^2 - 2*A*a*b^3 - 8*A*a^3*b + (208*B*a*b^3)/5 + 48*B*a^3*b))/(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)^8 - 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^ 
12 + 1))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 922, normalized size of antiderivative = 2.76 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx =\text {Too large to display} \] Input:

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

Output:

( - 48*cos(c + d*x)*sin(c + d*x)**5*a**5 - 320*cos(c + d*x)*sin(c + d*x)** 
5*a**3*b**2 - 128*cos(c + d*x)*sin(c + d*x)**5*a*b**4 + 96*cos(c + d*x)*si 
n(c + d*x)**3*a**5 + 800*cos(c + d*x)*sin(c + d*x)**3*a**3*b**2 + 320*cos( 
c + d*x)*sin(c + d*x)**3*a*b**4 - 48*cos(c + d*x)*sin(c + d*x)*a**5 - 480* 
cos(c + d*x)*sin(c + d*x)*a**3*b**2 - 240*cos(c + d*x)*sin(c + d*x)*a*b**4 
 - 120*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**4*b - 180*log(tan((c + 
 d*x)/2) - 1)*sin(c + d*x)**6*a**2*b**3 - 15*log(tan((c + d*x)/2) - 1)*sin 
(c + d*x)**6*b**5 + 360*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**4*b + 
 540*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**2*b**3 + 45*log(tan((c + 
 d*x)/2) - 1)*sin(c + d*x)**4*b**5 - 360*log(tan((c + d*x)/2) - 1)*sin(c + 
 d*x)**2*a**4*b - 540*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**2*b**3 
- 45*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*b**5 + 120*log(tan((c + d*x 
)/2) - 1)*a**4*b + 180*log(tan((c + d*x)/2) - 1)*a**2*b**3 + 15*log(tan((c 
 + d*x)/2) - 1)*b**5 + 120*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a**4* 
b + 180*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a**2*b**3 + 15*log(tan(( 
c + d*x)/2) + 1)*sin(c + d*x)**6*b**5 - 360*log(tan((c + d*x)/2) + 1)*sin( 
c + d*x)**4*a**4*b - 540*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a**2*b* 
*3 - 45*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*b**5 + 360*log(tan((c + 
d*x)/2) + 1)*sin(c + d*x)**2*a**4*b + 540*log(tan((c + d*x)/2) + 1)*sin(c 
+ d*x)**2*a**2*b**3 + 45*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*b**5...