Integrand size = 33, antiderivative size = 306 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a \left (2 a^4 C+24 b^4 (5 A+4 C)+a^2 b^2 (30 A+17 C)\right ) \tan (c+d x)}{60 b^2 d}+\frac {\left (4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right ) \sec (c+d x) \tan (c+d x)}{240 b d}+\frac {a \left (30 A b^2+2 a^2 C+21 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}-\frac {a C (a+b \sec (c+d x))^4 \tan (c+d x)}{15 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d} \]
1/16*b*(6*a^2*(4*A+3*C)+b^2*(6*A+5*C))*arctanh(sin(d*x+c))/d+1/60*a*(2*a^4 *C+24*b^4*(5*A+4*C)+a^2*b^2*(30*A+17*C))*tan(d*x+c)/b^2/d+1/240*(4*a^4*C+1 2*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*a*(3 0*A*b^2+2*C*a^2+21*C*b^2)*(a+b*sec(d*x+c))^2*tan(d*x+c)/b^2/d+1/120*(2*C*a ^2+5*b^2*(6*A+5*C))*(a+b*sec(d*x+c))^3*tan(d*x+c)/b^2/d-1/15*a*C*(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
Time = 5.50 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.61 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \sec (c+d x)+10 b \left (6 A b^2+18 a^2 C+5 b^2 C\right ) \sec ^3(c+d x)+40 b^3 C \sec ^5(c+d x)+16 a \left (15 \left (a^2+3 b^2\right ) (A+C)+5 \left (3 A b^2+a^2 C+6 b^2 C\right ) \tan ^2(c+d x)+9 b^2 C \tan ^4(c+d x)\right )\right )}{240 d} \]
(15*b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*Sec[c + d*x] + 10*b*(6* A*b^2 + 18*a^2*C + 5*b^2*C)*Sec[c + d*x]^3 + 40*b^3*C*Sec[c + d*x]^5 + 16* a*(15*(a^2 + 3*b^2)*(A + C) + 5*(3*A*b^2 + a^2*C + 6*b^2*C)*Tan[c + d*x]^2 + 9*b^2*C*Tan[c + d*x]^4)))/(240*d)
Time = 1.87 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.06, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.606, Rules used = {3042, 4581, 3042, 4570, 25, 3042, 4490, 27, 3042, 4490, 25, 3042, 4485, 25, 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.
| \(\displaystyle \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4581 |
| \(\displaystyle \frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (-2 a C \sec ^2(c+d x)+b (6 A+5 C) \sec (c+d x)+a C\right )dx}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-2 a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+b (6 A+5 C) \csc \left (c+d x+\frac {\pi }{2}\right )+a C\right )dx}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {\frac {\int -\sec (c+d x) (a+b \sec (c+d x))^3 \left (3 a b C-\left (2 C a^2+5 b^2 (6 A+5 C)\right ) \sec (c+d x)\right )dx}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (3 a b C-\left (2 C a^2+5 b^2 (6 A+5 C)\right ) \sec (c+d x)\right )dx}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (3 a b C+\left (-2 C a^2-5 b^2 (6 A+5 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {-\frac {\frac {1}{4} \int -3 \sec (c+d x) (a+b \sec (c+d x))^2 \left (b \left (-2 C a^2+30 A b^2+25 b^2 C\right )+a \left (2 C a^2+30 A b^2+21 b^2 C\right ) \sec (c+d x)\right )dx-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \int \sec (c+d x) (a+b \sec (c+d x))^2 \left (b \left (-2 C a^2+30 A b^2+25 b^2 C\right )+a \left (2 C a^2+30 A b^2+21 b^2 C\right ) \sec (c+d x)\right )dx-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b \left (-2 C a^2+30 A b^2+25 b^2 C\right )+a \left (2 C a^2+30 A b^2+21 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \left (\frac {1}{3} \int -\sec (c+d x) (a+b \sec (c+d x)) \left (a b \left (2 a^2 C-3 b^2 (50 A+39 C)\right )-\left (4 C a^4+12 b^2 (5 A+3 C) a^2+15 b^4 (6 A+5 C)\right ) \sec (c+d x)\right )dx+\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \left (\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}-\frac {1}{3} \int \sec (c+d x) (a+b \sec (c+d x)) \left (a b \left (2 a^2 C-3 b^2 (50 A+39 C)\right )-\left (4 C a^4+12 b^2 (5 A+3 C) a^2+15 b^4 (6 A+5 C)\right ) \sec (c+d x)\right )dx\right )-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \left (\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}-\frac {1}{3} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a b \left (2 a^2 C-3 b^2 (50 A+39 C)\right )+\left (-4 C a^4-12 b^2 (5 A+3 C) a^2-15 b^4 (6 A+5 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\right )-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 4485 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \left (\frac {1}{3} \left (\frac {b \left (4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right ) \tan (c+d x) \sec (c+d x)}{2 d}-\frac {1}{2} \int -\sec (c+d x) \left (15 \left (6 (4 A+3 C) a^2+b^2 (6 A+5 C)\right ) b^3+4 a \left (2 C a^4+b^2 (30 A+17 C) a^2+24 b^4 (5 A+4 C)\right ) \sec (c+d x)\right )dx\right )+\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \sec (c+d x) \left (15 \left (6 (4 A+3 C) a^2+b^2 (6 A+5 C)\right ) b^3+4 a \left (2 C a^4+b^2 (30 A+17 C) a^2+24 b^4 (5 A+4 C)\right ) \sec (c+d x)\right )dx+\frac {b \left (4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (15 \left (6 (4 A+3 C) a^2+b^2 (6 A+5 C)\right ) b^3+4 a \left (2 C a^4+b^2 (30 A+17 C) a^2+24 b^4 (5 A+4 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {b \left (4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b^3 \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \int \sec (c+d x)dx+4 a \left (2 a^4 C+a^2 b^2 (30 A+17 C)+24 b^4 (5 A+4 C)\right ) \int \sec ^2(c+d x)dx\right )+\frac {b \left (4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b^3 \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+4 a \left (2 a^4 C+a^2 b^2 (30 A+17 C)+24 b^4 (5 A+4 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {b \left (4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b^3 \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {4 a \left (2 a^4 C+a^2 b^2 (30 A+17 C)+24 b^4 (5 A+4 C)\right ) \int 1d(-\tan (c+d x))}{d}\right )+\frac {b \left (4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {-\frac {-\frac {3}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 b^3 \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {4 a \left (2 a^4 C+a^2 b^2 (30 A+17 C)+24 b^4 (5 A+4 C)\right ) \tan (c+d x)}{d}\right )+\frac {b \left (4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}\right )-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {-\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}-\frac {3}{4} \left (\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {1}{2} \left (\frac {15 b^3 \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {4 a \left (2 a^4 C+a^2 b^2 (30 A+17 C)+24 b^4 (5 A+4 C)\right ) \tan (c+d x)}{d}\right )+\frac {b \left (4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )\right )}{5 b}-\frac {2 a C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}}{6 b}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}\) |
(C*Sec[c + d*x]*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(6*b*d) + ((-2*a*C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d) - (-1/4*((2*a^2*C + 5*b^2*(6*A + 5*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/d - (3*((a*(30*A*b^2 + 2*a^2* C + 21*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + ((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])/(2 *d) + ((15*b^3*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*ArcTanh[Sin[c + d*x]] )/d + (4*a*(2*a^4*C + 24*b^4*(5*A + 4*C) + a^2*b^2*(30*A + 17*C))*Tan[c + d*x])/d)/2)/3))/4)/(5*b))/(6*b)
3.7.54.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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]
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]
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]
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]
Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(cs c[(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*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Time = 1.51 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.83
| method | result | size |
| parts | \(\frac {\left (A \,b^{3}+3 a^{2} b C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (3 a A \,b^{2}+a^{3} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {a^{3} A \tan \left (d x +c \right )}{d}+\frac {3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {3 C a \,b^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(254\) |
| derivativedivides | \(\frac {a^{3} A \tan \left (d x +c \right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{2} b C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a A \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-3 C a \,b^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+A \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(298\) |
| default | \(\frac {a^{3} A \tan \left (d x +c \right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 A \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{2} b C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 a A \,b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-3 C a \,b^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+A \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(298\) |
| parallelrisch | \(\frac {-3 b \left (\frac {\left (A +\frac {5 C}{6}\right ) b^{2}}{4}+a^{2} \left (A +\frac {3 C}{4}\right )\right ) \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+3 b \left (\frac {\left (A +\frac {5 C}{6}\right ) b^{2}}{4}+a^{2} \left (A +\frac {3 C}{4}\right )\right ) \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+10 a \left (\frac {6 b^{2} \left (3 A +4 C \right )}{5}+a^{2} \left (A +\frac {6 C}{5}\right )\right ) \sin \left (2 d x +2 c \right )+18 \left (\frac {17 \left (A +\frac {5 C}{6}\right ) b^{2}}{36}+a^{2} \left (A +\frac {17 C}{12}\right )\right ) b \sin \left (3 d x +3 c \right )+8 a \left (3 b^{2} \left (A +\frac {4 C}{5}\right )+\left (A +C \right ) a^{2}\right ) \sin \left (4 d x +4 c \right )+6 b \left (\frac {\left (A +\frac {5 C}{6}\right ) b^{2}}{4}+a^{2} \left (A +\frac {3 C}{4}\right )\right ) \sin \left (5 d x +5 c \right )+2 a \left (2 b^{2} \left (A +\frac {4 C}{5}\right )+a^{2} \left (A +\frac {2 C}{3}\right )\right ) \sin \left (6 d x +6 c \right )+12 \left (\frac {\left (\frac {7 A}{3}+\frac {11 C}{2}\right ) b^{2}}{4}+a^{2} \left (A +\frac {7 C}{4}\right )\right ) \sin \left (d x +c \right ) b}{2 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(369\) |
| norman | \(\frac {-\frac {\left (16 a^{3} A -24 A \,a^{2} b +48 a A \,b^{2}-10 A \,b^{3}+16 a^{3} C -30 a^{2} b C +48 C a \,b^{2}-11 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (16 a^{3} A +24 A \,a^{2} b +48 a A \,b^{2}+10 A \,b^{3}+16 a^{3} C +30 a^{2} b C +48 C a \,b^{2}+11 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (240 a^{3} A -216 A \,a^{2} b +528 a A \,b^{2}-42 A \,b^{3}+176 a^{3} C -126 a^{2} b C +336 C a \,b^{2}+5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {\left (240 a^{3} A +216 A \,a^{2} b +528 a A \,b^{2}+42 A \,b^{3}+176 a^{3} C +126 a^{2} b C +336 C a \,b^{2}-5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}-\frac {\left (400 a^{3} A -120 A \,a^{2} b +720 a A \,b^{2}-10 A \,b^{3}+240 a^{3} C -30 a^{2} b C +624 C a \,b^{2}-75 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (400 a^{3} A +120 A \,a^{2} b +720 a A \,b^{2}+10 A \,b^{3}+240 a^{3} C +30 a^{2} b C +624 C a \,b^{2}+75 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {b \left (24 a^{2} A +6 A \,b^{2}+18 C \,a^{2}+5 C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {b \left (24 a^{2} A +6 A \,b^{2}+18 C \,a^{2}+5 C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(517\) |
| risch | \(-\frac {i \left (-160 a^{3} C -384 C a \,b^{2}-240 a^{3} A -480 a A \,b^{2}-720 A \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-1260 C \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-360 a^{2} A b \,{\mathrm e}^{i \left (d x +c \right )}-270 C \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}+360 A \,a^{2} b \,{\mathrm e}^{11 i \left (d x +c \right )}+270 C \,a^{2} b \,{\mathrm e}^{11 i \left (d x +c \right )}+1080 A \,a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}-510 A \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-425 C \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-1920 C \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-990 C \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-2400 A \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-420 A \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-2400 A \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-1600 C \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+420 A \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+990 C \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-480 C \,a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+510 A \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+425 C \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-1200 A \,a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+75 C \,b^{3} {\mathrm e}^{11 i \left (d x +c \right )}-240 A \,a^{3} {\mathrm e}^{10 i \left (d x +c \right )}+90 A \,b^{3} {\mathrm e}^{11 i \left (d x +c \right )}-1200 a^{3} A \,{\mathrm e}^{2 i \left (d x +c \right )}-960 C \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-90 A \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-75 C \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-5760 A a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-5760 C a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-1080 A \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-1530 C \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-2880 A a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2304 C a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+1530 C \,a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}-1440 A a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+720 A \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+1260 C \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-4800 A a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-3840 C a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} A}{2 d}-\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{8 d}-\frac {9 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,a^{2}}{8 d}-\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{16 d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} A}{2 d}+\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{8 d}+\frac {9 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,a^{2}}{8 d}+\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{16 d}\) | \(847\) |
(A*b^3+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+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)+3*A*a^2*b/d*(1/2 *sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))-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)
Time = 0.29 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.86 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (5 \, {\left (3 \, A + 2 \, C\right )} a^{3} + 6 \, {\left (5 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{5} + 144 \, C a b^{2} \cos \left (d x + c\right ) + 15 \, {\left (6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} + 40 \, C b^{3} + 16 \, {\left (5 \, C a^{3} + 3 \, {\left (5 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (18 \, C a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
1/480*(15*(6*(4*A + 3*C)*a^2*b + (6*A + 5*C)*b^3)*cos(d*x + c)^6*log(sin(d *x + c) + 1) - 15*(6*(4*A + 3*C)*a^2*b + (6*A + 5*C)*b^3)*cos(d*x + c)^6*l og(-sin(d*x + c) + 1) + 2*(16*(5*(3*A + 2*C)*a^3 + 6*(5*A + 4*C)*a*b^2)*co s(d*x + c)^5 + 144*C*a*b^2*cos(d*x + c) + 15*(6*(4*A + 3*C)*a^2*b + (6*A + 5*C)*b^3)*cos(d*x + c)^4 + 40*C*b^3 + 16*(5*C*a^3 + 3*(5*A + 4*C)*a*b^2)* cos(d*x + c)^3 + 10*(18*C*a^2*b + (6*A + 5*C)*b^3)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^6)
\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3} \sec ^{2}{\left (c + d x \right )}\, dx \]
Time = 0.21 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.26 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a b^{2} + 96 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a b^{2} - 5 \, C b^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, C a^{2} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, A b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, A a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, A a^{3} \tan \left (d x + c\right )}{480 \, d} \]
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))*A*a*b^2 + 96*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*t an(d*x + c))*C*a*b^2 - 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)) - 30*A*b^3*(2*(3*si n(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)) - 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
Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (292) = 584\).
Time = 0.39 (sec) , antiderivative size = 932, normalized size of antiderivative = 3.05 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
1/240*(15*(24*A*a^2*b + 18*C*a^2*b + 6*A*b^3 + 5*C*b^3)*log(abs(tan(1/2*d* x + 1/2*c) + 1)) - 15*(24*A*a^2*b + 18*C*a^2*b + 6*A*b^3 + 5*C*b^3)*log(ab s(tan(1/2*d*x + 1/2*c) - 1)) - 2*(240*A*a^3*tan(1/2*d*x + 1/2*c)^11 + 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)^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 + 72 0*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 - 16 5*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 - 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 + 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 - 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 - 25*C* b^3*tan(1/2*d*x + 1/2*c)^9 + 2400*A*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 - 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 + 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 - 450*C*b^3*t an(1/2*d*x + 1/2*c)^7 - 2400*A*a^3*tan(1/2*d*x + 1/2*c)^5 - 1440*C*a^3*tan (1/2*d*x + 1/2*c)^5 - 720*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 180*C*a^2*b*tan (1/2*d*x + 1/2*c)^5 - 4320*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 3744*C*a*b^2*t an(1/2*d*x + 1/2*c)^5 - 60*A*b^3*tan(1/2*d*x + 1/2*c)^5 - 450*C*b^3*tan(1/ 2*d*x + 1/2*c)^5 + 1200*A*a^3*tan(1/2*d*x + 1/2*c)^3 + 880*C*a^3*tan(1/2*d *x + 1/2*c)^3 + 1080*A*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 630*C*a^2*b*tan(1...
Time = 19.13 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.88 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {5\,A\,b^3}{4}-2\,A\,a^3-2\,C\,a^3+\frac {11\,C\,b^3}{8}-6\,A\,a\,b^2+3\,A\,a^2\,b-6\,C\,a\,b^2+\frac {15\,C\,a^2\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (10\,A\,a^3-\frac {7\,A\,b^3}{4}+\frac {22\,C\,a^3}{3}+\frac {5\,C\,b^3}{24}+22\,A\,a\,b^2-9\,A\,a^2\,b+14\,C\,a\,b^2-\frac {21\,C\,a^2\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {A\,b^3}{2}-20\,A\,a^3-12\,C\,a^3+\frac {15\,C\,b^3}{4}-36\,A\,a\,b^2+6\,A\,a^2\,b-\frac {156\,C\,a\,b^2}{5}+\frac {3\,C\,a^2\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (20\,A\,a^3+\frac {A\,b^3}{2}+12\,C\,a^3+\frac {15\,C\,b^3}{4}+36\,A\,a\,b^2+6\,A\,a^2\,b+\frac {156\,C\,a\,b^2}{5}+\frac {3\,C\,a^2\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {5\,C\,b^3}{24}-\frac {7\,A\,b^3}{4}-\frac {22\,C\,a^3}{3}-10\,A\,a^3-22\,A\,a\,b^2-9\,A\,a^2\,b-14\,C\,a\,b^2-\frac {21\,C\,a^2\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^3+\frac {5\,A\,b^3}{4}+2\,C\,a^3+\frac {11\,C\,b^3}{8}+6\,A\,a\,b^2+3\,A\,a^2\,b+6\,C\,a\,b^2+\frac {15\,C\,a^2\,b}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {b\,\mathrm {atanh}\left (\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,A\,a^2+6\,A\,b^2+18\,C\,a^2+5\,C\,b^2\right )}{4\,\left (\frac {3\,A\,b^3}{2}+\frac {5\,C\,b^3}{4}+6\,A\,a^2\,b+\frac {9\,C\,a^2\,b}{2}\right )}\right )\,\left (24\,A\,a^2+6\,A\,b^2+18\,C\,a^2+5\,C\,b^2\right )}{8\,d} \]
(tan(c/2 + (d*x)/2)*(2*A*a^3 + (5*A*b^3)/4 + 2*C*a^3 + (11*C*b^3)/8 + 6*A* a*b^2 + 3*A*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 + 2*C*a^3 - (11*C*b^3)/8 + 6*A*a*b^2 - 3*A*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 + (22*C*a^3)/3 - (5*C*b^3)/24 + 22*A*a*b^2 + 9*A*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 + (22*C*a^3)/3 + (5*C*b^3)/24 + 22*A*a*b^2 - 9*A*a^2*b + 14*C*a*b^2 - (21*C*a^2*b)/4) + ta n(c/2 + (d*x)/2)^5*(20*A*a^3 + (A*b^3)/2 + 12*C*a^3 + (15*C*b^3)/4 + 36*A* a*b^2 + 6*A*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 + 12*C*a^3 - (15*C*b^3)/4 + 36*A*a*b^2 - 6*A*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)^8 - 6*ta n(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (b*atanh((b*tan(c/2 + (d*x)/2)*(24*A*a^2 + 6*A*b^2 + 18*C*a^2 + 5*C*b^2))/(4*((3*A*b^3)/2 + (5*C *b^3)/4 + 6*A*a^2*b + (9*C*a^2*b)/2)))*(24*A*a^2 + 6*A*b^2 + 18*C*a^2 + 5* C*b^2))/(8*d)