Integrand size = 40, antiderivative size = 278 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (8 a^3 B+18 a b^2 B+18 a^2 b C+5 b^3 C\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (15 a^2 b B+4 b^3 B+5 a^3 C+12 a b^2 C\right ) \tan (c+d x)}{5 d}+\frac {\left (8 a^3 B+18 a b^2 B+18 a^2 b C+5 b^3 C\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {b \left (18 a b B+14 a^2 C+5 b^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (3 b B+4 a C) \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {b C \sec ^3(c+d x) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {\left (15 a^2 b B+4 b^3 B+5 a^3 C+12 a b^2 C\right ) \tan ^3(c+d x)}{15 d} \]
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Time = 0.67 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4157, 4111, 4161, 4132, 3852, 4131, 3853, 3855} \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {b \left (14 a^2 C+18 a b B+5 b^2 C\right ) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {\left (8 a^3 B+18 a^2 b C+18 a b^2 B+5 b^3 C\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (5 a^3 C+15 a^2 b B+12 a b^2 C+4 b^3 B\right ) \tan ^3(c+d x)}{15 d}+\frac {\left (5 a^3 C+15 a^2 b B+12 a b^2 C+4 b^3 B\right ) \tan (c+d x)}{5 d}+\frac {\left (8 a^3 B+18 a^2 b C+18 a b^2 B+5 b^3 C\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {b^2 (4 a C+3 b B) \tan (c+d x) \sec ^4(c+d x)}{15 d}+\frac {b C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^2}{6 d} \]
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Rule 3852
Rule 3853
Rule 3855
Rule 4111
Rule 4131
Rule 4132
Rule 4157
Rule 4161
Rubi steps \begin{align*} \text {integral}& = \int \sec ^3(c+d x) (a+b \sec (c+d x))^3 (B+C \sec (c+d x)) \, dx \\ & = \frac {b C \sec ^3(c+d x) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {1}{6} \int \sec ^3(c+d x) (a+b \sec (c+d x)) \left (3 a (2 a B+b C)+\left (5 b^2 C+6 a (2 b B+a C)\right ) \sec (c+d x)+2 b (3 b B+4 a C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {b^2 (3 b B+4 a C) \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {b C \sec ^3(c+d x) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {1}{30} \int \sec ^3(c+d x) \left (15 a^2 (2 a B+b C)+6 \left (15 a^2 b B+4 b^3 B+5 a^3 C+12 a b^2 C\right ) \sec (c+d x)+5 b \left (18 a b B+14 a^2 C+5 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {b^2 (3 b B+4 a C) \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {b C \sec ^3(c+d x) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {1}{30} \int \sec ^3(c+d x) \left (15 a^2 (2 a B+b C)+5 b \left (18 a b B+14 a^2 C+5 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{5} \left (15 a^2 b B+4 b^3 B+5 a^3 C+12 a b^2 C\right ) \int \sec ^4(c+d x) \, dx \\ & = \frac {b \left (18 a b B+14 a^2 C+5 b^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (3 b B+4 a C) \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {b C \sec ^3(c+d x) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {1}{8} \left (8 a^3 B+18 a b^2 B+18 a^2 b C+5 b^3 C\right ) \int \sec ^3(c+d x) \, dx-\frac {\left (15 a^2 b B+4 b^3 B+5 a^3 C+12 a b^2 C\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d} \\ & = \frac {\left (15 a^2 b B+4 b^3 B+5 a^3 C+12 a b^2 C\right ) \tan (c+d x)}{5 d}+\frac {\left (8 a^3 B+18 a b^2 B+18 a^2 b C+5 b^3 C\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {b \left (18 a b B+14 a^2 C+5 b^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (3 b B+4 a C) \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {b C \sec ^3(c+d x) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {\left (15 a^2 b B+4 b^3 B+5 a^3 C+12 a b^2 C\right ) \tan ^3(c+d x)}{15 d}+\frac {1}{16} \left (8 a^3 B+18 a b^2 B+18 a^2 b C+5 b^3 C\right ) \int \sec (c+d x) \, dx \\ & = \frac {\left (8 a^3 B+18 a b^2 B+18 a^2 b C+5 b^3 C\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (15 a^2 b B+4 b^3 B+5 a^3 C+12 a b^2 C\right ) \tan (c+d x)}{5 d}+\frac {\left (8 a^3 B+18 a b^2 B+18 a^2 b C+5 b^3 C\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {b \left (18 a b B+14 a^2 C+5 b^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {b^2 (3 b B+4 a C) \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {b C \sec ^3(c+d x) (a+b \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {\left (15 a^2 b B+4 b^3 B+5 a^3 C+12 a b^2 C\right ) \tan ^3(c+d x)}{15 d} \\ \end{align*}
Time = 3.99 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.77 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \left (8 a^3 B+18 a b^2 B+18 a^2 b C+5 b^3 C\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (240 \left (3 a^2 b B+b^3 B+a^3 C+3 a b^2 C\right )+15 \left (8 a^3 B+18 a b^2 B+18 a^2 b C+5 b^3 C\right ) \sec (c+d x)+10 b \left (18 a b B+18 a^2 C+5 b^2 C\right ) \sec ^3(c+d x)+40 b^3 C \sec ^5(c+d x)+80 \left (3 a^2 b B+2 b^3 B+a^3 C+6 a b^2 C\right ) \tan ^2(c+d x)+48 b^2 (b B+3 a C) \tan ^4(c+d x)\right )}{240 d} \]
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Time = 1.42 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.89
method | result | size |
parts | \(-\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 B a \,b^{2}+3 a^{2} b C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (3 B \,a^{2} b +a^{3} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(248\) |
derivativedivides | \(\frac {B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-3 B \,a^{2} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{2} b C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 B a \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 C a \,b^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-B \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(321\) |
default | \(\frac {B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{3} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-3 B \,a^{2} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 a^{2} b C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+3 B a \,b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 C a \,b^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-B \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(321\) |
parallelrisch | \(\frac {-120 \left (B \,a^{3}+\frac {9}{4} B a \,b^{2}+\frac {9}{4} a^{2} b C +\frac {5}{8} C \,b^{3}\right ) \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+120 \left (B \,a^{3}+\frac {9}{4} B a \,b^{2}+\frac {9}{4} a^{2} b C +\frac {5}{8} C \,b^{3}\right ) \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (4320 B \,a^{2} b +1920 B \,b^{3}+1440 a^{3} C +5760 C a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (720 B \,a^{3}+3060 B a \,b^{2}+3060 a^{2} b C +850 C \,b^{3}\right ) \sin \left (3 d x +3 c \right )+\left (2880 B \,a^{2} b +768 B \,b^{3}+960 a^{3} C +2304 C a \,b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (240 B \,a^{3}+540 B a \,b^{2}+540 a^{2} b C +150 C \,b^{3}\right ) \sin \left (5 d x +5 c \right )+\left (480 B \,a^{2} b +128 B \,b^{3}+160 a^{3} C +384 C a \,b^{2}\right ) \sin \left (6 d x +6 c \right )+480 \left (B \,a^{3}+\frac {21}{4} B a \,b^{2}+\frac {21}{4} a^{2} b C +\frac {33}{8} C \,b^{3}\right ) \sin \left (d x +c \right )}{240 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(407\) |
norman | \(\frac {\frac {\left (8 B \,a^{3}-48 B \,a^{2} b +30 B a \,b^{2}-16 B \,b^{3}-16 a^{3} C +30 a^{2} b C -48 C a \,b^{2}+11 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (8 B \,a^{3}+48 B \,a^{2} b +30 B a \,b^{2}+16 B \,b^{3}+16 a^{3} C +30 a^{2} b C +48 C a \,b^{2}+11 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (40 B \,a^{3}-720 B \,a^{2} b +30 B a \,b^{2}-208 B \,b^{3}-240 a^{3} C +30 a^{2} b C -624 C a \,b^{2}+75 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (40 B \,a^{3}+720 B \,a^{2} b +30 B a \,b^{2}+208 B \,b^{3}+240 a^{3} C +30 a^{2} b C +624 C a \,b^{2}+75 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}-\frac {\left (72 B \,a^{3}-528 B \,a^{2} b +126 B a \,b^{2}-112 B \,b^{3}-176 a^{3} C +126 a^{2} b C -336 C a \,b^{2}-5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {\left (72 B \,a^{3}+528 B \,a^{2} b +126 B a \,b^{2}+112 B \,b^{3}+176 a^{3} C +126 a^{2} b C +336 C a \,b^{2}-5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {\left (8 B \,a^{3}+18 B a \,b^{2}+18 a^{2} b C +5 C \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {\left (8 B \,a^{3}+18 B a \,b^{2}+18 a^{2} b C +5 C \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(519\) |
risch | \(-\frac {i \left (-160 a^{3} C -384 C a \,b^{2}-128 B \,b^{3}-75 C \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-480 B \,a^{2} b +360 B \,a^{3} {\mathrm e}^{9 i \left (d x +c \right )}-960 C \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+120 B \,a^{3} {\mathrm e}^{11 i \left (d x +c \right )}-768 B \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-1920 B \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-360 B \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-120 a^{3} B \,{\mathrm e}^{i \left (d x +c \right )}+240 B \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-1280 B \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-240 B \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-1920 C \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-425 C \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-1600 C \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-990 C \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-480 C \,a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+990 C \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+75 C \,b^{3} {\mathrm e}^{11 i \left (d x +c \right )}+425 C \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-1260 B a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-5760 B \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-1530 B a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-2880 B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-3840 C a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-1260 C \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-5760 C a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-1530 C \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-2304 C a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+270 C \,a^{2} b \,{\mathrm e}^{11 i \left (d x +c \right )}+1530 C \,a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}+1260 C \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+1260 B a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-4800 B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+270 B a \,b^{2} {\mathrm e}^{11 i \left (d x +c \right )}+1530 B a \,b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-1440 B \,a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-270 B a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-270 C \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,a^{3}}{2 d}-\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{2}}{8 d}-\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} b C}{8 d}-\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{16 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,a^{3}}{2 d}+\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{2}}{8 d}+\frac {9 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} b C}{8 d}+\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{16 d}\) | \(817\) |
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Time = 0.27 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.03 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (8 \, B a^{3} + 18 \, C a^{2} b + 18 \, B a b^{2} + 5 \, C b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, B a^{3} + 18 \, C a^{2} b + 18 \, B a b^{2} + 5 \, C b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (32 \, {\left (5 \, C a^{3} + 15 \, B a^{2} b + 12 \, C a b^{2} + 4 \, B b^{3}\right )} \cos \left (d x + c\right )^{5} + 15 \, {\left (8 \, B a^{3} + 18 \, C a^{2} b + 18 \, B a b^{2} + 5 \, C b^{3}\right )} \cos \left (d x + c\right )^{4} + 40 \, C b^{3} + 16 \, {\left (5 \, C a^{3} + 15 \, B a^{2} b + 12 \, C a b^{2} + 4 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (18 \, C a^{2} b + 18 \, B a b^{2} + 5 \, C b^{3}\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
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\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3} \sec ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.47 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} b + 96 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a b^{2} + 32 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B b^{3} - 5 \, C b^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, C a^{2} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, B a b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (264) = 528\).
Time = 0.36 (sec) , antiderivative size = 932, normalized size of antiderivative = 3.35 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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Time = 20.90 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.05 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {B\,a^3}{2}+\frac {9\,C\,a^2\,b}{8}+\frac {9\,B\,a\,b^2}{8}+\frac {5\,C\,b^3}{16}\right )}{2\,B\,a^3+\frac {9\,C\,a^2\,b}{2}+\frac {9\,B\,a\,b^2}{2}+\frac {5\,C\,b^3}{4}}\right )\,\left (B\,a^3+\frac {9\,C\,a^2\,b}{4}+\frac {9\,B\,a\,b^2}{4}+\frac {5\,C\,b^3}{8}\right )}{d}+\frac {\left (B\,a^3-2\,B\,b^3-2\,C\,a^3+\frac {11\,C\,b^3}{8}+\frac {15\,B\,a\,b^2}{4}-6\,B\,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 (\frac {14\,B\,b^3}{3}-3\,B\,a^3+\frac {22\,C\,a^3}{3}+\frac {5\,C\,b^3}{24}-\frac {21\,B\,a\,b^2}{4}+22\,B\,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 (2\,B\,a^3-\frac {52\,B\,b^3}{5}-12\,C\,a^3+\frac {15\,C\,b^3}{4}+\frac {3\,B\,a\,b^2}{2}-36\,B\,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 (2\,B\,a^3+\frac {52\,B\,b^3}{5}+12\,C\,a^3+\frac {15\,C\,b^3}{4}+\frac {3\,B\,a\,b^2}{2}+36\,B\,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 {14\,B\,b^3}{3}-\frac {22\,C\,a^3}{3}-3\,B\,a^3-\frac {21\,B\,a\,b^2}{4}-22\,B\,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 (B\,a^3+2\,B\,b^3+2\,C\,a^3+\frac {11\,C\,b^3}{8}+\frac {15\,B\,a\,b^2}{4}+6\,B\,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 )} \]
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