Integrand size = 29, antiderivative size = 250 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {\left (8 a^4 A+24 a^2 A b^2+3 A b^4+16 a^3 b B+12 a b^3 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (95 a^3 A b+80 a A b^3+12 a^4 B+112 a^2 b^2 B+16 b^4 B\right ) \tan (c+d x)}{30 d}+\frac {b \left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d} \]
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Time = 0.55 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4087, 4082, 3872, 3855, 3852, 8} \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {\left (12 a^2 B+35 a A b+16 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{60 d}+\frac {b \left (24 a^3 B+130 a^2 A b+116 a b^2 B+45 A b^3\right ) \tan (c+d x) \sec (c+d x)}{120 d}+\frac {\left (8 a^4 A+16 a^3 b B+24 a^2 A b^2+12 a b^3 B+3 A b^4\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (12 a^4 B+95 a^3 A b+112 a^2 b^2 B+80 a A b^3+16 b^4 B\right ) \tan (c+d x)}{30 d}+\frac {(4 a B+5 A b) \tan (c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^4}{5 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4082
Rule 4087
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{5} \int \sec (c+d x) (a+b \sec (c+d x))^3 (5 a A+4 b B+(5 A b+4 a B) \sec (c+d x)) \, dx \\ & = \frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{20} \int \sec (c+d x) (a+b \sec (c+d x))^2 \left (20 a^2 A+15 A b^2+28 a b B+\left (35 a A b+12 a^2 B+16 b^2 B\right ) \sec (c+d x)\right ) \, dx \\ & = \frac {\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{60} \int \sec (c+d x) (a+b \sec (c+d x)) \left (60 a^3 A+115 a A b^2+108 a^2 b B+32 b^3 B+\left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \sec (c+d x)\right ) \, dx \\ & = \frac {b \left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{120} \int \sec (c+d x) \left (15 \left (8 a^4 A+24 a^2 A b^2+3 A b^4+16 a^3 b B+12 a b^3 B\right )+4 \left (95 a^3 A b+80 a A b^3+12 a^4 B+112 a^2 b^2 B+16 b^4 B\right ) \sec (c+d x)\right ) \, dx \\ & = \frac {b \left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{8} \left (8 a^4 A+24 a^2 A b^2+3 A b^4+16 a^3 b B+12 a b^3 B\right ) \int \sec (c+d x) \, dx+\frac {1}{30} \left (95 a^3 A b+80 a A b^3+12 a^4 B+112 a^2 b^2 B+16 b^4 B\right ) \int \sec ^2(c+d x) \, dx \\ & = \frac {\left (8 a^4 A+24 a^2 A b^2+3 A b^4+16 a^3 b B+12 a b^3 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}-\frac {\left (95 a^3 A b+80 a A b^3+12 a^4 B+112 a^2 b^2 B+16 b^4 B\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{30 d} \\ & = \frac {\left (8 a^4 A+24 a^2 A b^2+3 A b^4+16 a^3 b B+12 a b^3 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (95 a^3 A b+80 a A b^3+12 a^4 B+112 a^2 b^2 B+16 b^4 B\right ) \tan (c+d x)}{30 d}+\frac {b \left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{60 d}+\frac {(5 A b+4 a B) (a+b \sec (c+d x))^3 \tan (c+d x)}{20 d}+\frac {B (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d} \\ \end{align*}
Time = 3.37 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.79 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {15 \left (8 a^4 A+24 a^2 A b^2+3 A b^4+16 a^3 b B+12 a b^3 B\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (120 \left (4 a^3 A b+4 a A b^3+a^4 B+6 a^2 b^2 B+b^4 B\right )+15 b \left (24 a^2 A b+3 A b^3+16 a^3 B+12 a b^2 B\right ) \sec (c+d x)+30 b^3 (A b+4 a B) \sec ^3(c+d x)+80 b^2 \left (2 a A b+3 a^2 B+b^2 B\right ) \tan ^2(c+d x)+24 b^4 B \tan ^4(c+d x)\right )}{120 d} \]
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Time = 6.02 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.95
method | result | size |
parts | \(\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}-\frac {B \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(237\) |
derivativedivides | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \tan \left (d x +c \right )+4 A \,a^{3} b \tan \left (d x +c \right )+4 B \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 B \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 A a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 B a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+A \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(313\) |
default | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \tan \left (d x +c \right )+4 A \,a^{3} b \tan \left (d x +c \right )+4 B \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 B \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 A a \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 B a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+A \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(313\) |
parallelrisch | \(\frac {-120 \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right ) \left (a^{4} A +3 A \,a^{2} b^{2}+\frac {3}{8} A \,b^{4}+2 B \,a^{3} b +\frac {3}{2} B a \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+120 \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right ) \left (a^{4} A +3 A \,a^{2} b^{2}+\frac {3}{8} A \,b^{4}+2 B \,a^{3} b +\frac {3}{2} B a \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (1440 A \,a^{3} b +1600 A a \,b^{3}+360 B \,a^{4}+2400 B \,a^{2} b^{2}+320 B \,b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (480 A \,a^{3} b +320 A a \,b^{3}+120 B \,a^{4}+480 B \,a^{2} b^{2}+64 B \,b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (1440 A \,a^{2} b^{2}+420 A \,b^{4}+960 B \,a^{3} b +1680 B a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (720 A \,a^{2} b^{2}+90 A \,b^{4}+480 B \,a^{3} b +360 B a \,b^{3}\right ) \sin \left (4 d x +4 c \right )+960 \sin \left (d x +c \right ) \left (A \,a^{3} b +\frac {4}{3} A a \,b^{3}+\frac {1}{4} B \,a^{4}+2 B \,a^{2} b^{2}+\frac {2}{3} B \,b^{4}\right )}{120 d \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\) | \(409\) |
norman | \(\frac {-\frac {4 \left (180 A \,a^{3} b +100 A a \,b^{3}+45 B \,a^{4}+150 B \,a^{2} b^{2}+29 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}-\frac {\left (32 A \,a^{3} b -24 A \,a^{2} b^{2}+32 A a \,b^{3}-5 A \,b^{4}+8 B \,a^{4}-16 B \,a^{3} b +48 B \,a^{2} b^{2}-20 B a \,b^{3}+8 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {\left (32 A \,a^{3} b +24 A \,a^{2} b^{2}+32 A a \,b^{3}+5 A \,b^{4}+8 B \,a^{4}+16 B \,a^{3} b +48 B \,a^{2} b^{2}+20 B a \,b^{3}+8 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (192 A \,a^{3} b -72 A \,a^{2} b^{2}+128 A a \,b^{3}-3 A \,b^{4}+48 B \,a^{4}-48 B \,a^{3} b +192 B \,a^{2} b^{2}-12 B a \,b^{3}+16 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {\left (192 A \,a^{3} b +72 A \,a^{2} b^{2}+128 A a \,b^{3}+3 A \,b^{4}+48 B \,a^{4}+48 B \,a^{3} b +192 B \,a^{2} b^{2}+12 B a \,b^{3}+16 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}-\frac {\left (8 a^{4} A +24 A \,a^{2} b^{2}+3 A \,b^{4}+16 B \,a^{3} b +12 B a \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (8 a^{4} A +24 A \,a^{2} b^{2}+3 A \,b^{4}+16 B \,a^{3} b +12 B a \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\) | \(499\) |
risch | \(\frac {i \left (480 A \,a^{3} b +320 A a \,b^{3}+480 B \,a^{2} b^{2}+64 B \,b^{4}+640 B \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+45 A \,b^{4} {\mathrm e}^{i \left (d x +c \right )}-45 A \,b^{4} {\mathrm e}^{9 i \left (d x +c \right )}+320 B \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-210 A \,b^{4} {\mathrm e}^{7 i \left (d x +c \right )}+210 A \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+120 B \,a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+480 B \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+480 B \,a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+720 B \,a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-360 A \,a^{2} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-240 B \,a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}+2240 A a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3360 B \,a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+720 A \,a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+480 B \,a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+840 B a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-180 B a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}+480 A \,a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+960 A a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+1440 B \,a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+2880 A \,a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+360 A \,a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}+240 B \,a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+180 B a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+1920 A \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+1600 A a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2400 B \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-720 A \,a^{2} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-480 B \,a^{3} b \,{\mathrm e}^{7 i \left (d x +c \right )}-840 B a \,b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+1920 A \,a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+120 B \,a^{4}\right )}{60 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{4} A}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,a^{2} b^{2}}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{4}}{8 d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,a^{3} b}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{3}}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{4} A}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,a^{2} b^{2}}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{4}}{8 d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,a^{3} b}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{3}}{2 d}\) | \(802\) |
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Time = 0.29 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.12 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (8 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 12 \, B a b^{3} + 3 \, A b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 12 \, B a b^{3} + 3 \, A b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, B b^{4} + 8 \, {\left (15 \, B a^{4} + 60 \, A a^{3} b + 60 \, B a^{2} b^{2} + 40 \, A a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 12 \, B a b^{3} + 3 \, A b^{4}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (15 \, B a^{2} b^{2} + 10 \, A a b^{3} + 2 \, B b^{4}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
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\[ \int \sec (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{4} \sec {\left (c + d x \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.52 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} b^{2} + 320 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a b^{3} + 16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B b^{4} - 60 \, B a b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 15 \, A b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 240 \, B a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, A a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 240 \, B a^{4} \tan \left (d x + c\right ) + 960 \, A a^{3} b \tan \left (d x + c\right )}{240 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 850 vs. \(2 (238) = 476\).
Time = 0.38 (sec) , antiderivative size = 850, normalized size of antiderivative = 3.40 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Too large to display} \]
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Time = 18.03 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.22 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A\,a^4+2\,B\,a^3\,b+3\,A\,a^2\,b^2+\frac {3\,B\,a\,b^3}{2}+\frac {3\,A\,b^4}{8}\right )}{4\,A\,a^4+8\,B\,a^3\,b+12\,A\,a^2\,b^2+6\,B\,a\,b^3+\frac {3\,A\,b^4}{2}}\right )\,\left (2\,A\,a^4+4\,B\,a^3\,b+6\,A\,a^2\,b^2+3\,B\,a\,b^3+\frac {3\,A\,b^4}{4}\right )}{d}-\frac {\left (2\,B\,a^4-\frac {5\,A\,b^4}{4}+2\,B\,b^4-6\,A\,a^2\,b^2+12\,B\,a^2\,b^2+8\,A\,a\,b^3+8\,A\,a^3\,b-5\,B\,a\,b^3-4\,B\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {A\,b^4}{2}-8\,B\,a^4-\frac {8\,B\,b^4}{3}+12\,A\,a^2\,b^2-32\,B\,a^2\,b^2-\frac {64\,A\,a\,b^3}{3}-32\,A\,a^3\,b+2\,B\,a\,b^3+8\,B\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (12\,B\,a^4+48\,A\,a^3\,b+40\,B\,a^2\,b^2+\frac {80\,A\,a\,b^3}{3}+\frac {116\,B\,b^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {A\,b^4}{2}-8\,B\,a^4-\frac {8\,B\,b^4}{3}-12\,A\,a^2\,b^2-32\,B\,a^2\,b^2-\frac {64\,A\,a\,b^3}{3}-32\,A\,a^3\,b-2\,B\,a\,b^3-8\,B\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,A\,b^4}{4}+2\,B\,a^4+2\,B\,b^4+6\,A\,a^2\,b^2+12\,B\,a^2\,b^2+8\,A\,a\,b^3+8\,A\,a^3\,b+5\,B\,a\,b^3+4\,B\,a^3\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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