Integrand size = 31, antiderivative size = 334 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {\left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (24 a^4 A b+224 a^2 A b^3+32 A b^5-4 a^5 B+121 a^3 b^2 B+128 a b^4 B\right ) \tan (c+d x)}{60 b d}+\frac {\left (48 a^3 A b+232 a A b^3-8 a^4 B+178 a^2 b^2 B+75 b^4 B\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {\left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac {\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac {(6 A b-a B) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac {B (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d} \]
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Time = 0.73 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4095, 4087, 4082, 3872, 3855, 3852, 8} \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{120 b d}+\frac {\left (-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{120 b d}+\frac {\left (8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (-8 a^4 B+48 a^3 A b+178 a^2 b^2 B+232 a A b^3+75 b^4 B\right ) \tan (c+d x) \sec (c+d x)}{240 d}+\frac {\left (-4 a^5 B+24 a^4 A b+121 a^3 b^2 B+224 a^2 A b^3+128 a b^4 B+32 A b^5\right ) \tan (c+d x)}{60 b d}+\frac {(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d}+\frac {B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4082
Rule 4087
Rule 4095
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\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 {(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}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (3 b (8 A b+7 a B)+\left (24 a A b-4 a^2 B+25 b^2 B\right ) \sec (c+d x)\right ) \, dx}{30 b} \\ & = \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}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^2 \left (3 b \left (56 a A b+24 a^2 B+25 b^2 B\right )+3 \left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) \sec (c+d x)\right ) \, dx}{120 b} \\ & = \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}+\frac {\int \sec (c+d x) (a+b \sec (c+d x)) \left (3 b \left (216 a^2 A b+64 A b^3+64 a^3 B+181 a b^2 B\right )+3 \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)\right ) \, dx}{360 b} \\ & = \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}+\frac {\int \sec (c+d x) \left (45 b \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 )+12 \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 ) \sec (c+d x)\right ) \, dx}{720 b} \\ & = \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}+\frac {1}{16} \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 ) \int \sec (c+d x) \, dx+\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 ) \int \sec ^2(c+d x) \, dx}{60 b} \\ & = \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 (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}-\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 ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{60 b d} \\ & = \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} \\ \end{align*}
Time = 2.18 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.73 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {15 \left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (240 \left (a^4 A+6 a^2 A b^2+A b^4+4 a^3 b B+4 a b^3 B\right )+15 \left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right ) \sec (c+d x)+10 b^2 \left (24 a A b+36 a^2 B+5 b^2 B\right ) \sec ^3(c+d x)+40 b^4 B \sec ^5(c+d x)+160 b \left (3 a^2 A b+A b^3+2 a^3 B+4 a b^2 B\right ) \tan ^2(c+d x)+48 b^3 (A b+4 a B) \tan ^4(c+d x)\right )}{240 d} \]
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Time = 7.38 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.83
method | result | size |
parts | \(-\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {a^{4} A \tan \left (d x +c \right )}{d}\) | \(277\) |
derivativedivides | \(\frac {a^{4} A \tan \left (d x +c \right )+B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B \,a^{3} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-6 A \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 B \,a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 A a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 B a \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-A \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(375\) |
default | \(\frac {a^{4} A \tan \left (d x +c \right )+B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B \,a^{3} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-6 A \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 B \,a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 A a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 B a \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-A \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(375\) |
parallelrisch | \(\frac {-480 \left (\frac {1}{4} B \,a^{4}+\frac {3}{4} A a \,b^{3}+\frac {9}{8} B \,a^{2} b^{2}+A \,a^{3} b +\frac {5}{32} B \,b^{4}\right ) \left (10+\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+480 \left (\frac {1}{4} B \,a^{4}+\frac {3}{4} A a \,b^{3}+\frac {9}{8} B \,a^{2} b^{2}+A \,a^{3} b +\frac {5}{32} B \,b^{4}\right ) \left (10+\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (1200 a^{4} A +8640 A \,a^{2} b^{2}+1920 A \,b^{4}+5760 B \,a^{3} b +7680 B a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (2880 A \,a^{3} b +4080 A a \,b^{3}+720 B \,a^{4}+6120 B \,a^{2} b^{2}+850 B \,b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (960 a^{4} A +5760 A \,a^{2} b^{2}+768 A \,b^{4}+3840 B \,a^{3} b +3072 B a \,b^{3}\right ) \sin \left (4 d x +4 c \right )+\left (960 A \,a^{3} b +720 A a \,b^{3}+240 B \,a^{4}+1080 B \,a^{2} b^{2}+150 B \,b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (240 a^{4} A +960 A \,a^{2} b^{2}+128 A \,b^{4}+640 B \,a^{3} b +512 B a \,b^{3}\right ) \sin \left (6 d x +6 c \right )+1920 \left (A \,a^{3} b +\frac {7}{4} A a \,b^{3}+\frac {1}{4} B \,a^{4}+\frac {21}{8} B \,a^{2} b^{2}+\frac {33}{32} B \,b^{4}\right ) \sin \left (d x +c \right )}{240 d \left (10+\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )\right )}\) | \(479\) |
norman | \(\frac {-\frac {\left (16 a^{4} A -32 A \,a^{3} b +96 A \,a^{2} b^{2}-40 A a \,b^{3}+16 A \,b^{4}-8 B \,a^{4}+64 B \,a^{3} b -60 B \,a^{2} b^{2}+64 B a \,b^{3}-11 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (16 a^{4} A +32 A \,a^{3} b +96 A \,a^{2} b^{2}+40 A a \,b^{3}+16 A \,b^{4}+8 B \,a^{4}+64 B \,a^{3} b +60 B \,a^{2} b^{2}+64 B a \,b^{3}+11 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (240 a^{4} A -288 A \,a^{3} b +1056 A \,a^{2} b^{2}-168 A a \,b^{3}+112 A \,b^{4}-72 B \,a^{4}+704 B \,a^{3} b -252 B \,a^{2} b^{2}+448 B a \,b^{3}+5 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {\left (240 a^{4} A +288 A \,a^{3} b +1056 A \,a^{2} b^{2}+168 A a \,b^{3}+112 A \,b^{4}+72 B \,a^{4}+704 B \,a^{3} b +252 B \,a^{2} b^{2}+448 B a \,b^{3}-5 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}-\frac {\left (400 a^{4} A -160 A \,a^{3} b +1440 A \,a^{2} b^{2}-40 A a \,b^{3}+208 A \,b^{4}-40 B \,a^{4}+960 B \,a^{3} b -60 B \,a^{2} b^{2}+832 B a \,b^{3}-75 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (400 a^{4} A +160 A \,a^{3} b +1440 A \,a^{2} b^{2}+40 A a \,b^{3}+208 A \,b^{4}+40 B \,a^{4}+960 B \,a^{3} b +60 B \,a^{2} b^{2}+832 B a \,b^{3}+75 B \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {\left (32 A \,a^{3} b +24 A a \,b^{3}+8 B \,a^{4}+36 B \,a^{2} b^{2}+5 B \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {\left (32 A \,a^{3} b +24 A a \,b^{3}+8 B \,a^{4}+36 B \,a^{2} b^{2}+5 B \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(645\) |
risch | \(\text {Expression too large to display}\) | \(1071\) |
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Time = 0.29 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.98 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (15 \, A a^{4} + 40 \, B a^{3} b + 60 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{5} + 40 \, B b^{4} + 15 \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{4} + 32 \, {\left (10 \, B a^{3} b + 15 \, A a^{2} b^{2} + 8 \, B a b^{3} + 2 \, A b^{4}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
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\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\int \left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{4} \sec ^{2}{\left (c + d x \right )}\, dx \]
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Time = 0.22 (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} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1186 vs. \(2 (319) = 638\).
Time = 0.40 (sec) , antiderivative size = 1186, normalized size of antiderivative = 3.55 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Too large to display} \]
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Time = 18.35 (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=\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {B\,a^4}{2}+2\,A\,a^3\,b+\frac {9\,B\,a^2\,b^2}{4}+\frac {3\,A\,a\,b^3}{2}+\frac {5\,B\,b^4}{16}\right )}{2\,B\,a^4+8\,A\,a^3\,b+9\,B\,a^2\,b^2+6\,A\,a\,b^3+\frac {5\,B\,b^4}{4}}\right )\,\left (B\,a^4+4\,A\,a^3\,b+\frac {9\,B\,a^2\,b^2}{2}+3\,A\,a\,b^3+\frac {5\,B\,b^4}{8}\right )}{d}+\frac {\left (B\,a^4-2\,A\,b^4-2\,A\,a^4+\frac {11\,B\,b^4}{8}-12\,A\,a^2\,b^2+\frac {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\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (10\,A\,a^4+\frac {14\,A\,b^4}{3}-3\,B\,a^4+\frac {5\,B\,b^4}{24}+44\,A\,a^2\,b^2-\frac {21\,B\,a^2\,b^2}{2}-7\,A\,a\,b^3-12\,A\,a^3\,b+\frac {56\,B\,a\,b^3}{3}+\frac {88\,B\,a^3\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (2\,B\,a^4-\frac {52\,A\,b^4}{5}-20\,A\,a^4+\frac {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-\frac {208\,B\,a\,b^3}{5}-48\,B\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (20\,A\,a^4+\frac {52\,A\,b^4}{5}+2\,B\,a^4+\frac {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+\frac {208\,B\,a\,b^3}{5}+48\,B\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {5\,B\,b^4}{24}-\frac {14\,A\,b^4}{3}-3\,B\,a^4-10\,A\,a^4-44\,A\,a^2\,b^2-\frac {21\,B\,a^2\,b^2}{2}-7\,A\,a\,b^3-12\,A\,a^3\,b-\frac {56\,B\,a\,b^3}{3}-\frac {88\,B\,a^3\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^4+2\,A\,b^4+B\,a^4+\frac {11\,B\,b^4}{8}+12\,A\,a^2\,b^2+\frac {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\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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