Integrand size = 31, antiderivative size = 310 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {a \left (4 a^4 C-32 b^4 (5 A+4 C)-a^2 b^2 (190 A+121 C)\right ) \tan (c+d x)}{60 b d}-\frac {\left (8 a^4 C-15 b^4 (6 A+5 C)-2 a^2 b^2 (130 A+89 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {a \left (70 A b^2-4 a^2 C+53 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}-\frac {\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}-\frac {a C (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac {C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d} \]
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Time = 0.84 (sec) , antiderivative size = 310, 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 = {4168, 4087, 4082, 3872, 3855, 3852, 8} \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{120 b d}+\frac {a \left (-4 a^2 C+70 A b^2+53 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{120 b d}+\frac {\left (8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {a \left (4 a^4 C-a^2 b^2 (190 A+121 C)-32 b^4 (5 A+4 C)\right ) \tan (c+d x)}{60 b d}-\frac {\left (8 a^4 C-2 a^2 b^2 (130 A+89 C)-15 b^4 (6 A+5 C)\right ) \tan (c+d x) \sec (c+d x)}{240 d}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}-\frac {a C \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4082
Rule 4087
Rule 4168
Rubi steps \begin{align*} \text {integral}& = \frac {C (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 (b (6 A+5 C)-a C \sec (c+d x)) \, dx}{6 b} \\ & = -\frac {a C (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac {C (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 a b (10 A+7 C)-\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) \sec (c+d x)\right ) \, dx}{30 b} \\ & = -\frac {\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}-\frac {a C (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac {C (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 (8 a^2 (5 A+3 C)+5 b^2 (6 A+5 C)\right )+3 a \left (70 A b^2-4 a^2 C+53 b^2 C\right ) \sec (c+d x)\right ) \, dx}{120 b} \\ & = \frac {a \left (70 A b^2-4 a^2 C+53 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}-\frac {\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}-\frac {a C (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac {C (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 a b \left (8 a^2 (15 A+8 C)+b^2 (230 A+181 C)\right )-3 \left (8 a^4 C-15 b^4 (6 A+5 C)-2 a^2 b^2 (130 A+89 C)\right ) \sec (c+d x)\right ) \, dx}{360 b} \\ & = -\frac {\left (8 a^4 C-15 b^4 (6 A+5 C)-2 a^2 b^2 (130 A+89 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {a \left (70 A b^2-4 a^2 C+53 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}-\frac {\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}-\frac {a C (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac {C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac {\int \sec (c+d x) \left (45 b \left (8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right )-12 a \left (4 a^4 C-32 b^4 (5 A+4 C)-a^2 b^2 (190 A+121 C)\right ) \sec (c+d x)\right ) \, dx}{720 b} \\ & = -\frac {\left (8 a^4 C-15 b^4 (6 A+5 C)-2 a^2 b^2 (130 A+89 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {a \left (70 A b^2-4 a^2 C+53 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}-\frac {\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}-\frac {a C (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac {C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac {1}{16} \left (8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (a \left (4 a^4 C-32 b^4 (5 A+4 C)-a^2 b^2 (190 A+121 C)\right )\right ) \int \sec ^2(c+d x) \, dx}{60 b} \\ & = \frac {\left (8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {\left (8 a^4 C-15 b^4 (6 A+5 C)-2 a^2 b^2 (130 A+89 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {a \left (70 A b^2-4 a^2 C+53 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}-\frac {\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}-\frac {a C (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac {C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac {\left (a \left (4 a^4 C-32 b^4 (5 A+4 C)-a^2 b^2 (190 A+121 C)\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{60 b d} \\ & = \frac {\left (8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {a \left (4 a^4 C-32 b^4 (5 A+4 C)-a^2 b^2 (190 A+121 C)\right ) \tan (c+d x)}{60 b d}-\frac {\left (8 a^4 C-15 b^4 (6 A+5 C)-2 a^2 b^2 (130 A+89 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac {a \left (70 A b^2-4 a^2 C+53 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}-\frac {\left (4 a^2 C-5 b^2 (6 A+5 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}-\frac {a C (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac {C (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d} \\ \end{align*}
Time = 8.88 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.67 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \left (8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 \left (8 a^4 C+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right ) \sec (c+d x)+10 b^2 \left (6 A b^2+36 a^2 C+5 b^2 C\right ) \sec ^3(c+d x)+40 b^4 C \sec ^5(c+d x)+64 a b \left (15 \left (a^2+b^2\right ) (A+C)+5 \left (A b^2+a^2 C+2 b^2 C\right ) \tan ^2(c+d x)+3 b^2 C \tan ^4(c+d x)\right )\right )}{240 d} \]
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Time = 1.57 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.94
method | result | size |
parts | \(\frac {\left (A \,b^{4}+6 C \,a^{2} b^{2}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (4 a A \,b^{3}+4 a^{3} b C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {C \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}+\frac {4 A \,a^{3} b \tan \left (d x +c \right )}{d}-\frac {4 C a \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(291\) |
derivativedivides | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 A \,a^{3} b \tan \left (d x +c \right )-4 a^{3} b C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 C \,a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a A \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 C a \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+A \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(360\) |
default | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 A \,a^{3} b \tan \left (d x +c \right )-4 a^{3} b C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 C \,a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a A \,b^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 C a \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+A \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(360\) |
parallelrisch | \(\frac {-240 \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (\left (\frac {3 A}{8}+\frac {5 C}{16}\right ) b^{4}+3 \left (A +\frac {3 C}{4}\right ) a^{2} b^{2}+a^{4} \left (A +\frac {C}{2}\right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+240 \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (\left (\frac {3 A}{8}+\frac {5 C}{16}\right ) b^{4}+3 \left (A +\frac {3 C}{4}\right ) a^{2} b^{2}+a^{4} \left (A +\frac {C}{2}\right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (1020 A +850 C \right ) b^{4}+4320 a^{2} \left (A +\frac {17 C}{12}\right ) b^{2}+720 a^{4} C \right ) \sin \left (3 d x +3 c \right )+\left (\left (180 A +150 C \right ) b^{4}+1440 \left (A +\frac {3 C}{4}\right ) a^{2} b^{2}+240 a^{4} C \right ) \sin \left (5 d x +5 c \right )+4800 a b \left (\left (\frac {6 A}{5}+\frac {8 C}{5}\right ) b^{2}+a^{2} \left (A +\frac {6 C}{5}\right )\right ) \sin \left (2 d x +2 c \right )+3840 a b \left (b^{2} \left (A +\frac {4 C}{5}\right )+\left (A +C \right ) a^{2}\right ) \sin \left (4 d x +4 c \right )+960 a b \left (\left (\frac {2 A}{3}+\frac {8 C}{15}\right ) b^{2}+a^{2} \left (A +\frac {2 C}{3}\right )\right ) \sin \left (6 d x +6 c \right )+2880 \left (\left (\frac {7 A}{24}+\frac {11 C}{16}\right ) b^{4}+a^{2} \left (A +\frac {7 C}{4}\right ) b^{2}+\frac {a^{4} C}{6}\right ) \sin \left (d x +c \right )}{240 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(422\) |
norman | \(\frac {-\frac {\left (64 A \,a^{3} b -48 A \,a^{2} b^{2}+64 a A \,b^{3}-10 A \,b^{4}-8 a^{4} C +64 a^{3} b C -60 C \,a^{2} b^{2}+64 C a \,b^{3}-11 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (64 A \,a^{3} b +48 A \,a^{2} b^{2}+64 a A \,b^{3}+10 A \,b^{4}+8 a^{4} C +64 a^{3} b C +60 C \,a^{2} b^{2}+64 C a \,b^{3}+11 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (960 A \,a^{3} b -432 A \,a^{2} b^{2}+704 a A \,b^{3}-42 A \,b^{4}-72 a^{4} C +704 a^{3} b C -252 C \,a^{2} b^{2}+448 C a \,b^{3}+5 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {\left (960 A \,a^{3} b +432 A \,a^{2} b^{2}+704 a A \,b^{3}+42 A \,b^{4}+72 a^{4} C +704 a^{3} b C +252 C \,a^{2} b^{2}+448 C a \,b^{3}-5 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}-\frac {\left (1600 A \,a^{3} b -240 A \,a^{2} b^{2}+960 a A \,b^{3}-10 A \,b^{4}-40 a^{4} C +960 a^{3} b C -60 C \,a^{2} b^{2}+832 C a \,b^{3}-75 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (1600 A \,a^{3} b +240 A \,a^{2} b^{2}+960 a A \,b^{3}+10 A \,b^{4}+40 a^{4} C +960 a^{3} b C +60 C \,a^{2} b^{2}+832 C a \,b^{3}+75 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {\left (16 a^{4} A +48 A \,a^{2} b^{2}+6 A \,b^{4}+8 a^{4} C +36 C \,a^{2} b^{2}+5 C \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {\left (16 a^{4} A +48 A \,a^{2} b^{2}+6 A \,b^{4}+8 a^{4} C +36 C \,a^{2} b^{2}+5 C \,b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(623\) |
risch | \(\text {Expression too large to display}\) | \(1067\) |
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Time = 0.28 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.96 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (8 \, {\left (2 \, A + C\right )} a^{4} + 12 \, {\left (4 \, A + 3 \, C\right )} a^{2} b^{2} + {\left (6 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, {\left (2 \, A + C\right )} a^{4} + 12 \, {\left (4 \, A + 3 \, C\right )} a^{2} b^{2} + {\left (6 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (192 \, C a b^{3} \cos \left (d x + c\right ) + 64 \, {\left (5 \, {\left (3 \, A + 2 \, C\right )} a^{3} b + 2 \, {\left (5 \, A + 4 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{5} + 40 \, C b^{4} + 15 \, {\left (8 \, C a^{4} + 12 \, {\left (4 \, A + 3 \, C\right )} a^{2} b^{2} + {\left (6 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 64 \, {\left (5 \, C a^{3} b + {\left (5 \, A + 4 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (36 \, C a^{2} b^{2} + {\left (6 \, A + 5 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
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\[ \int \sec (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{4} \sec {\left (c + d x \right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.48 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} b + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a b^{3} + 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 )} C a b^{3} - 5 \, C 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 \, C 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 )} - 30 \, 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 )} - 120 \, C 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 )} - 720 \, 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 )} + 480 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 1920 \, A a^{3} b \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. 1100 vs. \(2 (296) = 592\).
Time = 0.41 (sec) , antiderivative size = 1100, normalized size of antiderivative = 3.55 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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Time = 19.12 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.23 \[ \int \sec (c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {5\,A\,b^4}{4}+C\,a^4+\frac {11\,C\,b^4}{8}+6\,A\,a^2\,b^2+\frac {15\,C\,a^2\,b^2}{2}-8\,A\,a\,b^3-8\,A\,a^3\,b-8\,C\,a\,b^3-8\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {5\,C\,b^4}{24}-3\,C\,a^4-\frac {7\,A\,b^4}{4}-18\,A\,a^2\,b^2-\frac {21\,C\,a^2\,b^2}{2}+\frac {88\,A\,a\,b^3}{3}+40\,A\,a^3\,b+\frac {56\,C\,a\,b^3}{3}+\frac {88\,C\,a^3\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {A\,b^4}{2}+2\,C\,a^4+\frac {15\,C\,b^4}{4}+12\,A\,a^2\,b^2+3\,C\,a^2\,b^2-48\,A\,a\,b^3-80\,A\,a^3\,b-\frac {208\,C\,a\,b^3}{5}-48\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {A\,b^4}{2}+2\,C\,a^4+\frac {15\,C\,b^4}{4}+12\,A\,a^2\,b^2+3\,C\,a^2\,b^2+48\,A\,a\,b^3+80\,A\,a^3\,b+\frac {208\,C\,a\,b^3}{5}+48\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {5\,C\,b^4}{24}-3\,C\,a^4-\frac {7\,A\,b^4}{4}-18\,A\,a^2\,b^2-\frac {21\,C\,a^2\,b^2}{2}-\frac {88\,A\,a\,b^3}{3}-40\,A\,a^3\,b-\frac {56\,C\,a\,b^3}{3}-\frac {88\,C\,a^3\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,A\,b^4}{4}+C\,a^4+\frac {11\,C\,b^4}{8}+6\,A\,a^2\,b^2+\frac {15\,C\,a^2\,b^2}{2}+8\,A\,a\,b^3+8\,A\,a^3\,b+8\,C\,a\,b^3+8\,C\,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 )}+\frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A\,a^4+\frac {3\,A\,b^4}{8}+\frac {C\,a^4}{2}+\frac {5\,C\,b^4}{16}+3\,A\,a^2\,b^2+\frac {9\,C\,a^2\,b^2}{4}\right )}{4\,A\,a^4+\frac {3\,A\,b^4}{2}+2\,C\,a^4+\frac {5\,C\,b^4}{4}+12\,A\,a^2\,b^2+9\,C\,a^2\,b^2}\right )\,\left (2\,A\,a^4+\frac {3\,A\,b^4}{4}+C\,a^4+\frac {5\,C\,b^4}{8}+6\,A\,a^2\,b^2+\frac {9\,C\,a^2\,b^2}{2}\right )}{d} \]
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