Integrand size = 33, antiderivative size = 306 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a \left (2 a^4 C+24 b^4 (5 A+4 C)+a^2 b^2 (30 A+17 C)\right ) \tan (c+d x)}{60 b^2 d}+\frac {\left (4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right ) \sec (c+d x) \tan (c+d x)}{240 b d}+\frac {a \left (30 A b^2+2 a^2 C+21 b^2 C\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}-\frac {a C (a+b \sec (c+d x))^4 \tan (c+d x)}{15 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d} \]
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Time = 0.79 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4178, 4167, 4087, 4082, 3872, 3855, 3852, 8} \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{120 b^2 d}+\frac {a \left (2 a^2 C+30 A b^2+21 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{120 b^2 d}+\frac {a \left (2 a^4 C+a^2 b^2 (30 A+17 C)+24 b^4 (5 A+4 C)\right ) \tan (c+d x)}{60 b^2 d}+\frac {\left (4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right ) \tan (c+d x) \sec (c+d x)}{240 b d}-\frac {a C \tan (c+d x) (a+b \sec (c+d x))^4}{15 b^2 d}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4082
Rule 4087
Rule 4167
Rule 4178
Rubi steps \begin{align*} \text {integral}& = \frac {C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (a C+b (6 A+5 C) \sec (c+d x)-2 a C \sec ^2(c+d x)\right ) \, dx}{6 b} \\ & = -\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}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (-3 a b C+\left (2 a^2 C+5 b^2 (6 A+5 C)\right ) \sec (c+d x)\right ) \, dx}{30 b^2} \\ & = \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}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^2 \left (3 b \left (30 A b^2-2 a^2 C+25 b^2 C\right )+3 a \left (30 A b^2+2 a^2 C+21 b^2 C\right ) \sec (c+d x)\right ) \, dx}{120 b^2} \\ & = \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}+\frac {\int \sec (c+d x) (a+b \sec (c+d x)) \left (-3 a b \left (2 a^2 C-3 b^2 (50 A+39 C)\right )+3 \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)\right ) \, dx}{360 b^2} \\ & = \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}+\frac {\int \sec (c+d x) \left (45 b^3 \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right )+12 a \left (2 a^4 C+24 b^4 (5 A+4 C)+a^2 b^2 (30 A+17 C)\right ) \sec (c+d x)\right ) \, dx}{720 b^2} \\ & = \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}+\frac {1}{16} \left (b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right )\right ) \int \sec (c+d x) \, dx+\frac {\left (a \left (2 a^4 C+24 b^4 (5 A+4 C)+a^2 b^2 (30 A+17 C)\right )\right ) \int \sec ^2(c+d x) \, dx}{60 b^2} \\ & = \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 {\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}-\frac {\left (a \left (2 a^4 C+24 b^4 (5 A+4 C)+a^2 b^2 (30 A+17 C)\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{60 b^2 d} \\ & = \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} \\ \end{align*}
Time = 5.50 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.61 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \sec (c+d x)+10 b \left (6 A b^2+18 a^2 C+5 b^2 C\right ) \sec ^3(c+d x)+40 b^3 C \sec ^5(c+d x)+16 a \left (15 \left (a^2+3 b^2\right ) (A+C)+5 \left (3 A b^2+a^2 C+6 b^2 C\right ) \tan ^2(c+d x)+9 b^2 C \tan ^4(c+d x)\right )\right )}{240 d} \]
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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}\) | \(847\) |
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Time = 0.29 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.86 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (5 \, {\left (3 \, A + 2 \, C\right )} a^{3} + 6 \, {\left (5 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{5} + 144 \, C a b^{2} \cos \left (d x + c\right ) + 15 \, {\left (6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} + 40 \, C b^{3} + 16 \, {\left (5 \, C a^{3} + 3 \, {\left (5 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (18 \, C a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
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\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3} \sec ^{2}{\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.26 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a b^{2} + 96 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a b^{2} - 5 \, C b^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, C a^{2} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, A b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, A a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, A a^{3} \tan \left (d x + c\right )}{480 \, d} \]
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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 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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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} \]
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