Integrand size = 41, antiderivative size = 381 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {\left (6 a^4 b B-104 a^2 b^3 B-32 b^5 B-2 a^5 C-24 a b^4 (5 A+4 C)-a^3 b^2 (30 A+17 C)\right ) \tan (c+d x)}{60 b^2 d}-\frac {\left (12 a^3 b B-142 a b^3 B-4 a^4 C-12 a^2 b^2 (5 A+3 C)-15 b^4 (6 A+5 C)\right ) \sec (c+d x) \tan (c+d x)}{240 b d}-\frac {\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac {\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac {(3 b B-a C) (a+b \sec (c+d x))^4 \tan (c+d x)}{15 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{6 b d} \]
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Time = 0.95 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {4177, 4167, 4087, 4082, 3872, 3855, 3852, 8} \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\tan (c+d x) \left (2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right ) (a+b \sec (c+d x))^3}{120 b^2 d}+\frac {\left (8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {\tan (c+d x) \left (-2 a^3 C+6 a^2 b B-3 a b^2 (10 A+7 C)-32 b^3 B\right ) (a+b \sec (c+d x))^2}{120 b^2 d}-\frac {\tan (c+d x) \sec (c+d x) \left (-4 a^4 C+12 a^3 b B-12 a^2 b^2 (5 A+3 C)-142 a b^3 B-15 b^4 (6 A+5 C)\right )}{240 b d}-\frac {\tan (c+d x) \left (-2 a^5 C+6 a^4 b B-a^3 b^2 (30 A+17 C)-104 a^2 b^3 B-24 a b^4 (5 A+4 C)-32 b^5 B\right )}{60 b^2 d}+\frac {(3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{15 b^2 d}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4082
Rule 4087
Rule 4167
Rule 4177
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 (3 b B-a C) \sec ^2(c+d x)\right ) \, dx}{6 b} \\ & = \frac {(3 b B-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 b (8 b B-a C)+\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) \sec (c+d x)\right ) \, dx}{30 b^2} \\ & = \frac {\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac {(3 b B-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+26 a b B-2 a^2 C+25 b^2 C\right )-3 \left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) \sec (c+d x)\right ) \, dx}{120 b^2} \\ & = -\frac {\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac {\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac {(3 b B-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 b \left (66 a^2 b B+64 b^3 B-2 a^3 C+3 a b^2 (50 A+39 C)\right )-3 \left (12 a^3 b B-142 a b^3 B-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 (12 a^3 b B-142 a b^3 B-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 {\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac {\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac {(3 b B-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^2 \left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right )-12 \left (6 a^4 b B-104 a^2 b^3 B-32 b^5 B-2 a^5 C-24 a b^4 (5 A+4 C)-a^3 b^2 (30 A+17 C)\right ) \sec (c+d x)\right ) \, dx}{720 b^2} \\ & = -\frac {\left (12 a^3 b B-142 a b^3 B-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 {\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac {\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac {(3 b B-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 (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (6 a^4 b B-104 a^2 b^3 B-32 b^5 B-2 a^5 C-24 a b^4 (5 A+4 C)-a^3 b^2 (30 A+17 C)\right ) \int \sec ^2(c+d x) \, dx}{60 b^2} \\ & = \frac {\left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {\left (12 a^3 b B-142 a b^3 B-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 {\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac {\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac {(3 b B-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 (6 a^4 b B-104 a^2 b^3 B-32 b^5 B-2 a^5 C-24 a b^4 (5 A+4 C)-a^3 b^2 (30 A+17 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{60 b^2 d} \\ & = \frac {\left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {\left (6 a^4 b B-104 a^2 b^3 B-32 b^5 B-2 a^5 C-24 a b^4 (5 A+4 C)-a^3 b^2 (30 A+17 C)\right ) \tan (c+d x)}{60 b^2 d}-\frac {\left (12 a^3 b B-142 a b^3 B-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 {\left (6 a^2 b B-32 b^3 B-2 a^3 C-3 a b^2 (10 A+7 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b^2 d}+\frac {\left (30 A b^2-6 a b B+2 a^2 C+25 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b^2 d}+\frac {(3 b B-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 = 11.19 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.86 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {C \sec ^2(c+d x) (a+b \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {1}{6} \left (\frac {3 (2 b B+a C) \sec ^2(c+d x) (a+b \sec (c+d x))^2 \tan (c+d x)}{5 d}+\frac {1}{5} \left (\frac {b \left (30 A b^2+42 a b B+6 a^2 C+25 b^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \left (\frac {\left (24 a^2 (15 a A+6 b B+8 a C)+48 \left (12 a^2 b B+4 b^3 B+a^3 C+3 a b^2 (5 A+4 C)\right )\right ) \tan (c+d x)}{3 d}+\frac {8 \left (12 a^2 b B+4 b^3 B+a^3 C+3 a b^2 (5 A+4 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{d}+15 \left (8 a^3 B+18 a b^2 B+6 a^2 b (4 A+3 C)+b^3 (6 A+5 C)\right ) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\sec (c+d x) \tan (c+d x)}{2 d}\right )\right )\right )\right ) \]
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Time = 1.72 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.74
method | result | size |
parts | \(\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (A \,b^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (3 a A \,b^{2}+3 B \,a^{2} b +a^{3} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {C \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {a^{3} A \tan \left (d x +c \right )}{d}\) | \(283\) |
derivativedivides | \(\frac {a^{3} A \tan \left (d x +c \right )+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 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 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 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 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 )+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 )-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}\) | \(444\) |
default | \(\frac {a^{3} A \tan \left (d x +c \right )+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 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 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 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 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 )+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 )-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}\) | \(444\) |
parallelrisch | \(\frac {-2160 \left (\frac {\cos \left (6 d x +6 c \right )}{6}+\cos \left (4 d x +4 c \right )+\frac {5 \cos \left (2 d x +2 c \right )}{2}+\frac {5}{3}\right ) \left (\left (\frac {A}{4}+\frac {5 C}{24}\right ) b^{3}+\frac {3 B a \,b^{2}}{4}+a^{2} \left (A +\frac {3 C}{4}\right ) b +\frac {B \,a^{3}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2160 \left (\frac {\cos \left (6 d x +6 c \right )}{6}+\cos \left (4 d x +4 c \right )+\frac {5 \cos \left (2 d x +2 c \right )}{2}+\frac {5}{3}\right ) \left (\left (\frac {A}{4}+\frac {5 C}{24}\right ) b^{3}+\frac {3 B a \,b^{2}}{4}+a^{2} \left (A +\frac {3 C}{4}\right ) b +\frac {B \,a^{3}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (1920 B \,b^{3}+4320 \left (A +\frac {4 C}{3}\right ) a \,b^{2}+4320 B \,a^{2} b +1200 a^{3} \left (A +\frac {6 C}{5}\right )\right ) \sin \left (2 d x +2 c \right )+\left (\left (1020 A +850 C \right ) b^{3}+3060 B a \,b^{2}+2160 a^{2} \left (A +\frac {17 C}{12}\right ) b +720 B \,a^{3}\right ) \sin \left (3 d x +3 c \right )+\left (768 B \,b^{3}+2880 a \left (A +\frac {4 C}{5}\right ) b^{2}+2880 B \,a^{2} b +960 a^{3} \left (A +C \right )\right ) \sin \left (4 d x +4 c \right )+\left (\left (180 A +150 C \right ) b^{3}+540 B a \,b^{2}+720 a^{2} \left (A +\frac {3 C}{4}\right ) b +240 B \,a^{3}\right ) \sin \left (5 d x +5 c \right )+\left (128 B \,b^{3}+480 a \left (A +\frac {4 C}{5}\right ) b^{2}+480 B \,a^{2} b +240 a^{3} \left (A +\frac {2 C}{3}\right )\right ) \sin \left (6 d x +6 c \right )+1440 \left (\left (\frac {7 A}{12}+\frac {11 C}{8}\right ) b^{3}+\frac {7 B a \,b^{2}}{4}+a^{2} \left (A +\frac {7 C}{4}\right ) b +\frac {B \,a^{3}}{3}\right ) \sin \left (d x +c \right )}{240 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(474\) |
norman | \(\frac {-\frac {\left (16 a^{3} A -24 A \,a^{2} b +48 a A \,b^{2}-10 A \,b^{3}-8 B \,a^{3}+48 B \,a^{2} b -30 B a \,b^{2}+16 B \,b^{3}+16 a^{3} C -30 a^{2} b C +48 C a \,b^{2}-11 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (16 a^{3} A +24 A \,a^{2} b +48 a A \,b^{2}+10 A \,b^{3}+8 B \,a^{3}+48 B \,a^{2} b +30 B a \,b^{2}+16 B \,b^{3}+16 a^{3} C +30 a^{2} b C +48 C a \,b^{2}+11 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (240 a^{3} A -216 A \,a^{2} b +528 a A \,b^{2}-42 A \,b^{3}-72 B \,a^{3}+528 B \,a^{2} b -126 B a \,b^{2}+112 B \,b^{3}+176 a^{3} C -126 a^{2} b C +336 C a \,b^{2}+5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {\left (240 a^{3} A +216 A \,a^{2} b +528 a A \,b^{2}+42 A \,b^{3}+72 B \,a^{3}+528 B \,a^{2} b +126 B a \,b^{2}+112 B \,b^{3}+176 a^{3} C +126 a^{2} b C +336 C a \,b^{2}-5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}-\frac {\left (400 a^{3} A -120 A \,a^{2} b +720 a A \,b^{2}-10 A \,b^{3}-40 B \,a^{3}+720 B \,a^{2} b -30 B a \,b^{2}+208 B \,b^{3}+240 a^{3} C -30 a^{2} b C +624 C a \,b^{2}-75 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (400 a^{3} A +120 A \,a^{2} b +720 a A \,b^{2}+10 A \,b^{3}+40 B \,a^{3}+720 B \,a^{2} b +30 B a \,b^{2}+208 B \,b^{3}+240 a^{3} C +30 a^{2} b C +624 C a \,b^{2}+75 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{6}}-\frac {\left (24 A \,a^{2} b +6 A \,b^{3}+8 B \,a^{3}+18 B a \,b^{2}+18 a^{2} b C +5 C \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {\left (24 A \,a^{2} b +6 A \,b^{3}+8 B \,a^{3}+18 B a \,b^{2}+18 a^{2} b C +5 C \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(701\) |
risch | \(\text {Expression too large to display}\) | \(1245\) |
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Time = 0.30 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.90 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (8 \, B a^{3} + 6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (8 \, B a^{3} + 6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (5 \, {\left (3 \, A + 2 \, C\right )} a^{3} + 30 \, B a^{2} b + 6 \, {\left (5 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )^{5} + 15 \, {\left (8 \, B a^{3} + 6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + 18 \, B a b^{2} + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} + 40 \, C b^{3} + 16 \, {\left (5 \, C a^{3} + 15 \, B a^{2} b + 3 \, {\left (5 \, A + 4 \, C\right )} a b^{2} + 4 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (18 \, C a^{2} b + 18 \, B a b^{2} + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
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\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.48 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+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 + 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} + 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 )} - 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 )} - 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 )} - 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. 1370 vs. \(2 (365) = 730\).
Time = 0.40 (sec) , antiderivative size = 1370, normalized size of antiderivative = 3.60 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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Time = 18.68 (sec) , antiderivative size = 769, normalized size of antiderivative = 2.02 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left (A+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 {3\,A\,b^3}{8}+\frac {B\,a^3}{2}+\frac {5\,C\,b^3}{16}+\frac {3\,A\,a^2\,b}{2}+\frac {9\,B\,a\,b^2}{8}+\frac {9\,C\,a^2\,b}{8}\right )}{\frac {3\,A\,b^3}{2}+2\,B\,a^3+\frac {5\,C\,b^3}{4}+6\,A\,a^2\,b+\frac {9\,B\,a\,b^2}{2}+\frac {9\,C\,a^2\,b}{2}}\right )\,\left (\frac {3\,A\,b^3}{4}+B\,a^3+\frac {5\,C\,b^3}{8}+3\,A\,a^2\,b+\frac {9\,B\,a\,b^2}{4}+\frac {9\,C\,a^2\,b}{4}\right )}{d}+\frac {\left (\frac {5\,A\,b^3}{4}-2\,A\,a^3+B\,a^3-2\,B\,b^3-2\,C\,a^3+\frac {11\,C\,b^3}{8}-6\,A\,a\,b^2+3\,A\,a^2\,b+\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 (10\,A\,a^3-\frac {7\,A\,b^3}{4}-3\,B\,a^3+\frac {14\,B\,b^3}{3}+\frac {22\,C\,a^3}{3}+\frac {5\,C\,b^3}{24}+22\,A\,a\,b^2-9\,A\,a^2\,b-\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 (\frac {A\,b^3}{2}-20\,A\,a^3+2\,B\,a^3-\frac {52\,B\,b^3}{5}-12\,C\,a^3+\frac {15\,C\,b^3}{4}-36\,A\,a\,b^2+6\,A\,a^2\,b+\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 (20\,A\,a^3+\frac {A\,b^3}{2}+2\,B\,a^3+\frac {52\,B\,b^3}{5}+12\,C\,a^3+\frac {15\,C\,b^3}{4}+36\,A\,a\,b^2+6\,A\,a^2\,b+\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 {7\,A\,b^3}{4}-3\,B\,a^3-\frac {14\,B\,b^3}{3}-\frac {22\,C\,a^3}{3}-10\,A\,a^3-22\,A\,a\,b^2-9\,A\,a^2\,b-\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 (2\,A\,a^3+\frac {5\,A\,b^3}{4}+B\,a^3+2\,B\,b^3+2\,C\,a^3+\frac {11\,C\,b^3}{8}+6\,A\,a\,b^2+3\,A\,a^2\,b+\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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