Integrand size = 31, antiderivative size = 210 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 (26 A+23 B) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (19 A+17 B) \tan (c+d x)}{5 d}+\frac {a^3 (26 A+23 B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^3 (22 A+21 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {(3 A+4 B) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {a^3 (19 A+17 B) \tan ^3(c+d x)}{15 d} \]
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Time = 0.47 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4103, 4082, 3872, 3853, 3855, 3852} \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 (26 A+23 B) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (19 A+17 B) \tan ^3(c+d x)}{15 d}+\frac {a^3 (19 A+17 B) \tan (c+d x)}{5 d}+\frac {a^3 (22 A+21 B) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac {(3 A+4 B) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac {a^3 (26 A+23 B) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {a B \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^2}{6 d} \]
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Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 4082
Rule 4103
Rubi steps \begin{align*} \text {integral}& = \frac {a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {1}{6} \int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (3 a (2 A+B)+2 a (3 A+4 B) \sec (c+d x)) \, dx \\ & = \frac {a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {(3 A+4 B) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {1}{30} \int \sec ^3(c+d x) (a+a \sec (c+d x)) \left (3 a^2 (16 A+13 B)+3 a^2 (22 A+21 B) \sec (c+d x)\right ) \, dx \\ & = \frac {a^3 (22 A+21 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {(3 A+4 B) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {1}{120} \int \sec ^3(c+d x) \left (15 a^3 (26 A+23 B)+24 a^3 (19 A+17 B) \sec (c+d x)\right ) \, dx \\ & = \frac {a^3 (22 A+21 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {(3 A+4 B) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {1}{5} \left (a^3 (19 A+17 B)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{8} \left (a^3 (26 A+23 B)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {a^3 (26 A+23 B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^3 (22 A+21 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {(3 A+4 B) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {1}{16} \left (a^3 (26 A+23 B)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^3 (19 A+17 B)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d} \\ & = \frac {a^3 (26 A+23 B) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (19 A+17 B) \tan (c+d x)}{5 d}+\frac {a^3 (26 A+23 B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^3 (22 A+21 B) \sec ^3(c+d x) \tan (c+d x)}{40 d}+\frac {a B \sec ^3(c+d x) (a+a \sec (c+d x))^2 \tan (c+d x)}{6 d}+\frac {(3 A+4 B) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {a^3 (19 A+17 B) \tan ^3(c+d x)}{15 d} \\ \end{align*}
Time = 3.65 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.55 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 \left (15 (26 A+23 B) \text {arctanh}(\sin (c+d x))+\sec (c+d x) \left (390 A+345 B+16 (19 A+17 B) (2+\cos (2 (c+d x))) \sec (c+d x)+10 (18 A+23 B) \sec ^2(c+d x)+48 (A+3 B) \sec ^3(c+d x)+40 B \sec ^4(c+d x)\right ) \tan (c+d x)\right )}{240 d} \]
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Time = 5.32 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.08
method | result | size |
norman | \(\frac {-\frac {33 a^{3} \left (26 A +23 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {17 a^{3} \left (26 A +23 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {a^{3} \left (26 A +23 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {3 a^{3} \left (34 A +35 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a^{3} \left (838 A +633 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {a^{3} \left (998 A +969 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {a^{3} \left (26 A +23 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {a^{3} \left (26 A +23 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(227\) |
parts | \(-\frac {\left (a^{3} A +3 B \,a^{3}\right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}-\frac {\left (3 a^{3} A +B \,a^{3}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 a^{3} A +3 B \,a^{3}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(244\) |
parallelrisch | \(\frac {25 \left (-\frac {39 \left (A +\frac {23 B}{26}\right ) \left (\frac {2}{3}+\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{20}+\frac {39 \left (A +\frac {23 B}{26}\right ) \left (\frac {2}{3}+\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{20}+\frac {4 \left (\frac {13 A}{5}+3 B \right ) \sin \left (2 d x +2 c \right )}{5}+\frac {\left (63 A +\frac {391 B}{6}\right ) \sin \left (3 d x +3 c \right )}{50}+\frac {8 \left (19 A +17 B \right ) \sin \left (4 d x +4 c \right )}{125}+\frac {\left (13 A +\frac {23 B}{2}\right ) \sin \left (5 d x +5 c \right )}{50}+\frac {4 \left (19 A +17 B \right ) \sin \left (6 d x +6 c \right )}{375}+\sin \left (d x +c \right ) \left (A +\frac {3 B}{2}\right )\right ) a^{3}}{2 d \left (10+\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )\right )}\) | \(251\) |
derivativedivides | \(\frac {-a^{3} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+3 a^{3} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 B \,a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-3 a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(317\) |
default | \(\frac {-a^{3} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+3 a^{3} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 B \,a^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-3 a^{3} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B \,a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(317\) |
risch | \(-\frac {i a^{3} \left (390 A \,{\mathrm e}^{11 i \left (d x +c \right )}+345 B \,{\mathrm e}^{11 i \left (d x +c \right )}+1890 A \,{\mathrm e}^{9 i \left (d x +c \right )}+1955 B \,{\mathrm e}^{9 i \left (d x +c \right )}-1440 A \,{\mathrm e}^{8 i \left (d x +c \right )}-480 B \,{\mathrm e}^{8 i \left (d x +c \right )}+1500 A \,{\mathrm e}^{7 i \left (d x +c \right )}+2250 B \,{\mathrm e}^{7 i \left (d x +c \right )}-6080 A \,{\mathrm e}^{6 i \left (d x +c \right )}-5440 B \,{\mathrm e}^{6 i \left (d x +c \right )}-1500 A \,{\mathrm e}^{5 i \left (d x +c \right )}-2250 B \,{\mathrm e}^{5 i \left (d x +c \right )}-7680 A \,{\mathrm e}^{4 i \left (d x +c \right )}-7680 B \,{\mathrm e}^{4 i \left (d x +c \right )}-1890 A \,{\mathrm e}^{3 i \left (d x +c \right )}-1955 B \,{\mathrm e}^{3 i \left (d x +c \right )}-3648 A \,{\mathrm e}^{2 i \left (d x +c \right )}-3264 B \,{\mathrm e}^{2 i \left (d x +c \right )}-390 \,{\mathrm e}^{i \left (d x +c \right )} A -345 B \,{\mathrm e}^{i \left (d x +c \right )}-608 A -544 B \right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{8 d}-\frac {23 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{16 d}+\frac {13 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{8 d}+\frac {23 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{16 d}\) | \(359\) |
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Time = 0.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (26 \, A + 23 \, B\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (26 \, A + 23 \, B\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (32 \, {\left (19 \, A + 17 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} + 15 \, {\left (26 \, A + 23 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} + 16 \, {\left (19 \, A + 17 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 10 \, {\left (18 \, A + 23 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 48 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right ) + 40 \, B a^{3}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
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\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=a^{3} \left (\int A \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{5}{\left (c + d x \right )}\, dx + \int A \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{5}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{7}{\left (c + d x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (196) = 392\).
Time = 0.22 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.93 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {32 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{3} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 96 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{3} + 160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} - 5 \, B a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, A a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, B a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, A a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.33 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (26 \, A a^{3} + 23 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (26 \, A a^{3} + 23 \, B a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (390 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 345 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 2210 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1955 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5148 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4554 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5988 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5814 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4190 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3165 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1530 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1575 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]
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Time = 16.42 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.25 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {\left (-\frac {13\,A\,a^3}{4}-\frac {23\,B\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {221\,A\,a^3}{12}+\frac {391\,B\,a^3}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {429\,A\,a^3}{10}-\frac {759\,B\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {499\,A\,a^3}{10}+\frac {969\,B\,a^3}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {419\,A\,a^3}{12}-\frac {211\,B\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,A\,a^3}{4}+\frac {105\,B\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (26\,A+23\,B\right )}{8\,d} \]
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