Integrand size = 41, antiderivative size = 274 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (26 A+23 B+21 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (133 A+119 B+108 C) \tan (c+d x)}{35 d}+\frac {a^3 (26 A+23 B+21 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^3 (154 A+147 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac {(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {a^3 (133 A+119 B+108 C) \tan ^3(c+d x)}{105 d} \]
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Time = 0.74 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4173, 4103, 4082, 3872, 3853, 3855, 3852} \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (26 A+23 B+21 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (133 A+119 B+108 C) \tan ^3(c+d x)}{105 d}+\frac {a^3 (133 A+119 B+108 C) \tan (c+d x)}{35 d}+\frac {a^3 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{280 d}+\frac {(3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac {a^3 (26 A+23 B+21 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{42 a d}+\frac {C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d} \]
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Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rule 4082
Rule 4103
Rule 4173
Rubi steps \begin{align*} \text {integral}& = \frac {C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac {\int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (a (7 A+3 C)+a (7 B+3 C) \sec (c+d x)) \, dx}{7 a} \\ & = \frac {C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac {(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac {\int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \left (3 a^2 (14 A+7 B+9 C)+14 a^2 (3 A+4 B+3 C) \sec (c+d x)\right ) \, dx}{42 a} \\ & = \frac {C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac {(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {\int \sec ^3(c+d x) (a+a \sec (c+d x)) \left (3 a^3 (112 A+91 B+87 C)+3 a^3 (154 A+147 B+129 C) \sec (c+d x)\right ) \, dx}{210 a} \\ & = \frac {a^3 (154 A+147 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac {(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {\int \sec ^3(c+d x) \left (105 a^4 (26 A+23 B+21 C)+24 a^4 (133 A+119 B+108 C) \sec (c+d x)\right ) \, dx}{840 a} \\ & = \frac {a^3 (154 A+147 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac {(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {1}{8} \left (a^3 (26 A+23 B+21 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{35} \left (a^3 (133 A+119 B+108 C)\right ) \int \sec ^4(c+d x) \, dx \\ & = \frac {a^3 (26 A+23 B+21 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^3 (154 A+147 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac {(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \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+21 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (a^3 (133 A+119 B+108 C)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 d} \\ & = \frac {a^3 (26 A+23 B+21 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (133 A+119 B+108 C) \tan (c+d x)}{35 d}+\frac {a^3 (26 A+23 B+21 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^3 (154 A+147 B+129 C) \sec ^3(c+d x) \tan (c+d x)}{280 d}+\frac {C \sec ^3(c+d x) (a+a \sec (c+d x))^3 \tan (c+d x)}{7 d}+\frac {(7 B+3 C) \sec ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \sec ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{15 d}+\frac {a^3 (133 A+119 B+108 C) \tan ^3(c+d x)}{105 d} \\ \end{align*}
Time = 5.98 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.55 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 \left (105 (26 A+23 B+21 C) \text {arctanh}(\sin (c+d x))+\sec (c+d x) \left (105 (26 A+23 B+21 C)+16 (133 A+119 B+108 C) (2+\cos (2 (c+d x))) \sec (c+d x)+70 (18 A+23 B+21 C) \sec ^2(c+d x)+48 (7 A+21 B+27 C) \sec ^3(c+d x)+280 (B+3 C) \sec ^4(c+d x)+240 C \sec ^5(c+d x)\right ) \tan (c+d x)\right )}{1680 d} \]
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Time = 1.03 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.02
method | result | size |
norman | \(\frac {-\frac {283 a^{3} \left (26 A +23 B +21 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}+\frac {5 a^{3} \left (26 A +23 B +21 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{6 d}-\frac {a^{3} \left (26 A +23 B +21 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}-\frac {a^{3} \left (102 A +105 B +107 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {16 a^{3} \left (203 A +189 B +163 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}+\frac {a^{3} \left (286 A +237 B +183 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}-\frac {a^{3} \left (10178 A +8979 B +9033 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{7}}-\frac {a^{3} \left (26 A +23 B +21 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {a^{3} \left (26 A +23 B +21 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(280\) |
parts | \(-\frac {\left (3 a^{3} A +B \,a^{3}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,a^{3}+3 a^{3} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (a^{3} A +3 B \,a^{3}+3 a^{3} C \right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 a^{3} A +3 B \,a^{3}+a^{3} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {a^{3} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {a^{3} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(310\) |
parallelrisch | \(-\frac {13 \left (\left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right ) \left (A +\frac {23 B}{26}+\frac {21 C}{26}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right ) \left (A +\frac {23 B}{26}+\frac {21 C}{26}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {345 C}{91}-\frac {226 A}{91}-\frac {841 B}{273}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {1744 B}{455}-\frac {1728 C}{455}-\frac {1648 A}{455}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {20 C}{13}-\frac {152 A}{91}-\frac {460 B}{273}\right ) \sin \left (4 d x +4 c \right )+\left (-\frac {272 B}{195}-\frac {304 A}{195}-\frac {576 C}{455}\right ) \sin \left (5 d x +5 c \right )+\left (-\frac {23 B}{91}-\frac {2 A}{7}-\frac {3 C}{13}\right ) \sin \left (6 d x +6 c \right )+\left (-\frac {576 C}{3185}-\frac {304 A}{1365}-\frac {272 B}{1365}\right ) \sin \left (7 d x +7 c \right )-\frac {16 \left (A +\frac {15 B}{13}+\frac {20 C}{13}\right ) \sin \left (d x +c \right )}{7}\right ) a^{3}}{8 d \left (\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )+\cos \left (5 d x +5 c \right )\right )}\) | \(311\) |
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 )-a^{3} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \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} C \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 (-\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 )-3 a^{3} C \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^{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 )+a^{3} 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 )}{d}\) | \(504\) |
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 )-a^{3} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \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} C \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 (-\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 )-3 a^{3} C \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^{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 )+a^{3} 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 )}{d}\) | \(504\) |
risch | \(-\frac {i a^{3} \left (-3456 C -4256 A -3808 B -52640 A \,{\mathrm e}^{8 i \left (d x +c \right )}-23730 A \,{\mathrm e}^{5 i \left (d x +c \right )}-15960 A \,{\mathrm e}^{3 i \left (d x +c \right )}-2730 A \,{\mathrm e}^{i \left (d x +c \right )}-76608 B \,{\mathrm e}^{4 i \left (d x +c \right )}-96320 A \,{\mathrm e}^{6 i \left (d x +c \right )}-94080 C \,{\mathrm e}^{6 i \left (d x +c \right )}-79296 A \,{\mathrm e}^{4 i \left (d x +c \right )}-72576 C \,{\mathrm e}^{4 i \left (d x +c \right )}-29792 A \,{\mathrm e}^{2 i \left (d x +c \right )}-24192 C \,{\mathrm e}^{2 i \left (d x +c \right )}-36225 C \,{\mathrm e}^{5 i \left (d x +c \right )}-14700 C \,{\mathrm e}^{3 i \left (d x +c \right )}-26656 B \,{\mathrm e}^{2 i \left (d x +c \right )}-2205 C \,{\mathrm e}^{i \left (d x +c \right )}+2205 C \,{\mathrm e}^{13 i \left (d x +c \right )}+2730 A \,{\mathrm e}^{13 i \left (d x +c \right )}+15960 A \,{\mathrm e}^{11 i \left (d x +c \right )}+14700 C \,{\mathrm e}^{11 i \left (d x +c \right )}-10080 A \,{\mathrm e}^{10 i \left (d x +c \right )}+23730 A \,{\mathrm e}^{9 i \left (d x +c \right )}+36225 C \,{\mathrm e}^{9 i \left (d x +c \right )}-26880 C \,{\mathrm e}^{8 i \left (d x +c \right )}-2415 B \,{\mathrm e}^{i \left (d x +c \right )}-29435 B \,{\mathrm e}^{5 i \left (d x +c \right )}-41440 B \,{\mathrm e}^{8 i \left (d x +c \right )}+16100 B \,{\mathrm e}^{11 i \left (d x +c \right )}+29435 B \,{\mathrm e}^{9 i \left (d x +c \right )}-3360 B \,{\mathrm e}^{10 i \left (d x +c \right )}-91840 B \,{\mathrm e}^{6 i \left (d x +c \right )}+2415 B \,{\mathrm e}^{13 i \left (d x +c \right )}-16100 B \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{840 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}-\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 {21 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{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}+\frac {21 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{16 d}\) | \(550\) |
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Time = 0.27 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.82 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (32 \, {\left (133 \, A + 119 \, B + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} + 105 \, {\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 16 \, {\left (133 \, A + 119 \, B + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 70 \, {\left (18 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 48 \, {\left (7 \, A + 21 \, B + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 280 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 240 \, C a^{3}\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \]
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\[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\int A \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 A \sec ^{5}{\left (c + d x \right )}\, dx + \int A \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{5}{\left (c + d x \right )}\, dx + \int 3 B \sec ^{6}{\left (c + d x \right )}\, dx + \int B \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{5}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{6}{\left (c + d x \right )}\, dx + \int 3 C \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{8}{\left (c + d x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (258) = 516\).
Time = 0.25 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.37 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {224 \, {\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} + 3360 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 672 \, {\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} + 1120 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} + 96 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} C a^{3} + 672 \, {\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^{3} - 35 \, 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 )} - 105 \, C 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 )} - 630 \, 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 )} - 630 \, 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 )} - 210 \, C 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 )} - 840 \, 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 )}}{3360 \, d} \]
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Time = 0.41 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.62 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (26 \, A a^{3} + 23 \, B a^{3} + 21 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (26 \, A a^{3} + 23 \, B a^{3} + 21 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2730 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2415 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2205 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 18200 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 16100 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 14700 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 51506 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 45563 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 41601 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 77952 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72576 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 62592 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 71246 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 62853 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 63231 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40040 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 33180 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 25620 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10710 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11025 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11235 \, C 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 )}^{7}}}{1680 \, d} \]
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Time = 20.61 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.39 \[ \int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3\,\mathrm {atanh}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (26\,A+23\,B+21\,C\right )}{4\,\left (\frac {13\,A\,a^3}{2}+\frac {23\,B\,a^3}{4}+\frac {21\,C\,a^3}{4}\right )}\right )\,\left (26\,A+23\,B+21\,C\right )}{8\,d}-\frac {\left (\frac {13\,A\,a^3}{4}+\frac {23\,B\,a^3}{8}+\frac {21\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (-\frac {65\,A\,a^3}{3}-\frac {115\,B\,a^3}{6}-\frac {35\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3679\,A\,a^3}{60}+\frac {6509\,B\,a^3}{120}+\frac {1981\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {464\,A\,a^3}{5}-\frac {432\,B\,a^3}{5}-\frac {2608\,C\,a^3}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {5089\,A\,a^3}{60}+\frac {2993\,B\,a^3}{40}+\frac {3011\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {143\,A\,a^3}{3}-\frac {79\,B\,a^3}{2}-\frac {61\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,A\,a^3}{4}+\frac {105\,B\,a^3}{8}+\frac {107\,C\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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