Integrand size = 33, antiderivative size = 136 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} a^2 (7 A+12 C) x+\frac {a^2 (7 A+12 C) \sin (c+d x)}{6 d}+\frac {a^2 (7 A+12 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{4 d} \]
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Time = 0.36 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4172, 4098, 3873, 2717, 4130, 8} \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 (7 A+12 C) \sin (c+d x)}{6 d}+\frac {a^2 (7 A+12 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} a^2 x (7 A+12 C)+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^2}{4 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{6 d} \]
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Rule 8
Rule 2717
Rule 3873
Rule 4098
Rule 4130
Rule 4172
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x))^2 (2 a A+a (A+4 C) \sec (c+d x)) \, dx}{4 a} \\ & = \frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {1}{12} (7 A+12 C) \int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \, dx \\ & = \frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {1}{12} (7 A+12 C) \int \cos ^2(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx+\frac {1}{6} \left (a^2 (7 A+12 C)\right ) \int \cos (c+d x) \, dx \\ & = \frac {a^2 (7 A+12 C) \sin (c+d x)}{6 d}+\frac {a^2 (7 A+12 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {1}{8} \left (a^2 (7 A+12 C)\right ) \int 1 \, dx \\ & = \frac {1}{8} a^2 (7 A+12 C) x+\frac {a^2 (7 A+12 C) \sin (c+d x)}{6 d}+\frac {a^2 (7 A+12 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{4 d} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.54 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 (84 A d x+144 C d x+48 (3 A+4 C) \sin (c+d x)+24 (2 A+C) \sin (2 (c+d x))+16 A \sin (3 (c+d x))+3 A \sin (4 (c+d x)))}{96 d} \]
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Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.54
method | result | size |
parallelrisch | \(\frac {a^{2} \left (\left (16 A +8 C \right ) \sin \left (2 d x +2 c \right )+\frac {16 A \sin \left (3 d x +3 c \right )}{3}+A \sin \left (4 d x +4 c \right )+\left (48 A +64 C \right ) \sin \left (d x +c \right )+28 x d \left (A +\frac {12 C}{7}\right )\right )}{32 d}\) | \(73\) |
risch | \(\frac {7 a^{2} A x}{8}+\frac {3 a^{2} x C}{2}+\frac {3 \sin \left (d x +c \right ) a^{2} A}{2 d}+\frac {2 \sin \left (d x +c \right ) C \,a^{2}}{d}+\frac {a^{2} A \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{2} A \sin \left (3 d x +3 c \right )}{6 d}+\frac {a^{2} A \sin \left (2 d x +2 c \right )}{2 d}+\frac {\sin \left (2 d x +2 c \right ) C \,a^{2}}{4 d}\) | \(118\) |
derivativedivides | \(\frac {a^{2} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+C \,a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 a^{2} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 C \,a^{2} \sin \left (d x +c \right )+a^{2} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{2} \left (d x +c \right )}{d}\) | \(142\) |
default | \(\frac {a^{2} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+C \,a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 a^{2} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 C \,a^{2} \sin \left (d x +c \right )+a^{2} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{2} \left (d x +c \right )}{d}\) | \(142\) |
norman | \(\frac {\frac {a^{2} \left (3 A -4 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {a^{2} \left (7 A +12 C \right ) x}{8}-\frac {5 a^{2} \left (5 A +4 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{2} \left (7 A +12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{6 d}+\frac {a^{2} \left (7 A +12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{4 d}-\frac {a^{2} \left (7 A +12 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {3 a^{2} \left (7 A +12 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8}+\frac {3 a^{2} \left (7 A +12 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8}-\frac {3 a^{2} \left (7 A +12 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}-\frac {3 a^{2} \left (7 A +12 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}+\frac {a^{2} \left (7 A +12 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{8}+\frac {a^{2} \left (7 A +12 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{8}-\frac {a^{2} \left (53 A -156 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}+\frac {a^{2} \left (71 A +12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}-\frac {a^{2} \left (85 A +132 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3}}\) | \(393\) |
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Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.63 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (7 \, A + 12 \, C\right )} a^{2} d x + {\left (6 \, A a^{2} \cos \left (d x + c\right )^{3} + 16 \, A a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (7 \, A + 4 \, C\right )} a^{2} \cos \left (d x + c\right ) + 16 \, {\left (2 \, A + 3 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{24 \, d} \]
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Timed out. \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.97 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 96 \, {\left (d x + c\right )} C a^{2} - 192 \, C a^{2} \sin \left (d x + c\right )}{96 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.29 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (7 \, A a^{2} + 12 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (21 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 36 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 77 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 132 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 83 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 156 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 75 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
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Time = 15.14 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.86 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {7\,A\,a^2\,x}{8}+\frac {3\,C\,a^2\,x}{2}+\frac {3\,A\,a^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {2\,C\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {A\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {A\,a^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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