Integrand size = 40, antiderivative size = 102 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} a^2 (2 B+3 C) x+\frac {2 a^2 (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a^2 (2 B+3 C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {B \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d} \]
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Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4157, 4098, 3873, 2717, 4130, 8} \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a^2 (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a^2 (2 B+3 C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {1}{2} a^2 x (2 B+3 C)+\frac {B \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d} \]
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Rule 8
Rule 2717
Rule 3873
Rule 4098
Rule 4130
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \cos ^3(c+d x) (a+a \sec (c+d x))^2 (B+C \sec (c+d x)) \, dx \\ & = \frac {B \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} (2 B+3 C) \int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \, dx \\ & = \frac {B \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} (2 B+3 C) \int \cos ^2(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx+\frac {1}{3} \left (2 a^2 (2 B+3 C)\right ) \int \cos (c+d x) \, dx \\ & = \frac {2 a^2 (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a^2 (2 B+3 C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {B \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{2} \left (a^2 (2 B+3 C)\right ) \int 1 \, dx \\ & = \frac {1}{2} a^2 (2 B+3 C) x+\frac {2 a^2 (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a^2 (2 B+3 C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {B \cos ^2(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{3 d} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.82 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 \sin (c+d x) \left (11 B+12 C+3 (2 B+C) \cos (c+d x)+B \cos (2 (c+d x))+\frac {6 (2 B+3 C) \arcsin \left (\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )}{\sqrt {\sin ^2(c+d x)}}\right )}{6 d} \]
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.59
method | result | size |
parallelrisch | \(\frac {a^{2} \left (\left (\frac {B}{2}+\frac {C}{4}\right ) \sin \left (2 d x +2 c \right )+\frac {B \sin \left (3 d x +3 c \right )}{12}+\left (\frac {7 B}{4}+2 C \right ) \sin \left (d x +c \right )+d x \left (B +\frac {3 C}{2}\right )\right )}{d}\) | \(60\) |
risch | \(a^{2} B x +\frac {3 a^{2} x C}{2}+\frac {7 a^{2} B \sin \left (d x +c \right )}{4 d}+\frac {2 \sin \left (d x +c \right ) C \,a^{2}}{d}+\frac {B \,a^{2} \sin \left (3 d x +3 c \right )}{12 d}+\frac {B \,a^{2} \sin \left (2 d x +2 c \right )}{2 d}+\frac {\sin \left (2 d x +2 c \right ) C \,a^{2}}{4 d}\) | \(99\) |
derivativedivides | \(\frac {B \,a^{2} \sin \left (d x +c \right )+C \,a^{2} \left (d x +c \right )+2 B \,a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 C \,a^{2} \sin \left (d x +c \right )+\frac {B \,a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C \,a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(116\) |
default | \(\frac {B \,a^{2} \sin \left (d x +c \right )+C \,a^{2} \left (d x +c \right )+2 B \,a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 C \,a^{2} \sin \left (d x +c \right )+\frac {B \,a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C \,a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(116\) |
norman | \(\frac {\frac {a^{2} \left (2 B +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}-\frac {a^{2} \left (2 B +3 C \right ) x}{2}-\frac {4 a^{2} \left (2 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}+\frac {13 a^{2} \left (2 B +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}+\frac {2 a^{2} \left (2 B +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}-\frac {a^{2} \left (2 B +3 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+\frac {3 a^{2} \left (2 B +3 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+\frac {3 a^{2} \left (2 B +3 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2}-\frac {3 a^{2} \left (2 B +3 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}-\frac {3 a^{2} \left (2 B +3 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}+\frac {a^{2} \left (2 B +3 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{2}+\frac {a^{2} \left (2 B +3 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{2}-\frac {a^{2} \left (6 B +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{2} \left (10 B +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {a^{2} \left (14 B +33 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3}}\) | \(391\) |
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Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (2 \, B + 3 \, C\right )} a^{2} d x + {\left (2 \, B a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, B + C\right )} a^{2} \cos \left (d x + c\right ) + 2 \, {\left (5 \, B + 6 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{6 \, d} \]
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Timed out. \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.08 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} - 6 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 12 \, {\left (d x + c\right )} C a^{2} - 12 \, B a^{2} \sin \left (d x + c\right ) - 24 \, C a^{2} \sin \left (d x + c\right )}{12 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.39 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (2 \, B a^{2} + 3 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
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Time = 15.60 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.96 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=B\,a^2\,x+\frac {3\,C\,a^2\,x}{2}+\frac {7\,B\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {2\,C\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {B\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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