Integrand size = 38, antiderivative size = 84 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} (b B+a C) x+\frac {(2 a B+3 b C) \sin (c+d x)}{3 d}+\frac {(b B+a C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d} \]
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Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4157, 4081, 3872, 2715, 8, 2717} \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(2 a B+3 b C) \sin (c+d x)}{3 d}+\frac {(a C+b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} x (a C+b B)+\frac {a B \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
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Rule 8
Rule 2715
Rule 2717
Rule 3872
Rule 4081
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \cos ^3(c+d x) (a+b \sec (c+d x)) (B+C \sec (c+d x)) \, dx \\ & = \frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac {1}{3} \int \cos ^2(c+d x) (-3 (b B+a C)-(2 a B+3 b C) \sec (c+d x)) \, dx \\ & = \frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d}-(-b B-a C) \int \cos ^2(c+d x) \, dx-\frac {1}{3} (-2 a B-3 b C) \int \cos (c+d x) \, dx \\ & = \frac {(2 a B+3 b C) \sin (c+d x)}{3 d}+\frac {(b B+a C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac {1}{2} (-b B-a C) \int 1 \, dx \\ & = \frac {1}{2} (b B+a C) x+\frac {(2 a B+3 b C) \sin (c+d x)}{3 d}+\frac {(b B+a C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {6 b B c+6 a c C+6 b B d x+6 a C d x+3 (3 a B+4 b C) \sin (c+d x)+3 (b B+a C) \sin (2 (c+d x))+a B \sin (3 (c+d x))}{12 d} \]
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Time = 0.38 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.77
method | result | size |
parallelrisch | \(\frac {3 \left (B b +C a \right ) \sin \left (2 d x +2 c \right )+a B \sin \left (3 d x +3 c \right )+3 \left (3 a B +4 C b \right ) \sin \left (d x +c \right )+6 \left (B b +C a \right ) x d}{12 d}\) | \(65\) |
derivativedivides | \(\frac {\frac {a B \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B b \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b}{d}\) | \(85\) |
default | \(\frac {\frac {a B \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B b \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b}{d}\) | \(85\) |
risch | \(\frac {B b x}{2}+\frac {a x C}{2}+\frac {3 a B \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (d x +c \right ) C b}{d}+\frac {a B \sin \left (3 d x +3 c \right )}{12 d}+\frac {\sin \left (2 d x +2 c \right ) B b}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C a}{4 d}\) | \(85\) |
norman | \(\frac {\left (\frac {B b}{2}+\frac {C a}{2}\right ) x +\left (-\frac {B b}{2}-\frac {C a}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {B b}{2}-\frac {C a}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {B b}{2}+\frac {C a}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-2 B b -2 C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (B b +C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (B b +C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\frac {\left (2 a B -B b -C a +2 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}+\frac {\left (2 a B +B b +C a +2 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {\left (2 a B -3 B b -3 C a -6 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}-\frac {2 \left (2 a B -3 B b -3 C a +6 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {\left (2 a B +3 B b +3 C a -6 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {2 \left (2 a B +3 B b +3 C a +6 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(364\) |
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Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (C a + B b\right )} d x + {\left (2 \, B a \cos \left (d x + c\right )^{2} + 4 \, B a + 6 \, C b + 3 \, {\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d} \]
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\[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b - 12 \, C b \sin \left (d x + c\right )}{12 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (76) = 152\).
Time = 0.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.14 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (C a + B b\right )} {\left (d x + c\right )} + \frac {2 \, {\left (6 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
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Time = 17.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B\,b\,x}{2}+\frac {C\,a\,x}{2}+\frac {3\,B\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {C\,b\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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