Integrand size = 38, antiderivative size = 52 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} (a B+2 b C) x+\frac {(b B+a C) \sin (c+d x)}{d}+\frac {a B \cos (c+d x) \sin (c+d x)}{2 d} \]
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Time = 0.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {4157, 4081, 3872, 2717, 8} \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(a C+b B) \sin (c+d x)}{d}+\frac {1}{2} x (a B+2 b C)+\frac {a B \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rule 8
Rule 2717
Rule 3872
Rule 4081
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \cos ^2(c+d x) (a+b \sec (c+d x)) (B+C \sec (c+d x)) \, dx \\ & = \frac {a B \cos (c+d x) \sin (c+d x)}{2 d}-\frac {1}{2} \int \cos (c+d x) (-2 (b B+a C)-(a B+2 b C) \sec (c+d x)) \, dx \\ & = \frac {a B \cos (c+d x) \sin (c+d x)}{2 d}-(-b B-a C) \int \cos (c+d x) \, dx-\frac {1}{2} (-a B-2 b C) \int 1 \, dx \\ & = \frac {1}{2} (a B+2 b C) x+\frac {(b B+a C) \sin (c+d x)}{d}+\frac {a B \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.98 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a B c+2 a B d x+4 b C d x+4 (b B+a C) \sin (c+d x)+a B \sin (2 (c+d x))}{4 d} \]
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Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {a B \sin \left (2 d x +2 c \right )+\left (4 B b +4 C a \right ) \sin \left (d x +c \right )+2 \left (a B +2 C b \right ) x d}{4 d}\) | \(47\) |
risch | \(\frac {a B x}{2}+x C b +\frac {\sin \left (d x +c \right ) B b}{d}+\frac {\sin \left (d x +c \right ) C a}{d}+\frac {a B \sin \left (2 d x +2 c \right )}{4 d}\) | \(51\) |
derivativedivides | \(\frac {a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B b \sin \left (d x +c \right )+C a \sin \left (d x +c \right )+C b \left (d x +c \right )}{d}\) | \(57\) |
default | \(\frac {a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B b \sin \left (d x +c \right )+C a \sin \left (d x +c \right )+C b \left (d x +c \right )}{d}\) | \(57\) |
norman | \(\frac {\left (\frac {a B}{2}+C b \right ) x +\left (\frac {a B}{2}+C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {a B}{2}+C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {a B}{2}+C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-a B -2 C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-a B -2 C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {\left (a B +2 B b +2 C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {4 \left (B b +C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {\left (a B -2 B b -2 C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}-\frac {2 a B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}+\frac {2 a B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(262\) |
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.81 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (B a + 2 \, C b\right )} d x + {\left (B a \cos \left (d x + c\right ) + 2 \, C a + 2 \, B b\right )} \sin \left (d x + c\right )}{2 \, d} \]
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\[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.06 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a + 4 \, {\left (d x + c\right )} C b + 4 \, C a \sin \left (d x + c\right ) + 4 \, B b \sin \left (d x + c\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (48) = 96\).
Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.33 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (B a + 2 \, C b\right )} {\left (d x + c\right )} - \frac {2 \, {\left (B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
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Time = 17.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B\,a\,x}{2}+C\,b\,x+\frac {B\,b\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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