Integrand size = 38, antiderivative size = 47 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} a (B+2 C) x+\frac {a (B+C) \sin (c+d x)}{d}+\frac {a B \cos (c+d x) \sin (c+d x)}{2 d} \]
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Time = 0.16 (sec) , antiderivative size = 47, 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} \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (B+C) \sin (c+d x)}{d}+\frac {a B \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} a x (B+2 C) \]
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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+a \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 a (B+C)-a (B+2 C) \sec (c+d x)) \, dx \\ & = \frac {a B \cos (c+d x) \sin (c+d x)}{2 d}+(a (B+C)) \int \cos (c+d x) \, dx+\frac {1}{2} (a (B+2 C)) \int 1 \, dx \\ & = \frac {1}{2} a (B+2 C) x+\frac {a (B+C) \sin (c+d x)}{d}+\frac {a B \cos (c+d x) \sin (c+d x)}{2 d} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (2 B c+2 B d x+4 C d x+4 (B+C) \sin (c+d x)+B \sin (2 (c+d x)))}{4 d} \]
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Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89
method | result | size |
parallelrisch | \(\frac {\left (\frac {B \sin \left (2 d x +2 c \right )}{2}+\left (2 B +2 C \right ) \sin \left (d x +c \right )+\left (B +2 C \right ) x d \right ) a}{2 d}\) | \(42\) |
risch | \(\frac {a B x}{2}+a x C +\frac {a B \sin \left (d x +c \right )}{d}+\frac {\sin \left (d x +c \right ) C a}{d}+\frac {a B \sin \left (2 d x +2 c \right )}{4 d}\) | \(51\) |
derivativedivides | \(\frac {a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \sin \left (d x +c \right )+C a \sin \left (d x +c \right )+C a \left (d x +c \right )}{d}\) | \(57\) |
default | \(\frac {a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \sin \left (d x +c \right )+C a \sin \left (d x +c \right )+C a \left (d x +c \right )}{d}\) | \(57\) |
norman | \(\frac {\frac {a \left (B +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {a \left (3 B +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (B +2 C \right ) x}{2}-\frac {2 a B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}+\frac {2 a B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {4 a \left (B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+\frac {a \left (B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-a \left (B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-a \left (B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {a \left (B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {a \left (B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(240\) |
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Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (B + 2 \, C\right )} a d x + {\left (B a \cos \left (d x + c\right ) + 2 \, {\left (B + C\right )} a\right )} \sin \left (d x + c\right )}{2 \, d} \]
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\[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a \left (\int B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.22 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.17 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a + 4 \, {\left (d x + c\right )} C a + 4 \, B a \sin \left (d x + c\right ) + 4 \, C a \sin \left (d x + c\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (43) = 86\).
Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.98 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (B a + 2 \, C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
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Time = 15.54 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.06 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B\,a\,x}{2}+C\,a\,x+\frac {B\,a\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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