Integrand size = 38, antiderivative size = 77 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} a (B+C) x+\frac {a (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a (B+C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d} \]
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Time = 0.20 (sec) , antiderivative size = 77, 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} \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a (B+C) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {a B \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {1}{2} a x (B+C) \]
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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+a \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 a (B+C)-a (2 B+3 C) \sec (c+d x)) \, dx \\ & = \frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d}+(a (B+C)) \int \cos ^2(c+d x) \, dx+\frac {1}{3} (a (2 B+3 C)) \int \cos (c+d x) \, dx \\ & = \frac {a (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a (B+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} (a (B+C)) \int 1 \, dx \\ & = \frac {1}{2} a (B+C) x+\frac {a (2 B+3 C) \sin (c+d x)}{3 d}+\frac {a (B+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 = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (6 B c+6 c C+6 B d x+6 C d x+3 (3 B+4 C) \sin (c+d x)+3 (B+C) \sin (2 (c+d x))+B \sin (3 (c+d x)))}{12 d} \]
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Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.70
method | result | size |
parallelrisch | \(\frac {a \left (\frac {\left (B +C \right ) \sin \left (2 d x +2 c \right )}{2}+\frac {B \sin \left (3 d x +3 c \right )}{6}+\left (\frac {3 B}{2}+2 C \right ) \sin \left (d x +c \right )+\left (B +C \right ) x d \right )}{2 d}\) | \(54\) |
derivativedivides | \(\frac {\frac {a B \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \sin \left (d x +c \right )}{d}\) | \(85\) |
default | \(\frac {\frac {a B \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \sin \left (d x +c \right )}{d}\) | \(85\) |
risch | \(\frac {a B x}{2}+\frac {a x C}{2}+\frac {3 a B \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (d x +c \right ) C a}{d}+\frac {a B \sin \left (3 d x +3 c \right )}{12 d}+\frac {a B \sin \left (2 d x +2 c \right )}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C a}{4 d}\) | \(85\) |
norman | \(\frac {\left (\frac {1}{2} a B +\frac {1}{2} C a \right ) x +\left (-\frac {1}{2} a B -\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {1}{2} a B -\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {1}{2} a B +\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-2 a B -2 C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (a B +C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (a B +C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\frac {a \left (B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}+\frac {2 a \left (B -3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}+\frac {3 a \left (B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (B +9 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}-\frac {a \left (5 B -3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {2 a \left (5 B +9 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(301\) |
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Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.73 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (B + C\right )} a d x + {\left (2 \, B a \cos \left (d x + c\right )^{2} + 3 \, {\left (B + C\right )} a \cos \left (d x + c\right ) + 2 \, {\left (2 \, B + 3 \, C\right )} a\right )} \sin \left (d x + c\right )}{6 \, d} \]
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\[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a \left (\int B \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 12 \, C a \sin \left (d x + c\right )}{12 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.61 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (B a + C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
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Time = 15.44 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {B\,a\,x}{2}+\frac {C\,a\,x}{2}+\frac {3\,B\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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