Integrand size = 39, antiderivative size = 82 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} a (A+B+2 C) x+\frac {a (2 A+3 (B+C)) \sin (c+d x)}{3 d}+\frac {a (A+B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d} \]
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Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {4159, 4132, 2717, 4130, 8} \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (2 A+3 (B+C)) \sin (c+d x)}{3 d}+\frac {a (A+B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} a x (A+B+2 C)+\frac {a A \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
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Rule 8
Rule 2717
Rule 4130
Rule 4132
Rule 4159
Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac {1}{3} \int \cos ^2(c+d x) \left (-3 a (A+B)-a (2 A+3 (B+C)) \sec (c+d x)-3 a C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac {1}{3} \int \cos ^2(c+d x) \left (-3 a (A+B)-3 a C \sec ^2(c+d x)\right ) \, dx+\frac {1}{3} (a (2 A+3 (B+C))) \int \cos (c+d x) \, dx \\ & = \frac {a (2 A+3 (B+C)) \sin (c+d x)}{3 d}+\frac {a (A+B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {1}{2} (a (A+B+2 C)) \int 1 \, dx \\ & = \frac {1}{2} a (A+B+2 C) x+\frac {a (2 A+3 (B+C)) \sin (c+d x)}{3 d}+\frac {a (A+B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a A \cos ^2(c+d x) \sin (c+d x)}{3 d} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.78 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (6 A d x+6 B d x+12 C d x+3 (3 A+4 (B+C)) \sin (c+d x)+3 (A+B) \sin (2 (c+d x))+A \sin (3 (c+d x)))}{12 d} \]
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Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.74
method | result | size |
parallelrisch | \(\frac {a \left (3 \left (A +B \right ) \sin \left (2 d x +2 c \right )+A \sin \left (3 d x +3 c \right )+3 \left (3 A +4 B +4 C \right ) \sin \left (d x +c \right )+6 \left (A +B +2 C \right ) x d \right )}{12 d}\) | \(61\) |
risch | \(\frac {a A x}{2}+\frac {a B x}{2}+a x C +\frac {3 a A \sin \left (d x +c \right )}{4 d}+\frac {a B \sin \left (d x +c \right )}{d}+\frac {\sin \left (d x +c \right ) C a}{d}+\frac {a A \sin \left (3 d x +3 c \right )}{12 d}+\frac {a A \sin \left (2 d x +2 c \right )}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a B}{4 d}\) | \(101\) |
derivativedivides | \(\frac {\frac {a A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \sin \left (d x +c \right )+C a \sin \left (d x +c \right )+C a \left (d x +c \right )}{d}\) | \(102\) |
default | \(\frac {\frac {a A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \sin \left (d x +c \right )+C a \sin \left (d x +c \right )+C a \left (d x +c \right )}{d}\) | \(102\) |
norman | \(\frac {\frac {a \left (A +B +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {a \left (3 A +3 B +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {a \left (A +B +2 C \right ) x}{2}-\frac {2 a \left (A -3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {2 a \left (7 A +3 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {4 a \left (A -3 B -3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}+\frac {a \left (A +B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-a \left (A +B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-a \left (A +B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {a \left (A +B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{2}+\frac {a \left (A +B +2 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(265\) |
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Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.76 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (A + B + 2 \, C\right )} a d x + {\left (2 \, A a \cos \left (d x + c\right )^{2} + 3 \, {\left (A + B\right )} a \cos \left (d x + c\right ) + 2 \, {\left (2 \, A + 3 \, B + 3 \, C\right )} a\right )} \sin \left (d x + c\right )}{6 \, d} \]
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\[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a \left (\int A \cos ^{3}{\left (c + d x \right )}\, dx + \int A \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.24 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.20 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 12 \, {\left (d x + c\right )} C a - 12 \, B a \sin \left (d x + c\right ) - 12 \, C a \sin \left (d x + c\right )}{12 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (75) = 150\).
Time = 0.31 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.09 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (A a + B a + 2 \, C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (3 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{6 \, d} \]
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Time = 16.68 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.22 \[ \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {A\,a\,x}{2}+\frac {B\,a\,x}{2}+C\,a\,x+\frac {3\,A\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a\,\sin \left (c+d\,x\right )}{d}+\frac {A\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
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