Integrand size = 39, antiderivative size = 102 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} a (3 A+4 (B+C)) x+\frac {a (A+B+C) \sin (c+d x)}{d}+\frac {a (3 A+4 (B+C)) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a (A+B) \sin ^3(c+d x)}{3 d} \]
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Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4159, 4132, 2715, 8, 4129, 3092} \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (A+B+C) \sin (c+d x)}{d}+\frac {a (3 A+4 (B+C)) \sin (c+d x) \cos (c+d x)}{8 d}-\frac {a (A+B) \sin ^3(c+d x)}{3 d}+\frac {1}{8} a x (3 A+4 (B+C))+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
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Rule 8
Rule 2715
Rule 3092
Rule 4129
Rule 4132
Rule 4159
Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {1}{4} \int \cos ^3(c+d x) \left (-4 a (A+B)-a (3 A+4 (B+C)) \sec (c+d x)-4 a C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {1}{4} \int \cos ^3(c+d x) \left (-4 a (A+B)-4 a C \sec ^2(c+d x)\right ) \, dx+\frac {1}{4} (a (3 A+4 (B+C))) \int \cos ^2(c+d x) \, dx \\ & = \frac {a (3 A+4 (B+C)) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {1}{4} \int \cos (c+d x) \left (-4 a C-4 a (A+B) \cos ^2(c+d x)\right ) \, dx+\frac {1}{8} (a (3 A+4 (B+C))) \int 1 \, dx \\ & = \frac {1}{8} a (3 A+4 (B+C)) x+\frac {a (3 A+4 (B+C)) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {\text {Subst}\left (\int \left (-4 a (A+B)-4 a C+4 a (A+B) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{4 d} \\ & = \frac {1}{8} a (3 A+4 (B+C)) x+\frac {a (A+B+C) \sin (c+d x)}{d}+\frac {a (3 A+4 (B+C)) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a (A+B) \sin ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a (24 A c+48 B c+36 A d x+48 B d x+48 C d x+24 (3 A+3 B+4 C) \sin (c+d x)+24 (A+B+C) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+8 B \sin (3 (c+d x))+3 A \sin (4 (c+d x)))}{96 d} \]
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Time = 0.38 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.76
method | result | size |
parallelrisch | \(\frac {\left (3 \left (A +B +C \right ) \sin \left (2 d x +2 c \right )+\left (A +B \right ) \sin \left (3 d x +3 c \right )+\frac {3 A \sin \left (4 d x +4 c \right )}{8}+3 \left (3 A +3 B +4 C \right ) \sin \left (d x +c \right )+\frac {9 x d \left (A +\frac {4 B}{3}+\frac {4 C}{3}\right )}{2}\right ) a}{12 d}\) | \(78\) |
derivativedivides | \(\frac {a A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {a B \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \sin \left (d x +c \right )}{d}\) | \(141\) |
default | \(\frac {a A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {a B \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a B \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \sin \left (d x +c \right )}{d}\) | \(141\) |
risch | \(\frac {3 a A x}{8}+\frac {a B x}{2}+\frac {a x C}{2}+\frac {3 a A \sin \left (d x +c \right )}{4 d}+\frac {3 a B \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (d x +c \right ) C a}{d}+\frac {a A \sin \left (4 d x +4 c \right )}{32 d}+\frac {a A \sin \left (3 d x +3 c \right )}{12 d}+\frac {\sin \left (3 d x +3 c \right ) a B}{12 d}+\frac {a A \sin \left (2 d x +2 c \right )}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a B}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C a}{4 d}\) | \(151\) |
norman | \(\frac {\left (\frac {3}{8} a A +\frac {1}{2} a B +\frac {1}{2} C a \right ) x +\left (-\frac {3}{2} a A -2 a B -2 C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {3}{8} a A -\frac {1}{2} a B -\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {3}{8} a A -\frac {1}{2} a B -\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {3}{4} a A +a B +C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {3}{4} a A +a B +C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {3}{8} a A +\frac {1}{2} a B +\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\frac {a \left (3 A +4 B +4 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d}+\frac {a \left (13 A -20 B -36 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 d}+\frac {a \left (13 A +12 B +12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {a \left (29 A -4 B +12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {a \left (31 A +4 B +36 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}-\frac {a \left (47 A +20 B -12 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(360\) |
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Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.85 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (3 \, A + 4 \, B + 4 \, C\right )} a d x + {\left (6 \, A a \cos \left (d x + c\right )^{3} + 8 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A + 4 \, B + 4 \, C\right )} a \cos \left (d x + c\right ) + 8 \, {\left (2 \, A + 2 \, B + 3 \, C\right )} a\right )} \sin \left (d x + c\right )}{24 \, d} \]
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\[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a \left (\int A \cos ^{4}{\left (c + d x \right )}\, dx + \int A \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.29 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a + 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 96 \, C a \sin \left (d x + c\right )}{96 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (96) = 192\).
Time = 0.31 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.14 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (3 \, A a + 4 \, B a + 4 \, C a\right )} {\left (d x + c\right )} + \frac {2 \, {\left (9 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 49 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 28 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 31 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 52 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 84 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 39 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
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Time = 19.29 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.05 \[ \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {3\,A\,a}{4}+B\,a+C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {49\,A\,a}{12}+\frac {7\,B\,a}{3}+5\,C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {31\,A\,a}{12}+\frac {13\,B\,a}{3}+7\,C\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {13\,A\,a}{4}+3\,B\,a+3\,C\,a\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,A+4\,B+4\,C\right )}{4\,\left (\frac {3\,A\,a}{4}+B\,a+C\,a\right )}\right )\,\left (3\,A+4\,B+4\,C\right )}{4\,d} \]
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