Integrand size = 31, antiderivative size = 95 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} a (3 A+4 C) x+\frac {b (A+C) \sin (c+d x)}{d}+\frac {a (3 A+4 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a A \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {A b \sin ^3(c+d x)}{3 d} \]
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Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4160, 4132, 2715, 8, 4129, 3092} \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{8} a x (3 A+4 C)+\frac {b (A+C) \sin (c+d x)}{d}-\frac {A b \sin ^3(c+d x)}{3 d} \]
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Rule 8
Rule 2715
Rule 3092
Rule 4129
Rule 4132
Rule 4160
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 b-a (3 A+4 C) \sec (c+d x)-4 b 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 b-4 b C \sec ^2(c+d x)\right ) \, dx+\frac {1}{4} (a (3 A+4 C)) \int \cos ^2(c+d x) \, dx \\ & = \frac {a (3 A+4 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 b C-4 A b \cos ^2(c+d x)\right ) \, dx+\frac {1}{8} (a (3 A+4 C)) \int 1 \, dx \\ & = \frac {1}{8} a (3 A+4 C) x+\frac {a (3 A+4 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 b-4 b C+4 A b x^2\right ) \, dx,x,-\sin (c+d x)\right )}{4 d} \\ & = \frac {1}{8} a (3 A+4 C) x+\frac {b (A+C) \sin (c+d x)}{d}+\frac {a (3 A+4 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 b \sin ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.88 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {36 a A c+48 a c C+36 a A d x+48 a C d x+24 b (3 A+4 C) \sin (c+d x)+24 a (A+C) \sin (2 (c+d x))+8 A b \sin (3 (c+d x))+3 a A \sin (4 (c+d x))}{96 d} \]
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Time = 0.52 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.73
method | result | size |
parallelrisch | \(\frac {24 a \left (A +C \right ) \sin \left (2 d x +2 c \right )+8 A b \sin \left (3 d x +3 c \right )+3 a A \sin \left (4 d x +4 c \right )+36 \left (A +\frac {4 C}{3}\right ) \left (a x d +2 b \sin \left (d x +c \right )\right )}{96 d}\) | \(69\) |
derivativedivides | \(\frac {a A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b}{d}\) | \(96\) |
default | \(\frac {a A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C a \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b}{d}\) | \(96\) |
risch | \(\frac {3 a A x}{8}+\frac {a x C}{2}+\frac {3 A b \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (d x +c \right ) C b}{d}+\frac {a A \sin \left (4 d x +4 c \right )}{32 d}+\frac {A b \sin \left (3 d x +3 c \right )}{12 d}+\frac {a A \sin \left (2 d x +2 c \right )}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C a}{4 d}\) | \(101\) |
norman | \(\frac {\left (\frac {3}{8} a A +\frac {1}{2} C a \right ) x +\left (-\frac {3}{2} a A -2 C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {3}{8} a A -\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {3}{8} a A -\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {3}{4} a A +C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {3}{4} a A +C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {3}{8} a A +\frac {1}{2} C a \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}-\frac {\left (5 a A -8 A b +4 C a -8 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d}+\frac {\left (5 a A +8 A b +4 C a +8 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (21 a A -8 A b -12 C a -24 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 d}-\frac {\left (21 a A +8 A b -12 C a +24 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {\left (39 a A -8 A b +12 C a +24 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}-\frac {\left (39 a A +8 A b +12 C a -24 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(370\) |
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Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.80 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (3 \, A + 4 \, C\right )} a d x + {\left (6 \, A a \cos \left (d x + c\right )^{3} + 8 \, A b \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A + 4 \, C\right )} a \cos \left (d x + c\right ) + 8 \, {\left (2 \, A + 3 \, C\right )} b\right )} \sin \left (d x + c\right )}{24 \, d} \]
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\[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right ) \cos ^{4}{\left (c + d x \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.95 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b + 96 \, C b \sin \left (d x + c\right )}{96 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (87) = 174\).
Time = 0.32 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.86 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (3 \, A a + 4 \, C a\right )} {\left (d x + c\right )} - \frac {2 \, {\left (15 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
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Time = 18.54 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.26 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (2\,A\,b-\frac {5\,A\,a}{4}-C\,a+2\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {3\,A\,a}{4}+\frac {10\,A\,b}{3}-C\,a+6\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {10\,A\,b}{3}-\frac {3\,A\,a}{4}+C\,a+6\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,A\,a}{4}+2\,A\,b+C\,a+2\,C\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,A+4\,C\right )}{4\,\left (\frac {3\,A\,a}{4}+C\,a\right )}\right )\,\left (3\,A+4\,C\right )}{4\,d} \]
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