Integrand size = 31, antiderivative size = 131 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} b (3 A+4 C) x+\frac {a (4 A+5 C) \sin (c+d x)}{5 d}+\frac {b (3 A+4 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {a (4 A+5 C) \sin ^3(c+d x)}{15 d} \]
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Time = 0.22 (sec) , antiderivative size = 131, 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, 2713, 4130, 2715, 8} \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {a (4 A+5 C) \sin ^3(c+d x)}{15 d}+\frac {a (4 A+5 C) \sin (c+d x)}{5 d}+\frac {a A \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac {b (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {A b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{8} b x (3 A+4 C) \]
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Rule 8
Rule 2713
Rule 2715
Rule 4130
Rule 4132
Rule 4160
Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {1}{5} \int \cos ^4(c+d x) \left (-5 A b-a (4 A+5 C) \sec (c+d x)-5 b C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {1}{5} \int \cos ^4(c+d x) \left (-5 A b-5 b C \sec ^2(c+d x)\right ) \, dx+\frac {1}{5} (a (4 A+5 C)) \int \cos ^3(c+d x) \, dx \\ & = \frac {A b \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a A \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{4} (b (3 A+4 C)) \int \cos ^2(c+d x) \, dx-\frac {(a (4 A+5 C)) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {a (4 A+5 C) \sin (c+d x)}{5 d}+\frac {b (3 A+4 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {a (4 A+5 C) \sin ^3(c+d x)}{15 d}+\frac {1}{8} (b (3 A+4 C)) \int 1 \, dx \\ & = \frac {1}{8} b (3 A+4 C) x+\frac {a (4 A+5 C) \sin (c+d x)}{5 d}+\frac {b (3 A+4 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {A b \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {a A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {a (4 A+5 C) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.68 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {480 a (A+C) \sin (c+d x)-160 a (2 A+C) \sin ^3(c+d x)+96 a A \sin ^5(c+d x)+15 b (4 (3 A+4 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+A \sin (4 (c+d x)))}{480 d} \]
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Time = 0.65 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.68
method | result | size |
parallelrisch | \(\frac {120 b \left (A +C \right ) \sin \left (2 d x +2 c \right )+50 a \left (A +\frac {4 C}{5}\right ) \sin \left (3 d x +3 c \right )+15 A b \sin \left (4 d x +4 c \right )+6 a A \sin \left (5 d x +5 c \right )+300 \left (A +\frac {6 C}{5}\right ) a \sin \left (d x +c \right )+180 \left (A +\frac {4 C}{3}\right ) x b d}{480 d}\) | \(89\) |
derivativedivides | \(\frac {\frac {a A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+A b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C a \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C b \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(117\) |
default | \(\frac {\frac {a A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+A b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C a \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C b \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(117\) |
risch | \(\frac {3 A b x}{8}+\frac {b x C}{2}+\frac {5 a A \sin \left (d x +c \right )}{8 d}+\frac {3 \sin \left (d x +c \right ) C a}{4 d}+\frac {a A \sin \left (5 d x +5 c \right )}{80 d}+\frac {A b \sin \left (4 d x +4 c \right )}{32 d}+\frac {5 a A \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) C a}{12 d}+\frac {A b \sin \left (2 d x +2 c \right )}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C b}{4 d}\) | \(134\) |
norman | \(\frac {\left (\frac {3}{8} A b +\frac {1}{2} C b \right ) x +\left (-\frac {15}{8} A b -\frac {5}{2} C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {15}{8} A b -\frac {5}{2} C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {3}{8} A b +\frac {1}{2} C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {3}{8} A b +\frac {1}{2} C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {3}{8} A b +\frac {1}{2} C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {9}{8} A b +\frac {3}{2} C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {9}{8} A b +\frac {3}{2} C b \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}-\frac {2 \left (2 a A -3 A b -2 C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}-\frac {2 \left (2 a A +3 A b -2 C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {\left (8 a A -5 A b +8 C a -4 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{4 d}+\frac {\left (8 a A +5 A b +8 C a +4 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (88 a A -5 A b -40 C a +60 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{20 d}+\frac {\left (88 a A +5 A b -40 C a -60 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}-\frac {8 a \left (19 A +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{15 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(410\) |
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Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.72 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (3 \, A + 4 \, C\right )} b d x + {\left (24 \, A a \cos \left (d x + c\right )^{4} + 30 \, A b \cos \left (d x + c\right )^{3} + 8 \, {\left (4 \, A + 5 \, C\right )} a \cos \left (d x + c\right )^{2} + 15 \, {\left (3 \, A + 4 \, C\right )} b \cos \left (d x + c\right ) + 16 \, {\left (4 \, A + 5 \, C\right )} a\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A b + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C b}{480 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (119) = 238\).
Time = 0.32 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.31 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (3 \, A b + 4 \, C b\right )} {\left (d x + c\right )} + \frac {2 \, {\left (120 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 60 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 160 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 320 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 120 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 464 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 400 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 320 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
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Time = 18.18 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.91 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (2\,A\,a-\frac {5\,A\,b}{4}+2\,C\,a-C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {8\,A\,a}{3}-\frac {A\,b}{2}+\frac {16\,C\,a}{3}-2\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {116\,A\,a}{15}+\frac {20\,C\,a}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {8\,A\,a}{3}+\frac {A\,b}{2}+\frac {16\,C\,a}{3}+2\,C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a+\frac {5\,A\,b}{4}+2\,C\,a+C\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,A+4\,C\right )}{4\,\left (\frac {3\,A\,b}{4}+C\,b\right )}\right )\,\left (3\,A+4\,C\right )}{4\,d} \]
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