Integrand size = 31, antiderivative size = 258 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {1}{8} \left (12 a^3 A b+16 a A b^3+3 a^4 B+24 a^2 b^2 B+8 b^4 B\right ) x+\frac {\left (8 a^4 A+60 a^2 A b^2+15 A b^4+40 a^3 b B+60 a b^3 B\right ) \sin (c+d x)}{15 d}+\frac {a \left (60 a^2 A b+56 A b^3+15 a^3 B+110 a b^2 B\right ) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {a^2 \left (8 a^2 A+18 A b^2+25 a b B\right ) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac {a (8 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 d} \]
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Time = 0.73 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4110, 4179, 4159, 4132, 2717, 4130, 8} \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {a^2 \left (8 a^2 A+25 a b B+18 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{30 d}+\frac {a \left (15 a^3 B+60 a^2 A b+110 a b^2 B+56 A b^3\right ) \sin (c+d x) \cos (c+d x)}{40 d}+\frac {\left (8 a^4 A+40 a^3 b B+60 a^2 A b^2+60 a b^3 B+15 A b^4\right ) \sin (c+d x)}{15 d}+\frac {1}{8} x \left (3 a^4 B+12 a^3 A b+24 a^2 b^2 B+16 a A b^3+8 b^4 B\right )+\frac {a (5 a B+8 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{20 d}+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d} \]
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Rule 8
Rule 2717
Rule 4110
Rule 4130
Rule 4132
Rule 4159
Rule 4179
Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 d}-\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \left (-a (8 A b+5 a B)-\left (4 a^2 A+5 A b^2+10 a b B\right ) \sec (c+d x)-b (a A+5 b B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a (8 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 d}-\frac {1}{20} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (-2 a \left (8 a^2 A+18 A b^2+25 a b B\right )-\left (44 a^2 A b+20 A b^3+15 a^3 B+60 a b^2 B\right ) \sec (c+d x)-b \left (12 a A b+5 a^2 B+20 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 \left (8 a^2 A+18 A b^2+25 a b B\right ) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac {a (8 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac {1}{60} \int \cos ^2(c+d x) \left (3 a \left (60 a^2 A b+56 A b^3+15 a^3 B+110 a b^2 B\right )+4 \left (8 a^4 A+60 a^2 A b^2+15 A b^4+40 a^3 b B+60 a b^3 B\right ) \sec (c+d x)+3 b^2 \left (12 a A b+5 a^2 B+20 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 \left (8 a^2 A+18 A b^2+25 a b B\right ) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac {a (8 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac {1}{60} \int \cos ^2(c+d x) \left (3 a \left (60 a^2 A b+56 A b^3+15 a^3 B+110 a b^2 B\right )+3 b^2 \left (12 a A b+5 a^2 B+20 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{15} \left (8 a^4 A+60 a^2 A b^2+15 A b^4+40 a^3 b B+60 a b^3 B\right ) \int \cos (c+d x) \, dx \\ & = \frac {\left (8 a^4 A+60 a^2 A b^2+15 A b^4+40 a^3 b B+60 a b^3 B\right ) \sin (c+d x)}{15 d}+\frac {a \left (60 a^2 A b+56 A b^3+15 a^3 B+110 a b^2 B\right ) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {a^2 \left (8 a^2 A+18 A b^2+25 a b B\right ) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac {a (8 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac {1}{8} \left (12 a^3 A b+16 a A b^3+3 a^4 B+24 a^2 b^2 B+8 b^4 B\right ) \int 1 \, dx \\ & = \frac {1}{8} \left (12 a^3 A b+16 a A b^3+3 a^4 B+24 a^2 b^2 B+8 b^4 B\right ) x+\frac {\left (8 a^4 A+60 a^2 A b^2+15 A b^4+40 a^3 b B+60 a b^3 B\right ) \sin (c+d x)}{15 d}+\frac {a \left (60 a^2 A b+56 A b^3+15 a^3 B+110 a b^2 B\right ) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {a^2 \left (8 a^2 A+18 A b^2+25 a b B\right ) \cos ^2(c+d x) \sin (c+d x)}{30 d}+\frac {a (8 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{5 d} \\ \end{align*}
Time = 1.79 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.02 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {720 a^3 A b c+960 a A b^3 c+180 a^4 B c+1440 a^2 b^2 B c+480 b^4 B c+720 a^3 A b d x+960 a A b^3 d x+180 a^4 B d x+1440 a^2 b^2 B d x+480 b^4 B d x+60 \left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) \sin (c+d x)+120 a \left (4 a^2 A b+4 A b^3+a^3 B+6 a b^2 B\right ) \sin (2 (c+d x))+50 a^4 A \sin (3 (c+d x))+240 a^2 A b^2 \sin (3 (c+d x))+160 a^3 b B \sin (3 (c+d x))+60 a^3 A b \sin (4 (c+d x))+15 a^4 B \sin (4 (c+d x))+6 a^4 A \sin (5 (c+d x))}{480 d} \]
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Time = 3.64 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.78
method | result | size |
parallelrisch | \(\frac {\left (480 A \,a^{3} b +480 A a \,b^{3}+120 B \,a^{4}+720 B \,a^{2} b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (50 a^{4} A +240 A \,a^{2} b^{2}+160 B \,a^{3} b \right ) \sin \left (3 d x +3 c \right )+\left (60 A \,a^{3} b +15 B \,a^{4}\right ) \sin \left (4 d x +4 c \right )+6 a^{4} A \sin \left (5 d x +5 c \right )+\left (300 a^{4} A +2160 A \,a^{2} b^{2}+480 A \,b^{4}+1440 B \,a^{3} b +1920 B a \,b^{3}\right ) \sin \left (d x +c \right )+720 \left (A \,a^{3} b +\frac {4}{3} A a \,b^{3}+\frac {1}{4} B \,a^{4}+2 B \,a^{2} b^{2}+\frac {2}{3} B \,b^{4}\right ) d x}{480 d}\) | \(201\) |
derivativedivides | \(\frac {\frac {a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 A \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 A \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {4 B \,a^{3} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 A a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 B \,a^{2} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,b^{4} \sin \left (d x +c \right )+4 B a \,b^{3} \sin \left (d x +c \right )+B \,b^{4} \left (d x +c \right )}{d}\) | \(258\) |
default | \(\frac {\frac {a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 A \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 A \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {4 B \,a^{3} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 A a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 B \,a^{2} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,b^{4} \sin \left (d x +c \right )+4 B a \,b^{3} \sin \left (d x +c \right )+B \,b^{4} \left (d x +c \right )}{d}\) | \(258\) |
risch | \(\frac {3 A \,a^{3} b x}{2}+2 A a \,b^{3} x +\frac {3 a^{4} x B}{8}+3 B \,a^{2} b^{2} x +x B \,b^{4}+\frac {5 \sin \left (d x +c \right ) a^{4} A}{8 d}+\frac {9 \sin \left (d x +c \right ) A \,a^{2} b^{2}}{2 d}+\frac {\sin \left (d x +c \right ) A \,b^{4}}{d}+\frac {3 \sin \left (d x +c \right ) B \,a^{3} b}{d}+\frac {4 \sin \left (d x +c \right ) B a \,b^{3}}{d}+\frac {a^{4} A \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (4 d x +4 c \right ) A \,a^{3} b}{8 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{4}}{32 d}+\frac {5 a^{4} A \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{2} b^{2}}{2 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3} b}{3 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3} b}{d}+\frac {\sin \left (2 d x +2 c \right ) A a \,b^{3}}{d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{4}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b^{2}}{2 d}\) | \(308\) |
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Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.76 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (3 \, B a^{4} + 12 \, A a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 8 \, B b^{4}\right )} d x + {\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 64 \, A a^{4} + 320 \, B a^{3} b + 480 \, A a^{2} b^{2} + 480 \, B a b^{3} + 120 \, A b^{4} + 30 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (2 \, A a^{4} + 10 \, B a^{3} b + 15 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (3 \, B a^{4} + 12 \, A a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.95 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} b - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{2} + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} + 480 \, {\left (d x + c\right )} B b^{4} + 1920 \, B a b^{3} \sin \left (d x + c\right ) + 480 \, A b^{4} \sin \left (d x + c\right )}{480 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 791 vs. \(2 (246) = 492\).
Time = 0.35 (sec) , antiderivative size = 791, normalized size of antiderivative = 3.07 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Too large to display} \]
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Time = 15.22 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.19 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {3\,B\,a^4\,x}{8}+B\,b^4\,x+2\,A\,a\,b^3\,x+\frac {3\,A\,a^3\,b\,x}{2}+\frac {5\,A\,a^4\,\sin \left (c+d\,x\right )}{8\,d}+\frac {A\,b^4\,\sin \left (c+d\,x\right )}{d}+3\,B\,a^2\,b^2\,x+\frac {5\,A\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {A\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {B\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {A\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {A\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {A\,a^3\,b\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {9\,A\,a^2\,b^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {B\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {A\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{2\,d}+\frac {3\,B\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {4\,B\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {3\,B\,a^3\,b\,\sin \left (c+d\,x\right )}{d} \]
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