Optimal. Leaf size=309 \[ \frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{120 d}-\frac {a \left (4 a^3 B+16 a^2 A b+27 a b^2 B+13 A b^3\right ) \sin ^3(c+d x)}{15 d}+\frac {\left (12 a^4 B+48 a^3 A b+87 a^2 b^2 B+53 a A b^3+15 b^4 B\right ) \sin (c+d x)}{15 d}+\frac {\left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} x \left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right )+\frac {a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{10 d}+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d} \]
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Rubi [A] time = 0.82, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4025, 4094, 4074, 4047, 2635, 8, 4044, 3013} \[ -\frac {a \left (16 a^2 A b+4 a^3 B+27 a b^2 B+13 A b^3\right ) \sin ^3(c+d x)}{15 d}+\frac {\left (48 a^3 A b+87 a^2 b^2 B+12 a^4 B+53 a A b^3+15 b^4 B\right ) \sin (c+d x)}{15 d}+\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{120 d}+\frac {\left (36 a^2 A b^2+5 a^4 A+24 a^3 b B+32 a b^3 B+8 A b^4\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} x \left (36 a^2 A b^2+5 a^4 A+24 a^3 b B+32 a b^3 B+8 A b^4\right )+\frac {a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{10 d}+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3013
Rule 4025
Rule 4044
Rule 4047
Rule 4074
Rule 4094
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {1}{6} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (-3 a (3 A b+2 a B)-\left (5 a^2 A+6 A b^2+12 a b B\right ) \sec (c+d x)-2 b (a A+3 b B) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {1}{30} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (-a \left (25 a^2 A+48 A b^2+72 a b B\right )-\left (71 a^2 A b+30 A b^3+24 a^3 B+90 a b^2 B\right ) \sec (c+d x)-2 b \left (14 a A b+6 a^2 B+15 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{120} \int \cos ^3(c+d x) \left (24 a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right )+15 \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 ) \sec (c+d x)+8 b^2 \left (14 a A b+6 a^2 B+15 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{120} \int \cos ^3(c+d x) \left (24 a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right )+8 b^2 \left (14 a A b+6 a^2 B+15 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{8} \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 ) \int \cos ^2(c+d x) \, dx\\ &=\frac {\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 ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{120} \int \cos (c+d x) \left (8 b^2 \left (14 a A b+6 a^2 B+15 b^2 B\right )+24 a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right ) \cos ^2(c+d x)\right ) \, dx+\frac {1}{16} \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 ) \int 1 \, dx\\ &=\frac {1}{16} \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 ) x+\frac {\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 ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {\operatorname {Subst}\left (\int \left (8 b^2 \left (14 a A b+6 a^2 B+15 b^2 B\right )+24 a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right )-24 a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right ) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{120 d}\\ &=\frac {1}{16} \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 ) x+\frac {\left (48 a^3 A b+53 a A b^3+12 a^4 B+87 a^2 b^2 B+15 b^4 B\right ) \sin (c+d x)}{15 d}+\frac {\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 ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right ) \sin ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A] time = 1.24, size = 333, normalized size = 1.08 \[ \frac {45 a^4 A \sin (4 (c+d x))+5 a^4 A \sin (6 (c+d x))+300 a^4 A c+300 a^4 A d x+100 a^4 B \sin (3 (c+d x))+12 a^4 B \sin (5 (c+d x))+400 a^3 A b \sin (3 (c+d x))+48 a^3 A b \sin (5 (c+d x))+120 a^3 b B \sin (4 (c+d x))+1440 a^3 b B c+1440 a^3 b B d x+180 a^2 A b^2 \sin (4 (c+d x))+2160 a^2 A b^2 c+2160 a^2 A b^2 d x+480 a^2 b^2 B \sin (3 (c+d x))+120 \left (5 a^4 B+20 a^3 A b+36 a^2 b^2 B+24 a A b^3+8 b^4 B\right ) \sin (c+d x)+15 \left (15 a^4 A+64 a^3 b B+96 a^2 A b^2+64 a b^3 B+16 A b^4\right ) \sin (2 (c+d x))+320 a A b^3 \sin (3 (c+d x))+1920 a b^3 B c+1920 a b^3 B d x+480 A b^4 c+480 A b^4 d x}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 243, normalized size = 0.79 \[ \frac {15 \, {\left (5 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} d x + {\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 128 \, B a^{4} + 512 \, A a^{3} b + 960 \, B a^{2} b^{2} + 640 \, A a b^{3} + 240 \, B b^{4} + 48 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (5 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 32 \, {\left (2 \, B a^{4} + 8 \, A a^{3} b + 15 \, B a^{2} b^{2} + 10 \, A a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.05, size = 1127, normalized size = 3.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.85, size = 316, normalized size = 1.02 \[ \frac {A \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{4} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 A \,a^{3} b \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 B \,a^{3} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 A \,a^{2} b^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{2} b^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {4 a A \,b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 B a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,b^{4} \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 307, normalized size = 0.99 \[ -\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} b - 120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} b - 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} + 1920 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} b^{2} + 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{3} - 960 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{3} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{4} - 960 \, B b^{4} \sin \left (d x + c\right )}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.18, size = 403, normalized size = 1.30 \[ \frac {5\,A\,a^4\,x}{16}+\frac {A\,b^4\,x}{2}+2\,B\,a\,b^3\,x+\frac {3\,B\,a^3\,b\,x}{2}+\frac {5\,B\,a^4\,\sin \left (c+d\,x\right )}{8\,d}+\frac {B\,b^4\,\sin \left (c+d\,x\right )}{d}+\frac {9\,A\,a^2\,b^2\,x}{4}+\frac {15\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,A\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {A\,a^4\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {A\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,B\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {A\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {5\,A\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {A\,a^3\,b\,\sin \left (5\,c+5\,d\,x\right )}{20\,d}+\frac {B\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {B\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {B\,a^3\,b\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {9\,B\,a^2\,b^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {B\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{2\,d}+\frac {3\,A\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {5\,A\,a^3\,b\,\sin \left (c+d\,x\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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