Optimal. Leaf size=320 \[ -\frac{\sin ^3(c+d x) \left (3 a^2 b (4 A+5 C)+4 a^3 B+12 a b^2 B+A b^3\right )}{15 d}+\frac{\sin (c+d x) \left (9 a^2 b (4 A+5 C)+12 a^3 B+42 a b^2 B+b^3 (11 A+15 C)\right )}{15 d}+\frac{a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{120 d}+\frac{\sin (c+d x) \cos (c+d x) \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right )}{16 d}+\frac{1}{16} x \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right )+\frac{(2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{10 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d} \]
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Rubi [A] time = 0.895196, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4094, 4074, 4047, 2635, 8, 4044, 3013} \[ -\frac{\sin ^3(c+d x) \left (3 a^2 b (4 A+5 C)+4 a^3 B+12 a b^2 B+A b^3\right )}{15 d}+\frac{\sin (c+d x) \left (9 a^2 b (4 A+5 C)+12 a^3 B+42 a b^2 B+b^3 (11 A+15 C)\right )}{15 d}+\frac{a \sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right )}{120 d}+\frac{\sin (c+d x) \cos (c+d x) \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right )}{16 d}+\frac{1}{16} x \left (a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right )+\frac{(2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{10 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d} \]
Antiderivative was successfully verified.
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Rule 4094
Rule 4074
Rule 4047
Rule 2635
Rule 8
Rule 4044
Rule 3013
Rubi steps
\begin{align*} \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{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 b+2 a B)+(5 a A+6 b B+6 a C) \sec (c+d x)+2 b (A+3 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{(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 \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 (6 A b^2+42 a b B+5 a^2 (5 A+6 C)+\left (24 a^2 B+30 b^2 B+a b (47 A+60 C)\right ) \sec (c+d x)+2 b (8 A b+6 a B+15 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac{(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 \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 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right )-15 \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \sec (c+d x)-8 b^2 (8 A b+6 a B+15 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac{(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 \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 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right )-8 b^2 (8 A b+6 a B+15 b C) \sec ^2(c+d x)\right ) \, dx-\frac{1}{8} \left (-18 a^2 b B-8 b^3 B-6 a b^2 (3 A+4 C)-a^3 (5 A+6 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac{(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 \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 (8 A b+6 a B+15 b C)-24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) \cos ^2(c+d x)\right ) \, dx-\frac{1}{16} \left (-18 a^2 b B-8 b^3 B-6 a b^2 (3 A+4 C)-a^3 (5 A+6 C)\right ) \int 1 \, dx\\ &=\frac{1}{16} \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) x+\frac{\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac{(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 \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 (8 A b+6 a B+15 b C)-24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right )+24 \left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{120 d}\\ &=\frac{1}{16} \left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) x+\frac{\left (12 a^3 B+42 a b^2 B+9 a^2 b (4 A+5 C)+b^3 (11 A+15 C)\right ) \sin (c+d x)}{15 d}+\frac{\left (18 a^2 b B+8 b^3 B+6 a b^2 (3 A+4 C)+a^3 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a \left (6 A b^2+42 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac{(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 \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac{\left (A b^3+4 a^3 B+12 a b^2 B+3 a^2 b (4 A+5 C)\right ) \sin ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 1.18818, size = 369, normalized size = 1.15 \[ \frac{120 \sin (c+d x) \left (3 a^2 b (5 A+6 C)+5 a^3 B+18 a b^2 B+2 b^3 (3 A+4 C)\right )+15 \sin (2 (c+d x)) \left (a^3 (15 A+16 C)+48 a^2 b B+48 a b^2 (A+C)+16 b^3 B\right )+300 a^2 A b \sin (3 (c+d x))+36 a^2 A b \sin (5 (c+d x))+45 a^3 A \sin (4 (c+d x))+5 a^3 A \sin (6 (c+d x))+300 a^3 A c+300 a^3 A d x+90 a^2 b B \sin (4 (c+d x))+1080 a^2 b B c+1080 a^2 b B d x+240 a^2 b C \sin (3 (c+d x))+100 a^3 B \sin (3 (c+d x))+12 a^3 B \sin (5 (c+d x))+30 a^3 C \sin (4 (c+d x))+360 a^3 c C+360 a^3 C d x+90 a A b^2 \sin (4 (c+d x))+1080 a A b^2 c+1080 a A b^2 d x+240 a b^2 B \sin (3 (c+d x))+1440 a b^2 c C+1440 a b^2 C d x+80 A b^3 \sin (3 (c+d x))+480 b^3 B c+480 b^3 B d x}{960 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 370, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( A{a}^{3} \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +{\frac{B{a}^{3}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+{a}^{3}C \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{3\,A{a}^{2}b\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+3\,B{a}^{2}b \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{a}^{2}bC \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,Aa{b}^{2} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +Ba{b}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,Ca{b}^{2} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{\frac{A{b}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+B{b}^{3} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +C{b}^{3}\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05398, size = 486, normalized size = 1.52 \begin{align*} -\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^{3} - 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^{3} - 30 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 192 \,{\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^{2} b - 90 \,{\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^{2} b + 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b - 90 \,{\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 b^{2} + 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{2} - 720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{2} + 320 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{3} - 240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{3} - 960 \, C b^{3} \sin \left (d x + c\right )}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.577018, size = 610, normalized size = 1.91 \begin{align*} \frac{15 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 6 \,{\left (3 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} d x +{\left (40 \, A a^{3} \cos \left (d x + c\right )^{5} + 48 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{4} + 128 \, B a^{3} + 96 \,{\left (4 \, A + 5 \, C\right )} a^{2} b + 480 \, B a b^{2} + 80 \,{\left (2 \, A + 3 \, C\right )} b^{3} + 10 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 18 \, A a b^{2}\right )} \cos \left (d x + c\right )^{3} + 16 \,{\left (4 \, B a^{3} + 3 \,{\left (4 \, A + 5 \, C\right )} a^{2} b + 15 \, B a b^{2} + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, B a^{2} b + 6 \,{\left (3 \, A + 4 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.3343, size = 1764, normalized size = 5.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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