Optimal. Leaf size=220 \[ \frac{a^4 (72 A+83 B) \sin (c+d x)}{15 d}+\frac{a^4 (159 A+176 B) \sin (c+d x) \cos ^2(c+d x)}{120 d}+\frac{7 a^4 (7 A+8 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(3 A+2 B) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{10 d}+\frac{(73 A+72 B) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{120 d}+\frac{7}{16} a^4 x (7 A+8 B)+\frac{a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d} \]
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Rubi [A] time = 0.532254, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {4017, 3996, 3787, 2635, 8, 2637} \[ \frac{a^4 (72 A+83 B) \sin (c+d x)}{15 d}+\frac{a^4 (159 A+176 B) \sin (c+d x) \cos ^2(c+d x)}{120 d}+\frac{7 a^4 (7 A+8 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(3 A+2 B) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{10 d}+\frac{(73 A+72 B) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{120 d}+\frac{7}{16} a^4 x (7 A+8 B)+\frac{a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d} \]
Antiderivative was successfully verified.
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Rule 4017
Rule 3996
Rule 3787
Rule 2635
Rule 8
Rule 2637
Rubi steps
\begin{align*} \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac{a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{1}{6} \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (3 a (3 A+2 B)+2 a (A+3 B) \sec (c+d x)) \, dx\\ &=\frac{a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac{1}{30} \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (a^2 (73 A+72 B)+14 a^2 (2 A+3 B) \sec (c+d x)\right ) \, dx\\ &=\frac{a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac{(73 A+72 B) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac{1}{120} \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (3 a^3 (159 A+176 B)+6 a^3 (43 A+52 B) \sec (c+d x)\right ) \, dx\\ &=\frac{a^4 (159 A+176 B) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac{a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac{(73 A+72 B) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}-\frac{1}{360} \int \cos ^2(c+d x) \left (-315 a^4 (7 A+8 B)-24 a^4 (72 A+83 B) \sec (c+d x)\right ) \, dx\\ &=\frac{a^4 (159 A+176 B) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac{a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac{(73 A+72 B) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac{1}{8} \left (7 a^4 (7 A+8 B)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{15} \left (a^4 (72 A+83 B)\right ) \int \cos (c+d x) \, dx\\ &=\frac{a^4 (72 A+83 B) \sin (c+d x)}{15 d}+\frac{7 a^4 (7 A+8 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^4 (159 A+176 B) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac{a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac{(73 A+72 B) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac{1}{16} \left (7 a^4 (7 A+8 B)\right ) \int 1 \, dx\\ &=\frac{7}{16} a^4 (7 A+8 B) x+\frac{a^4 (72 A+83 B) \sin (c+d x)}{15 d}+\frac{7 a^4 (7 A+8 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^4 (159 A+176 B) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac{a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac{(73 A+72 B) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}\\ \end{align*}
Mathematica [A] time = 0.585357, size = 134, normalized size = 0.61 \[ \frac{a^4 (120 (44 A+49 B) \sin (c+d x)+15 (127 A+128 B) \sin (2 (c+d x))+720 A \sin (3 (c+d x))+225 A \sin (4 (c+d x))+48 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+2940 A c+2940 A d x+580 B \sin (3 (c+d x))+120 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+3360 B d x)}{960 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 306, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( A{a}^{4} \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}^{4}\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 ) }+{\frac{4\,A{a}^{4}\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 ) }+4\,B{a}^{4} \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 ) +6\,A{a}^{4} \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 ) +2\,B{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +{\frac{4\,A{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+4\,B{a}^{4} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +A{a}^{4} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +B{a}^{4}\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 = 1.01621, size = 401, normalized size = 1.82 \begin{align*} \frac{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^{4} - 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} - 1280 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 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^{4} + 240 \,{\left (2 \, d x + 2 \, c + \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} - 1920 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 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^{4} + 960 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 960 \, B a^{4} \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 = 0.492452, size = 329, normalized size = 1.5 \begin{align*} \frac{105 \,{\left (7 \, A + 8 \, B\right )} a^{4} d x +{\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 48 \,{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{4} + 10 \,{\left (41 \, A + 24 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 32 \,{\left (18 \, A + 17 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \,{\left (7 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right ) + 16 \,{\left (72 \, A + 83 \, B\right )} a^{4}\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 [A] time = 1.40133, size = 329, normalized size = 1.5 \begin{align*} \frac{105 \,{\left (7 \, A a^{4} + 8 \, B a^{4}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (735 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 840 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 4165 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 4760 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 9702 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 11088 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 11802 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 13488 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 7355 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9320 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3105 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3000 \, B a^{4} \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 )}^{6}}}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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