Optimal. Leaf size=235 \[ \frac{a^3 (34 A+38 B+45 C) \sin (c+d x)}{15 d}+\frac{a^3 (73 A+86 B+90 C) \sin (c+d x) \cos ^2(c+d x)}{120 d}+\frac{a^3 (23 A+26 B+30 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(31 A+42 B+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{120 d}+\frac{(A+2 B) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{10 a d}+\frac{1}{16} a^3 x (23 A+26 B+30 C)+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d} \]
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Rubi [A] time = 0.591015, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4086, 4017, 3996, 3787, 2635, 8, 2637} \[ \frac{a^3 (34 A+38 B+45 C) \sin (c+d x)}{15 d}+\frac{a^3 (73 A+86 B+90 C) \sin (c+d x) \cos ^2(c+d x)}{120 d}+\frac{a^3 (23 A+26 B+30 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(31 A+42 B+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{120 d}+\frac{(A+2 B) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{10 a d}+\frac{1}{16} a^3 x (23 A+26 B+30 C)+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d} \]
Antiderivative was successfully verified.
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Rule 4086
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))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (3 a (A+2 B)+2 a (A+3 C) \sec (c+d x)) \, dx}{6 a}\\ &=\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac{\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (a^2 (31 A+42 B+30 C)+2 a^2 (8 A+6 B+15 C) \sec (c+d x)\right ) \, dx}{30 a}\\ &=\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac{(31 A+42 B+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac{\int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (3 a^3 (73 A+86 B+90 C)+6 a^3 (21 A+22 B+30 C) \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac{a^3 (73 A+86 B+90 C) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac{(31 A+42 B+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d}-\frac{\int \cos ^2(c+d x) \left (-45 a^4 (23 A+26 B+30 C)-24 a^4 (34 A+38 B+45 C) \sec (c+d x)\right ) \, dx}{360 a}\\ &=\frac{a^3 (73 A+86 B+90 C) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac{(31 A+42 B+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac{1}{8} \left (a^3 (23 A+26 B+30 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{15} \left (a^3 (34 A+38 B+45 C)\right ) \int \cos (c+d x) \, dx\\ &=\frac{a^3 (34 A+38 B+45 C) \sin (c+d x)}{15 d}+\frac{a^3 (23 A+26 B+30 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^3 (73 A+86 B+90 C) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac{(31 A+42 B+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac{1}{16} \left (a^3 (23 A+26 B+30 C)\right ) \int 1 \, dx\\ &=\frac{1}{16} a^3 (23 A+26 B+30 C) x+\frac{a^3 (34 A+38 B+45 C) \sin (c+d x)}{15 d}+\frac{a^3 (23 A+26 B+30 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^3 (73 A+86 B+90 C) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac{(A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac{(31 A+42 B+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d}\\ \end{align*}
Mathematica [A] time = 0.752729, size = 170, normalized size = 0.72 \[ \frac{a^3 (120 (21 A+23 B+26 C) \sin (c+d x)+15 (63 A+64 (B+C)) \sin (2 (c+d x))+380 A \sin (3 (c+d x))+135 A \sin (4 (c+d x))+36 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+900 A c+1380 A d x+340 B \sin (3 (c+d x))+90 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+1560 B c+1560 B d x+240 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+1800 C d x)}{960 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 364, normalized size = 1.6 \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}^{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 ) }+3\,B{a}^{3} \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}^{3}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,A{a}^{3} \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 ) +B{a}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,{a}^{3}C \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{\frac{A{a}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+B{a}^{3} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{a}^{3}C\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 = 0.971601, size = 478, normalized size = 2.03 \begin{align*} \frac{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^{3} - 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} - 320 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 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^{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} - 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 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^{3} + 240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C 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} + 720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 960 \, C a^{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 = 0.584421, size = 378, normalized size = 1.61 \begin{align*} \frac{15 \,{\left (23 \, A + 26 \, B + 30 \, C\right )} a^{3} d x +{\left (40 \, A a^{3} \cos \left (d x + c\right )^{5} + 48 \,{\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{4} + 10 \,{\left (23 \, A + 18 \, B + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \,{\left (17 \, A + 19 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \,{\left (23 \, A + 26 \, B + 30 \, C\right )} a^{3} \cos \left (d x + c\right ) + 16 \,{\left (34 \, A + 38 \, B + 45 \, C\right )} a^{3}\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 = 1.32686, size = 473, normalized size = 2.01 \begin{align*} \frac{15 \,{\left (23 \, A a^{3} + 26 \, B a^{3} + 30 \, C a^{3}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (345 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 390 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 450 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 1955 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 2210 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 2550 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 4554 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 5148 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 5940 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 5814 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 5988 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 7500 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3165 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4190 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5130 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1575 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1530 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1470 \, C a^{3} \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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