Optimal. Leaf size=213 \[ -\frac{2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac{a^4 (7 A+8 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (7 A+8 B+10 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (7 A+8 B+10 C)+\frac{(2 A+3 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{15 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d} \]
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Rubi [A] time = 0.40923, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4086, 4013, 3791, 2637, 2635, 8, 2633} \[ -\frac{2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac{a^4 (7 A+8 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (7 A+8 B+10 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (7 A+8 B+10 C)+\frac{(2 A+3 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{15 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d} \]
Antiderivative was successfully verified.
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Rule 4086
Rule 4013
Rule 3791
Rule 2637
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \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))^4 \sin (c+d x)}{6 d}+\frac{\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 (2 a (2 A+3 B)+a (A+6 C) \sec (c+d x)) \, dx}{6 a}\\ &=\frac{(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac{1}{10} (7 A+8 B+10 C) \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \, dx\\ &=\frac{(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac{1}{10} (7 A+8 B+10 C) \int \left (a^4+4 a^4 \cos (c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^4(c+d x)\right ) \, dx\\ &=\frac{1}{10} a^4 (7 A+8 B+10 C) x+\frac{(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac{1}{10} \left (a^4 (7 A+8 B+10 C)\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{5} \left (2 a^4 (7 A+8 B+10 C)\right ) \int \cos (c+d x) \, dx+\frac{1}{5} \left (2 a^4 (7 A+8 B+10 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{5} \left (3 a^4 (7 A+8 B+10 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{1}{10} a^4 (7 A+8 B+10 C) x+\frac{2 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac{3 a^4 (7 A+8 B+10 C) \cos (c+d x) \sin (c+d x)}{10 d}+\frac{a^4 (7 A+8 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac{1}{40} \left (3 a^4 (7 A+8 B+10 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{10} \left (3 a^4 (7 A+8 B+10 C)\right ) \int 1 \, dx-\frac{\left (2 a^4 (7 A+8 B+10 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac{2}{5} a^4 (7 A+8 B+10 C) x+\frac{4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac{27 a^4 (7 A+8 B+10 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac{a^4 (7 A+8 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac{2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d}+\frac{1}{80} \left (3 a^4 (7 A+8 B+10 C)\right ) \int 1 \, dx\\ &=\frac{7}{16} a^4 (7 A+8 B+10 C) x+\frac{4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac{27 a^4 (7 A+8 B+10 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac{a^4 (7 A+8 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac{A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac{2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.496599, size = 163, normalized size = 0.77 \[ \frac{a^4 (120 (44 A+49 B+56 C) \sin (c+d x)+15 (127 A+128 B+112 C) \sin (2 (c+d x))+720 A \sin (3 (c+d x))+225 A \sin (4 (c+d x))+48 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+2940 A d x+580 B \sin (3 (c+d x))+120 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+3360 B d x+320 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+4200 C d x)}{960 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.13, size = 416, normalized size = 2. \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{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 ) }+{\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 ) }+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 ) +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 ) +{a}^{4}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{4\,A{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,B{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +{\frac{4\,{a}^{4}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+A{a}^{4} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +4\,B{a}^{4} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +6\,{a}^{4}C \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +B{a}^{4}\sin \left ( dx+c \right ) +4\,{a}^{4}C\sin \left ( dx+c \right ) +{a}^{4}C \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.972799, size = 540, normalized size = 2.54 \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} - 1280 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 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^{4} + 1440 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 960 \,{\left (d x + c\right )} C a^{4} + 960 \, B a^{4} \sin \left (d x + c\right ) + 3840 \, C 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 = 0.528742, size = 378, normalized size = 1.77 \begin{align*} \frac{105 \,{\left (7 \, A + 8 \, B + 10 \, C\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 + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 32 \,{\left (18 \, A + 17 \, B + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \,{\left (49 \, A + 56 \, B + 54 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \,{\left (72 \, A + 83 \, B + 100 \, C\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 = 1.3357, size = 473, normalized size = 2.22 \begin{align*} \frac{105 \,{\left (7 \, A a^{4} + 8 \, B a^{4} + 10 \, C 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} + 1050 \, C 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} + 5950 \, C 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} + 13860 \, C 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} + 16860 \, C 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} + 10690 \, C 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 ) + 2790 \, C 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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