Optimal. Leaf size=179 \[ -\frac{a^3 (13 A+15 B+20 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (13 A+15 B+20 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (13 A+15 B+20 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (13 A+15 B+20 C)+\frac{(3 A+5 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^3}{5 d} \]
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Rubi [A] time = 0.365264, antiderivative size = 179, 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 = {4086, 4013, 3791, 2637, 2635, 8, 2633} \[ -\frac{a^3 (13 A+15 B+20 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (13 A+15 B+20 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (13 A+15 B+20 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (13 A+15 B+20 C)+\frac{(3 A+5 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^3}{5 d} \]
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 ^5(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 ^4(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac{\int \cos ^4(c+d x) (a+a \sec (c+d x))^3 (a (3 A+5 B)+a (A+5 C) \sec (c+d x)) \, dx}{5 a}\\ &=\frac{(3 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac{1}{20} (13 A+15 B+20 C) \int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \, dx\\ &=\frac{(3 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac{1}{20} (13 A+15 B+20 C) \int \left (a^3+3 a^3 \cos (c+d x)+3 a^3 \cos ^2(c+d x)+a^3 \cos ^3(c+d x)\right ) \, dx\\ &=\frac{1}{20} a^3 (13 A+15 B+20 C) x+\frac{(3 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac{1}{20} \left (a^3 (13 A+15 B+20 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{20} \left (3 a^3 (13 A+15 B+20 C)\right ) \int \cos (c+d x) \, dx+\frac{1}{20} \left (3 a^3 (13 A+15 B+20 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{1}{20} a^3 (13 A+15 B+20 C) x+\frac{3 a^3 (13 A+15 B+20 C) \sin (c+d x)}{20 d}+\frac{3 a^3 (13 A+15 B+20 C) \cos (c+d x) \sin (c+d x)}{40 d}+\frac{(3 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}+\frac{1}{40} \left (3 a^3 (13 A+15 B+20 C)\right ) \int 1 \, dx-\frac{\left (a^3 (13 A+15 B+20 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{20 d}\\ &=\frac{1}{8} a^3 (13 A+15 B+20 C) x+\frac{a^3 (13 A+15 B+20 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (13 A+15 B+20 C) \cos (c+d x) \sin (c+d x)}{40 d}+\frac{(3 A+5 B) \cos ^3(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac{A \cos ^4(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{5 d}-\frac{a^3 (13 A+15 B+20 C) \sin ^3(c+d x)}{60 d}\\ \end{align*}
Mathematica [A] time = 0.430481, size = 130, normalized size = 0.73 \[ \frac{a^3 (60 (23 A+26 B+30 C) \sin (c+d x)+120 (4 A+4 B+3 C) \sin (2 (c+d x))+170 A \sin (3 (c+d x))+45 A \sin (4 (c+d x))+6 A \sin (5 (c+d x))+780 A d x+120 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+900 B d x+40 C \sin (3 (c+d x))+1200 C d x)}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.109, size = 295, normalized size = 1.7 \begin{align*}{\frac{1}{d} \left ({\frac{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\,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 ({\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 ) +A{a}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +B{a}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +{\frac{{a}^{3}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+A{a}^{3} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +3\,B{a}^{3} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +3\,{a}^{3}C \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +B{a}^{3}\sin \left ( dx+c \right ) +3\,{a}^{3}C\sin \left ( dx+c \right ) +{a}^{3}C \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961152, size = 381, normalized size = 2.13 \begin{align*} \frac{32 \,{\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} - 480 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 45 \,{\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} + 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 480 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 15 \,{\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} + 360 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 360 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 480 \,{\left (d x + c\right )} C a^{3} + 480 \, B a^{3} \sin \left (d x + c\right ) + 1440 \, C a^{3} \sin \left (d x + c\right )}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.529727, size = 315, normalized size = 1.76 \begin{align*} \frac{15 \,{\left (13 \, A + 15 \, B + 20 \, C\right )} a^{3} d x +{\left (24 \, A a^{3} \cos \left (d x + c\right )^{4} + 30 \,{\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \,{\left (19 \, A + 15 \, B + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \,{\left (13 \, A + 15 \, B + 12 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \,{\left (38 \, A + 45 \, B + 55 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{120 \, 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 = 1.29428, size = 404, normalized size = 2.26 \begin{align*} \frac{15 \,{\left (13 \, A a^{3} + 15 \, B a^{3} + 20 \, C a^{3}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (195 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 225 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 300 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 910 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1050 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1400 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1664 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1920 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2560 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1330 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1830 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2120 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 765 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 735 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 660 \, 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 )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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