Optimal. Leaf size=166 \[ -\frac{a^3 (20 A+15 B+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (20 A+15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+15 B+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (20 A+15 B+13 C)+\frac{(5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d} \]
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Rubi [A] time = 0.226378, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {3023, 2751, 2645, 2637, 2635, 8, 2633} \[ -\frac{a^3 (20 A+15 B+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (20 A+15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+15 B+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (20 A+15 B+13 C)+\frac{(5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d} \]
Antiderivative was successfully verified.
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Rule 3023
Rule 2751
Rule 2645
Rule 2637
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac{\int (a+a \cos (c+d x))^3 (a (5 A+4 C)+a (5 B-C) \cos (c+d x)) \, dx}{5 a}\\ &=\frac{(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac{1}{20} (20 A+15 B+13 C) \int (a+a \cos (c+d x))^3 \, dx\\ &=\frac{(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac{1}{20} (20 A+15 B+13 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 (20 A+15 B+13 C) x+\frac{(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac{1}{20} \left (a^3 (20 A+15 B+13 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{20} \left (3 a^3 (20 A+15 B+13 C)\right ) \int \cos (c+d x) \, dx+\frac{1}{20} \left (3 a^3 (20 A+15 B+13 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{1}{20} a^3 (20 A+15 B+13 C) x+\frac{3 a^3 (20 A+15 B+13 C) \sin (c+d x)}{20 d}+\frac{3 a^3 (20 A+15 B+13 C) \cos (c+d x) \sin (c+d x)}{40 d}+\frac{(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}+\frac{1}{40} \left (3 a^3 (20 A+15 B+13 C)\right ) \int 1 \, dx-\frac{\left (a^3 (20 A+15 B+13 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 (20 A+15 B+13 C) x+\frac{a^3 (20 A+15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+15 B+13 C) \cos (c+d x) \sin (c+d x)}{40 d}+\frac{(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}-\frac{a^3 (20 A+15 B+13 C) \sin ^3(c+d x)}{60 d}\\ \end{align*}
Mathematica [A] time = 0.481844, size = 129, normalized size = 0.78 \[ \frac{a^3 (60 (30 A+26 B+23 C) \sin (c+d x)+120 (3 A+4 (B+C)) \sin (2 (c+d x))+40 A \sin (3 (c+d x))+1200 A d x+120 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+900 B d x+170 C \sin (3 (c+d x))+45 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+780 C d x)}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 295, normalized size = 1.8 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{3}C\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}B \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 ) +3\,{a}^{3}C \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 ) +{\frac{A{a}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{a}^{3}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+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 ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +3\,{a}^{3}B \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{a}^{3}C \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +3\,A{a}^{3}\sin \left ( dx+c \right ) +{a}^{3}B\sin \left ( dx+c \right ) +A{a}^{3} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00335, size = 381, normalized size = 2.3 \begin{align*} -\frac{160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 360 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 480 \,{\left (d x + c\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} - 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 )} C a^{3} + 480 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C 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 )} C a^{3} - 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 1440 \, A a^{3} \sin \left (d x + c\right ) - 480 \, B 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 = 1.92371, size = 315, normalized size = 1.9 \begin{align*} \frac{15 \,{\left (20 \, A + 15 \, B + 13 \, C\right )} a^{3} d x +{\left (24 \, C a^{3} \cos \left (d x + c\right )^{4} + 30 \,{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \,{\left (5 \, A + 15 \, B + 19 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \,{\left (12 \, A + 15 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \,{\left (55 \, A + 45 \, B + 38 \, 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 [A] time = 3.46027, size = 658, normalized size = 3.96 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14808, size = 220, normalized size = 1.33 \begin{align*} \frac{C a^{3} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{1}{8} \,{\left (20 \, A a^{3} + 15 \, B a^{3} + 13 \, C a^{3}\right )} x + \frac{{\left (B a^{3} + 3 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{{\left (4 \, A a^{3} + 12 \, B a^{3} + 17 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (3 \, A a^{3} + 4 \, B a^{3} + 4 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (30 \, A a^{3} + 26 \, B a^{3} + 23 \, C a^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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