Optimal. Leaf size=154 \[ -\frac{a^3 (15 B+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (15 B+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (15 B+13 C)+\frac{(5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d} \]
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Rubi [A] time = 0.194015, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {3023, 2751, 2645, 2637, 2635, 8, 2633} \[ -\frac{a^3 (15 B+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (15 B+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (15 B+13 C)+\frac{(5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d} \]
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 (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 (4 a 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} (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} (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 (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 (15 B+13 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{20} \left (3 a^3 (15 B+13 C)\right ) \int \cos (c+d x) \, dx+\frac{1}{20} \left (3 a^3 (15 B+13 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{1}{20} a^3 (15 B+13 C) x+\frac{3 a^3 (15 B+13 C) \sin (c+d x)}{20 d}+\frac{3 a^3 (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 (15 B+13 C)\right ) \int 1 \, dx-\frac{\left (a^3 (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 (15 B+13 C) x+\frac{a^3 (15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (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 (15 B+13 C) \sin ^3(c+d x)}{60 d}\\ \end{align*}
Mathematica [A] time = 0.431695, size = 108, normalized size = 0.7 \[ \frac{a^3 (60 (26 B+23 C) \sin (c+d x)+480 (B+C) \sin (2 (c+d x))+120 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+900 B d x+170 C \sin (3 (c+d x))+45 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+780 c C+780 C d x)}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 223, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({a}^{3}C \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{a}^{3}B\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}^{3}B \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +3\,{a}^{3}C \left ( 1/4\, \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}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +{\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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20633, size = 288, normalized size = 1.87 \begin{align*} -\frac{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} - 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 = 1.62282, size = 278, normalized size = 1.81 \begin{align*} \frac{15 \,{\left (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 (15 \, B + 19 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \,{\left (15 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \,{\left (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 = 3.37347, size = 532, normalized size = 3.45 \begin{align*} \begin{cases} \frac{3 B a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 B a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 B a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{3 B a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 B a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{3 B a^{3} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{2 B a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac{5 B a^{3} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{3 B a^{3} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 B a^{3} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{B a^{3} \sin{\left (c + d x \right )}}{d} + \frac{9 C a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{9 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{C a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{9 C a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{C a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{8 C a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{9 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{2 C a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac{C a^{3} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{15 C a^{3} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{3 C a^{3} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{C a^{3} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (B \cos{\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos{\left (c \right )} + a\right )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.55717, size = 184, normalized size = 1.19 \begin{align*} \frac{C a^{3} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{1}{8} \,{\left (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 (12 \, B a^{3} + 17 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (B a^{3} + C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{d} + \frac{{\left (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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