Optimal. Leaf size=160 \[ -\frac{a^2 (10 B+9 C) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 B+9 C) \sin (c+d x)}{5 d}+\frac{a^2 (5 B+6 C) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (7 B+6 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (7 B+6 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{5 d} \]
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Rubi [A] time = 0.338862, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3029, 2976, 2968, 3023, 2748, 2635, 8, 2633} \[ -\frac{a^2 (10 B+9 C) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 B+9 C) \sin (c+d x)}{5 d}+\frac{a^2 (5 B+6 C) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (7 B+6 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (7 B+6 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{5 d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2976
Rule 2968
Rule 3023
Rule 2748
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 (B+C \cos (c+d x)) \, dx\\ &=\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{5} \int \cos ^2(c+d x) (a+a \cos (c+d x)) (a (5 B+3 C)+a (5 B+6 C) \cos (c+d x)) \, dx\\ &=\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{5} \int \cos ^2(c+d x) \left (a^2 (5 B+3 C)+\left (a^2 (5 B+3 C)+a^2 (5 B+6 C)\right ) \cos (c+d x)+a^2 (5 B+6 C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 (5 B+6 C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{20} \int \cos ^2(c+d x) \left (5 a^2 (7 B+6 C)+4 a^2 (10 B+9 C) \cos (c+d x)\right ) \, dx\\ &=\frac{a^2 (5 B+6 C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{4} \left (a^2 (7 B+6 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{5} \left (a^2 (10 B+9 C)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac{a^2 (7 B+6 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^2 (5 B+6 C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{8} \left (a^2 (7 B+6 C)\right ) \int 1 \, dx-\frac{\left (a^2 (10 B+9 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac{1}{8} a^2 (7 B+6 C) x+\frac{a^2 (10 B+9 C) \sin (c+d x)}{5 d}+\frac{a^2 (7 B+6 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^2 (5 B+6 C) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}-\frac{a^2 (10 B+9 C) \sin ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.324909, size = 104, normalized size = 0.65 \[ \frac{a^2 (60 (12 B+11 C) \sin (c+d x)+240 (B+C) \sin (2 (c+d x))+80 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+420 B d x+90 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+360 C d x)}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 186, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{2}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{a}^{2}B \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +2\,{a}^{2}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{2\,{a}^{2}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{\frac{{a}^{2}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}^{2}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 = 1.08221, size = 240, normalized size = 1.5 \begin{align*} -\frac{320 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} - 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^{2} - 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 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^{2} + 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} - 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^{2}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65918, size = 271, normalized size = 1.69 \begin{align*} \frac{15 \,{\left (7 \, B + 6 \, C\right )} a^{2} d x +{\left (24 \, C a^{2} \cos \left (d x + c\right )^{4} + 30 \,{\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \,{\left (10 \, B + 9 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \,{\left (7 \, B + 6 \, C\right )} a^{2} \cos \left (d x + c\right ) + 16 \,{\left (10 \, B + 9 \, C\right )} a^{2}\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 = 3.35701, size = 462, normalized size = 2.89 \begin{align*} \begin{cases} \frac{3 B a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{B a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{3 B a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{B a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{3 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{4 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{5 B a^{2} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{2 B a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{B a^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{3 C a^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac{3 C a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac{3 C a^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac{8 C a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{3 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{4 d} + \frac{2 C a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{C a^{2} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{5 C a^{2} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac{C a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{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 )^{2} \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.64717, size = 185, normalized size = 1.16 \begin{align*} \frac{C a^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{1}{8} \,{\left (7 \, B a^{2} + 6 \, C a^{2}\right )} x + \frac{{\left (B a^{2} + 2 \, C a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{{\left (8 \, B a^{2} + 9 \, C a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (B a^{2} + C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac{{\left (12 \, B a^{2} + 11 \, C a^{2}\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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