Optimal. Leaf size=194 \[ -\frac{2 a^2 (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{2 a^2 (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a^2 (10 A+9 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^2 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (14 A+11 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{15 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d} \]
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Rubi [A] time = 0.472005, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {3046, 2976, 2968, 3023, 2748, 2635, 8, 2633} \[ -\frac{2 a^2 (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{2 a^2 (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a^2 (10 A+9 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^2 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (14 A+11 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{15 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d} \]
Antiderivative was successfully verified.
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Rule 3046
Rule 2976
Rule 2968
Rule 3023
Rule 2748
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 (3 a (2 A+C)+2 a C \cos (c+d x)) \, dx}{6 a}\\ &=\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^2 (10 A+7 C)+3 a^2 (10 A+9 C) \cos (c+d x)\right ) \, dx}{30 a}\\ &=\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{\int \cos ^2(c+d x) \left (3 a^3 (10 A+7 C)+\left (3 a^3 (10 A+7 C)+3 a^3 (10 A+9 C)\right ) \cos (c+d x)+3 a^3 (10 A+9 C) \cos ^2(c+d x)\right ) \, dx}{30 a}\\ &=\frac{a^2 (10 A+9 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{\int \cos ^2(c+d x) \left (15 a^3 (14 A+11 C)+48 a^3 (5 A+4 C) \cos (c+d x)\right ) \, dx}{120 a}\\ &=\frac{a^2 (10 A+9 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{5} \left (2 a^2 (5 A+4 C)\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{8} \left (a^2 (14 A+11 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{a^2 (14 A+11 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^2 (10 A+9 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{16} \left (a^2 (14 A+11 C)\right ) \int 1 \, dx-\frac{\left (2 a^2 (5 A+4 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac{1}{16} a^2 (14 A+11 C) x+\frac{2 a^2 (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a^2 (14 A+11 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^2 (10 A+9 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac{2 a^2 (5 A+4 C) \sin ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.482144, size = 123, normalized size = 0.63 \[ \frac{a^2 (240 (6 A+5 C) \sin (c+d x)+15 (32 A+31 C) \sin (2 (c+d x))+160 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+840 A d x+200 C \sin (3 (c+d x))+75 C \sin (4 (c+d x))+24 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+420 c C+660 C d x)}{960 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 211, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( A{a}^{2} \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}^{2}C \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{2\,A{a}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{\frac{2\,{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{a}^{2} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{a}^{2}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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13132, size = 275, normalized size = 1.42 \begin{align*} -\frac{640 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A 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 )} A a^{2} - 240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 128 \,{\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} + 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 )} 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}}{960 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44155, size = 315, normalized size = 1.62 \begin{align*} \frac{15 \,{\left (14 \, A + 11 \, C\right )} a^{2} d x +{\left (40 \, C a^{2} \cos \left (d x + c\right )^{5} + 96 \, C a^{2} \cos \left (d x + c\right )^{4} + 10 \,{\left (6 \, A + 11 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 32 \,{\left (5 \, A + 4 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \,{\left (14 \, A + 11 \, C\right )} a^{2} \cos \left (d x + c\right ) + 64 \,{\left (5 \, A + 4 \, C\right )} a^{2}\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 [A] time = 6.31262, size = 592, normalized size = 3.05 \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 = 1.16631, size = 213, normalized size = 1.1 \begin{align*} \frac{C a^{2} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{C a^{2} \sin \left (5 \, d x + 5 \, c\right )}{40 \, d} + \frac{1}{16} \,{\left (14 \, A a^{2} + 11 \, C a^{2}\right )} x + \frac{{\left (2 \, A a^{2} + 5 \, C a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (4 \, A a^{2} + 5 \, C a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{24 \, d} + \frac{{\left (32 \, A a^{2} + 31 \, C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (6 \, A a^{2} + 5 \, C a^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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