Optimal. Leaf size=237 \[ -\frac{a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (154 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(A+C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}+\frac{a^3 (26 A+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+21 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{14 a d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]
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Rubi [A] time = 0.606788, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (154 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(A+C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{5 d}+\frac{a^3 (26 A+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+21 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{14 a d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]
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))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (a (7 A+3 C)+3 a C \cos (c+d x)) \, dx}{7 a}\\ &=\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac{\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (3 a^2 (14 A+9 C)+42 a^2 (A+C) \cos (c+d x)\right ) \, dx}{42 a}\\ &=\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac{(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^3 (112 A+87 C)+3 a^3 (154 A+129 C) \cos (c+d x)\right ) \, dx}{210 a}\\ &=\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac{(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{\int \cos ^2(c+d x) \left (3 a^4 (112 A+87 C)+\left (3 a^4 (112 A+87 C)+3 a^4 (154 A+129 C)\right ) \cos (c+d x)+3 a^4 (154 A+129 C) \cos ^2(c+d x)\right ) \, dx}{210 a}\\ &=\frac{a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac{(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{\int \cos ^2(c+d x) \left (105 a^4 (26 A+21 C)+24 a^4 (133 A+108 C) \cos (c+d x)\right ) \, dx}{840 a}\\ &=\frac{a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac{(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{8} \left (a^3 (26 A+21 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{35} \left (a^3 (133 A+108 C)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac{a^3 (26 A+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac{(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{16} \left (a^3 (26 A+21 C)\right ) \int 1 \, dx-\frac{\left (a^3 (133 A+108 C)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac{1}{16} a^3 (26 A+21 C) x+\frac{a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (26 A+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^3 (154 A+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac{C \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{14 a d}+\frac{(A+C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{5 d}-\frac{a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d}\\ \end{align*}
Mathematica [A] time = 0.673984, size = 145, normalized size = 0.61 \[ \frac{a^3 (105 (184 A+155 C) \sin (c+d x)+105 (64 A+61 C) \sin (2 (c+d x))+2380 A \sin (3 (c+d x))+630 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+10920 A d x+2835 C \sin (3 (c+d x))+1155 C \sin (4 (c+d x))+399 C \sin (5 (c+d x))+105 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+5460 c C+8820 C d x)}{6720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 286, normalized size = 1.2 \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 ) }+{\frac{{a}^{3}C\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \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 ) +3\,{a}^{3}C \left ( 1/6\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +A{a}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +{\frac{3\,{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{a}^{3} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{a}^{3}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 = 1.0631, size = 383, normalized size = 1.62 \begin{align*} \frac{448 \,{\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} - 6720 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 630 \,{\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} + 1680 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 192 \,{\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{3} + 1344 \,{\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} - 105 \,{\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^{3} + 210 \,{\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}}{6720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51285, size = 382, normalized size = 1.61 \begin{align*} \frac{105 \,{\left (26 \, A + 21 \, C\right )} a^{3} d x +{\left (240 \, C a^{3} \cos \left (d x + c\right )^{6} + 840 \, C a^{3} \cos \left (d x + c\right )^{5} + 48 \,{\left (7 \, A + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 210 \,{\left (6 \, A + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \,{\left (133 \, A + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 105 \,{\left (26 \, A + 21 \, C\right )} a^{3} \cos \left (d x + c\right ) + 32 \,{\left (133 \, A + 108 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{1680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.449, size = 750, normalized size = 3.16 \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 = 1.23773, size = 250, normalized size = 1.05 \begin{align*} \frac{C a^{3} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{C a^{3} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} + \frac{1}{16} \,{\left (26 \, A a^{3} + 21 \, C a^{3}\right )} x + \frac{{\left (4 \, A a^{3} + 19 \, C a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (6 \, A a^{3} + 11 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (68 \, A a^{3} + 81 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{{\left (64 \, A a^{3} + 61 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (184 \, A a^{3} + 155 \, C a^{3}\right )} \sin \left (d x + c\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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