Optimal. Leaf size=265 \[ -\frac{a^3 (133 A+119 B+108 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (133 A+119 B+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{15 d}+\frac{a^3 (26 A+23 B+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+23 B+21 C)+\frac{(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{42 a d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]
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Rubi [A] time = 0.678233, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195, Rules used = {3045, 2976, 2968, 3023, 2748, 2635, 8, 2633} \[ -\frac{a^3 (133 A+119 B+108 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (133 A+119 B+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{15 d}+\frac{a^3 (26 A+23 B+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+23 B+21 C)+\frac{(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{42 a d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]
Antiderivative was successfully verified.
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Rule 3045
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+B \cos (c+d x)+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)+a (7 B+3 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{(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (3 a^2 (14 A+7 B+9 C)+14 a^2 (3 A+4 B+3 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{(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \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^3 (112 A+91 B+87 C)+3 a^3 (154 A+147 B+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{(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{\int \cos ^2(c+d x) \left (3 a^4 (112 A+91 B+87 C)+\left (3 a^4 (112 A+91 B+87 C)+3 a^4 (154 A+147 B+129 C)\right ) \cos (c+d x)+3 a^4 (154 A+147 B+129 C) \cos ^2(c+d x)\right ) \, dx}{210 a}\\ &=\frac{a^3 (154 A+147 B+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{(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{\int \cos ^2(c+d x) \left (105 a^4 (26 A+23 B+21 C)+24 a^4 (133 A+119 B+108 C) \cos (c+d x)\right ) \, dx}{840 a}\\ &=\frac{a^3 (154 A+147 B+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{(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{8} \left (a^3 (26 A+23 B+21 C)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{35} \left (a^3 (133 A+119 B+108 C)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac{a^3 (26 A+23 B+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^3 (154 A+147 B+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{(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{16} \left (a^3 (26 A+23 B+21 C)\right ) \int 1 \, dx-\frac{\left (a^3 (133 A+119 B+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+23 B+21 C) x+\frac{a^3 (133 A+119 B+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (26 A+23 B+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^3 (154 A+147 B+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{(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac{(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac{a^3 (133 A+119 B+108 C) \sin ^3(c+d x)}{105 d}\\ \end{align*}
Mathematica [A] time = 1.06557, size = 204, normalized size = 0.77 \[ \frac{a^3 (105 (184 A+168 B+155 C) \sin (c+d x)+105 (64 A+63 B+61 C) \sin (2 (c+d x))+2380 A \sin (3 (c+d x))+630 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+10920 A d x+2660 B \sin (3 (c+d x))+945 B \sin (4 (c+d x))+252 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+9660 B c+9660 B d x+2835 C \sin (3 (c+d x))+1155 C \sin (4 (c+d x))+399 C \sin (5 (c+d x))+105 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+5460 c C+8820 C d x)}{6720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 427, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01304, size = 574, normalized size = 2.17 \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} + 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 )} B a^{3} - 35 \,{\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 )} B a^{3} - 2240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B 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 )} B 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 = 2.14476, size = 454, normalized size = 1.71 \begin{align*} \frac{105 \,{\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} d x +{\left (240 \, C a^{3} \cos \left (d x + c\right )^{6} + 280 \,{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 48 \,{\left (7 \, A + 21 \, B + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 70 \,{\left (18 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \,{\left (133 \, A + 119 \, B + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 105 \,{\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right ) + 32 \,{\left (133 \, A + 119 \, B + 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 = 11.1243, size = 1149, normalized size = 4.34 \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 = 1.24902, size = 309, normalized size = 1.17 \begin{align*} \frac{C a^{3} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{1}{16} \,{\left (26 \, A a^{3} + 23 \, B a^{3} + 21 \, C a^{3}\right )} x + \frac{{\left (B a^{3} + 3 \, C a^{3}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{{\left (4 \, A a^{3} + 12 \, B a^{3} + 19 \, C a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (6 \, A a^{3} + 9 \, B a^{3} + 11 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{{\left (68 \, A a^{3} + 76 \, B a^{3} + 81 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{{\left (64 \, A a^{3} + 63 \, B a^{3} + 61 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (184 \, A a^{3} + 168 \, B 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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