Optimal. Leaf size=129 \[ \frac{a^2 (8 B+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 B+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 B+7 C)+\frac{(4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d} \]
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Rubi [A] time = 0.140573, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {3023, 2751, 2644} \[ \frac{a^2 (8 B+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 B+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 B+7 C)+\frac{(4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d} \]
Antiderivative was successfully verified.
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Rule 3023
Rule 2751
Rule 2644
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d}+\frac{\int (a+a \cos (c+d x))^2 (3 a C+a (4 B-C) \cos (c+d x)) \, dx}{4 a}\\ &=\frac{(4 B-C) (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d}+\frac{1}{12} (8 B+7 C) \int (a+a \cos (c+d x))^2 \, dx\\ &=\frac{1}{8} a^2 (8 B+7 C) x+\frac{a^2 (8 B+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 B+7 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac{(4 B-C) (a+a \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{C (a+a \cos (c+d x))^3 \sin (c+d x)}{4 a d}\\ \end{align*}
Mathematica [A] time = 0.330804, size = 86, normalized size = 0.67 \[ \frac{a^2 (24 (7 B+6 C) \sin (c+d x)+48 (B+C) \sin (2 (c+d x))+8 B \sin (3 (c+d x))+96 B d x+16 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+84 c C+84 C d x)}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 154, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{2}C \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{a}^{2}B\sin \left ( dx+c \right ) +{\frac{2\,{a}^{2}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,{a}^{2}B \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+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 ) +{\frac{{a}^{2}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13956, size = 194, normalized size = 1.5 \begin{align*} -\frac{32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} - 48 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 64 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} - 3 \,{\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} - 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 96 \, B a^{2} \sin \left (d x + c\right )}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65236, size = 213, normalized size = 1.65 \begin{align*} \frac{3 \,{\left (8 \, B + 7 \, C\right )} a^{2} d x +{\left (6 \, C a^{2} \cos \left (d x + c\right )^{3} + 8 \,{\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \,{\left (8 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \,{\left (5 \, B + 4 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.51834, size = 340, normalized size = 2.64 \begin{align*} \begin{cases} B a^{2} x \sin ^{2}{\left (c + d x \right )} + B a^{2} x \cos ^{2}{\left (c + d x \right )} + \frac{2 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{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 )}}{d} + \frac{B a^{2} \sin{\left (c + d x \right )}}{d} + \frac{3 C a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 C a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{C a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{3 C a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{C a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{3 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{4 C a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{5 C a^{2} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{2 C a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{C a^{2} \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 )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.84607, size = 149, normalized size = 1.16 \begin{align*} \frac{C a^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{1}{8} \,{\left (8 \, B a^{2} + 7 \, C a^{2}\right )} x + \frac{{\left (B a^{2} + 2 \, C a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{{\left (B a^{2} + C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac{{\left (7 \, B a^{2} + 6 \, 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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