Optimal. Leaf size=118 \[ \frac{a (3 A+2 (B+C)) \sin (c+d x)}{3 d}+\frac{a (4 A+4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+4 B+3 C)+\frac{a (B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.138148, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used = {3033, 3023, 2734} \[ \frac{a (3 A+2 (B+C)) \sin (c+d x)}{3 d}+\frac{a (4 A+4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+4 B+3 C)+\frac{a (B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3033
Rule 3023
Rule 2734
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{a C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{4} \int \cos (c+d x) \left (4 a A+a (4 A+4 B+3 C) \cos (c+d x)+4 a (B+C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{a (B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac{a C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{12} \int \cos (c+d x) (4 a (3 A+2 (B+C))+3 a (4 A+4 B+3 C) \cos (c+d x)) \, dx\\ &=\frac{1}{8} a (4 A+4 B+3 C) x+\frac{a (3 A+2 (B+C)) \sin (c+d x)}{3 d}+\frac{a (4 A+4 B+3 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a (B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac{a C \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.395352, size = 96, normalized size = 0.81 \[ \frac{a (24 (4 A+3 (B+C)) \sin (c+d x)+24 (A+B+C) \sin (2 (c+d x))+48 A d x+8 B \sin (3 (c+d x))+48 B c+48 B d x+8 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+24 c C+36 C d x)}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 141, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( aC \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{Ba \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{\frac{aC \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+aA \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +Ba \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +aA\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987671, size = 178, normalized size = 1.51 \begin{align*} \frac{24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a + 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a + 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 + 96 \, A a \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 = 1.89475, size = 221, normalized size = 1.87 \begin{align*} \frac{3 \,{\left (4 \, A + 4 \, B + 3 \, C\right )} a d x +{\left (6 \, C a \cos \left (d x + c\right )^{3} + 8 \,{\left (B + C\right )} a \cos \left (d x + c\right )^{2} + 3 \,{\left (4 \, A + 4 \, B + 3 \, C\right )} a \cos \left (d x + c\right ) + 8 \,{\left (3 \, A + 2 \, B + 2 \, C\right )} a\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 = 1.42431, size = 320, normalized size = 2.71 \begin{align*} \begin{cases} \frac{A a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{A a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{A a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{A a \sin{\left (c + d x \right )}}{d} + \frac{B a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{B a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{2 B a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{B a \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{B a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{3 C a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 C a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 C a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 C a \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{2 C a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{5 C a \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{C a \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + a\right ) \left (A + B \cos{\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \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 = 1.23747, size = 138, normalized size = 1.17 \begin{align*} \frac{1}{8} \,{\left (4 \, A a + 4 \, B a + 3 \, C a\right )} x + \frac{C a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{{\left (B a + C a\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{{\left (A a + B a + C a\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (4 \, A a + 3 \, B a + 3 \, C a\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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