Optimal. Leaf size=123 \[ \frac{a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (12 A+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (12 A+7 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d} \]
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Rubi [A] time = 0.139384, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3024, 2751, 2644} \[ \frac{a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (12 A+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (12 A+7 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d} \]
Antiderivative was successfully verified.
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Rule 3024
Rule 2751
Rule 2644
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 \left (A+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 (a (4 A+3 C)-a C \cos (c+d x)) \, dx}{4 a}\\ &=-\frac{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} (12 A+7 C) \int (a+a \cos (c+d x))^2 \, dx\\ &=\frac{1}{8} a^2 (12 A+7 C) x+\frac{a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (12 A+7 C) \cos (c+d x) \sin (c+d x)}{24 d}-\frac{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 = 0.24651, size = 73, normalized size = 0.59 \[ \frac{a^2 (48 (4 A+3 C) \sin (c+d x)+24 (A+2 C) \sin (2 (c+d x))+144 A d x+16 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+84 C d x)}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 142, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({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{2\,{a}^{2}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+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{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +2\,A{a}^{2}\sin \left ( dx+c \right ) +A{a}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12641, size = 178, normalized size = 1.45 \begin{align*} \frac{24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 96 \,{\left (d x + c\right )} A 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} + 192 \, A 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 = 1.43682, size = 207, normalized size = 1.68 \begin{align*} \frac{3 \,{\left (12 \, A + 7 \, C\right )} a^{2} d x +{\left (6 \, C a^{2} \cos \left (d x + c\right )^{3} + 16 \, C a^{2} \cos \left (d x + c\right )^{2} + 3 \,{\left (4 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right ) + 16 \,{\left (3 \, A + 2 \, 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 = 1.55956, size = 309, normalized size = 2.51 \begin{align*} \begin{cases} \frac{A a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{A a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + A a^{2} x + \frac{A a^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{2 A 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 (A + 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 = 1.15564, size = 139, normalized size = 1.13 \begin{align*} \frac{C a^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{C a^{2} \sin \left (3 \, d x + 3 \, c\right )}{6 \, d} + \frac{1}{8} \,{\left (12 \, A a^{2} + 7 \, C a^{2}\right )} x + \frac{{\left (A a^{2} + 2 \, C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (4 \, A a^{2} + 3 \, C a^{2}\right )} \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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