Optimal. Leaf size=97 \[ -\frac{a (A+B) \sin ^3(c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x)}{d}+\frac{a (4 A+3 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.149015, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2968, 3023, 2748, 2635, 8, 2633} \[ -\frac{a (A+B) \sin ^3(c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x)}{d}+\frac{a (4 A+3 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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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)) (A+B \cos (c+d x)) \, dx &=\int \cos ^2(c+d x) \left (a A+(a A+a B) \cos (c+d x)+a B \cos ^2(c+d x)\right ) \, dx\\ &=\frac{a B \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{4} \int \cos ^2(c+d x) (a (4 A+3 B)+4 a (A+B) \cos (c+d x)) \, dx\\ &=\frac{a B \cos ^3(c+d x) \sin (c+d x)}{4 d}+(a (A+B)) \int \cos ^3(c+d x) \, dx+\frac{1}{4} (a (4 A+3 B)) \int \cos ^2(c+d x) \, dx\\ &=\frac{a (4 A+3 B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a B \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{8} (a (4 A+3 B)) \int 1 \, dx-\frac{(a (A+B)) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{1}{8} a (4 A+3 B) x+\frac{a (A+B) \sin (c+d x)}{d}+\frac{a (4 A+3 B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a B \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{a (A+B) \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.223432, size = 75, normalized size = 0.77 \[ \frac{a \left (-32 (A+B) \sin ^3(c+d x)+96 (A+B) \sin (c+d x)+24 (A+B) \sin (2 (c+d x))+48 A c+48 A d x+3 B \sin (4 (c+d x))+36 B c+36 B d x\right )}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 107, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( aB \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{aA \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{\frac{aB \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00651, size = 136, normalized size = 1.4 \begin{align*} -\frac{32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 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 - 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 )} B a}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43322, size = 193, normalized size = 1.99 \begin{align*} \frac{3 \,{\left (4 \, A + 3 \, B\right )} a d x +{\left (6 \, B a \cos \left (d x + c\right )^{3} + 8 \,{\left (A + B\right )} a \cos \left (d x + c\right )^{2} + 3 \,{\left (4 \, A + 3 \, B\right )} a \cos \left (d x + c\right ) + 16 \,{\left (A + B\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 = 1.80525, size = 252, normalized size = 2.6 \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{2 A a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{A a \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{A a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{3 B a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 B a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 B a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 B a \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{2 B a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{5 B a \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{B a \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (A + B \cos{\left (c \right )}\right ) \left (a \cos{\left (c \right )} + a\right ) \cos ^{2}{\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 = 1.14739, size = 120, normalized size = 1.24 \begin{align*} \frac{1}{8} \,{\left (4 \, A a + 3 \, B a\right )} x + \frac{B a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{{\left (A a + B a\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{{\left (A a + B a\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{3 \,{\left (A a + B 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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