Optimal. Leaf size=54 \[ \frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d} \]
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Rubi [A] time = 0.047293, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2748, 2635, 8, 2633} \[ \frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+b \cos (c+d x)) \, dx &=a \int \cos ^2(c+d x) \, dx+b \int \cos ^3(c+d x) \, dx\\ &=\frac{a \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} a \int 1 \, dx-\frac{b \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{a x}{2}+\frac{b \sin (c+d x)}{d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}-\frac{b \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.064984, size = 57, normalized size = 1.06 \[ \frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 49, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{b \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+a \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 = 0.945717, size = 62, normalized size = 1.15 \begin{align*} \frac{3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a - 4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} b}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90571, size = 105, normalized size = 1.94 \begin{align*} \frac{3 \, a d x +{\left (2 \, b \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) + 4 \, b\right )} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.601227, size = 92, normalized size = 1.7 \begin{align*} \begin{cases} \frac{a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{2 b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{b \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 ) \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 = 1.30865, size = 63, normalized size = 1.17 \begin{align*} \frac{1}{2} \, a x + \frac{b \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{3 \, b \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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