Optimal. Leaf size=65 \[ -\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a x}{8} \]
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Rubi [A] time = 0.0929874, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2838, 2565, 30, 2568, 2635, 8} \[ -\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a x}{8} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2565
Rule 30
Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^2(c+d x) \sin (c+d x) \, dx+a \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx\\ &=-\frac{a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{4} a \int \cos ^2(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x) \sin (c+d x)}{8 d}-\frac{a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{8} a \int 1 \, dx\\ &=\frac{a x}{8}-\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x) \sin (c+d x)}{8 d}-\frac{a \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0978183, size = 42, normalized size = 0.65 \[ -\frac{a (3 (\sin (4 (c+d x))-4 d x)+24 \cos (c+d x)+8 \cos (3 (c+d x)))}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 57, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{4}}+{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8}}+{\frac{dx}{8}}+{\frac{c}{8}} \right ) -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24129, size = 53, normalized size = 0.82 \begin{align*} -\frac{32 \, a \cos \left (d x + c\right )^{3} - 3 \,{\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70671, size = 128, normalized size = 1.97 \begin{align*} -\frac{8 \, a \cos \left (d x + c\right )^{3} - 3 \, a d x + 3 \,{\left (2 \, a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )\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 = 1.32814, size = 119, normalized size = 1.83 \begin{align*} \begin{cases} \frac{a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{a \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} - \frac{a \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac{a \cos ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \sin{\left (c \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 = 1.27498, size = 63, normalized size = 0.97 \begin{align*} \frac{1}{8} \, a x - \frac{a \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac{a \cos \left (d x + c\right )}{4 \, d} - \frac{a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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