Optimal. Leaf size=65 \[ -\frac{a \cos ^5(c+d x)}{5 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8} \]
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Rubi [A] time = 0.0453441, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2669, 2635, 8} \[ -\frac{a \cos ^5(c+d x)}{5 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx &=-\frac{a \cos ^5(c+d x)}{5 d}+a \int \cos ^4(c+d x) \, dx\\ &=-\frac{a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{4} (3 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac{a \cos ^5(c+d x)}{5 d}+\frac{3 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} (3 a) \int 1 \, dx\\ &=\frac{3 a x}{8}-\frac{a \cos ^5(c+d x)}{5 d}+\frac{3 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 = 0.0822159, size = 62, normalized size = 0.95 \[ \frac{3 a (c+d x)}{8 d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \sin (4 (c+d x))}{32 d}-\frac{a \cos ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 52, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5}}+a \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950086, size = 65, normalized size = 1. \begin{align*} -\frac{32 \, a \cos \left (d x + c\right )^{5} - 5 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a}{160 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64397, size = 132, normalized size = 2.03 \begin{align*} -\frac{8 \, a \cos \left (d x + c\right )^{5} - 15 \, a d x - 5 \,{\left (2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.34263, size = 124, normalized size = 1.91 \begin{align*} \begin{cases} \frac{3 a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 a \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{5 a \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac{a \cos ^{5}{\left (c + d x \right )}}{5 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \cos ^{4}{\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 = 1.19006, size = 104, normalized size = 1.6 \begin{align*} \frac{3}{8} \, a x - \frac{a \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac{a \cos \left (3 \, d x + 3 \, c\right )}{16 \, d} - \frac{a \cos \left (d x + c\right )}{8 \, d} + \frac{a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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