Optimal. Leaf size=64 \[ \frac{(a \sin (c+d x)+a)^6}{6 a^5 d}-\frac{4 (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac{(a \sin (c+d x)+a)^4}{a^3 d} \]
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Rubi [A] time = 0.0444546, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 43} \[ \frac{(a \sin (c+d x)+a)^6}{6 a^5 d}-\frac{4 (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac{(a \sin (c+d x)+a)^4}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^2 (a+x)^3-4 a (a+x)^4+(a+x)^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{(a+a \sin (c+d x))^4}{a^3 d}-\frac{4 (a+a \sin (c+d x))^5}{5 a^4 d}+\frac{(a+a \sin (c+d x))^6}{6 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.0226898, size = 60, normalized size = 0.94 \[ \frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{a \cos ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 46, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ( -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6}}+{\frac{a\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952699, size = 95, normalized size = 1.48 \begin{align*} \frac{5 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} - 15 \, a \sin \left (d x + c\right )^{4} - 20 \, a \sin \left (d x + c\right )^{3} + 15 \, a \sin \left (d x + c\right )^{2} + 30 \, a \sin \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62355, size = 128, normalized size = 2. \begin{align*} -\frac{5 \, a \cos \left (d x + c\right )^{6} - 2 \,{\left (3 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right )}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.31667, size = 83, normalized size = 1.3 \begin{align*} \begin{cases} \frac{8 a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{a \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac{a \cos ^{6}{\left (c + d x \right )}}{6 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \cos ^{5}{\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 = 1.18751, size = 119, normalized size = 1.86 \begin{align*} -\frac{a \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac{a \cos \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac{5 \, a \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{a \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{5 \, a \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{5 \, a \sin \left (d x + c\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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