Optimal. Leaf size=45 \[ \frac{(a \sin (c+d x)+a)^6}{6 a^2 d}-\frac{(a \sin (c+d x)+a)^5}{5 a d} \]
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Rubi [A] time = 0.0448015, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 43} \[ \frac{(a \sin (c+d x)+a)^6}{6 a^2 d}-\frac{(a \sin (c+d x)+a)^5}{5 a d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x (a+x)^4}{a} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int x (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a (a+x)^4+(a+x)^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=-\frac{(a+a \sin (c+d x))^5}{5 a d}+\frac{(a+a \sin (c+d x))^6}{6 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.0946878, size = 30, normalized size = 0.67 \[ \frac{a^4 (\sin (c+d x)+1)^5 (5 \sin (c+d x)-1)}{30 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 71, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6}}+{\frac{4\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{3\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{2}}+{\frac{4\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09017, size = 96, normalized size = 2.13 \begin{align*} \frac{5 \, a^{4} \sin \left (d x + c\right )^{6} + 24 \, a^{4} \sin \left (d x + c\right )^{5} + 45 \, a^{4} \sin \left (d x + c\right )^{4} + 40 \, a^{4} \sin \left (d x + c\right )^{3} + 15 \, a^{4} \sin \left (d x + c\right )^{2}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.36852, size = 207, normalized size = 4.6 \begin{align*} -\frac{5 \, a^{4} \cos \left (d x + c\right )^{6} - 60 \, a^{4} \cos \left (d x + c\right )^{4} + 120 \, a^{4} \cos \left (d x + c\right )^{2} - 8 \,{\left (3 \, a^{4} \cos \left (d x + c\right )^{4} - 11 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4}\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 = 5.22655, size = 121, normalized size = 2.69 \begin{align*} \begin{cases} \frac{a^{4} \sin ^{6}{\left (c + d x \right )}}{6 d} + \frac{4 a^{4} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac{4 a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{3 a^{4} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac{3 a^{4} \cos ^{4}{\left (c + d x \right )}}{2 d} - \frac{a^{4} \cos ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{4} \sin{\left (c \right )} \cos{\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 = 1.25758, size = 96, normalized size = 2.13 \begin{align*} \frac{5 \, a^{4} \sin \left (d x + c\right )^{6} + 24 \, a^{4} \sin \left (d x + c\right )^{5} + 45 \, a^{4} \sin \left (d x + c\right )^{4} + 40 \, a^{4} \sin \left (d x + c\right )^{3} + 15 \, a^{4} \sin \left (d x + c\right )^{2}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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