Optimal. Leaf size=73 \[ \frac{a^3 \sin ^6(c+d x)}{6 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{3 a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0706032, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ \frac{a^3 \sin ^6(c+d x)}{6 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{3 a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cos (c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 (a+x)^3}{a^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int x^2 (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^3 x^2+3 a^2 x^3+3 a x^4+x^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \sin ^4(c+d x)}{4 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{a^3 \sin ^6(c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 0.284775, size = 70, normalized size = 0.96 \[ -\frac{a^3 (-1200 \sin (c+d x)+520 \sin (3 (c+d x))-72 \sin (5 (c+d x))+870 \cos (2 (c+d x))-240 \cos (4 (c+d x))+10 \cos (6 (c+d x))-45)}{1920 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 58, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}}+{\frac{{a}^{3} \left ( \sin \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 = 1.12206, size = 78, normalized size = 1.07 \begin{align*} \frac{10 \, a^{3} \sin \left (d x + c\right )^{6} + 36 \, a^{3} \sin \left (d x + c\right )^{5} + 45 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64717, size = 209, normalized size = 2.86 \begin{align*} -\frac{10 \, a^{3} \cos \left (d x + c\right )^{6} - 75 \, a^{3} \cos \left (d x + c\right )^{4} + 120 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \,{\left (9 \, a^{3} \cos \left (d x + c\right )^{4} - 23 \, a^{3} \cos \left (d x + c\right )^{2} + 14 \, a^{3}\right )} \sin \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.55424, size = 107, normalized size = 1.47 \begin{align*} \begin{cases} \frac{a^{3} \sin ^{6}{\left (c + d x \right )}}{6 d} + \frac{3 a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac{a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{3 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} - \frac{3 a^{3} \cos ^{4}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{3} \sin ^{2}{\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 = 1.20233, size = 78, normalized size = 1.07 \begin{align*} \frac{10 \, a^{3} \sin \left (d x + c\right )^{6} + 36 \, a^{3} \sin \left (d x + c\right )^{5} + 45 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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