Optimal. Leaf size=55 \[ \frac{\sin ^6(c+d x)}{6 a^2 d}-\frac{2 \sin ^5(c+d x)}{5 a^2 d}+\frac{\sin ^4(c+d x)}{4 a^2 d} \]
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Rubi [A] time = 0.106426, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 43} \[ \frac{\sin ^6(c+d x)}{6 a^2 d}-\frac{2 \sin ^5(c+d x)}{5 a^2 d}+\frac{\sin ^4(c+d x)}{4 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 x^3}{a^3} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int (a-x)^2 x^3 \, dx,x,a \sin (c+d x)\right )}{a^8 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 x^3-2 a x^4+x^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^8 d}\\ &=\frac{\sin ^4(c+d x)}{4 a^2 d}-\frac{2 \sin ^5(c+d x)}{5 a^2 d}+\frac{\sin ^6(c+d x)}{6 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.528169, size = 38, normalized size = 0.69 \[ \frac{\sin ^4(c+d x) \left (10 \sin ^2(c+d x)-24 \sin (c+d x)+15\right )}{60 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.083, size = 39, normalized size = 0.7 \begin{align*}{\frac{1}{d{a}^{2}} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6}}-{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20915, size = 53, normalized size = 0.96 \begin{align*} \frac{10 \, \sin \left (d x + c\right )^{6} - 24 \, \sin \left (d x + c\right )^{5} + 15 \, \sin \left (d x + c\right )^{4}}{60 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06009, size = 180, normalized size = 3.27 \begin{align*} -\frac{10 \, \cos \left (d x + c\right )^{6} - 45 \, \cos \left (d x + c\right )^{4} + 60 \, \cos \left (d x + c\right )^{2} + 24 \,{\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right )}{60 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29208, size = 53, normalized size = 0.96 \begin{align*} \frac{10 \, \sin \left (d x + c\right )^{6} - 24 \, \sin \left (d x + c\right )^{5} + 15 \, \sin \left (d x + c\right )^{4}}{60 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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