Optimal. Leaf size=30 \[ \frac{\sin ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.065052, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 37} \[ \frac{\sin ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 37
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \sin ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{a^2 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{(a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\sin ^3(c+d x)}{3 a d (a+a \sin (c+d x))^3}\\ \end{align*}
Mathematica [A] time = 0.183911, size = 53, normalized size = 1.77 \[ \frac{-6 \sin (c+d x)+3 \cos (2 (c+d x))-5}{6 a^4 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 43, normalized size = 1.4 \begin{align*}{\frac{1}{d{a}^{4}} \left ( \left ( 1+\sin \left ( dx+c \right ) \right ) ^{-2}-{\frac{1}{3\, \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}- \left ( 1+\sin \left ( dx+c \right ) \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08461, size = 90, normalized size = 3. \begin{align*} -\frac{3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) + 1}{3 \,{\left (a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.36662, size = 174, normalized size = 5.8 \begin{align*} -\frac{3 \, \cos \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) - 4}{3 \,{\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d +{\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.98311, size = 76, normalized size = 2.53 \begin{align*} \begin{cases} \frac{\sin ^{3}{\left (c + d x \right )}}{3 a^{4} d \sin ^{3}{\left (c + d x \right )} + 9 a^{4} d \sin ^{2}{\left (c + d x \right )} + 9 a^{4} d \sin{\left (c + d x \right )} + 3 a^{4} d} & \text{for}\: d \neq 0 \\\frac{x \sin ^{2}{\left (c \right )} \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18622, size = 51, normalized size = 1.7 \begin{align*} -\frac{3 \, \sin \left (d x + c\right )^{2} + 3 \, \sin \left (d x + c\right ) + 1}{3 \, a^{4} d{\left (\sin \left (d x + c\right ) + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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