Optimal. Leaf size=74 \[ \frac{\sin (c+d x)}{a^3 d}-\frac{3}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{3 \log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.0896497, antiderivative size = 74, 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} \[ \frac{\sin (c+d x)}{a^3 d}-\frac{3}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{3 \log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{a^3 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{a^3}{(a+x)^3}+\frac{3 a^2}{(a+x)^2}-\frac{3 a}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=-\frac{3 \log (1+\sin (c+d x))}{a^3 d}+\frac{\sin (c+d x)}{a^3 d}+\frac{1}{2 a d (a+a \sin (c+d x))^2}-\frac{3}{d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.402829, size = 70, normalized size = 0.95 \[ \frac{\frac{\sin ^2(c+d x)}{(\sin (c+d x)+1)^2}+4 \sin (c+d x)+\frac{-10 \sin (c+d x)-9}{(\sin (c+d x)+1)^2}-12 \log (\sin (c+d x)+1)}{4 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 68, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{{a}^{3}d}}+{\frac{1}{2\,{a}^{3}d \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-3\,{\frac{1}{{a}^{3}d \left ( 1+\sin \left ( dx+c \right ) \right ) }}-3\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{3}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12152, size = 96, normalized size = 1.3 \begin{align*} -\frac{\frac{6 \, \sin \left (d x + c\right ) + 5}{a^{3} \sin \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) + a^{3}} + \frac{6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} - \frac{2 \, \sin \left (d x + c\right )}{a^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41297, size = 251, normalized size = 3.39 \begin{align*} \frac{4 \, \cos \left (d x + c\right )^{2} - 6 \,{\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (\cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 1}{2 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.69133, size = 303, normalized size = 4.09 \begin{align*} \begin{cases} - \frac{6 \log{\left (\sin{\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin{\left (c + d x \right )} + 2 a^{3} d} - \frac{12 \log{\left (\sin{\left (c + d x \right )} + 1 \right )} \sin{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin{\left (c + d x \right )} + 2 a^{3} d} - \frac{6 \log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin{\left (c + d x \right )} + 2 a^{3} d} + \frac{2 \sin ^{3}{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin{\left (c + d x \right )} + 2 a^{3} d} - \frac{12 \sin{\left (c + d x \right )}}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin{\left (c + d x \right )} + 2 a^{3} d} - \frac{9}{2 a^{3} d \sin ^{2}{\left (c + d x \right )} + 4 a^{3} d \sin{\left (c + d x \right )} + 2 a^{3} d} & \text{for}\: d \neq 0 \\\frac{x \sin ^{3}{\left (c \right )} \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2596, size = 76, normalized size = 1.03 \begin{align*} -\frac{\frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac{2 \, \sin \left (d x + c\right )}{a^{3}} + \frac{6 \, \sin \left (d x + c\right ) + 5}{a^{3}{\left (\sin \left (d x + c\right ) + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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