Optimal. Leaf size=70 \[ \frac{\sin ^2(c+d x)}{2 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{3 \log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.0801053, antiderivative size = 70, 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} \[ \frac{\sin ^2(c+d x)}{2 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{3 \log (\sin (c+d x)+1)}{a^2 d} \]
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))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{a^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-2 a+x-\frac{a^3}{(a+x)^2}+\frac{3 a^2}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac{3 \log (1+\sin (c+d x))}{a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{\sin ^2(c+d x)}{2 a^2 d}+\frac{1}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.17847, size = 71, normalized size = 1.01 \[ \frac{\sin ^3(c+d x)-3 \sin ^2(c+d x)+\sin (c+d x) (6 \log (\sin (c+d x)+1)-4)+6 \log (\sin (c+d x)+1)+2}{2 a^2 d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 66, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,{a}^{2}d}}-2\,{\frac{\sin \left ( dx+c \right ) }{{a}^{2}d}}+{\frac{1}{{a}^{2}d \left ( 1+\sin \left ( dx+c \right ) \right ) }}+3\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13884, size = 80, normalized size = 1.14 \begin{align*} \frac{\frac{2}{a^{2} \sin \left (d x + c\right ) + a^{2}} + \frac{\sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )}{a^{2}} + \frac{6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41799, size = 190, normalized size = 2.71 \begin{align*} \frac{6 \, \cos \left (d x + c\right )^{2} + 12 \,{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (2 \, \cos \left (d x + c\right )^{2} + 7\right )} \sin \left (d x + c\right ) - 3}{4 \,{\left (a^{2} d \sin \left (d x + c\right ) + a^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.67568, size = 209, normalized size = 2.99 \begin{align*} \begin{cases} \frac{6 \log{\left (\sin{\left (c + d x \right )} + 1 \right )} \sin{\left (c + d x \right )}}{2 a^{2} d \sin{\left (c + d x \right )} + 2 a^{2} d} + \frac{6 \log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{2 a^{2} d \sin{\left (c + d x \right )} + 2 a^{2} d} + \frac{4 \sin ^{3}{\left (c + d x \right )}}{2 a^{2} d \sin{\left (c + d x \right )} + 2 a^{2} d} + \frac{3 \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 a^{2} d \sin{\left (c + d x \right )} + 2 a^{2} d} + \frac{3 \cos ^{2}{\left (c + d x \right )}}{2 a^{2} d \sin{\left (c + d x \right )} + 2 a^{2} d} + \frac{6}{2 a^{2} d \sin{\left (c + d x \right )} + 2 a^{2} 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 )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33541, size = 122, normalized size = 1.74 \begin{align*} -\frac{\frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}{\left (\frac{6 \, a}{a \sin \left (d x + c\right ) + a} - 1\right )}}{a^{4}} + \frac{6 \, \log \left (\frac{{\left | a \sin \left (d x + c\right ) + a \right |}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}{\left | a \right |}}\right )}{a^{2}} - \frac{2}{{\left (a \sin \left (d x + c\right ) + a\right )} a}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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