Optimal. Leaf size=52 \[ \frac{\sin (c+d x)}{a^2 d}-\frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}-\frac{2 \log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.0725106, antiderivative size = 52, 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} \[ \frac{\sin (c+d x)}{a^2 d}-\frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}-\frac{2 \log (\sin (c+d x)+1)}{a^2 d} \]
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 ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{a^2 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{a^2}{(a+x)^2}-\frac{2 a}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac{2 \log (1+\sin (c+d x))}{a^2 d}+\frac{\sin (c+d x)}{a^2 d}-\frac{1}{d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.187079, size = 55, normalized size = 1.06 \[ \frac{4 \sin (c+d x)-8 \log (\sin (c+d x)+1)-\frac{4}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}}{4 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 50, normalized size = 1. \begin{align*}{\frac{\sin \left ( dx+c \right ) }{{a}^{2}d}}-{\frac{1}{{a}^{2}d \left ( 1+\sin \left ( dx+c \right ) \right ) }}-2\,{\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 = 1.08563, size = 63, normalized size = 1.21 \begin{align*} -\frac{\frac{1}{a^{2} \sin \left (d x + c\right ) + a^{2}} + \frac{2 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} - \frac{\sin \left (d x + c\right )}{a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4947, size = 146, normalized size = 2.81 \begin{align*} -\frac{\cos \left (d x + c\right )^{2} + 2 \,{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - \sin \left (d x + c\right )}{a^{2} d \sin \left (d x + c\right ) + a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.55332, size = 185, normalized size = 3.56 \begin{align*} \begin{cases} - \frac{2 \log{\left (\sin{\left (c + d x \right )} + 1 \right )} \sin{\left (c + d x \right )}}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} - \frac{2 \log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} - \frac{\sin ^{3}{\left (c + d x \right )}}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} - \frac{\sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} - \frac{\cos ^{2}{\left (c + d x \right )}}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} - \frac{2}{a^{2} d \sin{\left (c + d x \right )} + a^{2} 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 )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29634, size = 95, normalized size = 1.83 \begin{align*} \frac{\frac{2 \, \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{a \sin \left (d x + c\right ) + a}{a^{3}} - \frac{1}{{\left (a \sin \left (d x + c\right ) + a\right )} a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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