Optimal. Leaf size=32 \[ \frac{2 \log (\sin (c+d x)+1)}{a^2 d}-\frac{\sin (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.0494738, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac{2 \log (\sin (c+d x)+1)}{a^2 d}-\frac{\sin (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a-x}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1+\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}\\ \end{align*}
Mathematica [A] time = 0.0331975, size = 26, normalized size = 0.81 \[ -\frac{\sin (c+d x)-2 \log (\sin (c+d x)+1)}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 33, normalized size = 1. \begin{align*} 2\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{2}d}}-{\frac{\sin \left ( dx+c \right ) }{{a}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.924143, size = 41, normalized size = 1.28 \begin{align*} \frac{\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 = 1.91854, size = 68, normalized size = 2.12 \begin{align*} \frac{2 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - \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 = 1.57043, size = 180, normalized size = 5.62 \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 ^{2}{\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{2}{a^{2} d \sin{\left (c + d x \right )} + a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \cos ^{3}{\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 = 1.15992, size = 73, normalized size = 2.28 \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}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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