Optimal. Leaf size=30 \[ -\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.0792449, antiderivative size = 30, 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 = {2836, 12, 43} \[ -\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^2 (a-x)}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{a-x}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}-\frac{1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac{\csc (c+d x)}{a d}-\frac{\log (\sin (c+d x))}{a d}\\ \end{align*}
Mathematica [A] time = 0.0370506, size = 22, normalized size = 0.73 \[ -\frac{\csc (c+d x)+\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 30, normalized size = 1. \begin{align*} -{\frac{\csc \left ( dx+c \right ) }{da}}+{\frac{\ln \left ( \csc \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970025, size = 39, normalized size = 1.3 \begin{align*} -\frac{\frac{\log \left (\sin \left (d x + c\right )\right )}{a} + \frac{1}{a \sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26051, size = 84, normalized size = 2.8 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 1}{a d \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39437, size = 41, normalized size = 1.37 \begin{align*} -\frac{\frac{\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac{1}{a \sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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