Optimal. Leaf size=52 \[ \frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{\log (\sin (c+d x))}{a^2 d}-\frac{\log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.0485757, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2707, 44} \[ \frac{1}{d \left (a^2 \sin (c+d x)+a^2\right )}+\frac{\log (\sin (c+d x))}{a^2 d}-\frac{\log (\sin (c+d x)+1)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 44
Rubi steps
\begin{align*} \int \frac{\cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{1}{a (a+x)^2}-\frac{1}{a^2 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\log (\sin (c+d x))}{a^2 d}-\frac{\log (1+\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 = 0.0542632, size = 36, normalized size = 0.69 \[ \frac{\frac{1}{\sin (c+d x)+1}+\log (\sin (c+d x))-\log (\sin (c+d x)+1)}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 50, normalized size = 1. \begin{align*}{\frac{1}{d{a}^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{2}}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08983, size = 62, normalized size = 1.19 \begin{align*} \frac{\frac{1}{a^{2} \sin \left (d x + c\right ) + a^{2}} - \frac{\log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac{\log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44698, size = 162, normalized size = 3.12 \begin{align*} \frac{{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) -{\left (\sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 1}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cos{\left (c + d x \right )} \csc{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24873, size = 61, normalized size = 1.17 \begin{align*} \frac{a{\left (\frac{\log \left ({\left | -\frac{a}{a \sin \left (d x + c\right ) + a} + 1 \right |}\right )}{a^{3}} + \frac{1}{{\left (a \sin \left (d x + c\right ) + a\right )} a^{2}}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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