Optimal. Leaf size=22 \[ -\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{a} \]
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Rubi [A] time = 0.0715916, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2839, 3770, 8} \[ -\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 2839
Rule 3770
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int 1 \, dx}{a}+\frac{\int \csc (c+d x) \, dx}{a}\\ &=-\frac{x}{a}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}\\ \end{align*}
Mathematica [A] time = 0.088778, size = 37, normalized size = 1.68 \[ -\frac{-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+c+d x}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.088, size = 37, normalized size = 1.7 \begin{align*} -2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}}+{\frac{1}{da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.09784, size = 70, normalized size = 3.18 \begin{align*} -\frac{\frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79255, size = 109, normalized size = 4.95 \begin{align*} -\frac{2 \, d x + \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \, a 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 ^{2}{\left (c + d x \right )} \csc{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36141, size = 42, normalized size = 1.91 \begin{align*} -\frac{\frac{d x + c}{a} - \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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