Optimal. Leaf size=15 \[ -\frac{x}{a}-\frac{\tanh ^{-1}(\cos (x))}{a} \]
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Rubi [A] time = 0.0397653, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3888, 3770} \[ -\frac{x}{a}-\frac{\tanh ^{-1}(\cos (x))}{a} \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3770
Rubi steps
\begin{align*} \int \frac{\cot ^2(x)}{a+a \csc (x)} \, dx &=\frac{\int (-a+a \csc (x)) \, dx}{a^2}\\ &=-\frac{x}{a}+\frac{\int \csc (x) \, dx}{a}\\ &=-\frac{x}{a}-\frac{\tanh ^{-1}(\cos (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0432713, size = 30, normalized size = 2. \[ -\frac{x}{a}+\frac{\log \left (\sin \left (\frac{x}{2}\right )\right )}{a}-\frac{\log \left (\cos \left (\frac{x}{2}\right )\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 21, normalized size = 1.4 \begin{align*} -2\,{\frac{\arctan \left ( \tan \left ( x/2 \right ) \right ) }{a}}+{\frac{1}{a}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.46531, size = 41, normalized size = 2.73 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} + \frac{\log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.496634, size = 85, normalized size = 5.67 \begin{align*} -\frac{2 \, x + \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{2 \, a} \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{\cot ^{2}{\left (x \right )}}{\csc{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31193, size = 23, normalized size = 1.53 \begin{align*} -\frac{x}{a} + \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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