Optimal. Leaf size=17 \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{a \cot ^2(x)}} \]
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Rubi [A] time = 0.0117983, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3658, 3475} \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{a \cot ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \cot ^2(x)}} \, dx &=\frac{\cot (x) \int \tan (x) \, dx}{\sqrt{a \cot ^2(x)}}\\ &=-\frac{\cot (x) \log (\cos (x))}{\sqrt{a \cot ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.007814, size = 17, normalized size = 1. \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{a \cot ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 28, normalized size = 1.7 \begin{align*} -{\frac{\cot \left ( x \right ) \left ( -\ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}+1 \right ) +2\,\ln \left ( \cot \left ( x \right ) \right ) \right ) }{2}{\frac{1}{\sqrt{a \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55324, size = 16, normalized size = 0.94 \begin{align*} \frac{\log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.67252, size = 128, normalized size = 7.53 \begin{align*} -\frac{\sqrt{-\frac{a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right ) - 1}} \log \left (\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) \sin \left (2 \, x\right )}{2 \,{\left (a \cos \left (2 \, x\right ) + a\right )}} \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*} \int \frac{1}{\sqrt{a \cot ^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26688, size = 26, normalized size = 1.53 \begin{align*} -\frac{\log \left ({\left | \cos \left (x\right ) \right |}\right )}{\sqrt{a} \mathrm{sgn}\left (\cos \left (x\right )\right ) \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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