Optimal. Leaf size=16 \[ \tan (x) \sqrt{a \cot ^2(x)} \log (\sin (x)) \]
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Rubi [A] time = 0.0205653, antiderivative size = 16, 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} \[ \tan (x) \sqrt{a \cot ^2(x)} \log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \sqrt{a \cot ^2(x)} \, dx &=\left (\sqrt{a \cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx\\ &=\sqrt{a \cot ^2(x)} \log (\sin (x)) \tan (x)\\ \end{align*}
Mathematica [A] time = 0.0066614, size = 16, normalized size = 1. \[ \tan (x) \sqrt{a \cot ^2(x)} \log (\sin (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 22, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}+1 \right ) }{2\,\cot \left ( x \right ) }\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 = 1.62628, size = 27, normalized size = 1.69 \begin{align*} -\frac{1}{2} \, \sqrt{a} \log \left (\tan \left (x\right )^{2} + 1\right ) + \sqrt{a} \log \left (\tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.64949, size = 126, normalized size = 7.88 \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 (\cos \left (2 \, x\right ) + 1\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 \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 = 1.24426, size = 27, normalized size = 1.69 \begin{align*} \frac{1}{2} \, \sqrt{a} \log \left (-\cos \left (x\right )^{2} + 1\right ) \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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