Optimal. Leaf size=13 \[ -2 \cot (x) \sqrt{\sin (x) \tan (x)} \]
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Rubi [A] time = 0.0321317, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4400, 2589} \[ -2 \cot (x) \sqrt{\sin (x) \tan (x)} \]
Antiderivative was successfully verified.
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Rule 4400
Rule 2589
Rubi steps
\begin{align*} \int \sqrt{\sin (x) \tan (x)} \, dx &=\frac{\sqrt{\sin (x) \tan (x)} \int \sqrt{\sin (x)} \sqrt{\tan (x)} \, dx}{\sqrt{\sin (x)} \sqrt{\tan (x)}}\\ &=-2 \cot (x) \sqrt{\sin (x) \tan (x)}\\ \end{align*}
Mathematica [A] time = 0.0817373, size = 13, normalized size = 1. \[ -2 \cot (x) \sqrt{\sin (x) \tan (x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.217, size = 177, normalized size = 13.6 \begin{align*}{\frac{\sqrt{4} \left ( -1+\cos \left ( x \right ) \right ) \cos \left ( x \right ) }{4\, \left ( \sin \left ( x \right ) \right ) ^{3}} \left ( 4\,\cos \left ( x \right ) \sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}+4\,\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}-\ln \left ( -{\frac{1}{ \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( 2\, \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}- \left ( \cos \left ( x \right ) \right ) ^{2}+2\,\cos \left ( x \right ) -2\,\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}-1 \right ) } \right ) +\ln \left ( -2\,{\frac{1}{ \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( 2\, \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}- \left ( \cos \left ( x \right ) \right ) ^{2}+2\,\cos \left ( x \right ) -2\,\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}-1 \right ) } \right ) \right ) \sqrt{-{\frac{-1+ \left ( \cos \left ( x \right ) \right ) ^{2}}{\cos \left ( x \right ) }}}{\frac{1}{\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.53871, size = 77, normalized size = 5.92 \begin{align*} \frac{2 \,{\left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )}}{\sqrt{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1} \sqrt{-\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1} \sqrt{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43167, size = 63, normalized size = 4.85 \begin{align*} -\frac{2 \, \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{\sin \left (x\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{\sin{\left (x \right )} \tan{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sin \left (x\right ) \tan \left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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