Optimal. Leaf size=16 \[ -2 \sqrt{x}-2 \cot \left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0184914, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3748, 3473, 8} \[ -2 \sqrt{x}-2 \cot \left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 3748
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{\cot ^2\left (\sqrt{x}\right )}{\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \cot ^2(x) \, dx,x,\sqrt{x}\right )\\ &=-2 \cot \left (\sqrt{x}\right )-2 \operatorname{Subst}\left (\int 1 \, dx,x,\sqrt{x}\right )\\ &=-2 \sqrt{x}-2 \cot \left (\sqrt{x}\right )\\ \end{align*}
Mathematica [C] time = 0.0443974, size = 26, normalized size = 1.62 \[ -2 \cot \left (\sqrt{x}\right ) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2\left (\sqrt{x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 14, normalized size = 0.9 \begin{align*} -2\,\cot \left ( \sqrt{x} \right ) +\pi -2\,\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49848, size = 19, normalized size = 1.19 \begin{align*} -2 \, \sqrt{x} - \frac{2}{\tan \left (\sqrt{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.38831, size = 88, normalized size = 5.5 \begin{align*} -\frac{2 \,{\left (\sqrt{x} \sin \left (2 \, \sqrt{x}\right ) + \cos \left (2 \, \sqrt{x}\right ) + 1\right )}}{\sin \left (2 \, \sqrt{x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.709173, size = 15, normalized size = 0.94 \begin{align*} - 2 \sqrt{x} - 2 \cot{\left (\sqrt{x} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12142, size = 30, normalized size = 1.88 \begin{align*} -2 \, \sqrt{x} - \frac{1}{\tan \left (\frac{1}{2} \, \sqrt{x}\right )} + \tan \left (\frac{1}{2} \, \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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