Optimal. Leaf size=16 \[ -2 \sqrt {x}-2 \cot \left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 8
Rule 3473
Rule 3748
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*}
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Mathematica [C] time = 0.05, size = 26, normalized size = 1.62 \[ -2 \cot \left (\sqrt {x}\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2\left (\sqrt {x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.22, size = 28, normalized size = 1.75 \[ -\frac {2 \, {\left (\sqrt {x} \sin \left (2 \, \sqrt {x}\right ) + \cos \left (2 \, \sqrt {x}\right ) + 1\right )}}{\sin \left (2 \, \sqrt {x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 22, normalized size = 1.38 \[ -2 \, \sqrt {x} - \frac {1}{\tan \left (\frac {1}{2} \, \sqrt {x}\right )} + \tan \left (\frac {1}{2} \, \sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 14, normalized size = 0.88 \[ -2 \cot \left (\sqrt {x}\right )+\pi -2 \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 14, normalized size = 0.88 \[ -2 \, \sqrt {x} - \frac {2}{\tan \left (\sqrt {x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.55, size = 20, normalized size = 1.25 \[ -2\,\sqrt {x}-\frac {4{}\mathrm {i}}{{\mathrm {e}}^{\sqrt {x}\,2{}\mathrm {i}}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 15, normalized size = 0.94 \[ - 2 \sqrt {x} - 2 \cot {\left (\sqrt {x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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