Optimal. Leaf size=22 \[ \sqrt {x}-\tan ^{-1}\left (\sqrt {x}\right )+x \cot ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5028, 50, 63, 203} \[ \sqrt {x}-\tan ^{-1}\left (\sqrt {x}\right )+x \cot ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 5028
Rubi steps
\begin {align*} \int \cot ^{-1}\left (\sqrt {x}\right ) \, dx &=x \cot ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} \int \frac {\sqrt {x}}{1+x} \, dx\\ &=\sqrt {x}+x \cot ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=\sqrt {x}+x \cot ^{-1}\left (\sqrt {x}\right )-\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x}+x \cot ^{-1}\left (\sqrt {x}\right )-\tan ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \sqrt {x}-\tan ^{-1}\left (\sqrt {x}\right )+x \cot ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 12, normalized size = 0.55 \[ {\left (x + 1\right )} \operatorname {arccot}\left (\sqrt {x}\right ) + \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 14, normalized size = 0.64 \[ x \arctan \left (\frac {1}{\sqrt {x}}\right ) + \sqrt {x} + \arctan \left (\frac {1}{\sqrt {x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 17, normalized size = 0.77 \[ x \,\mathrm {arccot}\left (\sqrt {x}\right )-\arctan \left (\sqrt {x}\right )+\sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 16, normalized size = 0.73 \[ x \operatorname {arccot}\left (\sqrt {x}\right ) + \sqrt {x} - \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 16, normalized size = 0.73 \[ x\,\mathrm {acot}\left (\sqrt {x}\right )-\mathrm {atan}\left (\sqrt {x}\right )+\sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.18, size = 19, normalized size = 0.86 \[ \sqrt {x} + x \operatorname {acot}{\left (\sqrt {x} \right )} - \operatorname {atan}{\left (\sqrt {x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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