Optimal. Leaf size=22 \[ \sqrt {x}+x \cot ^{-1}\left (\sqrt {x}\right )-\text {ArcTan}\left (\sqrt {x}\right ) \]
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Rubi [A]
time = 0.00, 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 = {4931, 52, 65,
209} \begin {gather*} -\text {ArcTan}\left (\sqrt {x}\right )+\sqrt {x}+x \cot ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 4931
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 )-\text {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 \begin {gather*} \sqrt {x}+x \cot ^{-1}\left (\sqrt {x}\right )-\text {ArcTan}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 17, normalized size = 0.77
method | result | size |
derivativedivides | \(x \,\mathrm {arccot}\left (\sqrt {x}\right )-\arctan \left (\sqrt {x}\right )+\sqrt {x}\) | \(17\) |
default | \(x \,\mathrm {arccot}\left (\sqrt {x}\right )-\arctan \left (\sqrt {x}\right )+\sqrt {x}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 16, normalized size = 0.73 \begin {gather*} x \operatorname {arccot}\left (\sqrt {x}\right ) + \sqrt {x} - \arctan \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.13, size = 12, normalized size = 0.55 \begin {gather*} {\left (x + 1\right )} \operatorname {arccot}\left (\sqrt {x}\right ) + \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.76, size = 19, normalized size = 0.86 \begin {gather*} \sqrt {x} + x \operatorname {acot}{\left (\sqrt {x} \right )} - \operatorname {atan}{\left (\sqrt {x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 14, normalized size = 0.64 \begin {gather*} x \arctan \left (\frac {1}{\sqrt {x}}\right ) + \sqrt {x} + \arctan \left (\frac {1}{\sqrt {x}}\right ) \end {gather*}
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 \begin {gather*} x\,\mathrm {acot}\left (\sqrt {x}\right )-\mathrm {atan}\left (\sqrt {x}\right )+\sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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