Optimal. Leaf size=22 \[ -\sqrt {x}+x \tan ^{-1}\left (\sqrt {x}\right )+\tan ^{-1}\left (\sqrt {x}\right ) \]
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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 = {5027, 50, 63, 203} \[ -\sqrt {x}+x \tan ^{-1}\left (\sqrt {x}\right )+\tan ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 5027
Rubi steps
\begin {align*} \int \tan ^{-1}\left (\sqrt {x}\right ) \, dx &=x \tan ^{-1}\left (\sqrt {x}\right )-\frac {1}{2} \int \frac {\sqrt {x}}{1+x} \, dx\\ &=-\sqrt {x}+x \tan ^{-1}\left (\sqrt {x}\right )+\frac {1}{2} \int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=-\sqrt {x}+x \tan ^{-1}\left (\sqrt {x}\right )+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=-\sqrt {x}+\tan ^{-1}\left (\sqrt {x}\right )+x \tan ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.82 \[ (x+1) \tan ^{-1}\left (\sqrt {x}\right )-\sqrt {x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 14, normalized size = 0.64 \[ {\left (x + 1\right )} \arctan \left (\sqrt {x}\right ) - \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.02, size = 16, normalized size = 0.73 \[ x \arctan \left (\sqrt {x}\right ) - \sqrt {x} + \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.77 \[ x \arctan \left (\sqrt {x}\right )+\arctan \left (\sqrt {x}\right )-\sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 16, normalized size = 0.73 \[ x \arctan \left (\sqrt {x}\right ) - \sqrt {x} + \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 16, normalized size = 0.73 \[ \mathrm {atan}\left (\sqrt {x}\right )+x\,\mathrm {atan}\left (\sqrt {x}\right )-\sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.52, size = 19, normalized size = 0.86 \[ - \sqrt {x} + x \operatorname {atan}{\left (\sqrt {x} \right )} + \operatorname {atan}{\left (\sqrt {x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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