3.95 \(\int \tan ^{-1}\left (\sqrt{x}\right ) \, dx\)

Optimal. Leaf size=22 \[ -\sqrt{x}+x \tan ^{-1}\left (\sqrt{x}\right )+\tan ^{-1}\left (\sqrt{x}\right ) \]

[Out]

-Sqrt[x] + ArcTan[Sqrt[x]] + x*ArcTan[Sqrt[x]]

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Rubi [A]  time = 0.0125779, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\sqrt{x}+x \tan ^{-1}\left (\sqrt{x}\right )+\tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[ArcTan[Sqrt[x]],x]

[Out]

-Sqrt[x] + ArcTan[Sqrt[x]] + x*ArcTan[Sqrt[x]]

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Rubi in Sympy [A]  time = 1.3685, 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.

[In]  rubi_integrate(atan(x**(1/2)),x)

[Out]

-sqrt(x) + x*atan(sqrt(x)) + atan(sqrt(x))

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Mathematica [A]  time = 0.0106311, size = 18, normalized size = 0.82 \[ (x+1) \tan ^{-1}\left (\sqrt{x}\right )-\sqrt{x} \]

Antiderivative was successfully verified.

[In]  Integrate[ArcTan[Sqrt[x]],x]

[Out]

-Sqrt[x] + (1 + x)*ArcTan[Sqrt[x]]

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Maple [A]  time = 0.003, size = 17, normalized size = 0.8 \[ \arctan \left ( \sqrt{x} \right ) +x\arctan \left ( \sqrt{x} \right ) -\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arctan(x^(1/2)),x)

[Out]

arctan(x^(1/2))+x*arctan(x^(1/2))-x^(1/2)

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Maxima [A]  time = 1.53953, size = 22, normalized size = 1. \[ x \arctan \left (\sqrt{x}\right ) - \sqrt{x} + \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(sqrt(x)),x, algorithm="maxima")

[Out]

x*arctan(sqrt(x)) - sqrt(x) + arctan(sqrt(x))

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Fricas [A]  time = 0.217494, size = 19, normalized size = 0.86 \[{\left (x + 1\right )} \arctan \left (\sqrt{x}\right ) - \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(sqrt(x)),x, algorithm="fricas")

[Out]

(x + 1)*arctan(sqrt(x)) - sqrt(x)

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Sympy [A]  time = 1.13534, 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.

[In]  integrate(atan(x**(1/2)),x)

[Out]

-sqrt(x) + x*atan(sqrt(x)) + atan(sqrt(x))

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GIAC/XCAS [A]  time = 0.214596, size = 22, normalized size = 1. \[ x \arctan \left (\sqrt{x}\right ) - \sqrt{x} + \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(sqrt(x)),x, algorithm="giac")

[Out]

x*arctan(sqrt(x)) - sqrt(x) + arctan(sqrt(x))