3.224 \(\int \frac{\sqrt{x}}{1+x} \, dx\)

Optimal. Leaf size=16 \[ 2 \sqrt{x}-2 \tan ^{-1}\left (\sqrt{x}\right ) \]

[Out]

2*Sqrt[x] - 2*ArcTan[Sqrt[x]]

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Rubi [A]  time = 0.0111232, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ 2 \sqrt{x}-2 \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(1 + x),x]

[Out]

2*Sqrt[x] - 2*ArcTan[Sqrt[x]]

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Rubi in Sympy [A]  time = 1.16002, size = 14, normalized size = 0.88 \[ 2 \sqrt{x} - 2 \operatorname{atan}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) - 2*atan(sqrt(x))

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Mathematica [A]  time = 0.00647774, size = 16, normalized size = 1. \[ 2 \sqrt{x}-2 \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(1 + x),x]

[Out]

2*Sqrt[x] - 2*ArcTan[Sqrt[x]]

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Maple [A]  time = 0.005, size = 13, normalized size = 0.8 \[ -2\,\arctan \left ( \sqrt{x} \right ) +2\,\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(1+x),x)

[Out]

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

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Maxima [A]  time = 1.49315, size = 16, normalized size = 1. \[ 2 \, \sqrt{x} - 2 \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x + 1),x, algorithm="maxima")

[Out]

2*sqrt(x) - 2*arctan(sqrt(x))

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Fricas [A]  time = 0.205837, size = 16, normalized size = 1. \[ 2 \, \sqrt{x} - 2 \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x + 1),x, algorithm="fricas")

[Out]

2*sqrt(x) - 2*arctan(sqrt(x))

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Sympy [A]  time = 2.03, size = 51, normalized size = 3.19 \[ \begin{cases} 2 \sqrt{x} + 2 \operatorname{asin}{\left (\frac{1}{\sqrt{x + 1}} \right )} & \text{for}\: \left |{x + 1}\right | > 1 \\2 i \sqrt{- x} + i \log{\left (x + 1 \right )} - 2 i \log{\left (\sqrt{- x} + 1 \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(1+x),x)

[Out]

Piecewise((2*sqrt(x) + 2*asin(1/sqrt(x + 1)), Abs(x + 1) > 1), (2*I*sqrt(-x) + I
*log(x + 1) - 2*I*log(sqrt(-x) + 1), True))

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GIAC/XCAS [A]  time = 0.209568, size = 16, normalized size = 1. \[ 2 \, \sqrt{x} - 2 \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x + 1),x, algorithm="giac")

[Out]

2*sqrt(x) - 2*arctan(sqrt(x))