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

Optimal. Leaf size=18 \[ 2 \sqrt{x}-2 \log \left (\sqrt{x}+1\right ) \]

[Out]

2*Sqrt[x] - 2*Log[1 + Sqrt[x]]

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Rubi [A]  time = 0.0146088, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ 2 \sqrt{x}-2 \log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + Sqrt[x])^(-1),x]

[Out]

2*Sqrt[x] - 2*Log[1 + Sqrt[x]]

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Rubi in Sympy [A]  time = 1.09657, size = 15, normalized size = 0.83 \[ 2 \sqrt{x} - 2 \log{\left (\sqrt{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1)

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Mathematica [A]  time = 0.00604224, size = 18, normalized size = 1. \[ 2 \sqrt{x}-2 \log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Sqrt[x])^(-1),x]

[Out]

2*Sqrt[x] - 2*Log[1 + Sqrt[x]]

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Maple [A]  time = 0.003, size = 27, normalized size = 1.5 \[ 2\,\sqrt{x}+\ln \left ( \sqrt{x}-1 \right ) -\ln \left ( 1+\sqrt{x} \right ) -\ln \left ( -1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.35673, size = 20, normalized size = 1.11 \[ 2 \, \sqrt{x} - 2 \, \log \left (\sqrt{x} + 1\right ) + 2 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1) + 2

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Fricas [A]  time = 0.19982, size = 19, normalized size = 1.06 \[ 2 \, \sqrt{x} - 2 \, \log \left (\sqrt{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1)

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Sympy [A]  time = 0.125218, size = 15, normalized size = 0.83 \[ 2 \sqrt{x} - 2 \log{\left (\sqrt{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+x**(1/2)),x)

[Out]

2*sqrt(x) - 2*log(sqrt(x) + 1)

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GIAC/XCAS [A]  time = 0.207748, size = 19, normalized size = 1.06 \[ 2 \, \sqrt{x} - 2 \,{\rm ln}\left (\sqrt{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x) - 2*ln(sqrt(x) + 1)