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

Optimal. Leaf size=22 \[ 2 \sqrt{x+1}-2 \log \left (\sqrt{x+1}+1\right ) \]

[Out]

2*Sqrt[1 + x] - 2*Log[1 + Sqrt[1 + x]]

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Rubi [A]  time = 0.0209189, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ 2 \sqrt{x+1}-2 \log \left (\sqrt{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + Sqrt[1 + x])^(-1),x]

[Out]

2*Sqrt[1 + x] - 2*Log[1 + Sqrt[1 + x]]

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Rubi in Sympy [A]  time = 1.13092, size = 19, normalized size = 0.86 \[ 2 \sqrt{x + 1} - 2 \log{\left (\sqrt{x + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + 1) - 2*log(sqrt(x + 1) + 1)

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Mathematica [A]  time = 0.00710266, size = 22, normalized size = 1. \[ 2 \sqrt{x+1}-2 \log \left (\sqrt{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Sqrt[1 + x])^(-1),x]

[Out]

2*Sqrt[1 + x] - 2*Log[1 + Sqrt[1 + x]]

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Maple [A]  time = 0.003, size = 31, normalized size = 1.4 \[ 2\,\sqrt{1+x}+\ln \left ( -1+\sqrt{1+x} \right ) -\ln \left ( 1+\sqrt{1+x} \right ) -\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+(1+x)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.45334, size = 24, normalized size = 1.09 \[ 2 \, \sqrt{x + 1} - 2 \, \log \left (\sqrt{x + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + 1) - 2*log(sqrt(x + 1) + 1)

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Fricas [A]  time = 0.201436, size = 24, normalized size = 1.09 \[ 2 \, \sqrt{x + 1} - 2 \, \log \left (\sqrt{x + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + 1) - 2*log(sqrt(x + 1) + 1)

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Sympy [A]  time = 0.129975, size = 19, normalized size = 0.86 \[ 2 \sqrt{x + 1} - 2 \log{\left (\sqrt{x + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+(1+x)**(1/2)),x)

[Out]

2*sqrt(x + 1) - 2*log(sqrt(x + 1) + 1)

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GIAC/XCAS [A]  time = 0.199365, size = 24, normalized size = 1.09 \[ 2 \, \sqrt{x + 1} - 2 \,{\rm ln}\left (\sqrt{x + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x + 1) - 2*ln(sqrt(x + 1) + 1)