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

Optimal. Leaf size=31 \[ 2 \sqrt{x-1}-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right ) \]

[Out]

2*Sqrt[-1 + x] - 2*Sqrt[2]*ArcTan[Sqrt[-1 + x]/Sqrt[2]]

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Rubi [A]  time = 0.026014, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ 2 \sqrt{x-1}-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-1 + x]/(1 + x),x]

[Out]

2*Sqrt[-1 + x] - 2*Sqrt[2]*ArcTan[Sqrt[-1 + x]/Sqrt[2]]

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Rubi in Sympy [A]  time = 1.62914, size = 29, normalized size = 0.94 \[ 2 \sqrt{x - 1} - 2 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{x - 1}}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x - 1) - 2*sqrt(2)*atan(sqrt(2)*sqrt(x - 1)/2)

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Mathematica [A]  time = 0.0137113, size = 31, normalized size = 1. \[ 2 \sqrt{x-1}-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[-1 + x] - 2*Sqrt[2]*ArcTan[Sqrt[-1 + x]/Sqrt[2]]

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Maple [A]  time = 0.009, size = 25, normalized size = 0.8 \[ -2\,\arctan \left ( 1/2\,\sqrt{-1+x}\sqrt{2} \right ) \sqrt{2}+2\,\sqrt{-1+x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+x)^(1/2)/(1+x),x)

[Out]

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

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Maxima [A]  time = 1.49829, size = 32, normalized size = 1.03 \[ -2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right ) + 2 \, \sqrt{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.204941, size = 32, normalized size = 1.03 \[ -2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right ) + 2 \, \sqrt{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.08826, size = 76, normalized size = 2.45 \[ \begin{cases} 2 \sqrt{x - 1} + 2 \sqrt{2} \operatorname{asin}{\left (\frac{\sqrt{2}}{\sqrt{x + 1}} \right )} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\2 i \sqrt{- x + 1} + \sqrt{2} i \log{\left (x + 1 \right )} - 2 \sqrt{2} i \log{\left (\sqrt{- \frac{x}{2} + \frac{1}{2}} + 1 \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209863, size = 32, normalized size = 1.03 \[ -2 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x - 1}\right ) + 2 \, \sqrt{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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