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

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

[Out]

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

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Rubi [A]  time = 0.0209026, antiderivative size = 37, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\pi x}{4}+\frac{\sqrt{x}}{2}-\frac{1}{2} x \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]

Warning: Unable to verify antiderivative.

[In]  Int[-ArcTan[Sqrt[x] - Sqrt[1 + x]],x]

[Out]

Sqrt[x]/2 + (Pi*x)/4 - ArcTan[Sqrt[x]]/2 - (x*ArcTan[Sqrt[x]])/2

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.0576872, size = 39, normalized size = 1.26 \[ \frac{\sqrt{x}}{2}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right )-x \tan ^{-1}\left (\sqrt{x}-\sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[-ArcTan[Sqrt[x] - Sqrt[1 + x]],x]

[Out]

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

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Maple [A]  time = 0.023, size = 28, normalized size = 0.9 \[ -x\arctan \left ( \sqrt{x}-\sqrt{1+x} \right ) +{\frac{1}{2}\sqrt{x}}-{\frac{1}{2}\arctan \left ( \sqrt{x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(-arctan(x^(1/2)-(1+x)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.74175, size = 35, normalized size = 1.13 \[ x \arctan \left (\sqrt{x + 1} - \sqrt{x}\right ) + \frac{1}{2} \, \sqrt{x} - \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.235846, size = 30, normalized size = 0.97 \[{\left (x + 1\right )} \arctan \left (\sqrt{x + 1} - \sqrt{x}\right ) + \frac{1}{2} \, \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

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GIAC/XCAS [A]  time = 0.213064, size = 36, normalized size = 1.16 \[ -x \arctan \left (-\sqrt{x + 1} + \sqrt{x}\right ) + \frac{1}{2} \, \sqrt{x} - \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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