3.804 \(\int \frac{1}{2 \sqrt{x} (1+x)} \, dx\)

Optimal. Leaf size=6 \[ \tan ^{-1}\left (\sqrt{x}\right ) \]

[Out]

ArcTan[Sqrt[x]]

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Rubi [A]  time = 0.00837332, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(2*Sqrt[x]*(1 + x)),x]

[Out]

ArcTan[Sqrt[x]]

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Rubi in Sympy [A]  time = 1.03338, size = 5, normalized size = 0.83 \[ \operatorname{atan}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(x))

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Mathematica [A]  time = 0.00637886, size = 6, normalized size = 1. \[ \tan ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Sqrt[x]]

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Maple [A]  time = 0.005, size = 5, normalized size = 0.8 \[ \arctan \left ( \sqrt{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/2/(1+x)/x^(1/2),x)

[Out]

arctan(x^(1/2))

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Maxima [A]  time = 0.774768, size = 5, normalized size = 0.83 \[ \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2/((x + 1)*sqrt(x)),x, algorithm="maxima")

[Out]

arctan(sqrt(x))

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Fricas [A]  time = 0.270482, size = 5, normalized size = 0.83 \[ \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2/((x + 1)*sqrt(x)),x, algorithm="fricas")

[Out]

arctan(sqrt(x))

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Sympy [A]  time = 3.30272, size = 32, normalized size = 5.33 \[ \frac{\begin{cases} 2 i \operatorname{acosh}{\left (\frac{1}{\sqrt{x + 1}} \right )} & \text{for}\: \left |{\frac{1}{x + 1}}\right | > 1 \\- 2 \operatorname{asin}{\left (\frac{1}{\sqrt{x + 1}} \right )} & \text{otherwise} \end{cases}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2/(1+x)/x**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.259751, size = 5, normalized size = 0.83 \[ \arctan \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2/((x + 1)*sqrt(x)),x, algorithm="giac")

[Out]

arctan(sqrt(x))