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

Optimal. Leaf size=8 \[ \tan ^{-1}\left (\sqrt{x}\right )^2 \]

[Out]

ArcTan[Sqrt[x]]^2

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Rubi [A]  time = 0.0553465, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \tan ^{-1}\left (\sqrt{x}\right )^2 \]

Antiderivative was successfully verified.

[In]  Int[ArcTan[Sqrt[x]]/(Sqrt[x]*(1 + x)),x]

[Out]

ArcTan[Sqrt[x]]^2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 \operatorname{atan}^{2}{\left (\sqrt{x} \right )} - \int \frac{\operatorname{atan}{\left (\sqrt{x} \right )}}{\sqrt{x} \left (x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atan(sqrt(x))**2 - Integral(atan(sqrt(x))/(sqrt(x)*(x + 1)), x)

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Mathematica [A]  time = 0.00404746, size = 8, normalized size = 1. \[ \tan ^{-1}\left (\sqrt{x}\right )^2 \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Sqrt[x]]^2

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Maple [A]  time = 0.006, size = 7, normalized size = 0.9 \[ \left ( \arctan \left ( \sqrt{x} \right ) \right ) ^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x^(1/2))^2

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Maxima [A]  time = 1.37559, size = 8, normalized size = 1. \[ \arctan \left (\sqrt{x}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(sqrt(x))^2

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Fricas [A]  time = 0.231464, size = 8, normalized size = 1. \[ \arctan \left (\sqrt{x}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(sqrt(x))^2

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Sympy [A]  time = 3.55055, size = 7, normalized size = 0.88 \[ \operatorname{atan}^{2}{\left (\sqrt{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(x))**2

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GIAC/XCAS [A]  time = 0.218283, size = 8, normalized size = 1. \[ \arctan \left (\sqrt{x}\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(sqrt(x))^2