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

Optimal. Leaf size=12 \[ \tan ^{-1}\left (\sqrt{x^2+2 x}\right ) \]

[Out]

ArcTan[Sqrt[2*x + x^2]]

_______________________________________________________________________________________

Rubi [A]  time = 0.0224967, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \tan ^{-1}\left (\sqrt{x^2+2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Sqrt[2*x + x^2]]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.18074, size = 10, normalized size = 0.83 \[ \operatorname{atan}{\left (\sqrt{x^{2} + 2 x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(x**2 + 2*x))

_______________________________________________________________________________________

Mathematica [B]  time = 0.0343143, size = 37, normalized size = 3.08 \[ \frac{2 \sqrt{x} \sqrt{x+2} \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+2}}\right )}{\sqrt{x (x+2)}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 13, normalized size = 1.1 \[ -\arctan \left ({\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-1}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x)/(x^2+2*x)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.76532, size = 12, normalized size = 1. \[ -\arcsin \left (\frac{1}{{\left | x + 1 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsin(1/abs(x + 1))

_______________________________________________________________________________________

Fricas [A]  time = 0.26409, size = 23, normalized size = 1.92 \[ 2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(-x + sqrt(x^2 + 2*x) - 1)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.266872, size = 23, normalized size = 1.92 \[ 2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(-x + sqrt(x^2 + 2*x) - 1)