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

Optimal. Leaf size=29 \[ -\frac{\sqrt{x^2+2 x+2}}{x+1}+\frac{1}{x+1}+\sinh ^{-1}(x+1) \]

[Out]

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

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Rubi [A]  time = 0.0667507, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{x^2+2 x+2}}{x+1}+\frac{1}{x+1}+\sinh ^{-1}(x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.72814, size = 32, normalized size = 1.1 \[ \log{\left (x + \sqrt{x^{2} + 2 x + 2} + 1 \right )} + \frac{2}{x + \sqrt{x^{2} + 2 x + 2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x + sqrt(x**2 + 2*x + 2) + 1) + 2/(x + sqrt(x**2 + 2*x + 2) + 2)

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Mathematica [A]  time = 0.0308931, size = 29, normalized size = 1. \[ -\frac{\sqrt{x^2+2 x+2}}{x+1}+\frac{1}{x+1}+\sinh ^{-1}(x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 40, normalized size = 1.4 \[ -{\frac{1}{1+x} \left ( \left ( 1+x \right ) ^{2}+1 \right ) ^{{\frac{3}{2}}}}+ \left ( 1+x \right ) \sqrt{ \left ( 1+x \right ) ^{2}+1}+{\it Arcsinh} \left ( 1+x \right ) + \left ( 1+x \right ) ^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+(x^2+2*x+2)^(1/2)),x)

[Out]

-1/(1+x)*((1+x)^2+1)^(3/2)+(1+x)*((1+x)^2+1)^(1/2)+arcsinh(1+x)+1/(1+x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} + 2 \, x + 2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^2 + 2*x + 2) + 1), x)

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Fricas [A]  time = 0.200665, size = 111, normalized size = 3.83 \[ -\frac{{\left (x^{2} - \sqrt{x^{2} + 2 \, x + 2}{\left (x + 1\right )} + 2 \, x + 1\right )} \log \left (-x + \sqrt{x^{2} + 2 \, x + 2} - 1\right ) - x + \sqrt{x^{2} + 2 \, x + 2} - 2}{x^{2} - \sqrt{x^{2} + 2 \, x + 2}{\left (x + 1\right )} + 2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-((x^2 - sqrt(x^2 + 2*x + 2)*(x + 1) + 2*x + 1)*log(-x + sqrt(x^2 + 2*x + 2) - 1
) - x + sqrt(x^2 + 2*x + 2) - 2)/(x^2 - sqrt(x^2 + 2*x + 2)*(x + 1) + 2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.202553, size = 81, normalized size = 2.79 \[ \frac{2}{{\left (x - \sqrt{x^{2} + 2 \, x + 2}\right )}^{2} + 2 \, x - 2 \, \sqrt{x^{2} + 2 \, x + 2}} + \frac{1}{x + 1} -{\rm ln}\left (-x + \sqrt{x^{2} + 2 \, x + 2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/((x - sqrt(x^2 + 2*x + 2))^2 + 2*x - 2*sqrt(x^2 + 2*x + 2)) + 1/(x + 1) - ln(-
x + sqrt(x^2 + 2*x + 2) - 1)