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