3.254 \(\int \frac{1-x+x^2}{\left (1+x^2\right )^{3/2}} \, dx\)

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

[Out]

1/Sqrt[1 + x^2] + ArcSinh[x]

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Rubi [A]  time = 0.0161735, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{\sqrt{x^2+1}}+\sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x + x^2)/(1 + x^2)^(3/2),x]

[Out]

1/Sqrt[1 + x^2] + ArcSinh[x]

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Rubi in Sympy [A]  time = 2.03418, size = 12, normalized size = 1. \[ \operatorname{asinh}{\left (x \right )} + \frac{1}{\sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2-x+1)/(x**2+1)**(3/2),x)

[Out]

asinh(x) + 1/sqrt(x**2 + 1)

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Mathematica [A]  time = 0.0193417, size = 12, normalized size = 1. \[ \frac{1}{\sqrt{x^2+1}}+\sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x + x^2)/(1 + x^2)^(3/2),x]

[Out]

1/Sqrt[1 + x^2] + ArcSinh[x]

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Maple [A]  time = 0.009, size = 11, normalized size = 0.9 \[{\it Arcsinh} \left ( x \right ) +{\frac{1}{\sqrt{{x}^{2}+1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-x+1)/(x^2+1)^(3/2),x)

[Out]

arcsinh(x)+1/(x^2+1)^(1/2)

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Maxima [A]  time = 1.48769, size = 14, normalized size = 1.17 \[ \frac{1}{\sqrt{x^{2} + 1}} + \operatorname{arsinh}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - x + 1)/(x^2 + 1)^(3/2),x, algorithm="maxima")

[Out]

1/sqrt(x^2 + 1) + arcsinh(x)

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Fricas [A]  time = 0.25672, size = 78, normalized size = 6.5 \[ -\frac{{\left (x^{2} - \sqrt{x^{2} + 1} x + 1\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) + x - \sqrt{x^{2} + 1}}{x^{2} - \sqrt{x^{2} + 1} x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - x + 1)/(x^2 + 1)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 9.90964, size = 29, normalized size = 2.42 \[ \frac{x^{2} \operatorname{asinh}{\left (x \right )}}{x^{2} + 1} + \frac{\operatorname{asinh}{\left (x \right )}}{x^{2} + 1} + \frac{1}{\sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2-x+1)/(x**2+1)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215245, size = 30, normalized size = 2.5 \[ \frac{1}{\sqrt{x^{2} + 1}} -{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - x + 1)/(x^2 + 1)^(3/2),x, algorithm="giac")

[Out]

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