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