3.99 \(\int \frac{x^2}{1+(-1+x)^2} \, dx\)

Optimal. Leaf size=10 \[ x+\log \left ((x-1)^2+1\right ) \]

[Out]

x + Log[1 + (-1 + x)^2]

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Rubi [A]  time = 0.0287703, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ x+\log \left ((x-1)^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(1 + (-1 + x)^2),x]

[Out]

x + Log[1 + (-1 + x)^2]

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Rubi in Sympy [A]  time = 5.57177, size = 10, normalized size = 1. \[ x + \log{\left (\left (x - 1\right )^{2} + 1 \right )} - 1 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + log((x - 1)**2 + 1) - 1

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Mathematica [A]  time = 0.0106058, size = 11, normalized size = 1.1 \[ \log \left (x^2-2 x+2\right )+x \]

Antiderivative was successfully verified.

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

[Out]

x + Log[2 - 2*x + x^2]

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Maple [A]  time = 0.003, size = 12, normalized size = 1.2 \[ x+\ln \left ({x}^{2}-2\,x+2 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(1+(-1+x)^2),x)

[Out]

x+ln(x^2-2*x+2)

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Maxima [A]  time = 0.801978, size = 15, normalized size = 1.5 \[ x + \log \left (x^{2} - 2 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + log(x^2 - 2*x + 2)

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Fricas [A]  time = 0.289087, size = 15, normalized size = 1.5 \[ x + \log \left (x^{2} - 2 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + log(x^2 - 2*x + 2)

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Sympy [A]  time = 0.147849, size = 10, normalized size = 1. \[ x + \log{\left (x^{2} - 2 x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(1+(-1+x)**2),x)

[Out]

x + log(x**2 - 2*x + 2)

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GIAC/XCAS [A]  time = 0.265363, size = 15, normalized size = 1.5 \[ x +{\rm ln}\left (x^{2} - 2 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + ln(x^2 - 2*x + 2)