3.268 \(\int \log \left (1+x^2\right ) \, dx\)

Optimal. Leaf size=16 \[ x \log \left (x^2+1\right )-2 x+2 \tan ^{-1}(x) \]

[Out]

-2*x + 2*ArcTan[x] + x*Log[1 + x^2]

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Rubi [A]  time = 0.0143375, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ x \log \left (x^2+1\right )-2 x+2 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Log[1 + x^2],x]

[Out]

-2*x + 2*ArcTan[x] + x*Log[1 + x^2]

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Rubi in Sympy [A]  time = 1.61095, size = 15, normalized size = 0.94 \[ x \log{\left (x^{2} + 1 \right )} - 2 x + 2 \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(x**2+1),x)

[Out]

x*log(x**2 + 1) - 2*x + 2*atan(x)

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Mathematica [A]  time = 0.0038302, size = 16, normalized size = 1. \[ x \log \left (x^2+1\right )-2 x+2 \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Log[1 + x^2],x]

[Out]

-2*x + 2*ArcTan[x] + x*Log[1 + x^2]

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Maple [A]  time = 0.002, size = 17, normalized size = 1.1 \[ -2\,x+2\,\arctan \left ( x \right ) +x\ln \left ({x}^{2}+1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(x^2+1),x)

[Out]

-2*x+2*arctan(x)+x*ln(x^2+1)

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Maxima [A]  time = 1.51649, size = 22, normalized size = 1.38 \[ x \log \left (x^{2} + 1\right ) - 2 \, x + 2 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*log(x^2 + 1) - 2*x + 2*arctan(x)

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Fricas [A]  time = 0.208972, size = 22, normalized size = 1.38 \[ x \log \left (x^{2} + 1\right ) - 2 \, x + 2 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*log(x^2 + 1) - 2*x + 2*arctan(x)

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Sympy [A]  time = 0.127364, size = 15, normalized size = 0.94 \[ x \log{\left (x^{2} + 1 \right )} - 2 x + 2 \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(x**2+1),x)

[Out]

x*log(x**2 + 1) - 2*x + 2*atan(x)

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GIAC/XCAS [A]  time = 0.202274, size = 22, normalized size = 1.38 \[ x{\rm ln}\left (x^{2} + 1\right ) - 2 \, x + 2 \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*ln(x^2 + 1) - 2*x + 2*arctan(x)