3.71 \(\int x \log \left (a^2+x^2\right ) \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{2} \left (a^2+x^2\right ) \log \left (a^2+x^2\right )-\frac{x^2}{2} \]

[Out]

-x^2/2 + ((a^2 + x^2)*Log[a^2 + x^2])/2

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Rubi [A]  time = 0.0706951, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{1}{2} \left (a^2+x^2\right ) \log \left (a^2+x^2\right )-\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[x*Log[a^2 + x^2],x]

[Out]

-x^2/2 + ((a^2 + x^2)*Log[a^2 + x^2])/2

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Rubi in Sympy [A]  time = 1.22678, size = 31, normalized size = 1.15 \[ \frac{a^{2} \log{\left (a^{2} + x^{2} \right )}}{2} + \frac{x^{2} \log{\left (a^{2} + x^{2} \right )}}{2} - \frac{x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*ln(a**2+x**2),x)

[Out]

a**2*log(a**2 + x**2)/2 + x**2*log(a**2 + x**2)/2 - x**2/2

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Mathematica [A]  time = 0.00324879, size = 38, normalized size = 1.41 \[ \frac{1}{2} a^2 \log \left (a^2+x^2\right )+\frac{1}{2} x^2 \log \left (a^2+x^2\right )-\frac{x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Log[a^2 + x^2],x]

[Out]

-x^2/2 + (a^2*Log[a^2 + x^2])/2 + (x^2*Log[a^2 + x^2])/2

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Maple [A]  time = 0.001, size = 29, normalized size = 1.1 \[{\frac{ \left ({a}^{2}+{x}^{2} \right ) \ln \left ({a}^{2}+{x}^{2} \right ) }{2}}-{\frac{{x}^{2}}{2}}-{\frac{{a}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*ln(a^2+x^2),x)

[Out]

1/2*(a^2+x^2)*ln(a^2+x^2)-1/2*x^2-1/2*a^2

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Maxima [A]  time = 1.38547, size = 38, normalized size = 1.41 \[ -\frac{1}{2} \, a^{2} - \frac{1}{2} \, x^{2} + \frac{1}{2} \,{\left (a^{2} + x^{2}\right )} \log \left (a^{2} + x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.215803, size = 31, normalized size = 1.15 \[ -\frac{1}{2} \, x^{2} + \frac{1}{2} \,{\left (a^{2} + x^{2}\right )} \log \left (a^{2} + x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.553694, size = 31, normalized size = 1.15 \[ \frac{a^{2} \log{\left (a^{2} + x^{2} \right )}}{2} + \frac{x^{2} \log{\left (a^{2} + x^{2} \right )}}{2} - \frac{x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*ln(a**2+x**2),x)

[Out]

a**2*log(a**2 + x**2)/2 + x**2*log(a**2 + x**2)/2 - x**2/2

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GIAC/XCAS [A]  time = 0.198737, size = 38, normalized size = 1.41 \[ -\frac{1}{2} \, a^{2} - \frac{1}{2} \, x^{2} + \frac{1}{2} \,{\left (a^{2} + x^{2}\right )}{\rm ln}\left (a^{2} + x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a^2 - 1/2*x^2 + 1/2*(a^2 + x^2)*ln(a^2 + x^2)