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

Optimal. Leaf size=44 \[ -\frac{2}{3} a^3 \tan ^{-1}\left (\frac{x}{a}\right )+\frac{1}{3} x^3 \log \left (a^2+x^2\right )+\frac{2 a^2 x}{3}-\frac{2 x^3}{9} \]

[Out]

(2*a^2*x)/3 - (2*x^3)/9 - (2*a^3*ArcTan[x/a])/3 + (x^3*Log[a^2 + x^2])/3

_______________________________________________________________________________________

Rubi [A]  time = 0.046763, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2}{3} a^3 \tan ^{-1}\left (\frac{x}{a}\right )+\frac{1}{3} x^3 \log \left (a^2+x^2\right )+\frac{2 a^2 x}{3}-\frac{2 x^3}{9} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Log[a^2 + x^2],x]

[Out]

(2*a^2*x)/3 - (2*x^3)/9 - (2*a^3*ArcTan[x/a])/3 + (x^3*Log[a^2 + x^2])/3

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 a^{3} \operatorname{atan}{\left (\frac{x}{a} \right )}}{3} + \frac{x^{3} \log{\left (a^{2} + x^{2} \right )}}{3} - \frac{2 x^{3}}{9} + \frac{2 \int a^{2}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**3*atan(x/a)/3 + x**3*log(a**2 + x**2)/3 - 2*x**3/9 + 2*Integral(a**2, x)/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.00437289, size = 44, normalized size = 1. \[ -\frac{2}{3} a^3 \tan ^{-1}\left (\frac{x}{a}\right )+\frac{1}{3} x^3 \log \left (a^2+x^2\right )+\frac{2 a^2 x}{3}-\frac{2 x^3}{9} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Log[a^2 + x^2],x]

[Out]

(2*a^2*x)/3 - (2*x^3)/9 - (2*a^3*ArcTan[x/a])/3 + (x^3*Log[a^2 + x^2])/3

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 37, normalized size = 0.8 \[{\frac{2\,{a}^{2}x}{3}}-{\frac{2\,{x}^{3}}{9}}-{\frac{2\,{a}^{3}}{3}\arctan \left ({\frac{x}{a}} \right ) }+{\frac{{x}^{3}\ln \left ({a}^{2}+{x}^{2} \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*ln(a^2+x^2),x)

[Out]

2/3*a^2*x-2/9*x^3-2/3*a^3*arctan(x/a)+1/3*x^3*ln(a^2+x^2)

_______________________________________________________________________________________

Maxima [A]  time = 1.57122, size = 49, normalized size = 1.11 \[ -\frac{2}{3} \, a^{3} \arctan \left (\frac{x}{a}\right ) + \frac{1}{3} \, x^{3} \log \left (a^{2} + x^{2}\right ) + \frac{2}{3} \, a^{2} x - \frac{2}{9} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*a^3*arctan(x/a) + 1/3*x^3*log(a^2 + x^2) + 2/3*a^2*x - 2/9*x^3

_______________________________________________________________________________________

Fricas [A]  time = 0.228245, size = 49, normalized size = 1.11 \[ -\frac{2}{3} \, a^{3} \arctan \left (\frac{x}{a}\right ) + \frac{1}{3} \, x^{3} \log \left (a^{2} + x^{2}\right ) + \frac{2}{3} \, a^{2} x - \frac{2}{9} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*a^3*arctan(x/a) + 1/3*x^3*log(a^2 + x^2) + 2/3*a^2*x - 2/9*x^3

_______________________________________________________________________________________

Sympy [A]  time = 0.583052, size = 53, normalized size = 1.2 \[ - 2 a^{3} \left (- \frac{i \log{\left (- i a + x \right )}}{6} + \frac{i \log{\left (i a + x \right )}}{6}\right ) + \frac{2 a^{2} x}{3} + \frac{x^{3} \log{\left (a^{2} + x^{2} \right )}}{3} - \frac{2 x^{3}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*ln(a**2+x**2),x)

[Out]

-2*a**3*(-I*log(-I*a + x)/6 + I*log(I*a + x)/6) + 2*a**2*x/3 + x**3*log(a**2 + x
**2)/3 - 2*x**3/9

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.201867, size = 49, normalized size = 1.11 \[ -\frac{2}{3} \, a^{3} \arctan \left (\frac{x}{a}\right ) + \frac{1}{3} \, x^{3}{\rm ln}\left (a^{2} + x^{2}\right ) + \frac{2}{3} \, a^{2} x - \frac{2}{9} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*a^3*arctan(x/a) + 1/3*x^3*ln(a^2 + x^2) + 2/3*a^2*x - 2/9*x^3