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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]