Optimal. Leaf size=248 \[ \frac{(-1)^{2/3} \tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{27\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac{(-1)^{2/3} \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{81 \sqrt [3]{2} \sqrt [6]{3} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{81\ 2^{5/6} \sqrt [6]{3} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]
[Out]
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Rubi [A] time = 1.32345, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(-1)^{2/3} \tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{27\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac{(-1)^{2/3} \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{81 \sqrt [3]{2} \sqrt [6]{3} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{81\ 2^{5/6} \sqrt [6]{3} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(x**6+18*x**4+324*x**3+108*x**2+216),x)
[Out]
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Mathematica [C] time = 0.019311, size = 59, normalized size = 0.24 \[ \frac{1}{6} \text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^4+12 \text{$\#$1}^2+162 \text{$\#$1}+36}\&\right ] \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
[Out]
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Maple [C] time = 0.008, size = 56, normalized size = 0.2 \[{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(x^6+18*x^4+324*x^3+108*x^2+216),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.485465, size = 48, normalized size = 0.19 \[ \operatorname{RootSum}{\left (732274264442769408 t^{6} - 2677850419968 t^{4} + 2834352 t^{2} - 1, \left ( t \mapsto t \log{\left (10170475895038464 t^{5} - 5231726283456 t^{4} - 31809932496 t^{3} + 19131876 t^{2} + 19683 t + x - \frac{27}{2} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(x**6+18*x**4+324*x**3+108*x**2+216),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216),x, algorithm="giac")
[Out]