Optimal. Leaf size=377 \[ -\frac{\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{\sqrt [3]{-\frac{1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{648\ 2^{2/3}}+\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{648\ 2^{2/3} \sqrt [3]{3}}+\frac{(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{324 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac{\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{972 \sqrt{3 \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac{\left (9-2^{2/3} \sqrt [3]{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{972 \sqrt{6 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}} \]
[Out]
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Rubi [A] time = 2.73424, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{216\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{\sqrt [3]{-\frac{1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{648\ 2^{2/3}}+\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{648\ 2^{2/3} \sqrt [3]{3}}+\frac{(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{324 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac{\left ((-2)^{2/3}-3\ 3^{2/3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{972 \sqrt [6]{3} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac{\left (2^{2/3}-3\ 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{972 \sqrt [6]{3} \sqrt{2 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}} \]
Antiderivative was successfully verified.
[In] Int[(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^(-1),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(1/(x**6+18*x**4+324*x**3+108*x**2+216),x)
[Out]
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Mathematica [C] time = 0.0180006, size = 62, normalized size = 0.16 \[ \frac{1}{6} \text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\&,\frac{\log (x-\text{$\#$1})}{\text{$\#$1}^5+12 \text{$\#$1}^3+162 \text{$\#$1}^2+36 \text{$\#$1}}\&\right ] \]
Antiderivative was successfully verified.
[In] Integrate[(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^(-1),x]
[Out]
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Maple [C] time = 0.007, size = 53, normalized size = 0.1 \[{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(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. \[ \int \frac{1}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(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(1/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.689366, size = 65, normalized size = 0.17 \[ \operatorname{RootSum}{\left (34164988081841849499648 t^{6} - 3470494144278528 t^{4} - 86087932019712 t^{3} - 1530550080 t^{2} + 69984 t - 1, \left ( t \mapsto t \log{\left (\frac{185904446699109611410573787136 t^{5}}{57121295165} + \frac{6377301253267917382766592 t^{4}}{57121295165} - \frac{18904636002388564311552 t^{3}}{57121295165} - \frac{469080552915181723968 t^{2}}{57121295165} - \frac{24358640509989936 t}{57121295165} + x + \frac{152427895956}{57121295165} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(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. \[ \int \frac{1}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216),x, algorithm="giac")
[Out]