Optimal. Leaf size=415 \[ -\frac{\left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt{3}\right )\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{46656}-\frac{\left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{23328}-\frac{\left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{23328}+\frac{\log (x)}{216}+\frac{(-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{216 \sqrt [3]{2} 3^{5/6} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac{(-1)^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{216 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (1-\sqrt [3]{2} 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{216 \sqrt [6]{2} 3^{5/6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]
[Out]
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Rubi [A] time = 3.06564, antiderivative size = 415, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\left (36+2^{2/3} \sqrt [3]{3} \left (1+i \sqrt{3}\right )\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{46656}-\frac{\left (18-(-2)^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{23328}-\frac{\left (18-2^{2/3} \sqrt [3]{3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{23328}+\frac{\log (x)}{216}+\frac{(-1)^{2/3} \left ((-2)^{2/3}-2\ 3^{2/3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{216 \sqrt [3]{2} 3^{5/6} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac{(-1)^{2/3} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\right ) \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{216 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\left (1-\sqrt [3]{2} 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{216 \sqrt [6]{2} 3^{5/6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(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(1/x/(x**6+18*x**4+324*x**3+108*x**2+216),x)
[Out]
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Mathematica [C] time = 0.0288183, size = 103, normalized size = 0.25 \[ \frac{\log (x)}{216}-\frac{\text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})+18 \text{$\#$1}^2 \log (x-\text{$\#$1})+324 \text{$\#$1} \log (x-\text{$\#$1})+108 \log (x-\text{$\#$1})}{\text{$\#$1}^4+12 \text{$\#$1}^2+162 \text{$\#$1}+36}\&\right ]}{1296} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)),x]
[Out]
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Maple [C] time = 0.013, size = 75, normalized size = 0.2 \[ -{\frac{1}{1296}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{ \left ({{\it \_R}}^{5}+18\,{{\it \_R}}^{3}+324\,{{\it \_R}}^{2}+108\,{\it \_R} \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}}+{\frac{\ln \left ( x \right ) }{216}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(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. \[ -\frac{1}{216} \, \int \frac{x^{5} + 18 \, x^{3} + 324 \, x^{2} + 108 \, x}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} + \frac{1}{216} \, \log \left (x\right ) \]
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),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),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.06293, size = 82, normalized size = 0.2 \[ \frac{\log{\left (x \right )}}{216} + \operatorname{RootSum}{\left (7379637425677839491923968 t^{6} + 34164988081841849499648 t^{5} + 52598809250685370368 t^{4} + 26673506015311872 t^{3} - 309171116160 t^{2} + 944784 t - 1, \left ( t \mapsto t \log{\left (\frac{8145570099668817936783362115119297360560128 t^{6}}{143425799309052440063} + \frac{977068766770806381087358257564745728 t^{5}}{143425799309052440063} - \frac{116529526608851264288400971539061538816 t^{4}}{143425799309052440063} - \frac{239359794985242202542501440710766592 t^{3}}{143425799309052440063} - \frac{136678312638137094439887341418240 t^{2}}{143425799309052440063} + \frac{1563115569067663795735413696 t}{143425799309052440063} + x - \frac{3164446315075236190044}{143425799309052440063} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(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. \[ \int \frac{1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} x}\,{d x} \]
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),x, algorithm="giac")
[Out]