Optimal. Leaf size=377 \[ \frac{\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\sqrt [3]{-\frac{1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{18\ 2^{2/3}}-\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{18\ 2^{2/3} \sqrt [3]{3}}+\frac{(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{9 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac{\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{27 \sqrt{3 \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac{\left (9-2^{2/3} \sqrt [3]{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{27 \sqrt{6 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}} \]
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Rubi [A] time = 0.90978, antiderivative size = 377, 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, Rules used = {2097, 634, 618, 204, 628, 206} \[ \frac{\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\sqrt [3]{-\frac{1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{18\ 2^{2/3}}-\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{18\ 2^{2/3} \sqrt [3]{3}}+\frac{(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{9 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac{\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{27 \sqrt{3 \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac{\left (9-2^{2/3} \sqrt [3]{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{27 \sqrt{6 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}} \]
Antiderivative was successfully verified.
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Rule 2097
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx &=1259712 \int \left (\frac{(-1)^{2/3} \left (-2+\sqrt [3]{-3} 2^{2/3} x\right )}{7558272 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )}+\frac{(-1)^{2/3} \left (2+(-2)^{2/3} \sqrt [3]{3} x\right )}{7558272 \sqrt [3]{2} 3^{2/3} \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac{\sqrt [3]{2}+\sqrt [3]{3} x}{11337408\ 6^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=-\frac{(-1)^{2/3} \int \frac{2+(-2)^{2/3} \sqrt [3]{3} x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{18 \sqrt [3]{2} 3^{2/3}}-\frac{\int \frac{\sqrt [3]{2}+\sqrt [3]{3} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{9\ 6^{2/3}}+\frac{(-1)^{2/3} \int \frac{-2+\sqrt [3]{-3} 2^{2/3} x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{6 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}\\ &=\frac{\sqrt [3]{-\frac{1}{3}} \int \frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{18\ 2^{2/3}}-\frac{\int \frac{3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{18\ 2^{2/3} \sqrt [3]{3}}+\frac{\int \frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{\left (\sqrt [3]{-1} \left (9+\sqrt [3]{-3} 2^{2/3}\right )\right ) \int \frac{1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{18 \left (1+\sqrt [3]{-1}\right )^2}-\frac{1}{54} \left (-9+(-2)^{2/3} \sqrt [3]{3}\right ) \int \frac{1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx-\frac{1}{54} \left (-9+2^{2/3} \sqrt [3]{3}\right ) \int \frac{1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx\\ &=\frac{\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\sqrt [3]{-\frac{1}{3}} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{18\ 2^{2/3}}-\frac{\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{18\ 2^{2/3} \sqrt [3]{3}}+\frac{\left (\sqrt [3]{-1} \left (9+\sqrt [3]{-3} 2^{2/3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )-x^2} \, dx,x,3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{9 \left (1+\sqrt [3]{-1}\right )^2}-\frac{1}{27} \left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )-x^2} \, dx,x,3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )-\frac{1}{27} \left (9-2^{2/3} \sqrt [3]{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-6 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )-x^2} \, dx,x,3\ 2^{2/3} \sqrt [3]{3}+2 x\right )\\ &=-\frac{\sqrt [3]{-1} \left (9+\sqrt [3]{-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 )}{9 \left (1+\sqrt [3]{-1}\right )^2 \sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac{\left ((-2)^{2/3}-3\ 3^{2/3}\right ) \tan ^{-1}\left (\frac{3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{27 \sqrt [6]{3} \sqrt{2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}+\frac{\left (2^{2/3}-3\ 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt{3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{27 \sqrt [6]{3} \sqrt{2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}+\frac{\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\sqrt [3]{-\frac{1}{3}} \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{18\ 2^{2/3}}-\frac{\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{18\ 2^{2/3} \sqrt [3]{3}}\\ \end{align*}
Mathematica [C] time = 0.0131222, size = 61, normalized size = 0.16 \[ \frac{1}{6} \text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\& ,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{\text{$\#$1}^4+12 \text{$\#$1}^2+162 \text{$\#$1}+36}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.006, size = 56, normalized size = 0.2 \begin{align*}{\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}}^{4}\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.23073, size = 65, normalized size = 0.17 \begin{align*} \operatorname{RootSum}{\left (15695178850368 t^{6} - 2066242608 t^{4} + 1845163152 t^{3} - 1180980 t^{2} - 1944 t - 1, \left ( t \mapsto t \log{\left (\frac{614714526178551746208 t^{5}}{57121295165} - \frac{1270857362386176 t^{4}}{57121295165} - \frac{80483053187684376 t^{3}}{57121295165} + \frac{72431318325103884 t^{2}}{57121295165} - \frac{45358602689088 t}{57121295165} + x - \frac{44532180783}{57121295165} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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