Optimal. Leaf size=986 \[ -\frac {27 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right )-\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{104976\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac {\left (1+i \sqrt {3}+3 \sqrt [3]{2} 3^{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 )}{8748\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (i+\sqrt {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 )}{34992 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {\left (3 (-3)^{2/3}+\sqrt [3]{-1} 2^{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 )}{17496\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}+\frac {i \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 )}{17496 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{157464 \sqrt [6]{2} 3^{5/6} \sqrt {-4+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 (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{17496\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (i+\sqrt {3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{419904 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{209952 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{1889568 \sqrt [3]{2} 3^{2/3}}-\frac {27\ 2^{2/3} \left (1+\sqrt [3]{-2} 3^{2/3}\right )-\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{472392\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {9 \left (6-2^{2/3} \sqrt [3]{3}\right )-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{314928\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )} \]
[Out]
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Rubi [A] time = 1.93, antiderivative size = 986, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2097, 638, 618, 204, 634, 628, 206} \[ -\frac {27 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right )-\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{104976\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac {\left (1+i \sqrt {3}+3 \sqrt [3]{2} 3^{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 )}{8748\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (i+\sqrt {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 )}{34992 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {\left (3 (-3)^{2/3}+\sqrt [3]{-1} 2^{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 )}{17496\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}+\frac {i \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 )}{17496 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{157464 \sqrt [6]{2} 3^{5/6} \sqrt {-4+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 (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{17496\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (i+\sqrt {3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{419904 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{209952 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{1889568 \sqrt [3]{2} 3^{2/3}}-\frac {27\ 2^{2/3} \left (1+\sqrt [3]{-2} 3^{2/3}\right )-\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{472392\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {9 \left (6-2^{2/3} \sqrt [3]{3}\right )-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{314928\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 206
Rule 618
Rule 628
Rule 634
Rule 638
Rule 2097
Rubi steps
\begin {align*} \int \frac {x^2}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx &=1586874322944 \int \left (\frac {2 \sqrt [3]{-1} 3^{2/3}+18 \sqrt [3]{6}+3 (-2)^{2/3} x}{55527906308456448\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2}-\frac {i \left (18\ 3^{5/6}+\sqrt [3]{2} \left (3 i-\sqrt {3}\right ) x\right )}{333167437850738688\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {2 \sqrt [3]{-1} 3^{2/3}+18 (-1)^{2/3} \sqrt [3]{6}+3\ 2^{2/3} x}{55527906308456448\ 2^{2/3} \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2}+\frac {9+9 \sqrt [3]{-1}-i \sqrt [3]{2} \sqrt [6]{3} x}{166583718925369344\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {-2+6 \sqrt [3]{2} 3^{2/3}+2^{2/3} \sqrt [3]{3} x}{18509302102818816\ 2^{2/3} \sqrt [3]{3} \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2}+\frac {9 \sqrt [3]{3}+\sqrt [3]{2} x}{1499253470328324096\ 6^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {2 \sqrt [3]{-1} 3^{2/3}+18 (-1)^{2/3} \sqrt [3]{6}+3\ 2^{2/3} x}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{314928\ 2^{2/3}}+\frac {\int \frac {-2+6 \sqrt [3]{2} 3^{2/3}+2^{2/3} \sqrt [3]{3} x}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{104976\ 2^{2/3} \sqrt [3]{3}}+\frac {\int \frac {9 \sqrt [3]{3}+\sqrt [3]{2} x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{944784\ 6^{2/3}}+\frac {\int \frac {9+9 \sqrt [3]{-1}-i \sqrt [3]{2} \sqrt [6]{3} x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{104976\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \int \frac {18\ 3^{5/6}+\sqrt [3]{2} \left (3 i-\sqrt {3}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{209952\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\int \frac {2 \sqrt [3]{-1} 3^{2/3}+18 \sqrt [3]{6}+3 (-2)^{2/3} x}{\left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2} \, dx}{34992\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}\\ &=-\frac {27 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right )-\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{104976\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}-\frac {27\ 2^{2/3} \left (1+\sqrt [3]{-2} 3^{2/3}\right )-\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{944784\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {9 \left (6-2^{2/3} \sqrt [3]{3}\right )-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{314928\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\int \frac {3\ 2^{2/3} \sqrt [3]{3}+2 x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1889568 \sqrt [3]{2} 3^{2/3}}+\frac {\int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{157464\ 2^{2/3} \sqrt [3]{3}}-\frac {i \int \frac {1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{5832\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \int \frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{209952 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (i+\sqrt {3}\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{11664\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (i+\sqrt {3}\right ) \int \frac {-3 \sqrt [3]{-3} 2^{2/3}+2 x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{419904 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (2\ 3^{2/3}-9 \sqrt [3]{6}\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1889568 \left (2\ 2^{2/3}-3\ 3^{2/3}\right )}--\frac {\left (-18 \sqrt [3]{-6} (-1)^{2/3}-2 \left (2 \sqrt [3]{-1} 3^{2/3}+18 \sqrt [3]{6}\right )\right ) \int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{34992\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (24-18 (-3)^{2/3} \sqrt [3]{2}\right )}+\frac {\left (-18 (-1)^{2/3} \sqrt [3]{6}+2 \left (2 \sqrt [3]{-1} 3^{2/3}+18 (-1)^{2/3} \sqrt [3]{6}\right )\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{314928\ 2^{2/3} \left (24+18 \sqrt [3]{-2} 3^{2/3}\right )}\\ &=-\frac {27 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right )-\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{104976\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}-\frac {27\ 2^{2/3} \left (1+\sqrt [3]{-2} 3^{2/3}\right )-\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{944784\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {9 \left (6-2^{2/3} \sqrt [3]{3}\right )-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{314928\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\left (i+\sqrt {3}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{419904 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{209952 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{1889568 \sqrt [3]{2} 3^{2/3}}-\frac {\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 )}{78732\ 2^{2/3} \sqrt [3]{3}}+\frac {i \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 )}{2916\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (i+\sqrt {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 )}{5832\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (2\ 3^{2/3}-9 \sqrt [3]{6}\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 )}{944784 \left (2\ 2^{2/3}-3\ 3^{2/3}\right )}+-\frac {\left (-18 \sqrt [3]{-6} (-1)^{2/3}-2 \left (2 \sqrt [3]{-1} 3^{2/3}+18 \sqrt [3]{6}\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 )}{17496\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (24-18 (-3)^{2/3} \sqrt [3]{2}\right )}-\frac {\left (-18 (-1)^{2/3} \sqrt [3]{6}+2 \left (2 \sqrt [3]{-1} 3^{2/3}+18 (-1)^{2/3} \sqrt [3]{6}\right )\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 )}{157464\ 2^{2/3} \left (24+18 \sqrt [3]{-2} 3^{2/3}\right )}\\ &=-\frac {27 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right )-\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{104976\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}-\frac {27\ 2^{2/3} \left (1+\sqrt [3]{-2} 3^{2/3}\right )-\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{944784\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {9 \left (6-2^{2/3} \sqrt [3]{3}\right )-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{314928\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {\left (2 \sqrt [3]{-1}+3 \sqrt [3]{2} 3^{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 )}{34992 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}+\frac {\left (3 (-3)^{2/3} \sqrt [6]{2}+\sqrt [3]{-1} 2^{5/6}\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 )}{314928\ 3^{5/6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}+\frac {\left (i+\sqrt {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 )}{34992 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {i \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 )}{17496 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\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 )}{157464\ 6^{5/6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\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 )}{157464 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\left (i+\sqrt {3}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{419904 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{209952 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{1889568 \sqrt [3]{2} 3^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 167, normalized size = 0.17 \[ \frac {-9 x^5+8 x^4-216 x^3-2724 x^2+324 x-7884}{7383312 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )}-\frac {\text {RootSum}\left [\text {$\#$1}^6+18 \text {$\#$1}^4+324 \text {$\#$1}^3+108 \text {$\#$1}^2+216\& ,\frac {9 \text {$\#$1}^4 \log (x-\text {$\#$1})-16 \text {$\#$1}^3 \log (x-\text {$\#$1})+324 \text {$\#$1}^2 \log (x-\text {$\#$1})+2436 \text {$\#$1} \log (x-\text {$\#$1})+324 \log (x-\text {$\#$1})}{\text {$\#$1}^5+12 \text {$\#$1}^3+162 \text {$\#$1}^2+36 \text {$\#$1}}\& \right ]}{44299872} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 122, normalized size = 0.12 \[ \frac {\left (-9 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{4}+16 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}-324 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}-2436 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )-324\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )+x \right )}{44299872 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{5}+531598464 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{3}+7176579264 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )^{2}+1594795392 \RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}+\frac {-\frac {1}{820368} x^{5}+\frac {1}{922914} x^{4}-\frac {1}{34182} x^{3}-\frac {227}{615276} x^{2}+\frac {1}{22788} x -\frac {73}{68364}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {9 \, x^{5} - 8 \, x^{4} + 216 \, x^{3} + 2724 \, x^{2} - 324 \, x + 7884}{7383312 \, {\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} - \frac {1}{7383312} \, \int \frac {9 \, x^{4} - 16 \, x^{3} + 324 \, x^{2} + 2436 \, x + 324}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.48, size = 388, normalized size = 0.39 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 112, normalized size = 0.11 \[ \operatorname {RootSum} {\left (8658597397620778437929792538933565560629231616 t^{6} + 109068095871770168248838645612544 t^{4} - 492655707593366915713499136 t^{3} + 40378331745144603648 t^{2} - 695635011360 t + 4513, \left (t \mapsto t \log {\left (\frac {101442531561804181113161287039859349851881619653631712165888 t^{5}}{356900697070792948475845} - \frac {149796550082359335112709434971975088967050210050048 t^{4}}{356900697070792948475845} + \frac {1222409754458272818505898777768670783617236992 t^{3}}{356900697070792948475845} - \frac {5775055524251595723022901938558261453824 t^{2}}{356900697070792948475845} + \frac {96165242200260265765603930470432 t}{71380139414158589695169} + x - \frac {17059152341129698120545584}{1070702091212378845427535} \right )} \right )\right )} + \frac {- 9 x^{5} + 8 x^{4} - 216 x^{3} - 2724 x^{2} + 324 x - 7884}{7383312 x^{6} + 132899616 x^{4} + 2392193088 x^{3} + 797397696 x^{2} + 1594795392} \]
Verification of antiderivative is not currently implemented for this CAS.
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