Integrand size = 26, antiderivative size = 850 \[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {\sqrt [3]{-\frac {1}{3}} \left (3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{5832\ 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 {\sqrt [3]{-\frac {1}{3}} \left (3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )}{26244\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {3 \sqrt [3]{3}+\sqrt [3]{2} x}{52488 \left (9 \sqrt [3]{2}-4 \sqrt [3]{3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\sqrt [3]{-1} \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{729\ 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 {\sqrt [3]{-1} \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{2916 \sqrt [6]{2} 3^{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 ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{11664 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {i \arctan \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 )}{5832 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\text {arctanh}\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 )}{26244 \sqrt [6]{2} 3^{5/6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\text {arctanh}\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 )}{52488 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{34992 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{34992 \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 )}{314928 \sqrt [3]{2} 3^{2/3}} \]
[Out]
Time = 1.45 (sec) , antiderivative size = 850, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2122, 628, 632, 210, 648, 642, 212} \[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {\sqrt [3]{-\frac {1}{3}} \left (3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{5832\ 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 {\sqrt [3]{-1} \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{729\ 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 ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{11664 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\sqrt [3]{-1} \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{2916 \sqrt [6]{2} 3^{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 \arctan \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 )}{5832 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\text {arctanh}\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 )}{52488 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\text {arctanh}\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 )}{26244 \sqrt [6]{2} 3^{5/6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{34992 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac {i \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{34992 \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 )}{314928 \sqrt [3]{2} 3^{2/3}}-\frac {\sqrt [3]{-\frac {1}{3}} \left (2 x+3 (-2)^{2/3} \sqrt [3]{3}\right )}{26244\ 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 {\sqrt [3]{2} x+3 \sqrt [3]{3}}{52488 \left (9 \sqrt [3]{2}-4 \sqrt [3]{3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )} \]
[In]
[Out]
Rule 210
Rule 212
Rule 628
Rule 632
Rule 642
Rule 648
Rule 2122
Rubi steps \begin{align*} \text {integral}& = 1586874322944 \int \left (-\frac {\sqrt [3]{-\frac {1}{3}}}{1542441841901568\ 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 {6 i 3^{5/6}-\left (\sqrt [3]{-2}+\sqrt [3]{2}\right ) x}{27763953154228224\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}-\frac {\sqrt [3]{-\frac {1}{3}}}{1542441841901568\ 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 {i \left (9 i-3 \sqrt {3}+\sqrt [3]{2} \sqrt [6]{3} x\right )}{27763953154228224\ 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 {1}{1542441841901568\ 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 {3\ 2^{2/3} \sqrt [3]{3}+x}{249875578388054016 \sqrt [3]{2} 3^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx \\ & = -\frac {\sqrt [3]{-\frac {1}{3}} \int \frac {1}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{8748\ 2^{2/3}}-\frac {\int \frac {3\ 2^{2/3} \sqrt [3]{3}+x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{157464 \sqrt [3]{2} 3^{2/3}}+\frac {\int \frac {1}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{8748\ 2^{2/3} \sqrt [3]{3}}+\frac {\int \frac {6 i 3^{5/6}-\left (\sqrt [3]{-2}+\sqrt [3]{2}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{17496\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\sqrt [3]{-\frac {1}{3}} \int \frac {1}{\left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2} \, dx}{972\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac {\int \frac {9 i-3 \sqrt {3}+\sqrt [3]{2} \sqrt [6]{3} x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{78732\ 2^{2/3} \sqrt [3]{3} \left (3 i+\sqrt {3}\right )} \\ & = \frac {\sqrt [3]{-\frac {1}{3}} \left (3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{5832\ 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 {\sqrt [3]{-\frac {1}{3}} \left (3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )}{52488\ 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 {3 \sqrt [3]{3}+\sqrt [3]{2} x}{52488 \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}{314928 \sqrt [3]{2} 3^{2/3}}-\frac {\int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{52488\ 2^{2/3} \sqrt [3]{3}}+\frac {i \int \frac {1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{1944\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\int \frac {-3 \sqrt [3]{-3} 2^{2/3}+2 x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{34992 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac {\sqrt [3]{-\frac {1}{3}} \int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{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}{157464 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right )}-\frac {\left (1-i \sqrt {3}\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{17496\ 2^{2/3} 3^{5/6} \left (3 i+\sqrt {3}\right )}-\frac {\sqrt [3]{-\frac {1}{3}} \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{26244\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}+\frac {\int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{26244\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right )} \\ & = \frac {\sqrt [3]{-\frac {1}{3}} \left (3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{5832\ 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 {\sqrt [3]{-\frac {1}{3}} \left (3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )}{52488\ 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 {3 \sqrt [3]{3}+\sqrt [3]{2} x}{52488 \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 {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{34992 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac {\log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{157464 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right )}-\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{314928 \sqrt [3]{2} 3^{2/3}}+\frac {\text {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 )}{26244\ 2^{2/3} \sqrt [3]{3}}-\frac {i \text {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 )}{972\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\sqrt [3]{-\frac {1}{3}} \text {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 )}{1458\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}+\frac {\left (1-i \sqrt {3}\right ) \text {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 )}{8748\ 2^{2/3} 3^{5/6} \left (3 i+\sqrt {3}\right )}+\frac {\sqrt [3]{-\frac {1}{3}} \text {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 )}{13122\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}-\frac {\text {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 )}{13122\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right )} \\ & = \frac {\sqrt [3]{-\frac {1}{3}} \left (3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{5832\ 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 {\sqrt [3]{-\frac {1}{3}} \left (3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )}{52488\ 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 {3 \sqrt [3]{3}+\sqrt [3]{2} x}{52488 \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 {\sqrt [3]{-1} \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 )}{2916 \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 {\sqrt [3]{-1} \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 )}{26244 \sqrt [6]{2} 3^{5/6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (1-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 )}{52488 \sqrt [6]{2} \sqrt [3]{3} \left (3 i+\sqrt {3}\right ) \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 )}{5832 \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 (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{26244 \sqrt [6]{2} 3^{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 )}{52488 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{34992 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac {\log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{157464 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right )}-\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{314928 \sqrt [3]{2} 3^{2/3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.20 \[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {-288+324 x-1458 x^2-216 x^3+8 x^4-9 x^5}{1230552 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {324 \log (x-\text {$\#$1})-2628 \log (x-\text {$\#$1}) \text {$\#$1}+324 \log (x-\text {$\#$1}) \text {$\#$1}^2-16 \log (x-\text {$\#$1}) \text {$\#$1}^3+9 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{7383312} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.14
method | result | size |
default | \(\frac {-\frac {1}{136728} x^{5}+\frac {1}{153819} x^{4}-\frac {1}{5697} x^{3}-\frac {1}{844} x^{2}+\frac {1}{3798} x -\frac {4}{17091}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-9 \textit {\_R}^{4}+16 \textit {\_R}^{3}-324 \textit {\_R}^{2}+2628 \textit {\_R} -324\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{7383312}\) | \(122\) |
risch | \(\frac {-\frac {1}{136728} x^{5}+\frac {1}{153819} x^{4}-\frac {1}{5697} x^{3}-\frac {1}{844} x^{2}+\frac {1}{3798} x -\frac {4}{17091}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-9 \textit {\_R}^{4}+16 \textit {\_R}^{3}-324 \textit {\_R}^{2}+2628 \textit {\_R} -324\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{7383312}\) | \(122\) |
[In]
[Out]
Timed out. \[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.13 \[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\operatorname {RootSum} {\left (185583791958607219605834030755606257729536 t^{6} - 1309367357962223565522033377280 t^{4} + 4356336487052294744666112 t^{3} - 4052982845480387328 t^{2} + 303890718384 t - 880007, \left ( t \mapsto t \log {\left (\frac {39083462657955593476841044707333565976412952759280634691584 t^{5}}{49797855396139900267573395695} + \frac {8836979346223785538912817601414711102396804462575616 t^{4}}{49797855396139900267573395695} - \frac {264930581348308532588844249597134695706805067776 t^{3}}{49797855396139900267573395695} + \frac {886135333547363185201515109826158376250624 t^{2}}{49797855396139900267573395695} - \frac {682321479574909906511394635855601936 t}{49797855396139900267573395695} + x - \frac {21375560770846486224291519568}{49797855396139900267573395695} \right )} \right )\right )} + \frac {- 9 x^{5} + 8 x^{4} - 216 x^{3} - 1458 x^{2} + 324 x - 288}{1230552 x^{6} + 22149936 x^{4} + 398698848 x^{3} + 132899616 x^{2} + 265799232} \]
[In]
[Out]
\[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{4}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{4}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
[In]
[Out]
Time = 9.10 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.46 \[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Too large to display} \]
[In]
[Out]