Optimal. Leaf size=1064 \[ -\frac {\sqrt [3]{-\frac {1}{3}} \left (9 \left (6+\sqrt [3]{-3} 2^{2/3}\right )+\left (2-2^{2/3} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}\right )\right ) x\right )}{162\ 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 (9 \left (6-(-2)^{2/3} \sqrt [3]{3}\right )+\left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{729\ 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 {9 \left (6-2^{2/3} \sqrt [3]{3}\right )+\left (2+2^{2/3} \left (27 \sqrt [3]{3}-6\ 6^{2/3}\right )\right ) x}{1458\ 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 {i \left ((-2)^{2/3}+6\ 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 )}{162\ 2^{5/6} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\sqrt [3]{-1} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 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 )}{162 \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 {\sqrt [3]{-1} \left (6 (-6)^{2/3}+27 \sqrt [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 )}{81 \sqrt {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 (i 2^{2/3}-9 \sqrt [6]{3}-3 i 3^{2/3}\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 )}{162\ 2^{5/6} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (1+3 \sqrt [3]{2} 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 )}{1458 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{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 )}{81 \sqrt {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 {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{972 \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 )}{972 \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 )}{8748 \sqrt [3]{2} 3^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.27, antiderivative size = 1064, 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 = {2122, 652,
632, 210, 648, 642, 212} \begin {gather*} -\frac {\sqrt [3]{-\frac {1}{3}} \left (\left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) x+9 \left (6-(-2)^{2/3} \sqrt [3]{3}\right )\right )}{729\ 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]{-1} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) \text {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 )}{162 \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 \left ((-2)^{2/3}+6\ 3^{2/3}\right ) \text {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 )}{162\ 2^{5/6} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {\left (i 2^{2/3}-9 \sqrt [6]{3}-3 i 3^{2/3}\right ) \text {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 )}{162\ 2^{5/6} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\sqrt [3]{-1} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right ) \text {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 )}{81 \sqrt {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 (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{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 )}{81 \sqrt {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 (1+3 \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 (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{1458 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{972 \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 )}{972 \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 )}{8748 \sqrt [3]{2} 3^{2/3}}-\frac {\sqrt [3]{-\frac {1}{3}} \left (\left (2-3\ 2^{2/3} \left (2 (-6)^{2/3}+9 \sqrt [3]{-3}\right )\right ) x+9 \left (6+\sqrt [3]{-3} 2^{2/3}\right )\right )}{162\ 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 (2+2^{2/3} \left (27 \sqrt [3]{3}-6\ 6^{2/3}\right )\right ) x+9 \left (6-2^{2/3} \sqrt [3]{3}\right )}{1458\ 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 )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 212
Rule 632
Rule 642
Rule 648
Rule 652
Rule 2122
Rubi steps
\begin {align*} \int \frac {x^8}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx &=1586874322944 \int \left (\frac {\sqrt [3]{-\frac {1}{3}} \left (-1+3 (-3)^{2/3} \sqrt [3]{2}+\left (9+\sqrt [3]{-3} 2^{2/3}\right ) x\right )}{42845606719488\ 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 {27 \left (2+(-1)^{2/3}\right )-\left (1+\sqrt [3]{-1}\right ) x}{771220920950784 \sqrt [3]{2} 3^{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}} \left (-1-3 \sqrt [3]{-2} 3^{2/3}+\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) x\right )}{42845606719488\ 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 (-27+x)}{771220920950784 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {1-3 \sqrt [3]{2} 3^{2/3}-\left (9-2^{2/3} \sqrt [3]{3}\right ) x}{42845606719488\ 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 {-27+x}{6940988288557056 \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 {-1-3 \sqrt [3]{-2} 3^{2/3}+\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) x}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{243\ 2^{2/3}}-\frac {\int \frac {-27+x}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{4374 \sqrt [3]{2} 3^{2/3}}+\frac {\int \frac {1-3 \sqrt [3]{2} 3^{2/3}-\left (9-2^{2/3} \sqrt [3]{3}\right ) x}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{243\ 2^{2/3} \sqrt [3]{3}}+\frac {\int \frac {27 \left (2+(-1)^{2/3}\right )-\left (1+\sqrt [3]{-1}\right ) x}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{486 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {i \int \frac {-27+x}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{486 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\sqrt [3]{-\frac {1}{3}} \int \frac {-1+3 (-3)^{2/3} \sqrt [3]{2}+\left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{\left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )^2} \, dx}{27\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}\\ &=-\frac {\sqrt [3]{-\frac {1}{3}} \left (9 \left (6+\sqrt [3]{-3} 2^{2/3}\right )+\left (2-3\ 2^{2/3} \left (2 (-6)^{2/3}+9 \sqrt [3]{-3}\right )\right ) x\right )}{162\ 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 (9 \left (6-(-2)^{2/3} \sqrt [3]{3}\right )+\left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{1458\ 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+2^{2/3} \left (27 \sqrt [3]{3}-6\ 6^{2/3}\right )\right ) x}{1458\ 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}{8748 \sqrt [3]{2} 3^{2/3}}+\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}{972 \sqrt [3]{2} \sqrt [6]{3} \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}{972 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac {\left (\sqrt [3]{-\frac {1}{3}} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right )\right ) \int \frac {1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{162 \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}-\frac {\left (i \left ((-2)^{2/3}+6\ 3^{2/3}\right )\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{108 \sqrt [3]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (\sqrt [3]{-\frac {1}{3}} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right )\right ) \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1458\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}+\frac {\left (1+3 \sqrt [3]{2} 3^{2/3}\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{1458\ 2^{2/3} \sqrt [3]{3}}+\frac {\left (3 \sqrt [3]{-3} 2^{2/3} \left (-1-\sqrt [3]{-1}\right )+54 \left (2+(-1)^{2/3}\right )\right ) \int \frac {1}{6-3 \sqrt [3]{-3} 2^{2/3} x+x^2} \, dx}{972 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (81+3^{2/3} \left (\sqrt [3]{2}-6\ 6^{2/3}\right )\right ) \int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{4374 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}\\ &=-\frac {\sqrt [3]{-\frac {1}{3}} \left (9 \left (6+\sqrt [3]{-3} 2^{2/3}\right )+\left (2-3\ 2^{2/3} \left (2 (-6)^{2/3}+9 \sqrt [3]{-3}\right )\right ) x\right )}{162\ 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 (9 \left (6-(-2)^{2/3} \sqrt [3]{3}\right )+\left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{1458\ 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+2^{2/3} \left (27 \sqrt [3]{3}-6\ 6^{2/3}\right )\right ) x}{1458\ 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 {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{972 \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 )}{972 \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 )}{8748 \sqrt [3]{2} 3^{2/3}}-\frac {\left (\sqrt [3]{-\frac {1}{3}} \left (6 (-6)^{2/3}+27 \sqrt [3]{-3}-\sqrt [3]{2}\right )\right ) \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 )}{81 \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}+\frac {\left (i \left ((-2)^{2/3}+6\ 3^{2/3}\right )\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 )}{54 \sqrt [3]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (\sqrt [3]{-\frac {1}{3}} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right )\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 )}{729\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}-\frac {\left (1+3 \sqrt [3]{2} 3^{2/3}\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 )}{729\ 2^{2/3} \sqrt [3]{3}}-\frac {\left (3 \sqrt [3]{-3} 2^{2/3} \left (-1-\sqrt [3]{-1}\right )+54 \left (2+(-1)^{2/3}\right )\right ) \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 )}{486 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (81+3^{2/3} \left (\sqrt [3]{2}-6\ 6^{2/3}\right )\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 )}{2187 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}\\ &=-\frac {\sqrt [3]{-\frac {1}{3}} \left (9 \left (6+\sqrt [3]{-3} 2^{2/3}\right )+\left (2-3\ 2^{2/3} \left (2 (-6)^{2/3}+9 \sqrt [3]{-3}\right )\right ) x\right )}{162\ 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 (9 \left (6-(-2)^{2/3} \sqrt [3]{3}\right )+\left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) x\right )}{1458\ 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+2^{2/3} \left (27 \sqrt [3]{3}-6\ 6^{2/3}\right )\right ) x}{1458\ 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 {i \left ((-2)^{2/3}+6\ 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 )}{162\ 2^{5/6} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\sqrt [3]{-1} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 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 )}{1458 \sqrt [6]{2} 3^{5/6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\sqrt [3]{-1} \left (6 (-6)^{2/3}+27 \sqrt [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 )}{81 \sqrt {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 (i 3^{5/6}-9 \sqrt [3]{2} \left (2+(-1)^{2/3}\right )\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 )}{486 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\left (1+3 \sqrt [3]{2} 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 )}{1458 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\left (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{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 )}{729 \sqrt {2} 3^{5/6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{972 \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 )}{972 \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 )}{8748 \sqrt [3]{2} 3^{2/3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.03, size = 167, normalized size = 0.16 \begin {gather*} \frac {-7884+324 x-3990 x^2-11610 x^3-203 x^4-9 x^5}{34182 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {324 \log (x-\text {$\#$1})-96 \log (x-\text {$\#$1}) \text {$\#$1}+324 \log (x-\text {$\#$1}) \text {$\#$1}^2+406 \log (x-\text {$\#$1}) \text {$\#$1}^3+9 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{205092} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.03, size = 122, normalized size = 0.11
method | result | size |
default | \(\frac {-\frac {1}{3798} x^{5}-\frac {203}{34182} x^{4}-\frac {215}{633} x^{3}-\frac {665}{5697} x^{2}+\frac {2}{211} x -\frac {146}{633}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-9 \textit {\_R}^{4}-406 \textit {\_R}^{3}-324 \textit {\_R}^{2}+96 \textit {\_R} -324\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{205092}\) | \(122\) |
risch | \(\frac {-\frac {1}{3798} x^{5}-\frac {203}{34182} x^{4}-\frac {215}{633} x^{3}-\frac {665}{5697} x^{2}+\frac {2}{211} x -\frac {146}{633}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-9 \textit {\_R}^{4}-406 \textit {\_R}^{3}-324 \textit {\_R}^{2}+96 \textit {\_R} -324\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{205092}\) | \(122\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 112, normalized size = 0.11 \begin {gather*} \operatorname {RootSum} {\left (85256017052964187415123360664576 t^{6} + 50105191533385434568704 t^{4} + 48885748051277486016 t^{3} + 865447782603408 t^{2} + 3220532460 t + 4513, \left ( t \mapsto t \log {\left (\frac {35492036204084174404119193135483487466590764032 t^{5}}{356900697070792948475845} - \frac {19474160067218837086826809631017022308224 t^{4}}{71380139414158589695169} + \frac {20779963076545132233894582764903396544 t^{3}}{356900697070792948475845} + \frac {20265219154367004972162198012037344 t^{2}}{356900697070792948475845} + \frac {275192468949210532049075145372 t}{356900697070792948475845} + x + \frac {1290285191292177289622012}{1070702091212378845427535} \right )} \right )\right )} + \frac {- 9 x^{5} - 203 x^{4} - 11610 x^{3} - 3990 x^{2} + 324 x - 7884}{34182 x^{6} + 615276 x^{4} + 11074968 x^{3} + 3691656 x^{2} + 7383312} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.34, size = 388, normalized size = 0.36 \begin {gather*} \left (\sum _{k=1}^6\ln \left (-\frac {3848128\,\mathrm {root}\left (z^6+\frac {326\,z^4}{554702231619}+\frac {8113597\,z^3}{14149992416343982992}+\frac {5171\,z^2}{509399726988383387712}+\frac {505\,z}{13368686435083133627113728}+\frac {4513}{85256017052964187415123360664576},z,k\right )}{3606201}-\frac {275536\,x}{638827688547}+\frac {\mathrm {root}\left (z^6+\frac {326\,z^4}{554702231619}+\frac {8113597\,z^3}{14149992416343982992}+\frac {5171\,z^2}{509399726988383387712}+\frac {505\,z}{13368686435083133627113728}+\frac {4513}{85256017052964187415123360664576},z,k\right )\,x\,239491904}{876306843}-\frac {{\mathrm {root}\left (z^6+\frac {326\,z^4}{554702231619}+\frac {8113597\,z^3}{14149992416343982992}+\frac {5171\,z^2}{509399726988383387712}+\frac {505\,z}{13368686435083133627113728}+\frac {4513}{85256017052964187415123360664576},z,k\right )}^2\,x\,152363520}{44521}-\frac {{\mathrm {root}\left (z^6+\frac {326\,z^4}{554702231619}+\frac {8113597\,z^3}{14149992416343982992}+\frac {5171\,z^2}{509399726988383387712}+\frac {505\,z}{13368686435083133627113728}+\frac {4513}{85256017052964187415123360664576},z,k\right )}^3\,x\,698075283456}{44521}+\frac {{\mathrm {root}\left (z^6+\frac {326\,z^4}{554702231619}+\frac {8113597\,z^3}{14149992416343982992}+\frac {5171\,z^2}{509399726988383387712}+\frac {505\,z}{13368686435083133627113728}+\frac {4513}{85256017052964187415123360664576},z,k\right )}^4\,x\,130789789876224}{211}-{\mathrm {root}\left (z^6+\frac {326\,z^4}{554702231619}+\frac {8113597\,z^3}{14149992416343982992}+\frac {5171\,z^2}{509399726988383387712}+\frac {505\,z}{13368686435083133627113728}+\frac {4513}{85256017052964187415123360664576},z,k\right )}^5\,x\,6940988288557056-\frac {4264220928\,{\mathrm {root}\left (z^6+\frac {326\,z^4}{554702231619}+\frac {8113597\,z^3}{14149992416343982992}+\frac {5171\,z^2}{509399726988383387712}+\frac {505\,z}{13368686435083133627113728}+\frac {4513}{85256017052964187415123360664576},z,k\right )}^2}{44521}-\frac {5086414725120\,{\mathrm {root}\left (z^6+\frac {326\,z^4}{554702231619}+\frac {8113597\,z^3}{14149992416343982992}+\frac {5171\,z^2}{509399726988383387712}+\frac {505\,z}{13368686435083133627113728}+\frac {4513}{85256017052964187415123360664576},z,k\right )}^3}{44521}+\frac {243585208571904\,{\mathrm {root}\left (z^6+\frac {326\,z^4}{554702231619}+\frac {8113597\,z^3}{14149992416343982992}+\frac {5171\,z^2}{509399726988383387712}+\frac {505\,z}{13368686435083133627113728}+\frac {4513}{85256017052964187415123360664576},z,k\right )}^4}{211}-168897381688221696\,{\mathrm {root}\left (z^6+\frac {326\,z^4}{554702231619}+\frac {8113597\,z^3}{14149992416343982992}+\frac {5171\,z^2}{509399726988383387712}+\frac {505\,z}{13368686435083133627113728}+\frac {4513}{85256017052964187415123360664576},z,k\right )}^5-\frac {48160}{23660284761}\right )\,\mathrm {root}\left (z^6+\frac {326\,z^4}{554702231619}+\frac {8113597\,z^3}{14149992416343982992}+\frac {5171\,z^2}{509399726988383387712}+\frac {505\,z}{13368686435083133627113728}+\frac {4513}{85256017052964187415123360664576},z,k\right )\right )-\frac {\frac {x^5}{3798}+\frac {203\,x^4}{34182}+\frac {215\,x^3}{633}+\frac {665\,x^2}{5697}-\frac {2\,x}{211}+\frac {146}{633}}{x^6+18\,x^4+324\,x^3+108\,x^2+216} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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