Integrand size = 24, antiderivative size = 206 \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=-\frac {\arctan \left (\frac {-2 3^{5/6}+3\ 3^{5/6} x}{-2 \sqrt [3]{3}+3 \sqrt [3]{3} x+2 \sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}}\right )}{3^{5/6}}+\frac {\log \left (6-9 x+3^{2/3} \sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (12-36 x+27 x^2+\left (-2 3^{2/3}+3\ 3^{2/3} x\right ) \sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}+\sqrt [3]{3} \left (-8+12 x+54 x^2-135 x^3+81 x^4\right )^{2/3}\right )}{6 \sqrt [3]{3}} \]
-1/3*arctan((-2*3^(5/6)+3*x*3^(5/6))/(-2*3^(1/3)+3*3^(1/3)*x+2*(81*x^4-135 *x^3+54*x^2+12*x-8)^(1/3)))*3^(1/6)+1/9*ln(6-9*x+3^(2/3)*(81*x^4-135*x^3+5 4*x^2+12*x-8)^(1/3))*3^(2/3)-1/18*ln(12-36*x+27*x^2+(-2*3^(2/3)+3*3^(2/3)* x)*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)+3^(1/3)*(81*x^4-135*x^3+54*x^2+12* x-8)^(2/3))*3^(2/3)
Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\frac {(-2+3 x) \sqrt [3]{1+3 x} \left (6 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+3 x}}{3^{5/6}}\right )+\sqrt {3} \left (2 \log \left (-3+3^{2/3} \sqrt [3]{1+3 x}\right )-\log \left (3+3^{2/3} \sqrt [3]{1+3 x}+\sqrt [3]{3} (1+3 x)^{2/3}\right )\right )\right )}{6\ 3^{5/6} \sqrt [3]{(-2+3 x)^3 (1+3 x)}} \]
((-2 + 3*x)*(1 + 3*x)^(1/3)*(6*ArcTan[1/Sqrt[3] + (2*(1 + 3*x)^(1/3))/3^(5 /6)] + Sqrt[3]*(2*Log[-3 + 3^(2/3)*(1 + 3*x)^(1/3)] - Log[3 + 3^(2/3)*(1 + 3*x)^(1/3) + 3^(1/3)*(1 + 3*x)^(2/3)])))/(6*3^(5/6)*((-2 + 3*x)^3*(1 + 3* x))^(1/3))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt [3]{81 x^4-135 x^3+54 x^2+12 x-8}} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {1}{\sqrt [3]{81 x^4-135 x^3+54 x^2+12 x-8}}dx\) |
3.25.91.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.48 (sec) , antiderivative size = 1366, normalized size of antiderivative = 6.63
1/9*RootOf(_Z^3-9)*ln((729*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9) +81*_Z^2)^2*RootOf(_Z^3-9)^2*x^3-810*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*Root Of(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x^3-972*RootOf(16*RootOf(_Z^3-9)^2+36 *_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x^2+1080*RootOf(16*RootOf(_ Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x^2+324*RootOf(16* RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x-360*Ro otOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x+ 270*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9 )^2*(81*x^4-135*x^3+54*x^2+12*x-8)^(2/3)-1134*RootOf(16*RootOf(_Z^3-9)^2+3 6*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)*(81*x^4-135*x^3+54*x^2+12*x-8) ^(1/3)*x-972*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^3- 1080*RootOf(_Z^3-9)^2*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)*x+1080*RootOf(_ Z^3-9)*x^3+756*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*Ro otOf(_Z^3-9)*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)-2916*RootOf(16*RootOf(_Z ^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^2+720*RootOf(_Z^3-9)^2*(81*x^4-135 *x^3+54*x^2+12*x-8)^(1/3)+3240*RootOf(_Z^3-9)*x^2+5184*RootOf(16*RootOf(_Z ^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x-5760*RootOf(_Z^3-9)*x+576*(81*x^4- 135*x^3+54*x^2+12*x-8)^(2/3)-1872*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf( _Z^3-9)+81*_Z^2)+2080*RootOf(_Z^3-9))/(-2+3*x)^3)+1/4*RootOf(16*RootOf(_Z^ 3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*ln((3645*RootOf(16*RootOf(_Z^3-9)^...
Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=-\frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {2}{3}} {\left (9 \, x^{2} - 12 \, x + 4\right )} + 3^{\frac {1}{3}} {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}} {\left (3 \, x - 2\right )} + {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {2}{3}}}{9 \, x^{2} - 12 \, x + 4}\right ) + \frac {1}{9} \cdot 3^{\frac {2}{3}} \log \left (-\frac {3^{\frac {1}{3}} {\left (3 \, x - 2\right )} - {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}}{3 \, x - 2}\right ) + \frac {1}{3} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} {\left (3 \, x - 2\right )} + 2 \, {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (3 \, x - 2\right )}}\right ) \]
-1/18*3^(2/3)*log((3^(2/3)*(9*x^2 - 12*x + 4) + 3^(1/3)*(81*x^4 - 135*x^3 + 54*x^2 + 12*x - 8)^(1/3)*(3*x - 2) + (81*x^4 - 135*x^3 + 54*x^2 + 12*x - 8)^(2/3))/(9*x^2 - 12*x + 4)) + 1/9*3^(2/3)*log(-(3^(1/3)*(3*x - 2) - (81 *x^4 - 135*x^3 + 54*x^2 + 12*x - 8)^(1/3))/(3*x - 2)) + 1/3*3^(1/6)*arctan (1/3*3^(1/6)*(3^(1/3)*(3*x - 2) + 2*(81*x^4 - 135*x^3 + 54*x^2 + 12*x - 8) ^(1/3))/(3*x - 2))
\[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\int \frac {1}{\sqrt [3]{81 x^{4} - 135 x^{3} + 54 x^{2} + 12 x - 8}}\, dx \]
\[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\int { \frac {1}{{\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\int { \frac {1}{{\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\int \frac {1}{{\left (81\,x^4-135\,x^3+54\,x^2+12\,x-8\right )}^{1/3}} \,d x \]