\(\int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx\) [2491]

Optimal result

Integrand size = 24, antiderivative size = 206 \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=-\frac {\arctan \left (\frac {-2 3^{5/6}+3\ 3^{5/6} x}{-2 \sqrt [3]{3}+3 \sqrt [3]{3} x+2 \sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}}\right )}{3^{5/6}}+\frac {\log \left (6-9 x+3^{2/3} \sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}\right )}{3 \sqrt [3]{3}}-\frac {\log \left (12-36 x+27 x^2+\left (-2 3^{2/3}+3\ 3^{2/3} x\right ) \sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}+\sqrt [3]{3} \left (-8+12 x+54 x^2-135 x^3+81 x^4\right )^{2/3}\right )}{6 \sqrt [3]{3}} \] Output:

-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)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\frac {(-2+3 x) \sqrt [3]{1+3 x} \left (6 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+3 x}}{3^{5/6}}\right )+\sqrt {3} \left (2 \log \left (-3+3^{2/3} \sqrt [3]{1+3 x}\right )-\log \left (3+3^{2/3} \sqrt [3]{1+3 x}+\sqrt [3]{3} (1+3 x)^{2/3}\right )\right )\right )}{6\ 3^{5/6} \sqrt [3]{(-2+3 x)^3 (1+3 x)}} \] Input:

Integrate[(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(-1/3),x]
 

Output:

((-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))
 

Rubi [F]

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

\(\displaystyle \int \frac {1}{\sqrt [3]{81 x^4-135 x^3+54 x^2+12 x-8}}dx\)

Input:

Int[(-8 + 12*x + 54*x^2 - 135*x^3 + 81*x^4)^(-1/3),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.42 (sec) , antiderivative size = 2061, normalized size of antiderivative = 10.00

method	result	size
trager	\(\text {Expression too large to display}\)	\(2061\)

Input:

int(1/(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

-1/9*ln(-(5103*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)^2* 
RootOf(_Z^3-9)^2*x^3+1620*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+ 
81*_Z^2)*RootOf(_Z^3-9)^3*x^3-6804*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf 
(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x^2-2160*RootOf(16*RootOf(_Z^3-9)^2+3 
6*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x^2+2268*RootOf(16*RootOf(_Z 
^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x+720*RootOf(16*R 
ootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x+540*(81*x 
^4-135*x^3+54*x^2+12*x-8)^(2/3)*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z 
^3-9)+81*_Z^2)*RootOf(_Z^3-9)^2+4860*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)* 
RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)*x+ 
27216*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^3+1008*(8 
1*x^4-135*x^3+54*x^2+12*x-8)^(1/3)*RootOf(_Z^3-9)^2*x+8640*RootOf(_Z^3-9)* 
x^3-3240*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)*RootOf(16*RootOf(_Z^3-9)^2+3 
6*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)-6804*RootOf(16*RootOf(_Z^3-9)^ 
2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^2-672*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/ 
3)*RootOf(_Z^3-9)^2-2160*RootOf(_Z^3-9)*x^2-27216*RootOf(16*RootOf(_Z^3-9) 
^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x-8640*RootOf(_Z^3-9)*x+1008*(81*x^4-135* 
x^3+54*x^2+12*x-8)^(2/3)+13104*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^ 
3-9)+81*_Z^2)+4160*RootOf(_Z^3-9))/(-2+3*x)^3)*RootOf(_Z^3-9)-1/4*ln(-(510 
3*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^...
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=-\frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {2}{3}} {\left (9 \, x^{2} - 12 \, x + 4\right )} + 3^{\frac {1}{3}} {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}} {\left (3 \, x - 2\right )} + {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {2}{3}}}{9 \, x^{2} - 12 \, x + 4}\right ) + \frac {1}{9} \cdot 3^{\frac {2}{3}} \log \left (-\frac {3^{\frac {1}{3}} {\left (3 \, x - 2\right )} - {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}}{3 \, x - 2}\right ) + \frac {1}{3} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} {\left (3 \, x - 2\right )} + 2 \, {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (3 \, x - 2\right )}}\right ) \] Input:

integrate(1/(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3),x, algorithm="fricas")
 

Output:

-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))
 

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\int \frac {1}{\sqrt [3]{81 x^{4} - 135 x^{3} + 54 x^{2} + 12 x - 8}}\, dx \] Input:

integrate(1/(81*x**4-135*x**3+54*x**2+12*x-8)**(1/3),x)
 

Output:

Integral((81*x**4 - 135*x**3 + 54*x**2 + 12*x - 8)**(-1/3), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\int { \frac {1}{{\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3),x, algorithm="maxima")
 

Output:

integrate((81*x^4 - 135*x^3 + 54*x^2 + 12*x - 8)^(-1/3), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\int { \frac {1}{{\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3),x, algorithm="giac")
 

Output:

integrate((81*x^4 - 135*x^3 + 54*x^2 + 12*x - 8)^(-1/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\int \frac {1}{{\left (81\,x^4-135\,x^3+54\,x^2+12\,x-8\right )}^{1/3}} \,d x \] Input:

int(1/(12*x + 54*x^2 - 135*x^3 + 81*x^4 - 8)^(1/3),x)
 

Output:

int(1/(12*x + 54*x^2 - 135*x^3 + 81*x^4 - 8)^(1/3), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\frac {\left (-6 \sqrt {3}\, \mathit {atan} \left (\frac {\left (2 \left (3 x +1\right )^{\frac {1}{6}}+3^{\frac {1}{6}}\right ) 3^{\frac {5}{6}}}{3 \sqrt {3}}\right )+6 \sqrt {3}\, \mathit {atan} \left (\frac {\left (2 \left (3 x +1\right )^{\frac {1}{6}}-3^{\frac {1}{6}}\right ) 3^{\frac {5}{6}}}{3 \sqrt {3}}\right )+6 \,\mathrm {log}\left (\left (3 x +1\right )^{\frac {1}{6}}+3^{\frac {1}{6}}\right )+6 \,\mathrm {log}\left (\left (3 x +1\right )^{\frac {1}{6}}-3^{\frac {1}{6}}\right )-3 \,\mathrm {log}\left (-\left (3 x +1\right )^{\frac {1}{6}} 3^{\frac {1}{6}}+\left (3 x +1\right )^{\frac {1}{3}}+3^{\frac {1}{3}}\right )-3 \,\mathrm {log}\left (\left (3 x +1\right )^{\frac {1}{6}} 3^{\frac {1}{6}}+\left (3 x +1\right )^{\frac {1}{3}}+3^{\frac {1}{3}}\right )\right ) 3^{\frac {2}{3}}}{54} \] Input:

int(1/(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3),x)
 

Output:

( - 2*sqrt(3)*3**(1/3)*3**(1/6)*3**(4/9)*3**(1/18)*atan((2*(3*x + 1)**(1/6 
) + 3**(1/6))/(sqrt(3)*3**(1/6))) + 2*sqrt(3)*3**(1/3)*3**(1/6)*3**(4/9)*3 
**(1/18)*atan((2*(3*x + 1)**(1/6) - 3**(1/6))/(sqrt(3)*3**(1/6))) + 6*log( 
(3*x + 1)**(1/6) + 3**(1/6)) + 6*log((3*x + 1)**(1/6) - 3**(1/6)) - 3*log( 
 - (3*x + 1)**(1/6)*3**(1/6) + (3*x + 1)**(1/3) + 3**(1/3)) - 3*log((3*x + 
 1)**(1/6)*3**(1/6) + (3*x + 1)**(1/3) + 3**(1/3)))/(18*3**(1/3))