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

Optimal result

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

[Out]

-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+54*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)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.83, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6820, 6851, 57, 631, 210, 31} \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=-\frac {(2-3 x) \sqrt [3]{3 x+1} \arctan \left (\frac {2 \sqrt [3]{3 x+1}+\sqrt [3]{3}}{3^{5/6}}\right )}{3^{5/6} \sqrt [3]{-(2-3 x)^3 (3 x+1)}}+\frac {(2-3 x) \sqrt [3]{3 x+1} \log (2-3 x)}{6 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (3 x+1)}}-\frac {(2-3 x) \sqrt [3]{3 x+1} \log \left (\sqrt [3]{3}-\sqrt [3]{3 x+1}\right )}{2 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (3 x+1)}} \]

[In]

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

[Out]

-(((2 - 3*x)*(1 + 3*x)^(1/3)*ArcTan[(3^(1/3) + 2*(1 + 3*x)^(1/3))/3^(5/6)])/(3^(5/6)*(-((2 - 3*x)^3*(1 + 3*x))
)^(1/3))) + ((2 - 3*x)*(1 + 3*x)^(1/3)*Log[2 - 3*x])/(6*3^(1/3)*(-((2 - 3*x)^3*(1 + 3*x)))^(1/3)) - ((2 - 3*x)
*(1 + 3*x)^(1/3)*Log[3^(1/3) - (1 + 3*x)^(1/3)])/(2*3^(1/3)*(-((2 - 3*x)^3*(1 + 3*x)))^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[p])*w^(n*FracPart[p]))), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt [3]{(-2+3 x)^3 (1+3 x)}} \, dx \\ & = \frac {\left ((-2+3 x) \sqrt [3]{1+3 x}\right ) \int \frac {1}{(-2+3 x) \sqrt [3]{1+3 x}} \, dx}{\sqrt [3]{(-2+3 x)^3 (1+3 x)}} \\ & = \frac {(2-3 x) \sqrt [3]{1+3 x} \log (2-3 x)}{6 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (1+3 x)}}+\frac {\left ((-2+3 x) \sqrt [3]{1+3 x}\right ) \text {Subst}\left (\int \frac {1}{3^{2/3}+\sqrt [3]{3} x+x^2} \, dx,x,\sqrt [3]{1+3 x}\right )}{2 \sqrt [3]{(-2+3 x)^3 (1+3 x)}}-\frac {\left ((-2+3 x) \sqrt [3]{1+3 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{3}-x} \, dx,x,\sqrt [3]{1+3 x}\right )}{2 \sqrt [3]{3} \sqrt [3]{(-2+3 x)^3 (1+3 x)}} \\ & = \frac {(2-3 x) \sqrt [3]{1+3 x} \log (2-3 x)}{6 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (1+3 x)}}-\frac {(2-3 x) \sqrt [3]{1+3 x} \log \left (\sqrt [3]{3}-\sqrt [3]{1+3 x}\right )}{2 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (1+3 x)}}-\frac {\left ((-2+3 x) \sqrt [3]{1+3 x}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\frac {1}{3}+x}\right )}{\sqrt [3]{3} \sqrt [3]{(-2+3 x)^3 (1+3 x)}} \\ & = -\frac {(2-3 x) \sqrt [3]{1+3 x} \arctan \left (\frac {1}{3} \left (\sqrt {3}+2 \sqrt [6]{3} \sqrt [3]{1+3 x}\right )\right )}{3^{5/6} \sqrt [3]{-(2-3 x)^3 (1+3 x)}}+\frac {(2-3 x) \sqrt [3]{1+3 x} \log (2-3 x)}{6 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (1+3 x)}}-\frac {(2-3 x) \sqrt [3]{1+3 x} \log \left (\sqrt [3]{3}-\sqrt [3]{1+3 x}\right )}{2 \sqrt [3]{3} \sqrt [3]{-(2-3 x)^3 (1+3 x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=\frac {(-2+3 x) \sqrt [3]{1+3 x} \left (6 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+3 x}}{3^{5/6}}\right )+\sqrt {3} \left (2 \log \left (-3+3^{2/3} \sqrt [3]{1+3 x}\right )-\log \left (3+3^{2/3} \sqrt [3]{1+3 x}+\sqrt [3]{3} (1+3 x)^{2/3}\right )\right )\right )}{6\ 3^{5/6} \sqrt [3]{(-2+3 x)^3 (1+3 x)}} \]

[In]

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

[Out]

((-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))

Maple [C] (warning: unable to verify)

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

method	result	size
trager	\(\text {Expression too large to display}\)	\(1366\)

[In]

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

[Out]

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*RootOf(_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-36
0*RootOf(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+3
6*_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+36*_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)*RootOf(_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+3
6*_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)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x^3-648*Root
Of(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x^3-4860*RootOf(16*RootOf(_Z^3-9)^2+36*_
Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x^2+864*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*
RootOf(_Z^3-9)^3*x^2+1620*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)^2*RootOf(_Z^3-9)^2*x-288*Ro
otOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*RootOf(_Z^3-9)^3*x-540*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)+2592*RootOf(16*RootOf(_Z^3-9)^2
+36*_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+19440*RootOf(16*RootOf(_Z
^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^3+2160*RootOf(_Z^3-9)^2*(81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)*x-3456*Ro
otOf(_Z^3-9)*x^3-1728*RootOf(16*RootOf(_Z^3-9)^2+36*_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)-4860*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-9)+81*_Z^2)*x^2-1440*RootOf(_Z^3-9)^2*(
81*x^4-135*x^3+54*x^2+12*x-8)^(1/3)+864*RootOf(_Z^3-9)*x^2-19440*RootOf(16*RootOf(_Z^3-9)^2+36*_Z*RootOf(_Z^3-
9)+81*_Z^2)*x+3456*RootOf(_Z^3-9)*x-1008*(81*x^4-135*x^3+54*x^2+12*x-8)^(2/3)+9360*RootOf(16*RootOf(_Z^3-9)^2+
36*_Z*RootOf(_Z^3-9)+81*_Z^2)-1664*RootOf(_Z^3-9))/(-2+3*x)^3)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt [3]{-8+12 x+54 x^2-135 x^3+81 x^4}} \, dx=-\frac {1}{18} \cdot 3^{\frac {2}{3}} \log \left (\frac {3^{\frac {2}{3}} {\left (9 \, x^{2} - 12 \, x + 4\right )} + 3^{\frac {1}{3}} {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}} {\left (3 \, x - 2\right )} + {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {2}{3}}}{9 \, x^{2} - 12 \, x + 4}\right ) + \frac {1}{9} \cdot 3^{\frac {2}{3}} \log \left (-\frac {3^{\frac {1}{3}} {\left (3 \, x - 2\right )} - {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}}{3 \, x - 2}\right ) + \frac {1}{3} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} {\left (3 \, x - 2\right )} + 2 \, {\left (81 \, x^{4} - 135 \, x^{3} + 54 \, x^{2} + 12 \, x - 8\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (3 \, x - 2\right )}}\right ) \]

[In]

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

[Out]

-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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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