\(\int \frac {1}{\sqrt [3]{-x^2+x^3} (-1+x^4)} \, dx\) [2499]

Optimal result

Integrand size = 21, antiderivative size = 208 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=-\frac {3 \left (-x^2+x^3\right )^{2/3}}{4 (-1+x) x}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{4 \sqrt [3]{2}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{4 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x^2+x^3}+\sqrt [3]{2} \left (-x^2+x^3\right )^{2/3}\right )}{8 \sqrt [3]{2}}+\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 538, normalized size of antiderivative = 2.59, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2081, 6857, 862, 98, 93, 926} \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1-i} \sqrt {3} \sqrt [3]{x}}\right )}{4 \sqrt [3]{1-i} \sqrt [3]{x^3-x^2}}+\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1+i} \sqrt {3} \sqrt [3]{x}}\right )}{4 \sqrt [3]{1+i} \sqrt [3]{x^3-x^2}}+\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {3 x}{4 \sqrt [3]{x^3-x^2}}+\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{x-1}}{\sqrt [3]{1-i}}\right )}{8 \sqrt [3]{1-i} \sqrt [3]{x^3-x^2}}+\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{x-1}}{\sqrt [3]{1+i}}\right )}{8 \sqrt [3]{1+i} \sqrt [3]{x^3-x^2}}+\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (-x-1)}{8 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (-x+i)}{8 \sqrt [3]{1+i} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x+i)}{8 \sqrt [3]{1-i} \sqrt [3]{x^3-x^2}} \]

[In]

Int[1/((-x^2 + x^3)^(1/3)*(-1 + x^4)),x]

[Out]

(-3*x)/(4*(-x^2 + x^3)^(1/3)) + (Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/((1 - I)
^(1/3)*Sqrt[3]*x^(1/3))])/(4*(1 - I)^(1/3)*(-x^2 + x^3)^(1/3)) + (Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt
[3] + (2*(-1 + x)^(1/3))/((1 + I)^(1/3)*Sqrt[3]*x^(1/3))])/(4*(1 + I)^(1/3)*(-x^2 + x^3)^(1/3)) + (Sqrt[3]*(-1
+ x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2^(2/3)*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(4*2^(1/3)*(-x^2 + x^3)^(1
/3)) + (3*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/(1 - I)^(1/3) - x^(1/3)])/(8*(1 - I)^(1/3)*(-x^2 + x^3)^(1
/3)) + (3*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/(1 + I)^(1/3) - x^(1/3)])/(8*(1 + I)^(1/3)*(-x^2 + x^3)^(1
/3)) + (3*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/2^(1/3) - x^(1/3)])/(8*2^(1/3)*(-x^2 + x^3)^(1/3)) - ((-1
+ x)^(1/3)*x^(2/3)*Log[-1 - x])/(8*2^(1/3)*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*Log[I - x])/(8*(1 + I
)^(1/3)*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*Log[I + x])/(8*(1 - I)^(1/3)*(-x^2 + x^3)^(1/3))

Rule 93

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])*q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1
/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q*(a + b*x)^(1/3) - (c +
d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 862

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 926

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (-1+x^4\right )} \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (-\frac {1}{2 \sqrt [3]{-1+x} x^{2/3} \left (1-x^2\right )}-\frac {1}{2 \sqrt [3]{-1+x} x^{2/3} \left (1+x^2\right )}\right ) \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = -\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (1-x^2\right )} \, dx}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} \left (1+x^2\right )} \, dx}{2 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(-1-x) (-1+x)^{4/3} x^{2/3}} \, dx}{2 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (\frac {i}{2 (i-x) \sqrt [3]{-1+x} x^{2/3}}+\frac {i}{2 \sqrt [3]{-1+x} x^{2/3} (i+x)}\right ) \, dx}{2 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {3 x}{4 \sqrt [3]{-x^2+x^3}}-\frac {\left (i \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(i-x) \sqrt [3]{-1+x} x^{2/3}} \, dx}{4 \sqrt [3]{-x^2+x^3}}-\frac {\left (i \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} (i+x)} \, dx}{4 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(-1-x) \sqrt [3]{-1+x} x^{2/3}} \, dx}{4 \sqrt [3]{-x^2+x^3}} \\ & = -\frac {3 x}{4 \sqrt [3]{-x^2+x^3}}+\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt [3]{1-i} \sqrt {3} \sqrt [3]{x}}\right )}{4 \sqrt [3]{1-i} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt [3]{1+i} \sqrt {3} \sqrt [3]{x}}\right )}{4 \sqrt [3]{1+i} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{4 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}+\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-i}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{1-i} \sqrt [3]{-x^2+x^3}}+\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+i}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{1+i} \sqrt [3]{-x^2+x^3}}+\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (-1-x)}{8 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (i-x)}{8 \sqrt [3]{1+i} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (i+x)}{8 \sqrt [3]{1-i} \sqrt [3]{-x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=-\frac {x^{2/3} \left (12 \sqrt [3]{x}+2\ 2^{2/3} \sqrt {3} \sqrt [3]{-1+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2^{2/3} \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2\ 2^{2/3} \sqrt [3]{-1+x} \log \left (2^{2/3} \sqrt [3]{-1+x}-2 \sqrt [3]{x}\right )+2^{2/3} \sqrt [3]{-1+x} \log \left (\sqrt [3]{2} (-1+x)^{2/3}+2^{2/3} \sqrt [3]{-1+x} \sqrt [3]{x}+2 x^{2/3}\right )-4 \sqrt [3]{-1+x} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{16 \sqrt [3]{(-1+x) x^2}} \]

[In]

Integrate[1/((-x^2 + x^3)^(1/3)*(-1 + x^4)),x]

[Out]

-1/16*(x^(2/3)*(12*x^(1/3) + 2*2^(2/3)*Sqrt[3]*(-1 + x)^(1/3)*ArcTan[(Sqrt[3]*x^(1/3))/(2^(2/3)*(-1 + x)^(1/3)
+ x^(1/3))] - 2*2^(2/3)*(-1 + x)^(1/3)*Log[2^(2/3)*(-1 + x)^(1/3) - 2*x^(1/3)] + 2^(2/3)*(-1 + x)^(1/3)*Log[2
^(1/3)*(-1 + x)^(2/3) + 2^(2/3)*(-1 + x)^(1/3)*x^(1/3) + 2*x^(2/3)] - 4*(-1 + x)^(1/3)*RootSum[2 - 2*#1^3 + #1
^6 & , (-Log[x^(1/3)] + Log[(-1 + x)^(1/3) - x^(1/3)*#1])/#1 & ]))/((-1 + x)*x^2)^(1/3)

Maple [N/A] (verified)

Time = 55.42 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.91

method	result	size
pseudoelliptic	\(-\frac {-2 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-2 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}+12 x}{16 \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}\)	\(189\)
trager	\(\text {Expression too large to display}\)	\(18221\)
risch	\(\text {Expression too large to display}\)	\(19795\)

[In]

int(1/(x^3-x^2)^(1/3)/(x^4-1),x,method=_RETURNVERBOSE)

[Out]

-1/16*(-2*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(2^(2/3)*((-1+x)*x^2)^(1/3)+x)/x)*((-1+x)*x^2)^(1/3)-2*2^(2/3)*ln
((-2^(1/3)*x+((-1+x)*x^2)^(1/3))/x)*((-1+x)*x^2)^(1/3)+2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*((-1+x)*x^2)^(1/3)*x+((
-1+x)*x^2)^(2/3))/x^2)*((-1+x)*x^2)^(1/3)-4*sum(ln((-_R*x+((-1+x)*x^2)^(1/3))/x)/_R,_R=RootOf(_Z^6-2*_Z^3+2))*
((-1+x)*x^2)^(1/3)+12*x)/((-1+x)*x^2)^(1/3)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.27 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.40 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=\frac {2 \, \sqrt {3} 2^{\frac {2}{3}} {\left (x^{2} - x\right )} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} x + 2 \, \sqrt {2} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) - 2^{\frac {2}{3}} \left (-i + 1\right )^{\frac {1}{3}} {\left (x^{2} - \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} \log \left (\frac {2^{\frac {1}{3}} \left (-i + 1\right )^{\frac {2}{3}} {\left (\left (i + 1\right ) \, \sqrt {-3} x + \left (i + 1\right ) \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2^{\frac {2}{3}} \left (i + 1\right )^{\frac {1}{3}} {\left (x^{2} - \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} \log \left (\frac {2^{\frac {1}{3}} \left (i + 1\right )^{\frac {2}{3}} {\left (-\left (i - 1\right ) \, \sqrt {-3} x - \left (i - 1\right ) \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2^{\frac {2}{3}} \left (i + 1\right )^{\frac {1}{3}} {\left (x^{2} + \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} \log \left (\frac {2^{\frac {1}{3}} \left (i + 1\right )^{\frac {2}{3}} {\left (\left (i - 1\right ) \, \sqrt {-3} x - \left (i - 1\right ) \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2^{\frac {2}{3}} \left (-i + 1\right )^{\frac {1}{3}} {\left (x^{2} + \sqrt {-3} {\left (x^{2} - x\right )} - x\right )} \log \left (\frac {2^{\frac {1}{3}} \left (-i + 1\right )^{\frac {2}{3}} {\left (-\left (i + 1\right ) \, \sqrt {-3} x + \left (i + 1\right ) \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 2 \cdot 2^{\frac {2}{3}} \left (i + 1\right )^{\frac {1}{3}} {\left (x^{2} - x\right )} \log \left (\frac {\left (i - 1\right ) \cdot 2^{\frac {1}{3}} \left (i + 1\right )^{\frac {2}{3}} x + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 2 \cdot 2^{\frac {2}{3}} \left (-i + 1\right )^{\frac {1}{3}} {\left (x^{2} - x\right )} \log \left (\frac {-\left (i + 1\right ) \cdot 2^{\frac {1}{3}} \left (-i + 1\right )^{\frac {2}{3}} x + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 2 \cdot 2^{\frac {2}{3}} {\left (x^{2} - x\right )} \log \left (-\frac {2^{\frac {1}{3}} x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2^{\frac {2}{3}} {\left (x^{2} - x\right )} \log \left (\frac {2^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 12 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{16 \, {\left (x^{2} - x\right )}} \]

[In]

integrate(1/(x^3-x^2)^(1/3)/(x^4-1),x, algorithm="fricas")

[Out]

1/16*(2*sqrt(3)*2^(2/3)*(x^2 - x)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*x + 2*sqrt(2)*(x^3 - x^2)^(1/3))/x) - 2^
(2/3)*(-I + 1)^(1/3)*(x^2 - sqrt(-3)*(x^2 - x) - x)*log((2^(1/3)*(-I + 1)^(2/3)*((I + 1)*sqrt(-3)*x + (I + 1)*
x) + 4*(x^3 - x^2)^(1/3))/x) - 2^(2/3)*(I + 1)^(1/3)*(x^2 - sqrt(-3)*(x^2 - x) - x)*log((2^(1/3)*(I + 1)^(2/3)
*(-(I - 1)*sqrt(-3)*x - (I - 1)*x) + 4*(x^3 - x^2)^(1/3))/x) - 2^(2/3)*(I + 1)^(1/3)*(x^2 + sqrt(-3)*(x^2 - x)
- x)*log((2^(1/3)*(I + 1)^(2/3)*((I - 1)*sqrt(-3)*x - (I - 1)*x) + 4*(x^3 - x^2)^(1/3))/x) - 2^(2/3)*(-I + 1)
^(1/3)*(x^2 + sqrt(-3)*(x^2 - x) - x)*log((2^(1/3)*(-I + 1)^(2/3)*(-(I + 1)*sqrt(-3)*x + (I + 1)*x) + 4*(x^3 -
x^2)^(1/3))/x) + 2*2^(2/3)*(I + 1)^(1/3)*(x^2 - x)*log(((I - 1)*2^(1/3)*(I + 1)^(2/3)*x + 2*(x^3 - x^2)^(1/3)
)/x) + 2*2^(2/3)*(-I + 1)^(1/3)*(x^2 - x)*log((-(I + 1)*2^(1/3)*(-I + 1)^(2/3)*x + 2*(x^3 - x^2)^(1/3))/x) + 2
*2^(2/3)*(x^2 - x)*log(-(2^(1/3)*x - (x^3 - x^2)^(1/3))/x) - 2^(2/3)*(x^2 - x)*log((2^(2/3)*x^2 + 2^(1/3)*(x^3
- x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2) - 12*(x^3 - x^2)^(2/3))/(x^2 - x)

Sympy [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.12 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]

[In]

integrate(1/(x**3-x**2)**(1/3)/(x**4-1),x)

[Out]

Integral(1/((x**2*(x - 1))**(1/3)*(x - 1)*(x + 1)*(x**2 + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.10 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {1}{{\left (x^{4} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(1/(x^3-x^2)^(1/3)/(x^4-1),x, algorithm="maxima")

[Out]

integrate(1/((x^4 - 1)*(x^3 - x^2)^(1/3)), x)

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.10 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=\int { \frac {1}{{\left (x^{4} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(1/(x^3-x^2)^(1/3)/(x^4-1),x, algorithm="giac")

[Out]

integrate(1/((x^4 - 1)*(x^3 - x^2)^(1/3)), x)

Mupad [N/A]

Not integrable

Time = 6.95 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.10 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx=\int \frac {1}{\left (x^4-1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \]

[In]

int(1/((x^4 - 1)*(x^3 - x^2)^(1/3)),x)

[Out]

int(1/((x^4 - 1)*(x^3 - x^2)^(1/3)), x)