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

Optimal. Leaf size=208 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3+2\& ,\frac {\log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\frac {3 \left (x^3-x^2\right )^{2/3}}{4 (x-1) x}+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x^2}-2 x\right )}{4 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3-x^2} x+\sqrt [3]{2} \left (x^3-x^2\right )^{2/3}\right )}{8 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x^2}+x}\right )}{4 \sqrt [3]{2}} \]

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Rubi [C]  time = 0.35, antiderivative size = 538, normalized size of antiderivative = 2.59, number of steps used = 10, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2056, 6725, 848, 96, 91, 912} \begin {gather*} -\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}}+\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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 91

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 96

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 848

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*x)/e)^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 912

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 2056

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 6725

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*} \int \frac {1}{\sqrt [3]{-x^2+x^3} \left (-1+x^4\right )} \, dx &=\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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*}

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Mathematica [C]  time = 0.21, size = 392, normalized size = 1.88 \begin {gather*} \frac {x \left (-12-\frac {-\frac {4 \log \left (1-\sqrt [3]{1-i} \sqrt [3]{\frac {x}{x-1}}\right )}{\sqrt [3]{1-i}}-\frac {4 \log \left (1-\sqrt [3]{1+i} \sqrt [3]{\frac {x}{x-1}}\right )}{\sqrt [3]{1+i}}-2\ 2^{2/3} \log \left (1-\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}\right )+\frac {2 \log \left ((1-i)^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{1-i} \sqrt [3]{\frac {x}{x-1}}+1\right )}{\sqrt [3]{1-i}}+\frac {2 \log \left ((1+i)^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{1+i} \sqrt [3]{\frac {x}{x-1}}+1\right )}{\sqrt [3]{1+i}}+2^{2/3} \log \left (2^{2/3} \left (\frac {x}{x-1}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}+1\right )+\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-i} \sqrt [3]{\frac {x}{x-1}}}{\sqrt {3}}\right )}{\sqrt [3]{1-i}}+\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+i} \sqrt [3]{\frac {x}{x-1}}}{\sqrt {3}}\right )}{\sqrt [3]{1+i}}+2\ 2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{2} \sqrt [3]{\frac {x}{x-1}}+1}{\sqrt {3}}\right )}{\sqrt [3]{\frac {x}{x-1}}}\right )}{16 \sqrt [3]{(x-1) x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-12 - ((4*Sqrt[3]*ArcTan[(1 + 2*(1 - I)^(1/3)*(x/(-1 + x))^(1/3))/Sqrt[3]])/(1 - I)^(1/3) + (4*Sqrt[3]*Arc
Tan[(1 + 2*(1 + I)^(1/3)*(x/(-1 + x))^(1/3))/Sqrt[3]])/(1 + I)^(1/3) + 2*2^(2/3)*Sqrt[3]*ArcTan[(1 + 2*2^(1/3)
*(x/(-1 + x))^(1/3))/Sqrt[3]] - (4*Log[1 - (1 - I)^(1/3)*(x/(-1 + x))^(1/3)])/(1 - I)^(1/3) - (4*Log[1 - (1 +
I)^(1/3)*(x/(-1 + x))^(1/3)])/(1 + I)^(1/3) - 2*2^(2/3)*Log[1 - 2^(1/3)*(x/(-1 + x))^(1/3)] + (2*Log[1 + (1 -
I)^(1/3)*(x/(-1 + x))^(1/3) + (1 - I)^(2/3)*(x/(-1 + x))^(2/3)])/(1 - I)^(1/3) + (2*Log[1 + (1 + I)^(1/3)*(x/(
-1 + x))^(1/3) + (1 + I)^(2/3)*(x/(-1 + x))^(2/3)])/(1 + I)^(1/3) + 2^(2/3)*Log[1 + 2^(1/3)*(x/(-1 + x))^(1/3)
+ 2^(2/3)*(x/(-1 + x))^(2/3)])/(x/(-1 + x))^(1/3)))/(16*((-1 + x)*x^2)^(1/3))

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IntegrateAlgebraic [A]  time = 0.00, size = 208, normalized size = 1.00 \begin {gather*} -\frac {3 \left (-x^2+x^3\right )^{2/3}}{4 (-1+x) x}-\frac {\sqrt {3} \tan ^{-1}\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-x^2 + x^3)^(2/3))/(4*(-1 + x)*x) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))])/(4*2^(
1/3)) + Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)]/(4*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3
)*(-x^2 + x^3)^(2/3)]/(8*2^(1/3)) + RootSum[2 - 2*#1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#
1 & ]/4

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fricas [B]  time = 0.92, size = 1790, normalized size = 8.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*2^(5/6)*(x^2 - x)*cos(2/3*arctan(sqrt(2) - 1))*log(-4*(2*2^(2/3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sq
rt(2) - 1))^2 + 2*2^(2/3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(2) - 1))*sin(2/3*arctan(sqrt(2) - 1)) - 2^(1
/3)*x^2 - 2^(2/3)*(x^3 - x^2)^(1/3)*x - (x^3 - x^2)^(2/3))/x^2) + 8*2^(5/6)*(x^2 - x)*arctan(-(2*x*cos(2/3*arc
tan(sqrt(2) - 1))^2 + 2*x*cos(2/3*arctan(sqrt(2) - 1))*sin(2/3*arctan(sqrt(2) - 1)) + 2^(1/3)*x*sqrt(-(2*2^(2/
3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(2) - 1))^2 + 2*2^(2/3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(2) -
1))*sin(2/3*arctan(sqrt(2) - 1)) - 2^(1/3)*x^2 - 2^(2/3)*(x^3 - x^2)^(1/3)*x - (x^3 - x^2)^(2/3))/x^2) - x -
2^(1/3)*(x^3 - x^2)^(1/3))/(2*x*cos(2/3*arctan(sqrt(2) - 1))^2 - 2*x*cos(2/3*arctan(sqrt(2) - 1))*sin(2/3*arct
an(sqrt(2) - 1)) - x))*sin(2/3*arctan(sqrt(2) - 1)) + 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^(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) - 4*(sqrt(3)*2^(5/6)*
(x^2 - x)*cos(2/3*arctan(sqrt(2) - 1)) + 2^(5/6)*(x^2 - x)*sin(2/3*arctan(sqrt(2) - 1)))*arctan(1/2*(32*x*cos(
2/3*arctan(sqrt(2) - 1))^4 - 4*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3) + 2^(1/3))*cos(2/3*arctan(sqrt(2) - 1))*sin(
2/3*arctan(sqrt(2) - 1)) - 4*((x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3) - 2^(1/3)) + 8*x)*cos(2/3*arctan(sqrt(2) - 1)
)^2 + sqrt(2)*(2*(sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*cos(2/3*arctan(sqrt(2) - 1))^2 + 2*(sqrt(3)*2^(1/3)*x + 2^(1/
3)*x)*cos(2/3*arctan(sqrt(2) - 1))*sin(2/3*arctan(sqrt(2) - 1)) - sqrt(3)*2^(1/3)*x + 2^(1/3)*x)*sqrt(-(2*(x^3
- x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x)*cos(2/3*arctan(sqrt(2) - 1))^2 - 2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^
(2/3)*x + 2^(2/3)*x)*cos(2/3*arctan(sqrt(2) - 1))*sin(2/3*arctan(sqrt(2) - 1)) - 2*2^(1/3)*x^2 - (x^3 - x^2)^(
1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x) - 2*(x^3 - x^2)^(2/3))/x^2) - 2*sqrt(3)*x + 2*(x^3 - x^2)^(1/3)*(sqrt(3)*
2^(1/3) - 2^(1/3)) + 4*x)/(8*(2*x*cos(2/3*arctan(sqrt(2) - 1))^3 - x*cos(2/3*arctan(sqrt(2) - 1)))*sin(2/3*arc
tan(sqrt(2) - 1)) + x)) - 4*(sqrt(3)*2^(5/6)*(x^2 - x)*cos(2/3*arctan(sqrt(2) - 1)) - 2^(5/6)*(x^2 - x)*sin(2/
3*arctan(sqrt(2) - 1)))*arctan(-1/2*(32*x*cos(2/3*arctan(sqrt(2) - 1))^4 + 4*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3
) - 2^(1/3))*cos(2/3*arctan(sqrt(2) - 1))*sin(2/3*arctan(sqrt(2) - 1)) + 4*((x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3)
+ 2^(1/3)) - 8*x)*cos(2/3*arctan(sqrt(2) - 1))^2 - sqrt(2)*(2*(sqrt(3)*2^(1/3)*x + 2^(1/3)*x)*cos(2/3*arctan(
sqrt(2) - 1))^2 + 2*(sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*cos(2/3*arctan(sqrt(2) - 1))*sin(2/3*arctan(sqrt(2) - 1))
- sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*sqrt((2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2^(2/3)*x)*cos(2/3*arctan(sqrt
(2) - 1))^2 - 2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x)*cos(2/3*arctan(sqrt(2) - 1))*sin(2/3*arctan(
sqrt(2) - 1)) + 2*2^(1/3)*x^2 - (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2^(2/3)*x) + 2*(x^3 - x^2)^(2/3))/x^2)
+ 2*sqrt(3)*x - 2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3) + 2^(1/3)) + 4*x)/(8*(2*x*cos(2/3*arctan(sqrt(2) - 1))^3
- x*cos(2/3*arctan(sqrt(2) - 1)))*sin(2/3*arctan(sqrt(2) - 1)) + x)) - (sqrt(3)*2^(5/6)*(x^2 - x)*sin(2/3*arct
an(sqrt(2) - 1)) + 2^(5/6)*(x^2 - x)*cos(2/3*arctan(sqrt(2) - 1)))*log(8*(2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)
*x + 2^(2/3)*x)*cos(2/3*arctan(sqrt(2) - 1))^2 - 2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x)*cos(2/3*a
rctan(sqrt(2) - 1))*sin(2/3*arctan(sqrt(2) - 1)) + 2*2^(1/3)*x^2 - (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2^(2
/3)*x) + 2*(x^3 - x^2)^(2/3))/x^2) + (sqrt(3)*2^(5/6)*(x^2 - x)*sin(2/3*arctan(sqrt(2) - 1)) - 2^(5/6)*(x^2 -
x)*cos(2/3*arctan(sqrt(2) - 1)))*log(-8*(2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x)*cos(2/3*arctan(sq
rt(2) - 1))^2 - 2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2^(2/3)*x)*cos(2/3*arctan(sqrt(2) - 1))*sin(2/3*arcta
n(sqrt(2) - 1)) - 2*2^(1/3)*x^2 - (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x) - 2*(x^3 - x^2)^(2/3))/x^2
) - 12*(x^3 - x^2)^(2/3))/(x^2 - x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{4} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 151.21, size = 20141, normalized size = 96.83

method	result	size
risch	\(\text {Expression too large to display}\)	\(20141\)
trager	\(\text {Expression too large to display}\)	\(27937\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{4} - 1\right )} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\left (x^4-1\right )\,{\left (x^3-x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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