3.26.22 \(\int \frac {1}{x^3 (1+x^3) \sqrt [3]{-x^2+x^3}} \, dx\)
Optimal. Leaf size=211 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {\log \left (2^{2/3} \sqrt [3]{x^3-x^2}-2 x\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3-x^2} x+\sqrt [3]{2} \left (x^3-x^2\right )^{2/3}\right )}{6 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x^2}+x}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {3 \left (x^3-x^2\right )^{2/3} \left (9 x^2+6 x+5\right )}{40 x^4} \]
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Rubi [C] time = 0.78, antiderivative size = 774, normalized size of antiderivative = 3.67,
number of steps used = 18, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used
= {2056, 6725, 129, 155, 12, 91} \begin {gather*} -\frac {\left (3-4 (-1)^{2/3}\right ) (1-x)}{20 x \sqrt [3]{x^3-x^2}}-\frac {\left (3+4 \sqrt [3]{-1}\right ) (1-x)}{20 x \sqrt [3]{x^3-x^2}}+\frac {1-x}{20 x \sqrt [3]{x^3-x^2}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{x^3-x^2}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{x^3-x^2}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{x^3-x^2}}-\frac {17 (1-x)}{40 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (-x-1)}{6 \sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\sqrt [3]{-1} x-1\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (-(-1)^{2/3} x-1\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}}+\frac {\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 )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3-x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^3*(1 + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
(-17*(1 - x))/(40*(-x^2 + x^3)^(1/3)) - ((5 - (16*I)*Sqrt[3])*(1 - x))/(40*(-x^2 + x^3)^(1/3)) - ((5 + (16*I)*
Sqrt[3])*(1 - x))/(40*(-x^2 + x^3)^(1/3)) - (3*(1 - x))/(8*x^2*(-x^2 + x^3)^(1/3)) + (1 - x)/(20*x*(-x^2 + x^3
)^(1/3)) - ((3 + 4*(-1)^(1/3))*(1 - x))/(20*x*(-x^2 + x^3)^(1/3)) - ((3 - 4*(-1)^(2/3))*(1 - x))/(20*x*(-x^2 +
x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2^(2/3)*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(2^(1/3
)*Sqrt[3]*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*(1 - (-
1)^(1/3))^(1/3)*x^(1/3))])/(Sqrt[3]*(1 - (-1)^(1/3))^(1/3)*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)*ArcTa
n[1/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*(1 + (-1)^(2/3))^(1/3)*x^(1/3))])/(Sqrt[3]*(1 + (-1)^(2/3))^(1/3)*(-
x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/2^(1/3) - x^(1/3)])/(2*2^(1/3)*(-x^2 + x^3)^(1/
3)) + ((-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/(1 - (-1)^(1/3))^(1/3) - x^(1/3)])/(2*(1 - (-1)^(1/3))^(1/3)*
(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/(1 + (-1)^(2/3))^(1/3) - x^(1/3)])/(2*(1 + (-
1)^(2/3))^(1/3)*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*Log[-1 - x])/(6*2^(1/3)*(-x^2 + x^3)^(1/3)) - ((
-1 + x)^(1/3)*x^(2/3)*Log[-1 + (-1)^(1/3)*x])/(6*(1 - (-1)^(1/3))^(1/3)*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*
x^(2/3)*Log[-1 - (-1)^(2/3)*x])/(6*(1 + (-1)^(2/3))^(1/3)*(-x^2 + x^3)^(1/3))
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 129
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[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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && S
umSimplerQ[p, 1])))
Rule 155
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) && !(NeQ[p, -1] && SumSimplerQ[p, 1])))
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}{x^3 \left (1+x^3\right ) \sqrt [3]{-x^2+x^3}} \, dx &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{11/3} \left (1+x^3\right )} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \left (-\frac {1}{3 (-1-x) \sqrt [3]{-1+x} x^{11/3}}-\frac {1}{3 \sqrt [3]{-1+x} x^{11/3} \left (-1+\sqrt [3]{-1} x\right )}-\frac {1}{3 \sqrt [3]{-1+x} x^{11/3} \left (-1-(-1)^{2/3} x\right )}\right ) \, dx}{\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^{11/3}} \, dx}{3 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{11/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{-x^2+x^3}}-\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{11/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\frac {2}{3}-2 x}{(-1-x) \sqrt [3]{-1+x} x^{8/3}} \, dx}{8 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {-\frac {2}{3} \left (3+4 \sqrt [3]{-1}\right )+2 \sqrt [3]{-1} x}{\sqrt [3]{-1+x} x^{8/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{8 \sqrt [3]{-x^2+x^3}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {-\frac {2}{3} \left (3-4 (-1)^{2/3}\right )-2 (-1)^{2/3} x}{\sqrt [3]{-1+x} x^{8/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{8 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3+4 \sqrt [3]{-1}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3-4 (-1)^{2/3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\frac {34}{9}-\frac {2 x}{3}}{(-1-x) \sqrt [3]{-1+x} x^{5/3}} \, dx}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\frac {2}{9} \left (5+16 i \sqrt {3}\right )-\frac {2}{3} \sqrt [3]{-1} \left (3+4 \sqrt [3]{-1}\right ) x}{\sqrt [3]{-1+x} x^{5/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (3 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {\frac {2}{9} \left (5-16 i \sqrt {3}\right )+\frac {2}{3} (-1)^{2/3} \left (3-4 (-1)^{2/3}\right ) x}{\sqrt [3]{-1+x} x^{5/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{40 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {17 (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3+4 \sqrt [3]{-1}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3-4 (-1)^{2/3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}+\frac {\left (9 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {80}{27 (-1-x) \sqrt [3]{-1+x} x^{2/3}} \, dx}{80 \sqrt [3]{-x^2+x^3}}+\frac {\left (9 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {80}{27 \sqrt [3]{-1+x} x^{2/3} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{80 \sqrt [3]{-x^2+x^3}}+\frac {\left (9 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {80}{27 \sqrt [3]{-1+x} x^{2/3} \left (-1-(-1)^{2/3} x\right )} \, dx}{80 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {17 (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3+4 \sqrt [3]{-1}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3-4 (-1)^{2/3}\right ) (1-x)}{20 x \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}{3 \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+\sqrt [3]{-1} x\right )} \, dx}{3 \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-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {17 (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5-16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {\left (5+16 i \sqrt {3}\right ) (1-x)}{40 \sqrt [3]{-x^2+x^3}}-\frac {3 (1-x)}{8 x^2 \sqrt [3]{-x^2+x^3}}+\frac {1-x}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3+4 \sqrt [3]{-1}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}-\frac {\left (3-4 (-1)^{2/3}\right ) (1-x)}{20 x \sqrt [3]{-x^2+x^3}}+\frac {\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 )}{\sqrt [3]{2} \sqrt {3} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1-\sqrt [3]{-1}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}+\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+(-1)^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (-1-x)}{6 \sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\sqrt [3]{-1} x\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1-(-1)^{2/3} x\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{-x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.31, size = 173, normalized size = 0.82 \begin {gather*} -\frac {(x-1)^2 \left (10 x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x-1}{2 x}\right )+10 \left (1-i \sqrt {3}\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {2 i (x-1)}{\left (-i+\sqrt {3}\right ) x}\right )+10 i \sqrt {3} x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 i (x-1)}{\left (i+\sqrt {3}\right ) x}\right )+10 x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {2 i (x-1)}{\left (i+\sqrt {3}\right ) x}\right )-27 x^2-18 x-15\right )}{40 \left ((x-1) x^2\right )^{4/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x^3*(1 + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
-1/40*((-1 + x)^2*(-15 - 18*x - 27*x^2 + 10*x^2*Hypergeometric2F1[2/3, 1, 5/3, (-1 + x)/(2*x)] + 10*(1 - I*Sqr
t[3])*x^2*Hypergeometric2F1[2/3, 1, 5/3, ((-2*I)*(-1 + x))/((-I + Sqrt[3])*x)] + 10*x^2*Hypergeometric2F1[2/3,
1, 5/3, ((2*I)*(-1 + x))/((I + Sqrt[3])*x)] + (10*I)*Sqrt[3]*x^2*Hypergeometric2F1[2/3, 1, 5/3, ((2*I)*(-1 +
x))/((I + Sqrt[3])*x)]))/((-1 + x)*x^2)^(4/3)
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IntegrateAlgebraic [A] time = 0.53, size = 211, normalized size = 1.00 \begin {gather*} \frac {3 \left (5+6 x+9 x^2\right ) \left (-x^2+x^3\right )^{2/3}}{40 x^4}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )}{3 \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 )}{6 \sqrt [3]{2}}+\frac {1}{3} \text {RootSum}\left [1-\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^3*(1 + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
(3*(5 + 6*x + 9*x^2)*(-x^2 + x^3)^(2/3))/(40*x^4) - ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2 + x^3)^(1/3))]/(2^(1
/3)*Sqrt[3]) + Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)]/(3*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)]/(6*2^(1/3)) + RootSum[1 - #1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#
1])/#1 & ]/3
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fricas [B] time = 0.86, size = 895, normalized size = 4.24
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(x^3+1)/(x^3-x^2)^(1/3),x, algorithm="fricas")
[Out]
1/120*(40*x^4*cos(1/9*pi)*log(16*(x^2 - (2*sqrt(3)*x*cos(1/9*pi)*sin(1/9*pi) + 2*x*cos(1/9*pi)^2 - x)*(x^3 - x
^2)^(1/3) + (x^3 - x^2)^(2/3))/x^2) - 160*x^4*arctan((8*(2*x*cos(1/9*pi)^3 - x*cos(1/9*pi))*sin(1/9*pi) + sqrt
(3)*x + 2*(2*sqrt(3)*x*cos(1/9*pi)^2 + 2*x*cos(1/9*pi)*sin(1/9*pi) - sqrt(3)*x)*sqrt((x^2 - (2*sqrt(3)*x*cos(1
/9*pi)*sin(1/9*pi) + 2*x*cos(1/9*pi)^2 - x)*(x^3 - x^2)^(1/3) + (x^3 - x^2)^(2/3))/x^2) - 2*(x^3 - x^2)^(1/3)*
(2*sqrt(3)*cos(1/9*pi)^2 + 2*cos(1/9*pi)*sin(1/9*pi) - sqrt(3)))/(16*x*cos(1/9*pi)^4 - 16*x*cos(1/9*pi)^2 + 3*
x))*sin(1/9*pi) + 20*sqrt(6)*2^(1/6)*x^4*arctan(1/6*2^(1/6)*(sqrt(6)*2^(1/3)*x + 2*sqrt(6)*(x^3 - x^2)^(1/3))/
x) + 20*2^(2/3)*x^4*log(-(2^(1/3)*x - (x^3 - x^2)^(1/3))/x) - 10*2^(2/3)*x^4*log((2^(2/3)*x^2 + 2^(1/3)*(x^3 -
x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2) + 80*(sqrt(3)*x^4*cos(1/9*pi) + x^4*sin(1/9*pi))*arctan((8*(2*x*cos(1/
9*pi)^3 - x*cos(1/9*pi))*sin(1/9*pi) - sqrt(3)*x - 2*(2*sqrt(3)*x*cos(1/9*pi)^2 - 2*x*cos(1/9*pi)*sin(1/9*pi)
- sqrt(3)*x)*sqrt((x^2 + (2*sqrt(3)*x*cos(1/9*pi)*sin(1/9*pi) - 2*x*cos(1/9*pi)^2 + x)*(x^3 - x^2)^(1/3) + (x^
3 - x^2)^(2/3))/x^2) + 2*(x^3 - x^2)^(1/3)*(2*sqrt(3)*cos(1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi) - sqrt(3)))/(1
6*x*cos(1/9*pi)^4 - 16*x*cos(1/9*pi)^2 + 3*x)) - 80*(sqrt(3)*x^4*cos(1/9*pi) - x^4*sin(1/9*pi))*arctan(-1/2*(2
*x*cos(1/9*pi)^2 - x*sqrt((x^2 + 2*(x^3 - x^2)^(1/3)*(2*x*cos(1/9*pi)^2 - x) + (x^3 - x^2)^(2/3))/x^2) - x + (
x^3 - x^2)^(1/3))/(x*cos(1/9*pi)*sin(1/9*pi))) + 20*(sqrt(3)*x^4*sin(1/9*pi) - x^4*cos(1/9*pi))*log(64*(x^2 +
(2*sqrt(3)*x*cos(1/9*pi)*sin(1/9*pi) - 2*x*cos(1/9*pi)^2 + x)*(x^3 - x^2)^(1/3) + (x^3 - x^2)^(2/3))/x^2) - 20
*(sqrt(3)*x^4*sin(1/9*pi) + x^4*cos(1/9*pi))*log(64*(x^2 + 2*(x^3 - x^2)^(1/3)*(2*x*cos(1/9*pi)^2 - x) + (x^3
- x^2)^(2/3))/x^2) + 9*(x^3 - x^2)^(2/3)*(9*x^2 + 6*x + 5))/x^4
________________________________________________________________________________________
giac [B] time = 67.89, size = 1007, normalized size = 4.77
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(x^3+1)/(x^3-x^2)^(1/3),x, algorithm="giac")
[Out]
3/8*(1/x - 1)^2*(-1/x + 1)^(2/3) + 1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x + 1)^(1/3
))) - 1/3*(sqrt(3)*cos(4/9*pi)^5 - 10*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi)^2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^
4 - 5*cos(4/9*pi)^4*sin(4/9*pi) + 10*cos(4/9*pi)^2*sin(4/9*pi)^3 - sin(4/9*pi)^5 + sqrt(3)*cos(4/9*pi)^2 - sqr
t(3)*sin(4/9*pi)^2 - 2*cos(4/9*pi)*sin(4/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*(-1/x + 1)^(1/3))
/((1/2*I*sqrt(3) + 1/2)*sin(4/9*pi))) - 1/3*(sqrt(3)*cos(2/9*pi)^5 - 10*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi)^2 +
5*sqrt(3)*cos(2/9*pi)*sin(2/9*pi)^4 - 5*cos(2/9*pi)^4*sin(2/9*pi) + 10*cos(2/9*pi)^2*sin(2/9*pi)^3 - sin(2/9*p
i)^5 + sqrt(3)*cos(2/9*pi)^2 - sqrt(3)*sin(2/9*pi)^2 - 2*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)
*cos(2/9*pi) + 2*(-1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) + 1/3*(sqrt(3)*cos(1/9*pi)^5 - 10*sqrt
(3)*cos(1/9*pi)^3*sin(1/9*pi)^2 + 5*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)^4 + 5*cos(1/9*pi)^4*sin(1/9*pi) - 10*cos(1
/9*pi)^2*sin(1/9*pi)^3 + sin(1/9*pi)^5 - sqrt(3)*cos(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9
*pi))*arctan(-1/2*((-I*sqrt(3) - 1)*cos(1/9*pi) - 2*(-1/x + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) - 1
/6*(5*sqrt(3)*cos(4/9*pi)^4*sin(4/9*pi) - 10*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^3 + sqrt(3)*sin(4/9*pi)^5 + cos
(4/9*pi)^5 - 10*cos(4/9*pi)^3*sin(4/9*pi)^2 + 5*cos(4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)
+ cos(4/9*pi)^2 - sin(4/9*pi)^2)*log((-I*sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*(-1/x + 1)^(1/3) + (-1/x + 1)^(2/3
) + 1) - 1/6*(5*sqrt(3)*cos(2/9*pi)^4*sin(2/9*pi) - 10*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^3 + sqrt(3)*sin(2/9*p
i)^5 + cos(2/9*pi)^5 - 10*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5*cos(2/9*pi)*sin(2/9*pi)^4 + 2*sqrt(3)*cos(2/9*pi)*si
n(2/9*pi) + cos(2/9*pi)^2 - sin(2/9*pi)^2)*log((-I*sqrt(3)*cos(2/9*pi) - cos(2/9*pi))*(-1/x + 1)^(1/3) + (-1/x
+ 1)^(2/3) + 1) - 1/6*(5*sqrt(3)*cos(1/9*pi)^4*sin(1/9*pi) - 10*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^3 + sqrt(3)
*sin(1/9*pi)^5 - cos(1/9*pi)^5 + 10*cos(1/9*pi)^3*sin(1/9*pi)^2 - 5*cos(1/9*pi)*sin(1/9*pi)^4 - 2*sqrt(3)*cos(
1/9*pi)*sin(1/9*pi) + cos(1/9*pi)^2 - sin(1/9*pi)^2)*log((I*sqrt(3)*cos(1/9*pi) + cos(1/9*pi))*(-1/x + 1)^(1/3
) + (-1/x + 1)^(2/3) + 1) - 6/5*(-1/x + 1)^(5/3) - 1/12*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-1/x + 1)^(1/3) + (-1/x
+ 1)^(2/3)) + 1/6*2^(2/3)*log(abs(-2^(1/3) + (-1/x + 1)^(1/3))) + 3/2*(-1/x + 1)^(2/3)
________________________________________________________________________________________
maple [B] time = 28.15, size = 2405, normalized size = 11.40
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(2405\) |
| risch |
\(\text {Expression too large to display}\) |
\(4131\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^3/(x^3+1)/(x^3-x^2)^(1/3),x,method=_RETURNVERBOSE)
[Out]
3/40*(9*x^2+6*x+5)*(x^3-x^2)^(2/3)/x^4+644972544*RootOf(139314069504*_Z^6-373248*_Z^3+1)^5*ln(-(-1289945088*Ro
otOf(139314069504*_Z^6-373248*_Z^3+1)^5*x^2-26873856*RootOf(139314069504*_Z^6-373248*_Z^3+1)^4*(x^3-x^2)^(1/3)
*x+644972544*RootOf(139314069504*_Z^6-373248*_Z^3+1)^5*x+373248*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3*(x^3
-x^2)^(2/3)+6912*RootOf(139314069504*_Z^6-373248*_Z^3+1)^2*x^2-3456*RootOf(139314069504*_Z^6-373248*_Z^3+1)^2*
x-(x^3-x^2)^(2/3))/(373248*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3*x-746496*RootOf(139314069504*_Z^6-373248*
_Z^3+1)^3+x+1)/x)-644972544*RootOf(139314069504*_Z^6-373248*_Z^3+1)^5*ln((-644972544*RootOf(139314069504*_Z^6-
373248*_Z^3+1)^5*x^2-26873856*RootOf(139314069504*_Z^6-373248*_Z^3+1)^4*(x^3-x^2)^(1/3)*x+1289945088*RootOf(13
9314069504*_Z^6-373248*_Z^3+1)^5*x+373248*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3*(x^3-x^2)^(2/3)+3456*RootO
f(139314069504*_Z^6-373248*_Z^3+1)^2*x^2-1728*RootOf(139314069504*_Z^6-373248*_Z^3+1)^2*x-(x^3-x^2)^(2/3))/(37
3248*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3*x-746496*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3+x+1)/x)+1728
*RootOf(139314069504*_Z^6-373248*_Z^3+1)^2*ln((-644972544*RootOf(139314069504*_Z^6-373248*_Z^3+1)^5*x^2-268738
56*RootOf(139314069504*_Z^6-373248*_Z^3+1)^4*(x^3-x^2)^(1/3)*x+1289945088*RootOf(139314069504*_Z^6-373248*_Z^3
+1)^5*x+373248*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3*(x^3-x^2)^(2/3)+3456*RootOf(139314069504*_Z^6-373248*
_Z^3+1)^2*x^2-1728*RootOf(139314069504*_Z^6-373248*_Z^3+1)^2*x-(x^3-x^2)^(2/3))/(373248*RootOf(139314069504*_Z
^6-373248*_Z^3+1)^3*x-746496*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3+x+1)/x)+24*RootOf(139314069504*_Z^6-373
248*_Z^3+1)*ln((80621568*RootOf(139314069504*_Z^6-373248*_Z^3+1)^4*(x^3-x^2)^(1/3)*x-1492992*RootOf(1393140695
04*_Z^6-373248*_Z^3+1)^3*x^2+15552*(x^3-x^2)^(2/3)*RootOf(139314069504*_Z^6-373248*_Z^3+1)^2+746496*RootOf(139
314069504*_Z^6-373248*_Z^3+1)^3*x-216*RootOf(139314069504*_Z^6-373248*_Z^3+1)*(x^3-x^2)^(1/3)*x+2*x^2-x)/(3732
48*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3*x-746496*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3-2*x+1)/x)+8957
952*RootOf(139314069504*_Z^6-373248*_Z^3+1)^4*ln((-644972544*RootOf(139314069504*_Z^6-373248*_Z^3+1)^5*x^2+128
9945088*RootOf(139314069504*_Z^6-373248*_Z^3+1)^5*x+373248*(x^3-x^2)^(1/3)*RootOf(139314069504*_Z^6-373248*_Z^
3+1)^3*x-1728*RootOf(139314069504*_Z^6-373248*_Z^3+1)^2*x^2+72*RootOf(139314069504*_Z^6-373248*_Z^3+1)*(x^3-x^
2)^(2/3)-1728*RootOf(139314069504*_Z^6-373248*_Z^3+1)^2*x-x*(x^3-x^2)^(1/3))/(373248*RootOf(139314069504*_Z^6-
373248*_Z^3+1)^3*x-746496*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3-2*x+1)/x)-24*RootOf(139314069504*_Z^6-3732
48*_Z^3+1)*ln((-644972544*RootOf(139314069504*_Z^6-373248*_Z^3+1)^5*x^2+1289945088*RootOf(139314069504*_Z^6-37
3248*_Z^3+1)^5*x+373248*(x^3-x^2)^(1/3)*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3*x-1728*RootOf(139314069504*_
Z^6-373248*_Z^3+1)^2*x^2+72*RootOf(139314069504*_Z^6-373248*_Z^3+1)*(x^3-x^2)^(2/3)-1728*RootOf(139314069504*_
Z^6-373248*_Z^3+1)^2*x-x*(x^3-x^2)^(1/3))/(373248*RootOf(139314069504*_Z^6-373248*_Z^3+1)^3*x-746496*RootOf(13
9314069504*_Z^6-373248*_Z^3+1)^3-2*x+1)/x)+24*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*ln(-(-
72*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*RootOf(_Z^3-4)^3*x^2-31104*RootOf(RootOf(_Z^3-4)^
2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+1728*(x^3-x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+144*_Z
*RootOf(_Z^3-4)+20736*_Z^2)*RootOf(_Z^3-4)^2+144*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*Roo
tOf(_Z^3-4)^3*x+62208*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)^2*RootOf(_Z^3-4)^2*x+15*RootOf
(_Z^3-4)^2*(x^3-x^2)^(1/3)*x+3456*(x^3-x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*Ro
otOf(_Z^3-4)*x+11*RootOf(_Z^3-4)*x^2+4752*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*x^2+30*(x^
3-x^2)^(2/3)-RootOf(_Z^3-4)*x-432*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*x)/x/(1+x))-1/6*ln
((144*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*RootOf(_Z^3-4)^3*x^2+31104*RootOf(RootOf(_Z^3-
4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+1728*(x^3-x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+144
*_Z*RootOf(_Z^3-4)+20736*_Z^2)*RootOf(_Z^3-4)^2-288*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*
RootOf(_Z^3-4)^3*x-62208*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)^2*RootOf(_Z^3-4)^2*x+9*Root
Of(_Z^3-4)^2*(x^3-x^2)^(1/3)*x+3456*(x^3-x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*
RootOf(_Z^3-4)*x+26*RootOf(_Z^3-4)*x^2+5616*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*x^2+18*(
x^3-x^2)^(2/3)-10*RootOf(_Z^3-4)*x-2160*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*x)/x/(1+x))*
RootOf(_Z^3-4)-24*ln((144*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*RootOf(_Z^3-4)^3*x^2+31104
*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+1728*(x^3-x^2)^(2/3)*RootOf(
RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)*RootOf(_Z^3-4)^2-288*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(
_Z^3-4)+20736*_Z^2)*RootOf(_Z^3-4)^3*x-62208*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)^2*RootO
f(_Z^3-4)^2*x+9*RootOf(_Z^3-4)^2*(x^3-x^2)^(1/3)*x+3456*(x^3-x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(
_Z^3-4)+20736*_Z^2)*RootOf(_Z^3-4)*x+26*RootOf(_Z^3-4)*x^2+5616*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+
20736*_Z^2)*x^2+18*(x^3-x^2)^(2/3)-10*RootOf(_Z^3-4)*x-2160*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+2073
6*_Z^2)*x)/x/(1+x))*RootOf(RootOf(_Z^3-4)^2+144*_Z*RootOf(_Z^3-4)+20736*_Z^2)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(x^3+1)/(x^3-x^2)^(1/3),x, algorithm="maxima")
[Out]
integrate(1/((x^3 - x^2)^(1/3)*(x^3 + 1)*x^3), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^3\,\left (x^3+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^3*(x^3 + 1)*(x^3 - x^2)^(1/3)),x)
[Out]
int(1/(x^3*(x^3 + 1)*(x^3 - x^2)^(1/3)), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (x - 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**3/(x**3+1)/(x**3-x**2)**(1/3),x)
[Out]
Integral(1/(x**3*(x**2*(x - 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)
________________________________________________________________________________________