3.25.51 \(\int \frac {1}{x^3 (-1+x^3) \sqrt [3]{x^2+x^3}} \, dx\)
Optimal. Leaf size=199 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {\log \left (2^{2/3} \sqrt [3]{x^3+x^2}-2 x\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3+x^2} x+\sqrt [3]{2} \left (x^3+x^2\right )^{2/3}\right )}{6 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+x^2}+x}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {3 \left (x^3+x^2\right )^{2/3} \left (9 x^2-6 x+5\right )}{40 x^4} \]
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Rubi [C] time = 0.64, antiderivative size = 728, normalized size of antiderivative = 3.66,
number of steps used = 18, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used
= {2056, 6725, 129, 155, 12, 91} \begin {gather*} -\frac {\left (3-4 (-1)^{2/3}\right ) (x+1)}{20 x \sqrt [3]{x^3+x^2}}-\frac {\left (3+4 \sqrt [3]{-1}\right ) (x+1)}{20 x \sqrt [3]{x^3+x^2}}+\frac {x+1}{20 x \sqrt [3]{x^3+x^2}}+\frac {3 (x+1)}{8 x^2 \sqrt [3]{x^3+x^2}}+\frac {\left (5+16 i \sqrt {3}\right ) (x+1)}{40 \sqrt [3]{x^3+x^2}}+\frac {\left (5-16 i \sqrt {3}\right ) (x+1)}{40 \sqrt [3]{x^3+x^2}}+\frac {17 (x+1)}{40 \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \sqrt [3]{x+1} \log (1-x)}{6 \sqrt [3]{2} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \sqrt [3]{x+1} \log \left (\sqrt [3]{-1} x+1\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \sqrt [3]{x+1} \log \left (1-(-1)^{2/3} x\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^3+x^2}}+\frac {x^{2/3} \sqrt [3]{x+1} \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 {x^{2/3} \sqrt [3]{x+1} \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 {x^{2/3} \sqrt [3]{x+1} \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 {x^{2/3} \sqrt [3]{x+1} \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 {x^{2/3} \sqrt [3]{x+1} \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 {x^{2/3} \sqrt [3]{x+1} \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)*Sqr
t[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)) + (x^(2/3)*(1 + x)^(1/3)*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)) + (x^(2/3)*(1 + x)^(1/3)*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)) + (x^(2/3)*(1 + x)^(1/3)*ArcTan[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))
- (x^(2/3)*(1 + x)^(1/3)*Log[1 - x])/(6*2^(1/3)*(x^2 + x^3)^(1/3)) - (x^(2/3)*(1 + x)^(1/3)*Log[1 + (-1)^(1/3
)*x])/(6*(1 - (-1)^(1/3))^(1/3)*(x^2 + x^3)^(1/3)) - (x^(2/3)*(1 + x)^(1/3)*Log[1 - (-1)^(2/3)*x])/(6*(1 + (-1
)^(2/3))^(1/3)*(x^2 + x^3)^(1/3)) + (x^(2/3)*(1 + x)^(1/3)*Log[-x^(1/3) + (1 + x)^(1/3)/2^(1/3)])/(2*2^(1/3)*(
x^2 + x^3)^(1/3)) + (x^(2/3)*(1 + x)^(1/3)*Log[-x^(1/3) + (1 + x)^(1/3)/(1 - (-1)^(1/3))^(1/3)])/(2*(1 - (-1)^
(1/3))^(1/3)*(x^2 + x^3)^(1/3)) + (x^(2/3)*(1 + x)^(1/3)*Log[-x^(1/3) + (1 + x)^(1/3)/(1 + (-1)^(2/3))^(1/3)])
/(2*(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 (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{1+x} \left (-1+x^3\right )} \, dx}{\sqrt [3]{x^2+x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \left (-\frac {1}{3 (1-x) x^{11/3} \sqrt [3]{1+x}}-\frac {1}{3 x^{11/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )}-\frac {1}{3 x^{11/3} \sqrt [3]{1+x} \left (1-(-1)^{2/3} x\right )}\right ) \, dx}{\sqrt [3]{x^2+x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{(1-x) x^{11/3} \sqrt [3]{1+x}} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{1+x} \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 (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {-\frac {2}{3}-2 x}{(1-x) x^{8/3} \sqrt [3]{1+x}} \, dx}{8 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{3} \left (3+4 \sqrt [3]{-1}\right )+2 \sqrt [3]{-1} x}{x^{8/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{8 \sqrt [3]{x^2+x^3}}+\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {2}{3} \left (3-4 (-1)^{2/3}\right )-2 (-1)^{2/3} x}{x^{8/3} \sqrt [3]{1+x} \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 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {\frac {34}{9}+\frac {2 x}{3}}{(1-x) x^{5/3} \sqrt [3]{1+x}} \, dx}{40 \sqrt [3]{x^2+x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\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}{x^{5/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{40 \sqrt [3]{x^2+x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x}\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}{x^{5/3} \sqrt [3]{1+x} \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 x^{2/3} \sqrt [3]{1+x}\right ) \int -\frac {80}{27 (1-x) x^{2/3} \sqrt [3]{1+x}} \, dx}{80 \sqrt [3]{x^2+x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{1+x}\right ) \int -\frac {80}{27 x^{2/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{80 \sqrt [3]{x^2+x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{1+x}\right ) \int -\frac {80}{27 x^{2/3} \sqrt [3]{1+x} \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 (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{(1-x) x^{2/3} \sqrt [3]{1+x}} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{1+x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{x^2+x^3}}-\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{1+x} \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 {x^{2/3} \sqrt [3]{1+x} \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 {x^{2/3} \sqrt [3]{1+x} \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 {x^{2/3} \sqrt [3]{1+x} \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 {x^{2/3} \sqrt [3]{1+x} \log (1-x)}{6 \sqrt [3]{2} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log \left (1+\sqrt [3]{-1} x\right )}{6 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log \left (1-(-1)^{2/3} x\right )}{6 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{2}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-\sqrt [3]{-1}}}\right )}{2 \sqrt [3]{1-\sqrt [3]{-1}} \sqrt [3]{x^2+x^3}}+\frac {x^{2/3} \sqrt [3]{1+x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1+(-1)^{2/3}}}\right )}{2 \sqrt [3]{1+(-1)^{2/3}} \sqrt [3]{x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 120, normalized size = 0.60 \begin {gather*} \frac {(x+1) \left (-40 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 x}{x+1}\right )-40 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x-i \sqrt {3} x}{2 x+2}\right )-40 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {i \sqrt {3} x+x}{2 x+2}\right )+27 x^3+9 x^2-3 x+15\right )}{40 \left (x^2 (x+1)\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 + x)*(15 - 3*x + 9*x^2 + 27*x^3 - 40*x^3*Hypergeometric2F1[1/3, 1, 4/3, (2*x)/(1 + x)] - 40*x^3*Hypergeome
tric2F1[1/3, 1, 4/3, (x - I*Sqrt[3]*x)/(2 + 2*x)] - 40*x^3*Hypergeometric2F1[1/3, 1, 4/3, (x + I*Sqrt[3]*x)/(2
+ 2*x)]))/(40*(x^2*(1 + x))^(4/3))
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IntegrateAlgebraic [A] time = 0.00, size = 199, 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 = 0.89, size = 855, normalized size = 4.30
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 = 60.72, size = 968, normalized size = 4.86
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)^(8/3) + 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*(sqr
t(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 - sqrt(3)*sin(4/9*p
i)^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*pi)^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*c
os(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*sqr
t(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*pi)^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)*sin(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 = 79.58, size = 3857, normalized size = 19.38
| | |
| method |
result |
size |
| | |
| risch |
\(\text {Expression too large to display}\) |
\(3857\) |
| trager |
\(\text {Expression too large to display}\) |
\(4513\) |
| | |
|
|
|
|
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^3+3*x^2-x+5)/x^2/(x^2*(1+x))^(1/3)+1/2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*ln((-3*Ro
otOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x^2+90*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(
_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+360*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)
*RootOf(_Z^3-4)^2+6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x-180*RootOf(RootOf(_
Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*x-240*(x^3+x^2)^(1/3)*RootOf(_Z^3-4)^2*x-126*(x^3+x^2)
^(1/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)*x+30*RootOf(_Z^3-4)*x^2-900*RootOf(R
ootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x^2+396*(x^3+x^2)^(2/3)+14*RootOf(_Z^3-4)*x-420*RootOf(RootOf(_Z^3
-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x)/x/(-1+x))+1/6*RootOf(_Z^3-4)*ln(-(30*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(
_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x^2-9*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^2*
x^2+360*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-60*RootOf(RootOf(
_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^3*x+18*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2
)^2*RootOf(_Z^3-4)^2*x-240*(x^3+x^2)^(1/3)*RootOf(_Z^3-4)^2*x-594*(x^3+x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z
*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)*x+340*RootOf(_Z^3-4)*x^2-102*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4
)+9*_Z^2)*x^2+84*(x^3+x^2)^(2/3)+60*RootOf(_Z^3-4)*x-18*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x)
/x/(-1+x))-1/6*ln((-27*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2
*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*(x^3+x^2)^(1/3)*x-156*RootOf(_Z^3-6*Root
Of(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4
)+9*_Z^2)*RootOf(_Z^3-4)^2*x^2+108*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(
_Z^3-4)^2-78*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*RootOf(Root
Of(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x+60*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*Root
Of(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*(x^3+x^2)^(1/3)*x-176*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*Root
Of(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*x^2-240*(x^3+x^2)^(2/3)-88*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*R
ootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*x)/(3*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9
*_Z^2)*x-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2+12*x+4)/x)*RootOf(_Z^3-6*RootO
f(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)-1/8*ln((-27*RootOf(_Z^3-6*RootOf(RootOf(_Z^
3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*R
ootOf(_Z^3-4)^2*(x^3+x^2)^(1/3)*x-156*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf
(_Z^3-4)^2-8)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x^2+108*(x^3+x^2)^(2/3)*Roo
tOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-78*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_
Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4
)^2*x+60*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*(x^3+x^2)^(1/
3)*x-176*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*x^2-240*(x^3+x^
2)^(2/3)-88*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*x)/(3*RootOf
(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)
+9*_Z^2)*RootOf(_Z^3-4)^2+12*x+4)/x)*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*Root
Of(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)+1/6*RootOf(_Z^3-6*RootOf(Roo
tOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*ln(-(-9*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*
_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^
3-4)^4*x^2+18*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*RootOf(Roo
tOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^4*x-126*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*
RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4
)^2*(x^3+x^2)^(1/3)*x+72*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)
*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x^2+504*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z
^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-60*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3
-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x-168*Roo
tOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*(x^3+x^2)^(1/3)*x-128*Roo
tOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*x^2+672*(x^3+x^2)^(2/3)-192
*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*x)/(3*RootOf(_Z^3-4)^2*
RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*Ro
otOf(_Z^3-4)^2+12*x+4)/x)-1/16*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4
)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*ln(-(78*RootOf(_Z^3-6*RootOf(Roo
tOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*
_Z^2)*RootOf(_Z^3-4)^2*x^2-144*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4
)^2-8)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*(x^3+x^2)^(1/3)*x+39*RootOf(_Z^3-6
*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf
(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x+108*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*Ro
otOf(_Z^3-4)^2+16*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*x^2-
120*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*(x^3+x^2)^(1/3)*x+8*
RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*x+384*(x^3+x^2)^(2/3))
/(3*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*Root
Of(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8*x-12)/x)-1/12*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+
9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*ln((-9*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(
_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^4*x^2+18*RootOf(_Z^3-6*Roo
tOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^
3-4)+9*_Z^2)^2*RootOf(_Z^3-4)^4*x-96*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(
_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x^2+108*RootOf(_Z^3-6*Root
Of(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3
-4)+9*_Z^2)*RootOf(_Z^3-4)^2*x+1008*(x^3+x^2)^(2/3)*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf
(_Z^3-4)^2-240*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*x^2+672
*RootOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)*(x^3+x^2)^(1/3)*x-80*Ro
otOf(_Z^3-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*RootOf(_Z^3-4)^2-8)^2*x)/(3*RootOf(_Z^3-4)^2*R
ootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*x-6*RootOf(RootOf(_Z^3-4)^2+3*_Z*RootOf(_Z^3-4)+9*_Z^2)*Roo
tOf(_Z^3-4)^2-8*x-12)/x)
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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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^3-x^6\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^3*(x^2 + x^3)^(1/3)*(x^3 - 1)),x)
[Out]
-int(1/((x^2 + x^3)^(1/3)*(x^3 - x^6)), x)
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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)
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