3.10.70 \(\int \frac {1}{(-1+x^3) \sqrt [3]{-x^2+x^3}} \, dx\)
Optimal. Leaf size=74 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6-3 \text {$\#$1}^3+3\& ,\frac {\log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {\left (x^3-x^2\right )^{2/3}}{(1-x) x} \]
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Rubi [B] time = 0.34, antiderivative size = 422, normalized size of antiderivative = 5.70,
number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used =
{2056, 6725, 21, 37, 91} \begin {gather*} -\frac {x}{\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 \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-(-1)^{2/3} x\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 \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/((-1 + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
-(x/(-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)*ArcTan[1/Sq
rt[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)/(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 + (-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 21
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])
Rule 37
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]
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 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}{\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^{2/3} \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^{2/3}}-\frac {1}{3 \sqrt [3]{-1+x} x^{2/3} \left (1+\sqrt [3]{-1} x\right )}-\frac {1}{3 \sqrt [3]{-1+x} x^{2/3} \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^{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 {\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]{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 \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}}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{(-1+x)^{4/3} x^{2/3}} \, dx}{3 \sqrt [3]{-x^2+x^3}}\\ &=-\frac {x}{\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]{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 \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 [A] time = 0.05, size = 106, normalized size = 1.43 \begin {gather*} \frac {x \left (\frac {\text {RootSum}\left [3 \text {$\#$1}^6-3 \text {$\#$1}^3+1\&,\frac {3 \text {$\#$1}^3 \log \left (\sqrt [3]{\frac {1}{x-1}+1}-\text {$\#$1}\right )-2 \log \left (\sqrt [3]{\frac {1}{x-1}+1}-\text {$\#$1}\right )}{2 \text {$\#$1}^5-\text {$\#$1}^2}\&\right ]}{\sqrt [3]{\frac {x}{x-1}}}-9\right )}{9 \sqrt [3]{(x-1) x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((-1 + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
(x*(-9 + RootSum[1 - 3*#1^3 + 3*#1^6 & , (-2*Log[(1 + (-1 + x)^(-1))^(1/3) - #1] + 3*Log[(1 + (-1 + x)^(-1))^(
1/3) - #1]*#1^3)/(-#1^2 + 2*#1^5) & ]/(x/(-1 + x))^(1/3)))/(9*((-1 + x)*x^2)^(1/3))
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IntegrateAlgebraic [A] time = 0.00, size = 74, normalized size = 1.00 \begin {gather*} \frac {\left (-x^2+x^3\right )^{2/3}}{(1-x) x}+\frac {1}{3} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((-1 + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
(-x^2 + x^3)^(2/3)/((1 - x)*x) + RootSum[3 - 3*#1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1 &
]/3
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fricas [B] time = 0.63, size = 1437, normalized size = 19.42
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^3-1)/(x^3-x^2)^(1/3),x, algorithm="fricas")
[Out]
1/36*(2*12^(1/6)*6^(2/3)*(x^2 - x)*cos(2/3*arctan(sqrt(3) - 2))*log(12*(4*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 - x^2)
^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)) - 12*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x*c
os(2/3*arctan(sqrt(3) - 2))^2 + 12^(2/3)*6^(2/3)*x^2 + 6*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x + 12*(x^3 - x^2)
^(2/3))/x^2) - 8*12^(1/6)*6^(2/3)*(x^2 - x)*arctan(1/108*(12*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 - x^2)^(1/3)*cos(2/
3*arctan(sqrt(3) - 2))^2 - 6*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 - x^2)^(1/3) - sqrt(3)*(2*12^(2/3)*6^(2/3)*sqrt(3)*
x*cos(2/3*arctan(sqrt(3) - 2))^2 - 6*12^(2/3)*6^(2/3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) -
2)) - 12^(2/3)*6^(2/3)*sqrt(3)*x)*sqrt((4*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3)
- 2))*sin(2/3*arctan(sqrt(3) - 2)) - 12*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 +
12^(2/3)*6^(2/3)*x^2 + 6*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x + 12*(x^3 - x^2)^(2/3))/x^2) + 36*(48*x*cos(2/3*
arctan(sqrt(3) - 2))^3 - (12^(2/3)*6^(2/3)*(x^3 - x^2)^(1/3) + 24*x)*cos(2/3*arctan(sqrt(3) - 2)))*sin(2/3*arc
tan(sqrt(3) - 2)) - 108*sqrt(3)*x)/(16*x*cos(2/3*arctan(sqrt(3) - 2))^4 - 16*x*cos(2/3*arctan(sqrt(3) - 2))^2
+ x))*sin(2/3*arctan(sqrt(3) - 2)) - 4*(12^(1/6)*6^(2/3)*sqrt(3)*(x^2 - x)*cos(2/3*arctan(sqrt(3) - 2)) + 12^(
1/6)*6^(2/3)*(x^2 - x)*sin(2/3*arctan(sqrt(3) - 2)))*arctan(-1/108*(12*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 - x^2)^(1
/3)*cos(2/3*arctan(sqrt(3) - 2))^2 - 6*12^(2/3)*6^(2/3)*sqrt(3)*(x^3 - x^2)^(1/3) - sqrt(3)*(2*12^(2/3)*6^(2/3
)*sqrt(3)*x*cos(2/3*arctan(sqrt(3) - 2))^2 + 6*12^(2/3)*6^(2/3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(
sqrt(3) - 2)) - 12^(2/3)*6^(2/3)*sqrt(3)*x)*sqrt((4*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arcta
n(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)) + 12*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3)
- 2))^2 + 12^(2/3)*6^(2/3)*x^2 - 6*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x + 12*(x^3 - x^2)^(2/3))/x^2) + 36*(48*
x*cos(2/3*arctan(sqrt(3) - 2))^3 + (12^(2/3)*6^(2/3)*(x^3 - x^2)^(1/3) - 24*x)*cos(2/3*arctan(sqrt(3) - 2)))*s
in(2/3*arctan(sqrt(3) - 2)) + 108*sqrt(3)*x)/(16*x*cos(2/3*arctan(sqrt(3) - 2))^4 - 16*x*cos(2/3*arctan(sqrt(3
) - 2))^2 + x)) - 4*(12^(1/6)*6^(2/3)*sqrt(3)*(x^2 - x)*cos(2/3*arctan(sqrt(3) - 2)) - 12^(1/6)*6^(2/3)*(x^2 -
x)*sin(2/3*arctan(sqrt(3) - 2)))*arctan(1/72*(144*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2))
+ 12^(2/3)*6^(2/3)*x*sqrt(-(8*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2
/3*arctan(sqrt(3) - 2)) - 12^(2/3)*6^(2/3)*x^2 - 12*(x^3 - x^2)^(2/3))/x^2) - 2*12^(2/3)*6^(2/3)*sqrt(3)*(x^3
- x^2)^(1/3))/(2*x*cos(2/3*arctan(sqrt(3) - 2))^2 - x)) - (12^(1/6)*6^(2/3)*sqrt(3)*(x^2 - x)*sin(2/3*arctan(s
qrt(3) - 2)) + 12^(1/6)*6^(2/3)*(x^2 - x)*cos(2/3*arctan(sqrt(3) - 2)))*log(-48*(8*12^(1/3)*6^(1/3)*sqrt(3)*(x
^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))*sin(2/3*arctan(sqrt(3) - 2)) - 12^(2/3)*6^(2/3)*x^2 - 12*(x^3 -
x^2)^(2/3))/x^2) + (12^(1/6)*6^(2/3)*sqrt(3)*(x^2 - x)*sin(2/3*arctan(sqrt(3) - 2)) - 12^(1/6)*6^(2/3)*(x^2 -
x)*cos(2/3*arctan(sqrt(3) - 2)))*log(48*(4*12^(1/3)*6^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3
) - 2))*sin(2/3*arctan(sqrt(3) - 2)) + 12*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(3) - 2))^2
+ 12^(2/3)*6^(2/3)*x^2 - 6*12^(1/3)*6^(1/3)*(x^3 - x^2)^(1/3)*x + 12*(x^3 - x^2)^(2/3))/x^2) - 36*(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^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^3-1)/(x^3-x^2)^(1/3),x, algorithm="giac")
[Out]
integrate(1/((x^3 - x^2)^(1/3)*(x^3 - 1)), x)
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maple [B] time = 8.14, size = 2083, normalized size = 28.15
| | |
| method |
result |
size |
| | |
| risch |
\(\text {Expression too large to display}\) |
\(2083\) |
| trager |
\(\text {Expression too large to display}\) |
\(2579\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^3-1)/(x^3-x^2)^(1/3),x,method=_RETURNVERBOSE)
[Out]
-x/((-1+x)*x^2)^(1/3)+1/6*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*ln(-(69*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^
3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^6*x^2-138*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3
+64)^6*x+183*(x^3-x^2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^3*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)^2*x-1268*R
ootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^3*x^2+228*(x^3-x^2)^(2/3)*RootOf(3*_Z^6-24
*_Z^3+64)^3+1340*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^3*x-1920*(x^3-x^2)^(1/3)
*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)^2*x+4640*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*x^2+1952*(x^3-
x^2)^(2/3)-3248*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*x)/(3*x*RootOf(3*_Z^6-24*_Z^3+64)^3-6*RootOf(3*_Z^6
-24*_Z^3+64)^3+8*x+40)/x)+1/6*RootOf(3*_Z^6-24*_Z^3+64)*ln((-69*x^2*RootOf(3*_Z^6-24*_Z^3+64)^7+138*x*RootOf(3
*_Z^6-24*_Z^3+64)^7+183*(x^3-x^2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^5*x-164*RootOf(3*_Z^6-24*_Z^3+64)^4*x^2+228*
(x^3-x^2)^(2/3)*RootOf(3*_Z^6-24*_Z^3+64)^3-868*RootOf(3*_Z^6-24*_Z^3+64)^4*x+456*(x^3-x^2)^(1/3)*RootOf(3*_Z^
6-24*_Z^3+64)^2*x+1088*x^2*RootOf(3*_Z^6-24*_Z^3+64)-3776*(x^3-x^2)^(2/3)+1360*x*RootOf(3*_Z^6-24*_Z^3+64))/(3
*x*RootOf(3*_Z^6-24*_Z^3+64)^3-6*RootOf(3*_Z^6-24*_Z^3+64)^3-32*x+8)/x)+1/16*ln((-183*RootOf(RootOf(3*_Z^6-24*
_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^6*x^2+366*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^
6-24*_Z^3+64)^6*x+276*(x^3-x^2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^3*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)^2
*x-4136*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^3*x^2+3120*(x^3-x^2)^(2/3)*RootOf
(3*_Z^6-24*_Z^3+64)^3-1000*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^3*x-8448*(x^3-
x^2)^(1/3)*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)^2*x+19840*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*x^2
+2944*(x^3-x^2)^(2/3)-1984*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*x)/(3*x*RootOf(3*_Z^6-24*_Z^3+64)^3-6*Ro
otOf(3*_Z^6-24*_Z^3+64)^3+8*x+40)/x)*RootOf(3*_Z^6-24*_Z^3+64)^3*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)-1/
3*ln((-183*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^6*x^2+366*RootOf(RootOf(3*_Z^6
-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^6*x+276*(x^3-x^2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^3*RootOf(Ro
otOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)^2*x-4136*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64
)^3*x^2+3120*(x^3-x^2)^(2/3)*RootOf(3*_Z^6-24*_Z^3+64)^3-1000*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootO
f(3*_Z^6-24*_Z^3+64)^3*x-8448*(x^3-x^2)^(1/3)*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)^2*x+19840*RootOf(Root
Of(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*x^2+2944*(x^3-x^2)^(2/3)-1984*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*x)/(3
*x*RootOf(3*_Z^6-24*_Z^3+64)^3-6*RootOf(3*_Z^6-24*_Z^3+64)^3+8*x+40)/x)*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^
3-8)-1/16*ln(-(183*x^2*RootOf(3*_Z^6-24*_Z^3+64)^7-366*x*RootOf(3*_Z^6-24*_Z^3+64)^7+276*(x^3-x^2)^(1/3)*RootO
f(3*_Z^6-24*_Z^3+64)^5*x-7064*RootOf(3*_Z^6-24*_Z^3+64)^4*x^2+3120*(x^3-x^2)^(2/3)*RootOf(3*_Z^6-24*_Z^3+64)^3
+4856*RootOf(3*_Z^6-24*_Z^3+64)^4*x+6240*(x^3-x^2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^2*x+24960*x^2*RootOf(3*_Z^6
-24*_Z^3+64)-27904*(x^3-x^2)^(2/3)-13440*x*RootOf(3*_Z^6-24*_Z^3+64))/(3*x*RootOf(3*_Z^6-24*_Z^3+64)^3-6*RootO
f(3*_Z^6-24*_Z^3+64)^3-32*x+8)/x)*RootOf(3*_Z^6-24*_Z^3+64)^4+1/6*ln(-(183*x^2*RootOf(3*_Z^6-24*_Z^3+64)^7-366
*x*RootOf(3*_Z^6-24*_Z^3+64)^7+276*(x^3-x^2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^5*x-7064*RootOf(3*_Z^6-24*_Z^3+64
)^4*x^2+3120*(x^3-x^2)^(2/3)*RootOf(3*_Z^6-24*_Z^3+64)^3+4856*RootOf(3*_Z^6-24*_Z^3+64)^4*x+6240*(x^3-x^2)^(1/
3)*RootOf(3*_Z^6-24*_Z^3+64)^2*x+24960*x^2*RootOf(3*_Z^6-24*_Z^3+64)-27904*(x^3-x^2)^(2/3)-13440*x*RootOf(3*_Z
^6-24*_Z^3+64))/(3*x*RootOf(3*_Z^6-24*_Z^3+64)^3-6*RootOf(3*_Z^6-24*_Z^3+64)^3-32*x+8)/x)*RootOf(3*_Z^6-24*_Z^
3+64)
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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 )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\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 - 1)*(x^3 - x^2)^(1/3)),x)
[Out]
int(1/((x^3 - 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^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x**3-1)/(x**3-x**2)**(1/3),x)
[Out]
Integral(1/((x**2*(x - 1))**(1/3)*(x - 1)*(x**2 + x + 1)), x)
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