3.10.70 \(\int \frac {1}{(-1+x^3) \sqrt [3]{-x^2+x^3}} \, dx\) [970]
Optimal. Leaf size=74 \[ \frac {\left (-x^2+x^3\right )^{2/3}}{(1-x) x}+\frac {1}{3} \text {RootSum}\left [3-3 \text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]
[Out]
Unintegrable
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(422\) vs. \(2(74)=148\).
time = 0.20, 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 = {2081, 6857, 21,
37, 93} \begin {gather*} \frac {\sqrt [3]{x-1} x^{2/3} \text {ArcTan}\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} \text {ArcTan}\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}}-\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}} \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 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 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*} \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 = 0.00, size = 79, normalized size = 1.07 \begin {gather*} \frac {-3 x+\sqrt [3]{-1+x} x^{2/3} \text {RootSum}\left [3-3 \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 ]}{3 \sqrt [3]{(-1+x) x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((-1 + x^3)*(-x^2 + x^3)^(1/3)),x]
[Out]
(-3*x + (-1 + x)^(1/3)*x^(2/3)*RootSum[3 - 3*#1^3 + #1^6 & , (-Log[x^(1/3)] + Log[(-1 + x)^(1/3) - x^(1/3)*#1]
)/#1 & ])/(3*((-1 + x)*x^2)^(1/3))
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 15.01, size = 2136, normalized size = 28.86
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(2136\) |
| risch |
\(\text {Expression too large to display}\) |
\(2784\) |
| | |
|
|
|
|
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^3-x^2)^(2/3)/(-1+x)/x-27*ln((21870*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*RootOf(2187*_Z^6-81*_Z^
3+1)^6*x^2-43740*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*RootOf(2187*_Z^6-81*_Z^3+1)^6*x+14499*(x^3-x^
2)^(1/3)*RootOf(2187*_Z^6-81*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)^2*x-4104*RootOf(_Z^3+27
*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*RootOf(2187*_Z^6-81*_Z^3+1)^3*x^2+6453*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z
^3+1)^3-1)*RootOf(2187*_Z^6-81*_Z^3+1)^3*x-1539*(x^3-x^2)^(2/3)*RootOf(2187*_Z^6-81*_Z^3+1)^3-297*(x^3-x^2)^(1
/3)*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)^2*x+170*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*x^
2-119*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*x-61*(x^3-x^2)^(2/3))/(81*RootOf(2187*_Z^6-81*_Z^3+1)^3*
x-162*RootOf(2187*_Z^6-81*_Z^3+1)^3+x+5)/x)*RootOf(2187*_Z^6-81*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_
Z^3+1)^3-1)+1/3*ln((21870*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*RootOf(2187*_Z^6-81*_Z^3+1)^6*x^2-43
740*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*RootOf(2187*_Z^6-81*_Z^3+1)^6*x+14499*(x^3-x^2)^(1/3)*Root
Of(2187*_Z^6-81*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)^2*x-4104*RootOf(_Z^3+27*RootOf(2187*
_Z^6-81*_Z^3+1)^3-1)*RootOf(2187*_Z^6-81*_Z^3+1)^3*x^2+6453*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*Ro
otOf(2187*_Z^6-81*_Z^3+1)^3*x-1539*(x^3-x^2)^(2/3)*RootOf(2187*_Z^6-81*_Z^3+1)^3-297*(x^3-x^2)^(1/3)*RootOf(_Z
^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)^2*x+170*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*x^2-119*RootOf(
_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*x-61*(x^3-x^2)^(2/3))/(81*RootOf(2187*_Z^6-81*_Z^3+1)^3*x-162*RootOf(
2187*_Z^6-81*_Z^3+1)^3+x+5)/x)*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)+RootOf(2187*_Z^6-81*_Z^3+1)*ln(
(190269*RootOf(2187*_Z^6-81*_Z^3+1)^7*x^2-380538*RootOf(2187*_Z^6-81*_Z^3+1)^7*x+175689*(x^3-x^2)^(1/3)*RootOf
(2187*_Z^6-81*_Z^3+1)^5*x-30294*RootOf(2187*_Z^6-81*_Z^3+1)^4*x^2+15957*RootOf(2187*_Z^6-81*_Z^3+1)^4*x-5265*(
x^3-x^2)^(2/3)*RootOf(2187*_Z^6-81*_Z^3+1)^3-2376*(x^3-x^2)^(1/3)*RootOf(2187*_Z^6-81*_Z^3+1)^2*x-39*RootOf(21
87*_Z^6-81*_Z^3+1)*x^2+21*RootOf(2187*_Z^6-81*_Z^3+1)*x+218*(x^3-x^2)^(2/3))/(81*RootOf(2187*_Z^6-81*_Z^3+1)^3
*x-162*RootOf(2187*_Z^6-81*_Z^3+1)^3-4*x+1)/x)+81*RootOf(2187*_Z^6-81*_Z^3+1)^4*ln((65610*RootOf(2187*_Z^6-81*
_Z^3+1)^7*x^2-131220*RootOf(2187*_Z^6-81*_Z^3+1)^7*x-130491*(x^3-x^2)^(1/3)*RootOf(2187*_Z^6-81*_Z^3+1)^5*x+74
52*RootOf(2187*_Z^6-81*_Z^3+1)^4*x^2+1539*(x^3-x^2)^(2/3)*RootOf(2187*_Z^6-81*_Z^3+1)^3-9639*RootOf(2187*_Z^6-
81*_Z^3+1)^4*x+2160*(x^3-x^2)^(1/3)*RootOf(2187*_Z^6-81*_Z^3+1)^2*x+144*RootOf(2187*_Z^6-81*_Z^3+1)*x^2-118*(x
^3-x^2)^(2/3)+180*RootOf(2187*_Z^6-81*_Z^3+1)*x)/(81*RootOf(2187*_Z^6-81*_Z^3+1)^3*x-162*RootOf(2187*_Z^6-81*_
Z^3+1)^3-4*x+1)/x)-2*RootOf(2187*_Z^6-81*_Z^3+1)*ln((65610*RootOf(2187*_Z^6-81*_Z^3+1)^7*x^2-131220*RootOf(218
7*_Z^6-81*_Z^3+1)^7*x-130491*(x^3-x^2)^(1/3)*RootOf(2187*_Z^6-81*_Z^3+1)^5*x+7452*RootOf(2187*_Z^6-81*_Z^3+1)^
4*x^2+1539*(x^3-x^2)^(2/3)*RootOf(2187*_Z^6-81*_Z^3+1)^3-9639*RootOf(2187*_Z^6-81*_Z^3+1)^4*x+2160*(x^3-x^2)^(
1/3)*RootOf(2187*_Z^6-81*_Z^3+1)^2*x+144*RootOf(2187*_Z^6-81*_Z^3+1)*x^2-118*(x^3-x^2)^(2/3)+180*RootOf(2187*_
Z^6-81*_Z^3+1)*x)/(81*RootOf(2187*_Z^6-81*_Z^3+1)^3*x-162*RootOf(2187*_Z^6-81*_Z^3+1)^3-4*x+1)/x)+1/3*RootOf(_
Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*ln((63423*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*RootOf(2187*
_Z^6-81*_Z^3+1)^6*x^2-126846*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*RootOf(2187*_Z^6-81*_Z^3+1)^6*x-1
9521*(x^3-x^2)^(1/3)*RootOf(2187*_Z^6-81*_Z^3+1)^3*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)^2*x+5400*Ro
otOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*RootOf(2187*_Z^6-81*_Z^3+1)^3*x^2+4077*RootOf(_Z^3+27*RootOf(218
7*_Z^6-81*_Z^3+1)^3-1)*RootOf(2187*_Z^6-81*_Z^3+1)^3*x+5265*(x^3-x^2)^(2/3)*RootOf(2187*_Z^6-81*_Z^3+1)^3+459*
(x^3-x^2)^(1/3)*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)^2*x-300*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^
3+1)^3-1)*x^2+30*RootOf(_Z^3+27*RootOf(2187*_Z^6-81*_Z^3+1)^3-1)*x+23*(x^3-x^2)^(2/3))/(81*RootOf(2187*_Z^6-81
*_Z^3+1)^3*x-162*RootOf(2187*_Z^6-81*_Z^3+1)^3+x+5)/x)
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 0.51, size = 1437, normalized size = 19.42 \begin {gather*} \text {Too large to display} \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="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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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