3.10.69 \(\int \frac {1}{(-1+x^3) \sqrt [3]{-x^2+x^3}} \, dx\) [969]

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 = 0.30, 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 = 0.14, 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 = 16.84, size = 2357, normalized size = 31.85

method	result	size
risch	\(\text {Expression too large to display}\)	\(2357\)
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)-1/16*ln((123*x^2*RootOf(3*_Z^6-24*_Z^3+64)^7-246*x*RootOf(3*_Z^6-24*_Z^3+64)^7+708*(x^3-
x^2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^5*x-40*RootOf(3*_Z^6-24*_Z^3+64)^4*x^2+456*(x^3-x^2)^(2/3)*RootOf(3*_Z^6-
24*_Z^3+64)^3+2212*RootOf(3*_Z^6-24*_Z^3+64)^4*x-4752*(x^3-x^2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^2*x-3712*RootO
f(3*_Z^6-24*_Z^3+64)*x^2-7552*(x^3-x^2)^(2/3)-4640*RootOf(3*_Z^6-24*_Z^3+64)*x)/(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)^4+1/6*ln((123*x^2*RootOf(3*_Z^6-24*_Z^3+
64)^7-246*x*RootOf(3*_Z^6-24*_Z^3+64)^7+708*(x^3-x^2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^5*x-40*RootOf(3*_Z^6-24*
_Z^3+64)^4*x^2+456*(x^3-x^2)^(2/3)*RootOf(3*_Z^6-24*_Z^3+64)^3+2212*RootOf(3*_Z^6-24*_Z^3+64)^4*x-4752*(x^3-x^
2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^2*x-3712*RootOf(3*_Z^6-24*_Z^3+64)*x^2-7552*(x^3-x^2)^(2/3)-4640*RootOf(3*_
Z^6-24*_Z^3+64)*x)/(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)+1/16*ln(-(-12*x^2*RootOf(3*_Z^6-24*_Z^3+64)^7+24*x*RootOf(3*_Z^6-24*_Z^3+64)^7+327*(x^3-x^2)^(1/3)*Ro
otOf(3*_Z^6-24*_Z^3+64)^5*x+509*RootOf(3*_Z^6-24*_Z^3+64)^4*x^2+390*(x^3-x^2)^(2/3)*RootOf(3*_Z^6-24*_Z^3+64)^
3-410*RootOf(3*_Z^6-24*_Z^3+64)^4*x-1836*(x^3-x^2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^2*x-3224*RootOf(3*_Z^6-24*_
Z^3+64)*x^2-3488*(x^3-x^2)^(2/3)+1736*RootOf(3*_Z^6-24*_Z^3+64)*x)/(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)^4-1/3*ln(-(-12*x^2*RootOf(3*_Z^6-24*_Z^3+64)^7+24*x*R
ootOf(3*_Z^6-24*_Z^3+64)^7+327*(x^3-x^2)^(1/3)*RootOf(3*_Z^6-24*_Z^3+64)^5*x+509*RootOf(3*_Z^6-24*_Z^3+64)^4*x
^2+390*(x^3-x^2)^(2/3)*RootOf(3*_Z^6-24*_Z^3+64)^3-410*RootOf(3*_Z^6-24*_Z^3+64)^4*x-1836*(x^3-x^2)^(1/3)*Root
Of(3*_Z^6-24*_Z^3+64)^2*x-3224*RootOf(3*_Z^6-24*_Z^3+64)*x^2-3488*(x^3-x^2)^(2/3)+1736*RootOf(3*_Z^6-24*_Z^3+6
4)*x)/(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)-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^6-2
4*_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(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3
-8)^2*RootOf(3*_Z^6-24*_Z^3+64)^3*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^2-1000*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^3*x+3120*(x^3-x^2)^(2/3)*RootO
f(3*_Z^6-24*_Z^3+64)^3-8448*(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^2-1984*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*x+2944*(x^3-x^2)^(2/3))/(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*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(RootOf(3*_Z^6-24*_Z^3+64
)^3+_Z^3-8)^2*RootOf(3*_Z^6-24*_Z^3+64)^3*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^2-1000*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^3*x+3120*(x^3-x^2)^(2/
3)*RootOf(3*_Z^6-24*_Z^3+64)^3-8448*(x^3-x^2)^(1/3)*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)^2*x+19840*RootO
f(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*x^2-1984*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*x+2944*(x^3-x^2)^(2/
3))/(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(3*_Z^6-24*_Z^3+64)^3*Root
Of(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)+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)*Root
Of(3*_Z^6-24*_Z^3+64)^6*x+183*(x^3-x^2)^(1/3)*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)^2*RootOf(3*_Z^6-24*_Z
^3+64)^3*x-1268*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^3*x^2+1340*RootOf(RootOf(
3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*RootOf(3*_Z^6-24*_Z^3+64)^3*x+228*(x^3-x^2)^(2/3)*RootOf(3*_Z^6-24*_Z^3+64)^3-192
0*(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^2-3248*RootOf(RootOf(3*_Z^6-24*_Z^3+64)^3+_Z^3-8)*x+1952*(x^3-x^2)^(2/3))/(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)

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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 = 0.49, 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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