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

Optimal. Leaf size=217 \[ -\log \left (\sqrt [3]{x^3-x^2}-x\right )+2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3-x^2}-2 x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3-x^2} x+\sqrt [3]{2} \left (x^3-x^2\right )^{2/3}\right )}{\sqrt [3]{2}}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right )-2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x^2}+x}\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 291, normalized size of antiderivative = 1.34, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2056, 105, 59, 91} \begin {gather*} -\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3-x^2}}+\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{\sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x)}{2 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x+1)}{\sqrt [3]{2} \sqrt [3]{x^3-x^2}}-\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3-x^2}}+\frac {2^{2/3} \sqrt {3} \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]{x^3-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x)/((1 + x)*(-x^2 + x^3)^(1/3)),x]

[Out]

-((Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(-x^2 + x^3)^(1/3)
) + (2^(2/3)*Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2^(2/3)*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(-
x^2 + x^3)^(1/3) - (3*(-1 + x)^(1/3)*x^(2/3)*Log[-1 + (-1 + x)^(1/3)/x^(1/3)])/(2*(-x^2 + x^3)^(1/3)) + (3*(-1
+ x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/2^(1/3) - x^(1/3)])/(2^(1/3)*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*x^(2
/3)*Log[x])/(2*(-x^2 + x^3)^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*Log[1 + x])/(2^(1/3)*(-x^2 + x^3)^(1/3))

Rule 59

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, -Simp[(Sqrt
[3]*q*ArcTan[(2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]])/d, x] + (-Simp[(3*q*Log[(q*(a + b*x
)^(1/3))/(c + d*x)^(1/3) - 1])/(2*d), x] - Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[
b*c - a*d, 0] && PosQ[d/b]

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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -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]

Rubi steps

\begin {align*} \int \frac {-1+x}{(1+x) \sqrt [3]{-x^2+x^3}} \, dx &=\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {(-1+x)^{2/3}}{x^{2/3} (1+x)} \, dx}{\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}} \, dx}{\sqrt [3]{-x^2+x^3}}-\frac {\left (2 \sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3} (1+x)} \, dx}{\sqrt [3]{-x^2+x^3}}\\ &=-\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{-x^2+x^3}}+\frac {2^{2/3} \sqrt {3} \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]{-x^2+x^3}}-\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{-x^2+x^3}}+\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (\frac {\sqrt [3]{-1+x}}{\sqrt [3]{2}}-\sqrt [3]{x}\right )}{\sqrt [3]{2} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (x)}{2 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (1+x)}{\sqrt [3]{2} \sqrt [3]{-x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 60, normalized size = 0.28 \begin {gather*} \frac {3 \left ((x-1) x^2\right )^{2/3} \left (x^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};1-x\right )-\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x-1}{2 x}\right )\right )}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x)/((1 + x)*(-x^2 + x^3)^(1/3)),x]

[Out]

(3*((-1 + x)*x^2)^(2/3)*(x^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, 1 - x] - Hypergeometric2F1[2/3, 1, 5/3, (-1
+ x)/(2*x)]))/(2*x^2)

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IntegrateAlgebraic [A]  time = 0.51, size = 217, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right )-\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 )}{\sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x)/((1 + x)*(-x^2 + x^3)^(1/3)),x]

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-x^2 + x^3)^(1/3))] - 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-x^2
+ x^3)^(1/3))] - Log[-x + (-x^2 + x^3)^(1/3)] + 2^(2/3)*Log[-2*x + 2^(2/3)*(-x^2 + x^3)^(1/3)] + Log[x^2 + x*
(-x^2 + x^3)^(1/3) + (-x^2 + x^3)^(2/3)]/2 - Log[2*x^2 + 2^(2/3)*x*(-x^2 + x^3)^(1/3) + 2^(1/3)*(-x^2 + x^3)^(
2/3)]/2^(1/3)

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fricas [A]  time = 1.07, size = 206, normalized size = 0.95 \begin {gather*} 4^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 4^{\frac {1}{3}} \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 4^{\frac {1}{3}} \log \left (-\frac {4^{\frac {2}{3}} x - 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{2} \cdot 4^{\frac {1}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + 2 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(1+x)/(x^3-x^2)^(1/3),x, algorithm="fricas")

[Out]

4^(1/3)*sqrt(3)*arctan(1/3*(sqrt(3)*x + 4^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3))/x) - sqrt(3)*arctan(1/3*(sqrt(3)*x
+ 2*sqrt(3)*(x^3 - x^2)^(1/3))/x) + 4^(1/3)*log(-(4^(2/3)*x - 2*(x^3 - x^2)^(1/3))/x) - 1/2*4^(1/3)*log((2*4^(
1/3)*x^2 + 4^(2/3)*(x^3 - x^2)^(1/3)*x + 2*(x^3 - x^2)^(2/3))/x^2) - log(-(x - (x^3 - x^2)^(1/3))/x) + 1/2*log
((x^2 + (x^3 - x^2)^(1/3)*x + (x^3 - x^2)^(2/3))/x^2)

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giac [A]  time = 0.25, size = 147, normalized size = 0.68 \begin {gather*} \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{2} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}}\right ) + 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(1+x)/(x^3-x^2)^(1/3),x, algorithm="giac")

[Out]

sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x + 1)^(1/3))) - 1/2*2^(2/3)*log(2^(2/3) + 2^(1/3)
*(-1/x + 1)^(1/3) + (-1/x + 1)^(2/3)) + 2^(2/3)*log(abs(-2^(1/3) + (-1/x + 1)^(1/3))) - sqrt(3)*arctan(1/3*sqr
t(3)*(2*(-1/x + 1)^(1/3) + 1)) + 1/2*log((-1/x + 1)^(2/3) + (-1/x + 1)^(1/3) + 1) - log(abs((-1/x + 1)^(1/3) -
1))

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maple [C]  time = 3.68, size = 1587, normalized size = 7.31

method	result	size
trager	\(\text {Expression too large to display}\)	\(1587\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+x)/(1+x)/(x^3-x^2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^2-2*_Z+4)*ln((5*RootOf(_Z^2-2*_Z+4)^2*x^2+48*RootOf(_Z^2-2*_Z+4)*(x^3-x^2)^(2/3)+48*RootOf(_Z^2-
2*_Z+4)*(x^3-x^2)^(1/3)*x-10*RootOf(_Z^2-2*_Z+4)^2*x+38*RootOf(_Z^2-2*_Z+4)*x^2-36*(x^3-x^2)^(2/3)-36*x*(x^3-x
^2)^(1/3)-6*RootOf(_Z^2-2*_Z+4)*x-16*x^2+4*x)/x)-1/2*ln((5*RootOf(_Z^2-2*_Z+4)^2*x^2-10*RootOf(_Z^2-2*_Z+4)^2*
x-48*RootOf(_Z^2-2*_Z+4)*(x^3-x^2)^(2/3)-48*RootOf(_Z^2-2*_Z+4)*(x^3-x^2)^(1/3)*x-58*RootOf(_Z^2-2*_Z+4)*x^2+4
6*RootOf(_Z^2-2*_Z+4)*x+60*(x^3-x^2)^(2/3)+60*x*(x^3-x^2)^(1/3)+80*x^2-48*x)/x)*RootOf(_Z^2-2*_Z+4)+ln((5*Root
Of(_Z^2-2*_Z+4)^2*x^2-10*RootOf(_Z^2-2*_Z+4)^2*x-48*RootOf(_Z^2-2*_Z+4)*(x^3-x^2)^(2/3)-48*RootOf(_Z^2-2*_Z+4)
*(x^3-x^2)^(1/3)*x-58*RootOf(_Z^2-2*_Z+4)*x^2+46*RootOf(_Z^2-2*_Z+4)*x+60*(x^3-x^2)^(2/3)+60*x*(x^3-x^2)^(1/3)
+80*x^2-48*x)/x)-ln((4*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^2+3*RootOf(4*Roo
tOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^2-8*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)
+_Z^2)*RootOf(_Z^3-4)^3*x-6*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x+48*(x^3-x
^2)^(2/3)*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2+72*RootOf(_Z^3-4)^2*(x^3-x^2)^(
1/3)*x+96*(x^3-x^2)^(1/3)*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)*x+208*RootOf(_Z^3
-4)*x^2+156*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x^2-80*RootOf(_Z^3-4)*x-60*RootOf(4*RootOf(_Z^
3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x+144*(x^3-x^2)^(2/3))/x/(1+x))*RootOf(_Z^3-4)-1/2*ln((4*RootOf(4*RootOf(_Z^3
-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^2+3*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*R
ootOf(_Z^3-4)^2*x^2-8*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x-6*RootOf(4*RootOf
(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x+48*(x^3-x^2)^(2/3)*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*Ro
otOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2+72*RootOf(_Z^3-4)^2*(x^3-x^2)^(1/3)*x+96*(x^3-x^2)^(1/3)*RootOf(4*RootOf(_
Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)*x+208*RootOf(_Z^3-4)*x^2+156*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*
RootOf(_Z^3-4)+_Z^2)*x^2-80*RootOf(_Z^3-4)*x-60*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x+144*(x^3
-x^2)^(2/3))/x/(1+x))*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)+1/2*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*R
ootOf(_Z^3-4)+_Z^2)*ln((2*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^2+3*RootOf(4*
RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^2-4*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3
-4)+_Z^2)*RootOf(_Z^3-4)^3*x-6*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x-48*(x^
3-x^2)^(2/3)*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2-120*RootOf(_Z^3-4)^2*(x^3-x^
2)^(1/3)*x-96*(x^3-x^2)^(1/3)*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)*x-88*RootOf(_
Z^3-4)*x^2-132*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x^2+8*RootOf(_Z^3-4)*x+12*RootOf(4*RootOf(_
Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x-240*(x^3-x^2)^(2/3))/x/(1+x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(1+x)/(x^3-x^2)^(1/3),x, algorithm="maxima")

[Out]

integrate((x - 1)/((x^3 - x^2)^(1/3)*(x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x-1}{{\left (x^3-x^2\right )}^{1/3}\,\left (x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)/((x^3 - x^2)^(1/3)*(x + 1)),x)

[Out]

int((x - 1)/((x^3 - x^2)^(1/3)*(x + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt [3]{x^{2} \left (x - 1\right )} \left (x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(1+x)/(x**3-x**2)**(1/3),x)

[Out]

Integral((x - 1)/((x**2*(x - 1))**(1/3)*(x + 1)), x)

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