3.5.61 \(\int \frac {-1+x}{\sqrt {-1+4 x+2 x^2-4 x^3+x^4}} \, dx\)

Optimal. Leaf size=37 \[ -\frac {1}{2} \log \left (-x^2+\sqrt {x^4-4 x^3+2 x^2+4 x-1}+2 x+1\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 33, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1680, 1107, 621, 206} \begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\frac {2-(x-1)^2}{\sqrt {(x-1)^4-4 (x-1)^2+2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x)/Sqrt[-1 + 4*x + 2*x^2 - 4*x^3 + x^4],x]

[Out]

-1/2*ArcTanh[(2 - (-1 + x)^2)/Sqrt[2 - 4*(-1 + x)^2 + (-1 + x)^4]]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {-1+x}{\sqrt {-1+4 x+2 x^2-4 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {2-4 x^2+x^4}} \, dx,x,-1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-4 x+x^2}} \, dx,x,(-1+x)^2\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2 \left (-2+(-1+x)^2\right )}{\sqrt {2-4 (-1+x)^2+(-1+x)^4}}\right )\\ &=\frac {1}{2} \tanh ^{-1}\left (\frac {-2+(-1+x)^2}{\sqrt {2-4 (-1+x)^2+(-1+x)^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 1.09, size = 716, normalized size = 19.35 \begin {gather*} -\frac {2 \left (-x+\sqrt {2-\sqrt {2}}+1\right )^2 \sqrt {\frac {-x+\sqrt {2+\sqrt {2}}+1}{-x+\sqrt {2-\sqrt {2}}+1}} \sqrt {-\frac {x+\sqrt {2+\sqrt {2}}-1}{\left (\sqrt {2-\sqrt {2}}-\sqrt {2+\sqrt {2}}\right ) \left (-x+\sqrt {2-\sqrt {2}}+1\right )}} \left (\left (-\sqrt {2+\sqrt {2}} \sqrt {-\frac {x+\sqrt {2-\sqrt {2}}-1}{-x+\sqrt {2-\sqrt {2}}+1}}+\sqrt {2} \sqrt {\frac {\left (\sqrt {2}-2\right ) \left (x+\sqrt {2-\sqrt {2}}-1\right )}{-x+\sqrt {2-\sqrt {2}}+1}}+\sqrt {\frac {\left (\sqrt {2}-2\right ) \left (x+\sqrt {2-\sqrt {2}}-1\right )}{-x+\sqrt {2-\sqrt {2}}+1}}+2 \sqrt {\frac {\left (2 \sqrt {2}-3\right ) \left (x+\sqrt {2-\sqrt {2}}-1\right )}{-x+\sqrt {2-\sqrt {2}}+1}}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {2-\sqrt {2}}-\sqrt {2+\sqrt {2}}\right ) \left (x+\sqrt {2-\sqrt {2}}-1\right )}{\left (\sqrt {2-\sqrt {2}}+\sqrt {2+\sqrt {2}}\right ) \left (-x+\sqrt {2-\sqrt {2}}+1\right )}}\right )|3+2 \sqrt {2}\right )-4 \sqrt {\frac {\left (2 \sqrt {2}-3\right ) \left (x+\sqrt {2-\sqrt {2}}-1\right )}{-x+\sqrt {2-\sqrt {2}}+1}} \Pi \left (\frac {\sqrt {2-\sqrt {2}}+\sqrt {2+\sqrt {2}}}{-\sqrt {2-\sqrt {2}}+\sqrt {2+\sqrt {2}}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {2-\sqrt {2}}-\sqrt {2+\sqrt {2}}\right ) \left (x+\sqrt {2-\sqrt {2}}-1\right )}{\left (\sqrt {2-\sqrt {2}}+\sqrt {2+\sqrt {2}}\right ) \left (-x+\sqrt {2-\sqrt {2}}+1\right )}}\right )|3+2 \sqrt {2}\right )\right )}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right )^{3/2} \sqrt {x^4-4 x^3+2 x^2+4 x-1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-1 + x)/Sqrt[-1 + 4*x + 2*x^2 - 4*x^3 + x^4],x]

[Out]

(-2*(1 + Sqrt[2 - Sqrt[2]] - x)^2*Sqrt[(1 + Sqrt[2 + Sqrt[2]] - x)/(1 + Sqrt[2 - Sqrt[2]] - x)]*Sqrt[-((-1 + S
qrt[2 + Sqrt[2]] + x)/((Sqrt[2 - Sqrt[2]] - Sqrt[2 + Sqrt[2]])*(1 + Sqrt[2 - Sqrt[2]] - x)))]*((-(Sqrt[2 + Sqr
t[2]]*Sqrt[-((-1 + Sqrt[2 - Sqrt[2]] + x)/(1 + Sqrt[2 - Sqrt[2]] - x))]) + Sqrt[((-2 + Sqrt[2])*(-1 + Sqrt[2 -
Sqrt[2]] + x))/(1 + Sqrt[2 - Sqrt[2]] - x)] + Sqrt[2]*Sqrt[((-2 + Sqrt[2])*(-1 + Sqrt[2 - Sqrt[2]] + x))/(1 +
Sqrt[2 - Sqrt[2]] - x)] + 2*Sqrt[((-3 + 2*Sqrt[2])*(-1 + Sqrt[2 - Sqrt[2]] + x))/(1 + Sqrt[2 - Sqrt[2]] - x)]
)*EllipticF[ArcSin[Sqrt[((Sqrt[2 - Sqrt[2]] - Sqrt[2 + Sqrt[2]])*(-1 + Sqrt[2 - Sqrt[2]] + x))/((Sqrt[2 - Sqrt
[2]] + Sqrt[2 + Sqrt[2]])*(1 + Sqrt[2 - Sqrt[2]] - x))]], 3 + 2*Sqrt[2]] - 4*Sqrt[((-3 + 2*Sqrt[2])*(-1 + Sqrt
[2 - Sqrt[2]] + x))/(1 + Sqrt[2 - Sqrt[2]] - x)]*EllipticPi[(Sqrt[2 - Sqrt[2]] + Sqrt[2 + Sqrt[2]])/(-Sqrt[2 -
Sqrt[2]] + Sqrt[2 + Sqrt[2]]), ArcSin[Sqrt[((Sqrt[2 - Sqrt[2]] - Sqrt[2 + Sqrt[2]])*(-1 + Sqrt[2 - Sqrt[2]] +
x))/((Sqrt[2 - Sqrt[2]] + Sqrt[2 + Sqrt[2]])*(1 + Sqrt[2 - Sqrt[2]] - x))]], 3 + 2*Sqrt[2]]))/((-Sqrt[2 - Sqr
t[2]] + Sqrt[2 + Sqrt[2]])^(3/2)*Sqrt[-1 + 4*x + 2*x^2 - 4*x^3 + x^4])

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IntegrateAlgebraic [A]  time = 0.15, size = 37, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \log \left (1+2 x-x^2+\sqrt {-1+4 x+2 x^2-4 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x)/Sqrt[-1 + 4*x + 2*x^2 - 4*x^3 + x^4],x]

[Out]

-1/2*Log[1 + 2*x - x^2 + Sqrt[-1 + 4*x + 2*x^2 - 4*x^3 + x^4]]

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fricas [A]  time = 0.51, size = 31, normalized size = 0.84 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 2 \, x + \sqrt {x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x - 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x^4-4*x^3+2*x^2+4*x-1)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.34, size = 35, normalized size = 0.95 \begin {gather*} -\frac {1}{2} \, \log \left ({\left | -x^{2} + 2 \, x + \sqrt {{\left (x^{2} - 2 \, x\right )}^{2} - 2 \, x^{2} + 4 \, x - 1} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x^4-4*x^3+2*x^2+4*x-1)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.40, size = 34, normalized size = 0.92

method	result	size
trager	\(-\frac {\ln \left (1+2 x -x^{2}+\sqrt {x^{4}-4 x^{3}+2 x^{2}+4 x -1}\right )}{2}\)	\(34\)
default	\(-\frac {2 \left (-\sqrt {2-\sqrt {2}}-\sqrt {2+\sqrt {2}}\right ) \sqrt {\frac {\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2-\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \left (x -1-\sqrt {2-\sqrt {2}}\right )^{2} \sqrt {-\frac {\left (x -1+\sqrt {2+\sqrt {2}}\right ) \sqrt {2-\sqrt {2}}}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \sqrt {\frac {\sqrt {2-\sqrt {2}}\, \left (x -1-\sqrt {2+\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2-\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}, \sqrt {\frac {\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right )^{2}}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right )^{2}}}\right )}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \sqrt {2-\sqrt {2}}\, \sqrt {\left (x -1+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2+\sqrt {2}}\right ) \left (x -1-\sqrt {2+\sqrt {2}}\right )}}+\frac {2 \left (-\sqrt {2-\sqrt {2}}-\sqrt {2+\sqrt {2}}\right ) \sqrt {\frac {\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2-\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \left (x -1-\sqrt {2-\sqrt {2}}\right )^{2} \sqrt {-\frac {\left (x -1+\sqrt {2+\sqrt {2}}\right ) \sqrt {2-\sqrt {2}}}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \sqrt {\frac {\sqrt {2-\sqrt {2}}\, \left (x -1-\sqrt {2+\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \left (\left (1+\sqrt {2-\sqrt {2}}\right ) \EllipticF \left (\sqrt {\frac {\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2-\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}, \sqrt {\frac {\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right )^{2}}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right )^{2}}}\right )-2 \sqrt {2-\sqrt {2}}\, \EllipticPi \left (\sqrt {\frac {\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2-\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}, \frac {\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}}, \sqrt {\frac {\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right )^{2}}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right )^{2}}}\right )\right )}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \sqrt {2-\sqrt {2}}\, \sqrt {\left (x -1+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2+\sqrt {2}}\right ) \left (x -1-\sqrt {2+\sqrt {2}}\right )}}\)	\(1020\)
elliptic	\(-\frac {2 \left (-\sqrt {2-\sqrt {2}}-\sqrt {2+\sqrt {2}}\right ) \sqrt {\frac {\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2-\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \left (x -1-\sqrt {2-\sqrt {2}}\right )^{2} \sqrt {-\frac {\left (x -1+\sqrt {2+\sqrt {2}}\right ) \sqrt {2-\sqrt {2}}}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \sqrt {\frac {\sqrt {2-\sqrt {2}}\, \left (x -1-\sqrt {2+\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2-\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}, \sqrt {\frac {\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right )^{2}}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right )^{2}}}\right )}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \sqrt {2-\sqrt {2}}\, \sqrt {\left (x -1+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2+\sqrt {2}}\right ) \left (x -1-\sqrt {2+\sqrt {2}}\right )}}+\frac {2 \left (-\sqrt {2-\sqrt {2}}-\sqrt {2+\sqrt {2}}\right ) \sqrt {\frac {\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2-\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \left (x -1-\sqrt {2-\sqrt {2}}\right )^{2} \sqrt {-\frac {\left (x -1+\sqrt {2+\sqrt {2}}\right ) \sqrt {2-\sqrt {2}}}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \sqrt {\frac {\sqrt {2-\sqrt {2}}\, \left (x -1-\sqrt {2+\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}\, \left (\left (1+\sqrt {2-\sqrt {2}}\right ) \EllipticF \left (\sqrt {\frac {\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2-\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}, \sqrt {\frac {\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right )^{2}}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right )^{2}}}\right )-2 \sqrt {2-\sqrt {2}}\, \EllipticPi \left (\sqrt {\frac {\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2-\sqrt {2}}\right )}{\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right )}}, \frac {\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}}, \sqrt {\frac {\left (\sqrt {2+\sqrt {2}}+\sqrt {2-\sqrt {2}}\right )^{2}}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right )^{2}}}\right )\right )}{\left (\sqrt {2+\sqrt {2}}-\sqrt {2-\sqrt {2}}\right ) \sqrt {2-\sqrt {2}}\, \sqrt {\left (x -1+\sqrt {2-\sqrt {2}}\right ) \left (x -1-\sqrt {2-\sqrt {2}}\right ) \left (x -1+\sqrt {2+\sqrt {2}}\right ) \left (x -1-\sqrt {2+\sqrt {2}}\right )}}\)	\(1020\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+x)/(x^4-4*x^3+2*x^2+4*x-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*ln(1+2*x-x^2+(x^4-4*x^3+2*x^2+4*x-1)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x - 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x^4-4*x^3+2*x^2+4*x-1)^(1/2),x, algorithm="maxima")

[Out]

integrate((x - 1)/sqrt(x^4 - 4*x^3 + 2*x^2 + 4*x - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x-1}{\sqrt {x^4-4\,x^3+2\,x^2+4\,x-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)/(4*x + 2*x^2 - 4*x^3 + x^4 - 1)^(1/2),x)

[Out]

int((x - 1)/(4*x + 2*x^2 - 4*x^3 + x^4 - 1)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {x^{4} - 4 x^{3} + 2 x^{2} + 4 x - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x**4-4*x**3+2*x**2+4*x-1)**(1/2),x)

[Out]

Integral((x - 1)/sqrt(x**4 - 4*x**3 + 2*x**2 + 4*x - 1), x)

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