3.4.80 \(\int \frac {-1+x}{\sqrt {-5+4 x^2-4 x^3+x^4}} \, dx\)

Optimal. Leaf size=31 \[ \frac {1}{2} \log \left (x^2+\sqrt {x^4-4 x^3+4 x^2-5}-2 x\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1680, 1107, 621, 206} \begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\frac {1-(x-1)^2}{\sqrt {(x-1)^4-2 (x-1)^2-4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x)/Sqrt[-5 + 4*x^2 - 4*x^3 + x^4],x]

[Out]

-1/2*ArcTanh[(1 - (-1 + x)^2)/Sqrt[-4 - 2*(-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 {-5+4 x^2-4 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {-4-2 x^2+x^4}} \, dx,x,-1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-4-2 x+x^2}} \, dx,x,(-1+x)^2\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {2 (-2+x) x}{\sqrt {-4-2 (-1+x)^2+(-1+x)^4}}\right )\\ &=\frac {1}{2} \tanh ^{-1}\left (\frac {(-2+x) x}{\sqrt {-4-2 (-1+x)^2+(-1+x)^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 1.56, size = 701, normalized size = 22.61 \begin {gather*} \frac {2 \sqrt {\sqrt {5}-1} \left (-x+i \sqrt {\sqrt {5}-1}+1\right ) \left (-x+\sqrt {1+\sqrt {5}}+1\right ) \sqrt {\frac {\left (\sqrt {1+\sqrt {5}}-i \sqrt {\sqrt {5}-1}\right ) \left (x+i \sqrt {\sqrt {5}-1}-1\right )}{\left (\sqrt {1+\sqrt {5}}+i \sqrt {\sqrt {5}-1}\right ) \left (x-i \sqrt {\sqrt {5}-1}-1\right )}} \sqrt {\frac {i \left (x+\sqrt {1+\sqrt {5}}-1\right )}{\left (\sqrt {\sqrt {5}-1}+i \sqrt {1+\sqrt {5}}\right ) \left (i x+\sqrt {\sqrt {5}-1}-i\right )}} \left (F\left (\sin ^{-1}\left (\sqrt {\frac {\sqrt {1+\sqrt {5}} x-i \sqrt {-1+\sqrt {5}} x-\sqrt {1+\sqrt {5}}+i \sqrt {-1+\sqrt {5}}+\sqrt {5}-(1-2 i)}{\sqrt {1+\sqrt {5}} x+i \sqrt {-1+\sqrt {5}} x-\sqrt {1+\sqrt {5}}-i \sqrt {-1+\sqrt {5}}+\sqrt {5}-(1+2 i)}}\right )|-\frac {3}{5}+\frac {4 i}{5}\right )-2 \Pi \left (-\frac {\sqrt {-1+\sqrt {5}}-i \sqrt {1+\sqrt {5}}}{\sqrt {-1+\sqrt {5}}+i \sqrt {1+\sqrt {5}}};\sin ^{-1}\left (\sqrt {\frac {\sqrt {1+\sqrt {5}} x-i \sqrt {-1+\sqrt {5}} x-\sqrt {1+\sqrt {5}}+i \sqrt {-1+\sqrt {5}}+\sqrt {5}-(1-2 i)}{\sqrt {1+\sqrt {5}} x+i \sqrt {-1+\sqrt {5}} x-\sqrt {1+\sqrt {5}}-i \sqrt {-1+\sqrt {5}}+\sqrt {5}-(1+2 i)}}\right )|-\frac {3}{5}+\frac {4 i}{5}\right )\right )}{\left (\sqrt {\sqrt {5}-1}+i \sqrt {1+\sqrt {5}}\right ) \sqrt {-\frac {i \left (-x+\sqrt {1+\sqrt {5}}+1\right )}{\left (\sqrt {\sqrt {5}-1}-i \sqrt {1+\sqrt {5}}\right ) \left (i x+\sqrt {\sqrt {5}-1}-i\right )}} \sqrt {x^4-4 x^3+4 x^2-5}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-1 + x)/Sqrt[-5 + 4*x^2 - 4*x^3 + x^4],x]

[Out]

(2*Sqrt[-1 + Sqrt[5]]*(1 + I*Sqrt[-1 + Sqrt[5]] - x)*(1 + Sqrt[1 + Sqrt[5]] - x)*Sqrt[(((-I)*Sqrt[-1 + Sqrt[5]
] + Sqrt[1 + Sqrt[5]])*(-1 + I*Sqrt[-1 + Sqrt[5]] + x))/((I*Sqrt[-1 + Sqrt[5]] + Sqrt[1 + Sqrt[5]])*(-1 - I*Sq
rt[-1 + Sqrt[5]] + x))]*Sqrt[(I*(-1 + Sqrt[1 + Sqrt[5]] + x))/((Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]])*(-I
+ Sqrt[-1 + Sqrt[5]] + I*x))]*(EllipticF[ArcSin[Sqrt[((-1 + 2*I) + Sqrt[5] + I*Sqrt[-1 + Sqrt[5]] - Sqrt[1 + S
qrt[5]] - I*Sqrt[-1 + Sqrt[5]]*x + Sqrt[1 + Sqrt[5]]*x)/((-1 - 2*I) + Sqrt[5] - I*Sqrt[-1 + Sqrt[5]] - Sqrt[1
+ Sqrt[5]] + I*Sqrt[-1 + Sqrt[5]]*x + Sqrt[1 + Sqrt[5]]*x)]], -3/5 + (4*I)/5] - 2*EllipticPi[-((Sqrt[-1 + Sqrt
[5]] - I*Sqrt[1 + Sqrt[5]])/(Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]])), ArcSin[Sqrt[((-1 + 2*I) + Sqrt[5] + I
*Sqrt[-1 + Sqrt[5]] - Sqrt[1 + Sqrt[5]] - I*Sqrt[-1 + Sqrt[5]]*x + Sqrt[1 + Sqrt[5]]*x)/((-1 - 2*I) + Sqrt[5]
- I*Sqrt[-1 + Sqrt[5]] - Sqrt[1 + Sqrt[5]] + I*Sqrt[-1 + Sqrt[5]]*x + Sqrt[1 + Sqrt[5]]*x)]], -3/5 + (4*I)/5])
)/((Sqrt[-1 + Sqrt[5]] + I*Sqrt[1 + Sqrt[5]])*Sqrt[((-I)*(1 + Sqrt[1 + Sqrt[5]] - x))/((Sqrt[-1 + Sqrt[5]] - I
*Sqrt[1 + Sqrt[5]])*(-I + Sqrt[-1 + Sqrt[5]] + I*x))]*Sqrt[-5 + 4*x^2 - 4*x^3 + x^4])

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IntegrateAlgebraic [A]  time = 0.06, size = 31, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (-2 x+x^2+\sqrt {-5+4 x^2-4 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x)/Sqrt[-5 + 4*x^2 - 4*x^3 + x^4],x]

[Out]

Log[-2*x + x^2 + Sqrt[-5 + 4*x^2 - 4*x^3 + x^4]]/2

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fricas [A]  time = 0.51, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 2 \, x + \sqrt {x^{4} - 4 \, x^{3} + 4 \, x^{2} - 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x^4-4*x^3+4*x^2-5)^(1/2),x, algorithm="fricas")

[Out]

1/2*log(x^2 - 2*x + sqrt(x^4 - 4*x^3 + 4*x^2 - 5))

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giac [A]  time = 0.46, size = 26, normalized size = 0.84 \begin {gather*} -\frac {1}{2} \, \log \left ({\left | -x^{2} + 2 \, x + \sqrt {{\left (x^{2} - 2 \, x\right )}^{2} - 5} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x^4-4*x^3+4*x^2-5)^(1/2),x, algorithm="giac")

[Out]

-1/2*log(abs(-x^2 + 2*x + sqrt((x^2 - 2*x)^2 - 5)))

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maple [A]  time = 0.58, size = 30, normalized size = 0.97

method	result	size
trager	\(-\frac {\ln \left (-x^{2}+\sqrt {x^{4}-4 x^{3}+4 x^{2}-5}+2 x \right )}{2}\)	\(30\)
default	\(-\frac {2 \left (-\sqrt {\sqrt {5}+1}-i \sqrt {\sqrt {5}-1}\right ) \sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (x -1+\sqrt {\sqrt {5}+1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \left (x -1-\sqrt {\sqrt {5}+1}\right )^{2} \sqrt {\frac {\sqrt {\sqrt {5}+1}\, \left (x -1+i \sqrt {\sqrt {5}-1}\right )}{\left (-i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \sqrt {\frac {\sqrt {\sqrt {5}+1}\, \left (x -1-i \sqrt {\sqrt {5}-1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (x -1+\sqrt {\sqrt {5}+1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}, \sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (-\sqrt {\sqrt {5}+1}-i \sqrt {\sqrt {5}-1}\right )}{\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (-i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right )}}\right )}{\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \sqrt {\sqrt {5}+1}\, \sqrt {\left (x -1+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right ) \left (x -1+i \sqrt {\sqrt {5}-1}\right ) \left (x -1-i \sqrt {\sqrt {5}-1}\right )}}+\frac {2 \left (-\sqrt {\sqrt {5}+1}-i \sqrt {\sqrt {5}-1}\right ) \sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (x -1+\sqrt {\sqrt {5}+1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \left (x -1-\sqrt {\sqrt {5}+1}\right )^{2} \sqrt {\frac {\sqrt {\sqrt {5}+1}\, \left (x -1+i \sqrt {\sqrt {5}-1}\right )}{\left (-i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \sqrt {\frac {\sqrt {\sqrt {5}+1}\, \left (x -1-i \sqrt {\sqrt {5}-1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \left (\left (1+\sqrt {\sqrt {5}+1}\right ) \EllipticF \left (\sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (x -1+\sqrt {\sqrt {5}+1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}, \sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (-\sqrt {\sqrt {5}+1}-i \sqrt {\sqrt {5}-1}\right )}{\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (-i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right )}}\right )-2 \sqrt {\sqrt {5}+1}\, \EllipticPi \left (\sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (x -1+\sqrt {\sqrt {5}+1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}, \frac {i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}}{i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}}, \sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (-\sqrt {\sqrt {5}+1}-i \sqrt {\sqrt {5}-1}\right )}{\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (-i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right )}}\right )\right )}{\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \sqrt {\sqrt {5}+1}\, \sqrt {\left (x -1+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right ) \left (x -1+i \sqrt {\sqrt {5}-1}\right ) \left (x -1-i \sqrt {\sqrt {5}-1}\right )}}\)	\(1110\)
elliptic	\(-\frac {2 \left (-\sqrt {\sqrt {5}+1}-i \sqrt {\sqrt {5}-1}\right ) \sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (x -1+\sqrt {\sqrt {5}+1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \left (x -1-\sqrt {\sqrt {5}+1}\right )^{2} \sqrt {\frac {\sqrt {\sqrt {5}+1}\, \left (x -1+i \sqrt {\sqrt {5}-1}\right )}{\left (-i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \sqrt {\frac {\sqrt {\sqrt {5}+1}\, \left (x -1-i \sqrt {\sqrt {5}-1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (x -1+\sqrt {\sqrt {5}+1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}, \sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (-\sqrt {\sqrt {5}+1}-i \sqrt {\sqrt {5}-1}\right )}{\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (-i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right )}}\right )}{\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \sqrt {\sqrt {5}+1}\, \sqrt {\left (x -1+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right ) \left (x -1+i \sqrt {\sqrt {5}-1}\right ) \left (x -1-i \sqrt {\sqrt {5}-1}\right )}}+\frac {2 \left (-\sqrt {\sqrt {5}+1}-i \sqrt {\sqrt {5}-1}\right ) \sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (x -1+\sqrt {\sqrt {5}+1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \left (x -1-\sqrt {\sqrt {5}+1}\right )^{2} \sqrt {\frac {\sqrt {\sqrt {5}+1}\, \left (x -1+i \sqrt {\sqrt {5}-1}\right )}{\left (-i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \sqrt {\frac {\sqrt {\sqrt {5}+1}\, \left (x -1-i \sqrt {\sqrt {5}-1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}\, \left (\left (1+\sqrt {\sqrt {5}+1}\right ) \EllipticF \left (\sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (x -1+\sqrt {\sqrt {5}+1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}, \sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (-\sqrt {\sqrt {5}+1}-i \sqrt {\sqrt {5}-1}\right )}{\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (-i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right )}}\right )-2 \sqrt {\sqrt {5}+1}\, \EllipticPi \left (\sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (x -1+\sqrt {\sqrt {5}+1}\right )}{\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right )}}, \frac {i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}}{i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}}, \sqrt {\frac {\left (i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right ) \left (-\sqrt {\sqrt {5}+1}-i \sqrt {\sqrt {5}-1}\right )}{\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \left (-i \sqrt {\sqrt {5}-1}+\sqrt {\sqrt {5}+1}\right )}}\right )\right )}{\left (i \sqrt {\sqrt {5}-1}-\sqrt {\sqrt {5}+1}\right ) \sqrt {\sqrt {5}+1}\, \sqrt {\left (x -1+\sqrt {\sqrt {5}+1}\right ) \left (x -1-\sqrt {\sqrt {5}+1}\right ) \left (x -1+i \sqrt {\sqrt {5}-1}\right ) \left (x -1-i \sqrt {\sqrt {5}-1}\right )}}\)	\(1110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+x)/(x^4-4*x^3+4*x^2-5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*ln(-x^2+(x^4-4*x^3+4*x^2-5)^(1/2)+2*x)

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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} + 4 \, x^{2} - 5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x^4-4*x^3+4*x^2-5)^(1/2),x, algorithm="maxima")

[Out]

integrate((x - 1)/sqrt(x^4 - 4*x^3 + 4*x^2 - 5), 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+4\,x^2-5}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)/(4*x^2 - 4*x^3 + x^4 - 5)^(1/2),x)

[Out]

int((x - 1)/(4*x^2 - 4*x^3 + x^4 - 5)^(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} + 4 x^{2} - 5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(x**4-4*x**3+4*x**2-5)**(1/2),x)

[Out]

Integral((x - 1)/sqrt(x**4 - 4*x**3 + 4*x**2 - 5), x)

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