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

Optimal. Leaf size=33 \[ -\log \left (-x^2+\sqrt {x^4-2 x^3+3 x^2-2 x-2}+x-1\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1680, 12, 1107, 621, 206} \begin {gather*} \tanh ^{-1}\left (\frac {4 \left (x-\frac {1}{2}\right )^2+3}{\sqrt {16 \left (x-\frac {1}{2}\right )^4+24 \left (x-\frac {1}{2}\right )^2-39}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[(3 + 4*(-1/2 + x)^2)/Sqrt[-39 + 24*(-1/2 + x)^2 + 16*(-1/2 + x)^4]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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+2 x}{\sqrt {-2-2 x+3 x^2-2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {-39+24 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {-39+24 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-39+24 x+16 x^2}} \, dx,x,\left (-\frac {1}{2}+x\right )^2\right )\\ &=8 \operatorname {Subst}\left (\int \frac {1}{64-x^2} \, dx,x,\frac {8 \left (3+4 \left (-\frac {1}{2}+x\right )^2\right )}{\sqrt {-39+(1-2 x)^4+24 \left (-\frac {1}{2}+x\right )^2}}\right )\\ &=\tanh ^{-1}\left (\frac {3+(-1+2 x)^2}{\sqrt {-39+6 (1-2 x)^2+(1-2 x)^4}}\right )\\ \end {align*}

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Mathematica [C]  time = 3.20, size = 702, normalized size = 21.27 \begin {gather*} \frac {\sqrt {4 \sqrt {3}-3} \left (-2 x+\sqrt {4 \sqrt {3}-3}+1\right )^2 \left (\frac {-2 x-i \sqrt {3+4 \sqrt {3}}+1}{\left (\sqrt {4 \sqrt {3}-3}-i \sqrt {3+4 \sqrt {3}}\right ) \left (-2 x+\sqrt {4 \sqrt {3}-3}+1\right )}\right )^{3/2} \left (-2 x+i \sqrt {3+4 \sqrt {3}}+1\right ) \sqrt {\frac {\left (\sqrt {4 \sqrt {3}-3}-i \sqrt {3+4 \sqrt {3}}\right ) \left (2 x+\sqrt {4 \sqrt {3}-3}-1\right )}{\left (\sqrt {4 \sqrt {3}-3}+i \sqrt {3+4 \sqrt {3}}\right ) \left (-2 x+\sqrt {4 \sqrt {3}-3}+1\right )}} \left (F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-3+4 \sqrt {3}}-i \sqrt {3+4 \sqrt {3}}\right ) \left (2 x+\sqrt {-3+4 \sqrt {3}}-1\right )}{\left (\sqrt {-3+4 \sqrt {3}}+i \sqrt {3+4 \sqrt {3}}\right ) \left (-2 x+\sqrt {-3+4 \sqrt {3}}+1\right )}}\right )|\frac {3 i+\sqrt {39}}{3 i-\sqrt {39}}\right )-2 \Pi \left (-\frac {\sqrt {-3+4 \sqrt {3}}+i \sqrt {3+4 \sqrt {3}}}{\sqrt {-3+4 \sqrt {3}}-i \sqrt {3+4 \sqrt {3}}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {-3+4 \sqrt {3}}-i \sqrt {3+4 \sqrt {3}}\right ) \left (2 x+\sqrt {-3+4 \sqrt {3}}-1\right )}{\left (\sqrt {-3+4 \sqrt {3}}+i \sqrt {3+4 \sqrt {3}}\right ) \left (-2 x+\sqrt {-3+4 \sqrt {3}}+1\right )}}\right )|\frac {3 i+\sqrt {39}}{3 i-\sqrt {39}}\right )\right )}{\left (-2 x-i \sqrt {3+4 \sqrt {3}}+1\right ) \sqrt {\frac {-2 x+i \sqrt {3+4 \sqrt {3}}+1}{\left (\sqrt {4 \sqrt {3}-3}+i \sqrt {3+4 \sqrt {3}}\right ) \left (-2 x+\sqrt {4 \sqrt {3}-3}+1\right )}} \sqrt {x^4-2 x^3+3 x^2-2 x-2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[-3 + 4*Sqrt[3]]*(1 + Sqrt[-3 + 4*Sqrt[3]] - 2*x)^2*((1 - I*Sqrt[3 + 4*Sqrt[3]] - 2*x)/((Sqrt[-3 + 4*Sqrt
[3]] - I*Sqrt[3 + 4*Sqrt[3]])*(1 + Sqrt[-3 + 4*Sqrt[3]] - 2*x)))^(3/2)*(1 + I*Sqrt[3 + 4*Sqrt[3]] - 2*x)*Sqrt[
((Sqrt[-3 + 4*Sqrt[3]] - I*Sqrt[3 + 4*Sqrt[3]])*(-1 + Sqrt[-3 + 4*Sqrt[3]] + 2*x))/((Sqrt[-3 + 4*Sqrt[3]] + I*
Sqrt[3 + 4*Sqrt[3]])*(1 + Sqrt[-3 + 4*Sqrt[3]] - 2*x))]*(EllipticF[ArcSin[Sqrt[((Sqrt[-3 + 4*Sqrt[3]] - I*Sqrt
[3 + 4*Sqrt[3]])*(-1 + Sqrt[-3 + 4*Sqrt[3]] + 2*x))/((Sqrt[-3 + 4*Sqrt[3]] + I*Sqrt[3 + 4*Sqrt[3]])*(1 + Sqrt[
-3 + 4*Sqrt[3]] - 2*x))]], (3*I + Sqrt[39])/(3*I - Sqrt[39])] - 2*EllipticPi[-((Sqrt[-3 + 4*Sqrt[3]] + I*Sqrt[
3 + 4*Sqrt[3]])/(Sqrt[-3 + 4*Sqrt[3]] - I*Sqrt[3 + 4*Sqrt[3]])), ArcSin[Sqrt[((Sqrt[-3 + 4*Sqrt[3]] - I*Sqrt[3
+ 4*Sqrt[3]])*(-1 + Sqrt[-3 + 4*Sqrt[3]] + 2*x))/((Sqrt[-3 + 4*Sqrt[3]] + I*Sqrt[3 + 4*Sqrt[3]])*(1 + Sqrt[-3
+ 4*Sqrt[3]] - 2*x))]], (3*I + Sqrt[39])/(3*I - Sqrt[39])]))/((1 - I*Sqrt[3 + 4*Sqrt[3]] - 2*x)*Sqrt[(1 + I*S
qrt[3 + 4*Sqrt[3]] - 2*x)/((Sqrt[-3 + 4*Sqrt[3]] + I*Sqrt[3 + 4*Sqrt[3]])*(1 + Sqrt[-3 + 4*Sqrt[3]] - 2*x))]*S
qrt[-2 - 2*x + 3*x^2 - 2*x^3 + x^4])

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IntegrateAlgebraic [A]  time = 0.13, size = 33, normalized size = 1.00 \begin {gather*} -\log \left (-1+x-x^2+\sqrt {-2-2 x+3 x^2-2 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.85, size = 31, normalized size = 0.94 \begin {gather*} \log \left (-x^{2} + x - \sqrt {x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x - 2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.39, size = 34, normalized size = 1.03 \begin {gather*} -\log \left (x^{2} - x - \sqrt {{\left (x^{2} - x\right )}^{2} + 2 \, x^{2} - 2 \, x - 2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x^2 - x - sqrt((x^2 - x)^2 + 2*x^2 - 2*x - 2) + 1)

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maple [A]  time = 0.76, size = 34, normalized size = 1.03

method	result	size
trager	\(-\ln \left (x^{2}-\sqrt {x^{4}-2 x^{3}+3 x^{2}-2 x -2}-x +1\right )\)	\(34\)
default	\(-\frac {2 \left (-\frac {\sqrt {-3+4 \sqrt {3}}}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {-3+4 \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (-\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {-3+4 \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {-3+4 \sqrt {3}}}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \sqrt {-3+4 \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {-3+4 \sqrt {3}}}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {-3+4 \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (-\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {-3+4 \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {-3+4 \sqrt {3}}}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\right )-\sqrt {-3+4 \sqrt {3}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}, \frac {\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}}{\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}}, \sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {-3+4 \sqrt {3}}}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\right )\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \sqrt {-3+4 \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}}\)	\(1358\)
elliptic	\(-\frac {2 \left (-\frac {\sqrt {-3+4 \sqrt {3}}}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {-3+4 \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (-\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {-3+4 \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {-3+4 \sqrt {3}}}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \sqrt {-3+4 \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {-3+4 \sqrt {3}}}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {-3+4 \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (-\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {-3+4 \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {-3+4 \sqrt {3}}}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\right )-\sqrt {-3+4 \sqrt {3}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}, \frac {\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}}{\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}}, \sqrt {\frac {\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {-3+4 \sqrt {3}}}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (-\frac {i \sqrt {3+4 \sqrt {3}}}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right )}}\right )\right )}{\left (\frac {i \sqrt {3+4 \sqrt {3}}}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \sqrt {-3+4 \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {-3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3+4 \sqrt {3}}}{2}\right )}}\)	\(1358\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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