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

Optimal. Leaf size=30 \[ -\log \left (-2 x^2+2 \sqrt {x^4-2 x^3+x+1}+2 x+1\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 23, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1680, 12, 1107, 619, 215} \begin {gather*} -\sinh ^{-1}\left (\frac {3-4 \left (x-\frac {1}{2}\right )^2}{2 \sqrt {3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/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 {1+x-2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {21-24 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {21-24 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {21-24 x+16 x^2}} \, dx,x,\left (-\frac {1}{2}+x\right )^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{768}}} \, dx,x,8 \left (-3+4 \left (-\frac {1}{2}+x\right )^2\right )\right )}{16 \sqrt {3}}\\ &=-\sinh ^{-1}\left (\frac {3-(-1+2 x)^2}{2 \sqrt {3}}\right )\\ \end {align*}

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Mathematica [C]  time = 3.37, size = 717, normalized size = 23.90 \begin {gather*} \frac {\left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right ) \sqrt {\frac {\sqrt {1+4 \sqrt [3]{-1}} \left (-2 x+\sqrt {1-4 (-1)^{2/3}}+1\right )}{\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right ) \left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right )}} \left (2 x+\sqrt {1+4 \sqrt [3]{-1}}-1\right ) \sqrt {-\frac {\sqrt {1+4 \sqrt [3]{-1}} \left (2 x+\sqrt {1-4 (-1)^{2/3}}-1\right )}{\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right ) \left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right )}} \left (F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right ) \left (2 x+\sqrt {1+4 \sqrt [3]{-1}}-1\right )}{\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right ) \left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right )}}\right )|\frac {\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right )^2}{\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right )^2}\right )-2 \Pi \left (-\frac {\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}}{\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right ) \left (2 x+\sqrt {1+4 \sqrt [3]{-1}}-1\right )}{\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right ) \left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right )}}\right )|\frac {\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right )^2}{\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right )^2}\right )\right )}{\sqrt {\frac {\left (\sqrt {1+4 \sqrt [3]{-1}}-\sqrt {1-4 (-1)^{2/3}}\right ) \left (2 x+\sqrt {1+4 \sqrt [3]{-1}}-1\right )}{\left (\sqrt {1+4 \sqrt [3]{-1}}+\sqrt {1-4 (-1)^{2/3}}\right ) \left (-2 x+\sqrt {1+4 \sqrt [3]{-1}}+1\right )}} \sqrt {x^4-2 x^3+x+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.37, size = 30, normalized size = 1.00 \begin {gather*} -\log \left (1+2 x-2 x^2+2 \sqrt {1+x-2 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.50, size = 26, normalized size = 0.87 \begin {gather*} \log \left (2 \, x^{2} - 2 \, x + 2 \, \sqrt {x^{4} - 2 \, x^{3} + x + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.32, size = 34, normalized size = 1.13 \begin {gather*} -\log \left (-2 \, x^{2} + 2 \, x + 2 \, \sqrt {{\left (x^{2} - x\right )}^{2} - x^{2} + x + 1} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.61, size = 29, normalized size = 0.97

method	result	size
trager	\(-\ln \left (1+2 x -2 x^{2}+2 \sqrt {x^{4}-2 x^{3}+x +1}\right )\)	\(29\)
default	\(-\frac {2 \left (-\frac {\sqrt {3-2 i \sqrt {3}}}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \sqrt {3-2 i \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {3-2 i \sqrt {3}}}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )-\sqrt {3-2 i \sqrt {3}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \frac {\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}}{\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \sqrt {3-2 i \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}}\)	\(1352\)
elliptic	\(-\frac {2 \left (-\frac {\sqrt {3-2 i \sqrt {3}}}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \sqrt {3-2 i \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {3-2 i \sqrt {3}}}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2} \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \sqrt {\frac {\sqrt {3-2 i \sqrt {3}}\, \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )-\sqrt {3-2 i \sqrt {3}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}, \frac {\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}}{\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}}, \sqrt {-\frac {\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )^{2}}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (-\frac {\sqrt {3+2 i \sqrt {3}}}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {3+2 i \sqrt {3}}}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \sqrt {3-2 i \sqrt {3}}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3-2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {3+2 i \sqrt {3}}}{2}\right )}}\)	\(1352\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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} + x + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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