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

Optimal. Leaf size=25 \[ \log \left (x^2+\sqrt {x^4-2 x^3+x^2+4}-x\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 9, normalized size of antiderivative = 0.36, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1680, 12, 1107, 619, 215} \begin {gather*} \sinh ^{-1}\left (\frac {1}{2} (x-1) x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSinh[((-1 + x)*x)/2]

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 {4+x^2-2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {65-8 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {65-8 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {65-8 x+16 x^2}} \, dx,x,\left (-\frac {1}{2}+x\right )^2\right )\\ &=\frac {1}{64} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4096}}} \, dx,x,32 (-1+x) x\right )\\ &=-\sinh ^{-1}\left (\frac {1}{2} (1-x) x\right )\\ \end {align*}

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Mathematica [C]  time = 2.36, size = 613, normalized size = 24.52 \begin {gather*} \frac {\left (-2 x+\sqrt {1-8 i}+1\right ) \sqrt {\frac {\sqrt {1-8 i} \left (-2 x+\sqrt {1+8 i}+1\right )}{\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}} \left (2 x+\sqrt {1-8 i}-1\right ) \sqrt {\frac {\sqrt {1-8 i} \left (2 x+\sqrt {1+8 i}-1\right )}{\left (\sqrt {1+8 i}-\sqrt {1-8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}} \left (\left (1+\sqrt {1-8 i}\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right ) \left (2 x+\sqrt {1-8 i}-1\right )}{\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}}\right )|\frac {\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right )^2}{\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right )^2}\right )-F\left (\sin ^{-1}\left (\sqrt {-\frac {2 \sqrt {1+8 i} x-2 \sqrt {1-8 i} x+\sqrt {65}-\sqrt {1+8 i}+\sqrt {1-8 i}-(1-8 i)}{\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}}\right )|\frac {\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right )^2}{\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right )^2}\right )-2 \sqrt {1-8 i} \Pi \left (-\frac {\sqrt {1-8 i}+\sqrt {1+8 i}}{\sqrt {1-8 i}-\sqrt {1+8 i}};\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right ) \left (2 x+\sqrt {1-8 i}-1\right )}{\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}}\right )|\frac {\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right )^2}{\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right )^2}\right )\right )}{\sqrt {1-8 i} \sqrt {\frac {\left (\sqrt {1-8 i}-\sqrt {1+8 i}\right ) \left (2 x+\sqrt {1-8 i}-1\right )}{\left (\sqrt {1-8 i}+\sqrt {1+8 i}\right ) \left (-2 x+\sqrt {1-8 i}+1\right )}} \sqrt {x^4-2 x^3+x^2+4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 + Sqrt[1 - 8*I] - 2*x)*Sqrt[(Sqrt[1 - 8*I]*(1 + Sqrt[1 + 8*I] - 2*x))/((Sqrt[1 - 8*I] + Sqrt[1 + 8*I])*(1
+ Sqrt[1 - 8*I] - 2*x))]*(-1 + Sqrt[1 - 8*I] + 2*x)*Sqrt[(Sqrt[1 - 8*I]*(-1 + Sqrt[1 + 8*I] + 2*x))/((-Sqrt[1
- 8*I] + Sqrt[1 + 8*I])*(1 + Sqrt[1 - 8*I] - 2*x))]*((1 + Sqrt[1 - 8*I])*EllipticF[ArcSin[Sqrt[((Sqrt[1 - 8*I]
- Sqrt[1 + 8*I])*(-1 + Sqrt[1 - 8*I] + 2*x))/((Sqrt[1 - 8*I] + Sqrt[1 + 8*I])*(1 + Sqrt[1 - 8*I] - 2*x))]], (
Sqrt[1 - 8*I] + Sqrt[1 + 8*I])^2/(Sqrt[1 - 8*I] - Sqrt[1 + 8*I])^2] - EllipticF[ArcSin[Sqrt[-(((-1 + 8*I) + Sq
rt[1 - 8*I] - Sqrt[1 + 8*I] + Sqrt[65] - 2*Sqrt[1 - 8*I]*x + 2*Sqrt[1 + 8*I]*x)/((Sqrt[1 - 8*I] + Sqrt[1 + 8*I
])*(1 + Sqrt[1 - 8*I] - 2*x)))]], (Sqrt[1 - 8*I] + Sqrt[1 + 8*I])^2/(Sqrt[1 - 8*I] - Sqrt[1 + 8*I])^2] - 2*Sqr
t[1 - 8*I]*EllipticPi[-((Sqrt[1 - 8*I] + Sqrt[1 + 8*I])/(Sqrt[1 - 8*I] - Sqrt[1 + 8*I])), ArcSin[Sqrt[((Sqrt[1
- 8*I] - Sqrt[1 + 8*I])*(-1 + Sqrt[1 - 8*I] + 2*x))/((Sqrt[1 - 8*I] + Sqrt[1 + 8*I])*(1 + Sqrt[1 - 8*I] - 2*x
))]], (Sqrt[1 - 8*I] + Sqrt[1 + 8*I])^2/(Sqrt[1 - 8*I] - Sqrt[1 + 8*I])^2]))/(Sqrt[1 - 8*I]*Sqrt[((Sqrt[1 - 8*
I] - Sqrt[1 + 8*I])*(-1 + Sqrt[1 - 8*I] + 2*x))/((Sqrt[1 - 8*I] + Sqrt[1 + 8*I])*(1 + Sqrt[1 - 8*I] - 2*x))]*S
qrt[4 + x^2 - 2*x^3 + x^4])

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IntegrateAlgebraic [A]  time = 0.11, size = 25, normalized size = 1.00 \begin {gather*} \log \left (-x+x^2+\sqrt {4+x^2-2 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 23, normalized size = 0.92 \begin {gather*} \log \left (x^{2} - x + \sqrt {x^{4} - 2 \, x^{3} + x^{2} + 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.68, size = 23, normalized size = 0.92 \begin {gather*} -\log \left (-x^{2} + x + \sqrt {{\left (x^{2} - x\right )}^{2} + 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(-x^2 + x + sqrt((x^2 - x)^2 + 4))

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maple [A]  time = 0.47, size = 26, normalized size = 1.04

method	result	size
trager	\(-\ln \left (-x^{2}+\sqrt {x^{4}-2 x^{3}+x^{2}+4}+x \right )\)	\(26\)
default	\(-\frac {2 \left (-\frac {\sqrt {1+8 i}}{2}-\frac {\sqrt {1-8 i}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right )}{\left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \sqrt {1+8 i}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {1+8 i}}{2}-\frac {\sqrt {1-8 i}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right )}{\left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )-\sqrt {1+8 i}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \frac {\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}}{\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \sqrt {1+8 i}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}}\)	\(882\)
elliptic	\(-\frac {2 \left (-\frac {\sqrt {1+8 i}}{2}-\frac {\sqrt {1-8 i}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right )}{\left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \sqrt {1+8 i}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}}+\frac {4 \left (-\frac {\sqrt {1+8 i}}{2}-\frac {\sqrt {1-8 i}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )^{2} \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right )}{\left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \sqrt {\frac {\sqrt {1+8 i}\, \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}\, \left (\left (\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )-\sqrt {1+8 i}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right )}{\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right )}}, \frac {\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}}{\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}}, \sqrt {-\frac {\left (\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )^{2}}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (-\frac {\sqrt {1-8 i}}{2}+\frac {\sqrt {1+8 i}}{2}\right )}}\right )\right )}{\left (\frac {\sqrt {1-8 i}}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \sqrt {1+8 i}\, \sqrt {\left (x -\frac {1}{2}+\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1+8 i}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {1-8 i}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {1-8 i}}{2}\right )}}\)	\(882\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{2} + 4}}\,{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^2+4)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,x-1}{\sqrt {x^4-2\,x^3+x^2+4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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