3.4.92 \(\int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx\)

Optimal. Leaf size=32 \[ -\log \left (-x^2+\sqrt {x^4-2 x^3+9 x^2-8 x}+x-4\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1680, 12, 1107, 621, 206} \begin {gather*} \tanh ^{-1}\left (\frac {4 \left (x-\frac {1}{2}\right )^2+15}{\sqrt {16 \left (x-\frac {1}{2}\right )^4+120 \left (x-\frac {1}{2}\right )^2-31}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 2*x)/Sqrt[-8*x + 9*x^2 - 2*x^3 + x^4],x]

[Out]

ArcTanh[(15 + 4*(-1/2 + x)^2)/Sqrt[-31 + 120*(-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 {-8 x+9 x^2-2 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {8 x}{\sqrt {-31+120 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x}{\sqrt {-31+120 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-31+120 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 {2 \left (15+4 \left (-\frac {1}{2}+x\right )^2\right )}{\sqrt {x \left (-8+9 x-2 x^2+x^3\right )}}\right )\\ &=\tanh ^{-1}\left (\frac {15+(-1+2 x)^2}{4 \sqrt {-x \left (8-9 x+2 x^2-x^3\right )}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.54, size = 292, normalized size = 9.12 \begin {gather*} \frac {\sqrt {-\frac {i (x-1)}{\left (\sqrt {31}-15 i\right ) x}} x \left (64 \sqrt {62} \sqrt {-\frac {16 i}{x}+\sqrt {31}+i} x \sqrt {\frac {x^2-x+8}{x^2}} \Pi \left (\frac {2 \sqrt {31}}{i+\sqrt {31}};\sin ^{-1}\left (\frac {\sqrt {\sqrt {31}+i-\frac {16 i}{x}}}{\sqrt {2} \sqrt [4]{31}}\right )|\frac {2 \sqrt {31}}{-15 i+\sqrt {31}}\right )+\sqrt {\frac {16 i}{x}+\sqrt {31}-i} \left (\left (-31+15 i \sqrt {31}\right ) x+8 i \sqrt {31}+248\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {31}+i-\frac {16 i}{x}}}{\sqrt {2} \sqrt [4]{31}}\right )|\frac {2 \sqrt {31}}{-15 i+\sqrt {31}}\right )\right )}{\sqrt {31} \left (\sqrt {31}+i\right ) \sqrt {-\frac {16 i}{x}+\sqrt {31}+i} \sqrt {x \left (x^3-2 x^2+9 x-8\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-1 + 2*x)/Sqrt[-8*x + 9*x^2 - 2*x^3 + x^4],x]

[Out]

(Sqrt[((-I)*(-1 + x))/((-15*I + Sqrt[31])*x)]*x*(Sqrt[-I + Sqrt[31] + (16*I)/x]*(248 + (8*I)*Sqrt[31] + (-31 +
(15*I)*Sqrt[31])*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[31] - (16*I)/x]/(Sqrt[2]*31^(1/4))], (2*Sqrt[31])/(-15*I +
Sqrt[31])] + 64*Sqrt[62]*Sqrt[I + Sqrt[31] - (16*I)/x]*x*Sqrt[(8 - x + x^2)/x^2]*EllipticPi[(2*Sqrt[31])/(I +
Sqrt[31]), ArcSin[Sqrt[I + Sqrt[31] - (16*I)/x]/(Sqrt[2]*31^(1/4))], (2*Sqrt[31])/(-15*I + Sqrt[31])]))/(Sqrt
[31]*(I + Sqrt[31])*Sqrt[I + Sqrt[31] - (16*I)/x]*Sqrt[x*(-8 + 9*x - 2*x^2 + x^3)])

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IntegrateAlgebraic [A]  time = 0.14, size = 32, normalized size = 1.00 \begin {gather*} -\log \left (-4+x-x^2+\sqrt {-8 x+9 x^2-2 x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + 2*x)/Sqrt[-8*x + 9*x^2 - 2*x^3 + x^4],x]

[Out]

-Log[-4 + x - x^2 + Sqrt[-8*x + 9*x^2 - 2*x^3 + x^4]]

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fricas [A]  time = 0.51, size = 30, normalized size = 0.94 \begin {gather*} \log \left (-x^{2} + x - \sqrt {x^{4} - 2 \, x^{3} + 9 \, x^{2} - 8 \, x} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(x^4-2*x^3+9*x^2-8*x)^(1/2),x, algorithm="fricas")

[Out]

log(-x^2 + x - sqrt(x^4 - 2*x^3 + 9*x^2 - 8*x) - 4)

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giac [A]  time = 0.73, size = 33, normalized size = 1.03 \begin {gather*} -\log \left (x^{2} - x - \sqrt {{\left (x^{2} - x\right )}^{2} + 8 \, x^{2} - 8 \, x} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(x^4-2*x^3+9*x^2-8*x)^(1/2),x, algorithm="giac")

[Out]

-log(x^2 - x - sqrt((x^2 - x)^2 + 8*x^2 - 8*x) + 4)

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maple [A]  time = 0.35, size = 33, normalized size = 1.03

method	result	size
trager	\(-\ln \left (x^{2}-\sqrt {x^{4}-2 x^{3}+9 x^{2}-8 x}-x +4\right )\)	\(33\)
default	\(-\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \EllipticF \left (\sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}+\frac {4 \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {31}}{2}}{-\frac {1}{2}+\frac {i \sqrt {31}}{2}}, \sqrt {\frac {\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\right )\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\)	\(487\)
elliptic	\(-\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \EllipticF \left (\sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}+\frac {4 \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {31}}{2}}{-\frac {1}{2}+\frac {i \sqrt {31}}{2}}, \sqrt {\frac {\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\right )\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\)	\(487\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+2*x)/(x^4-2*x^3+9*x^2-8*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-ln(x^2-(x^4-2*x^3+9*x^2-8*x)^(1/2)-x+4)

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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} + 9 \, x^{2} - 8 \, x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(x^4-2*x^3+9*x^2-8*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((2*x - 1)/sqrt(x^4 - 2*x^3 + 9*x^2 - 8*x), 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+9\,x^2-8\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 1)/(9*x^2 - 8*x - 2*x^3 + x^4)^(1/2),x)

[Out]

int((2*x - 1)/(9*x^2 - 8*x - 2*x^3 + x^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 \left (x - 1\right ) \left (x^{2} - x + 8\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+2*x)/(x**4-2*x**3+9*x**2-8*x)**(1/2),x)

[Out]

Integral((2*x - 1)/sqrt(x*(x - 1)*(x**2 - x + 8)), x)

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