\(\int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx\) [392]

Optimal result

Integrand size = 27, antiderivative size = 32 \[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=-\log \left (-4+x-x^2+\sqrt {-8 x+9 x^2-2 x^3+x^4}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1694, 12, 1121, 635, 212} \[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=\text {arctanh}\left (\frac {4 \left (x-\frac {1}{2}\right )^2+15}{\sqrt {16 \left (x-\frac {1}{2}\right )^4+120 \left (x-\frac {1}{2}\right )^2-31}}\right ) \]

[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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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 1121

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 1694

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*} \text {integral}& = \text {Subst}\left (\int \frac {8 x}{\sqrt {-31+120 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right ) \\ & = 8 \text {Subst}\left (\int \frac {x}{\sqrt {-31+120 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right ) \\ & = 4 \text {Subst}\left (\int \frac {1}{\sqrt {-31+120 x+16 x^2}} \, dx,x,\left (-\frac {1}{2}+x\right )^2\right ) \\ & = 8 \text {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 ) \\ & = \text {arctanh}\left (\frac {15+(-1+2 x)^2}{4 \sqrt {-x \left (8-9 x+2 x^2-x^3\right )}}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(32)=64\).

Time = 3.84 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.34 \[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=-\frac {\sqrt {x} \sqrt {-8+9 x-2 x^2+x^3} \log \left (-4+x-x^2+\sqrt {x} \sqrt {-8+9 x-2 x^2+x^3}\right )}{\sqrt {x \left (-8+9 x-2 x^2+x^3\right )}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.57 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78

method	result	size
default	\(\ln \left (x^{2}-x +4+\sqrt {\left (x^{2}-x +8\right ) x \left (x -1\right )}\right )\)	\(25\)
pseudoelliptic	\(\ln \left (x^{2}-x +4+\sqrt {\left (x^{2}-x +8\right ) x \left (x -1\right )}\right )\)	\(25\)
trager	\(-\ln \left (x^{2}-\sqrt {x^{4}-2 x^{3}+9 x^{2}-8 x}-x +4\right )\)	\(33\)
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 (x -1\right )}}\, \left (x -1\right )^{2} \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}\, \operatorname {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 (x -1\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 (x -1\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 (x -1\right )}}\, \left (x -1\right )^{2} \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}\, \left (\operatorname {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 (x -1\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 )-\operatorname {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 (x -1\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 (x -1\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\)	\(487\)

[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+((x^2-x+8)*x*(x-1))^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=\log \left (-x^{2} + x - \sqrt {x^{4} - 2 \, x^{3} + 9 \, x^{2} - 8 \, x} - 4\right ) \]

[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)

Sympy [F]

\[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=\int \frac {2 x - 1}{\sqrt {x \left (x - 1\right ) \left (x^{2} - x + 8\right )}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=\int { \frac {2 \, x - 1}{\sqrt {x^{4} - 2 \, x^{3} + 9 \, x^{2} - 8 \, x}} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (30) = 60\).

Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.00 \[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=\frac {1}{2} \, \sqrt {{\left (x^{2} - x\right )}^{2} + 8 \, x^{2} - 8 \, x} {\left (x^{2} - x + 4\right )} + 8 \, \log \left (x^{2} - x - \sqrt {{\left (x^{2} - x\right )}^{2} + 8 \, x^{2} - 8 \, x} + 4\right ) \]

[In]

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

[Out]

1/2*sqrt((x^2 - x)^2 + 8*x^2 - 8*x)*(x^2 - x + 4) + 8*log(x^2 - x - sqrt((x^2 - x)^2 + 8*x^2 - 8*x) + 4)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=\int \frac {2\,x-1}{\sqrt {x^4-2\,x^3+9\,x^2-8\,x}} \,d x \]

[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)