\(\int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx\) [726]

Optimal result

Integrand size = 26, antiderivative size = 56 \[ \int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx=-\frac {1}{4} \log \left (36-56 x+44 x^2-12 x^3+x^4+\left (-14+8 x-x^2\right ) \sqrt {4-16 x+12 x^2-8 x^3+x^4}\right ) \]

[Out]

-1/4*ln(36-56*x+44*x^2-12*x^3+x^4+(-x^2+8*x-14)*(x^4-8*x^3+12*x^2-16*x+4)^(1/2))

Rubi [F]

\[ \int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx=\int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx \]

[In]

Int[(-1 + x)/Sqrt[4 - 16*x + 12*x^2 - 8*x^3 + x^4],x]

[Out]

-Defer[Int][1/Sqrt[4 - 16*x + 12*x^2 - 8*x^3 + x^4], x] + Defer[Int][x/Sqrt[4 - 16*x + 12*x^2 - 8*x^3 + x^4],
x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}}+\frac {x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}}\right ) \, dx \\ & = -\int \frac {1}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.98 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx=-\frac {1}{4} \log \left (36-56 x+44 x^2-12 x^3+x^4+\left (-14+8 x-x^2\right ) \sqrt {4-16 x+12 x^2-8 x^3+x^4}\right ) \]

[In]

Integrate[(-1 + x)/Sqrt[4 - 16*x + 12*x^2 - 8*x^3 + x^4],x]

[Out]

-1/4*Log[36 - 56*x + 44*x^2 - 12*x^3 + x^4 + (-14 + 8*x - x^2)*Sqrt[4 - 16*x + 12*x^2 - 8*x^3 + x^4]]

Maple [A] (verified)

Time = 1.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.64

method	result	size
trager	\(-\frac {\ln \left (x^{4}-\sqrt {x^{4}-8 x^{3}+12 x^{2}-16 x +4}\, x^{2}-12 x^{3}+8 x \sqrt {x^{4}-8 x^{3}+12 x^{2}-16 x +4}+44 x^{2}-14 \sqrt {x^{4}-8 x^{3}+12 x^{2}-16 x +4}-56 x +36\right )}{4}\)	\(92\)
default	\(\text {Expression too large to display}\)	\(2700\)
elliptic	\(\text {Expression too large to display}\)	\(2700\)

[In]

int((x-1)/(x^4-8*x^3+12*x^2-16*x+4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*ln(x^4-(x^4-8*x^3+12*x^2-16*x+4)^(1/2)*x^2-12*x^3+8*x*(x^4-8*x^3+12*x^2-16*x+4)^(1/2)+44*x^2-14*(x^4-8*x^
3+12*x^2-16*x+4)^(1/2)-56*x+36)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95 \[ \int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx=\frac {1}{4} \, \log \left (-x^{4} + 12 \, x^{3} - 44 \, x^{2} - \sqrt {x^{4} - 8 \, x^{3} + 12 \, x^{2} - 16 \, x + 4} {\left (x^{2} - 8 \, x + 14\right )} + 56 \, x - 36\right ) \]

[In]

integrate((-1+x)/(x^4-8*x^3+12*x^2-16*x+4)^(1/2),x, algorithm="fricas")

[Out]

1/4*log(-x^4 + 12*x^3 - 44*x^2 - sqrt(x^4 - 8*x^3 + 12*x^2 - 16*x + 4)*(x^2 - 8*x + 14) + 56*x - 36)

Sympy [F]

\[ \int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx=\int \frac {x - 1}{\sqrt {x^{4} - 8 x^{3} + 12 x^{2} - 16 x + 4}}\, dx \]

[In]

integrate((-1+x)/(x**4-8*x**3+12*x**2-16*x+4)**(1/2),x)

[Out]

Integral((x - 1)/sqrt(x**4 - 8*x**3 + 12*x**2 - 16*x + 4), x)

Maxima [F]

\[ \int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx=\int { \frac {x - 1}{\sqrt {x^{4} - 8 \, x^{3} + 12 \, x^{2} - 16 \, x + 4}} \,d x } \]

[In]

integrate((-1+x)/(x^4-8*x^3+12*x^2-16*x+4)^(1/2),x, algorithm="maxima")

[Out]

integrate((x - 1)/sqrt(x^4 - 8*x^3 + 12*x^2 - 16*x + 4), x)

Giac [F]

\[ \int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx=\int { \frac {x - 1}{\sqrt {x^{4} - 8 \, x^{3} + 12 \, x^{2} - 16 \, x + 4}} \,d x } \]

[In]

integrate((-1+x)/(x^4-8*x^3+12*x^2-16*x+4)^(1/2),x, algorithm="giac")

[Out]

integrate((x - 1)/sqrt(x^4 - 8*x^3 + 12*x^2 - 16*x + 4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx=\int \frac {x-1}{\sqrt {x^4-8\,x^3+12\,x^2-16\,x+4}} \,d x \]

[In]

int((x - 1)/(12*x^2 - 16*x - 8*x^3 + x^4 + 4)^(1/2),x)

[Out]

int((x - 1)/(12*x^2 - 16*x - 8*x^3 + x^4 + 4)^(1/2), x)