\(\int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx\) [452]

Optimal result

Integrand size = 26, antiderivative size = 36 \[ \int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx=\frac {2}{3} \text {arctanh}\left (\frac {1-2 x+x^2}{\sqrt {-2-x+6 x^2-4 x^3+x^4}}\right ) \]

[Out]

2/3*arctanh((x^2-2*x+1)/(x^4-4*x^3+6*x^2-x-2)^(1/2))

Rubi [F]

\[ \int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx=\int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx \]

[In]

Int[(-1 + x)/Sqrt[-2 - x + 6*x^2 - 4*x^3 + x^4],x]

[Out]

-Defer[Int][1/Sqrt[-2 - x + 6*x^2 - 4*x^3 + x^4], x] + Defer[Int][x/Sqrt[-2 - x + 6*x^2 - 4*x^3 + x^4], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{\sqrt {-2-x+6 x^2-4 x^3+x^4}}+\frac {x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}}\right ) \, dx \\ & = -\int \frac {1}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx=\frac {2}{3} \text {arctanh}\left (\frac {1-2 x+x^2}{\sqrt {-2-x+6 x^2-4 x^3+x^4}}\right ) \]

[In]

Integrate[(-1 + x)/Sqrt[-2 - x + 6*x^2 - 4*x^3 + x^4],x]

[Out]

(2*ArcTanh[(1 - 2*x + x^2)/Sqrt[-2 - x + 6*x^2 - 4*x^3 + x^4]])/3

Maple [A] (verified)

Time = 4.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.78

method	result	size
trager	\(-\frac {\ln \left (-2 x^{3}+2 x \sqrt {x^{4}-4 x^{3}+6 x^{2}-x -2}+6 x^{2}-2 \sqrt {x^{4}-4 x^{3}+6 x^{2}-x -2}-6 x -1\right )}{3}\)	\(64\)
default	\(\text {Expression too large to display}\)	\(740\)
elliptic	\(\text {Expression too large to display}\)	\(740\)

[In]

int((x-1)/(x^4-4*x^3+6*x^2-x-2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*ln(-2*x^3+2*x*(x^4-4*x^3+6*x^2-x-2)^(1/2)+6*x^2-2*(x^4-4*x^3+6*x^2-x-2)^(1/2)-6*x-1)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19 \[ \int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx=\frac {1}{3} \, \log \left (2 \, x^{3} - 6 \, x^{2} + 2 \, \sqrt {x^{4} - 4 \, x^{3} + 6 \, x^{2} - x - 2} {\left (x - 1\right )} + 6 \, x + 1\right ) \]

[In]

integrate((-1+x)/(x^4-4*x^3+6*x^2-x-2)^(1/2),x, algorithm="fricas")

[Out]

1/3*log(2*x^3 - 6*x^2 + 2*sqrt(x^4 - 4*x^3 + 6*x^2 - x - 2)*(x - 1) + 6*x + 1)

Sympy [F]

\[ \int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx=\int \frac {x - 1}{\sqrt {\left (x - 1\right ) \left (x^{3} - 3 x^{2} + 3 x + 2\right )}}\, dx \]

[In]

integrate((-1+x)/(x**4-4*x**3+6*x**2-x-2)**(1/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx=\int { \frac {x - 1}{\sqrt {x^{4} - 4 \, x^{3} + 6 \, x^{2} - x - 2}} \,d x } \]

[In]

integrate((-1+x)/(x^4-4*x^3+6*x^2-x-2)^(1/2),x, algorithm="maxima")

[Out]

integrate((x - 1)/sqrt(x^4 - 4*x^3 + 6*x^2 - x - 2), x)

Giac [F]

\[ \int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx=\int { \frac {x - 1}{\sqrt {x^{4} - 4 \, x^{3} + 6 \, x^{2} - x - 2}} \,d x } \]

[In]

integrate((-1+x)/(x^4-4*x^3+6*x^2-x-2)^(1/2),x, algorithm="giac")

[Out]

integrate((x - 1)/sqrt(x^4 - 4*x^3 + 6*x^2 - x - 2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx=\int \frac {x-1}{\sqrt {x^4-4\,x^3+6\,x^2-x-2}} \,d x \]

[In]

int((x - 1)/(6*x^2 - x - 4*x^3 + x^4 - 2)^(1/2),x)

[Out]

int((x - 1)/(6*x^2 - x - 4*x^3 + x^4 - 2)^(1/2), x)