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

Optimal. Leaf size=36 \[ \frac {2}{3} \tanh ^{-1}\left (\frac {x^2-2 x+1}{\sqrt {x^4-4 x^3+6 x^2-x-2}}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.14, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \int \frac {-1+x}{\sqrt {-2-x+6 x^2-4 x^3+x^4}} \, dx &=\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 [C]  time = 0.83, size = 640, normalized size = 17.78 \begin {gather*} \frac {2 \sqrt [3]{3} (x-1) \sqrt {\frac {-x+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,2\right ]}{\left (x+\sqrt [3]{3}-1\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,2\right ]\right )}} \left (x-\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right ) \left (F\left (\sin ^{-1}\left (\sqrt {\frac {(x-1) \left (-1+\sqrt [3]{3}+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right )}{\left (x+\sqrt [3]{3}-1\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right )}}\right )|\frac {\left (-1+\sqrt [3]{3}+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,2\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,2\right ]\right ) \left (-1+\sqrt [3]{3}+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right )}\right )-\Pi \left (\frac {-1+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]}{-1+\sqrt [3]{3}+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]};\sin ^{-1}\left (\sqrt {\frac {(x-1) \left (-1+\sqrt [3]{3}+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right )}{\left (x+\sqrt [3]{3}-1\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right )}}\right )|\frac {\left (-1+\sqrt [3]{3}+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,2\right ]\right ) \left (-1+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right )}{\left (-1+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,2\right ]\right ) \left (-1+\sqrt [3]{3}+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right )}\right )\right )}{\sqrt {x^4-4 x^3+6 x^2-x-2} \left (-1+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right ) \sqrt {-\frac {(x-1) \left (\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]-1+\sqrt [3]{3}\right ) \left (x-\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right )}{\left (x+\sqrt [3]{3}-1\right )^2 \left (-1+\text {Root}\left [\text {$\#$1}^3-3 \text {$\#$1}^2+3 \text {$\#$1}+2\&,3\right ]\right )^2}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*3^(1/3)*(-1 + x)*(EllipticF[ArcSin[Sqrt[((-1 + x)*(-1 + 3^(1/3) + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0]))
/((-1 + 3^(1/3) + x)*(-1 + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0]))]], ((-1 + 3^(1/3) + Root[2 + 3*#1 - 3*#1^
2 + #1^3 & , 2, 0])*(-1 + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0]))/((-1 + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 2
, 0])*(-1 + 3^(1/3) + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0]))] - EllipticPi[(-1 + Root[2 + 3*#1 - 3*#1^2 + #
1^3 & , 3, 0])/(-1 + 3^(1/3) + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[((-1 + x)*(-1 + 3^(1/3) +
Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0]))/((-1 + 3^(1/3) + x)*(-1 + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0]))
]], ((-1 + 3^(1/3) + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 2, 0])*(-1 + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0]))/
((-1 + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 2, 0])*(-1 + 3^(1/3) + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0]))])*Sq
rt[(-x + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 2, 0])/((-1 + 3^(1/3) + x)*(-1 + Root[2 + 3*#1 - 3*#1^2 + #1^3 & ,
2, 0]))]*(x - Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0]))/(Sqrt[-2 - x + 6*x^2 - 4*x^3 + x^4]*(-1 + Root[2 + 3*#
1 - 3*#1^2 + #1^3 & , 3, 0])*Sqrt[-(((-1 + x)*(x - Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0])*(-1 + 3^(1/3) + Ro
ot[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0]))/((-1 + 3^(1/3) + x)^2*(-1 + Root[2 + 3*#1 - 3*#1^2 + #1^3 & , 3, 0])^2
))])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.14, size = 36, normalized size = 1.00 \begin {gather*} \frac {2}{3} \tanh ^{-1}\left (\frac {1-2 x+x^2}{\sqrt {-2-x+6 x^2-4 x^3+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-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

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 43, normalized size = 1.19 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {x^{4} - 4 \, x^{3} + 6 \, x^{2} - x - 2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.46, size = 64, normalized size = 1.78

method	result	size
trager	\(-\frac {\ln \left (-2 x^{3}+2 \sqrt {x^{4}-4 x^{3}+6 x^{2}-x -2}\, x +6 x^{2}-2 \sqrt {x^{4}-4 x^{3}+6 x^{2}-x -2}-6 x -1\right )}{3}\)	\(64\)
default	\(\frac {2 \left (-\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) \sqrt {\frac {\left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-1+x \right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, \left (x +3^{\frac {1}{3}}-1\right )^{2} \sqrt {-\frac {3^{\frac {1}{3}} \left (x -\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}-1\right )}{\left (\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, \sqrt {-\frac {3^{\frac {1}{3}} \left (x -\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}-1\right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, 3^{\frac {2}{3}} \EllipticF \left (\sqrt {\frac {\left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-1+x \right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}, \sqrt {\frac {\left (-\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}{\left (-\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}}\right )}{3 \left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \sqrt {\left (-1+x \right ) \left (x +3^{\frac {1}{3}}-1\right ) \left (x -\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}-1\right ) \left (x -\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}-1\right )}}-\frac {2 \left (-\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) \sqrt {\frac {\left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-1+x \right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, \left (x +3^{\frac {1}{3}}-1\right )^{2} \sqrt {-\frac {3^{\frac {1}{3}} \left (x -\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}-1\right )}{\left (\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, \sqrt {-\frac {3^{\frac {1}{3}} \left (x -\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}-1\right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, 3^{\frac {2}{3}} \left (\left (-3^{\frac {1}{3}}+1\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-1+x \right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}, \sqrt {\frac {\left (-\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}{\left (-\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}}\right )+3^{\frac {1}{3}} \EllipticPi \left (\sqrt {\frac {\left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-1+x \right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}, \frac {\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}}{\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}}, \sqrt {\frac {\left (-\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}{\left (-\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}}\right )\right )}{3 \left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \sqrt {\left (-1+x \right ) \left (x +3^{\frac {1}{3}}-1\right ) \left (x -\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}-1\right ) \left (x -\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}-1\right )}}\)	\(740\)
elliptic	\(\frac {2 \left (-\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) \sqrt {\frac {\left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-1+x \right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, \left (x +3^{\frac {1}{3}}-1\right )^{2} \sqrt {-\frac {3^{\frac {1}{3}} \left (x -\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}-1\right )}{\left (\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, \sqrt {-\frac {3^{\frac {1}{3}} \left (x -\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}-1\right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, 3^{\frac {2}{3}} \EllipticF \left (\sqrt {\frac {\left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-1+x \right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}, \sqrt {\frac {\left (-\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}{\left (-\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}}\right )}{3 \left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \sqrt {\left (-1+x \right ) \left (x +3^{\frac {1}{3}}-1\right ) \left (x -\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}-1\right ) \left (x -\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}-1\right )}}-\frac {2 \left (-\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) \sqrt {\frac {\left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-1+x \right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, \left (x +3^{\frac {1}{3}}-1\right )^{2} \sqrt {-\frac {3^{\frac {1}{3}} \left (x -\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}-1\right )}{\left (\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, \sqrt {-\frac {3^{\frac {1}{3}} \left (x -\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}-1\right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}\, 3^{\frac {2}{3}} \left (\left (-3^{\frac {1}{3}}+1\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-1+x \right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}, \sqrt {\frac {\left (-\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}{\left (-\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}}\right )+3^{\frac {1}{3}} \EllipticPi \left (\sqrt {\frac {\left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-1+x \right )}{\left (\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (x +3^{\frac {1}{3}}-1\right )}}, \frac {\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}}{\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}}, \sqrt {\frac {\left (-\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}{\left (-\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \left (-\frac {3 \,3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}\right )}}\right )\right )}{3 \left (\frac {3 \,3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}\right ) \sqrt {\left (-1+x \right ) \left (x +3^{\frac {1}{3}}-1\right ) \left (x -\frac {3^{\frac {1}{3}}}{2}+\frac {i 3^{\frac {5}{6}}}{2}-1\right ) \left (x -\frac {3^{\frac {1}{3}}}{2}-\frac {i 3^{\frac {5}{6}}}{2}-1\right )}}\)	\(740\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {x^{4} - 4 \, x^{3} + 6 \, x^{2} - x - 2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x-1}{\sqrt {x^4-4\,x^3+6\,x^2-x-2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {\left (x - 1\right ) \left (x^{3} - 3 x^{2} + 3 x + 2\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________