∫ −1+x_______√ 4−16x+12x2−8x3+x4 dx

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

Optimal. Leaf size=56 \[ -\frac {1}{4} \log \left (x^4-12 x^3+44 x^2+\left (-x^2+8 x-14\right ) \sqrt {x^4-8 x^3+12 x^2-16 x+4}-56 x+36\right ) \]

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Rubi [F] time = 0.13, 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 {4-16 x+12 x^2-8 x^3+x^4}} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[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],

Rubi steps

\begin {align*} \int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx &=\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*}

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Mathematica [C] time = 6.08, size = 2609, normalized size = 46.59 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0])^2*(-(EllipticF[ArcSin[Sqrt[((x - Root[4 - 16*#1 +

12*#1^2 - 8*#1^3 + #1^4 & , 1, 0])*(Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] - Root[4 - 16*#1 + 12*#

1^2 - 8*#1^3 + #1^4 & , 4, 0]))/((x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0])*(Root[4 - 16*#1 + 12

*#1^2 - 8*#1^3 + #1^4 & , 1, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]))]], -(((Root[4 - 16*#1 +

12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 3, 0])*(Root[4 - 16*#1 + 12*

#1^2 - 8*#1^3 + #1^4 & , 1, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]))/((-Root[4 - 16*#1 + 12*#

1^2 - 8*#1^3 + #1^4 & , 1, 0] + Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 3, 0])*(Root[4 - 16*#1 + 12*#1^2

- 8*#1^3 + #1^4 & , 2, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0])))]*Root[4 - 16*#1 + 12*#1^2 -

8*#1^3 + #1^4 & , 2, 0]) + EllipticPi[(-Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0] + Root[4 - 16*#1 +

12*#1^2 - 8*#1^3 + #1^4 & , 4, 0])/(-Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] + Root[4 - 16*#1 + 12*

#1^2 - 8*#1^3 + #1^4 & , 4, 0]), ArcSin[Sqrt[((x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0])*(Root[4

- 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]))/((x - Root

[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0])*(Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0] - Root[4 -

16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]))]], -(((Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] - Root[

4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 3, 0])*(Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0] - Root[4 -

16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]))/((-Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0] + Root[4 - 1

6*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 3, 0])*(Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] - Root[4 - 16*#1

+ 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0])))]*(-Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0] + Root[4 - 16*#1

+ 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0]))*Sqrt[(x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 3, 0])/((x - Roo

t[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0])*(-Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0] + Root[4

- 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 3, 0]))]*(Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0] - Root[4 -

16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0])*Sqrt[((x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0])*(Ro

ot[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]))/((x -

Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0])*(Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0] - Root

[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]))]*Sqrt[(x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]

)/((x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0])*(-Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1,

0] + Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]))])/(Sqrt[4 - 16*x + 12*x^2 - 8*x^3 + x^4]*(Root[4 - 1

6*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0])) - (2*EllipticF

[ArcSin[Sqrt[((x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0])*(-Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #

1^4 & , 2, 0] + Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]))/((x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 +

#1^4 & , 2, 0])*(-Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0] + Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1

^4 & , 4, 0]))]], ((Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #

1^4 & , 3, 0])*(Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4

& , 4, 0]))/((Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 1, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 &

, 3, 0])*(Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4,

0]))]*(x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0])^2*Sqrt[(x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3

+ #1^4 & , 3, 0])/((x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0])*(-Root[4 - 16*#1 + 12*#1^2 - 8*#1^

3 + #1^4 & , 1, 0] + Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 3, 0]))]*(Root[4 - 16*#1 + 12*#1^2 - 8*#1^3

+ #1^4 & , 1, 0] - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0])*Sqrt[(x - Root[4 - 16*#1 + 12*#1^2 - 8*

#1^3 + #1^4 & , 4, 0])/((x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0])*(-Root[4 - 16*#1 + 12*#1^2 -

8*#1^3 + #1^4 & , 1, 0] + Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]))]*Sqrt[((x - Root[4 - 16*#1 + 12

*#1^2 - 8*#1^3 + #1^4 & , 1, 0])*(-Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] + Root[4 - 16*#1 + 12*#1

^2 - 8*#1^3 + #1^4 & , 4, 0]))/((x - Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0])*(-Root[4 - 16*#1 + 12

*#1^2 - 8*#1^3 + #1^4 & , 1, 0] + Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 4, 0]))])/(Sqrt[4 - 16*x + 12*x

^2 - 8*x^3 + x^4]*(-Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #1^4 & , 2, 0] + Root[4 - 16*#1 + 12*#1^2 - 8*#1^3 + #

1^4 & , 4, 0]))

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IntegrateAlgebraic [A] time = 4.79, size = 56, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \log \left (x^4-12 x^3+44 x^2+\left (-x^2+8 x-14\right ) \sqrt {x^4-8 x^3+12 x^2-16 x+4}-56 x+36\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

IntegrateAlgebraic[(-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]]

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fricas [A] time = 0.44, size = 53, normalized size = 0.95 \begin {gather*} \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 ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {x^{4} - 8 \, x^{3} + 12 \, x^{2} - 16 \, x + 4}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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maple [C] time = 1.16, size = 2700, normalized size = 48.21 \begin {gather*} \text {Expression too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4))*((RootOf(_Z^4-8*_Z

^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2))*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4

,index=1))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))/(x-RootOf

(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)))^(1/2)*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2))^2*((RootOf(_Z^4-

8*_Z^3+12*_Z^2-16*_Z+4,index=2)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*

_Z+4,index=3))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=3)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))/(x-Ro

otOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)))^(1/2)*((RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)-RootOf(_Z^4-8*

_Z^3+12*_Z^2-16*_Z+4,index=1))*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*

_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))/(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)))^(1

/2)/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2))/(RootOf(_Z^4-8*_

Z^3+12*_Z^2-16*_Z+4,index=2)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))/((x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z

+4,index=1))*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2))*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=3))*(

x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)))^(1/2)*(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)*EllipticF((

(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2))*(x-RootOf(_Z^4-8*_Z^

3+12*_Z^2-16*_Z+4,index=1))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,in

dex=1))/(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)))^(1/2),((RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)-R

ootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=3))*(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1)-RootOf(_Z^4-8*_Z^3+12

*_Z^2-16*_Z+4,index=4))/(-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=3)+RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index

=1))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)))^(1/2))+(RootOf

(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2))*EllipticPi(((RootOf(_Z^4-8*

_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2))*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z

+4,index=1))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))/(x-Root

Of(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)))^(1/2),(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^

3+12*_Z^2-16*_Z+4,index=1))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,in

dex=2)),((RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=3))*(RootOf(_Z^

4-8*_Z^3+12*_Z^2-16*_Z+4,index=1)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4))/(-RootOf(_Z^4-8*_Z^3+12*_Z^2-16

*_Z+4,index=3)+RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)-RootO

f(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)))^(1/2)))-2*(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1)-RootOf(_Z^4-8*

_Z^3+12*_Z^2-16*_Z+4,index=4))*((RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+

4,index=2))*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootO

f(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))/(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)))^(1/2)*(x-RootOf(_Z^4-

8*_Z^3+12*_Z^2-16*_Z+4,index=2))^2*((RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16

*_Z+4,index=1))*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=3))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=3)-R

ootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))/(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)))^(1/2)*((RootOf(_Z

^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-

16*_Z+4,index=4))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))/(x

-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)))^(1/2)/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-

8*_Z^3+12*_Z^2-16*_Z+4,index=2))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z

+4,index=1))/((x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1))*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2))*

(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=3))*(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)))^(1/2)*Ellipti

cF(((RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2))*(x-RootOf(_Z^4-8

*_Z^3+12*_Z^2-16*_Z+4,index=1))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+

4,index=1))/(x-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)))^(1/2),((RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=

2)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=3))*(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=1)-RootOf(_Z^4-8*_Z^

3+12*_Z^2-16*_Z+4,index=4))/(-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=3)+RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,i

ndex=1))/(RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=2)-RootOf(_Z^4-8*_Z^3+12*_Z^2-16*_Z+4,index=4)))^(1/2))

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {x^{4} - 8 \, x^{3} + 12 \, x^{2} - 16 \, x + 4}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x-1}{\sqrt {x^4-8\,x^3+12\,x^2-16\,x+4}} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 1}{\sqrt {x^{4} - 8 x^{3} + 12 x^{2} - 16 x + 4}}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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