3.8.26 \(\int \frac {-1+x}{\sqrt {4-16 x+12 x^2-8 x^3+x^4}} \, dx\)
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],
x]
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 = 4.82, size = 56, normalized size = 1.00 \begin {gather*} -\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 ) \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 = 0.51, 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 [A] time = 1.03, size = 94, normalized size = 1.68
| | |
| method |
result |
size |
| | |
| trager |
\(\frac {\ln \left (-x^{4}-\sqrt {x^{4}-8 x^{3}+12 x^{2}-16 x +4}\, x^{2}+12 x^{3}+8 \sqrt {x^{4}-8 x^{3}+12 x^{2}-16 x +4}\, x -44 x^{2}-14 \sqrt {x^{4}-8 x^{3}+12 x^{2}-16 x +4}+56 x -36\right )}{4}\) |
\(94\) |
| default |
\(\text {Expression too large to display}\) |
\(2700\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(2700\) |
| | |
|
|
|
|
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,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^4-8*x^3+12*x^2-16*x+4)^(1/2)*x-44*x^2-14*(x^4-8*x^
3+12*x^2-16*x+4)^(1/2)+56*x-36)
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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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