3.5.24 \(\int \frac {x}{\sqrt {-3+3 x-3 x^2+2 x^3+x^4}} \, dx\) [424]
Optimal. Leaf size=34 \[ \frac {2}{3} \tanh ^{-1}\left (\frac {-2+x+x^2}{\sqrt {-3+3 x-3 x^2+2 x^3+x^4}}\right ) \]
[Out]
2/3*arctanh((x^2+x-2)/(x^4+2*x^3-3*x^2+3*x-3)^(1/2))
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Rubi [F]
time = 0.04, 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 {x}{\sqrt {-3+3 x-3 x^2+2 x^3+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[x/Sqrt[-3 + 3*x - 3*x^2 + 2*x^3 + x^4],x]
[Out]
Defer[Int][x/Sqrt[-3 + 3*x - 3*x^2 + 2*x^3 + x^4], x]
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-3+3 x-3 x^2+2 x^3+x^4}} \, dx &=\int \frac {x}{\sqrt {-3+3 x-3 x^2+2 x^3+x^4}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 34, normalized size = 1.00 \begin {gather*} \frac {2}{3} \tanh ^{-1}\left (\frac {-2+x+x^2}{\sqrt {-3+3 x-3 x^2+2 x^3+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x/Sqrt[-3 + 3*x - 3*x^2 + 2*x^3 + x^4],x]
[Out]
(2*ArcTanh[(-2 + x + x^2)/Sqrt[-3 + 3*x - 3*x^2 + 2*x^3 + x^4]])/3
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 2.43, size = 1633, normalized size = 48.03
| | |
| method |
result |
size |
| | |
| trager |
\(-\frac {\ln \left (-2 x^{3}+2 x \sqrt {x^{4}+2 x^{3}-3 x^{2}+3 x -3}-6 x^{2}+4 \sqrt {x^{4}+2 x^{3}-3 x^{2}+3 x -3}+1\right )}{3}\) |
\(61\) |
| default |
\(\text {Expression too large to display}\) |
\(1633\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(1633\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(x^4+2*x^3-3*x^2+3*x-3)^(1/2),x,method=_RETURNVERBOSE)
[Out]
2*(2-1/4*(20+4*21^(1/2))^(1/3)-1/(20+4*21^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1
/2))^(1/3)))*((3/4*(20+4*21^(1/2))^(1/3)+3/(20+4*21^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(
20+4*21^(1/2))^(1/3)))*(-1+x)/(1/4*(20+4*21^(1/2))^(1/3)+1/(20+4*21^(1/2))^(1/3)-2-1/2*I*3^(1/2)*(-1/2*(20+4*2
1^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)))/(x+1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)+1))^(1/2)*(x+1/2
*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)+1)^2*(-(x-1/4*(20+4*21^(1/2))^(1/3)-1/(20+4*21^(1/2))^(1/3)+1-1
/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)))/(1/4*(20+4*21^(1/2))^(1/3)+1/(20+4*21^(1/2)
)^(1/3)-2+1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)))/(x+1/2*(20+4*21^(1/2))^(1/3)+2/(
20+4*21^(1/2))^(1/3)+1)*(1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)+2))^(1/2)*(-(x-1/4*(20+4*21^(1/2))^
(1/3)-1/(20+4*21^(1/2))^(1/3)+1+1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)))/(1/4*(20+4
*21^(1/2))^(1/3)+1/(20+4*21^(1/2))^(1/3)-2-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)))
/(x+1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)+1)*(1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)+2)
)^(1/2)/(3/4*(20+4*21^(1/2))^(1/3)+3/(20+4*21^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*2
1^(1/2))^(1/3)))/(-1/2*(20+4*21^(1/2))^(1/3)-2/(20+4*21^(1/2))^(1/3)-2)/((-1+x)*(x+1/2*(20+4*21^(1/2))^(1/3)+2
/(20+4*21^(1/2))^(1/3)+1)*(x-1/4*(20+4*21^(1/2))^(1/3)-1/(20+4*21^(1/2))^(1/3)+1-1/2*I*3^(1/2)*(-1/2*(20+4*21^
(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)))*(x-1/4*(20+4*21^(1/2))^(1/3)-1/(20+4*21^(1/2))^(1/3)+1+1/2*I*3^(1/2)*(-
1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3))))^(1/2)*((-1/2*(20+4*21^(1/2))^(1/3)-2/(20+4*21^(1/2))^(1/3
)-1)*EllipticF(((3/4*(20+4*21^(1/2))^(1/3)+3/(20+4*21^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2
/(20+4*21^(1/2))^(1/3)))*(-1+x)/(1/4*(20+4*21^(1/2))^(1/3)+1/(20+4*21^(1/2))^(1/3)-2-1/2*I*3^(1/2)*(-1/2*(20+4
*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)))/(x+1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)+1))^(1/2),((-3
/4*(20+4*21^(1/2))^(1/3)-3/(20+4*21^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(
1/3)))*(2-1/4*(20+4*21^(1/2))^(1/3)-1/(20+4*21^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*
21^(1/2))^(1/3)))/(2-1/4*(20+4*21^(1/2))^(1/3)-1/(20+4*21^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/
3)+2/(20+4*21^(1/2))^(1/3)))/(-3/4*(20+4*21^(1/2))^(1/3)-3/(20+4*21^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/2*(20+4*21^
(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3))))^(1/2))+(1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)+2)*EllipticP
i(((3/4*(20+4*21^(1/2))^(1/3)+3/(20+4*21^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/
2))^(1/3)))*(-1+x)/(1/4*(20+4*21^(1/2))^(1/3)+1/(20+4*21^(1/2))^(1/3)-2-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1
/3)+2/(20+4*21^(1/2))^(1/3)))/(x+1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)+1))^(1/2),(1/4*(20+4*21^(1/
2))^(1/3)+1/(20+4*21^(1/2))^(1/3)-2-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)))/(3/4*(
20+4*21^(1/2))^(1/3)+3/(20+4*21^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3)
)),((-3/4*(20+4*21^(1/2))^(1/3)-3/(20+4*21^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2/(20+4*21^(
1/2))^(1/3)))*(2-1/4*(20+4*21^(1/2))^(1/3)-1/(20+4*21^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/2))^(1/3)+2
/(20+4*21^(1/2))^(1/3)))/(2-1/4*(20+4*21^(1/2))^(1/3)-1/(20+4*21^(1/2))^(1/3)-1/2*I*3^(1/2)*(-1/2*(20+4*21^(1/
2))^(1/3)+2/(20+4*21^(1/2))^(1/3)))/(-3/4*(20+4*21^(1/2))^(1/3)-3/(20+4*21^(1/2))^(1/3)+1/2*I*3^(1/2)*(-1/2*(2
0+4*21^(1/2))^(1/3)+2/(20+4*21^(1/2))^(1/3))))^(1/2)))
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(x^4+2*x^3-3*x^2+3*x-3)^(1/2),x, algorithm="maxima")
[Out]
integrate(x/sqrt(x^4 + 2*x^3 - 3*x^2 + 3*x - 3), x)
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Fricas [A]
time = 0.36, size = 40, normalized size = 1.18 \begin {gather*} \frac {1}{3} \, \log \left (2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{4} + 2 \, x^{3} - 3 \, x^{2} + 3 \, x - 3} {\left (x + 2\right )} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(x^4+2*x^3-3*x^2+3*x-3)^(1/2),x, algorithm="fricas")
[Out]
1/3*log(2*x^3 + 6*x^2 + 2*sqrt(x^4 + 2*x^3 - 3*x^2 + 3*x - 3)*(x + 2) - 1)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {\left (x - 1\right ) \left (x^{3} + 3 x^{2} + 3\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(x**4+2*x**3-3*x**2+3*x-3)**(1/2),x)
[Out]
Integral(x/sqrt((x - 1)*(x**3 + 3*x**2 + 3)), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(x^4+2*x^3-3*x^2+3*x-3)^(1/2),x, algorithm="giac")
[Out]
integrate(x/sqrt(x^4 + 2*x^3 - 3*x^2 + 3*x - 3), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x}{\sqrt {x^4+2\,x^3-3\,x^2+3\,x-3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(3*x - 3*x^2 + 2*x^3 + x^4 - 3)^(1/2),x)
[Out]
int(x/(3*x - 3*x^2 + 2*x^3 + x^4 - 3)^(1/2), x)
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