3.5.8 \(\int \frac {-1+2 x}{\sqrt {-2-2 x+3 x^2-2 x^3+x^4}} \, dx\) [408]
Optimal. Leaf size=33 \[ -\log \left (-1+x-x^2+\sqrt {-2-2 x+3 x^2-2 x^3+x^4}\right ) \]
[Out]
-ln(-1+x-x^2+(x^4-2*x^3+3*x^2-2*x-2)^(1/2))
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.12, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1694, 12, 1121,
635, 212} \begin {gather*} \tanh ^{-1}\left (\frac {4 \left (x-\frac {1}{2}\right )^2+3}{\sqrt {16 \left (x-\frac {1}{2}\right )^4+24 \left (x-\frac {1}{2}\right )^2-39}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-1 + 2*x)/Sqrt[-2 - 2*x + 3*x^2 - 2*x^3 + x^4],x]
[Out]
ArcTanh[(3 + 4*(-1/2 + x)^2)/Sqrt[-39 + 24*(-1/2 + x)^2 + 16*(-1/2 + x)^4]]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 635
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 1121
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Rule 1694
Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -d/(4*e) + x)*(a + d^4/(256*e^3)
- b*(d/(8*e)) + (c - 3*(d^2/(8*e)))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0]
&& NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] && !IGtQ[p, 0]
Rubi steps
\begin {align*} \int \frac {-1+2 x}{\sqrt {-2-2 x+3 x^2-2 x^3+x^4}} \, dx &=\text {Subst}\left (\int \frac {8 x}{\sqrt {-39+24 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=8 \text {Subst}\left (\int \frac {x}{\sqrt {-39+24 x^2+16 x^4}} \, dx,x,-\frac {1}{2}+x\right )\\ &=4 \text {Subst}\left (\int \frac {1}{\sqrt {-39+24 x+16 x^2}} \, dx,x,\left (-\frac {1}{2}+x\right )^2\right )\\ &=8 \text {Subst}\left (\int \frac {1}{64-x^2} \, dx,x,\frac {8 \left (3+4 \left (-\frac {1}{2}+x\right )^2\right )}{\sqrt {-39+(1-2 x)^4+24 \left (-\frac {1}{2}+x\right )^2}}\right )\\ &=\tanh ^{-1}\left (\frac {3+(-1+2 x)^2}{\sqrt {-39+6 (1-2 x)^2+(1-2 x)^4}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 33, normalized size = 1.00 \begin {gather*} -\log \left (-1+x-x^2+\sqrt {-2-2 x+3 x^2-2 x^3+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-1 + 2*x)/Sqrt[-2 - 2*x + 3*x^2 - 2*x^3 + x^4],x]
[Out]
-Log[-1 + x - x^2 + Sqrt[-2 - 2*x + 3*x^2 - 2*x^3 + x^4]]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.73, size = 1358, normalized size = 41.15
| | |
| method |
result |
size |
| | |
| trager |
\(-\ln \left (x^{2}-\sqrt {x^{4}-2 x^{3}+3 x^{2}-2 x -2}-x +1\right )\) |
\(34\) |
| default |
\(\text {Expression too large to display}\) |
\(1358\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(1358\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-1+2*x)/(x^4-2*x^3+3*x^2-2*x-2)^(1/2),x,method=_RETURNVERBOSE)
[Out]
-2*(-1/2*(-3+4*3^(1/2))^(1/2)-1/2*I*(3+4*3^(1/2))^(1/2))*((1/2*I*(3+4*3^(1/2))^(1/2)-1/2*(-3+4*3^(1/2))^(1/2))
*(x-1/2+1/2*(-3+4*3^(1/2))^(1/2))/(1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))/(x-1/2-1/2*(-3+4*3^(1/2
))^(1/2)))^(1/2)*(x-1/2-1/2*(-3+4*3^(1/2))^(1/2))^2*((-3+4*3^(1/2))^(1/2)*(x-1/2+1/2*I*(3+4*3^(1/2))^(1/2))/(-
1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))/(x-1/2-1/2*(-3+4*3^(1/2))^(1/2)))^(1/2)*((-3+4*3^(1/2))^(1
/2)*(x-1/2-1/2*I*(3+4*3^(1/2))^(1/2))/(1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))/(x-1/2-1/2*(-3+4*3^
(1/2))^(1/2)))^(1/2)/(1/2*I*(3+4*3^(1/2))^(1/2)-1/2*(-3+4*3^(1/2))^(1/2))/(-3+4*3^(1/2))^(1/2)/((x-1/2+1/2*(-3
+4*3^(1/2))^(1/2))*(x-1/2-1/2*(-3+4*3^(1/2))^(1/2))*(x-1/2+1/2*I*(3+4*3^(1/2))^(1/2))*(x-1/2-1/2*I*(3+4*3^(1/2
))^(1/2)))^(1/2)*EllipticF(((1/2*I*(3+4*3^(1/2))^(1/2)-1/2*(-3+4*3^(1/2))^(1/2))*(x-1/2+1/2*(-3+4*3^(1/2))^(1/
2))/(1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))/(x-1/2-1/2*(-3+4*3^(1/2))^(1/2)))^(1/2),((1/2*I*(3+4*
3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))*(-1/2*(-3+4*3^(1/2))^(1/2)-1/2*I*(3+4*3^(1/2))^(1/2))/(1/2*I*(3+4*3^(
1/2))^(1/2)-1/2*(-3+4*3^(1/2))^(1/2))/(-1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2)))^(1/2))+4*(-1/2*(-
3+4*3^(1/2))^(1/2)-1/2*I*(3+4*3^(1/2))^(1/2))*((1/2*I*(3+4*3^(1/2))^(1/2)-1/2*(-3+4*3^(1/2))^(1/2))*(x-1/2+1/2
*(-3+4*3^(1/2))^(1/2))/(1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))/(x-1/2-1/2*(-3+4*3^(1/2))^(1/2)))^
(1/2)*(x-1/2-1/2*(-3+4*3^(1/2))^(1/2))^2*((-3+4*3^(1/2))^(1/2)*(x-1/2+1/2*I*(3+4*3^(1/2))^(1/2))/(-1/2*I*(3+4*
3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))/(x-1/2-1/2*(-3+4*3^(1/2))^(1/2)))^(1/2)*((-3+4*3^(1/2))^(1/2)*(x-1/2-
1/2*I*(3+4*3^(1/2))^(1/2))/(1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))/(x-1/2-1/2*(-3+4*3^(1/2))^(1/2
)))^(1/2)/(1/2*I*(3+4*3^(1/2))^(1/2)-1/2*(-3+4*3^(1/2))^(1/2))/(-3+4*3^(1/2))^(1/2)/((x-1/2+1/2*(-3+4*3^(1/2))
^(1/2))*(x-1/2-1/2*(-3+4*3^(1/2))^(1/2))*(x-1/2+1/2*I*(3+4*3^(1/2))^(1/2))*(x-1/2-1/2*I*(3+4*3^(1/2))^(1/2)))^
(1/2)*((1/2+1/2*(-3+4*3^(1/2))^(1/2))*EllipticF(((1/2*I*(3+4*3^(1/2))^(1/2)-1/2*(-3+4*3^(1/2))^(1/2))*(x-1/2+1
/2*(-3+4*3^(1/2))^(1/2))/(1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))/(x-1/2-1/2*(-3+4*3^(1/2))^(1/2))
)^(1/2),((1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))*(-1/2*(-3+4*3^(1/2))^(1/2)-1/2*I*(3+4*3^(1/2))^(
1/2))/(1/2*I*(3+4*3^(1/2))^(1/2)-1/2*(-3+4*3^(1/2))^(1/2))/(-1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2
)))^(1/2))-(-3+4*3^(1/2))^(1/2)*EllipticPi(((1/2*I*(3+4*3^(1/2))^(1/2)-1/2*(-3+4*3^(1/2))^(1/2))*(x-1/2+1/2*(-
3+4*3^(1/2))^(1/2))/(1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))/(x-1/2-1/2*(-3+4*3^(1/2))^(1/2)))^(1/
2),(1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))/(1/2*I*(3+4*3^(1/2))^(1/2)-1/2*(-3+4*3^(1/2))^(1/2)),(
(1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2))*(-1/2*(-3+4*3^(1/2))^(1/2)-1/2*I*(3+4*3^(1/2))^(1/2))/(1/
2*I*(3+4*3^(1/2))^(1/2)-1/2*(-3+4*3^(1/2))^(1/2))/(-1/2*I*(3+4*3^(1/2))^(1/2)+1/2*(-3+4*3^(1/2))^(1/2)))^(1/2)
))
________________________________________________________________________________________
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((-1+2*x)/(x^4-2*x^3+3*x^2-2*x-2)^(1/2),x, algorithm="maxima")
[Out]
integrate((2*x - 1)/sqrt(x^4 - 2*x^3 + 3*x^2 - 2*x - 2), x)
________________________________________________________________________________________
Fricas [A]
time = 0.43, size = 31, normalized size = 0.94 \begin {gather*} \log \left (-x^{2} + x - \sqrt {x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x - 2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-1+2*x)/(x^4-2*x^3+3*x^2-2*x-2)^(1/2),x, algorithm="fricas")
[Out]
log(-x^2 + x - sqrt(x^4 - 2*x^3 + 3*x^2 - 2*x - 2) - 1)
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x - 1}{\sqrt {x^{4} - 2 x^{3} + 3 x^{2} - 2 x - 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-1+2*x)/(x**4-2*x**3+3*x**2-2*x-2)**(1/2),x)
[Out]
Integral((2*x - 1)/sqrt(x**4 - 2*x**3 + 3*x**2 - 2*x - 2), x)
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (31) = 62\).
time = 0.40, size = 66, normalized size = 2.00 \begin {gather*} \frac {1}{2} \, \sqrt {{\left (x^{2} - x\right )}^{2} + 2 \, x^{2} - 2 \, x - 2} {\left (x^{2} - x + 1\right )} + \frac {3}{2} \, \log \left (x^{2} - x - \sqrt {{\left (x^{2} - x\right )}^{2} + 2 \, x^{2} - 2 \, x - 2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-1+2*x)/(x^4-2*x^3+3*x^2-2*x-2)^(1/2),x, algorithm="giac")
[Out]
1/2*sqrt((x^2 - x)^2 + 2*x^2 - 2*x - 2)*(x^2 - x + 1) + 3/2*log(x^2 - x - sqrt((x^2 - x)^2 + 2*x^2 - 2*x - 2)
+ 1)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {2\,x-1}{\sqrt {x^4-2\,x^3+3\,x^2-2\,x-2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*x - 1)/(3*x^2 - 2*x - 2*x^3 + x^4 - 2)^(1/2),x)
[Out]
int((2*x - 1)/(3*x^2 - 2*x - 2*x^3 + x^4 - 2)^(1/2), x)
________________________________________________________________________________________