Integrand size = 23, antiderivative size = 17 \[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=-\frac {\operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )}{\sqrt {3}} \] Output:
1/3*3^(1/2)*EllipticF(-1+x,1/3*I*3^(1/2))
Result contains complex when optimal does not.
Time = 32.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 9.18 \[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=\frac {\sqrt {-i+\sqrt {3}+\frac {4 i}{x}} \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} x \left (-4+x-i \sqrt {3} x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{\sqrt {2} \sqrt {i+\sqrt {3}-\frac {4 i}{x}} \sqrt {-x \left (-8+8 x-4 x^2+x^3\right )}} \] Input:
Integrate[1/Sqrt[8*x - 8*x^2 + 4*x^3 - x^4],x]
Output:
(Sqrt[-I + Sqrt[3] + (4*I)/x]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x*( -4 + x - I*Sqrt[3]*x)*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2 ]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/(Sqrt[2]*Sqrt[I + Sqrt[3] - (4*I )/x]*Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))])
Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2458, 1408, 27, 321}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt {-x^4+4 x^3-8 x^2+8 x}} \, dx\) |
\(\Big \downarrow \) 2458 |
\(\displaystyle \int \frac {1}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)\) |
\(\Big \downarrow \) 1408 |
\(\displaystyle 2 \int \frac {1}{2 \sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {1}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)\) |
\(\Big \downarrow \) 321 |
\(\displaystyle -\frac {\operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )}{\sqrt {3}}\) |
Input:
Int[1/Sqrt[8*x - 8*x^2 + 4*x^3 - x^4],x]
Output:
-(EllipticF[ArcSin[1 - x], -1/3]/Sqrt[3])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[2*Sqrt[-c] Int[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[ c, 0]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (15 ) = 30\).
Time = 0.55 (sec) , antiderivative size = 200, normalized size of antiderivative = 11.76
method | result | size |
default | \(\frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{\left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}\) | \(200\) |
elliptic | \(\frac {2 \left (-1-i \sqrt {3}\right ) \sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \left (x -2\right )^{2} \sqrt {\frac {x -1+i \sqrt {3}}{\left (1-i \sqrt {3}\right ) \left (x -2\right )}}\, \sqrt {\frac {x -1-i \sqrt {3}}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (-1+i \sqrt {3}\right ) x}{\left (1+i \sqrt {3}\right ) \left (x -2\right )}}, \sqrt {\frac {\left (1+i \sqrt {3}\right ) \left (-1-i \sqrt {3}\right )}{\left (-1+i \sqrt {3}\right ) \left (1-i \sqrt {3}\right )}}\right )}{\left (-1+i \sqrt {3}\right ) \sqrt {-x \left (x -2\right ) \left (x -1+i \sqrt {3}\right ) \left (x -1-i \sqrt {3}\right )}}\) | \(200\) |
Input:
int(1/(-x^4+4*x^3-8*x^2+8*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(-1-I*3^(1/2))*((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2)*(x-2)^2*((x- 1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)*((x-1-I*3^(1/2))/(1+I*3^(1/2))/(x- 2))^(1/2)/(-1+I*3^(1/2))/(-x*(x-2)*(x-1+I*3^(1/2))*(x-1-I*3^(1/2)))^(1/2)* EllipticF(((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2),((1+I*3^(1/2))*(-1- I*3^(1/2))/(-1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2))
Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=-\frac {1}{2} \, \sqrt {2} {\rm weierstrassPInverse}\left (-\frac {2}{3}, \frac {7}{54}, -\frac {x - 3}{3 \, x}\right ) \] Input:
integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(1/2),x, algorithm="fricas")
Output:
-1/2*sqrt(2)*weierstrassPInverse(-2/3, 7/54, -1/3*(x - 3)/x)
\[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=\int \frac {1}{\sqrt {- x^{4} + 4 x^{3} - 8 x^{2} + 8 x}}\, dx \] Input:
integrate(1/(-x**4+4*x**3-8*x**2+8*x)**(1/2),x)
Output:
Integral(1/sqrt(-x**4 + 4*x**3 - 8*x**2 + 8*x), x)
\[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}} \,d x } \] Input:
integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)
\[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}} \,d x } \] Input:
integrate(1/(-x^4+4*x^3-8*x^2+8*x)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)
Timed out. \[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=\int \frac {1}{\sqrt {-x^4+4\,x^3-8\,x^2+8\,x}} \,d x \] Input:
int(1/(8*x - 8*x^2 + 4*x^3 - x^4)^(1/2),x)
Output:
int(1/(8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x)
\[ \int \frac {1}{\sqrt {8 x-8 x^2+4 x^3-x^4}} \, dx=-\left (\int \frac {\sqrt {x}\, \sqrt {-x^{3}+4 x^{2}-8 x +8}}{x^{4}-4 x^{3}+8 x^{2}-8 x}d x \right ) \] Input:
int(1/(-x^4+4*x^3-8*x^2+8*x)^(1/2),x)
Output:
- int((sqrt(x)*sqrt( - x**3 + 4*x**2 - 8*x + 8))/(x**4 - 4*x**3 + 8*x**2 - 8*x),x)