Integrand size = 27, antiderivative size = 32 \[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=-\log \left (-4+x-x^2+\sqrt {-8 x+9 x^2-2 x^3+x^4}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(32)=64\).
Time = 3.84 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.34 \[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=-\frac {\sqrt {x} \sqrt {-8+9 x-2 x^2+x^3} \log \left (-4+x-x^2+\sqrt {x} \sqrt {-8+9 x-2 x^2+x^3}\right )}{\sqrt {x \left (-8+9 x-2 x^2+x^3\right )}} \]
-((Sqrt[x]*Sqrt[-8 + 9*x - 2*x^2 + x^3]*Log[-4 + x - x^2 + Sqrt[x]*Sqrt[-8 + 9*x - 2*x^2 + x^3]])/Sqrt[x*(-8 + 9*x - 2*x^2 + x^3)])
Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2459, 27, 27, 1432, 1092, 219}
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.
\(\displaystyle \int \frac {2 x-1}{\sqrt {x^4-2 x^3+9 x^2-8 x}} \, dx\) |
\(\Big \downarrow \) 2459 |
\(\displaystyle \int \frac {2 \left (x-\frac {1}{2}\right )}{\sqrt {\left (x-\frac {1}{2}\right )^4+\frac {15}{2} \left (x-\frac {1}{2}\right )^2-\frac {31}{16}}}d\left (x-\frac {1}{2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {4 \left (x-\frac {1}{2}\right )}{\sqrt {16 \left (x-\frac {1}{2}\right )^4+120 \left (x-\frac {1}{2}\right )^2-31}}d\left (x-\frac {1}{2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 8 \int \frac {x-\frac {1}{2}}{\sqrt {16 \left (x-\frac {1}{2}\right )^4+120 \left (x-\frac {1}{2}\right )^2-31}}d\left (x-\frac {1}{2}\right )\) |
\(\Big \downarrow \) 1432 |
\(\displaystyle 4 \int \frac {1}{\sqrt {16 \left (x-\frac {1}{2}\right )^4+120 \left (x-\frac {1}{2}\right )^2-31}}d\left (x-\frac {1}{2}\right )^2\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle 8 \int \frac {1}{64-\left (x-\frac {1}{2}\right )^4}d\frac {8 \left (4 \left (x-\frac {1}{2}\right )^2+15\right )}{\sqrt {16 \left (x-\frac {1}{2}\right )^4+120 \left (x-\frac {1}{2}\right )^2-31}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \text {arctanh}\left (\frac {4 \left (x-\frac {1}{2}\right )^2+15}{\sqrt {16 \left (x-\frac {1}{2}\right )^4+120 \left (x-\frac {1}{2}\right )^2-31}}\right )\) |
3.4.92.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Time = 2.57 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78
method | result | size |
default | \(\ln \left (x^{2}-x +4+\sqrt {\left (x^{2}-x +8\right ) x \left (x -1\right )}\right )\) | \(25\) |
pseudoelliptic | \(\ln \left (x^{2}-x +4+\sqrt {\left (x^{2}-x +8\right ) x \left (x -1\right )}\right )\) | \(25\) |
trager | \(-\ln \left (x^{2}-\sqrt {x^{4}-2 x^{3}+9 x^{2}-8 x}-x +4\right )\) | \(33\) |
elliptic | \(-\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}\, \left (x -1\right )^{2} \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}, \sqrt {\frac {\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \sqrt {x \left (x -1\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}+\frac {4 \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}\, \left (x -1\right )^{2} \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {31}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}, \sqrt {\frac {\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\right )-\operatorname {EllipticPi}\left (\sqrt {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -1\right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {31}}{2}}{-\frac {1}{2}+\frac {i \sqrt {31}}{2}}, \sqrt {\frac {\left (\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\right )\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \sqrt {x \left (x -1\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {31}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {31}}{2}\right )}}\) | \(487\) |
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=\log \left (-x^{2} + x - \sqrt {x^{4} - 2 \, x^{3} + 9 \, x^{2} - 8 \, x} - 4\right ) \]
\[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=\int \frac {2 x - 1}{\sqrt {x \left (x - 1\right ) \left (x^{2} - x + 8\right )}}\, dx \]
\[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=\int { \frac {2 \, x - 1}{\sqrt {x^{4} - 2 \, x^{3} + 9 \, x^{2} - 8 \, x}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (30) = 60\).
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.00 \[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=\frac {1}{2} \, \sqrt {{\left (x^{2} - x\right )}^{2} + 8 \, x^{2} - 8 \, x} {\left (x^{2} - x + 4\right )} + 8 \, \log \left (x^{2} - x - \sqrt {{\left (x^{2} - x\right )}^{2} + 8 \, x^{2} - 8 \, x} + 4\right ) \]
1/2*sqrt((x^2 - x)^2 + 8*x^2 - 8*x)*(x^2 - x + 4) + 8*log(x^2 - x - sqrt(( x^2 - x)^2 + 8*x^2 - 8*x) + 4)
Timed out. \[ \int \frac {-1+2 x}{\sqrt {-8 x+9 x^2-2 x^3+x^4}} \, dx=\int \frac {2\,x-1}{\sqrt {x^4-2\,x^3+9\,x^2-8\,x}} \,d x \]