Integrand size = 109, antiderivative size = 27 \[ \int \frac {e^{\sqrt [338]{e}-x+2 \sqrt [676]{e} \log \left (\frac {4-5 x+x^2}{x}\right )+\log ^2\left (\frac {4-5 x+x^2}{x}\right )} \left (-4 x+5 x^2-x^3+\sqrt [676]{e} \left (-8+2 x^2\right )+\left (-8+2 x^2\right ) \log \left (\frac {4-5 x+x^2}{x}\right )\right )}{4 x-5 x^2+x^3} \, dx=e^{-x+\left (\sqrt [676]{e}+\log \left (-4+\frac {4-x}{x}+x\right )\right )^2} \] Output:
exp((exp(1/676)+ln(x+(4-x)/x-4))^2-x)
Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\sqrt [338]{e}-x+2 \sqrt [676]{e} \log \left (\frac {4-5 x+x^2}{x}\right )+\log ^2\left (\frac {4-5 x+x^2}{x}\right )} \left (-4 x+5 x^2-x^3+\sqrt [676]{e} \left (-8+2 x^2\right )+\left (-8+2 x^2\right ) \log \left (\frac {4-5 x+x^2}{x}\right )\right )}{4 x-5 x^2+x^3} \, dx=e^{\sqrt [338]{e}-x+\log ^2\left (-5+\frac {4}{x}+x\right )} \left (-5+\frac {4}{x}+x\right )^{2 \sqrt [676]{e}} \] Input:
Integrate[(E^(E^(1/338) - x + 2*E^(1/676)*Log[(4 - 5*x + x^2)/x] + Log[(4 - 5*x + x^2)/x]^2)*(-4*x + 5*x^2 - x^3 + E^(1/676)*(-8 + 2*x^2) + (-8 + 2* x^2)*Log[(4 - 5*x + x^2)/x]))/(4*x - 5*x^2 + x^3),x]
Output:
E^(E^(1/338) - x + Log[-5 + 4/x + x]^2)*(-5 + 4/x + x)^(2*E^(1/676))
Leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(27)=54\).
Time = 3.16 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.41, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2026, 2704, 2726}
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.
\(\displaystyle \int \frac {\left (-x^3+5 x^2+\sqrt [676]{e} \left (2 x^2-8\right )+\left (2 x^2-8\right ) \log \left (\frac {x^2-5 x+4}{x}\right )-4 x\right ) \exp \left (\log ^2\left (\frac {x^2-5 x+4}{x}\right )+2 \sqrt [676]{e} \log \left (\frac {x^2-5 x+4}{x}\right )-x+\sqrt [338]{e}\right )}{x^3-5 x^2+4 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (-x^3+5 x^2+\sqrt [676]{e} \left (2 x^2-8\right )+\left (2 x^2-8\right ) \log \left (\frac {x^2-5 x+4}{x}\right )-4 x\right ) \exp \left (\log ^2\left (\frac {x^2-5 x+4}{x}\right )+2 \sqrt [676]{e} \log \left (\frac {x^2-5 x+4}{x}\right )-x+\sqrt [338]{e}\right )}{x \left (x^2-5 x+4\right )}dx\) |
\(\Big \downarrow \) 2704 |
\(\displaystyle \int \frac {\left (\frac {x^2-5 x+4}{x}\right )^{2 \sqrt [676]{e}} e^{\log ^2\left (\frac {x^2-5 x+4}{x}\right )-x+\sqrt [338]{e}} \left (-x^3+5 x^2+\sqrt [676]{e} \left (2 x^2-8\right )+\left (2 x^2-8\right ) \log \left (\frac {x^2-5 x+4}{x}\right )-4 x\right )}{x \left (x^2-5 x+4\right )}dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {\left (\frac {x^2-5 x+4}{x}\right )^{2 \sqrt [676]{e}} e^{\log ^2\left (\frac {x^2-5 x+4}{x}\right )-x+\sqrt [338]{e}} \left (x^3-5 x^2+2 \left (4-x^2\right ) \log \left (\frac {x^2-5 x+4}{x}\right )+4 x\right )}{x \left (x^2-5 x+4\right ) \left (\frac {2 x \left (\frac {x^2-5 x+4}{x^2}+\frac {5-2 x}{x}\right ) \log \left (\frac {x^2-5 x+4}{x}\right )}{x^2-5 x+4}+1\right )}\) |
Input:
Int[(E^(E^(1/338) - x + 2*E^(1/676)*Log[(4 - 5*x + x^2)/x] + Log[(4 - 5*x + x^2)/x]^2)*(-4*x + 5*x^2 - x^3 + E^(1/676)*(-8 + 2*x^2) + (-8 + 2*x^2)*L og[(4 - 5*x + x^2)/x]))/(4*x - 5*x^2 + x^3),x]
Output:
(E^(E^(1/338) - x + Log[(4 - 5*x + x^2)/x]^2)*((4 - 5*x + x^2)/x)^(2*E^(1/ 676))*(4*x - 5*x^2 + x^3 + 2*(4 - x^2)*Log[(4 - 5*x + x^2)/x]))/(x*(4 - 5* x + x^2)*(1 + (2*x*((5 - 2*x)/x + (4 - 5*x + x^2)/x^2)*Log[(4 - 5*x + x^2) /x])/(4 - 5*x + x^2)))
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)* z^(a*b*Log[F]), x] /; FreeQ[{F, a, b}, x]
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 0.81 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52
method | result | size |
risch | \(\left (\frac {x^{2}-5 x +4}{x}\right )^{2 \,{\mathrm e}^{\frac {1}{676}}} {\mathrm e}^{\ln \left (\frac {x^{2}-5 x +4}{x}\right )^{2}+{\mathrm e}^{\frac {1}{338}}-x}\) | \(41\) |
norman | \({\mathrm e}^{\ln \left (\frac {x^{2}-5 x +4}{x}\right )^{2}+2 \,{\mathrm e}^{\frac {1}{676}} \ln \left (\frac {x^{2}-5 x +4}{x}\right )+{\mathrm e}^{\frac {1}{338}}-x}\) | \(42\) |
parallelrisch | \({\mathrm e}^{\ln \left (\frac {x^{2}-5 x +4}{x}\right )^{2}+2 \,{\mathrm e}^{\frac {1}{676}} \ln \left (\frac {x^{2}-5 x +4}{x}\right )+{\mathrm e}^{\frac {1}{338}}-x}\) | \(42\) |
Input:
int(((2*x^2-8)*ln((x^2-5*x+4)/x)+(2*x^2-8)*exp(1/676)-x^3+5*x^2-4*x)*exp(l n((x^2-5*x+4)/x)^2+2*exp(1/676)*ln((x^2-5*x+4)/x)+exp(1/676)^2-x)/(x^3-5*x ^2+4*x),x,method=_RETURNVERBOSE)
Output:
((x^2-5*x+4)/x)^(2*exp(1/676))*exp(ln((x^2-5*x+4)/x)^2+exp(1/338)-x)
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\sqrt [338]{e}-x+2 \sqrt [676]{e} \log \left (\frac {4-5 x+x^2}{x}\right )+\log ^2\left (\frac {4-5 x+x^2}{x}\right )} \left (-4 x+5 x^2-x^3+\sqrt [676]{e} \left (-8+2 x^2\right )+\left (-8+2 x^2\right ) \log \left (\frac {4-5 x+x^2}{x}\right )\right )}{4 x-5 x^2+x^3} \, dx=e^{\left (2 \, e^{\frac {1}{676}} \log \left (\frac {x^{2} - 5 \, x + 4}{x}\right ) + \log \left (\frac {x^{2} - 5 \, x + 4}{x}\right )^{2} - x + e^{\frac {1}{338}}\right )} \] Input:
integrate(((2*x^2-8)*log((x^2-5*x+4)/x)+(2*x^2-8)*exp(1/676)-x^3+5*x^2-4*x )*exp(log((x^2-5*x+4)/x)^2+2*exp(1/676)*log((x^2-5*x+4)/x)+exp(1/676)^2-x) /(x^3-5*x^2+4*x),x, algorithm="fricas")
Output:
e^(2*e^(1/676)*log((x^2 - 5*x + 4)/x) + log((x^2 - 5*x + 4)/x)^2 - x + e^( 1/338))
Timed out. \[ \int \frac {e^{\sqrt [338]{e}-x+2 \sqrt [676]{e} \log \left (\frac {4-5 x+x^2}{x}\right )+\log ^2\left (\frac {4-5 x+x^2}{x}\right )} \left (-4 x+5 x^2-x^3+\sqrt [676]{e} \left (-8+2 x^2\right )+\left (-8+2 x^2\right ) \log \left (\frac {4-5 x+x^2}{x}\right )\right )}{4 x-5 x^2+x^3} \, dx=\text {Timed out} \] Input:
integrate(((2*x**2-8)*ln((x**2-5*x+4)/x)+(2*x**2-8)*exp(1/676)-x**3+5*x**2 -4*x)*exp(ln((x**2-5*x+4)/x)**2+2*exp(1/676)*ln((x**2-5*x+4)/x)+exp(1/676) **2-x)/(x**3-5*x**2+4*x),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.63 \[ \int \frac {e^{\sqrt [338]{e}-x+2 \sqrt [676]{e} \log \left (\frac {4-5 x+x^2}{x}\right )+\log ^2\left (\frac {4-5 x+x^2}{x}\right )} \left (-4 x+5 x^2-x^3+\sqrt [676]{e} \left (-8+2 x^2\right )+\left (-8+2 x^2\right ) \log \left (\frac {4-5 x+x^2}{x}\right )\right )}{4 x-5 x^2+x^3} \, dx=e^{\left (2 \, e^{\frac {1}{676}} \log \left (x - 1\right ) + \log \left (x - 1\right )^{2} + 2 \, e^{\frac {1}{676}} \log \left (x - 4\right ) + 2 \, \log \left (x - 1\right ) \log \left (x - 4\right ) + \log \left (x - 4\right )^{2} - 2 \, e^{\frac {1}{676}} \log \left (x\right ) - 2 \, \log \left (x - 1\right ) \log \left (x\right ) - 2 \, \log \left (x - 4\right ) \log \left (x\right ) + \log \left (x\right )^{2} - x + e^{\frac {1}{338}}\right )} \] Input:
integrate(((2*x^2-8)*log((x^2-5*x+4)/x)+(2*x^2-8)*exp(1/676)-x^3+5*x^2-4*x )*exp(log((x^2-5*x+4)/x)^2+2*exp(1/676)*log((x^2-5*x+4)/x)+exp(1/676)^2-x) /(x^3-5*x^2+4*x),x, algorithm="maxima")
Output:
e^(2*e^(1/676)*log(x - 1) + log(x - 1)^2 + 2*e^(1/676)*log(x - 4) + 2*log( x - 1)*log(x - 4) + log(x - 4)^2 - 2*e^(1/676)*log(x) - 2*log(x - 1)*log(x ) - 2*log(x - 4)*log(x) + log(x)^2 - x + e^(1/338))
\[ \int \frac {e^{\sqrt [338]{e}-x+2 \sqrt [676]{e} \log \left (\frac {4-5 x+x^2}{x}\right )+\log ^2\left (\frac {4-5 x+x^2}{x}\right )} \left (-4 x+5 x^2-x^3+\sqrt [676]{e} \left (-8+2 x^2\right )+\left (-8+2 x^2\right ) \log \left (\frac {4-5 x+x^2}{x}\right )\right )}{4 x-5 x^2+x^3} \, dx=\int { -\frac {{\left (x^{3} - 5 \, x^{2} - 2 \, {\left (x^{2} - 4\right )} e^{\frac {1}{676}} - 2 \, {\left (x^{2} - 4\right )} \log \left (\frac {x^{2} - 5 \, x + 4}{x}\right ) + 4 \, x\right )} e^{\left (2 \, e^{\frac {1}{676}} \log \left (\frac {x^{2} - 5 \, x + 4}{x}\right ) + \log \left (\frac {x^{2} - 5 \, x + 4}{x}\right )^{2} - x + e^{\frac {1}{338}}\right )}}{x^{3} - 5 \, x^{2} + 4 \, x} \,d x } \] Input:
integrate(((2*x^2-8)*log((x^2-5*x+4)/x)+(2*x^2-8)*exp(1/676)-x^3+5*x^2-4*x )*exp(log((x^2-5*x+4)/x)^2+2*exp(1/676)*log((x^2-5*x+4)/x)+exp(1/676)^2-x) /(x^3-5*x^2+4*x),x, algorithm="giac")
Output:
integrate(-(x^3 - 5*x^2 - 2*(x^2 - 4)*e^(1/676) - 2*(x^2 - 4)*log((x^2 - 5 *x + 4)/x) + 4*x)*e^(2*e^(1/676)*log((x^2 - 5*x + 4)/x) + log((x^2 - 5*x + 4)/x)^2 - x + e^(1/338))/(x^3 - 5*x^2 + 4*x), x)
Time = 3.44 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {e^{\sqrt [338]{e}-x+2 \sqrt [676]{e} \log \left (\frac {4-5 x+x^2}{x}\right )+\log ^2\left (\frac {4-5 x+x^2}{x}\right )} \left (-4 x+5 x^2-x^3+\sqrt [676]{e} \left (-8+2 x^2\right )+\left (-8+2 x^2\right ) \log \left (\frac {4-5 x+x^2}{x}\right )\right )}{4 x-5 x^2+x^3} \, dx={\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\ln \left (\frac {x^2-5\,x+4}{x}\right )}^2}\,{\mathrm {e}}^{{\mathrm {e}}^{1/338}}\,{\left (x+\frac {4}{x}-5\right )}^{2\,{\mathrm {e}}^{1/676}} \] Input:
int((exp(exp(1/338) - x + log((x^2 - 5*x + 4)/x)^2 + 2*exp(1/676)*log((x^2 - 5*x + 4)/x))*(exp(1/676)*(2*x^2 - 8) - 4*x + log((x^2 - 5*x + 4)/x)*(2* x^2 - 8) + 5*x^2 - x^3))/(4*x - 5*x^2 + x^3),x)
Output:
exp(-x)*exp(log((x^2 - 5*x + 4)/x)^2)*exp(exp(1/338))*(x + 4/x - 5)^(2*exp (1/676))
Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {e^{\sqrt [338]{e}-x+2 \sqrt [676]{e} \log \left (\frac {4-5 x+x^2}{x}\right )+\log ^2\left (\frac {4-5 x+x^2}{x}\right )} \left (-4 x+5 x^2-x^3+\sqrt [676]{e} \left (-8+2 x^2\right )+\left (-8+2 x^2\right ) \log \left (\frac {4-5 x+x^2}{x}\right )\right )}{4 x-5 x^2+x^3} \, dx=\frac {e^{2 e^{\frac {1}{676}} \mathrm {log}\left (\frac {x^{2}-5 x +4}{x}\right )+e^{\frac {1}{338}}+\mathrm {log}\left (\frac {x^{2}-5 x +4}{x}\right )^{2}}}{e^{x}} \] Input:
int(((2*x^2-8)*log((x^2-5*x+4)/x)+(2*x^2-8)*exp(1/676)-x^3+5*x^2-4*x)*exp( log((x^2-5*x+4)/x)^2+2*exp(1/676)*log((x^2-5*x+4)/x)+exp(1/676)^2-x)/(x^3- 5*x^2+4*x),x)
Output:
e**(2*e**(1/676)*log((x**2 - 5*x + 4)/x) + e**(1/338) + log((x**2 - 5*x + 4)/x)**2)/e**x