Integrand size = 54, antiderivative size = 21 \[ \int \frac {-6+x+x^2+e (3+x)+x \log \left (\frac {e^{-x}}{x^3}\right )}{4 x+e^2 x-4 x^2+x^3+e \left (-4 x+2 x^2\right )} \, dx=\frac {\log \left (\frac {e^{-x}}{x^3}\right )}{2-e-x} \] Output:
ln(exp(-x)/x^3)/(2-x-exp(1))
Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {-6+x+x^2+e (3+x)+x \log \left (\frac {e^{-x}}{x^3}\right )}{4 x+e^2 x-4 x^2+x^3+e \left (-4 x+2 x^2\right )} \, dx=-\frac {\log \left (\frac {e^{-x}}{x^3}\right )}{-2+e+x} \] Input:
Integrate[(-6 + x + x^2 + E*(3 + x) + x*Log[1/(E^x*x^3)])/(4*x + E^2*x - 4 *x^2 + x^3 + E*(-4*x + 2*x^2)),x]
Output:
-(Log[1/(E^x*x^3)]/(-2 + E + x))
Leaf count is larger than twice the leaf count of optimal. \(194\) vs. \(2(21)=42\).
Time = 0.63 (sec) , antiderivative size = 194, normalized size of antiderivative = 9.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6, 2026, 2007, 7293, 2009}
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.
\(\displaystyle \int \frac {x \log \left (\frac {e^{-x}}{x^3}\right )+x^2+x+e (x+3)-6}{x^3-4 x^2+e \left (2 x^2-4 x\right )+e^2 x+4 x} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x \log \left (\frac {e^{-x}}{x^3}\right )+x^2+x+e (x+3)-6}{x^3-4 x^2+e \left (2 x^2-4 x\right )+\left (4+e^2\right ) x}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {x \log \left (\frac {e^{-x}}{x^3}\right )+x^2+x+e (x+3)-6}{x \left (x^2-2 (2-e) x+(e-2)^2\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {x \log \left (\frac {e^{-x}}{x^3}\right )+x^2+x+e (x+3)-6}{x (x+e-2)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\log \left (\frac {e^{-x}}{x^3}\right )}{(x+e-2)^2}+\frac {x}{(x+e-2)^2}+\frac {1}{(x+e-2)^2}+\frac {e (x+3)}{(x+e-2)^2 x}-\frac {6}{(x+e-2)^2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\log \left (\frac {e^{-x}}{x^3}\right )}{-x-e+2}+\frac {(5-e) e}{(2-e) (-x-e+2)}+\frac {2-e}{-x-e+2}-\frac {6}{(2-e) (-x-e+2)}+\frac {1}{-x-e+2}-\frac {3 e \log (-x-e+2)}{(2-e)^2}-\frac {(5-e) \log (-x-e+2)}{2-e}+\frac {6 \log (-x-e+2)}{(2-e)^2}+\log (-x-e+2)+\frac {3 e \log (x)}{(2-e)^2}+\frac {3 \log (x)}{2-e}-\frac {6 \log (x)}{(2-e)^2}\) |
Input:
Int[(-6 + x + x^2 + E*(3 + x) + x*Log[1/(E^x*x^3)])/(4*x + E^2*x - 4*x^2 + x^3 + E*(-4*x + 2*x^2)),x]
Output:
(2 - E - x)^(-1) - 6/((2 - E)*(2 - E - x)) + (2 - E)/(2 - E - x) + ((5 - E )*E)/((2 - E)*(2 - E - x)) + Log[2 - E - x] + (6*Log[2 - E - x])/(2 - E)^2 - ((5 - E)*Log[2 - E - x])/(2 - E) - (3*E*Log[2 - E - x])/(2 - E)^2 + Log [1/(E^x*x^3)]/(2 - E - x) - (6*Log[x])/(2 - E)^2 + (3*Log[x])/(2 - E) + (3 *E*Log[x])/(2 - E)^2
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
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])
Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
method | result | size |
norman | \(-\frac {\ln \left (\frac {{\mathrm e}^{-x}}{x^{3}}\right )}{-2+{\mathrm e}+x}\) | \(19\) |
parallelrisch | \(-\frac {\ln \left (\frac {{\mathrm e}^{-x}}{x^{3}}\right )}{-2+{\mathrm e}+x}\) | \(19\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{-x}\right )}{-2+{\mathrm e}+x}+\frac {-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{x^{3}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x^{3}}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{x^{3}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x^{3}}\right )^{2}-i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x^{3}}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x^{3}}\right )^{3}-i \pi \operatorname {csgn}\left (i x^{3}\right )^{3}+6 \ln \left (x \right )}{-4+2 \,{\mathrm e}+2 x}\) | \(249\) |
default | \(\frac {-{\mathrm e}+2-\ln \left (\frac {{\mathrm e}^{-x}}{x^{3}}\right )-x -3 \ln \left (x \right )}{-2+{\mathrm e}+x}-\ln \left (-2+{\mathrm e}+x \right )-\frac {3 \ln \left (x \right ) \left (\ln \left (\frac {-{\mathrm e}+2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}-x}{-{\mathrm e}+2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}\right )-\ln \left (\frac {{\mathrm e}-2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}+x}{{\mathrm e}-2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}\right )\right )}{2 \sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}-\frac {3 \left (\operatorname {dilog}\left (\frac {-{\mathrm e}+2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}-x}{-{\mathrm e}+2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}\right )-\operatorname {dilog}\left (\frac {{\mathrm e}-2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}+x}{{\mathrm e}-2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}\right )\right )}{2 \sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}+\frac {3 \ln \left (x \right )}{{\mathrm e}-2}+\frac {\left ({\mathrm e}-5\right ) \ln \left (-2+{\mathrm e}+x \right )}{{\mathrm e}-2}\) | \(255\) |
parts | \(\frac {-{\mathrm e}+2-\ln \left (\frac {{\mathrm e}^{-x}}{x^{3}}\right )-x -3 \ln \left (x \right )}{-2+{\mathrm e}+x}-\ln \left (-2+{\mathrm e}+x \right )-\frac {3 \ln \left (x \right ) \left (\ln \left (\frac {-{\mathrm e}+2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}-x}{-{\mathrm e}+2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}\right )-\ln \left (\frac {{\mathrm e}-2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}+x}{{\mathrm e}-2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}\right )\right )}{2 \sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}-\frac {3 \left (\operatorname {dilog}\left (\frac {-{\mathrm e}+2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}-x}{-{\mathrm e}+2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}\right )-\operatorname {dilog}\left (\frac {{\mathrm e}-2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}+x}{{\mathrm e}-2+\sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}\right )\right )}{2 \sqrt {\left ({\mathrm e}\right )^{2}-{\mathrm e}^{2}}}+\frac {3 \ln \left (x \right )}{{\mathrm e}-2}+\frac {\left ({\mathrm e}-5\right ) \ln \left (-2+{\mathrm e}+x \right )}{{\mathrm e}-2}\) | \(255\) |
Input:
int((x*ln(exp(-x)/x^3)+(3+x)*exp(1)+x^2+x-6)/(x*exp(1)^2+(2*x^2-4*x)*exp(1 )+x^3-4*x^2+4*x),x,method=_RETURNVERBOSE)
Output:
-ln(exp(-x)/x^3)/(-2+exp(1)+x)
Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {-6+x+x^2+e (3+x)+x \log \left (\frac {e^{-x}}{x^3}\right )}{4 x+e^2 x-4 x^2+x^3+e \left (-4 x+2 x^2\right )} \, dx=-\frac {\log \left (\frac {e^{\left (-x\right )}}{x^{3}}\right )}{x + e - 2} \] Input:
integrate((x*log(exp(-x)/x^3)+(3+x)*exp(1)+x^2+x-6)/(x*exp(1)^2+(2*x^2-4*x )*exp(1)+x^3-4*x^2+4*x),x, algorithm="fricas")
Output:
-log(e^(-x)/x^3)/(x + e - 2)
Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-6+x+x^2+e (3+x)+x \log \left (\frac {e^{-x}}{x^3}\right )}{4 x+e^2 x-4 x^2+x^3+e \left (-4 x+2 x^2\right )} \, dx=- \frac {\log {\left (\frac {e^{- x}}{x^{3}} \right )}}{x - 2 + e} \] Input:
integrate((x*ln(exp(-x)/x**3)+(3+x)*exp(1)+x**2+x-6)/(x*exp(1)**2+(2*x**2- 4*x)*exp(1)+x**3-4*x**2+4*x),x)
Output:
-log(exp(-x)/x**3)/(x - 2 + E)
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (18) = 36\).
Time = 0.04 (sec) , antiderivative size = 189, normalized size of antiderivative = 9.00 \[ \int \frac {-6+x+x^2+e (3+x)+x \log \left (\frac {e^{-x}}{x^3}\right )}{4 x+e^2 x-4 x^2+x^3+e \left (-4 x+2 x^2\right )} \, dx=-3 \, {\left (\frac {\log \left (x + e - 2\right )}{e^{2} - 4 \, e + 4} - \frac {\log \left (x\right )}{e^{2} - 4 \, e + 4} - \frac {1}{x {\left (e - 2\right )} + e^{2} - 4 \, e + 4}\right )} e - \frac {{\left (e - 5\right )} \log \left (x + e - 2\right )}{e - 2} + \frac {e - 2}{x + e - 2} - \frac {e}{x + e - 2} + \frac {6 \, \log \left (x + e - 2\right )}{e^{2} - 4 \, e + 4} - \frac {6 \, \log \left (x\right )}{e^{2} - 4 \, e + 4} - \frac {3 \, \log \left (x\right )}{e - 2} - \frac {\log \left (\frac {e^{\left (-x\right )}}{x^{3}}\right )}{x + e - 2} - \frac {6}{x {\left (e - 2\right )} + e^{2} - 4 \, e + 4} - \frac {1}{x + e - 2} + \log \left (x + e - 2\right ) \] Input:
integrate((x*log(exp(-x)/x^3)+(3+x)*exp(1)+x^2+x-6)/(x*exp(1)^2+(2*x^2-4*x )*exp(1)+x^3-4*x^2+4*x),x, algorithm="maxima")
Output:
-3*(log(x + e - 2)/(e^2 - 4*e + 4) - log(x)/(e^2 - 4*e + 4) - 1/(x*(e - 2) + e^2 - 4*e + 4))*e - (e - 5)*log(x + e - 2)/(e - 2) + (e - 2)/(x + e - 2 ) - e/(x + e - 2) + 6*log(x + e - 2)/(e^2 - 4*e + 4) - 6*log(x)/(e^2 - 4*e + 4) - 3*log(x)/(e - 2) - log(e^(-x)/x^3)/(x + e - 2) - 6/(x*(e - 2) + e^ 2 - 4*e + 4) - 1/(x + e - 2) + log(x + e - 2)
Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-6+x+x^2+e (3+x)+x \log \left (\frac {e^{-x}}{x^3}\right )}{4 x+e^2 x-4 x^2+x^3+e \left (-4 x+2 x^2\right )} \, dx=-\frac {e - 3 \, \log \left (x\right ) - 2}{x + e - 2} \] Input:
integrate((x*log(exp(-x)/x^3)+(3+x)*exp(1)+x^2+x-6)/(x*exp(1)^2+(2*x^2-4*x )*exp(1)+x^3-4*x^2+4*x),x, algorithm="giac")
Output:
-(e - 3*log(x) - 2)/(x + e - 2)
Time = 3.83 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {-6+x+x^2+e (3+x)+x \log \left (\frac {e^{-x}}{x^3}\right )}{4 x+e^2 x-4 x^2+x^3+e \left (-4 x+2 x^2\right )} \, dx=\frac {x-\ln \left (\frac {1}{x^3}\right )}{x+\mathrm {e}-2} \] Input:
int((x + exp(1)*(x + 3) + x*log(exp(-x)/x^3) + x^2 - 6)/(4*x - exp(1)*(4*x - 2*x^2) + x*exp(2) - 4*x^2 + x^3),x)
Output:
(x - log(1/x^3))/(x + exp(1) - 2)
Time = 0.16 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {-6+x+x^2+e (3+x)+x \log \left (\frac {e^{-x}}{x^3}\right )}{4 x+e^2 x-4 x^2+x^3+e \left (-4 x+2 x^2\right )} \, dx=\frac {\mathrm {log}\left (e^{x} x^{3}\right )}{e +x -2} \] Input:
int((x*log(exp(-x)/x^3)+(3+x)*exp(1)+x^2+x-6)/(x*exp(1)^2+(2*x^2-4*x)*exp( 1)+x^3-4*x^2+4*x),x)
Output:
log(e**x*x**3)/(e + x - 2)