\(\int -\frac {2 e}{(e x-x^2) \log ^2(-\frac {54 x^2}{e^2-2 e x+x^2})} \, dx\) [9302]

Optimal result

Integrand size = 35, antiderivative size = 17 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=2+\frac {1}{\log \left (-\frac {54 x^2}{(e-x)^2}\right )} \]

[Out]

2+1/ln(-54*x^2/(exp(1)-x)^2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 1607, 6818} \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\log \left (-\frac {54 x^2}{x^2-2 e x+e^2}\right )} \]

[In]

Int[(-2*E)/((E*x - x^2)*Log[(-54*x^2)/(E^2 - 2*E*x + x^2)]^2),x]

[Out]

Log[(-54*x^2)/(E^2 - 2*E*x + x^2)]^(-1)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\left ((2 e) \int \frac {1}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx\right ) \\ & = -\left ((2 e) \int \frac {1}{(e-x) x \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx\right ) \\ & = \frac {1}{\log \left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\log \left (-\frac {54 x^2}{(e-x)^2}\right )} \]

[In]

Integrate[(-2*E)/((E*x - x^2)*Log[(-54*x^2)/(E^2 - 2*E*x + x^2)]^2),x]

[Out]

Log[(-54*x^2)/(E - x)^2]^(-1)

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29

method	result	size
risch	\(\frac {1}{\ln \left (-\frac {54 x^{2}}{{\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}}\right )}\)	\(22\)
norman	\(\frac {1}{\ln \left (-\frac {54 x^{2}}{{\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}}\right )}\)	\(24\)
parallelrisch	\(\frac {1}{\ln \left (-\frac {54 x^{2}}{{\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}}\right )}\)	\(24\)
default	\(\frac {1}{\ln \left (54\right )+\ln \left (\frac {x^{2}}{-{\mathrm e}^{2}+2 x \,{\mathrm e}-x^{2}}\right )}\)	\(30\)

[In]

int(-2*exp(1)/(x*exp(1)-x^2)/ln(-54*x^2/(exp(1)^2-2*x*exp(1)+x^2))^2,x,method=_RETURNVERBOSE)

[Out]

1/ln(-54*x^2/(exp(2)-2*x*exp(1)+x^2))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\log \left (-\frac {54 \, x^{2}}{x^{2} - 2 \, x e + e^{2}}\right )} \]

[In]

integrate(-2*exp(1)/(x*exp(1)-x^2)/log(-54*x^2/(exp(1)^2-2*x*exp(1)+x^2))^2,x, algorithm="fricas")

[Out]

1/log(-54*x^2/(x^2 - 2*x*e + e^2))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\log {\left (- \frac {54 x^{2}}{x^{2} - 2 e x + e^{2}} \right )}} \]

[In]

integrate(-2*exp(1)/(x*exp(1)-x**2)/ln(-54*x**2/(exp(1)**2-2*x*exp(1)+x**2))**2,x)

[Out]

1/log(-54*x**2/(x**2 - 2*E*x + exp(2)))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.12 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {e}{{\left (3 i \, \pi + 3 \, \log \left (3\right ) + \log \left (2\right )\right )} e - 2 \, e \log \left (x - e\right ) + 2 \, e \log \left (x\right )} \]

[In]

integrate(-2*exp(1)/(x*exp(1)-x^2)/log(-54*x^2/(exp(1)^2-2*x*exp(1)+x^2))^2,x, algorithm="maxima")

[Out]

e/((3*I*pi + 3*log(3) + log(2))*e - 2*e*log(x - e) + 2*e*log(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\log \left (-\frac {54 \, x^{2}}{x^{2} - 2 \, x e + e^{2}}\right )} \]

[In]

integrate(-2*exp(1)/(x*exp(1)-x^2)/log(-54*x^2/(exp(1)^2-2*x*exp(1)+x^2))^2,x, algorithm="giac")

[Out]

1/log(-54*x^2/(x^2 - 2*x*e + e^2))

Mupad [B] (verification not implemented)

Time = 13.12 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int -\frac {2 e}{\left (e x-x^2\right ) \log ^2\left (-\frac {54 x^2}{e^2-2 e x+x^2}\right )} \, dx=\frac {1}{\ln \left (\frac {54\,x^2}{{\left (x-\mathrm {e}\right )}^2}\right )+\pi \,1{}\mathrm {i}} \]

[In]

int(-(2*exp(1))/(log(-(54*x^2)/(exp(2) - 2*x*exp(1) + x^2))^2*(x*exp(1) - x^2)),x)

[Out]

1/(pi*1i + log((54*x^2)/(x - exp(1))^2))