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

Optimal. Leaf size=17 \[ 2+\frac {1}{\log \left (-\frac {54 x^2}{(e-x)^2}\right )} \]

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Rubi [A]  time = 0.08, antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 1593, 6686} \begin {gather*} \frac {1}{\log \left (-\frac {54 x^2}{x^2-2 e x+e^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[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 1593

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 6686

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{\log \left (-\frac {54 x^2}{(e-x)^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.95, size = 47, normalized size = 2.76 \begin {gather*} \frac {\log \left (\frac {54 \, x^{2}}{x^{2} - 2 \, x e + e^{2}}\right )}{\pi ^{2} + \log \left (\frac {54 \, x^{2}}{x^{2} - 2 \, x e + e^{2}}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

log(54*x^2/(x^2 - 2*x*e + e^2))/(pi^2 + log(54*x^2/(x^2 - 2*x*e + e^2))^2)

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giac [A]  time = 0.13, size = 21, normalized size = 1.24 \begin {gather*} \frac {1}{\log \left (-\frac {54 \, x^{2}}{x^{2} - 2 \, x e + e^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.54, size = 22, normalized size = 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	\(\frac {1}{\ln \left (-\frac {54 x^{2}}{{\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}}\right )}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [C]  time = 0.49, size = 36, normalized size = 2.12 \begin {gather*} \frac {e}{{\left (3 i \, \pi + 3 \, \log \relax (3) + \log \relax (2)\right )} e - 2 \, e \log \left (x - e\right ) + 2 \, e \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 9.05, size = 21, normalized size = 1.24 \begin {gather*} \frac {1}{\ln \left (\frac {54\,x^2}{{\left (x-\mathrm {e}\right )}^2}\right )+\pi \,1{}\mathrm {i}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.13, size = 22, normalized size = 1.29 \begin {gather*} \frac {1}{\log {\left (- \frac {54 x^{2}}{x^{2} - 2 e x + e^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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