3.16.73 \(\int \frac {12 e^2+6 e x}{(3 e^2 x+3 e x^2-x^3) \log ^3(\frac {3 e^2+3 e x-x^2}{3 x^2})} \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{\log ^2\left (\frac {e+\frac {e^2}{x}-\frac {x}{3}}{x}\right )} \]

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Rubi [A]  time = 0.20, antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1594, 6686} \begin {gather*} \frac {1}{\log ^2\left (\frac {-x^2+3 e x+3 e^2}{3 x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12*E^2 + 6*E*x)/((3*E^2*x + 3*E*x^2 - x^3)*Log[(3*E^2 + 3*E*x - x^2)/(3*x^2)]^3),x]

[Out]

Log[(3*E^2 + 3*E*x - x^2)/(3*x^2)]^(-2)

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - 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} &=\int \frac {12 e^2+6 e x}{x \left (3 e^2+3 e x-x^2\right ) \log ^3\left (\frac {3 e^2+3 e x-x^2}{3 x^2}\right )} \, dx\\ &=\frac {1}{\log ^2\left (\frac {3 e^2+3 e x-x^2}{3 x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 19, normalized size = 0.90 \begin {gather*} \frac {1}{\log ^2\left (-\frac {1}{3}+\frac {e^2}{x^2}+\frac {e}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12*E^2 + 6*E*x)/((3*E^2*x + 3*E*x^2 - x^3)*Log[(3*E^2 + 3*E*x - x^2)/(3*x^2)]^3),x]

[Out]

Log[-1/3 + E^2/x^2 + E/x]^(-2)

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fricas [A]  time = 0.82, size = 21, normalized size = 1.00 \begin {gather*} \frac {1}{\log \left (-\frac {x^{2} - 3 \, x e - 3 \, e^{2}}{3 \, x^{2}}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*exp(2)+6*x*exp(1))/(3*exp(2)*x+3*x^2*exp(1)-x^3)/log(1/3*(3*exp(2)+3*x*exp(1)-x^2)/x^2)^3,x, alg
orithm="fricas")

[Out]

log(-1/3*(x^2 - 3*x*e - 3*e^2)/x^2)^(-2)

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giac [A]  time = 0.40, size = 21, normalized size = 1.00 \begin {gather*} \frac {1}{\log \left (-\frac {x^{2} - 3 \, x e - 3 \, e^{2}}{3 \, x^{2}}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*exp(2)+6*x*exp(1))/(3*exp(2)*x+3*x^2*exp(1)-x^3)/log(1/3*(3*exp(2)+3*x*exp(1)-x^2)/x^2)^3,x, alg
orithm="giac")

[Out]

log(-1/3*(x^2 - 3*x*e - 3*e^2)/x^2)^(-2)

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maple [A]  time = 0.10, size = 24, normalized size = 1.14

method	result	size
norman	\(\frac {1}{\ln \left (\frac {3 \,{\mathrm e}^{2}+3 x \,{\mathrm e}-x^{2}}{3 x^{2}}\right )^{2}}\)	\(24\)
risch	\(\frac {1}{\ln \left (\frac {3 \,{\mathrm e}^{2}+3 x \,{\mathrm e}-x^{2}}{3 x^{2}}\right )^{2}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*exp(2)+6*x*exp(1))/(3*exp(2)*x+3*x^2*exp(1)-x^3)/ln(1/3*(3*exp(2)+3*x*exp(1)-x^2)/x^2)^3,x,method=_RET
URNVERBOSE)

[Out]

1/ln(1/3*(3*exp(2)+3*x*exp(1)-x^2)/x^2)^2

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maxima [B]  time = 0.58, size = 62, normalized size = 2.95 \begin {gather*} \frac {1}{\log \relax (3)^{2} - 2 \, {\left (\log \relax (3) + 2 \, \log \relax (x)\right )} \log \left (-x^{2} + 3 \, x e + 3 \, e^{2}\right ) + \log \left (-x^{2} + 3 \, x e + 3 \, e^{2}\right )^{2} + 4 \, \log \relax (3) \log \relax (x) + 4 \, \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*exp(2)+6*x*exp(1))/(3*exp(2)*x+3*x^2*exp(1)-x^3)/log(1/3*(3*exp(2)+3*x*exp(1)-x^2)/x^2)^3,x, alg
orithm="maxima")

[Out]

1/(log(3)^2 - 2*(log(3) + 2*log(x))*log(-x^2 + 3*x*e + 3*e^2) + log(-x^2 + 3*x*e + 3*e^2)^2 + 4*log(3)*log(x)
+ 4*log(x)^2)

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mupad [B]  time = 1.92, size = 23, normalized size = 1.10 \begin {gather*} \frac {1}{{\ln \left (\frac {-x^2+3\,\mathrm {e}\,x+3\,{\mathrm {e}}^2}{3\,x^2}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*exp(2) + 6*x*exp(1))/(log((exp(2) + x*exp(1) - x^2/3)/x^2)^3*(3*x*exp(2) + 3*x^2*exp(1) - x^3)),x)

[Out]

1/log((3*exp(2) + 3*x*exp(1) - x^2)/(3*x^2))^2

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sympy [A]  time = 0.17, size = 20, normalized size = 0.95 \begin {gather*} \frac {1}{\log {\left (\frac {- \frac {x^{2}}{3} + e x + e^{2}}{x^{2}} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*exp(2)+6*x*exp(1))/(3*exp(2)*x+3*x**2*exp(1)-x**3)/ln(1/3*(3*exp(2)+3*x*exp(1)-x**2)/x**2)**3,x)

[Out]

log((-x**2/3 + E*x + exp(2))/x**2)**(-2)

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