\(\int \frac {e-e \log (x)}{3 e x^2+e^{5+4 e^7} x^2} \, dx\) [7652]

Optimal result

Integrand size = 30, antiderivative size = 19 \[ \int \frac {e-e \log (x)}{3 e x^2+e^{5+4 e^7} x^2} \, dx=\frac {\log (x)}{\left (3+e^{4+4 e^7}\right ) x} \]

[Out]

ln(x)/x/(exp(5)/exp(1)*exp(exp(7))^4+3)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6, 12, 2340} \[ \int \frac {e-e \log (x)}{3 e x^2+e^{5+4 e^7} x^2} \, dx=\frac {\log (x)}{\left (3+e^{4+4 e^7}\right ) x} \]

[In]

Int[(E - E*Log[x])/(3*E*x^2 + E^(5 + 4*E^7)*x^2),x]

[Out]

Log[x]/((3 + E^(4 + 4*E^7))*x)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

Rule 2340

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[b*(d*x)^(m + 1)*(Log[c*x^n]/(d
*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && EqQ[a*(m + 1) - b*n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e-e \log (x)}{\left (3 e+e^{5+4 e^7}\right ) x^2} \, dx \\ & = \frac {\int \frac {e-e \log (x)}{x^2} \, dx}{3 e+e^{5+4 e^7}} \\ & = \frac {\log (x)}{\left (3+e^{4+4 e^7}\right ) x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e-e \log (x)}{3 e x^2+e^{5+4 e^7} x^2} \, dx=\frac {\log (x)}{\left (3+e^{4+4 e^7}\right ) x} \]

[In]

Integrate[(E - E*Log[x])/(3*E*x^2 + E^(5 + 4*E^7)*x^2),x]

[Out]

Log[x]/((3 + E^(4 + 4*E^7))*x)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
risch	\(\frac {\ln \left (x \right )}{x \left ({\mathrm e}^{4+4 \,{\mathrm e}^{7}}+3\right )}\)	\(18\)
norman	\(\frac {{\mathrm e} \ln \left (x \right )}{\left ({\mathrm e}^{5} {\mathrm e}^{4 \,{\mathrm e}^{7}}+3 \,{\mathrm e}\right ) x}\)	\(24\)
parallelrisch	\(\frac {{\mathrm e} \ln \left (x \right )}{\left ({\mathrm e}^{5} {\mathrm e}^{4 \,{\mathrm e}^{7}}+3 \,{\mathrm e}\right ) x}\)	\(24\)
default	\(-\frac {{\mathrm e} \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )}{{\mathrm e}^{5} {\mathrm e}^{4 \,{\mathrm e}^{7}}+3 \,{\mathrm e}}-\frac {{\mathrm e}}{\left ({\mathrm e}^{5} {\mathrm e}^{4 \,{\mathrm e}^{7}}+3 \,{\mathrm e}\right ) x}\)	\(56\)
parts	\(-\frac {{\mathrm e} \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )}{{\mathrm e}^{5} {\mathrm e}^{4 \,{\mathrm e}^{7}}+3 \,{\mathrm e}}-\frac {{\mathrm e}}{\left ({\mathrm e}^{5} {\mathrm e}^{4 \,{\mathrm e}^{7}}+3 \,{\mathrm e}\right ) x}\)	\(56\)

[In]

int((-exp(1)*ln(x)+exp(1))/(x^2*exp(5)*exp(exp(7))^4+3*x^2*exp(1)),x,method=_RETURNVERBOSE)

[Out]

1/x/(exp(4+4*exp(7))+3)*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {e-e \log (x)}{3 e x^2+e^{5+4 e^7} x^2} \, dx=\frac {e \log \left (x\right )}{3 \, x e + x e^{\left (4 \, e^{7} + 5\right )}} \]

[In]

integrate((-exp(1)*log(x)+exp(1))/(x^2*exp(5)*exp(exp(7))^4+3*x^2*exp(1)),x, algorithm="fricas")

[Out]

e*log(x)/(3*x*e + x*e^(4*e^7 + 5))

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {e-e \log (x)}{3 e x^2+e^{5+4 e^7} x^2} \, dx=\frac {\log {\left (x \right )}}{3 x + x e^{4} e^{4 e^{7}}} \]

[In]

integrate((-exp(1)*ln(x)+exp(1))/(x**2*exp(5)*exp(exp(7))**4+3*x**2*exp(1)),x)

[Out]

log(x)/(3*x + x*exp(4)*exp(4*exp(7)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {e-e \log (x)}{3 e x^2+e^{5+4 e^7} x^2} \, dx=\frac {\log \left (x\right ) + 1}{x {\left (e^{\left (4 \, e^{7} + 4\right )} + 3\right )}} - \frac {1}{x {\left (e^{\left (4 \, e^{7} + 4\right )} + 3\right )}} \]

[In]

integrate((-exp(1)*log(x)+exp(1))/(x^2*exp(5)*exp(exp(7))^4+3*x^2*exp(1)),x, algorithm="maxima")

[Out]

(log(x) + 1)/(x*(e^(4*e^7 + 4) + 3)) - 1/(x*(e^(4*e^7 + 4) + 3))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {e-e \log (x)}{3 e x^2+e^{5+4 e^7} x^2} \, dx=\frac {e \log \left (x\right )}{3 \, x e + x e^{\left (4 \, e^{7} + 5\right )}} \]

[In]

integrate((-exp(1)*log(x)+exp(1))/(x^2*exp(5)*exp(exp(7))^4+3*x^2*exp(1)),x, algorithm="giac")

[Out]

e*log(x)/(3*x*e + x*e^(4*e^7 + 5))

Mupad [B] (verification not implemented)

Time = 12.98 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {e-e \log (x)}{3 e x^2+e^{5+4 e^7} x^2} \, dx=\frac {\mathrm {e}\,\ln \left (x\right )}{x\,\left (3\,\mathrm {e}+{\mathrm {e}}^{4\,{\mathrm {e}}^7+5}\right )} \]

[In]

int((exp(1) - exp(1)*log(x))/(3*x^2*exp(1) + x^2*exp(4*exp(7))*exp(5)),x)

[Out]

(exp(1)*log(x))/(x*(3*exp(1) + exp(4*exp(7) + 5)))