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

Optimal. Leaf size=19 \[ \frac {\log (x)}{\left (3+e^{4+4 e^7}\right ) x} \]

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Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6, 12, 2303} \begin {gather*} \frac {\log (x)}{\left (3+e^{4+4 e^7}\right ) x} \end {gather*}

Antiderivative was successfully verified.

[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 2303

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

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{\left (3+e^{4+4 e^7}\right ) x} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 1.88, size = 22, normalized size = 1.16 \begin {gather*} \frac {e \log \relax (x)}{3 \, x e + x e^{\left (4 \, e^{7} + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.12, size = 22, normalized size = 1.16 \begin {gather*} \frac {e \log \relax (x)}{3 \, x e + x e^{\left (4 \, e^{7} + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.22, size = 18, normalized size = 0.95

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [B]  time = 0.35, size = 36, normalized size = 1.89 \begin {gather*} \frac {\log \relax (x) + 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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 5.33, size = 22, normalized size = 1.16 \begin {gather*} \frac {\mathrm {e}\,\ln \relax (x)}{x\,\left (3\,\mathrm {e}+{\mathrm {e}}^{4\,{\mathrm {e}}^7+5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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sympy [A]  time = 0.16, size = 17, normalized size = 0.89 \begin {gather*} \frac {\log {\relax (x )}}{3 x + x e^{4} e^{4 e^{7}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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