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

Optimal. Leaf size=18 \[ \frac {e^{2 e^{x^2}}}{(-x+\log (x))^2} \]

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Rubi [B]  time = 0.27, antiderivative size = 70, normalized size of antiderivative = 3.89, number of steps used = 1, number of rules used = 1, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {2288} \begin {gather*} \frac {e^{2 e^{x^2}-x^2} \left (e^{x^2} x^3-e^{x^2} x^2 \log (x)\right )}{x \left (x^4-3 x^3 \log (x)+3 x^2 \log ^2(x)-x \log ^3(x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(2*E^x^2 - x^2)*(E^x^2*x^3 - E^x^2*x^2*Log[x]))/(x*(x^4 - 3*x^3*Log[x] + 3*x^2*Log[x]^2 - x*Log[x]^3))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{2 e^{x^2}-x^2} \left (e^{x^2} x^3-e^{x^2} x^2 \log (x)\right )}{x \left (x^4-3 x^3 \log (x)+3 x^2 \log ^2(x)-x \log ^3(x)\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 18, normalized size = 1.00 \begin {gather*} \frac {e^{2 e^{x^2}}}{(x-\log (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2*E^x^2)/(x - Log[x])^2

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fricas [A]  time = 0.51, size = 23, normalized size = 1.28 \begin {gather*} \frac {e^{\left (2 \, e^{\left (x^{2}\right )}\right )}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (2 \, x^{3} e^{\left (x^{2}\right )} - 2 \, x^{2} e^{\left (x^{2}\right )} \log \relax (x) - x + 1\right )} e^{\left (2 \, e^{\left (x^{2}\right )}\right )}}{x^{4} - 3 \, x^{3} \log \relax (x) + 3 \, x^{2} \log \relax (x)^{2} - x \log \relax (x)^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (4 x^{2} {\mathrm e}^{x^{2}} \ln \relax (x )-4 x^{3} {\mathrm e}^{x^{2}}+2 x -2\right ) {\mathrm e}^{2 \,{\mathrm e}^{x^{2}}}}{x \ln \relax (x )^{3}-3 x^{2} \ln \relax (x )^{2}+3 x^{3} \ln \relax (x )-x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2*exp(x^2)*ln(x)-4*x^3*exp(x^2)+2*x-2)*exp(exp(x^2))^2/(x*ln(x)^3-3*x^2*ln(x)^2+3*x^3*ln(x)-x^4),x)

[Out]

int((4*x^2*exp(x^2)*ln(x)-4*x^3*exp(x^2)+2*x-2)*exp(exp(x^2))^2/(x*ln(x)^3-3*x^2*ln(x)^2+3*x^3*ln(x)-x^4),x)

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maxima [A]  time = 0.41, size = 23, normalized size = 1.28 \begin {gather*} \frac {e^{\left (2 \, e^{\left (x^{2}\right )}\right )}}{x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.63, size = 16, normalized size = 0.89 \begin {gather*} \frac {{\mathrm {e}}^{2\,{\mathrm {e}}^{x^2}}}{{\left (x-\ln \relax (x)\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*exp(x^2))*(2*x - 4*x^3*exp(x^2) + 4*x^2*exp(x^2)*log(x) - 2))/(x*log(x)^3 + 3*x^3*log(x) - 3*x^2*lo
g(x)^2 - x^4),x)

[Out]

exp(2*exp(x^2))/(x - log(x))^2

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sympy [A]  time = 0.37, size = 22, normalized size = 1.22 \begin {gather*} \frac {e^{2 e^{x^{2}}}}{x^{2} - 2 x \log {\relax (x )} + \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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