3.92.93 \(\int \frac {e^{-e^x} (-4 \log (\frac {7}{2})-4 e^x x \log (\frac {7}{2})+e^{4 x} (16 \log (\frac {7}{2})+4 e^x \log (\frac {7}{2})))}{e^{8 x}-2 e^{4 x} x+x^2} \, dx\)

Optimal. Leaf size=24 \[ \frac {4 e^{-e^x} \log \left (\frac {7}{2}\right )}{-e^{4 x}+x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.54, antiderivative size = 40, normalized size of antiderivative = 1.67, number of steps used = 3, number of rules used = 3, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6688, 12, 2288} \begin {gather*} -\frac {4 e^{-x-e^x} \left (e^{5 x}-e^x x\right ) \log \left (\frac {7}{2}\right )}{\left (e^{4 x}-x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*Log[7/2] - 4*E^x*x*Log[7/2] + E^(4*x)*(16*Log[7/2] + 4*E^x*Log[7/2]))/(E^E^x*(E^(8*x) - 2*E^(4*x)*x +
x^2)),x]

[Out]

(-4*E^(-E^x - x)*(E^(5*x) - E^x*x)*Log[7/2])/(E^(4*x) - x)^2

Rule 12

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

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]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^{-e^x} \left (-1+4 e^{4 x}+e^{5 x}-e^x x\right ) \log \left (\frac {7}{2}\right )}{\left (e^{4 x}-x\right )^2} \, dx\\ &=\left (4 \log \left (\frac {7}{2}\right )\right ) \int \frac {e^{-e^x} \left (-1+4 e^{4 x}+e^{5 x}-e^x x\right )}{\left (e^{4 x}-x\right )^2} \, dx\\ &=-\frac {4 e^{-e^x-x} \left (e^{5 x}-e^x x\right ) \log \left (\frac {7}{2}\right )}{\left (e^{4 x}-x\right )^2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 24, normalized size = 1.00 \begin {gather*} -\frac {4 e^{-e^x} \log \left (\frac {7}{2}\right )}{e^{4 x}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*Log[7/2] - 4*E^x*x*Log[7/2] + E^(4*x)*(16*Log[7/2] + 4*E^x*Log[7/2]))/(E^E^x*(E^(8*x) - 2*E^(4*x
)*x + x^2)),x]

[Out]

(-4*Log[7/2])/(E^E^x*(E^(4*x) - x))

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 19, normalized size = 0.79 \begin {gather*} \frac {4 \, e^{\left (-e^{x}\right )} \log \left (\frac {7}{2}\right )}{x - e^{\left (4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(7/2)*exp(x)+16*log(7/2))*exp(2*x)^2-4*x*log(7/2)*exp(x)-4*log(7/2))/(exp(2*x)^4-2*x*exp(2*x)
^2+x^2)/exp(exp(x)),x, algorithm="fricas")

[Out]

4*e^(-e^x)*log(7/2)/(x - e^(4*x))

________________________________________________________________________________________

giac [B]  time = 0.18, size = 39, normalized size = 1.62 \begin {gather*} \frac {4 \, {\left (e^{\left (4 \, x\right )} \log \relax (7) - e^{\left (4 \, x\right )} \log \relax (2)\right )}}{x e^{\left (4 \, x + e^{x}\right )} - e^{\left (8 \, x + e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(7/2)*exp(x)+16*log(7/2))*exp(2*x)^2-4*x*log(7/2)*exp(x)-4*log(7/2))/(exp(2*x)^4-2*x*exp(2*x)
^2+x^2)/exp(exp(x)),x, algorithm="giac")

[Out]

4*(e^(4*x)*log(7) - e^(4*x)*log(2))/(x*e^(4*x + e^x) - e^(8*x + e^x))

________________________________________________________________________________________

maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (4 \ln \left (\frac {7}{2}\right ) {\mathrm e}^{x}+16 \ln \left (\frac {7}{2}\right )\right ) {\mathrm e}^{4 x}-4 x \ln \left (\frac {7}{2}\right ) {\mathrm e}^{x}-4 \ln \left (\frac {7}{2}\right )\right ) {\mathrm e}^{-{\mathrm e}^{x}}}{{\mathrm e}^{8 x}-2 x \,{\mathrm e}^{4 x}+x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*ln(7/2)*exp(x)+16*ln(7/2))*exp(2*x)^2-4*x*ln(7/2)*exp(x)-4*ln(7/2))/(exp(2*x)^4-2*x*exp(2*x)^2+x^2)/ex
p(exp(x)),x)

[Out]

int(((4*ln(7/2)*exp(x)+16*ln(7/2))*exp(2*x)^2-4*x*ln(7/2)*exp(x)-4*ln(7/2))/(exp(2*x)^4-2*x*exp(2*x)^2+x^2)/ex
p(exp(x)),x)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 24, normalized size = 1.00 \begin {gather*} \frac {4 \, {\left (\log \relax (7) - \log \relax (2)\right )} e^{\left (-e^{x}\right )}}{x - e^{\left (4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(7/2)*exp(x)+16*log(7/2))*exp(2*x)^2-4*x*log(7/2)*exp(x)-4*log(7/2))/(exp(2*x)^4-2*x*exp(2*x)
^2+x^2)/exp(exp(x)),x, algorithm="maxima")

[Out]

4*(log(7) - log(2))*e^(-e^x)/(x - e^(4*x))

________________________________________________________________________________________

mupad [B]  time = 0.39, size = 19, normalized size = 0.79 \begin {gather*} \frac {4\,{\mathrm {e}}^{-{\mathrm {e}}^x}\,\ln \left (\frac {7}{2}\right )}{x-{\mathrm {e}}^{4\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-exp(x))*(4*log(7/2) - exp(4*x)*(16*log(7/2) + 4*exp(x)*log(7/2)) + 4*x*exp(x)*log(7/2)))/(exp(8*x)
- 2*x*exp(4*x) + x^2),x)

[Out]

(4*exp(-exp(x))*log(7/2))/(x - exp(4*x))

________________________________________________________________________________________

sympy [A]  time = 0.23, size = 20, normalized size = 0.83 \begin {gather*} \frac {\left (- 4 \log {\relax (2 )} + 4 \log {\relax (7 )}\right ) e^{- e^{x}}}{x - e^{4 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*ln(7/2)*exp(x)+16*ln(7/2))*exp(2*x)**2-4*x*ln(7/2)*exp(x)-4*ln(7/2))/(exp(2*x)**4-2*x*exp(2*x)**
2+x**2)/exp(exp(x)),x)

[Out]

(-4*log(2) + 4*log(7))*exp(-exp(x))/(x - exp(4*x))

________________________________________________________________________________________