3.101.64 \(\int \frac {e^{3-e^{625}-x} (-1+2 x+e^{-3+e^{625}+x} x^2+(-1-x) \log (\frac {e^{2 x}}{x}))}{x^2} \, dx\)

Optimal. Leaf size=28 \[ x+\frac {e^{3-e^{625}-x} \log \left (\frac {e^{2 x}}{x}\right )}{x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.51, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6742, 2288} \begin {gather*} x+\frac {e^{-x-e^{625}+3} \log \left (\frac {e^{2 x}}{x}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(3 - E^625 - x)*(-1 + 2*x + E^(-3 + E^625 + x)*x^2 + (-1 - x)*Log[E^(2*x)/x]))/x^2,x]

[Out]

x + (E^(3 - E^625 - x)*Log[E^(2*x)/x])/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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {e^{3-e^{625}-x} \left (-1+2 x-\log \left (\frac {e^{2 x}}{x}\right )-x \log \left (\frac {e^{2 x}}{x}\right )\right )}{x^2}\right ) \, dx\\ &=x+\int \frac {e^{3-e^{625}-x} \left (-1+2 x-\log \left (\frac {e^{2 x}}{x}\right )-x \log \left (\frac {e^{2 x}}{x}\right )\right )}{x^2} \, dx\\ &=x+\frac {e^{3-e^{625}-x} \log \left (\frac {e^{2 x}}{x}\right )}{x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.21, size = 28, normalized size = 1.00 \begin {gather*} x+\frac {e^{3-e^{625}-x} \log \left (\frac {e^{2 x}}{x}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3 - E^625 - x)*(-1 + 2*x + E^(-3 + E^625 + x)*x^2 + (-1 - x)*Log[E^(2*x)/x]))/x^2,x]

[Out]

x + (E^(3 - E^625 - x)*Log[E^(2*x)/x])/x

________________________________________________________________________________________

fricas [A]  time = 0.93, size = 34, normalized size = 1.21 \begin {gather*} \frac {{\left (x^{2} e^{\left (x + e^{625} - 3\right )} + \log \left (\frac {e^{\left (2 \, x\right )}}{x}\right )\right )} e^{\left (-x - e^{625} + 3\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-1)*log(exp(x)^2/x)+x^2*exp(exp(625)+x-3)+2*x-1)/x^2/exp(exp(625)+x-3),x, algorithm="fricas")

[Out]

(x^2*e^(x + e^625 - 3) + log(e^(2*x)/x))*e^(-x - e^625 + 3)/x

________________________________________________________________________________________

giac [A]  time = 0.16, size = 35, normalized size = 1.25 \begin {gather*} \frac {x^{2} + 2 \, x e^{\left (-x - e^{625} + 3\right )} - e^{\left (-x - e^{625} + 3\right )} \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-1)*log(exp(x)^2/x)+x^2*exp(exp(625)+x-3)+2*x-1)/x^2/exp(exp(625)+x-3),x, algorithm="giac")

[Out]

(x^2 + 2*x*e^(-x - e^625 + 3) - e^(-x - e^625 + 3)*log(x))/x

________________________________________________________________________________________

maple [A]  time = 0.19, size = 24, normalized size = 0.86

method	result	size
default	\(x +\frac {\ln \left (\frac {{\mathrm e}^{2 x}}{x}\right ) {\mathrm e}^{-{\mathrm e}^{625}-x +3}}{x}\)	\(24\)
norman	\(\frac {\left ({\mathrm e}^{x} x^{2}+{\mathrm e}^{-{\mathrm e}^{625}} {\mathrm e}^{3} \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )\right ) {\mathrm e}^{-x}}{x}\)	\(35\)
risch	\(\frac {2 \ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{-{\mathrm e}^{625}-x +3}}{x}-\frac {\left (-2 x^{2} {\mathrm e}^{{\mathrm e}^{625}+x -3}+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )-2 i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}+i \pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{3}+2 \ln \relax (x )\right ) {\mathrm e}^{-{\mathrm e}^{625}-x +3}}{2 x}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x-1)*ln(exp(x)^2/x)+x^2*exp(exp(625)+x-3)+2*x-1)/x^2/exp(exp(625)+x-3),x,method=_RETURNVERBOSE)

[Out]

x+ln(exp(x)^2/x)/exp(exp(625)+x-3)/x

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, {\rm Ei}\left (-x\right ) e^{\left (-e^{625} + 3\right )} + e^{\left (-e^{625} + 3\right )} \Gamma \left (-1, x\right ) + x - \frac {e^{\left (-x - e^{625} + 3\right )} \log \relax (x)}{x} - \int \frac {{\left (2 \, x^{2} e^{3} + 2 \, x e^{3} - e^{3}\right )} e^{\left (-x - e^{625}\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-1)*log(exp(x)^2/x)+x^2*exp(exp(625)+x-3)+2*x-1)/x^2/exp(exp(625)+x-3),x, algorithm="maxima")

[Out]

2*Ei(-x)*e^(-e^625 + 3) + e^(-e^625 + 3)*gamma(-1, x) + x - e^(-x - e^625 + 3)*log(x)/x - integrate((2*x^2*e^3
+ 2*x*e^3 - e^3)*e^(-x - e^625)/x^2, x)

________________________________________________________________________________________

mupad [B]  time = 6.21, size = 34, normalized size = 1.21 \begin {gather*} x+2\,{\mathrm {e}}^{-{\mathrm {e}}^{625}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3+\frac {\ln \left (\frac {1}{x}\right )\,{\mathrm {e}}^{-{\mathrm {e}}^{625}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3 - exp(625) - x)*(2*x - log(exp(2*x)/x)*(x + 1) + x^2*exp(x + exp(625) - 3) - 1))/x^2,x)

[Out]

x + 2*exp(-exp(625))*exp(-x)*exp(3) + (log(1/x)*exp(-exp(625))*exp(-x)*exp(3))/x

________________________________________________________________________________________

sympy [A]  time = 0.29, size = 27, normalized size = 0.96 \begin {gather*} x + \frac {e^{3} \log {\left (\frac {e^{2 x}}{x} \right )}}{x \sqrt {e^{2 x}} e^{e^{625}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-1)*ln(exp(x)**2/x)+x**2*exp(exp(625)+x-3)+2*x-1)/x**2/exp(exp(625)+x-3),x)

[Out]

x + exp(3)*exp(-exp(625))*log(exp(2*x)/x)/(x*sqrt(exp(2*x)))

________________________________________________________________________________________