\(\int \frac {e^{\frac {-75-e^{e^3}+4 x+2 \log (4)}{x}} (75+e^{e^3}+x-2 \log (4))}{x} \, dx\) [7470]

Optimal result

Integrand size = 38, antiderivative size = 27 \[ \int \frac {e^{\frac {-75-e^{e^3}+4 x+2 \log (4)}{x}} \left (75+e^{e^3}+x-2 \log (4)\right )}{x} \, dx=e^{\frac {-25-e^{e^3}+2 x+2 (-25+x+\log (4))}{x}} x \]

[Out]

exp((-exp(exp(3))+4*ln(2)+4*x-75)/x)*x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(27)=54\).

Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2326} \[ \int \frac {e^{\frac {-75-e^{e^3}+4 x+2 \log (4)}{x}} \left (75+e^{e^3}+x-2 \log (4)\right )}{x} \, dx=\frac {4^{2/x} e^{-\frac {-4 x+e^{e^3}+75}{x}} \left (75+e^{e^3}-2 \log (4)\right )}{x \left (\frac {-4 x+e^{e^3}+75-\log (16)}{x^2}+\frac {4}{x}\right )} \]

[In]

Int[(E^((-75 - E^E^3 + 4*x + 2*Log[4])/x)*(75 + E^E^3 + x - 2*Log[4]))/x,x]

[Out]

(4^(2/x)*(75 + E^E^3 - 2*Log[4]))/(E^((75 + E^E^3 - 4*x)/x)*x*(4/x + (75 + E^E^3 - 4*x - Log[16])/x^2))

Rule 2326

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{align*} \text {integral}& = \frac {4^{2/x} e^{-\frac {75+e^{e^3}-4 x}{x}} \left (75+e^{e^3}-2 \log (4)\right )}{x \left (\frac {4}{x}+\frac {75+e^{e^3}-4 x-\log (16)}{x^2}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\frac {-75-e^{e^3}+4 x+2 \log (4)}{x}} \left (75+e^{e^3}+x-2 \log (4)\right )}{x} \, dx=16^{\frac {1}{x}} e^{-\frac {75+e^{e^3}-4 x}{x}} x \]

[In]

Integrate[(E^((-75 - E^E^3 + 4*x + 2*Log[4])/x)*(75 + E^E^3 + x - 2*Log[4]))/x,x]

[Out]

(16^x^(-1)*x)/E^((75 + E^E^3 - 4*x)/x)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81

method	result	size
gosper	\({\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )+4 x -75}{x}} x\)	\(22\)
norman	\({\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )+4 x -75}{x}} x\)	\(22\)
risch	\({\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )+4 x -75}{x}} x\)	\(22\)
parallelrisch	\({\mathrm e}^{\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )+4 x -75}{x}} x\)	\(22\)
derivativedivides	\({\mathrm e}^{{\mathrm e}^{3}} {\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}\right )-\frac {75 \,{\mathrm e}^{4+\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}} x}{-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}+{\mathrm e}^{{\mathrm e}^{3}} \left (-\frac {{\mathrm e}^{4+\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}} x}{-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}-{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}\right )\right )-4 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{4+\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}} x}{-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}-{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}\right )\right )-4 \ln \left (2\right ) {\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}\right )\)	\(206\)
default	\({\mathrm e}^{{\mathrm e}^{3}} {\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}\right )-\frac {75 \,{\mathrm e}^{4+\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}} x}{-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}+{\mathrm e}^{{\mathrm e}^{3}} \left (-\frac {{\mathrm e}^{4+\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}} x}{-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}-{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}\right )\right )-4 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{4+\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}} x}{-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}-{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}\right )\right )-4 \ln \left (2\right ) {\mathrm e}^{4} \operatorname {Ei}_{1}\left (-\frac {-{\mathrm e}^{{\mathrm e}^{3}}+4 \ln \left (2\right )-75}{x}\right )\)	\(206\)
meijerg	\(-{\mathrm e}^{4+{\mathrm e}^{3}} \left (-\ln \left (x \right )+\ln \left ({\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75\right )-\ln \left (\frac {{\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75}{x}\right )-\operatorname {Ei}_{1}\left (\frac {{\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75}{x}\right )\right )+4 \ln \left (2\right ) {\mathrm e}^{4} \left (-\ln \left (x \right )+\ln \left ({\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75\right )-\ln \left (\frac {{\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75}{x}\right )-\operatorname {Ei}_{1}\left (\frac {{\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75}{x}\right )\right )-{\mathrm e}^{4} \left ({\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75\right ) \left (-\frac {x}{{\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75}+1+\ln \left (x \right )-\ln \left ({\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75\right )+\frac {x \left (2-\frac {2 \left ({\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75\right )}{x}\right )}{2 \,{\mathrm e}^{{\mathrm e}^{3}}-8 \ln \left (2\right )+150}-\frac {x \,{\mathrm e}^{-\frac {{\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75}{x}}}{{\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75}+\ln \left (\frac {{\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75}{x}\right )+\operatorname {Ei}_{1}\left (\frac {{\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75}{x}\right )\right )-75 \,{\mathrm e}^{4} \left (-\ln \left (x \right )+\ln \left ({\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75\right )-\ln \left (\frac {{\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75}{x}\right )-\operatorname {Ei}_{1}\left (\frac {{\mathrm e}^{{\mathrm e}^{3}}-4 \ln \left (2\right )+75}{x}\right )\right )\)	\(294\)

[In]

int((exp(exp(3))-4*ln(2)+x+75)*exp((-exp(exp(3))+4*ln(2)+4*x-75)/x)/x,x,method=_RETURNVERBOSE)

[Out]

exp((-exp(exp(3))+4*ln(2)+4*x-75)/x)*x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\frac {-75-e^{e^3}+4 x+2 \log (4)}{x}} \left (75+e^{e^3}+x-2 \log (4)\right )}{x} \, dx=x e^{\left (\frac {4 \, x - e^{\left (e^{3}\right )} + 4 \, \log \left (2\right ) - 75}{x}\right )} \]

[In]

integrate((exp(exp(3))-4*log(2)+x+75)*exp((-exp(exp(3))+4*log(2)+4*x-75)/x)/x,x, algorithm="fricas")

[Out]

x*e^((4*x - e^(e^3) + 4*log(2) - 75)/x)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\frac {-75-e^{e^3}+4 x+2 \log (4)}{x}} \left (75+e^{e^3}+x-2 \log (4)\right )}{x} \, dx=x e^{\frac {4 x - e^{e^{3}} - 75 + 4 \log {\left (2 \right )}}{x}} \]

[In]

integrate((exp(exp(3))-4*ln(2)+x+75)*exp((-exp(exp(3))+4*ln(2)+4*x-75)/x)/x,x)

[Out]

x*exp((4*x - exp(exp(3)) - 75 + 4*log(2))/x)

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {-75-e^{e^3}+4 x+2 \log (4)}{x}} \left (75+e^{e^3}+x-2 \log (4)\right )}{x} \, dx=x e^{\left (-\frac {e^{\left (e^{3}\right )}}{x} + \frac {4 \, \log \left (2\right )}{x} - \frac {75}{x} + 4\right )} \]

[In]

integrate((exp(exp(3))-4*log(2)+x+75)*exp((-exp(exp(3))+4*log(2)+4*x-75)/x)/x,x, algorithm="maxima")

[Out]

x*e^(-e^(e^3)/x + 4*log(2)/x - 75/x + 4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (21) = 42\).

Time = 0.70 (sec) , antiderivative size = 179, normalized size of antiderivative = 6.63 \[ \int \frac {e^{\frac {-75-e^{e^3}+4 x+2 \log (4)}{x}} \left (75+e^{e^3}+x-2 \log (4)\right )}{x} \, dx=-\frac {16 \, e^{\left (\frac {4 \, x - e^{\left (e^{3}\right )} + 4 \, \log \left (2\right ) - 75}{x}\right )} \log \left (2\right )^{2} - 8 \, e^{\left (\frac {4 \, x - e^{\left (e^{3}\right )} + 4 \, \log \left (2\right ) - 75}{x} + e^{3}\right )} \log \left (2\right ) - 600 \, e^{\left (\frac {4 \, x - e^{\left (e^{3}\right )} + 4 \, \log \left (2\right ) - 75}{x}\right )} \log \left (2\right ) + e^{\left (\frac {4 \, x - e^{\left (e^{3}\right )} + 4 \, \log \left (2\right ) - 75}{x} + 2 \, e^{3}\right )} + 150 \, e^{\left (\frac {4 \, x - e^{\left (e^{3}\right )} + 4 \, \log \left (2\right ) - 75}{x} + e^{3}\right )} + 5625 \, e^{\left (\frac {4 \, x - e^{\left (e^{3}\right )} + 4 \, \log \left (2\right ) - 75}{x}\right )}}{{\left (\frac {4 \, x - e^{\left (e^{3}\right )} + 4 \, \log \left (2\right ) - 75}{x} - 4\right )} {\left (e^{\left (e^{3}\right )} - 4 \, \log \left (2\right ) + 75\right )}} \]

[In]

integrate((exp(exp(3))-4*log(2)+x+75)*exp((-exp(exp(3))+4*log(2)+4*x-75)/x)/x,x, algorithm="giac")

[Out]

-(16*e^((4*x - e^(e^3) + 4*log(2) - 75)/x)*log(2)^2 - 8*e^((4*x - e^(e^3) + 4*log(2) - 75)/x + e^3)*log(2) - 6
00*e^((4*x - e^(e^3) + 4*log(2) - 75)/x)*log(2) + e^((4*x - e^(e^3) + 4*log(2) - 75)/x + 2*e^3) + 150*e^((4*x
- e^(e^3) + 4*log(2) - 75)/x + e^3) + 5625*e^((4*x - e^(e^3) + 4*log(2) - 75)/x))/(((4*x - e^(e^3) + 4*log(2)
- 75)/x - 4)*(e^(e^3) - 4*log(2) + 75))

Mupad [B] (verification not implemented)

Time = 13.43 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.11 \[ \int \frac {e^{\frac {-75-e^{e^3}+4 x+2 \log (4)}{x}} \left (75+e^{e^3}+x-2 \log (4)\right )}{x} \, dx=75\,{\mathrm {e}}^4\,\mathrm {expint}\left (\frac {{\mathrm {e}}^{{\mathrm {e}}^3}-\ln \left (16\right )+75}{x}\right )+{\mathrm {e}}^{{\mathrm {e}}^3+4}\,\mathrm {expint}\left (\frac {{\mathrm {e}}^{{\mathrm {e}}^3}-\ln \left (16\right )+75}{x}\right )-{\mathrm {e}}^4\,\mathrm {expint}\left (\frac {{\mathrm {e}}^{{\mathrm {e}}^3}-\ln \left (16\right )+75}{x}\right )\,\left ({\mathrm {e}}^{{\mathrm {e}}^3}-\ln \left (16\right )+75\right )+2^{4/x}\,x\,{\mathrm {e}}^{4-\frac {75}{x}-\frac {{\mathrm {e}}^{{\mathrm {e}}^3}}{x}}-4\,{\mathrm {e}}^4\,\ln \left (2\right )\,\mathrm {expint}\left (\frac {{\mathrm {e}}^{{\mathrm {e}}^3}-4\,\ln \left (2\right )+75}{x}\right ) \]

[In]

int((exp((4*x + 4*log(2) - exp(exp(3)) - 75)/x)*(x - 4*log(2) + exp(exp(3)) + 75))/x,x)

[Out]

75*exp(4)*expint((exp(exp(3)) - log(16) + 75)/x) + exp(exp(3) + 4)*expint((exp(exp(3)) - log(16) + 75)/x) - ex
p(4)*expint((exp(exp(3)) - log(16) + 75)/x)*(exp(exp(3)) - log(16) + 75) + 2^(4/x)*x*exp(4 - 75/x - exp(exp(3)
)/x) - 4*exp(4)*log(2)*expint((exp(exp(3)) - 4*log(2) + 75)/x)