\(\int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} (x^2+x^{\frac {e^3+4 x}{x}} (-e^3-4 x+e^3 \log (x)))}{x^2} \, dx\) [6926]

Optimal result

Integrand size = 56, antiderivative size = 21 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=-e^{-x+x^{\frac {e^3+4 x}{x}}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (e^{-x+x^{\frac {e^3+4 x}{x}}}-e^{-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \left (e^3+4 x-e^3 \log (x)\right )\right ) \, dx \\ & = \int e^{-x+x^{\frac {e^3+4 x}{x}}} \, dx-\int e^{-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \left (e^3+4 x-e^3 \log (x)\right ) \, dx \\ & = \int e^{-x+x^{\frac {e^3+4 x}{x}}} \, dx-\int \left (e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}}+4 e^{-x+x^{\frac {e^3+4 x}{x}}} x^{3+\frac {e^3}{x}}-e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \log (x)\right ) \, dx \\ & = -\left (4 \int e^{-x+x^{\frac {e^3+4 x}{x}}} x^{3+\frac {e^3}{x}} \, dx\right )+\int e^{-x+x^{\frac {e^3+4 x}{x}}} \, dx-\int e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \, dx+\int e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \log (x) \, dx \\ & = -\left (4 \int e^{-x+x^{\frac {e^3+4 x}{x}}} x^{3+\frac {e^3}{x}} \, dx\right )+\log (x) \int e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \, dx+\int e^{-x+x^{\frac {e^3+4 x}{x}}} \, dx-\int e^{3-x+x^{\frac {e^3+4 x}{x}}} x^{2+\frac {e^3}{x}} \, dx-\int \frac {\int e^{3-x+x^{4+\frac {e^3}{x}}} x^{2+\frac {e^3}{x}} \, dx}{x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=-e^{-x+x^{4+\frac {e^3}{x}}} \]

[In]

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

[Out]

-E^(-x + x^(4 + E^3/x))

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95

method	result	size
risch	\(-{\mathrm e}^{x^{\frac {{\mathrm e}^{3}+4 x}{x}}-x}\)	\(20\)
parallelrisch	\(-{\mathrm e}^{{\mathrm e}^{\frac {\left ({\mathrm e}^{3}+4 x \right ) \ln \left (x \right )}{x}}-x}\)	\(21\)

[In]

int(((ln(x)*exp(3)-exp(3)-4*x)*exp((exp(3)+4*x)*ln(x)/x)+x^2)*exp(exp((exp(3)+4*x)*ln(x)/x)-x)/x^2,x,method=_R
ETURNVERBOSE)

[Out]

-exp(x^((exp(3)+4*x)/x)-x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=-e^{\left (x^{\frac {4 \, x + e^{3}}{x}} - x\right )} \]

[In]

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

[Out]

-e^(x^((4*x + e^3)/x) - x)

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=- e^{- x + e^{\frac {\left (4 x + e^{3}\right ) \log {\left (x \right )}}{x}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=-e^{\left (x^{4} x^{\frac {e^{3}}{x}} - x\right )} \]

[In]

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

[Out]

-e^(x^4*x^(e^3/x) - x)

Giac [F]

\[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=\int { \frac {{\left ({\left (e^{3} \log \left (x\right ) - 4 \, x - e^{3}\right )} x^{\frac {4 \, x + e^{3}}{x}} + x^{2}\right )} e^{\left (x^{\frac {4 \, x + e^{3}}{x}} - x\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-x+x^{\frac {e^3+4 x}{x}}} \left (x^2+x^{\frac {e^3+4 x}{x}} \left (-e^3-4 x+e^3 \log (x)\right )\right )}{x^2} \, dx=-{\mathrm {e}}^{x^{\frac {{\mathrm {e}}^3}{x}+4}}\,{\mathrm {e}}^{-x} \]

[In]

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

[Out]

-exp(x^(exp(3)/x + 4))*exp(-x)