3.4.49 \(\int \frac {e^{\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}} (-15+e^{\frac {2 (3+e^x x)}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} (-39-2 x+3 e^x x^2))}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 (3+e^x x)}{x}} x^3} \, dx\)

Optimal. Leaf size=37 \[ \frac {e^{\frac {3 \left (5-\frac {x}{1+e^{\frac {3+e^x x}{x}}}\right )}{x}}-x}{x} \]

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Rubi [F]  time = 37.93, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (\frac {15+15 e^{\frac {3+e^x x}{x}}-3 x}{x+e^{\frac {3+e^x x}{x}} x}\right ) \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{x^3+2 e^{\frac {3+e^x x}{x}} x^3+e^{\frac {2 \left (3+e^x x\right )}{x}} x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((15 + 15*E^((3 + E^x*x)/x) - 3*x)/(x + E^((3 + E^x*x)/x)*x))*(-15 + E^((2*(3 + E^x*x))/x)*(-15 - x) -
x + E^((3 + E^x*x)/x)*(-39 - 2*x + 3*E^x*x^2)))/(x^3 + 2*E^((3 + E^x*x)/x)*x^3 + E^((2*(3 + E^x*x))/x)*x^3),x]

[Out]

-15*Defer[Int][E^((3*(5 + 5*E^(E^x + 3/x) - x))/(x + E^((3 + E^x*x)/x)*x))/((1 + E^(E^x + 3/x))^2*x^3), x] - 3
9*Defer[Int][E^(E^x + 3/x + (3*(5 + 5*E^(E^x + 3/x) - x))/(x + E^((3 + E^x*x)/x)*x))/((1 + E^(E^x + 3/x))^2*x^
3), x] - 15*Defer[Int][E^((2*(3 + E^x*x))/x + (3*(5 + 5*E^(E^x + 3/x) - x))/(x + E^((3 + E^x*x)/x)*x))/((1 + E
^(E^x + 3/x))^2*x^3), x] - Defer[Int][E^((3*(5 + 5*E^(E^x + 3/x) - x))/(x + E^((3 + E^x*x)/x)*x))/((1 + E^(E^x
+ 3/x))^2*x^2), x] - 2*Defer[Int][E^(E^x + 3/x + (3*(5 + 5*E^(E^x + 3/x) - x))/(x + E^((3 + E^x*x)/x)*x))/((1
+ E^(E^x + 3/x))^2*x^2), x] - Defer[Int][E^((2*(3 + E^x*x))/x + (3*(5 + 5*E^(E^x + 3/x) - x))/(x + E^((3 + E^
x*x)/x)*x))/((1 + E^(E^x + 3/x))^2*x^2), x] + 3*Defer[Int][E^(E^x + 3/x + x + (3*(5 + 5*E^(E^x + 3/x) - x))/(x
+ E^((3 + E^x*x)/x)*x))/((1 + E^(E^x + 3/x))^2*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right ) \left (-15+e^{\frac {2 \left (3+e^x x\right )}{x}} (-15-x)-x+e^{\frac {3+e^x x}{x}} \left (-39-2 x+3 e^x x^2\right )\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx\\ &=\int \left (-\frac {15 \exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3}-\frac {39 \exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3}+\frac {\exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right ) (-15-x)}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3}-\frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2}-\frac {2 \exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2}+\frac {3 \exp \left (e^x+\frac {3}{x}+x+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x}\right ) \, dx\\ &=-\left (2 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx\right )+3 \int \frac {\exp \left (e^x+\frac {3}{x}+x+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x} \, dx-15 \int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-39 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx+\int \frac {\exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right ) (-15-x)}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-\int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx\\ &=-\left (2 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx\right )+3 \int \frac {\exp \left (e^x+\frac {3}{x}+x+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x} \, dx-15 \int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-39 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx+\int \left (-\frac {15 \exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3}-\frac {\exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2}\right ) \, dx-\int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx\\ &=-\left (2 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx\right )+3 \int \frac {\exp \left (e^x+\frac {3}{x}+x+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x} \, dx-15 \int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-15 \int \frac {\exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-39 \int \frac {\exp \left (e^x+\frac {3}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^3} \, dx-\int \frac {\exp \left (\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx-\int \frac {\exp \left (\frac {2 \left (3+e^x x\right )}{x}+\frac {3 \left (5+5 e^{e^x+\frac {3}{x}}-x\right )}{x+e^{\frac {3+e^x x}{x}} x}\right )}{\left (1+e^{e^x+\frac {3}{x}}\right )^2 x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 29, normalized size = 0.78 \begin {gather*} \frac {e^{-\frac {3}{1+e^{e^x+\frac {3}{x}}}+\frac {15}{x}}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((15 + 15*E^((3 + E^x*x)/x) - 3*x)/(x + E^((3 + E^x*x)/x)*x))*(-15 + E^((2*(3 + E^x*x))/x)*(-15 -
x) - x + E^((3 + E^x*x)/x)*(-39 - 2*x + 3*E^x*x^2)))/(x^3 + 2*E^((3 + E^x*x)/x)*x^3 + E^((2*(3 + E^x*x))/x)*x
^3),x]

[Out]

E^(-3/(1 + E^(E^x + 3/x)) + 15/x)/x

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fricas [A]  time = 0.59, size = 40, normalized size = 1.08 \begin {gather*} \frac {e^{\left (-\frac {3 \, {\left (x - 5 \, e^{\left (\frac {x e^{x} + 3}{x}\right )} - 5\right )}}{x e^{\left (\frac {x e^{x} + 3}{x}\right )} + x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-15)*exp((exp(x)*x+3)/x)^2+(3*exp(x)*x^2-2*x-39)*exp((exp(x)*x+3)/x)-x-15)*exp((15*exp((exp(x)*x
+3)/x)+15-3*x)/(x*exp((exp(x)*x+3)/x)+x))/(x^3*exp((exp(x)*x+3)/x)^2+2*x^3*exp((exp(x)*x+3)/x)+x^3),x, algorit
hm="fricas")

[Out]

e^(-3*(x - 5*e^((x*e^x + 3)/x) - 5)/(x*e^((x*e^x + 3)/x) + x))/x

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giac [B]  time = 1.09, size = 72, normalized size = 1.95 \begin {gather*} \frac {e^{\left (\frac {x e^{\left (x + \frac {x e^{x} + 3}{x}\right )} + x e^{x} - 3 \, x + 18 \, e^{\left (\frac {x e^{x} + 3}{x}\right )} + 18}{x e^{\left (\frac {x e^{x} + 3}{x}\right )} + x} - \frac {x e^{x} + 3}{x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-15)*exp((exp(x)*x+3)/x)^2+(3*exp(x)*x^2-2*x-39)*exp((exp(x)*x+3)/x)-x-15)*exp((15*exp((exp(x)*x
+3)/x)+15-3*x)/(x*exp((exp(x)*x+3)/x)+x))/(x^3*exp((exp(x)*x+3)/x)^2+2*x^3*exp((exp(x)*x+3)/x)+x^3),x, algorit
hm="giac")

[Out]

e^((x*e^(x + (x*e^x + 3)/x) + x*e^x - 3*x + 18*e^((x*e^x + 3)/x) + 18)/(x*e^((x*e^x + 3)/x) + x) - (x*e^x + 3)
/x)/x

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maple [A]  time = 0.43, size = 42, normalized size = 1.14

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {3 \left (-5 \,{\mathrm e}^{\frac {{\mathrm e}^{x} x +3}{x}}-5+x \right )}{x \left ({\mathrm e}^{\frac {{\mathrm e}^{x} x +3}{x}}+1\right )}}}{x}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x-15)*exp((exp(x)*x+3)/x)^2+(3*exp(x)*x^2-2*x-39)*exp((exp(x)*x+3)/x)-x-15)*exp((15*exp((exp(x)*x+3)/x)
+15-3*x)/(x*exp((exp(x)*x+3)/x)+x))/(x^3*exp((exp(x)*x+3)/x)^2+2*x^3*exp((exp(x)*x+3)/x)+x^3),x,method=_RETURN
VERBOSE)

[Out]

1/x*exp(-3*(-5*exp((exp(x)*x+3)/x)-5+x)/x/(exp((exp(x)*x+3)/x)+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left ({\left (x + 15\right )} e^{\left (\frac {2 \, {\left (x e^{x} + 3\right )}}{x}\right )} - {\left (3 \, x^{2} e^{x} - 2 \, x - 39\right )} e^{\left (\frac {x e^{x} + 3}{x}\right )} + x + 15\right )} e^{\left (-\frac {3 \, {\left (x - 5 \, e^{\left (\frac {x e^{x} + 3}{x}\right )} - 5\right )}}{x e^{\left (\frac {x e^{x} + 3}{x}\right )} + x}\right )}}{x^{3} e^{\left (\frac {2 \, {\left (x e^{x} + 3\right )}}{x}\right )} + 2 \, x^{3} e^{\left (\frac {x e^{x} + 3}{x}\right )} + x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-15)*exp((exp(x)*x+3)/x)^2+(3*exp(x)*x^2-2*x-39)*exp((exp(x)*x+3)/x)-x-15)*exp((15*exp((exp(x)*x
+3)/x)+15-3*x)/(x*exp((exp(x)*x+3)/x)+x))/(x^3*exp((exp(x)*x+3)/x)^2+2*x^3*exp((exp(x)*x+3)/x)+x^3),x, algorit
hm="maxima")

[Out]

-integrate(((x + 15)*e^(2*(x*e^x + 3)/x) - (3*x^2*e^x - 2*x - 39)*e^((x*e^x + 3)/x) + x + 15)*e^(-3*(x - 5*e^(
(x*e^x + 3)/x) - 5)/(x*e^((x*e^x + 3)/x) + x))/(x^3*e^(2*(x*e^x + 3)/x) + 2*x^3*e^((x*e^x + 3)/x) + x^3), x)

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mupad [B]  time = 0.86, size = 66, normalized size = 1.78 \begin {gather*} \frac {{\mathrm {e}}^{\frac {15\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}}{x+x\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}}}\,{\mathrm {e}}^{\frac {15}{x+x\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}}}\,{\mathrm {e}}^{-\frac {3}{{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{3/x}+1}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((15*exp((x*exp(x) + 3)/x) - 3*x + 15)/(x + x*exp((x*exp(x) + 3)/x)))*(x + exp((x*exp(x) + 3)/x)*(2*x
- 3*x^2*exp(x) + 39) + exp((2*(x*exp(x) + 3))/x)*(x + 15) + 15))/(x^3 + 2*x^3*exp((x*exp(x) + 3)/x) + x^3*exp
((2*(x*exp(x) + 3))/x)),x)

[Out]

(exp((15*exp(exp(x))*exp(3/x))/(x + x*exp(exp(x))*exp(3/x)))*exp(15/(x + x*exp(exp(x))*exp(3/x)))*exp(-3/(exp(
exp(x))*exp(3/x) + 1)))/x

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sympy [A]  time = 1.52, size = 32, normalized size = 0.86 \begin {gather*} \frac {e^{\frac {- 3 x + 15 e^{\frac {x e^{x} + 3}{x}} + 15}{x e^{\frac {x e^{x} + 3}{x}} + x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-15)*exp((exp(x)*x+3)/x)**2+(3*exp(x)*x**2-2*x-39)*exp((exp(x)*x+3)/x)-x-15)*exp((15*exp((exp(x)
*x+3)/x)+15-3*x)/(x*exp((exp(x)*x+3)/x)+x))/(x**3*exp((exp(x)*x+3)/x)**2+2*x**3*exp((exp(x)*x+3)/x)+x**3),x)

[Out]

exp((-3*x + 15*exp((x*exp(x) + 3)/x) + 15)/(x*exp((x*exp(x) + 3)/x) + x))/x

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