3.55.68 \(\int \frac {e^{\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}} (e^{2 x} (-1+x)-5 x-11 x^2-x^3+x^4+x^5+e^x (-7 x+2 x^3))}{e^{2 x} x^2+x^4+2 x^5+x^6+e^x (2 x^3+2 x^4)} \, dx\)

Optimal. Leaf size=24 \[ -6+e^{\frac {e^{x+\frac {5}{e^x+x+x^2}}}{x}} \]

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Rubi [F]  time = 139.03, 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 {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right ) \left (e^{2 x} (-1+x)-5 x-11 x^2-x^3+x^4+x^5+e^x \left (-7 x+2 x^3\right )\right )}{e^{2 x} x^2+x^4+2 x^5+x^6+e^x \left (2 x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^((5 + E^x*x + x^2 + x^3)/(E^x + x + x^2))/x + (5 + E^x*x + x^2 + x^3)/(E^x + x + x^2))*(E^(2*x)*(-1
+ x) - 5*x - 11*x^2 - x^3 + x^4 + x^5 + E^x*(-7*x + 2*x^3)))/(E^(2*x)*x^2 + x^4 + 2*x^5 + x^6 + E^x*(2*x^3 + 2
*x^4)),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right ) \left (e^{2 x} (-1+x)+e^x x \left (-7+2 x^2\right )+x \left (-5-11 x-x^2+x^3+x^4\right )\right )}{x^2 \left (e^x+x+x^2\right )^2} \, dx\\ &=\int \left (\frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right ) (-1+x)}{x^2}+\frac {5 \exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right ) \left (-1-x+x^2\right )}{x \left (e^x+x+x^2\right )^2}-\frac {5 \exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{x \left (e^x+x+x^2\right )}\right ) \, dx\\ &=5 \int \frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right ) \left (-1-x+x^2\right )}{x \left (e^x+x+x^2\right )^2} \, dx-5 \int \frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{x \left (e^x+x+x^2\right )} \, dx+\int \frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right ) (-1+x)}{x^2} \, dx\\ &=-\left (5 \int \frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{x \left (e^x+x+x^2\right )} \, dx\right )+5 \int \left (-\frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{\left (e^x+x+x^2\right )^2}-\frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{x \left (e^x+x+x^2\right )^2}+\frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right ) x}{\left (e^x+x+x^2\right )^2}\right ) \, dx+\int \left (-\frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{x^2}+\frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{x}\right ) \, dx\\ &=-\left (5 \int \frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{\left (e^x+x+x^2\right )^2} \, dx\right )-5 \int \frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{x \left (e^x+x+x^2\right )^2} \, dx+5 \int \frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right ) x}{\left (e^x+x+x^2\right )^2} \, dx-5 \int \frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{x \left (e^x+x+x^2\right )} \, dx-\int \frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{x^2} \, dx+\int \frac {\exp \left (\frac {e^{\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}}}{x}+\frac {5+e^x x+x^2+x^3}{e^x+x+x^2}\right )}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 22, normalized size = 0.92 \begin {gather*} e^{\frac {e^{x+\frac {5}{e^x+x+x^2}}}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^((5 + E^x*x + x^2 + x^3)/(E^x + x + x^2))/x + (5 + E^x*x + x^2 + x^3)/(E^x + x + x^2))*(E^(2*x
)*(-1 + x) - 5*x - 11*x^2 - x^3 + x^4 + x^5 + E^x*(-7*x + 2*x^3)))/(E^(2*x)*x^2 + x^4 + 2*x^5 + x^6 + E^x*(2*x
^3 + 2*x^4)),x]

[Out]

E^(E^(x + 5/(E^x + x + x^2))/x)

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fricas [B]  time = 0.57, size = 86, normalized size = 3.58 \begin {gather*} e^{\left (\frac {x^{4} + x^{3} + x^{2} e^{x} + {\left (x^{2} + x + e^{x}\right )} e^{\left (\frac {x^{3} + x^{2} + x e^{x} + 5}{x^{2} + x + e^{x}}\right )} + 5 \, x}{x^{3} + x^{2} + x e^{x}} - \frac {x^{3} + x^{2} + x e^{x} + 5}{x^{2} + x + e^{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-1)*exp(x)^2+(2*x^3-7*x)*exp(x)+x^5+x^4-x^3-11*x^2-5*x)*exp((exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))*
exp(exp((exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))/x)/(exp(x)^2*x^2+(2*x^4+2*x^3)*exp(x)+x^6+2*x^5+x^4),x, algorithm
="fricas")

[Out]

e^((x^4 + x^3 + x^2*e^x + (x^2 + x + e^x)*e^((x^3 + x^2 + x*e^x + 5)/(x^2 + x + e^x)) + 5*x)/(x^3 + x^2 + x*e^
x) - (x^3 + x^2 + x*e^x + 5)/(x^2 + x + e^x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{5} + x^{4} - x^{3} - 11 \, x^{2} + {\left (x - 1\right )} e^{\left (2 \, x\right )} + {\left (2 \, x^{3} - 7 \, x\right )} e^{x} - 5 \, x\right )} e^{\left (\frac {x^{3} + x^{2} + x e^{x} + 5}{x^{2} + x + e^{x}} + \frac {e^{\left (\frac {x^{3} + x^{2} + x e^{x} + 5}{x^{2} + x + e^{x}}\right )}}{x}\right )}}{x^{6} + 2 \, x^{5} + x^{4} + x^{2} e^{\left (2 \, x\right )} + 2 \, {\left (x^{4} + x^{3}\right )} e^{x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-1)*exp(x)^2+(2*x^3-7*x)*exp(x)+x^5+x^4-x^3-11*x^2-5*x)*exp((exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))*
exp(exp((exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))/x)/(exp(x)^2*x^2+(2*x^4+2*x^3)*exp(x)+x^6+2*x^5+x^4),x, algorithm
="giac")

[Out]

integrate((x^5 + x^4 - x^3 - 11*x^2 + (x - 1)*e^(2*x) + (2*x^3 - 7*x)*e^x - 5*x)*e^((x^3 + x^2 + x*e^x + 5)/(x
^2 + x + e^x) + e^((x^3 + x^2 + x*e^x + 5)/(x^2 + x + e^x))/x)/(x^6 + 2*x^5 + x^4 + x^2*e^(2*x) + 2*(x^4 + x^3
)*e^x), x)

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maple [A]  time = 0.16, size = 29, normalized size = 1.21

method	result	size
risch	\({\mathrm e}^{\frac {{\mathrm e}^{\frac {{\mathrm e}^{x} x +x^{3}+x^{2}+5}{{\mathrm e}^{x}+x^{2}+x}}}{x}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x-1)*exp(x)^2+(2*x^3-7*x)*exp(x)+x^5+x^4-x^3-11*x^2-5*x)*exp((exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))*exp(ex
p((exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))/x)/(exp(x)^2*x^2+(2*x^4+2*x^3)*exp(x)+x^6+2*x^5+x^4),x,method=_RETURNVE
RBOSE)

[Out]

exp(exp((exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))/x)

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maxima [A]  time = 0.90, size = 19, normalized size = 0.79 \begin {gather*} e^{\left (\frac {e^{\left (x + \frac {5}{x^{2} + x + e^{x}}\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-1)*exp(x)^2+(2*x^3-7*x)*exp(x)+x^5+x^4-x^3-11*x^2-5*x)*exp((exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))*
exp(exp((exp(x)*x+x^3+x^2+5)/(exp(x)+x^2+x))/x)/(exp(x)^2*x^2+(2*x^4+2*x^3)*exp(x)+x^6+2*x^5+x^4),x, algorithm
="maxima")

[Out]

e^(e^(x + 5/(x^2 + x + e^x))/x)

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mupad [B]  time = 3.80, size = 59, normalized size = 2.46 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^x}{x+{\mathrm {e}}^x+x^2}}\,{\mathrm {e}}^{\frac {x^2}{x+{\mathrm {e}}^x+x^2}}\,{\mathrm {e}}^{\frac {x^3}{x+{\mathrm {e}}^x+x^2}}\,{\mathrm {e}}^{\frac {5}{x+{\mathrm {e}}^x+x^2}}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp((x*exp(x) + x^2 + x^3 + 5)/(x + exp(x) + x^2))/x)*exp((x*exp(x) + x^2 + x^3 + 5)/(x + exp(x) + x
^2))*(5*x - exp(2*x)*(x - 1) + exp(x)*(7*x - 2*x^3) + 11*x^2 + x^3 - x^4 - x^5))/(exp(x)*(2*x^3 + 2*x^4) + x^2
*exp(2*x) + x^4 + 2*x^5 + x^6),x)

[Out]

exp((exp((x*exp(x))/(x + exp(x) + x^2))*exp(x^2/(x + exp(x) + x^2))*exp(x^3/(x + exp(x) + x^2))*exp(5/(x + exp
(x) + x^2)))/x)

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sympy [A]  time = 6.71, size = 26, normalized size = 1.08 \begin {gather*} e^{\frac {e^{\frac {x^{3} + x^{2} + x e^{x} + 5}{x^{2} + x + e^{x}}}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-1)*exp(x)**2+(2*x**3-7*x)*exp(x)+x**5+x**4-x**3-11*x**2-5*x)*exp((exp(x)*x+x**3+x**2+5)/(exp(x)+
x**2+x))*exp(exp((exp(x)*x+x**3+x**2+5)/(exp(x)+x**2+x))/x)/(exp(x)**2*x**2+(2*x**4+2*x**3)*exp(x)+x**6+2*x**5
+x**4),x)

[Out]

exp(exp((x**3 + x**2 + x*exp(x) + 5)/(x**2 + x + exp(x)))/x)

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