3.16.84 \(\int \frac {e^{10-6 e^{4/x}} (96 e^{4/x}+e^{-4+6 e^{4/x}-2 x+2 x^2+4 x \log (x)+2 \log ^2(x)} (2 x^2+4 x^3+(4 x+4 x^2) \log (x))+e^{-2+3 e^{4/x}-x+x^2+2 x \log (x)+\log ^2(x)} (48 e^{4/x}+4 x^2+8 x^3+(8 x+8 x^2) \log (x)))}{x^2} \, dx\)

Optimal. Leaf size=33 \[ 3+\left (2 e^{5-3 e^{4/x}}+e^{3-x+(x+\log (x))^2}\right )^2 \]

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Rubi [F]  time = 9.15, 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 {e^{10-6 e^{4/x}} \left (96 e^{4/x}+\exp \left (-4+6 e^{4/x}-2 x+2 x^2+4 x \log (x)+2 \log ^2(x)\right ) \left (2 x^2+4 x^3+\left (4 x+4 x^2\right ) \log (x)\right )+e^{-2+3 e^{4/x}-x+x^2+2 x \log (x)+\log ^2(x)} \left (48 e^{4/x}+4 x^2+8 x^3+\left (8 x+8 x^2\right ) \log (x)\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(10 - 6*E^(4/x))*(96*E^(4/x) + E^(-4 + 6*E^(4/x) - 2*x + 2*x^2 + 4*x*Log[x] + 2*Log[x]^2)*(2*x^2 + 4*x^
3 + (4*x + 4*x^2)*Log[x]) + E^(-2 + 3*E^(4/x) - x + x^2 + 2*x*Log[x] + Log[x]^2)*(48*E^(4/x) + 4*x^2 + 8*x^3 +
(8*x + 8*x^2)*Log[x])))/x^2,x]

[Out]

4*E^(10 - 6*E^(4/x)) + 4*Defer[Int][E^(8 - 3*E^(4/x) - x + x^2 + Log[x]^2)*x^(2*x), x] + 2*Defer[Int][E^(2*(3
- x + x^2 + Log[x]^2))*x^(4*x), x] + 48*Defer[Int][E^(8 - 3*E^(4/x) + 4/x - x + x^2 + Log[x]^2)*x^(-2 + 2*x),
x] + 8*Defer[Int][E^(8 - 3*E^(4/x) - x + x^2 + Log[x]^2)*x^(1 + 2*x), x] + 4*Defer[Int][E^(2*(3 - x + x^2 + Lo
g[x]^2))*x^(1 + 4*x), x] + 8*Defer[Int][E^(8 - 3*E^(4/x) - x + x^2 + Log[x]^2)*x^(2*x)*Log[x], x] + 4*Defer[In
t][E^(2*(3 - x + x^2 + Log[x]^2))*x^(4*x)*Log[x], x] + 8*Defer[Int][E^(8 - 3*E^(4/x) - x + x^2 + Log[x]^2)*x^(
-1 + 2*x)*Log[x], x] + 4*Defer[Int][E^(2*(3 - x + x^2 + Log[x]^2))*x^(-1 + 4*x)*Log[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-2 \left (-5+3 e^{4/x}\right )} \left (96 e^{4/x}+\exp \left (-4+6 e^{4/x}-2 x+2 x^2+4 x \log (x)+2 \log ^2(x)\right ) \left (2 x^2+4 x^3+\left (4 x+4 x^2\right ) \log (x)\right )+e^{-2+3 e^{4/x}-x+x^2+2 x \log (x)+\log ^2(x)} \left (48 e^{4/x}+4 x^2+8 x^3+\left (8 x+8 x^2\right ) \log (x)\right )\right )}{x^2} \, dx\\ &=\int \left (\frac {96 e^{-2 \left (-5+3 e^{4/x}\right )+\frac {4}{x}}}{x^2}+2 \exp \left (-2 \left (-5+3 e^{4/x}\right )+2 \left (-2+3 e^{4/x}-x+x^2+\log ^2(x)\right )\right ) x^{-1+4 x} \left (x+2 x^2+2 \log (x)+2 x \log (x)\right )+4 \exp \left (-2+3 e^{4/x}-2 \left (-5+3 e^{4/x}\right )-x+x^2+\log ^2(x)\right ) x^{-2+2 x} \left (12 e^{4/x}+x^2+2 x^3+2 x \log (x)+2 x^2 \log (x)\right )\right ) \, dx\\ &=2 \int \exp \left (-2 \left (-5+3 e^{4/x}\right )+2 \left (-2+3 e^{4/x}-x+x^2+\log ^2(x)\right )\right ) x^{-1+4 x} \left (x+2 x^2+2 \log (x)+2 x \log (x)\right ) \, dx+4 \int \exp \left (-2+3 e^{4/x}-2 \left (-5+3 e^{4/x}\right )-x+x^2+\log ^2(x)\right ) x^{-2+2 x} \left (12 e^{4/x}+x^2+2 x^3+2 x \log (x)+2 x^2 \log (x)\right ) \, dx+96 \int \frac {e^{-2 \left (-5+3 e^{4/x}\right )+\frac {4}{x}}}{x^2} \, dx\\ &=2 \int e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{-1+4 x} (x (1+2 x)+2 (1+x) \log (x)) \, dx+4 \int e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{-2+2 x} \left (12 e^{4/x}+x^2+2 x^3+2 x (1+x) \log (x)\right ) \, dx-96 \operatorname {Subst}\left (\int e^{-2 \left (-5+3 e^{4 x}\right )+4 x} \, dx,x,\frac {1}{x}\right )\\ &=2 \int \left (e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{4 x} (1+2 x)+2 e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{-1+4 x} (1+x) \log (x)\right ) \, dx+4 \int \left (e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{2 x}+12 e^{8-3 e^{4/x}+\frac {4}{x}-x+x^2+\log ^2(x)} x^{-2+2 x}+2 e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{1+2 x}+2 e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{-1+2 x} (1+x) \log (x)\right ) \, dx-24 \operatorname {Subst}\left (\int e^{10-6 x} \, dx,x,e^{4/x}\right )\\ &=4 e^{10-6 e^{4/x}}+2 \int e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{4 x} (1+2 x) \, dx+4 \int e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{2 x} \, dx+4 \int e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{-1+4 x} (1+x) \log (x) \, dx+8 \int e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{1+2 x} \, dx+8 \int e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{-1+2 x} (1+x) \log (x) \, dx+48 \int e^{8-3 e^{4/x}+\frac {4}{x}-x+x^2+\log ^2(x)} x^{-2+2 x} \, dx\\ &=4 e^{10-6 e^{4/x}}+2 \int \left (e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{4 x}+2 e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{1+4 x}\right ) \, dx+4 \int e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{2 x} \, dx+4 \int \left (e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{4 x} \log (x)+e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{-1+4 x} \log (x)\right ) \, dx+8 \int e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{1+2 x} \, dx+8 \int \left (e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{2 x} \log (x)+e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{-1+2 x} \log (x)\right ) \, dx+48 \int e^{8-3 e^{4/x}+\frac {4}{x}-x+x^2+\log ^2(x)} x^{-2+2 x} \, dx\\ &=4 e^{10-6 e^{4/x}}+2 \int e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{4 x} \, dx+4 \int e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{2 x} \, dx+4 \int e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{1+4 x} \, dx+4 \int e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{4 x} \log (x) \, dx+4 \int e^{2 \left (3-x+x^2+\log ^2(x)\right )} x^{-1+4 x} \log (x) \, dx+8 \int e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{1+2 x} \, dx+8 \int e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{2 x} \log (x) \, dx+8 \int e^{8-3 e^{4/x}-x+x^2+\log ^2(x)} x^{-1+2 x} \log (x) \, dx+48 \int e^{8-3 e^{4/x}+\frac {4}{x}-x+x^2+\log ^2(x)} x^{-2+2 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 52, normalized size = 1.58 \begin {gather*} e^{6-6 e^{4/x}-2 x} \left (2 e^{2+x}+e^{3 e^{4/x}+x^2+\log ^2(x)} x^{2 x}\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(10 - 6*E^(4/x))*(96*E^(4/x) + E^(-4 + 6*E^(4/x) - 2*x + 2*x^2 + 4*x*Log[x] + 2*Log[x]^2)*(2*x^2
+ 4*x^3 + (4*x + 4*x^2)*Log[x]) + E^(-2 + 3*E^(4/x) - x + x^2 + 2*x*Log[x] + Log[x]^2)*(48*E^(4/x) + 4*x^2 + 8
*x^3 + (8*x + 8*x^2)*Log[x])))/x^2,x]

[Out]

E^(6 - 6*E^(4/x) - 2*x)*(2*E^(2 + x) + E^(3*E^(4/x) + x^2 + Log[x]^2)*x^(2*x))^2

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fricas [B]  time = 0.74, size = 64, normalized size = 1.94 \begin {gather*} e^{\left (2 \, x^{2} + 4 \, x \log \relax (x) + 2 \, \log \relax (x)^{2} - 2 \, x + 6\right )} + 4 \, e^{\left (x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2} - x - 3 \, e^{\frac {4}{x}} + 8\right )} + 4 \, e^{\left (-6 \, e^{\frac {4}{x}} + 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+4*x)*log(x)+4*x^3+2*x^2)*exp(3*exp(4/x)-5)^2*exp(log(x)^2+2*x*log(x)+x^2-x+3)^2+((8*x^2+8*x
)*log(x)+48*exp(4/x)+8*x^3+4*x^2)*exp(3*exp(4/x)-5)*exp(log(x)^2+2*x*log(x)+x^2-x+3)+96*exp(4/x))/x^2/exp(3*ex
p(4/x)-5)^2,x, algorithm="fricas")

[Out]

e^(2*x^2 + 4*x*log(x) + 2*log(x)^2 - 2*x + 6) + 4*e^(x^2 + 2*x*log(x) + log(x)^2 - x - 3*e^(4/x) + 8) + 4*e^(-
6*e^(4/x) + 10)

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giac [B]  time = 0.42, size = 99, normalized size = 3.00 \begin {gather*} {\left (e^{\left (2 \, x^{2} + 4 \, x \log \relax (x) + 2 \, \log \relax (x)^{2} - 2 \, x + \frac {4}{x} + 6\right )} + 4 \, e^{\left (\frac {x^{3} + 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} - x^{2} - 3 \, x e^{\frac {4}{x}} + 8 \, x + 4}{x}\right )} + 4 \, e^{\left (-\frac {2 \, {\left (3 \, x e^{\frac {4}{x}} - 5 \, x - 2\right )}}{x}\right )}\right )} e^{\left (-\frac {4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+4*x)*log(x)+4*x^3+2*x^2)*exp(3*exp(4/x)-5)^2*exp(log(x)^2+2*x*log(x)+x^2-x+3)^2+((8*x^2+8*x
)*log(x)+48*exp(4/x)+8*x^3+4*x^2)*exp(3*exp(4/x)-5)*exp(log(x)^2+2*x*log(x)+x^2-x+3)+96*exp(4/x))/x^2/exp(3*ex
p(4/x)-5)^2,x, algorithm="giac")

[Out]

(e^(2*x^2 + 4*x*log(x) + 2*log(x)^2 - 2*x + 4/x + 6) + 4*e^((x^3 + 2*x^2*log(x) + x*log(x)^2 - x^2 - 3*x*e^(4/
x) + 8*x + 4)/x) + 4*e^(-2*(3*x*e^(4/x) - 5*x - 2)/x))*e^(-4/x)

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maple [B]  time = 0.11, size = 68, normalized size = 2.06

method	result	size
risch	\(4 \,{\mathrm e}^{-6 \,{\mathrm e}^{\frac {4}{x}}+10}+x^{4 x} {\mathrm e}^{2 \ln \relax (x )^{2}+6+2 x^{2}-2 x}+4 x^{2 x} {\mathrm e}^{-3 \,{\mathrm e}^{\frac {4}{x}}+8+\ln \relax (x )^{2}+x^{2}-x}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x^2+4*x)*ln(x)+4*x^3+2*x^2)*exp(3*exp(4/x)-5)^2*exp(ln(x)^2+2*x*ln(x)+x^2-x+3)^2+((8*x^2+8*x)*ln(x)+4
8*exp(4/x)+8*x^3+4*x^2)*exp(3*exp(4/x)-5)*exp(ln(x)^2+2*x*ln(x)+x^2-x+3)+96*exp(4/x))/x^2/exp(3*exp(4/x)-5)^2,
x,method=_RETURNVERBOSE)

[Out]

4*exp(-6*exp(4/x)+10)+(x^(2*x))^2*exp(2*ln(x)^2+6+2*x^2-2*x)+4*x^(2*x)*exp(-3*exp(4/x)+8+ln(x)^2+x^2-x)

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maxima [B]  time = 0.64, size = 65, normalized size = 1.97 \begin {gather*} {\left (e^{\left (2 \, x^{2} + 4 \, x \log \relax (x) + 2 \, \log \relax (x)^{2} + 6\right )} + 4 \, e^{\left (x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2} + x - 3 \, e^{\frac {4}{x}} + 8\right )}\right )} e^{\left (-2 \, x\right )} + 4 \, e^{\left (-6 \, e^{\frac {4}{x}} + 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+4*x)*log(x)+4*x^3+2*x^2)*exp(3*exp(4/x)-5)^2*exp(log(x)^2+2*x*log(x)+x^2-x+3)^2+((8*x^2+8*x
)*log(x)+48*exp(4/x)+8*x^3+4*x^2)*exp(3*exp(4/x)-5)*exp(log(x)^2+2*x*log(x)+x^2-x+3)+96*exp(4/x))/x^2/exp(3*ex
p(4/x)-5)^2,x, algorithm="maxima")

[Out]

(e^(2*x^2 + 4*x*log(x) + 2*log(x)^2 + 6) + 4*e^(x^2 + 2*x*log(x) + log(x)^2 + x - 3*e^(4/x) + 8))*e^(-2*x) + 4
*e^(-6*e^(4/x) + 10)

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mupad [B]  time = 1.35, size = 70, normalized size = 2.12 \begin {gather*} 4\,{\mathrm {e}}^{-6\,{\mathrm {e}}^{4/x}}\,{\mathrm {e}}^{10}+x^{4\,x}\,{\mathrm {e}}^{2\,{\ln \relax (x)}^2}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^6\,{\mathrm {e}}^{2\,x^2}+4\,x^{2\,x}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-3\,{\mathrm {e}}^{4/x}}\,{\mathrm {e}}^8\,{\mathrm {e}}^{{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(10 - 6*exp(4/x))*(96*exp(4/x) + exp(log(x)^2 - x + 2*x*log(x) + x^2 + 3)*exp(3*exp(4/x) - 5)*(48*exp(
4/x) + log(x)*(8*x + 8*x^2) + 4*x^2 + 8*x^3) + exp(2*log(x)^2 - 2*x + 4*x*log(x) + 2*x^2 + 6)*exp(6*exp(4/x) -
10)*(log(x)*(4*x + 4*x^2) + 2*x^2 + 4*x^3)))/x^2,x)

[Out]

4*exp(-6*exp(4/x))*exp(10) + x^(4*x)*exp(2*log(x)^2)*exp(-2*x)*exp(6)*exp(2*x^2) + 4*x^(2*x)*exp(-x)*exp(x^2)*
exp(-3*exp(4/x))*exp(8)*exp(log(x)^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x**2+4*x)*ln(x)+4*x**3+2*x**2)*exp(3*exp(4/x)-5)**2*exp(ln(x)**2+2*x*ln(x)+x**2-x+3)**2+((8*x**
2+8*x)*ln(x)+48*exp(4/x)+8*x**3+4*x**2)*exp(3*exp(4/x)-5)*exp(ln(x)**2+2*x*ln(x)+x**2-x+3)+96*exp(4/x))/x**2/e
xp(3*exp(4/x)-5)**2,x)

[Out]

Timed out

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