3.30.100 \(\int e^{-\frac {2 (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4))}{x}} (2 x+e^{\frac {2 (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4))}{x}} (-1+2 x)+e^x (-4+4 x) \log (2) \log (4)+4 \log ^2(2) \log (4)+e^{\frac {2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)}{x}} (-4 x+e^x (4-4 x) \log (2) \log (4)-4 \log ^2(2) \log (4))) \, dx\)

Optimal. Leaf size=33 \[ -x+\left (x-e^{-2 \left (1+\frac {\log (2) \left (-e^x+\log (2)\right ) \log (4)}{x}\right )} x\right )^2 \]

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Rubi [F]  time = 8.66, 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 \exp \left (-\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}\right ) \left (2 x+\exp \left (\frac {2 \left (2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)\right )}{x}\right ) (-1+2 x)+e^x (-4+4 x) \log (2) \log (4)+4 \log ^2(2) \log (4)+\exp \left (\frac {2 x-2 e^x \log (2) \log (4)+2 \log ^2(2) \log (4)}{x}\right ) \left (-4 x+e^x (4-4 x) \log (2) \log (4)-4 \log ^2(2) \log (4)\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x + E^((2*(2*x - 2*E^x*Log[2]*Log[4] + 2*Log[2]^2*Log[4]))/x)*(-1 + 2*x) + E^x*(-4 + 4*x)*Log[2]*Log[4]
+ 4*Log[2]^2*Log[4] + E^((2*x - 2*E^x*Log[2]*Log[4] + 2*Log[2]^2*Log[4])/x)*(-4*x + E^x*(4 - 4*x)*Log[2]*Log[
4] - 4*Log[2]^2*Log[4]))/E^((2*(2*x - 2*E^x*Log[2]*Log[4] + 2*Log[2]^2*Log[4]))/x),x]

[Out]

-x + x^2 - (2^(1 - (2*(E^x - Log[2])*Log[4])/x)*E^(2 - (4*(x - E^x*Log[2]*Log[4] + Log[2]^2*Log[4]))/x)*(E^x*L
og[2]*Log[4] - E^x*x*Log[2]*Log[4] - Log[2]^2*Log[4]))/((E^x*Log[2]*Log[4])/x - ((E^x - Log[2])*Log[2]*Log[4])
/x^2 + (2*(1 - E^x*Log[2]*Log[4]))/x - (2*(x - E^x*Log[2]*Log[4] + Log[2]^2*Log[4]))/x^2) + 4*Log[2]^2*Log[4]*
Defer[Int][E^((-4*(x - E^x*Log[2]*Log[4] + Log[2]^2*Log[4]))/x), x] - 4*Log[2]*Log[4]*Defer[Int][E^(x - (4*(x
- E^x*Log[2]*Log[4] + Log[2]^2*Log[4]))/x), x] + 2*Defer[Int][x/E^((4*(x - E^x*Log[2]*Log[4] + Log[2]^2*Log[4]
))/x), x] + 4*Log[2]*Log[4]*Defer[Int][E^(x - (4*(x - E^x*Log[2]*Log[4] + Log[2]^2*Log[4]))/x)*x, x]

Rubi steps

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

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Mathematica [A]  time = 0.59, size = 58, normalized size = 1.76 \begin {gather*} x \left (-1+\left (1+e^{-\frac {4 \left (x+2 \log ^3(2)-e^x \log (2) \log (4)\right )}{x}}-2 e^{\frac {-2 x-4 \log ^3(2)+e^x \log ^2(4)}{x}}\right ) x\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(2*x + E^((2*(2*x - 2*E^x*Log[2]*Log[4] + 2*Log[2]^2*Log[4]))/x)*(-1 + 2*x) + E^x*(-4 + 4*x)*Log[2]*
Log[4] + 4*Log[2]^2*Log[4] + E^((2*x - 2*E^x*Log[2]*Log[4] + 2*Log[2]^2*Log[4])/x)*(-4*x + E^x*(4 - 4*x)*Log[2
]*Log[4] - 4*Log[2]^2*Log[4]))/E^((2*(2*x - 2*E^x*Log[2]*Log[4] + 2*Log[2]^2*Log[4]))/x),x]

[Out]

x*(-1 + (1 + E^((-4*(x + 2*Log[2]^3 - E^x*Log[2]*Log[4]))/x) - 2*E^((-2*x - 4*Log[2]^3 + E^x*Log[4]^2)/x))*x)

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fricas [B]  time = 0.59, size = 94, normalized size = 2.85 \begin {gather*} -{\left (2 \, x^{2} e^{\left (-\frac {2 \, {\left (2 \, e^{x} \log \relax (2)^{2} - 2 \, \log \relax (2)^{3} - x\right )}}{x}\right )} - x^{2} - {\left (x^{2} - x\right )} e^{\left (-\frac {4 \, {\left (2 \, e^{x} \log \relax (2)^{2} - 2 \, \log \relax (2)^{3} - x\right )}}{x}\right )}\right )} e^{\left (\frac {4 \, {\left (2 \, e^{x} \log \relax (2)^{2} - 2 \, \log \relax (2)^{3} - x\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/x)^2+(2*(-4*x+4)*log(2)^2*exp(x)-8*log(2)^3-4*x)*ex
p((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/x)+2*(4*x-4)*log(2)^2*exp(x)+8*log(2)^3+2*x)/exp((-4*log(2)^2*exp(x)+4*l
og(2)^3+2*x)/x)^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (8 \, {\left (x - 1\right )} e^{x} \log \relax (2)^{2} + 8 \, \log \relax (2)^{3} - 4 \, {\left (2 \, {\left (x - 1\right )} e^{x} \log \relax (2)^{2} + 2 \, \log \relax (2)^{3} + x\right )} e^{\left (-\frac {2 \, {\left (2 \, e^{x} \log \relax (2)^{2} - 2 \, \log \relax (2)^{3} - x\right )}}{x}\right )} + {\left (2 \, x - 1\right )} e^{\left (-\frac {4 \, {\left (2 \, e^{x} \log \relax (2)^{2} - 2 \, \log \relax (2)^{3} - x\right )}}{x}\right )} + 2 \, x\right )} e^{\left (\frac {4 \, {\left (2 \, e^{x} \log \relax (2)^{2} - 2 \, \log \relax (2)^{3} - x\right )}}{x}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/x)^2+(2*(-4*x+4)*log(2)^2*exp(x)-8*log(2)^3-4*x)*ex
p((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/x)+2*(4*x-4)*log(2)^2*exp(x)+8*log(2)^3+2*x)/exp((-4*log(2)^2*exp(x)+4*l
og(2)^3+2*x)/x)^2,x, algorithm="giac")

[Out]

integrate((8*(x - 1)*e^x*log(2)^2 + 8*log(2)^3 - 4*(2*(x - 1)*e^x*log(2)^2 + 2*log(2)^3 + x)*e^(-2*(2*e^x*log(
2)^2 - 2*log(2)^3 - x)/x) + (2*x - 1)*e^(-4*(2*e^x*log(2)^2 - 2*log(2)^3 - x)/x) + 2*x)*e^(4*(2*e^x*log(2)^2 -
2*log(2)^3 - x)/x), x)

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maple [A]  time = 0.10, size = 65, normalized size = 1.97

method	result	size
risch	\(x^{2}-x -2 x^{2} {\mathrm e}^{\frac {4 \ln \relax (2)^{2} {\mathrm e}^{x}-4 \ln \relax (2)^{3}-2 x}{x}}+x^{2} {\mathrm e}^{\frac {8 \ln \relax (2)^{2} {\mathrm e}^{x}-8 \ln \relax (2)^{3}-4 x}{x}}\)	\(65\)
norman	\(\left (x^{2}+x^{2} {\mathrm e}^{\frac {-8 \ln \relax (2)^{2} {\mathrm e}^{x}+8 \ln \relax (2)^{3}+4 x}{x}}-x \,{\mathrm e}^{\frac {-8 \ln \relax (2)^{2} {\mathrm e}^{x}+8 \ln \relax (2)^{3}+4 x}{x}}-2 x^{2} {\mathrm e}^{\frac {-4 \ln \relax (2)^{2} {\mathrm e}^{x}+4 \ln \relax (2)^{3}+2 x}{x}}\right ) {\mathrm e}^{-\frac {2 \left (-4 \ln \relax (2)^{2} {\mathrm e}^{x}+4 \ln \relax (2)^{3}+2 x \right )}{x}}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x-1)*exp((-4*ln(2)^2*exp(x)+4*ln(2)^3+2*x)/x)^2+(2*(-4*x+4)*ln(2)^2*exp(x)-8*ln(2)^3-4*x)*exp((-4*ln(2
)^2*exp(x)+4*ln(2)^3+2*x)/x)+2*(4*x-4)*ln(2)^2*exp(x)+8*ln(2)^3+2*x)/exp((-4*ln(2)^2*exp(x)+4*ln(2)^3+2*x)/x)^
2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.94, size = 65, normalized size = 1.97 \begin {gather*} x^{2} + {\left (x^{2} e^{\left (\frac {8 \, e^{x} \log \relax (2)^{2}}{x}\right )} - 2 \, x^{2} e^{\left (\frac {4 \, e^{x} \log \relax (2)^{2}}{x} + \frac {4 \, \log \relax (2)^{3}}{x} + 2\right )}\right )} e^{\left (-\frac {8 \, \log \relax (2)^{3}}{x} - 4\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/x)^2+(2*(-4*x+4)*log(2)^2*exp(x)-8*log(2)^3-4*x)*ex
p((-4*log(2)^2*exp(x)+4*log(2)^3+2*x)/x)+2*(4*x-4)*log(2)^2*exp(x)+8*log(2)^3+2*x)/exp((-4*log(2)^2*exp(x)+4*l
og(2)^3+2*x)/x)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.02, size = 62, normalized size = 1.88 \begin {gather*} x^2-x-2\,x^2\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^x\,{\ln \relax (2)}^2}{x}-\frac {4\,{\ln \relax (2)}^3}{x}-2}+x^2\,{\mathrm {e}}^{\frac {8\,{\mathrm {e}}^x\,{\ln \relax (2)}^2}{x}-\frac {8\,{\ln \relax (2)}^3}{x}-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-(2*(2*x - 4*exp(x)*log(2)^2 + 4*log(2)^3))/x)*(2*x - exp((2*x - 4*exp(x)*log(2)^2 + 4*log(2)^3)/x)*(4
*x + 8*log(2)^3 + 2*exp(x)*log(2)^2*(4*x - 4)) + exp((2*(2*x - 4*exp(x)*log(2)^2 + 4*log(2)^3))/x)*(2*x - 1) +
8*log(2)^3 + 2*exp(x)*log(2)^2*(4*x - 4)),x)

[Out]

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

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sympy [B]  time = 0.28, size = 60, normalized size = 1.82 \begin {gather*} x^{2} - 2 x^{2} e^{- \frac {2 x - 4 e^{x} \log {\relax (2 )}^{2} + 4 \log {\relax (2 )}^{3}}{x}} + x^{2} e^{- \frac {2 \left (2 x - 4 e^{x} \log {\relax (2 )}^{2} + 4 \log {\relax (2 )}^{3}\right )}{x}} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*exp((-4*ln(2)**2*exp(x)+4*ln(2)**3+2*x)/x)**2+(2*(-4*x+4)*ln(2)**2*exp(x)-8*ln(2)**3-4*x)*e
xp((-4*ln(2)**2*exp(x)+4*ln(2)**3+2*x)/x)+2*(4*x-4)*ln(2)**2*exp(x)+8*ln(2)**3+2*x)/exp((-4*ln(2)**2*exp(x)+4*
ln(2)**3+2*x)/x)**2,x)

[Out]

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

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