3.79.48 \(\int \frac {-6-2 x+e^2 (9 x+20 x^2+5 x^3)+(2+e^2 (-6 x-6 x^2)) \log (4 e^{\frac {-1+e^2 (-x-x^2)}{e^2 x}})+e^2 x \log ^2(4 e^{\frac {-1+e^2 (-x-x^2)}{e^2 x}})}{e^2 x} \, dx\)

Optimal. Leaf size=32 \[ x \left (x+\left (-3-x+\log \left (4 e^{-2-x+\frac {-\frac {1}{e^2}+x}{x}}\right )\right )^2\right ) \]

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Rubi [F]  time = 0.32, 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 {-6-2 x+e^2 \left (9 x+20 x^2+5 x^3\right )+\left (2+e^2 \left (-6 x-6 x^2\right )\right ) \log \left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )+e^2 x \log ^2\left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )}{e^2 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 - 2*x + E^2*(9*x + 20*x^2 + 5*x^3) + (2 + E^2*(-6*x - 6*x^2))*Log[4*E^((-1 + E^2*(-x - x^2))/(E^2*x))]
+ E^2*x*Log[4*E^((-1 + E^2*(-x - x^2))/(E^2*x))]^2)/(E^2*x),x]

[Out]

2/(E^4*x) + x/E^2 - ((2 - 9*E^2)*x)/E^2 + 7*x^2 + (2*x^3)/3 - 6*x*Log[4*E^(-1 - 1/(E^2*x) - x)] - 3*x^2*Log[4*
E^(-1 - 1/(E^2*x) - x)] + (2*(1/(E^2*x) + x)*Log[x])/E^2 + (2*Log[4*E^(-1 - 1/(E^2*x) - x)]*Log[x])/E^2 + Defe
r[Int][Log[4*E^(-1 - 1/(E^2*x) - x)]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-6-2 x+e^2 \left (9 x+20 x^2+5 x^3\right )+\left (2+e^2 \left (-6 x-6 x^2\right )\right ) \log \left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )+e^2 x \log ^2\left (4 e^{\frac {-1+e^2 \left (-x-x^2\right )}{e^2 x}}\right )}{x} \, dx}{e^2}\\ &=\frac {\int \left (\frac {-6-\left (2-9 e^2\right ) x+20 e^2 x^2+5 e^2 x^3}{x}-\frac {2 \left (-1+3 e^2 x+3 e^2 x^2\right ) \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )}{x}+e^2 \log ^2\left (4 e^{-1-\frac {1}{e^2 x}-x}\right )\right ) \, dx}{e^2}\\ &=\frac {\int \frac {-6-\left (2-9 e^2\right ) x+20 e^2 x^2+5 e^2 x^3}{x} \, dx}{e^2}-\frac {2 \int \frac {\left (-1+3 e^2 x+3 e^2 x^2\right ) \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )}{x} \, dx}{e^2}+\int \log ^2\left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \, dx\\ &=\frac {\int \left (-2+9 e^2-\frac {6}{x}+20 e^2 x+5 e^2 x^2\right ) \, dx}{e^2}-\frac {2 \int \left (3 e^2 \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )-\frac {\log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )}{x}+3 e^2 x \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )\right ) \, dx}{e^2}+\int \log ^2\left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \, dx\\ &=-\frac {\left (2-9 e^2\right ) x}{e^2}+10 x^2+\frac {5 x^3}{3}-\frac {6 \log (x)}{e^2}-6 \int \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \, dx-6 \int x \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \, dx+\frac {2 \int \frac {\log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )}{x} \, dx}{e^2}+\int \log ^2\left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \, dx\\ &=-\frac {\left (2-9 e^2\right ) x}{e^2}+10 x^2+\frac {5 x^3}{3}-6 x \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )-3 x^2 \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )-\frac {6 \log (x)}{e^2}+\frac {2 \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \log (x)}{e^2}+3 \int \left (\frac {1}{e^2}-x^2\right ) \, dx+6 \int \left (-1+\frac {1}{e^2 x^2}\right ) x \, dx-\frac {2 \int \left (-1+\frac {1}{e^2 x^2}\right ) \log (x) \, dx}{e^2}+\int \log ^2\left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \, dx\\ &=\frac {3 x}{e^2}-\frac {\left (2-9 e^2\right ) x}{e^2}+10 x^2+\frac {2 x^3}{3}-6 x \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )-3 x^2 \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )-\frac {6 \log (x)}{e^2}+\frac {2 \left (\frac {1}{e^2 x}+x\right ) \log (x)}{e^2}+\frac {2 \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \log (x)}{e^2}+6 \int \left (\frac {1}{e^2 x}-x\right ) \, dx+\frac {2 \int \left (-1-\frac {1}{e^2 x^2}\right ) \, dx}{e^2}+\int \log ^2\left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \, dx\\ &=\frac {2}{e^4 x}+\frac {x}{e^2}-\frac {\left (2-9 e^2\right ) x}{e^2}+7 x^2+\frac {2 x^3}{3}-6 x \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )-3 x^2 \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )+\frac {2 \left (\frac {1}{e^2 x}+x\right ) \log (x)}{e^2}+\frac {2 \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \log (x)}{e^2}+\int \log ^2\left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.17, size = 80, normalized size = 2.50 \begin {gather*} \frac {2 (-3+x)}{e^2}+\frac {2}{e^4 x}+x \left (9+7 x+x^2\right )+\left (\frac {2}{e^2}-2 x (3+x)\right ) \log \left (4 e^{-1-\frac {1}{e^2 x}-x}\right )+x \log ^2\left (4 e^{-1-\frac {1}{e^2 x}-x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 2*x + E^2*(9*x + 20*x^2 + 5*x^3) + (2 + E^2*(-6*x - 6*x^2))*Log[4*E^((-1 + E^2*(-x - x^2))/(E^
2*x))] + E^2*x*Log[4*E^((-1 + E^2*(-x - x^2))/(E^2*x))]^2)/(E^2*x),x]

[Out]

(2*(-3 + x))/E^2 + 2/(E^4*x) + x*(9 + 7*x + x^2) + (2/E^2 - 2*x*(3 + x))*Log[4*E^(-1 - 1/(E^2*x) - x)] + x*Log
[4*E^(-1 - 1/(E^2*x) - x)]^2

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fricas [A]  time = 0.70, size = 60, normalized size = 1.88 \begin {gather*} \frac {{\left (4 \, x^{2} e^{4} \log \relax (2)^{2} + 4 \, x^{2} e^{2} - 8 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{4} \log \relax (2) + {\left (4 \, x^{4} + 17 \, x^{3} + 16 \, x^{2}\right )} e^{4} + 1\right )} e^{\left (-4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(2)*log(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))^2+((-6*x^2-6*x)*exp(2)+2)*log(4*exp(((-x^2-x)*exp
(2)-1)/exp(2)/x))+(5*x^3+20*x^2+9*x)*exp(2)-2*x-6)/exp(2)/x,x, algorithm="fricas")

[Out]

(4*x^2*e^4*log(2)^2 + 4*x^2*e^2 - 8*(x^3 + 2*x^2)*e^4*log(2) + (4*x^4 + 17*x^3 + 16*x^2)*e^4 + 1)*e^(-4)/x

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giac [B]  time = 0.18, size = 63, normalized size = 1.97 \begin {gather*} {\left ({\left (4 \, x^{3} e^{8} - 8 \, x^{2} e^{8} \log \relax (2) + 4 \, x e^{8} \log \relax (2)^{2} + 17 \, x^{2} e^{8} - 16 \, x e^{8} \log \relax (2) + 16 \, x e^{8} + 4 \, x e^{6}\right )} e^{\left (-6\right )} + \frac {e^{\left (-2\right )}}{x}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(2)*log(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))^2+((-6*x^2-6*x)*exp(2)+2)*log(4*exp(((-x^2-x)*exp
(2)-1)/exp(2)/x))+(5*x^3+20*x^2+9*x)*exp(2)-2*x-6)/exp(2)/x,x, algorithm="giac")

[Out]

((4*x^3*e^8 - 8*x^2*e^8*log(2) + 4*x*e^8*log(2)^2 + 17*x^2*e^8 - 16*x*e^8*log(2) + 16*x*e^8 + 4*x*e^6)*e^(-6)
+ e^(-2)/x)*e^(-2)

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maple [B]  time = 0.14, size = 94, normalized size = 2.94

method	result	size
risch	\(9 x +x^{3}+7 x^{2}+4 x \ln \relax (2)^{2}-12 x \ln \relax (2)-4 x^{2} \ln \relax (2)+x \ln \left ({\mathrm e}^{-\frac {\left (x^{2} {\mathrm e}^{2}+{\mathrm e}^{2} x +1\right ) {\mathrm e}^{-2}}{x}}\right )^{2}-\left (-4 x \ln \relax (2)+2 x^{2}+6 x \right ) \ln \left ({\mathrm e}^{-\frac {\left (x^{2} {\mathrm e}^{2}+{\mathrm e}^{2} x +1\right ) {\mathrm e}^{-2}}{x}}\right )\)	\(94\)
default	\({\mathrm e}^{-2} \left (x^{3} {\mathrm e}^{2}+7 x^{2} {\mathrm e}^{2}+9 \,{\mathrm e}^{2} x +2 x -2 \,{\mathrm e}^{2} x^{2} \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )+x \,{\mathrm e}^{2} \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )^{2}+2 \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right )-6 \ln \left (4 \,{\mathrm e}^{\frac {\left (\left (-x^{2}-x \right ) {\mathrm e}^{2}-1\right ) {\mathrm e}^{-2}}{x}}\right ) {\mathrm e}^{2} x +\frac {2 \,{\mathrm e}^{-2}}{x}\right )\)	\(161\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*exp(2)*ln(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))^2+((-6*x^2-6*x)*exp(2)+2)*ln(4*exp(((-x^2-x)*exp(2)-1)/e
xp(2)/x))+(5*x^3+20*x^2+9*x)*exp(2)-2*x-6)/exp(2)/x,x,method=_RETURNVERBOSE)

[Out]

9*x+x^3+7*x^2+4*x*ln(2)^2-12*x*ln(2)-4*x^2*ln(2)+x*ln(exp(-(x^2*exp(2)+exp(2)*x+1)*exp(-2)/x))^2-(-4*x*ln(2)+2
*x^2+6*x)*ln(exp(-(x^2*exp(2)+exp(2)*x+1)*exp(-2)/x))

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maxima [B]  time = 0.36, size = 220, normalized size = 6.88 \begin {gather*} \frac {1}{3} \, {\left (2 \, x^{3} e^{2} - 9 \, x^{2} e^{2} \log \left (4 \, e^{\left (-x - \frac {e^{\left (-2\right )}}{x} - 1\right )}\right ) + 3 \, x e^{2} \log \left (4 \, e^{\left (-x - \frac {e^{\left (-2\right )}}{x} - 1\right )}\right )^{2} + 30 \, x^{2} e^{2} - 18 \, x e^{2} \log \left (4 \, e^{\left (-x - \frac {e^{\left (-2\right )}}{x} - 1\right )}\right ) - 9 \, {\left (x^{2} - e^{\left (-2\right )} \log \left (x^{2}\right )\right )} e^{2} + {\left (3 \, {\left (x^{2} - e^{\left (-2\right )} \log \left (x^{2}\right )\right )} \log \left (4 \, e^{\left (-x - \frac {e^{\left (-2\right )}}{x} - 1\right )}\right ) + \frac {{\left (x^{4} e^{4} + 3 \, x^{2} e^{2} - 6 \, {\left (x^{2} e^{2} + 1\right )} \log \relax (x) - 6\right )} e^{\left (-4\right )}}{x}\right )} e^{2} + 27 \, x e^{2} + 6 \, {\left (x + \frac {e^{\left (-2\right )}}{x}\right )} \log \relax (x) + 6 \, \log \relax (x) \log \left (4 \, e^{\left (-x - \frac {e^{\left (-2\right )}}{x} - 1\right )}\right ) - 3 \, x + \frac {6 \, e^{\left (-2\right )}}{x} - 18 \, \log \relax (x)\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(2)*log(4*exp(((-x^2-x)*exp(2)-1)/exp(2)/x))^2+((-6*x^2-6*x)*exp(2)+2)*log(4*exp(((-x^2-x)*exp
(2)-1)/exp(2)/x))+(5*x^3+20*x^2+9*x)*exp(2)-2*x-6)/exp(2)/x,x, algorithm="maxima")

[Out]

1/3*(2*x^3*e^2 - 9*x^2*e^2*log(4*e^(-x - e^(-2)/x - 1)) + 3*x*e^2*log(4*e^(-x - e^(-2)/x - 1))^2 + 30*x^2*e^2
- 18*x*e^2*log(4*e^(-x - e^(-2)/x - 1)) - 9*(x^2 - e^(-2)*log(x^2))*e^2 + (3*(x^2 - e^(-2)*log(x^2))*log(4*e^(
-x - e^(-2)/x - 1)) + (x^4*e^4 + 3*x^2*e^2 - 6*(x^2*e^2 + 1)*log(x) - 6)*e^(-4)/x)*e^2 + 27*x*e^2 + 6*(x + e^(
-2)/x)*log(x) + 6*log(x)*log(4*e^(-x - e^(-2)/x - 1)) - 3*x + 6*e^(-2)/x - 18*log(x))*e^(-2)

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mupad [B]  time = 4.90, size = 41, normalized size = 1.28 \begin {gather*} \frac {{\mathrm {e}}^{-4}}{x}-x^2\,\left (8\,\ln \relax (2)-17\right )+4\,x^3+x\,\left (4\,{\mathrm {e}}^{-2}-16\,\ln \relax (2)+4\,{\ln \relax (2)}^2+16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-2)*(2*x - exp(2)*(9*x + 20*x^2 + 5*x^3) + log(4*exp(-(exp(-2)*(exp(2)*(x + x^2) + 1))/x))*(exp(2)*(
6*x + 6*x^2) - 2) - x*exp(2)*log(4*exp(-(exp(-2)*(exp(2)*(x + x^2) + 1))/x))^2 + 6))/x,x)

[Out]

exp(-4)/x - x^2*(8*log(2) - 17) + 4*x^3 + x*(4*exp(-2) - 16*log(2) + 4*log(2)^2 + 16)

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sympy [B]  time = 0.42, size = 61, normalized size = 1.91 \begin {gather*} \frac {4 x^{3} e^{4} + x^{2} \left (- 8 e^{4} \log {\relax (2 )} + 17 e^{4}\right ) + x \left (- 16 e^{4} \log {\relax (2 )} + 4 e^{2} + 4 e^{4} \log {\relax (2 )}^{2} + 16 e^{4}\right ) + \frac {1}{x}}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*exp(2)*ln(4*exp(((-x**2-x)*exp(2)-1)/exp(2)/x))**2+((-6*x**2-6*x)*exp(2)+2)*ln(4*exp(((-x**2-x)*e
xp(2)-1)/exp(2)/x))+(5*x**3+20*x**2+9*x)*exp(2)-2*x-6)/exp(2)/x,x)

[Out]

(4*x**3*exp(4) + x**2*(-8*exp(4)*log(2) + 17*exp(4)) + x*(-16*exp(4)*log(2) + 4*exp(2) + 4*exp(4)*log(2)**2 +
16*exp(4)) + 1/x)*exp(-4)

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