3.35.74 \(\int \frac {e^x x-2 x^2-\log (\frac {3}{2 x})+(-1-8 x^2+2 x^3+e^x (2 x+x^2)+x \log (\frac {3}{2 x})) \log (x)+x \log ^2(x)}{(-e^x x^2+2 x^3+x \log (\frac {3}{2 x})) \log (x)+x \log ^2(x)} \, dx\)

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

________________________________________________________________________________________

Rubi [F]  time = 3.28, 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^x x-2 x^2-\log \left (\frac {3}{2 x}\right )+\left (-1-8 x^2+2 x^3+e^x \left (2 x+x^2\right )+x \log \left (\frac {3}{2 x}\right )\right ) \log (x)+x \log ^2(x)}{\left (-e^x x^2+2 x^3+x \log \left (\frac {3}{2 x}\right )\right ) \log (x)+x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*x - 2*x^2 - Log[3/(2*x)] + (-1 - 8*x^2 + 2*x^3 + E^x*(2*x + x^2) + x*Log[3/(2*x)])*Log[x] + x*Log[x]^
2)/((-(E^x*x^2) + 2*x^3 + x*Log[3/(2*x)])*Log[x] + x*Log[x]^2),x]

[Out]

-x - 2*Log[x] - Log[Log[x]] - 4*Defer[Int][x/(-(E^x*x) + 2*x^2 + Log[3/(2*x)] + Log[x]), x] + 4*Defer[Int][x^2
/(-(E^x*x) + 2*x^2 + Log[3/(2*x)] + Log[x]), x] + 2*Defer[Int][Log[3/(2*x)]/(-(E^x*x) + 2*x^2 + Log[3/(2*x)] +
Log[x]), x] + 2*Defer[Int][Log[3/(2*x)]/(x*(-(E^x*x) + 2*x^2 + Log[3/(2*x)] + Log[x])), x] + 2*Defer[Int][Log
[x]/(-(E^x*x) + 2*x^2 + Log[3/(2*x)] + Log[x]), x] + 2*Defer[Int][Log[x]/(x*(-(E^x*x) + 2*x^2 + Log[3/(2*x)] +
Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )-\left (-1-8 x^2+2 x^3+e^x \left (2 x+x^2\right )+x \log \left (\frac {3}{2 x}\right )\right ) \log (x)-x \log ^2(x)}{x \left (e^x x-2 x^2-\log \left (\frac {3}{2 x}\right )-\log (x)\right ) \log (x)} \, dx\\ &=\int \left (\frac {-1-2 \log (x)-x \log (x)}{x \log (x)}+\frac {2 \left (-2 x^2+2 x^3+\log \left (\frac {3}{2 x}\right )+x \log \left (\frac {3}{2 x}\right )+\log (x)+x \log (x)\right )}{x \left (-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)\right )}\right ) \, dx\\ &=2 \int \frac {-2 x^2+2 x^3+\log \left (\frac {3}{2 x}\right )+x \log \left (\frac {3}{2 x}\right )+\log (x)+x \log (x)}{x \left (-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)\right )} \, dx+\int \frac {-1-2 \log (x)-x \log (x)}{x \log (x)} \, dx\\ &=2 \int \left (-\frac {2 x}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)}+\frac {2 x^2}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)}+\frac {\log \left (\frac {3}{2 x}\right )}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)}+\frac {\log \left (\frac {3}{2 x}\right )}{x \left (-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)\right )}+\frac {\log (x)}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)}+\frac {\log (x)}{x \left (-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)\right )}\right ) \, dx+\int \left (\frac {-2-x}{x}-\frac {1}{x \log (x)}\right ) \, dx\\ &=2 \int \frac {\log \left (\frac {3}{2 x}\right )}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx+2 \int \frac {\log \left (\frac {3}{2 x}\right )}{x \left (-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)\right )} \, dx+2 \int \frac {\log (x)}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx+2 \int \frac {\log (x)}{x \left (-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)\right )} \, dx-4 \int \frac {x}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx+4 \int \frac {x^2}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx+\int \frac {-2-x}{x} \, dx-\int \frac {1}{x \log (x)} \, dx\\ &=2 \int \frac {\log \left (\frac {3}{2 x}\right )}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx+2 \int \frac {\log \left (\frac {3}{2 x}\right )}{x \left (-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)\right )} \, dx+2 \int \frac {\log (x)}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx+2 \int \frac {\log (x)}{x \left (-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)\right )} \, dx-4 \int \frac {x}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx+4 \int \frac {x^2}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx+\int \left (-1-\frac {2}{x}\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-x-2 \log (x)-\log (\log (x))+2 \int \frac {\log \left (\frac {3}{2 x}\right )}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx+2 \int \frac {\log \left (\frac {3}{2 x}\right )}{x \left (-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)\right )} \, dx+2 \int \frac {\log (x)}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx+2 \int \frac {\log (x)}{x \left (-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)\right )} \, dx-4 \int \frac {x}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx+4 \int \frac {x^2}{-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 32, normalized size = 1.00 \begin {gather*} x-\log (\log (x))-2 \log \left (-e^x x+2 x^2+\log \left (\frac {3}{2 x}\right )+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*x - 2*x^2 - Log[3/(2*x)] + (-1 - 8*x^2 + 2*x^3 + E^x*(2*x + x^2) + x*Log[3/(2*x)])*Log[x] + x*L
og[x]^2)/((-(E^x*x^2) + 2*x^3 + x*Log[3/(2*x)])*Log[x] + x*Log[x]^2),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.67, size = 41, normalized size = 1.28 \begin {gather*} x - 2 \, \log \relax (x) - 2 \, \log \left (-\frac {2 \, x^{2} - x e^{x} + \log \left (\frac {3}{2}\right )}{x}\right ) - \log \left (-\log \left (\frac {3}{2}\right ) + \log \left (\frac {3}{2 \, x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.41, size = 26, normalized size = 0.81 \begin {gather*} x - 2 \, \log \left (-2 \, x^{2} + x e^{x} - \log \relax (3) + \log \relax (2)\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*log(-2*x^2 + x*e^x - log(3) + log(2)) - log(log(x))

________________________________________________________________________________________

maple [A]  time = 0.13, size = 30, normalized size = 0.94

method	result	size
default	\(x -\ln \left (\ln \relax (x )\right )-2 \ln \left (-{\mathrm e}^{x} x +2 x^{2}+\ln \relax (x )+\ln \left (\frac {3}{2 x}\right )\right )\)	\(30\)
risch	\(x -2 \ln \relax (x )-2 \ln \left ({\mathrm e}^{x}+\frac {i \left (4 i x^{2}+2 i \ln \relax (3)-2 i \ln \relax (2)\right )}{2 x}\right )-\ln \left (\ln \relax (x )\right )\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.67, size = 36, normalized size = 1.12 \begin {gather*} x - 2 \, \log \relax (x) - 2 \, \log \left (-\frac {2 \, x^{2} - x e^{x} + \log \relax (3) - \log \relax (2)}{x}\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 2.41, size = 37, normalized size = 1.16 \begin {gather*} x-\ln \left (\ln \relax (x)\right )-2\,\ln \left (\frac {\ln \left (\frac {3}{2\,x}\right )+\ln \relax (x)-x\,{\mathrm {e}}^x+2\,x^2}{x}\right )-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.48, size = 31, normalized size = 0.97 \begin {gather*} x - 2 \log {\relax (x )} - 2 \log {\left (e^{x} + \frac {- 2 x^{2} - \log {\relax (3 )} + \log {\relax (2 )}}{x} \right )} - \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________