∫ −x+2x2−8e2xx2−8x3+(2x−8e2xx−8x2) log(x)+(−2e2x−2x) log2(x)________________________________________________

Optimal. Leaf size=33 \[ \frac {11}{3}-e^{2 x}-x+(1-x) x+\frac {x}{2+\frac {\log (x)}{x}} \]

________________________________________________________________________________________

Rubi [F] time = 1.07, 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 {-x+2 x^2-8 e^{2 x} x^2-8 x^3+\left (2 x-8 e^{2 x} x-8 x^2\right ) \log (x)+\left (-2 e^{2 x}-2 x\right ) \log ^2(x)}{4 x^2+4 x \log (x)+\log ^2(x)} \, dx \end {gather*}

Int[(-x + 2*x^2 - 8*E^(2*x)*x^2 - 8*x^3 + (2*x - 8*E^(2*x)*x - 8*x^2)*Log[x] + (-2*E^(2*x) - 2*x)*Log[x]^2)/(4

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x+2 x^2-8 e^{2 x} x^2-8 x^3+\left (2 x-8 e^{2 x} x-8 x^2\right ) \log (x)+\left (-2 e^{2 x}-2 x\right ) \log ^2(x)}{(2 x+\log (x))^2} \, dx\\ &=\int \left (-2 e^{2 x}-\frac {x}{(2 x+\log (x))^2}+\frac {2 x^2}{(2 x+\log (x))^2}-\frac {8 x^3}{(2 x+\log (x))^2}+\frac {2 x \log (x)}{(2 x+\log (x))^2}-\frac {8 x^2 \log (x)}{(2 x+\log (x))^2}-\frac {2 x \log ^2(x)}{(2 x+\log (x))^2}\right ) \, dx\\ &=-\left (2 \int e^{2 x} \, dx\right )+2 \int \frac {x^2}{(2 x+\log (x))^2} \, dx+2 \int \frac {x \log (x)}{(2 x+\log (x))^2} \, dx-2 \int \frac {x \log ^2(x)}{(2 x+\log (x))^2} \, dx-8 \int \frac {x^3}{(2 x+\log (x))^2} \, dx-8 \int \frac {x^2 \log (x)}{(2 x+\log (x))^2} \, dx-\int \frac {x}{(2 x+\log (x))^2} \, dx\\ &=-e^{2 x}+2 \int \frac {x^2}{(2 x+\log (x))^2} \, dx+2 \int \left (-\frac {2 x^2}{(2 x+\log (x))^2}+\frac {x}{2 x+\log (x)}\right ) \, dx-2 \int \left (x+\frac {4 x^3}{(2 x+\log (x))^2}-\frac {4 x^2}{2 x+\log (x)}\right ) \, dx-8 \int \frac {x^3}{(2 x+\log (x))^2} \, dx-8 \int \left (-\frac {2 x^3}{(2 x+\log (x))^2}+\frac {x^2}{2 x+\log (x)}\right ) \, dx-\int \frac {x}{(2 x+\log (x))^2} \, dx\\ &=-e^{2 x}-x^2+2 \int \frac {x^2}{(2 x+\log (x))^2} \, dx+2 \int \frac {x}{2 x+\log (x)} \, dx-4 \int \frac {x^2}{(2 x+\log (x))^2} \, dx-2 \left (8 \int \frac {x^3}{(2 x+\log (x))^2} \, dx\right )+16 \int \frac {x^3}{(2 x+\log (x))^2} \, dx-\int \frac {x}{(2 x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A] time = 0.34, size = 25, normalized size = 0.76 \begin {gather*} -e^{2 x}-x^2+\frac {x^2}{2 x+\log (x)} \end {gather*}

Integrate[(-x + 2*x^2 - 8*E^(2*x)*x^2 - 8*x^3 + (2*x - 8*E^(2*x)*x - 8*x^2)*Log[x] + (-2*E^(2*x) - 2*x)*Log[x]

________________________________________________________________________________________

fricas [A] time = 0.94, size = 39, normalized size = 1.18 \begin {gather*} -\frac {2 \, x^{3} - x^{2} + 2 \, x e^{\left (2 \, x\right )} + {\left (x^{2} + e^{\left (2 \, x\right )}\right )} \log \relax (x)}{2 \, x + \log \relax (x)} \end {gather*}

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

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

________________________________________________________________________________________

giac [A] time = 0.26, size = 41, normalized size = 1.24 \begin {gather*} -\frac {2 \, x^{3} + x^{2} \log \relax (x) - x^{2} + 2 \, x e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )} \log \relax (x)}{2 \, x + \log \relax (x)} \end {gather*}

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

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

________________________________________________________________________________________


method	result	size

risch	\(-x^{2}-{\mathrm e}^{2 x}+\frac {x^{2}}{2 x +\ln \relax (x )}\)	\(25\)
default	\(\frac {x^{2}-2 x^{3}-x^{2} \ln \relax (x )}{2 x +\ln \relax (x )}+\frac {-2 x \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x} \ln \relax (x )}{2 x +\ln \relax (x )}\)	\(52\)

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

________________________________________________________________________________________

maxima [A] time = 0.51, size = 38, normalized size = 1.15 \begin {gather*} -\frac {2 \, x^{3} + x^{2} \log \relax (x) - x^{2} + {\left (2 \, x + \log \relax (x)\right )} e^{\left (2 \, x\right )}}{2 \, x + \log \relax (x)} \end {gather*}

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

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

________________________________________________________________________________________

mupad [B] time = 1.28, size = 28, normalized size = 0.85 \begin {gather*} \frac {x^3}{x\,\ln \relax (x)+2\,x^2}-x^2-{\mathrm {e}}^{2\,x} \end {gather*}

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

________________________________________________________________________________________

sympy [A] time = 0.31, size = 17, normalized size = 0.52 \begin {gather*} - x^{2} + \frac {x^{2}}{2 x + \log {\relax (x )}} - e^{2 x} \end {gather*}

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

________________________________________________________________________________________

3.19.12 \(\int \frac {-x+2 x^2-8 e^{2 x} x^2-8 x^3+(2 x-8 e^{2 x} x-8 x^2) \log (x)+(-2 e^{2 x}-2 x) \log ^2(x)}{4 x^2+4 x \log (x)+\log ^2(x)} \, dx\)