∫ −14+4e3+4e4+ex(−20+8e3+8e4−8x)+e2x(−6+4e3+4e4−4x)−4x+(2x+2e2xx) log(x)_____________________________________________________________

Optimal. Leaf size=33 \[ \frac {2 \left (-e^3-e^4+\frac {1}{2} \left (3+\frac {4}{1+e^x}\right )+x\right )}{\log ^2(x)} \]

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Rubi [F] time = 2.28, 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 {-14+4 e^3+4 e^4+e^x \left (-20+8 e^3+8 e^4-8 x\right )+e^{2 x} \left (-6+4 e^3+4 e^4-4 x\right )-4 x+\left (2 x+2 e^{2 x} x\right ) \log (x)}{\left (x+2 e^x x+e^{2 x} x\right ) \log ^3(x)} \, dx \end {gather*}

Int[(-14 + 4*E^3 + 4*E^4 + E^x*(-20 + 8*E^3 + 8*E^4 - 8*x) + E^(2*x)*(-6 + 4*E^3 + 4*E^4 - 4*x) - 4*x + (2*x +

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

(3 - 2*E^3 - 2*E^4)/Log[x]^2 + (2*x)/Log[x]^2 - 8*Defer[Int][1/((1 + E^x)*x*Log[x]^3), x] + 4*Defer[Int][1/((1

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

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-14 \left (1-\frac {2}{7} e^3 (1+e)\right )+e^x \left (-20+8 e^3+8 e^4-8 x\right )+e^{2 x} \left (-6+4 e^3+4 e^4-4 x\right )-4 x+\left (2 x+2 e^{2 x} x\right ) \log (x)}{\left (1+e^x\right )^2 x \log ^3(x)} \, dx\\ &=\int \left (\frac {4}{\left (1+e^x\right )^2 \log ^2(x)}-\frac {4 (2+x \log (x))}{\left (1+e^x\right ) x \log ^3(x)}+\frac {2 \left (-3 \left (1-\frac {2}{3} e^3 (1+e)\right )-2 x+x \log (x)\right )}{x \log ^3(x)}\right ) \, dx\\ &=2 \int \frac {-3 \left (1-\frac {2}{3} e^3 (1+e)\right )-2 x+x \log (x)}{x \log ^3(x)} \, dx+4 \int \frac {1}{\left (1+e^x\right )^2 \log ^2(x)} \, dx-4 \int \frac {2+x \log (x)}{\left (1+e^x\right ) x \log ^3(x)} \, dx\\ &=2 \int \left (\frac {-3+2 e^3+2 e^4-2 x}{x \log ^3(x)}+\frac {1}{\log ^2(x)}\right ) \, dx-4 \int \left (\frac {2}{\left (1+e^x\right ) x \log ^3(x)}+\frac {1}{\left (1+e^x\right ) \log ^2(x)}\right ) \, dx+4 \int \frac {1}{\left (1+e^x\right )^2 \log ^2(x)} \, dx\\ &=2 \int \frac {-3+2 e^3+2 e^4-2 x}{x \log ^3(x)} \, dx+2 \int \frac {1}{\log ^2(x)} \, dx+4 \int \frac {1}{\left (1+e^x\right )^2 \log ^2(x)} \, dx-4 \int \frac {1}{\left (1+e^x\right ) \log ^2(x)} \, dx-8 \int \frac {1}{\left (1+e^x\right ) x \log ^3(x)} \, dx\\ &=-\frac {2 x}{\log (x)}+2 \int \left (-\frac {2}{\log ^3(x)}+\frac {-3+2 e^3+2 e^4}{x \log ^3(x)}\right ) \, dx+2 \int \frac {1}{\log (x)} \, dx+4 \int \frac {1}{\left (1+e^x\right )^2 \log ^2(x)} \, dx-4 \int \frac {1}{\left (1+e^x\right ) \log ^2(x)} \, dx-8 \int \frac {1}{\left (1+e^x\right ) x \log ^3(x)} \, dx\\ &=-\frac {2 x}{\log (x)}+2 \text {li}(x)-4 \int \frac {1}{\log ^3(x)} \, dx+4 \int \frac {1}{\left (1+e^x\right )^2 \log ^2(x)} \, dx-4 \int \frac {1}{\left (1+e^x\right ) \log ^2(x)} \, dx-8 \int \frac {1}{\left (1+e^x\right ) x \log ^3(x)} \, dx-\left (2 \left (3-2 e^3-2 e^4\right )\right ) \int \frac {1}{x \log ^3(x)} \, dx\\ &=\frac {2 x}{\log ^2(x)}-\frac {2 x}{\log (x)}+2 \text {li}(x)-2 \int \frac {1}{\log ^2(x)} \, dx+4 \int \frac {1}{\left (1+e^x\right )^2 \log ^2(x)} \, dx-4 \int \frac {1}{\left (1+e^x\right ) \log ^2(x)} \, dx-8 \int \frac {1}{\left (1+e^x\right ) x \log ^3(x)} \, dx-\left (2 \left (3-2 e^3-2 e^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (x)\right )\\ &=\frac {3-2 e^3-2 e^4}{\log ^2(x)}+\frac {2 x}{\log ^2(x)}+2 \text {li}(x)-2 \int \frac {1}{\log (x)} \, dx+4 \int \frac {1}{\left (1+e^x\right )^2 \log ^2(x)} \, dx-4 \int \frac {1}{\left (1+e^x\right ) \log ^2(x)} \, dx-8 \int \frac {1}{\left (1+e^x\right ) x \log ^3(x)} \, dx\\ &=\frac {3-2 e^3-2 e^4}{\log ^2(x)}+\frac {2 x}{\log ^2(x)}+4 \int \frac {1}{\left (1+e^x\right )^2 \log ^2(x)} \, dx-4 \int \frac {1}{\left (1+e^x\right ) \log ^2(x)} \, dx-8 \int \frac {1}{\left (1+e^x\right ) x \log ^3(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.36, size = 53, normalized size = 1.61 \begin {gather*} -\frac {-7+2 e^3+2 e^4-3 e^x+2 e^{3+x}+2 e^{4+x}-2 x-2 e^x x}{\left (1+e^x\right ) \log ^2(x)} \end {gather*}

Integrate[(-14 + 4*E^3 + 4*E^4 + E^x*(-20 + 8*E^3 + 8*E^4 - 8*x) + E^(2*x)*(-6 + 4*E^3 + 4*E^4 - 4*x) - 4*x +

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

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

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fricas [A] time = 0.63, size = 40, normalized size = 1.21 \begin {gather*} \frac {{\left (2 \, x - 2 \, e^{4} - 2 \, e^{3} + 3\right )} e^{x} + 2 \, x - 2 \, e^{4} - 2 \, e^{3} + 7}{{\left (e^{x} + 1\right )} \log \relax (x)^{2}} \end {gather*}

integrate(((2*x*exp(x)^2+2*x)*log(x)+(4*exp(4)+4*exp(3)-4*x-6)*exp(x)^2+(8*exp(4)+8*exp(3)-8*x-20)*exp(x)+4*ex

p(4)+4*exp(3)-4*x-14)/(x*exp(x)^2+2*exp(x)*x+x)/log(x)^3,x, algorithm="fricas")

((2*x - 2*e^4 - 2*e^3 + 3)*e^x + 2*x - 2*e^4 - 2*e^3 + 7)/((e^x + 1)*log(x)^2)

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giac [A] time = 0.18, size = 49, normalized size = 1.48 \begin {gather*} \frac {2 \, x e^{x} + 2 \, x - 2 \, e^{4} - 2 \, e^{3} - 2 \, e^{\left (x + 4\right )} - 2 \, e^{\left (x + 3\right )} + 3 \, e^{x} + 7}{e^{x} \log \relax (x)^{2} + \log \relax (x)^{2}} \end {gather*}

integrate(((2*x*exp(x)^2+2*x)*log(x)+(4*exp(4)+4*exp(3)-4*x-6)*exp(x)^2+(8*exp(4)+8*exp(3)-8*x-20)*exp(x)+4*ex

p(4)+4*exp(3)-4*x-14)/(x*exp(x)^2+2*exp(x)*x+x)/log(x)^3,x, algorithm="giac")

(2*x*e^x + 2*x - 2*e^4 - 2*e^3 - 2*e^(x + 4) - 2*e^(x + 3) + 3*e^x + 7)/(e^x*log(x)^2 + log(x)^2)

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method	result	size

risch	\(-\frac {2 \,{\mathrm e}^{4+x}+2 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{3+x}+2 \,{\mathrm e}^{3}-2 \,{\mathrm e}^{x} x -2 x -3 \,{\mathrm e}^{x}-7}{\left ({\mathrm e}^{x}+1\right ) \ln \relax (x )^{2}}\)	\(47\)

int(((2*x*exp(x)^2+2*x)*ln(x)+(4*exp(4)+4*exp(3)-4*x-6)*exp(x)^2+(8*exp(4)+8*exp(3)-8*x-20)*exp(x)+4*exp(4)+4*

exp(3)-4*x-14)/(x*exp(x)^2+2*exp(x)*x+x)/ln(x)^3,x,method=_RETURNVERBOSE)

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

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maxima [A] time = 0.39, size = 44, normalized size = 1.33 \begin {gather*} \frac {{\left (2 \, x - 2 \, e^{4} - 2 \, e^{3} + 3\right )} e^{x} + 2 \, x - 2 \, e^{4} - 2 \, e^{3} + 7}{e^{x} \log \relax (x)^{2} + \log \relax (x)^{2}} \end {gather*}

integrate(((2*x*exp(x)^2+2*x)*log(x)+(4*exp(4)+4*exp(3)-4*x-6)*exp(x)^2+(8*exp(4)+8*exp(3)-8*x-20)*exp(x)+4*ex

p(4)+4*exp(3)-4*x-14)/(x*exp(x)^2+2*exp(x)*x+x)/log(x)^3,x, algorithm="maxima")

((2*x - 2*e^4 - 2*e^3 + 3)*e^x + 2*x - 2*e^4 - 2*e^3 + 7)/(e^x*log(x)^2 + log(x)^2)

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mupad [B] time = 4.62, size = 213, normalized size = 6.45 \begin {gather*} x-\frac {2\,x^2+2\,x}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}+\frac {2\,x+{\mathrm {e}}^{2\,x}\,\left (\frac {2\,x^2}{3}-2\,x+\frac {2}{3}\right )-{\mathrm {e}}^x\,\left (\frac {8\,x^2}{3}-\frac {4}{3}\right )+\frac {2\,x^2}{3}+\frac {2}{3}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}+\frac {\frac {4\,x^2}{3}-\frac {2}{3}}{{\mathrm {e}}^x+1}+\frac {\frac {2\,x-2\,{\mathrm {e}}^{x+3}-2\,{\mathrm {e}}^{x+4}-2\,{\mathrm {e}}^3-2\,{\mathrm {e}}^4+3\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^x+7}{{\mathrm {e}}^x+1}-\frac {x\,\ln \relax (x)\,\left ({\mathrm {e}}^{2\,x}+1\right )}{{\left ({\mathrm {e}}^x+1\right )}^2}}{{\ln \relax (x)}^2}+\frac {\frac {x\,\left ({\mathrm {e}}^{2\,x}+1\right )}{{\left ({\mathrm {e}}^x+1\right )}^2}-\frac {x\,\ln \relax (x)\,\left ({\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^{2\,x}-2\,x\,{\mathrm {e}}^x+1\right )}{{\left ({\mathrm {e}}^x+1\right )}^3}}{\ln \relax (x)} \end {gather*}

int(-(4*x - 4*exp(3) - 4*exp(4) + exp(2*x)*(4*x - 4*exp(3) - 4*exp(4) + 6) - log(x)*(2*x + 2*x*exp(2*x)) + exp

(x)*(8*x - 8*exp(3) - 8*exp(4) + 20) + 14)/(log(x)^3*(x + x*exp(2*x) + 2*x*exp(x))),x)

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

3) + (2*x^2)/3 + 2/3)/(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1) + ((4*x^2)/3 - 2/3)/(exp(x) + 1) + ((2*x - 2*exp(

x + 3) - 2*exp(x + 4) - 2*exp(3) - 2*exp(4) + 3*exp(x) + 2*x*exp(x) + 7)/(exp(x) + 1) - (x*log(x)*(exp(2*x) +

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

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

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sympy [A] time = 0.31, size = 34, normalized size = 1.03 \begin {gather*} \frac {2 x - 2 e^{4} - 2 e^{3} + 3}{\log {\relax (x )}^{2}} + \frac {4}{e^{x} \log {\relax (x )}^{2} + \log {\relax (x )}^{2}} \end {gather*}

integrate(((2*x*exp(x)**2+2*x)*ln(x)+(4*exp(4)+4*exp(3)-4*x-6)*exp(x)**2+(8*exp(4)+8*exp(3)-8*x-20)*exp(x)+4*e

xp(4)+4*exp(3)-4*x-14)/(x*exp(x)**2+2*exp(x)*x+x)/ln(x)**3,x)

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

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3.73.20 \(\int \frac {-14+4 e^3+4 e^4+e^x (-20+8 e^3+8 e^4-8 x)+e^{2 x} (-6+4 e^3+4 e^4-4 x)-4 x+(2 x+2 e^{2 x} x) \log (x)}{(x+2 e^x x+e^{2 x} x) \log ^3(x)} \, dx\)