\(\int \frac {e^x x^3+x^4+e^{e^x} (-6 x^2-6 e^x x^2)+(e^{2 e^x} (2 x+2 e^x x)+e^{e^x} (6 x^2-6 e^{2 x} x^2+e^x (6 x-6 x^3))) \log (2 e^x+2 x)+e^{2 e^x} (-2 x+2 e^{2 x} x+e^x (-2+2 x^2)) \log ^2(2 e^x+2 x)}{e^x x^3+x^4} \, dx\) [3976]

Optimal result

Integrand size = 153, antiderivative size = 24 \[ \int \frac {e^x x^3+x^4+e^{e^x} \left (-6 x^2-6 e^x x^2\right )+\left (e^{2 e^x} \left (2 x+2 e^x x\right )+e^{e^x} \left (6 x^2-6 e^{2 x} x^2+e^x \left (6 x-6 x^3\right )\right )\right ) \log \left (2 e^x+2 x\right )+e^{2 e^x} \left (-2 x+2 e^{2 x} x+e^x \left (-2+2 x^2\right )\right ) \log ^2\left (2 e^x+2 x\right )}{e^x x^3+x^4} \, dx=x+\left (3-\frac {e^{e^x} \log \left (2 \left (e^x+x\right )\right )}{x}\right )^2 \]

[Out]

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

Rubi [F]

\[ \int \frac {e^x x^3+x^4+e^{e^x} \left (-6 x^2-6 e^x x^2\right )+\left (e^{2 e^x} \left (2 x+2 e^x x\right )+e^{e^x} \left (6 x^2-6 e^{2 x} x^2+e^x \left (6 x-6 x^3\right )\right )\right ) \log \left (2 e^x+2 x\right )+e^{2 e^x} \left (-2 x+2 e^{2 x} x+e^x \left (-2+2 x^2\right )\right ) \log ^2\left (2 e^x+2 x\right )}{e^x x^3+x^4} \, dx=\int \frac {e^x x^3+x^4+e^{e^x} \left (-6 x^2-6 e^x x^2\right )+\left (e^{2 e^x} \left (2 x+2 e^x x\right )+e^{e^x} \left (6 x^2-6 e^{2 x} x^2+e^x \left (6 x-6 x^3\right )\right )\right ) \log \left (2 e^x+2 x\right )+e^{2 e^x} \left (-2 x+2 e^{2 x} x+e^x \left (-2+2 x^2\right )\right ) \log ^2\left (2 e^x+2 x\right )}{e^x x^3+x^4} \, dx \]

[In]

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

[Out]

x + 6*Log[2*(E^x + x)]*Defer[Int][E^E^x/x^2, x] + 2*Log[2*(E^x + x)]*Defer[Int][E^(2*E^x)/x^2, x] - 6*Defer[In
t][E^E^x/x, x] - 6*Log[2*(E^x + x)]*Defer[Int][E^(E^x + x)/x, x] + 6*Defer[Int][E^E^x/(E^x + x), x] + 2*Log[2*
(E^x + x)]*Defer[Int][E^(2*E^x)/(x^2*(E^x + x)), x] - 6*Defer[Int][E^E^x/(x*(E^x + x)), x] - 2*Log[2*(E^x + x)
]*Defer[Int][E^(2*E^x)/(x*(E^x + x)), x] - 2*Defer[Int][(E^(2*E^x)*Log[2*(E^x + x)]^2)/x^3, x] + 2*Defer[Int][
(E^(2*E^x + x)*Log[2*(E^x + x)]^2)/x^2, x] - 6*Defer[Int][Defer[Int][E^E^x/x^2, x], x] - 6*Defer[Int][Defer[In
t][E^E^x/x^2, x]/(E^x + x), x] + 6*Defer[Int][(x*Defer[Int][E^E^x/x^2, x])/(E^x + x), x] - 2*Defer[Int][Defer[
Int][E^(2*E^x)/x^2, x], x] - 2*Defer[Int][Defer[Int][E^(2*E^x)/x^2, x]/(E^x + x), x] + 2*Defer[Int][(x*Defer[I
nt][E^(2*E^x)/x^2, x])/(E^x + x), x] + 6*Defer[Int][Defer[Int][E^(E^x + x)/x, x], x] + 6*Defer[Int][Defer[Int]
[E^(E^x + x)/x, x]/(E^x + x), x] - 6*Defer[Int][(x*Defer[Int][E^(E^x + x)/x, x])/(E^x + x), x] - 2*Defer[Int][
Defer[Int][E^(2*E^x)/(x^2*(E^x + x)), x], x] - 2*Defer[Int][Defer[Int][E^(2*E^x)/(x^2*(E^x + x)), x]/(E^x + x)
, x] + 2*Defer[Int][(x*Defer[Int][E^(2*E^x)/(x^2*(E^x + x)), x])/(E^x + x), x] + 2*Defer[Int][Defer[Int][E^(2*
E^x)/(x*(E^x + x)), x], x] + 2*Defer[Int][Defer[Int][E^(2*E^x)/(x*(E^x + x)), x]/(E^x + x), x] - 2*Defer[Int][
(x*Defer[Int][E^(2*E^x)/(x*(E^x + x)), x])/(E^x + x), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 e^{e^x} (-1+x) \left (-3 x+e^{e^x} \log \left (2 \left (e^x+x\right )\right )\right )}{x^2 \left (e^x+x\right )}+\frac {2 e^{e^x+x} \log \left (2 \left (e^x+x\right )\right ) \left (-3 x+e^{e^x} \log \left (2 \left (e^x+x\right )\right )\right )}{x^2}+\frac {-6 e^{e^x} x^2+x^3+6 e^{e^x} x \log \left (2 \left (e^x+x\right )\right )+2 e^{2 e^x} x \log \left (2 \left (e^x+x\right )\right )-2 e^{2 e^x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^3}\right ) \, dx \\ & = -\left (2 \int \frac {e^{e^x} (-1+x) \left (-3 x+e^{e^x} \log \left (2 \left (e^x+x\right )\right )\right )}{x^2 \left (e^x+x\right )} \, dx\right )+2 \int \frac {e^{e^x+x} \log \left (2 \left (e^x+x\right )\right ) \left (-3 x+e^{e^x} \log \left (2 \left (e^x+x\right )\right )\right )}{x^2} \, dx+\int \frac {-6 e^{e^x} x^2+x^3+6 e^{e^x} x \log \left (2 \left (e^x+x\right )\right )+2 e^{2 e^x} x \log \left (2 \left (e^x+x\right )\right )-2 e^{2 e^x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^3} \, dx \\ & = 2 \int \left (-\frac {3 e^{e^x+x} \log \left (2 \left (e^x+x\right )\right )}{x}+\frac {e^{2 e^x+x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^2}\right ) \, dx-2 \int \left (\frac {e^{e^x} \left (3 x-e^{e^x} \log \left (2 \left (e^x+x\right )\right )\right )}{x^2 \left (e^x+x\right )}-\frac {e^{e^x} \left (3 x-e^{e^x} \log \left (2 \left (e^x+x\right )\right )\right )}{x \left (e^x+x\right )}\right ) \, dx+\int \left (1-\frac {6 e^{e^x} \left (x-\log \left (2 \left (e^x+x\right )\right )\right )}{x^2}+\frac {2 e^{2 e^x} \left (x-\log \left (2 \left (e^x+x\right )\right )\right ) \log \left (2 \left (e^x+x\right )\right )}{x^3}\right ) \, dx \\ & = x+2 \int \frac {e^{2 e^x} \left (x-\log \left (2 \left (e^x+x\right )\right )\right ) \log \left (2 \left (e^x+x\right )\right )}{x^3} \, dx+2 \int \frac {e^{2 e^x+x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^2} \, dx-2 \int \frac {e^{e^x} \left (3 x-e^{e^x} \log \left (2 \left (e^x+x\right )\right )\right )}{x^2 \left (e^x+x\right )} \, dx+2 \int \frac {e^{e^x} \left (3 x-e^{e^x} \log \left (2 \left (e^x+x\right )\right )\right )}{x \left (e^x+x\right )} \, dx-6 \int \frac {e^{e^x} \left (x-\log \left (2 \left (e^x+x\right )\right )\right )}{x^2} \, dx-6 \int \frac {e^{e^x+x} \log \left (2 \left (e^x+x\right )\right )}{x} \, dx \\ & = x+2 \int \frac {e^{2 e^x+x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^2} \, dx-2 \int \left (\frac {3 e^{e^x}}{x \left (e^x+x\right )}-\frac {e^{2 e^x} \log \left (2 \left (e^x+x\right )\right )}{x^2 \left (e^x+x\right )}\right ) \, dx+2 \int \left (\frac {3 e^{e^x}}{e^x+x}-\frac {e^{2 e^x} \log \left (2 \left (e^x+x\right )\right )}{x \left (e^x+x\right )}\right ) \, dx+2 \int \left (\frac {e^{2 e^x} \log \left (2 \left (e^x+x\right )\right )}{x^2}-\frac {e^{2 e^x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^3}\right ) \, dx-6 \int \left (\frac {e^{e^x}}{x}-\frac {e^{e^x} \log \left (2 \left (e^x+x\right )\right )}{x^2}\right ) \, dx+6 \int \frac {\left (1+e^x\right ) \int \frac {e^{e^x+x}}{x} \, dx}{e^x+x} \, dx-\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x+x}}{x} \, dx \\ & = x+2 \int \frac {e^{2 e^x} \log \left (2 \left (e^x+x\right )\right )}{x^2} \, dx+2 \int \frac {e^{2 e^x} \log \left (2 \left (e^x+x\right )\right )}{x^2 \left (e^x+x\right )} \, dx-2 \int \frac {e^{2 e^x} \log \left (2 \left (e^x+x\right )\right )}{x \left (e^x+x\right )} \, dx-2 \int \frac {e^{2 e^x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^3} \, dx+2 \int \frac {e^{2 e^x+x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^2} \, dx-6 \int \frac {e^{e^x}}{x} \, dx+6 \int \frac {e^{e^x}}{e^x+x} \, dx-6 \int \frac {e^{e^x}}{x \left (e^x+x\right )} \, dx+6 \int \frac {e^{e^x} \log \left (2 \left (e^x+x\right )\right )}{x^2} \, dx+6 \int \left (\int \frac {e^{e^x+x}}{x} \, dx-\frac {(-1+x) \int \frac {e^{e^x+x}}{x} \, dx}{e^x+x}\right ) \, dx-\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x+x}}{x} \, dx \\ & = x-2 \int \frac {e^{2 e^x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^3} \, dx+2 \int \frac {e^{2 e^x+x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^2} \, dx-2 \int \frac {\left (1+e^x\right ) \int \frac {e^{2 e^x}}{x^2} \, dx}{e^x+x} \, dx-2 \int \frac {\left (1+e^x\right ) \int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx}{e^x+x} \, dx+2 \int \frac {\left (1+e^x\right ) \int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx}{e^x+x} \, dx-6 \int \frac {e^{e^x}}{x} \, dx+6 \int \frac {e^{e^x}}{e^x+x} \, dx-6 \int \frac {e^{e^x}}{x \left (e^x+x\right )} \, dx-6 \int \frac {\left (1+e^x\right ) \int \frac {e^{e^x}}{x^2} \, dx}{e^x+x} \, dx+6 \int \left (\int \frac {e^{e^x+x}}{x} \, dx\right ) \, dx-6 \int \frac {(-1+x) \int \frac {e^{e^x+x}}{x} \, dx}{e^x+x} \, dx+\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x^2} \, dx+\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx-\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx+\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x}}{x^2} \, dx-\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x+x}}{x} \, dx \\ & = x-2 \int \frac {e^{2 e^x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^3} \, dx+2 \int \frac {e^{2 e^x+x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^2} \, dx-2 \int \left (\int \frac {e^{2 e^x}}{x^2} \, dx-\frac {(-1+x) \int \frac {e^{2 e^x}}{x^2} \, dx}{e^x+x}\right ) \, dx-2 \int \left (\int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx-\frac {(-1+x) \int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx}{e^x+x}\right ) \, dx+2 \int \left (\int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx-\frac {(-1+x) \int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx}{e^x+x}\right ) \, dx-6 \int \frac {e^{e^x}}{x} \, dx+6 \int \frac {e^{e^x}}{e^x+x} \, dx-6 \int \frac {e^{e^x}}{x \left (e^x+x\right )} \, dx-6 \int \left (\int \frac {e^{e^x}}{x^2} \, dx-\frac {(-1+x) \int \frac {e^{e^x}}{x^2} \, dx}{e^x+x}\right ) \, dx+6 \int \left (\int \frac {e^{e^x+x}}{x} \, dx\right ) \, dx-6 \int \left (-\frac {\int \frac {e^{e^x+x}}{x} \, dx}{e^x+x}+\frac {x \int \frac {e^{e^x+x}}{x} \, dx}{e^x+x}\right ) \, dx+\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x^2} \, dx+\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx-\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx+\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x}}{x^2} \, dx-\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x+x}}{x} \, dx \\ & = x-2 \int \frac {e^{2 e^x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^3} \, dx+2 \int \frac {e^{2 e^x+x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^2} \, dx-2 \int \left (\int \frac {e^{2 e^x}}{x^2} \, dx\right ) \, dx+2 \int \frac {(-1+x) \int \frac {e^{2 e^x}}{x^2} \, dx}{e^x+x} \, dx-2 \int \left (\int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx\right ) \, dx+2 \int \frac {(-1+x) \int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx}{e^x+x} \, dx+2 \int \left (\int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx\right ) \, dx-2 \int \frac {(-1+x) \int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx}{e^x+x} \, dx-6 \int \frac {e^{e^x}}{x} \, dx+6 \int \frac {e^{e^x}}{e^x+x} \, dx-6 \int \frac {e^{e^x}}{x \left (e^x+x\right )} \, dx-6 \int \left (\int \frac {e^{e^x}}{x^2} \, dx\right ) \, dx+6 \int \frac {(-1+x) \int \frac {e^{e^x}}{x^2} \, dx}{e^x+x} \, dx+6 \int \left (\int \frac {e^{e^x+x}}{x} \, dx\right ) \, dx+6 \int \frac {\int \frac {e^{e^x+x}}{x} \, dx}{e^x+x} \, dx-6 \int \frac {x \int \frac {e^{e^x+x}}{x} \, dx}{e^x+x} \, dx+\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x^2} \, dx+\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx-\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx+\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x}}{x^2} \, dx-\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x+x}}{x} \, dx \\ & = x-2 \int \frac {e^{2 e^x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^3} \, dx+2 \int \frac {e^{2 e^x+x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^2} \, dx-2 \int \left (\int \frac {e^{2 e^x}}{x^2} \, dx\right ) \, dx+2 \int \left (-\frac {\int \frac {e^{2 e^x}}{x^2} \, dx}{e^x+x}+\frac {x \int \frac {e^{2 e^x}}{x^2} \, dx}{e^x+x}\right ) \, dx-2 \int \left (\int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx\right ) \, dx+2 \int \left (-\frac {\int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx}{e^x+x}+\frac {x \int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx}{e^x+x}\right ) \, dx+2 \int \left (\int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx\right ) \, dx-2 \int \left (-\frac {\int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx}{e^x+x}+\frac {x \int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx}{e^x+x}\right ) \, dx-6 \int \frac {e^{e^x}}{x} \, dx+6 \int \frac {e^{e^x}}{e^x+x} \, dx-6 \int \frac {e^{e^x}}{x \left (e^x+x\right )} \, dx-6 \int \left (\int \frac {e^{e^x}}{x^2} \, dx\right ) \, dx+6 \int \left (-\frac {\int \frac {e^{e^x}}{x^2} \, dx}{e^x+x}+\frac {x \int \frac {e^{e^x}}{x^2} \, dx}{e^x+x}\right ) \, dx+6 \int \left (\int \frac {e^{e^x+x}}{x} \, dx\right ) \, dx+6 \int \frac {\int \frac {e^{e^x+x}}{x} \, dx}{e^x+x} \, dx-6 \int \frac {x \int \frac {e^{e^x+x}}{x} \, dx}{e^x+x} \, dx+\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x^2} \, dx+\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx-\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx+\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x}}{x^2} \, dx-\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x+x}}{x} \, dx \\ & = x-2 \int \frac {e^{2 e^x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^3} \, dx+2 \int \frac {e^{2 e^x+x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^2} \, dx-2 \int \left (\int \frac {e^{2 e^x}}{x^2} \, dx\right ) \, dx-2 \int \frac {\int \frac {e^{2 e^x}}{x^2} \, dx}{e^x+x} \, dx+2 \int \frac {x \int \frac {e^{2 e^x}}{x^2} \, dx}{e^x+x} \, dx-2 \int \left (\int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx\right ) \, dx-2 \int \frac {\int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx}{e^x+x} \, dx+2 \int \frac {x \int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx}{e^x+x} \, dx+2 \int \left (\int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx\right ) \, dx+2 \int \frac {\int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx}{e^x+x} \, dx-2 \int \frac {x \int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx}{e^x+x} \, dx-6 \int \frac {e^{e^x}}{x} \, dx+6 \int \frac {e^{e^x}}{e^x+x} \, dx-6 \int \frac {e^{e^x}}{x \left (e^x+x\right )} \, dx-6 \int \left (\int \frac {e^{e^x}}{x^2} \, dx\right ) \, dx-6 \int \frac {\int \frac {e^{e^x}}{x^2} \, dx}{e^x+x} \, dx+6 \int \frac {x \int \frac {e^{e^x}}{x^2} \, dx}{e^x+x} \, dx+6 \int \left (\int \frac {e^{e^x+x}}{x} \, dx\right ) \, dx+6 \int \frac {\int \frac {e^{e^x+x}}{x} \, dx}{e^x+x} \, dx-6 \int \frac {x \int \frac {e^{e^x+x}}{x} \, dx}{e^x+x} \, dx+\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x^2} \, dx+\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x^2 \left (e^x+x\right )} \, dx-\left (2 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{2 e^x}}{x \left (e^x+x\right )} \, dx+\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x}}{x^2} \, dx-\left (6 \log \left (2 \left (e^x+x\right )\right )\right ) \int \frac {e^{e^x+x}}{x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {e^x x^3+x^4+e^{e^x} \left (-6 x^2-6 e^x x^2\right )+\left (e^{2 e^x} \left (2 x+2 e^x x\right )+e^{e^x} \left (6 x^2-6 e^{2 x} x^2+e^x \left (6 x-6 x^3\right )\right )\right ) \log \left (2 e^x+2 x\right )+e^{2 e^x} \left (-2 x+2 e^{2 x} x+e^x \left (-2+2 x^2\right )\right ) \log ^2\left (2 e^x+2 x\right )}{e^x x^3+x^4} \, dx=x-\frac {6 e^{e^x} \log \left (2 \left (e^x+x\right )\right )}{x}+\frac {e^{2 e^x} \log ^2\left (2 \left (e^x+x\right )\right )}{x^2} \]

[In]

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

[Out]

x - (6*E^E^x*Log[2*(E^x + x)])/x + (E^(2*E^x)*Log[2*(E^x + x)]^2)/x^2

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67

method	result	size
risch	\(\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}} \ln \left (2 \,{\mathrm e}^{x}+2 x \right )^{2}}{x^{2}}-\frac {6 \ln \left (2 \,{\mathrm e}^{x}+2 x \right ) {\mathrm e}^{{\mathrm e}^{x}}}{x}+x\)	\(40\)
parallelrisch	\(-\frac {-2 \,{\mathrm e}^{2 \,{\mathrm e}^{x}} \ln \left (2 \,{\mathrm e}^{x}+2 x \right )^{2}-2 x^{3}+12 \,{\mathrm e}^{{\mathrm e}^{x}} \ln \left (2 \,{\mathrm e}^{x}+2 x \right ) x}{2 x^{2}}\)	\(45\)

[In]

int(((2*x*exp(x)^2+(2*x^2-2)*exp(x)-2*x)*exp(exp(x))^2*ln(2*exp(x)+2*x)^2+((2*exp(x)*x+2*x)*exp(exp(x))^2+(-6*
exp(x)^2*x^2+(-6*x^3+6*x)*exp(x)+6*x^2)*exp(exp(x)))*ln(2*exp(x)+2*x)+(-6*exp(x)*x^2-6*x^2)*exp(exp(x))+exp(x)
*x^3+x^4)/(exp(x)*x^3+x^4),x,method=_RETURNVERBOSE)

[Out]

1/x^2*exp(exp(x))^2*ln(2*exp(x)+2*x)^2-6*ln(2*exp(x)+2*x)/x*exp(exp(x))+x

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {e^x x^3+x^4+e^{e^x} \left (-6 x^2-6 e^x x^2\right )+\left (e^{2 e^x} \left (2 x+2 e^x x\right )+e^{e^x} \left (6 x^2-6 e^{2 x} x^2+e^x \left (6 x-6 x^3\right )\right )\right ) \log \left (2 e^x+2 x\right )+e^{2 e^x} \left (-2 x+2 e^{2 x} x+e^x \left (-2+2 x^2\right )\right ) \log ^2\left (2 e^x+2 x\right )}{e^x x^3+x^4} \, dx=\frac {x^{3} - 6 \, x e^{\left (e^{x}\right )} \log \left (2 \, x + 2 \, e^{x}\right ) + e^{\left (2 \, e^{x}\right )} \log \left (2 \, x + 2 \, e^{x}\right )^{2}}{x^{2}} \]

[In]

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

[Out]

(x^3 - 6*x*e^(e^x)*log(2*x + 2*e^x) + e^(2*e^x)*log(2*x + 2*e^x)^2)/x^2

Sympy [F(-1)]

Timed out. \[ \int \frac {e^x x^3+x^4+e^{e^x} \left (-6 x^2-6 e^x x^2\right )+\left (e^{2 e^x} \left (2 x+2 e^x x\right )+e^{e^x} \left (6 x^2-6 e^{2 x} x^2+e^x \left (6 x-6 x^3\right )\right )\right ) \log \left (2 e^x+2 x\right )+e^{2 e^x} \left (-2 x+2 e^{2 x} x+e^x \left (-2+2 x^2\right )\right ) \log ^2\left (2 e^x+2 x\right )}{e^x x^3+x^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (22) = 44\).

Time = 0.32 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.58 \[ \int \frac {e^x x^3+x^4+e^{e^x} \left (-6 x^2-6 e^x x^2\right )+\left (e^{2 e^x} \left (2 x+2 e^x x\right )+e^{e^x} \left (6 x^2-6 e^{2 x} x^2+e^x \left (6 x-6 x^3\right )\right )\right ) \log \left (2 e^x+2 x\right )+e^{2 e^x} \left (-2 x+2 e^{2 x} x+e^x \left (-2+2 x^2\right )\right ) \log ^2\left (2 e^x+2 x\right )}{e^x x^3+x^4} \, dx=\frac {x^{3} - 6 \, x e^{\left (e^{x}\right )} \log \left (2\right ) + e^{\left (2 \, e^{x}\right )} \log \left (2\right )^{2} + e^{\left (2 \, e^{x}\right )} \log \left (x + e^{x}\right )^{2} - 2 \, {\left (3 \, x e^{\left (e^{x}\right )} - e^{\left (2 \, e^{x}\right )} \log \left (2\right )\right )} \log \left (x + e^{x}\right )}{x^{2}} \]

[In]

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

[Out]

(x^3 - 6*x*e^(e^x)*log(2) + e^(2*e^x)*log(2)^2 + e^(2*e^x)*log(x + e^x)^2 - 2*(3*x*e^(e^x) - e^(2*e^x)*log(2))
*log(x + e^x))/x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (22) = 44\).

Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.67 \[ \int \frac {e^x x^3+x^4+e^{e^x} \left (-6 x^2-6 e^x x^2\right )+\left (e^{2 e^x} \left (2 x+2 e^x x\right )+e^{e^x} \left (6 x^2-6 e^{2 x} x^2+e^x \left (6 x-6 x^3\right )\right )\right ) \log \left (2 e^x+2 x\right )+e^{2 e^x} \left (-2 x+2 e^{2 x} x+e^x \left (-2+2 x^2\right )\right ) \log ^2\left (2 e^x+2 x\right )}{e^x x^3+x^4} \, dx=\frac {x^{3} - 6 \, x e^{\left (e^{x}\right )} \log \left (2\right ) + e^{\left (2 \, e^{x}\right )} \log \left (2\right )^{2} - 6 \, x e^{\left (e^{x}\right )} \log \left (x + e^{x}\right ) + 2 \, e^{\left (2 \, e^{x}\right )} \log \left (2\right ) \log \left (x + e^{x}\right ) + e^{\left (2 \, e^{x}\right )} \log \left (x + e^{x}\right )^{2}}{x^{2}} \]

[In]

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

[Out]

(x^3 - 6*x*e^(e^x)*log(2) + e^(2*e^x)*log(2)^2 - 6*x*e^(e^x)*log(x + e^x) + 2*e^(2*e^x)*log(2)*log(x + e^x) +
e^(2*e^x)*log(x + e^x)^2)/x^2

Mupad [B] (verification not implemented)

Time = 9.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {e^x x^3+x^4+e^{e^x} \left (-6 x^2-6 e^x x^2\right )+\left (e^{2 e^x} \left (2 x+2 e^x x\right )+e^{e^x} \left (6 x^2-6 e^{2 x} x^2+e^x \left (6 x-6 x^3\right )\right )\right ) \log \left (2 e^x+2 x\right )+e^{2 e^x} \left (-2 x+2 e^{2 x} x+e^x \left (-2+2 x^2\right )\right ) \log ^2\left (2 e^x+2 x\right )}{e^x x^3+x^4} \, dx=x+\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,{\ln \left (2\,x+2\,{\mathrm {e}}^x\right )}^2}{x^2}-\frac {6\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\ln \left (2\,x+2\,{\mathrm {e}}^x\right )}{x} \]

[In]

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

[Out]

x + (exp(2*exp(x))*log(2*x + 2*exp(x))^2)/x^2 - (6*exp(exp(x))*log(2*x + 2*exp(x)))/x