\(\int \frac {e^{e^x x} (5+e^x+x) (1+e^{2 x} (1+x)+e^x (6+6 x+x^2))+2 e^x x \log ^2(7)+(10 x+2 x^2) \log ^2(7)}{5+e^x+x} \, dx\) [6189]

Optimal result

Integrand size = 71, antiderivative size = 23 \[ \int \frac {e^{e^x x} \left (5+e^x+x\right ) \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right )+2 e^x x \log ^2(7)+\left (10 x+2 x^2\right ) \log ^2(7)}{5+e^x+x} \, dx=e^{e^x x} \left (5+e^x+x\right )+x^2 \log ^2(7) \]

[Out]

exp(ln(exp(x)+5+x)+exp(x)*x)+x^2*ln(7)^2

Rubi [F]

\[ \int \frac {e^{e^x x} \left (5+e^x+x\right ) \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right )+2 e^x x \log ^2(7)+\left (10 x+2 x^2\right ) \log ^2(7)}{5+e^x+x} \, dx=\int \frac {e^{e^x x} \left (5+e^x+x\right ) \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right )+2 e^x x \log ^2(7)+\left (10 x+2 x^2\right ) \log ^2(7)}{5+e^x+x} \, dx \]

[In]

Int[(E^(E^x*x)*(5 + E^x + x)*(1 + E^(2*x)*(1 + x) + E^x*(6 + 6*x + x^2)) + 2*E^x*x*Log[7]^2 + (10*x + 2*x^2)*L
og[7]^2)/(5 + E^x + x),x]

[Out]

x^2*Log[7]^2 + Defer[Int][E^(E^x*x), x] + 6*Defer[Int][E^(x + E^x*x), x] + Defer[Int][E^(2*x + E^x*x), x] + 6*
Defer[Int][E^(x + E^x*x)*x, x] + Defer[Int][E^(2*x + E^x*x)*x, x] + Defer[Int][E^(x + E^x*x)*x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (e^{e^x x} \left (1+6 e^x+e^{2 x}+6 e^x x+e^{2 x} x+e^x x^2\right )+2 x \log ^2(7)\right ) \, dx \\ & = x^2 \log ^2(7)+\int e^{e^x x} \left (1+6 e^x+e^{2 x}+6 e^x x+e^{2 x} x+e^x x^2\right ) \, dx \\ & = x^2 \log ^2(7)+\int e^{e^x x} \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right ) \, dx \\ & = x^2 \log ^2(7)+\int \left (e^{e^x x}+e^{2 x+e^x x} (1+x)+e^{x+e^x x} \left (6+6 x+x^2\right )\right ) \, dx \\ & = x^2 \log ^2(7)+\int e^{e^x x} \, dx+\int e^{2 x+e^x x} (1+x) \, dx+\int e^{x+e^x x} \left (6+6 x+x^2\right ) \, dx \\ & = x^2 \log ^2(7)+\int e^{e^x x} \, dx+\int \left (e^{2 x+e^x x}+e^{2 x+e^x x} x\right ) \, dx+\int \left (6 e^{x+e^x x}+6 e^{x+e^x x} x+e^{x+e^x x} x^2\right ) \, dx \\ & = x^2 \log ^2(7)+6 \int e^{x+e^x x} \, dx+6 \int e^{x+e^x x} x \, dx+\int e^{e^x x} \, dx+\int e^{2 x+e^x x} \, dx+\int e^{2 x+e^x x} x \, dx+\int e^{x+e^x x} x^2 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {e^{e^x x} \left (5+e^x+x\right ) \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right )+2 e^x x \log ^2(7)+\left (10 x+2 x^2\right ) \log ^2(7)}{5+e^x+x} \, dx=e^{x+e^x x}+e^{e^x x} (5+x)+x^2 \log ^2(7) \]

[In]

Integrate[(E^(E^x*x)*(5 + E^x + x)*(1 + E^(2*x)*(1 + x) + E^x*(6 + 6*x + x^2)) + 2*E^x*x*Log[7]^2 + (10*x + 2*
x^2)*Log[7]^2)/(5 + E^x + x),x]

[Out]

E^(x + E^x*x) + E^(E^x*x)*(5 + x) + x^2*Log[7]^2

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

method	result	size
risch	\(\left ({\mathrm e}^{x}+5+x \right ) {\mathrm e}^{{\mathrm e}^{x} x}+x^{2} \ln \left (7\right )^{2}\)	\(21\)
parallelrisch	\({\mathrm e}^{\ln \left ({\mathrm e}^{x}+5+x \right )+{\mathrm e}^{x} x}+x^{2} \ln \left (7\right )^{2}\)	\(22\)

[In]

int((((1+x)*exp(x)^2+(x^2+6*x+6)*exp(x)+1)*exp(ln(exp(x)+5+x)+exp(x)*x)+2*x*ln(7)^2*exp(x)+(2*x^2+10*x)*ln(7)^
2)/(exp(x)+5+x),x,method=_RETURNVERBOSE)

[Out]

(exp(x)+5+x)*exp(exp(x)*x)+x^2*ln(7)^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {e^{e^x x} \left (5+e^x+x\right ) \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right )+2 e^x x \log ^2(7)+\left (10 x+2 x^2\right ) \log ^2(7)}{5+e^x+x} \, dx=x^{2} \log \left (7\right )^{2} + e^{\left (x e^{x} + \log \left (x + e^{x} + 5\right )\right )} \]

[In]

integrate((((1+x)*exp(x)^2+(x^2+6*x+6)*exp(x)+1)*exp(log(exp(x)+5+x)+exp(x)*x)+2*x*log(7)^2*exp(x)+(2*x^2+10*x
)*log(7)^2)/(exp(x)+5+x),x, algorithm="fricas")

[Out]

x^2*log(7)^2 + e^(x*e^x + log(x + e^x + 5))

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{e^x x} \left (5+e^x+x\right ) \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right )+2 e^x x \log ^2(7)+\left (10 x+2 x^2\right ) \log ^2(7)}{5+e^x+x} \, dx=x^{2} \log {\left (7 \right )}^{2} + \left (x + e^{x} + 5\right ) e^{x e^{x}} \]

[In]

integrate((((1+x)*exp(x)**2+(x**2+6*x+6)*exp(x)+1)*exp(ln(exp(x)+5+x)+exp(x)*x)+2*x*ln(7)**2*exp(x)+(2*x**2+10
*x)*ln(7)**2)/(exp(x)+5+x),x)

[Out]

x**2*log(7)**2 + (x + exp(x) + 5)*exp(x*exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{e^x x} \left (5+e^x+x\right ) \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right )+2 e^x x \log ^2(7)+\left (10 x+2 x^2\right ) \log ^2(7)}{5+e^x+x} \, dx=x^{2} \log \left (7\right )^{2} + {\left (x + e^{x} + 5\right )} e^{\left (x e^{x}\right )} \]

[In]

integrate((((1+x)*exp(x)^2+(x^2+6*x+6)*exp(x)+1)*exp(log(exp(x)+5+x)+exp(x)*x)+2*x*log(7)^2*exp(x)+(2*x^2+10*x
)*log(7)^2)/(exp(x)+5+x),x, algorithm="maxima")

[Out]

x^2*log(7)^2 + (x + e^x + 5)*e^(x*e^x)

Giac [F]

\[ \int \frac {e^{e^x x} \left (5+e^x+x\right ) \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right )+2 e^x x \log ^2(7)+\left (10 x+2 x^2\right ) \log ^2(7)}{5+e^x+x} \, dx=\int { \frac {2 \, x e^{x} \log \left (7\right )^{2} + 2 \, {\left (x^{2} + 5 \, x\right )} \log \left (7\right )^{2} + {\left ({\left (x + 1\right )} e^{\left (2 \, x\right )} + {\left (x^{2} + 6 \, x + 6\right )} e^{x} + 1\right )} e^{\left (x e^{x} + \log \left (x + e^{x} + 5\right )\right )}}{x + e^{x} + 5} \,d x } \]

[In]

integrate((((1+x)*exp(x)^2+(x^2+6*x+6)*exp(x)+1)*exp(log(exp(x)+5+x)+exp(x)*x)+2*x*log(7)^2*exp(x)+(2*x^2+10*x
)*log(7)^2)/(exp(x)+5+x),x, algorithm="giac")

[Out]

integrate((2*x*e^x*log(7)^2 + 2*(x^2 + 5*x)*log(7)^2 + ((x + 1)*e^(2*x) + (x^2 + 6*x + 6)*e^x + 1)*e^(x*e^x +
log(x + e^x + 5)))/(x + e^x + 5), x)

Mupad [B] (verification not implemented)

Time = 11.65 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {e^{e^x x} \left (5+e^x+x\right ) \left (1+e^{2 x} (1+x)+e^x \left (6+6 x+x^2\right )\right )+2 e^x x \log ^2(7)+\left (10 x+2 x^2\right ) \log ^2(7)}{5+e^x+x} \, dx=5\,{\mathrm {e}}^{x\,{\mathrm {e}}^x}+{\mathrm {e}}^{x+x\,{\mathrm {e}}^x}+x^2\,{\ln \left (7\right )}^2+x\,{\mathrm {e}}^{x\,{\mathrm {e}}^x} \]

[In]

int((exp(log(x + exp(x) + 5) + x*exp(x))*(exp(x)*(6*x + x^2 + 6) + exp(2*x)*(x + 1) + 1) + log(7)^2*(10*x + 2*
x^2) + 2*x*exp(x)*log(7)^2)/(x + exp(x) + 5),x)

[Out]

5*exp(x*exp(x)) + exp(x + x*exp(x)) + x^2*log(7)^2 + x*exp(x*exp(x))