3.17.58 $\int \frac{12+e^{e^x}(2-20e^{2x}+e^x(20-2x))+e^x(80+40x)}{36+e^{2e^x}+24x+4x^2+e^{e^x}(12+4x)}dx$

Optimal. Leaf size=22 $$\frac{20e^x+2x}{6+e^{e^x}+2x}$$

________________________________________________________________________________________

Rubi [F]  time = 1.74, 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 = {} $$\begin{array}{c}\int \frac{12+e^{e^x}(2-20e^{2x}+e^x(20-2x))+e^x(80+40x)}{36+e^{2e^x}+24x+4x^2+e^{e^x}(12+4x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(12 + E^E^x*(2 - 20*E^(2*x) + E^x*(20 - 2*x)) + E^x*(80 + 40*x))/(36 + E^(2*E^x) + 24*x + 4*x^2 + E^E^x*(1
2 + 4*x)),x]

[Out]

-40*Defer[Int][E^x/(6 + E^E^x + 2*x)^2, x] - 20*Defer[Int][E^(E^x + 2*x)/(6 + E^E^x + 2*x)^2, x] - 4*Defer[Int
][x/(6 + E^E^x + 2*x)^2, x] + 12*Defer[Int][(E^x*x)/(6 + E^E^x + 2*x)^2, x] + 4*Defer[Int][(E^x*x^2)/(6 + E^E^
x + 2*x)^2, x] + 2*Defer[Int][(6 + E^E^x + 2*x)^(-1), x] + 20*Defer[Int][E^x/(6 + E^E^x + 2*x), x] - 2*Defer[I
nt][(E^x*x)/(6 + E^E^x + 2*x), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{12+e^{e^x}(2-20e^{2x}+e^x(20-2x))+e^x(80+40x)}{{(6+e^{e^x}+2x)}^2}dx\\ & =\int (-\frac{20e^{e^x+2x}}{{(6+e^{e^x}+2x)}^2}+\frac{2(6+e^{e^x})}{{(6+e^{e^x}+2x)}^2}-\frac{2e^x(-40-10e^{e^x}-20x+e^{e^x}x)}{{(6+e^{e^x}+2x)}^2})dx\\ & =2\int \frac{6+e^{e^x}}{{(6+e^{e^x}+2x)}^2}dx-2\int \frac{e^x(-40-10e^{e^x}-20x+e^{e^x}x)}{{(6+e^{e^x}+2x)}^2}dx-20\int \frac{e^{e^x+2x}}{{(6+e^{e^x}+2x)}^2}dx\\ & =2\int (-\frac{2x}{{(6+e^{e^x}+2x)}^2}+\frac{1}{6+e^{e^x}+2x})dx-2\int (\frac{e^x(-10+x)}{6+e^{e^x}+2x}-\frac{2e^x(-10+3x+x^2)}{{(6+e^{e^x}+2x)}^2})dx-20\int \frac{e^{e^x+2x}}{{(6+e^{e^x}+2x)}^2}dx\\ & =2\int \frac{1}{6+e^{e^x}+2x}dx-2\int \frac{e^x(-10+x)}{6+e^{e^x}+2x}dx-4\int \frac{x}{{(6+e^{e^x}+2x)}^2}dx+4\int \frac{e^x(-10+3x+x^2)}{{(6+e^{e^x}+2x)}^2}dx-20\int \frac{e^{e^x+2x}}{{(6+e^{e^x}+2x)}^2}dx\\ & =2\int \frac{1}{6+e^{e^x}+2x}dx-2\int (-\frac{10e^x}{6+e^{e^x}+2x}+\frac{e^{x}x}{6+e^{e^x}+2x})dx-4\int \frac{x}{{(6+e^{e^x}+2x)}^2}dx+4\int (-\frac{10e^x}{{(6+e^{e^x}+2x)}^2}+\frac{3e^{x}x}{{(6+e^{e^x}+2x)}^2}+\frac{e^x{x}^2}{{(6+e^{e^x}+2x)}^2})dx-20\int \frac{e^{e^x+2x}}{{(6+e^{e^x}+2x)}^2}dx\\ & =2\int \frac{1}{6+e^{e^x}+2x}dx-2\int \frac{e^{x}x}{6+e^{e^x}+2x}dx-4\int \frac{x}{{(6+e^{e^x}+2x)}^2}dx+4\int \frac{e^x{x}^2}{{(6+e^{e^x}+2x)}^2}dx+12\int \frac{e^{x}x}{{(6+e^{e^x}+2x)}^2}dx-20\int \frac{e^{e^x+2x}}{{(6+e^{e^x}+2x)}^2}dx+20\int \frac{e^x}{6+e^{e^x}+2x}dx-40\int \frac{e^x}{{(6+e^{e^x}+2x)}^2}dx\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.41, size = 21, normalized size = 0.95 $$\begin{array}{c}\frac{2(10e^x+x)}{6+e^{e^x}+2x}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(12 + E^E^x*(2 - 20*E^(2*x) + E^x*(20 - 2*x)) + E^x*(80 + 40*x))/(36 + E^(2*E^x) + 24*x + 4*x^2 + E^
E^x*(12 + 4*x)),x]

[Out]

(2*(10*E^x + x))/(6 + E^E^x + 2*x)

________________________________________________________________________________________

fricas [A]  time = 0.76, size = 18, normalized size = 0.82 $$\begin{array}{c}\frac{2(x+10e^x)}{2x+e^{\left(e^x\right)}+6}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(x)^2+(-2*x+20)*exp(x)+2)*exp(exp(x))+(40*x+80)*exp(x)+12)/(exp(exp(x))^2+(4*x+12)*exp(exp(
x))+4*x^2+24*x+36),x, algorithm="fricas")

[Out]

2*(x + 10*e^x)/(2*x + e^(e^x) + 6)

________________________________________________________________________________________

giac [A]  time = 0.24, size = 18, normalized size = 0.82 $$\begin{array}{c}\frac{2(x+10e^x)}{2x+e^{\left(e^x\right)}+6}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(x)^2+(-2*x+20)*exp(x)+2)*exp(exp(x))+(40*x+80)*exp(x)+12)/(exp(exp(x))^2+(4*x+12)*exp(exp(
x))+4*x^2+24*x+36),x, algorithm="giac")

[Out]

2*(x + 10*e^x)/(2*x + e^(e^x) + 6)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 19, normalized size = 0.86

method	result	size
risch	$\frac{2x+20{\mathrm{e}}^x}{2x+{\mathrm{e}}^{{\mathrm{e}}^x}+6}$	$19$
norman	$\frac{2x+20{\mathrm{e}}^x}{2x+{\mathrm{e}}^{{\mathrm{e}}^x}+6}$	$20$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-20*exp(x)^2+(-2*x+20)*exp(x)+2)*exp(exp(x))+(40*x+80)*exp(x)+12)/(exp(exp(x))^2+(4*x+12)*exp(exp(x))+4*
x^2+24*x+36),x,method=_RETURNVERBOSE)

[Out]

2*(10*exp(x)+x)/(2*x+exp(exp(x))+6)

________________________________________________________________________________________

maxima [A]  time = 0.67, size = 18, normalized size = 0.82 $$\begin{array}{c}\frac{2(x+10e^x)}{2x+e^{\left(e^x\right)}+6}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(x)^2+(-2*x+20)*exp(x)+2)*exp(exp(x))+(40*x+80)*exp(x)+12)/(exp(exp(x))^2+(4*x+12)*exp(exp(
x))+4*x^2+24*x+36),x, algorithm="maxima")

[Out]

2*(x + 10*e^x)/(2*x + e^(e^x) + 6)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.05 $$\begin{array}{c}\int \frac{{\mathrm{e}}^x(40x+80)-{\mathrm{e}}^{{\mathrm{e}}^x}(20{\mathrm{e}}^{2x}+{\mathrm{e}}^x(2x-20)-2)+12}{24x+{\mathrm{e}}^{2{\mathrm{e}}^x}+4x^2+{\mathrm{e}}^{{\mathrm{e}}^x}(4x+12)+36}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(40*x + 80) - exp(exp(x))*(20*exp(2*x) + exp(x)*(2*x - 20) - 2) + 12)/(24*x + exp(2*exp(x)) + 4*x^
2 + exp(exp(x))*(4*x + 12) + 36),x)

[Out]

int((exp(x)*(40*x + 80) - exp(exp(x))*(20*exp(2*x) + exp(x)*(2*x - 20) - 2) + 12)/(24*x + exp(2*exp(x)) + 4*x^
2 + exp(exp(x))*(4*x + 12) + 36), x)

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 17, normalized size = 0.77 $$\begin{array}{c}\frac{2x+20e^x}{2x+e^{e^x}+6}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(x)**2+(-2*x+20)*exp(x)+2)*exp(exp(x))+(40*x+80)*exp(x)+12)/(exp(exp(x))**2+(4*x+12)*exp(ex
p(x))+4*x**2+24*x+36),x)

[Out]

(2*x + 20*exp(x))/(2*x + exp(exp(x)) + 6)

________________________________________________________________________________________