3.73.91 $\int \frac{16x+4e^{4}x+16x^2+x^3+e^{2x-2e^{2}x}(16+x)+e^{x-e^{2}x}(-52x+2x^2+e^4(-4x+4e^{2}x)+e^2(16x-4x^2))}{e^{2x-2e^{2}x}x-2e^{x-e^{2}x}{x}^2+x^3}dx$

Optimal. Leaf size=32 $$x-\frac{4(4+e^4-x)}{-e^{x-e^{2}x}+x}+16\log (x)$$

________________________________________________________________________________________

Rubi [F]  time = 6.91, 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 = {} $$\begin{array}{c}\int \frac{16x+4e^{4}x+16x^2+x^3+e^{2x-2e^{2}x}(16+x)+e^{x-e^{2}x}(-52x+2x^2+e^4(-4x+4e^{2}x)+e^2(16x-4x^2))}{e^{2x-2e^{2}x}x-2e^{x-e^{2}x}{x}^2+x^3}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(16*x + 4*E^4*x + 16*x^2 + x^3 + E^(2*x - 2*E^2*x)*(16 + x) + E^(x - E^2*x)*(-52*x + 2*x^2 + E^4*(-4*x + 4
*E^2*x) + E^2*(16*x - 4*x^2)))/(E^(2*x - 2*E^2*x)*x - 2*E^(x - E^2*x)*x^2 + x^3),x]

[Out]

-4/(E^((1 - E^2)*x)*(1 - E^2)) + x + (4*(5 - 4*E^2 + E^4 - E^6 - (1 - E^2)*x))/(E^((1 - E^2)*x)*(1 - E^2)) + 1
6*Log[x] + 4*(4 + E^4)*Defer[Int][E^(2*E^2*x)/(E^x - E^(E^2*x)*x)^2, x] - 4*(5 - 4*E^2 + E^4 - E^6)*Defer[Int]
[(E^(2*E^2*x)*x)/(-E^x + E^(E^2*x)*x)^2, x] + 4*(1 - E)*(1 + E)*Defer[Int][(E^(2*E^2*x)*x^2)/(-E^x + E^(E^2*x)
*x)^2, x] + 4*(5 - 4*E^2 + E^4 - E^6)*Defer[Int][x/(E^((1 - 2*E^2)*x)*(-E^x + E^(E^2*x)*x)), x] - 4*(1 - E)*(1
+ E)*Defer[Int][x^2/(E^((1 - 2*E^2)*x)*(-E^x + E^(E^2*x)*x)), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{(16+4e^4)x+16x^2+x^3+e^{2x-2e^{2}x}(16+x)+e^{x-e^{2}x}(-52x+2x^2+e^4(-4x+4e^{2}x)+e^2(16x-4x^2))}{e^{2x-2e^{2}x}x-2e^{x-e^{2}x}{x}^2+x^3}dx\\ & =\int \frac{e^{2e^{2}x}((16+4e^4)x+16x^2+x^3+e^{2x-2e^{2}x}(16+x)+e^{x-e^{2}x}(-52x+2x^2+e^4(-4x+4e^{2}x)+e^2(16x-4x^2)))}{x{(e^x-e^{e^{2}x}x)}^2}dx\\ & =\int (\frac{16+x}{x}+4e^{2e^{2}x-(1+e^2)x}(-5+4e^2-e^4+e^6+(1-e^2)x)+\frac{4e^{-x+2e^{2}x}x(-5+4e^2-e^4+e^6+(1-e^2)x)}{e^x-e^{e^{2}x}x}+\frac{4e^{2e^{2}x}(4+e^4-(5-4e^2+e^4-e^6)x+(1-e^2)x^2)}{{(e^x-e^{e^{2}x}x)}^2})dx\\ & =4\int e^{2e^{2}x-(1+e^2)x}(-5+4e^2-e^4+e^6+(1-e^2)x)dx+4\int \frac{e^{-x+2e^{2}x}x(-5+4e^2-e^4+e^6+(1-e^2)x)}{e^x-e^{e^{2}x}x}dx+4\int \frac{e^{2e^{2}x}(4+e^4-(5-4e^2+e^4-e^6)x+(1-e^2)x^2)}{{(e^x-e^{e^{2}x}x)}^2}dx+\int \frac{16+x}{x}dx\\ & =4\int e^{-\left((1-e^2)x\right)}(-5+4e^2-e^4+e^6+(1-e^2)x)dx+4\int \frac{e^{-\left((1-2e^2)x\right)}x(-5+4e^2-e^4+e^6+(1-e^2)x)}{e^x-e^{e^{2}x}x}dx+4\int (\frac{4e^{2e^{2}x}(1+\frac{e^4}{4})}{{(e^x-e^{e^{2}x}x)}^2}+\frac{e^{2e^{2}x}(-5+4e^2-e^4+e^6)x}{{(-e^x+e^{e^{2}x}x)}^2}-\frac{(-1+e)e^{2e^{2}x}(1+e)x^2}{{(-e^x+e^{e^{2}x}x)}^2})dx+\int (1+\frac{16}{x})dx\\ & =x+\frac{4e^{-\left((1-e^2)x\right)}(5-4e^2+e^4-e^6-(1-e^2)x)}{1-e^2}+16\log (x)+4\int e^{(-1+e^2)x}dx+4\int (-\frac{e^{-\left((1-2e^2)x\right)}(-5+4e^2-e^4+e^6)x}{-e^x+e^{e^{2}x}x}+\frac{(-1+e)e^{-\left((1-2e^2)x\right)}(1+e)x^2}{-e^x+e^{e^{2}x}x})dx-(4(-1+e)(1+e))\int \frac{e^{2e^{2}x}{x}^2}{{(-e^x+e^{e^{2}x}x)}^2}dx+\left(4(4+e^4)\right)\int \frac{e^{2e^{2}x}}{{(e^x-e^{e^{2}x}x)}^2}dx-\left(4(5-4e^2+e^4-e^6)\right)\int \frac{e^{2e^{2}x}x}{{(-e^x+e^{e^{2}x}x)}^2}dx\\ & =-\frac{4e^{-\left((1-e^2)x\right)}}{1-e^2}+x+\frac{4e^{-\left((1-e^2)x\right)}(5-4e^2+e^4-e^6-(1-e^2)x)}{1-e^2}+16\log (x)-(4(-1+e)(1+e))\int \frac{e^{2e^{2}x}{x}^2}{{(-e^x+e^{e^{2}x}x)}^2}dx+(4(-1+e)(1+e))\int \frac{e^{-\left((1-2e^2)x\right)}{x}^2}{-e^x+e^{e^{2}x}x}dx+\left(4(4+e^4)\right)\int \frac{e^{2e^{2}x}}{{(e^x-e^{e^{2}x}x)}^2}dx-\left(4(5-4e^2+e^4-e^6)\right)\int \frac{e^{2e^{2}x}x}{{(-e^x+e^{e^{2}x}x)}^2}dx+\left(4(5-4e^2+e^4-e^6)\right)\int \frac{e^{-\left((1-2e^2)x\right)}x}{-e^x+e^{e^{2}x}x}dx\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 49, normalized size = 1.53 $$\begin{array}{c}-\frac{4(4+e^4)}{x}+x+\frac{4e^x(-4-e^4+x)}{x(-e^x+e^{e^{2}x}x)}+16\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(16*x + 4*E^4*x + 16*x^2 + x^3 + E^(2*x - 2*E^2*x)*(16 + x) + E^(x - E^2*x)*(-52*x + 2*x^2 + E^4*(-4
*x + 4*E^2*x) + E^2*(16*x - 4*x^2)))/(E^(2*x - 2*E^2*x)*x - 2*E^(x - E^2*x)*x^2 + x^3),x]

[Out]

(-4*(4 + E^4))/x + x + (4*E^x*(-4 - E^4 + x))/(x*(-E^x + E^(E^2*x)*x)) + 16*Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.92, size = 54, normalized size = 1.69 $$\begin{array}{c}\frac{x^2-x{e}^{(-x{e}^2+x)}+16(x-e^{(-x{e}^2+x)})\log (x)+4x-4e^4-16}{x-e^{(-x{e}^2+x)}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+16)*exp(-exp(2)*x+x)^2+((4*exp(2)*x-4*x)*exp(4)+(-4*x^2+16*x)*exp(2)+2*x^2-52*x)*exp(-exp(2)*x+x
)+4*x*exp(4)+x^3+16*x^2+16*x)/(x*exp(-exp(2)*x+x)^2-2*x^2*exp(-exp(2)*x+x)+x^3),x, algorithm="fricas")

[Out]

(x^2 - x*e^(-x*e^2 + x) + 16*(x - e^(-x*e^2 + x))*log(x) + 4*x - 4*e^4 - 16)/(x - e^(-x*e^2 + x))

________________________________________________________________________________________

giac [A]  time = 0.29, size = 55, normalized size = 1.72 $$\begin{array}{c}\frac{x^2-x{e}^{(-x{e}^2+x)}+16x\log (x)-16e^{(-x{e}^2+x)}\log (x)+4x-4e^4-16}{x-e^{(-x{e}^2+x)}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+16)*exp(-exp(2)*x+x)^2+((4*exp(2)*x-4*x)*exp(4)+(-4*x^2+16*x)*exp(2)+2*x^2-52*x)*exp(-exp(2)*x+x
)+4*x*exp(4)+x^3+16*x^2+16*x)/(x*exp(-exp(2)*x+x)^2-2*x^2*exp(-exp(2)*x+x)+x^3),x, algorithm="giac")

[Out]

(x^2 - x*e^(-x*e^2 + x) + 16*x*log(x) - 16*e^(-x*e^2 + x)*log(x) + 4*x - 4*e^4 - 16)/(x - e^(-x*e^2 + x))

________________________________________________________________________________________

maple [A]  time = 0.25, size = 30, normalized size = 0.94

method	result	size
risch	$x+16\ln (x)-\frac{4(4-x+{\mathrm{e}}^4)}{x-{\mathrm{e}}^{-({\mathrm{e}}^2-1)x}}$	$30$
norman	$\frac{x^2+4{\mathrm{e}}^{-{\mathrm{e}}^{2}x+x}-x{\mathrm{e}}^{-{\mathrm{e}}^{2}x+x}-16-4{\mathrm{e}}^4}{x-{\mathrm{e}}^{-{\mathrm{e}}^{2}x+x}}+16\ln (x)$	$51$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x+16)*exp(-exp(2)*x+x)^2+((4*exp(2)*x-4*x)*exp(4)+(-4*x^2+16*x)*exp(2)+2*x^2-52*x)*exp(-exp(2)*x+x)+4*x*
exp(4)+x^3+16*x^2+16*x)/(x*exp(-exp(2)*x+x)^2-2*x^2*exp(-exp(2)*x+x)+x^3),x,method=_RETURNVERBOSE)

[Out]

x+16*ln(x)-4*(4-x+exp(4))/(x-exp(-(exp(2)-1)*x))

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 43, normalized size = 1.34 $$\begin{array}{c}\frac{(x^2-4e^4-16)e^{\left(x{e}^2\right)}-(x-4)e^x}{x{e}^{\left(x{e}^2\right)}-e^x}+16\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+16)*exp(-exp(2)*x+x)^2+((4*exp(2)*x-4*x)*exp(4)+(-4*x^2+16*x)*exp(2)+2*x^2-52*x)*exp(-exp(2)*x+x
)+4*x*exp(4)+x^3+16*x^2+16*x)/(x*exp(-exp(2)*x+x)^2-2*x^2*exp(-exp(2)*x+x)+x^3),x, algorithm="maxima")

[Out]

((x^2 - 4*e^4 - 16)*e^(x*e^2) - (x - 4)*e^x)/(x*e^(x*e^2) - e^x) + 16*log(x)

________________________________________________________________________________________

mupad [B]  time = 0.27, size = 45, normalized size = 1.41 $$\begin{array}{c}16\ln (x)-\frac{4{\mathrm{e}}^4-4x+x{\mathrm{e}}^{x-x{\mathrm{e}}^2}-x^2+16}{x-{\mathrm{e}}^{x-x{\mathrm{e}}^2}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x + exp(2*x - 2*x*exp(2))*(x + 16) + 4*x*exp(4) - exp(x - x*exp(2))*(52*x - exp(2)*(16*x - 4*x^2) + ex
p(4)*(4*x - 4*x*exp(2)) - 2*x^2) + 16*x^2 + x^3)/(x*exp(2*x - 2*x*exp(2)) - 2*x^2*exp(x - x*exp(2)) + x^3),x)

[Out]

16*log(x) - (4*exp(4) - 4*x + x*exp(x - x*exp(2)) - x^2 + 16)/(x - exp(x - x*exp(2)))

________________________________________________________________________________________

sympy [A]  time = 0.20, size = 26, normalized size = 0.81 $$\begin{array}{c}x+16\log (x)+\frac{-4x+16+4e^4}{-x+e^{-x{e}^2+x}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+16)*exp(-exp(2)*x+x)**2+((4*exp(2)*x-4*x)*exp(4)+(-4*x**2+16*x)*exp(2)+2*x**2-52*x)*exp(-exp(2)*
x+x)+4*x*exp(4)+x**3+16*x**2+16*x)/(x*exp(-exp(2)*x+x)**2-2*x**2*exp(-exp(2)*x+x)+x**3),x)

[Out]

x + 16*log(x) + (-4*x + 16 + 4*exp(4))/(-x + exp(-x*exp(2) + x))

________________________________________________________________________________________