3.17.39 $\int \frac{e^{3e^{\frac{x}{1+x}}-x}(1+x+3e^{\frac{x}{1+x}}x-x^2-x^3)}{1+2x+x^2}dx$

Optimal. Leaf size=28 $$3+e^{-3+3(1+e^{\frac{x^2}{x+x^2}})-x}x$$

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Rubi [F]  time = 3.30, 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 = {} $$\begin{array}{c}\int \frac{e^{3e^{\frac{x}{1+x}}-x}(1+x+3e^{\frac{x}{1+x}}x-x^2-x^3)}{1+2x+x^2}dx\end{array}$$

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^{3e^{\frac{x}{1+x}}-x}(1+x+3e^{\frac{x}{1+x}}x-x^2-x^3)}{(1+x{)}^2}dx\\ & =\int (\frac{e^{3e^{\frac{x}{1+x}}-x}}{(1+x{)}^2}+\frac{e^{3e^{\frac{x}{1+x}}-x}x}{(1+x{)}^2}+\frac{3e^{3e^{\frac{x}{1+x}}-x+\frac{x}{1+x}}x}{(1+x{)}^2}-\frac{e^{3e^{\frac{x}{1+x}}-x}{x}^2}{(1+x{)}^2}-\frac{e^{3e^{\frac{x}{1+x}}-x}{x}^3}{(1+x{)}^2})dx\\ & =3\int \frac{e^{3e^{\frac{x}{1+x}}-x+\frac{x}{1+x}}x}{(1+x{)}^2}dx+\int \frac{e^{3e^{\frac{x}{1+x}}-x}}{(1+x{)}^2}dx+\int \frac{e^{3e^{\frac{x}{1+x}}-x}x}{(1+x{)}^2}dx-\int \frac{e^{3e^{\frac{x}{1+x}}-x}{x}^2}{(1+x{)}^2}dx-\int \frac{e^{3e^{\frac{x}{1+x}}-x}{x}^3}{(1+x{)}^2}dx\\ & =3\int (-\frac{e^{3e^{\frac{x}{1+x}}-x+\frac{x}{1+x}}}{(1+x{)}^2}+\frac{e^{3e^{\frac{x}{1+x}}-x+\frac{x}{1+x}}}{1+x})dx+\int \frac{e^{3e^{\frac{x}{1+x}}-x}}{(1+x{)}^2}dx-\int (e^{3e^{\frac{x}{1+x}}-x}+\frac{e^{3e^{\frac{x}{1+x}}-x}}{(1+x{)}^2}-\frac{2e^{3e^{\frac{x}{1+x}}-x}}{1+x})dx+\int (-\frac{e^{3e^{\frac{x}{1+x}}-x}}{(1+x{)}^2}+\frac{e^{3e^{\frac{x}{1+x}}-x}}{1+x})dx-\int (-2e^{3e^{\frac{x}{1+x}}-x}+e^{3e^{\frac{x}{1+x}}-x}x-\frac{e^{3e^{\frac{x}{1+x}}-x}}{(1+x{)}^2}+\frac{3e^{3e^{\frac{x}{1+x}}-x}}{1+x})dx\\ & =2\int e^{3e^{\frac{x}{1+x}}-x}dx+2\int \frac{e^{3e^{\frac{x}{1+x}}-x}}{1+x}dx-3\int \frac{e^{3e^{\frac{x}{1+x}}-x+\frac{x}{1+x}}}{(1+x{)}^2}dx-3\int \frac{e^{3e^{\frac{x}{1+x}}-x}}{1+x}dx+3\int \frac{e^{3e^{\frac{x}{1+x}}-x+\frac{x}{1+x}}}{1+x}dx-\int e^{3e^{\frac{x}{1+x}}-x}dx-\int e^{3e^{\frac{x}{1+x}}-x}xdx+\int \frac{e^{3e^{\frac{x}{1+x}}-x}}{1+x}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.64, size = 21, normalized size = 0.75 $$\begin{array}{c}{e}^{3e^{1-\frac{1}{1+x}}-x}x\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*E^(x/(1 + x)) - x)*(1 + x + 3*E^(x/(1 + x))*x - x^2 - x^3))/(1 + 2*x + x^2),x]

[Out]

E^(3*E^(1 - (1 + x)^(-1)) - x)*x

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fricas [A]  time = 0.79, size = 17, normalized size = 0.61 $$\begin{array}{c}x{e}^{(-x+3e^{\left(\frac{x}{x+1}\right)})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x*exp(x/(x+1))-x^3-x^2+x+1)/(x^2+2*x+1)/exp(-3*exp(x/(x+1))+x),x, algorithm="fricas")

[Out]

x*e^(-x + 3*e^(x/(x + 1)))

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giac [A]  time = 0.21, size = 44, normalized size = 1.57 $$\begin{array}{c}x{e}^{(-\frac{x^2-3x{e}^{\left(\frac{x}{x+1}\right)}-3e^{\left(\frac{x}{x+1}\right)}}{x+1}-\frac{x}{x+1})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x*exp(x/(x+1))-x^3-x^2+x+1)/(x^2+2*x+1)/exp(-3*exp(x/(x+1))+x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.32, size = 18, normalized size = 0.64

method	result	size
risch	$x{\mathrm{e}}^{3{\mathrm{e}}^{\frac{x}{x+1}}-x}$	$18$
norman	$\frac{(x^2+x){\mathrm{e}}^{3{\mathrm{e}}^{\frac{x}{x+1}}-x}}{x+1}$	$27$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x*exp(x/(x+1))-x^3-x^2+x+1)/(x^2+2*x+1)/exp(-3*exp(x/(x+1))+x),x,method=_RETURNVERBOSE)

[Out]

x*exp(3*exp(x/(x+1))-x)

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maxima [A]  time = 0.45, size = 19, normalized size = 0.68 $$\begin{array}{c}x{e}^{(-x+3e^{(-\frac{1}{x+1}+1)})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x*exp(x/(x+1))-x^3-x^2+x+1)/(x^2+2*x+1)/exp(-3*exp(x/(x+1))+x),x, algorithm="maxima")

[Out]

x*e^(-x + 3*e^(-1/(x + 1) + 1))

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mupad [B]  time = 1.26, size = 17, normalized size = 0.61 $$\begin{array}{c}x{\mathrm{e}}^{3{\mathrm{e}}^{\frac{x}{x+1}}}{\mathrm{e}}^{-x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*exp(3*exp(x/(x + 1)))*exp(-x)

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sympy [A]  time = 6.85, size = 12, normalized size = 0.43 $$\begin{array}{c}x{e}^{-x+3e^{\frac{x}{x+1}}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x*exp(x/(x+1))-x**3-x**2+x+1)/(x**2+2*x+1)/exp(-3*exp(x/(x+1))+x),x)

[Out]

x*exp(-x + 3*exp(x/(x + 1)))

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