3.85.2 $\int \frac{1}{5}(10x+e^{e^{\frac{1}{5}(10x-2e^{e^x}x-2e^{x}x+2x^2)}}(-5+e^{\frac{1}{5}(10x-2e^{e^x}x-2e^{x}x+2x^2)}(-10x-4x^2+e^x(2x+2x^2)+e^{e^x}(2x+2e^x{x}^2))))dx$

Optimal. Leaf size=30 $$x(-e^{e^{\frac{2}{5}x(5-e^{e^x}-e^x+x)}}+x)$$

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Rubi [B]  time = 0.30, antiderivative size = 105, normalized size of antiderivative = 3.50, number of steps used = 3, number of rules used = 2, integrand size = 112, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.018, Rules used = {12, 2288} $$\begin{array}{c}{x}^2-\frac{e^{e^{\frac{2}{5}(x^2-e^{e^x}x-e^{x}x+5x)}}(2x^2-e^x(x^2+x)-e^{e^x}(e^x{x}^2+x)+5x)}{-e^{x}x-e^{x+e^x}x+2x-e^{e^x}-e^x+5}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(10*x + E^E^((10*x - 2*E^E^x*x - 2*E^x*x + 2*x^2)/5)*(-5 + E^((10*x - 2*E^E^x*x - 2*E^x*x + 2*x^2)/5)*(-10
*x - 4*x^2 + E^x*(2*x + 2*x^2) + E^E^x*(2*x + 2*E^x*x^2))))/5,x]

[Out]

x^2 - (E^E^((2*(5*x - E^E^x*x - E^x*x + x^2))/5)*(5*x + 2*x^2 - E^x*(x + x^2) - E^E^x*(x + E^x*x^2)))/(5 - E^E
^x - E^x + 2*x - E^x*x - E^(E^x + x)*x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{5}\int (10x+e^{e^{\frac{1}{5}(10x-2e^{e^x}x-2e^{x}x+2x^2)}}(-5+e^{\frac{1}{5}(10x-2e^{e^x}x-2e^{x}x+2x^2)}(-10x-4x^2+e^x(2x+2x^2)+e^{e^x}(2x+2e^x{x}^2))))dx\\ & =x^2+\frac{1}{5}\int e^{e^{\frac{1}{5}(10x-2e^{e^x}x-2e^{x}x+2x^2)}}(-5+e^{\frac{1}{5}(10x-2e^{e^x}x-2e^{x}x+2x^2)}(-10x-4x^2+e^x(2x+2x^2)+e^{e^x}(2x+2e^x{x}^2)))dx\\ & =x^2-\frac{e^{e^{\frac{2}{5}(5x-e^{e^x}x-e^{x}x+x^2)}}(5x+2x^2-e^x(x+x^2)-e^{e^x}(x+e^x{x}^2))}{5-e^{e^x}-e^x+2x-e^{x}x-e^{e^x+x}x}\end{array}\end{array}$$

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Mathematica [A]  time = 0.47, size = 30, normalized size = 1.00 $$\begin{array}{c}x(-e^{e^{\frac{2}{5}x(5-e^{e^x}-e^x+x)}}+x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(10*x + E^E^((10*x - 2*E^E^x*x - 2*E^x*x + 2*x^2)/5)*(-5 + E^((10*x - 2*E^E^x*x - 2*E^x*x + 2*x^2)/5
)*(-10*x - 4*x^2 + E^x*(2*x + 2*x^2) + E^E^x*(2*x + 2*E^x*x^2))))/5,x]

[Out]

x*(-E^E^((2*x*(5 - E^E^x - E^x + x))/5) + x)

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fricas [A]  time = 0.67, size = 29, normalized size = 0.97 $$\begin{array}{c}{x}^2-x{e}^{\left(e^{(\frac{2}{5}{x}^2-\frac{2}{5}x{e}^x-\frac{2}{5}x{e}^{\left(e^x\right)}+2x)}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(((2*exp(x)*x^2+2*x)*exp(exp(x))+(2*x^2+2*x)*exp(x)-4*x^2-10*x)*exp(-2/5*x*exp(exp(x))-2/5*exp(x
)*x+2/5*x^2+2*x)-5)*exp(exp(-2/5*x*exp(exp(x))-2/5*exp(x)*x+2/5*x^2+2*x))+2*x,x, algorithm="fricas")

[Out]

x^2 - x*e^(e^(2/5*x^2 - 2/5*x*e^x - 2/5*x*e^(e^x) + 2*x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int -\frac{1}{5}(2(2x^2-(x^2+x)e^x-(x^2{e}^x+x)e^{\left(e^x\right)}+5x)e^{(\frac{2}{5}{x}^2-\frac{2}{5}x{e}^x-\frac{2}{5}x{e}^{\left(e^x\right)}+2x)}+5)e^{\left(e^{(\frac{2}{5}{x}^2-\frac{2}{5}x{e}^x-\frac{2}{5}x{e}^{\left(e^x\right)}+2x)}\right)}+2xdx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(((2*exp(x)*x^2+2*x)*exp(exp(x))+(2*x^2+2*x)*exp(x)-4*x^2-10*x)*exp(-2/5*x*exp(exp(x))-2/5*exp(x
)*x+2/5*x^2+2*x)-5)*exp(exp(-2/5*x*exp(exp(x))-2/5*exp(x)*x+2/5*x^2+2*x))+2*x,x, algorithm="giac")

[Out]

integrate(-1/5*(2*(2*x^2 - (x^2 + x)*e^x - (x^2*e^x + x)*e^(e^x) + 5*x)*e^(2/5*x^2 - 2/5*x*e^x - 2/5*x*e^(e^x)
+ 2*x) + 5)*e^(e^(2/5*x^2 - 2/5*x*e^x - 2/5*x*e^(e^x) + 2*x)) + 2*x, x)

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maple [A]  time = 0.13, size = 23, normalized size = 0.77

method	result	size
risch	$-x{\mathrm{e}}^{{\mathrm{e}}^{-\frac{2x({\mathrm{e}}^{{\mathrm{e}}^x}+{\mathrm{e}}^x-x-5)}{5}}}+x^2$	$23$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(((2*exp(x)*x^2+2*x)*exp(exp(x))+(2*x^2+2*x)*exp(x)-4*x^2-10*x)*exp(-2/5*x*exp(exp(x))-2/5*exp(x)*x+2/
5*x^2+2*x)-5)*exp(exp(-2/5*x*exp(exp(x))-2/5*exp(x)*x+2/5*x^2+2*x))+2*x,x,method=_RETURNVERBOSE)

[Out]

-x*exp(exp(-2/5*x*(exp(exp(x))+exp(x)-x-5)))+x^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}{x}^2-\frac{1}{5}\int -(2((x^2+x)e^{\left(3x\right)}-(2x^2+5x)e^{\left(2x\right)}+(x^2{e}^{\left(3x\right)}+x{e}^{\left(2x\right)})e^{\left(e^x\right)})e^{\left(\frac{2}{5}{x}^2\right)}-5e^{(\frac{2}{5}x{e}^x+\frac{2}{5}x{e}^{\left(e^x\right)})})e^{(-\frac{2}{5}x{e}^x-\frac{2}{5}x{e}^{\left(e^x\right)}+e^{(\frac{2}{5}{x}^2-\frac{2}{5}x{e}^x-\frac{2}{5}x{e}^{\left(e^x\right)}+2x)})}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(((2*exp(x)*x^2+2*x)*exp(exp(x))+(2*x^2+2*x)*exp(x)-4*x^2-10*x)*exp(-2/5*x*exp(exp(x))-2/5*exp(x
)*x+2/5*x^2+2*x)-5)*exp(exp(-2/5*x*exp(exp(x))-2/5*exp(x)*x+2/5*x^2+2*x))+2*x,x, algorithm="maxima")

[Out]

x^2 - 1/5*integrate(-(2*((x^2 + x)*e^(3*x) - (2*x^2 + 5*x)*e^(2*x) + (x^2*e^(3*x) + x*e^(2*x))*e^(e^x))*e^(2/5
*x^2) - 5*e^(2/5*x*e^x + 2/5*x*e^(e^x)))*e^(-2/5*x*e^x - 2/5*x*e^(e^x) + e^(2/5*x^2 - 2/5*x*e^x - 2/5*x*e^(e^x
) + 2*x)), x)

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mupad [B]  time = 5.67, size = 31, normalized size = 1.03 $$\begin{array}{c}x(x-{\mathrm{e}}^{{\mathrm{e}}^{-\frac{2x{\mathrm{e}}^x}{5}}{\mathrm{e}}^{2x}{\mathrm{e}}^{-\frac{2x{\mathrm{e}}^{{\mathrm{e}}^x}}{5}}{\mathrm{e}}^{\frac{2x^2}{5}}})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x - (exp(exp(2*x - (2*x*exp(exp(x)))/5 - (2*x*exp(x))/5 + (2*x^2)/5))*(exp(2*x - (2*x*exp(exp(x)))/5 - (
2*x*exp(x))/5 + (2*x^2)/5)*(10*x - exp(x)*(2*x + 2*x^2) + 4*x^2 - exp(exp(x))*(2*x + 2*x^2*exp(x))) + 5))/5,x)

[Out]

x*(x - exp(exp(-(2*x*exp(x))/5)*exp(2*x)*exp(-(2*x*exp(exp(x)))/5)*exp((2*x^2)/5)))

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sympy [A]  time = 11.52, size = 36, normalized size = 1.20 $$\begin{array}{c}{x}^2-x{e}^{e^{\frac{2x^2}{5}-\frac{2x{e}^x}{5}-\frac{2x{e}^{e^x}}{5}+2x}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(((2*exp(x)*x**2+2*x)*exp(exp(x))+(2*x**2+2*x)*exp(x)-4*x**2-10*x)*exp(-2/5*x*exp(exp(x))-2/5*ex
p(x)*x+2/5*x**2+2*x)-5)*exp(exp(-2/5*x*exp(exp(x))-2/5*exp(x)*x+2/5*x**2+2*x))+2*x,x)

[Out]

x**2 - x*exp(exp(2*x**2/5 - 2*x*exp(x)/5 - 2*x*exp(exp(x))/5 + 2*x))

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