3.17.93 $\int e^{e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x}}(2+3x-x^2-2x^3+e^{\frac{2+x+2x^2-\log (4)}{x}}(1+2x)+e^{e^4}(2+e^{\frac{2+x+2x^2-\log (4)}{x}}+x-2x^2-\log (4))+(-1-x)\log (4))dx$

Optimal. Leaf size=32 $$e^{e^{-2x-\frac{2+x-\log (4)}{x}}x}x(1+e^{e^4}+x)$$

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Rubi [F]  time = 26.23, 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 = {} $$\begin{array}{c}\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})(2+3x-x^2-2x^3+e^{\frac{2+x+2x^2-\log (4)}{x}}(1+2x)+e^{e^4}(2+e^{\frac{2+x+2x^2-\log (4)}{x}}+x-2x^2-\log (4))+(-1-x)\log (4))dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[E^(x/E^((2 + x + 2*x^2 - Log[4])/x) - (2 + x + 2*x^2 - Log[4])/x)*(2 + 3*x - x^2 - 2*x^3 + E^((2 + x + 2*x
^2 - Log[4])/x)*(1 + 2*x) + E^E^4*(2 + E^((2 + x + 2*x^2 - Log[4])/x) + x - 2*x^2 - Log[4]) + (-1 - x)*Log[4])
,x]

[Out]

Defer[Int][E^(4^x^(-1)*E^(-1 - 2/x - 2*x)*x), x] + Defer[Int][E^(E^4 + 4^x^(-1)*E^(-1 - 2/x - 2*x)*x), x] + (1
- Log[2])*Defer[Int][2^(1 + 2/x)*E^(-1 + E^4 - 2/x - 2*x + 4^x^(-1)*E^(-1 - 2/x - 2*x)*x), x] + 2*Defer[Int][
E^(x/E^((2 + x + 2*x^2 - Log[4])/x) - (2 + x + 2*x^2 - Log[4])/x), x] - Log[4]*Defer[Int][E^(x/E^((2 + x + 2*x
^2 - Log[4])/x) - (2 + x + 2*x^2 - Log[4])/x), x] + 2*Defer[Int][E^(4^x^(-1)*E^(-1 - 2/x - 2*x)*x)*x, x] + Def
er[Int][4^x^(-1)*E^(-1 + E^4 - 2/x - 2*x + 4^x^(-1)*E^(-1 - 2/x - 2*x)*x)*x, x] + 3*Defer[Int][E^(x/E^((2 + x
+ 2*x^2 - Log[4])/x) - (2 + x + 2*x^2 - Log[4])/x)*x, x] - Log[4]*Defer[Int][E^(x/E^((2 + x + 2*x^2 - Log[4])/
x) - (2 + x + 2*x^2 - Log[4])/x)*x, x] - Defer[Int][2^(1 + 2/x)*E^(-1 + E^4 - 2/x - 2*x + 4^x^(-1)*E^(-1 - 2/x
- 2*x)*x)*x^2, x] - Defer[Int][E^(x/E^((2 + x + 2*x^2 - Log[4])/x) - (2 + x + 2*x^2 - Log[4])/x)*x^2, x] - 2*
Defer[Int][E^(x/E^((2 + x + 2*x^2 - Log[4])/x) - (2 + x + 2*x^2 - Log[4])/x)*x^3, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (2\exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})+3\exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x-\exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^2-2\exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^3+4^{-1/x}\exp (1+\frac{2}{x}+2x+e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})(1+2x)+2^{-2/x}\exp (e^4+e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})(e^{1+\frac{2}{x}+2x}+2^{2/x}x-2^{1+\frac{2}{x}}{x}^2+2^{1+\frac{2}{x}}(1-\log (2)))-\exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})(1+x)\log (4))dx\\ & =2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})dx-2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{3}dx+3\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})xdx-\log (4)\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})(1+x)dx-\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{2}dx+\int 4^{-1/x}\exp (1+\frac{2}{x}+2x+e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})(1+2x)dx+\int 2^{-2/x}\exp (e^4+e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})(e^{1+\frac{2}{x}+2x}+2^{2/x}x-2^{1+\frac{2}{x}}{x}^2+2^{1+\frac{2}{x}}(1-\log (2)))dx\\ & =2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})dx-2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{3}dx+3\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})xdx-\log (4)\int (\exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})+\exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x)dx-\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{2}dx+\int e^{4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}(1+2x)dx+\int e^{-1+e^4-\frac{2}{x}-2x+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}(e^{1+\frac{2}{x}+2x}+4^{\frac{1}{x}}(2+x-2x^2-\log (4)))dx\\ & =2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})dx-2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{3}dx+3\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})xdx-\log (4)\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})dx-\log (4)\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})xdx-\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{2}dx+\int (e^{4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}+2e^{4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}x)dx+\int (e^{e^4+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}-4^{\frac{1}{x}}{e}^{-1+e^4-\frac{2}{x}-2x+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}(-2-x+2x^2+\log (4)))dx\\ & =2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})dx+2\int e^{4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}xdx-2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{3}dx+3\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})xdx-\log (4)\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})dx-\log (4)\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})xdx+\int e^{4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}dx+\int e^{e^4+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}dx-\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{2}dx-\int 4^{\frac{1}{x}}{e}^{-1+e^4-\frac{2}{x}-2x+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}(-2-x+2x^2+\log (4))dx\\ & =2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})dx+2\int e^{4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}xdx-2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{3}dx+3\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})xdx-\log (4)\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})dx-\log (4)\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})xdx+\int e^{4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}dx+\int e^{e^4+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}dx-\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{2}dx-\int (-4^{\frac{1}{x}}{e}^{-1+e^4-\frac{2}{x}-2x+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}x+2^{1+\frac{2}{x}}{e}^{-1+e^4-\frac{2}{x}-2x+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}{x}^2-2^{1+\frac{2}{x}}{e}^{-1+e^4-\frac{2}{x}-2x+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}(1-\log (2)))dx\\ & =2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})dx+2\int e^{4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}xdx-2\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{3}dx+3\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})xdx-(-1+\log (2))\int 2^{1+\frac{2}{x}}{e}^{-1+e^4-\frac{2}{x}-2x+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}dx-\log (4)\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})dx-\log (4)\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})xdx+\int e^{4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}dx+\int e^{e^4+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}dx+\int 4^{\frac{1}{x}}{e}^{-1+e^4-\frac{2}{x}-2x+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}xdx-\int 2^{1+\frac{2}{x}}{e}^{-1+e^4-\frac{2}{x}-2x+4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}{x}^{2}dx-\int \exp (e^{-\frac{2+x+2x^2-\log (4)}{x}}x-\frac{2+x+2x^2-\log (4)}{x})x^{2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.19, size = 31, normalized size = 0.97 $$\begin{array}{c}{e}^{4^{\frac{1}{x}}{e}^{-1-\frac{2}{x}-2x}x}x(1+e^{e^4}+x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[E^(x/E^((2 + x + 2*x^2 - Log[4])/x) - (2 + x + 2*x^2 - Log[4])/x)*(2 + 3*x - x^2 - 2*x^3 + E^((2 + x
+ 2*x^2 - Log[4])/x)*(1 + 2*x) + E^E^4*(2 + E^((2 + x + 2*x^2 - Log[4])/x) + x - 2*x^2 - Log[4]) + (-1 - x)*L
og[4]),x]

[Out]

E^(4^x^(-1)*E^(-1 - 2/x - 2*x)*x)*x*(1 + E^E^4 + x)

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fricas [B]  time = 0.96, size = 69, normalized size = 2.16 $$\begin{array}{c}(x^2+x{e}^{\left(e^4\right)}+x)e^{(\frac{x^2{e}^{(-\frac{2x^2+x-2\log (2)+2}{x})}-2x^2-x+2\log (2)-2}{x}+\frac{2x^2+x-2\log (2)+2}{x})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp((-2*log(2)+2*x^2+x+2)/x)-2*log(2)-2*x^2+x+2)*exp(exp(4))+(2*x+1)*exp((-2*log(2)+2*x^2+x+2)/x)+
2*(-x-1)*log(2)-2*x^3-x^2+3*x+2)*exp(x/exp((-2*log(2)+2*x^2+x+2)/x))/exp((-2*log(2)+2*x^2+x+2)/x),x, algorithm
="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int -(2x^3+x^2-(2x+1)e^{\left(\frac{2x^2+x-2\log (2)+2}{x}\right)}+(2x^2-x-e^{\left(\frac{2x^2+x-2\log (2)+2}{x}\right)}+2\log (2)-2)e^{\left(e^4\right)}+2(x+1)\log (2)-3x-2)e^{(x{e}^{(-\frac{2x^2+x-2\log (2)+2}{x})}-\frac{2x^2+x-2\log (2)+2}{x})}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp((-2*log(2)+2*x^2+x+2)/x)-2*log(2)-2*x^2+x+2)*exp(exp(4))+(2*x+1)*exp((-2*log(2)+2*x^2+x+2)/x)+
2*(-x-1)*log(2)-2*x^3-x^2+3*x+2)*exp(x/exp((-2*log(2)+2*x^2+x+2)/x))/exp((-2*log(2)+2*x^2+x+2)/x),x, algorithm
="giac")

[Out]

integrate(-(2*x^3 + x^2 - (2*x + 1)*e^((2*x^2 + x - 2*log(2) + 2)/x) + (2*x^2 - x - e^((2*x^2 + x - 2*log(2) +
2)/x) + 2*log(2) - 2)*e^(e^4) + 2*(x + 1)*log(2) - 3*x - 2)*e^(x*e^(-(2*x^2 + x - 2*log(2) + 2)/x) - (2*x^2 +
x - 2*log(2) + 2)/x), x)

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maple [A]  time = 0.28, size = 31, normalized size = 0.97

method	result	size
risch	$x({\mathrm{e}}^{{\mathrm{e}}^4}+x+1){\mathrm{e}}^{x{\mathrm{e}}^{\frac{-2x^2+2\ln (2)-x-2}{x}}}$	$31$
norman	$(x^2{\mathrm{e}}^{\frac{-2\ln (2)+2x^2+x+2}{x}}{\mathrm{e}}^{x{\mathrm{e}}^{-\frac{-2\ln (2)+2x^2+x+2}{x}}}+({\mathrm{e}}^{{\mathrm{e}}^4}+1)x{\mathrm{e}}^{\frac{-2\ln (2)+2x^2+x+2}{x}}{\mathrm{e}}^{x{\mathrm{e}}^{-\frac{-2\ln (2)+2x^2+x+2}{x}}}){\mathrm{e}}^{-\frac{-2\ln (2)+2x^2+x+2}{x}}$	$111$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp((-2*ln(2)+2*x^2+x+2)/x)-2*ln(2)-2*x^2+x+2)*exp(exp(4))+(2*x+1)*exp((-2*ln(2)+2*x^2+x+2)/x)+2*(-x-1)*
ln(2)-2*x^3-x^2+3*x+2)*exp(x/exp((-2*ln(2)+2*x^2+x+2)/x))/exp((-2*ln(2)+2*x^2+x+2)/x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.55, size = 33, normalized size = 1.03 $$\begin{array}{c}(x^2+x(e^{\left(e^4\right)}+1))e^{\left(x{e}^{(-2x+\frac{2\log (2)}{x}-\frac{2}{x}-1)}\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp((-2*log(2)+2*x^2+x+2)/x)-2*log(2)-2*x^2+x+2)*exp(exp(4))+(2*x+1)*exp((-2*log(2)+2*x^2+x+2)/x)+
2*(-x-1)*log(2)-2*x^3-x^2+3*x+2)*exp(x/exp((-2*log(2)+2*x^2+x+2)/x))/exp((-2*log(2)+2*x^2+x+2)/x),x, algorithm
="maxima")

[Out]

(x^2 + x*(e^(e^4) + 1))*e^(x*e^(-2*x + 2*log(2)/x - 2/x - 1))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 $$\begin{array}{c}\int {\mathrm{e}}^{-\frac{2x^2+x-2\ln (2)+2}{x}}{\mathrm{e}}^{x{\mathrm{e}}^{-\frac{2x^2+x-2\ln (2)+2}{x}}}(3x+{\mathrm{e}}^{\frac{2x^2+x-2\ln (2)+2}{x}}(2x+1)+{\mathrm{e}}^{{\mathrm{e}}^4}(x-2\ln (2)+{\mathrm{e}}^{\frac{2x^2+x-2\ln (2)+2}{x}}-2x^2+2)-2\ln (2)(x+1)-x^2-2x^3+2)dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-(x - 2*log(2) + 2*x^2 + 2)/x)*exp(x*exp(-(x - 2*log(2) + 2*x^2 + 2)/x))*(3*x + exp((x - 2*log(2) + 2*
x^2 + 2)/x)*(2*x + 1) + exp(exp(4))*(x - 2*log(2) + exp((x - 2*log(2) + 2*x^2 + 2)/x) - 2*x^2 + 2) - 2*log(2)*
(x + 1) - x^2 - 2*x^3 + 2),x)

[Out]

int(exp(-(x - 2*log(2) + 2*x^2 + 2)/x)*exp(x*exp(-(x - 2*log(2) + 2*x^2 + 2)/x))*(3*x + exp((x - 2*log(2) + 2*
x^2 + 2)/x)*(2*x + 1) + exp(exp(4))*(x - 2*log(2) + exp((x - 2*log(2) + 2*x^2 + 2)/x) - 2*x^2 + 2) - 2*log(2)*
(x + 1) - x^2 - 2*x^3 + 2), x)

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sympy [A]  time = 3.07, size = 31, normalized size = 0.97 $$\begin{array}{c}(x^2+x+x{e}^{e^4})e^{x{e}^{-\frac{2x^2+x-2\log (2)+2}{x}}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp((-2*ln(2)+2*x**2+x+2)/x)-2*ln(2)-2*x**2+x+2)*exp(exp(4))+(2*x+1)*exp((-2*ln(2)+2*x**2+x+2)/x)+
2*(-x-1)*ln(2)-2*x**3-x**2+3*x+2)*exp(x/exp((-2*ln(2)+2*x**2+x+2)/x))/exp((-2*ln(2)+2*x**2+x+2)/x),x)

[Out]

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

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