3.65.83 \(\int \frac {3}{4} e^{e^x-x} (16-16 x+16 e^x x+\frac {4}{3} e^{-e^x+x} (20+8 x)) \, dx\)

Optimal. Leaf size=22 \[ -5+4 \left (3+x \left (5+3 e^{e^x-x}+x\right )\right ) \]

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Rubi [F]  time = 0.36, 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 {gather*} \int \frac {3}{4} e^{e^x-x} \left (16-16 x+16 e^x x+\frac {4}{3} e^{-e^x+x} (20+8 x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*E^(E^x - x)*(16 - 16*x + 16*E^x*x + (4*E^(-E^x + x)*(20 + 8*x))/3))/4,x]

[Out]

-12*E^(E^x - x) + (5 + 2*x)^2 + 12*ExpIntegralEi[E^x] + 12*Defer[Int][E^E^x*x, x] - 12*Defer[Int][E^(E^x - x)*
x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {3}{4} \int e^{e^x-x} \left (16-16 x+16 e^x x+\frac {4}{3} e^{-e^x+x} (20+8 x)\right ) \, dx\\ &=\frac {3}{4} \int \left (16 e^{e^x-x}+16 e^{e^x} x-16 e^{e^x-x} x+\frac {16}{3} (5+2 x)\right ) \, dx\\ &=(5+2 x)^2+12 \int e^{e^x-x} \, dx+12 \int e^{e^x} x \, dx-12 \int e^{e^x-x} x \, dx\\ &=(5+2 x)^2+12 \int e^{e^x} x \, dx-12 \int e^{e^x-x} x \, dx+12 \operatorname {Subst}\left (\int \frac {e^x}{x^2} \, dx,x,e^x\right )\\ &=-12 e^{e^x-x}+(5+2 x)^2+12 \int e^{e^x} x \, dx-12 \int e^{e^x-x} x \, dx+12 \operatorname {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right )\\ &=-12 e^{e^x-x}+(5+2 x)^2+12 \text {Ei}\left (e^x\right )+12 \int e^{e^x} x \, dx-12 \int e^{e^x-x} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 17, normalized size = 0.77 \begin {gather*} 4 x \left (5+3 e^{e^x-x}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3*E^(E^x - x)*(16 - 16*x + 16*E^x*x + (4*E^(-E^x + x)*(20 + 8*x))/3))/4,x]

[Out]

4*x*(5 + 3*E^(E^x - x) + x)

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fricas [A]  time = 0.56, size = 21, normalized size = 0.95 \begin {gather*} 4 \, x^{2} + 16 \, x e^{\left (-x + e^{x} + \log \left (\frac {3}{4}\right )\right )} + 20 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+20)*exp(-log(3/4*exp(exp(x)))+x)+16*exp(x)*x-16*x+16)/exp(-log(3/4*exp(exp(x)))+x),x, algorith
m="fricas")

[Out]

4*x^2 + 16*x*e^(-x + e^x + log(3/4)) + 20*x

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giac [A]  time = 0.17, size = 19, normalized size = 0.86 \begin {gather*} 4 \, x^{2} + 12 \, x e^{\left (-x + e^{x}\right )} + 20 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+20)*exp(-log(3/4*exp(exp(x)))+x)+16*exp(x)*x-16*x+16)/exp(-log(3/4*exp(exp(x)))+x),x, algorith
m="giac")

[Out]

4*x^2 + 12*x*e^(-x + e^x) + 20*x

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maple [A]  time = 0.17, size = 38, normalized size = 1.73

method	result	size
norman	\(\left (12 x \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+20 x \,{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}+4 x^{2} {\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}\right ) {\mathrm e}^{-x} {\mathrm e}^{-{\mathrm e}^{x}}\)	\(38\)
default	\(\frac {3 \left (16 x +4 x^{2} {\mathrm e}^{-\ln \left (\frac {3 \,{\mathrm e}^{{\mathrm e}^{x}}}{4}\right )+x}+20 \,{\mathrm e}^{-\ln \left (\frac {3 \,{\mathrm e}^{{\mathrm e}^{x}}}{4}\right )+x} x \right ) {\mathrm e}^{{\mathrm e}^{x}-x}}{4}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x+20)*exp(-ln(3/4*exp(exp(x)))+x)+16*exp(x)*x-16*x+16)/exp(-ln(3/4*exp(exp(x)))+x),x,method=_RETURNVER
BOSE)

[Out]

(12*x*exp(exp(x))^2+20*x*exp(x)*exp(exp(x))+4*x^2*exp(x)*exp(exp(x)))/exp(x)/exp(exp(x))

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maxima [A]  time = 0.44, size = 19, normalized size = 0.86 \begin {gather*} 4 \, x^{2} + 12 \, x e^{\left (-x + e^{x}\right )} + 20 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+20)*exp(-log(3/4*exp(exp(x)))+x)+16*exp(x)*x-16*x+16)/exp(-log(3/4*exp(exp(x)))+x),x, algorith
m="maxima")

[Out]

4*x^2 + 12*x*e^(-x + e^x) + 20*x

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mupad [B]  time = 0.11, size = 15, normalized size = 0.68 \begin {gather*} 4\,x\,\left (x+3\,{\mathrm {e}}^{{\mathrm {e}}^x-x}+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(log((3*exp(exp(x)))/4) - x)*(exp(x - log((3*exp(exp(x)))/4))*(8*x + 20) - 16*x + 16*x*exp(x) + 16),x)

[Out]

4*x*(x + 3*exp(exp(x) - x) + 5)

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sympy [A]  time = 0.17, size = 19, normalized size = 0.86 \begin {gather*} 4 x^{2} + 20 x + 12 x e^{- x} e^{e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x+20)*exp(-ln(3/4*exp(exp(x)))+x)+16*exp(x)*x-16*x+16)/exp(-ln(3/4*exp(exp(x)))+x),x)

[Out]

4*x**2 + 20*x + 12*x*exp(-x)*exp(exp(x))

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