3.23.79 \(\int e^{-405 e^{e^{e^{-4 x} x}}-3 x} (e^{4 x}+e^{e^{e^{-4 x} x}+e^{-4 x} x} (-405+1620 x)) \, dx\)

Optimal. Leaf size=17 \[ e^{-405 e^{e^{e^{-4 x} x}}+x} \]

________________________________________________________________________________________

Rubi [F]  time = 1.24, 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 e^{-405 e^{e^{e^{-4 x} x}}-3 x} \left (e^{4 x}+e^{e^{e^{-4 x} x}+e^{-4 x} x} (-405+1620 x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-405*E^E^(x/E^(4*x)) - 3*x)*(E^(4*x) + E^(E^(x/E^(4*x)) + x/E^(4*x))*(-405 + 1620*x)),x]

[Out]

Defer[Int][E^(-405*E^E^(x/E^(4*x)) + x), x] - 405*Defer[Int][E^(E^(x/E^(4*x)) + x/E^(4*x) - 3*(135*E^E^(x/E^(4
*x)) + x)), x] + 1620*Defer[Int][E^(E^(x/E^(4*x)) + x/E^(4*x) - 3*(135*E^E^(x/E^(4*x)) + x))*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-3 \left (135 e^{e^{e^{-4 x} x}}+x\right )} \left (e^{4 x}+e^{e^{e^{-4 x} x}+e^{-4 x} x} (-405+1620 x)\right ) \, dx\\ &=\int \left (e^{4 x-3 \left (135 e^{e^{e^{-4 x} x}}+x\right )}+405 \exp \left (e^{e^{-4 x} x}+e^{-4 x} x-3 \left (135 e^{e^{e^{-4 x} x}}+x\right )\right ) (-1+4 x)\right ) \, dx\\ &=405 \int \exp \left (e^{e^{-4 x} x}+e^{-4 x} x-3 \left (135 e^{e^{e^{-4 x} x}}+x\right )\right ) (-1+4 x) \, dx+\int e^{4 x-3 \left (135 e^{e^{e^{-4 x} x}}+x\right )} \, dx\\ &=405 \int \left (-\exp \left (e^{e^{-4 x} x}+e^{-4 x} x-3 \left (135 e^{e^{e^{-4 x} x}}+x\right )\right )+4 \exp \left (e^{e^{-4 x} x}+e^{-4 x} x-3 \left (135 e^{e^{e^{-4 x} x}}+x\right )\right ) x\right ) \, dx+\int e^{-405 e^{e^{e^{-4 x} x}}+x} \, dx\\ &=-\left (405 \int \exp \left (e^{e^{-4 x} x}+e^{-4 x} x-3 \left (135 e^{e^{e^{-4 x} x}}+x\right )\right ) \, dx\right )+1620 \int \exp \left (e^{e^{-4 x} x}+e^{-4 x} x-3 \left (135 e^{e^{e^{-4 x} x}}+x\right )\right ) x \, dx+\int e^{-405 e^{e^{e^{-4 x} x}}+x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.57, size = 17, normalized size = 1.00 \begin {gather*} e^{-405 e^{e^{e^{-4 x} x}}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-405*E^E^(x/E^(4*x)) - 3*x)*(E^(4*x) + E^(E^(x/E^(4*x)) + x/E^(4*x))*(-405 + 1620*x)),x]

[Out]

E^(-405*E^E^(x/E^(4*x)) + x)

________________________________________________________________________________________

fricas [B]  time = 0.76, size = 46, normalized size = 2.71 \begin {gather*} e^{\left (-3 \, {\left (x e^{\left (x e^{\left (-4 \, x\right )}\right )} + 135 \, e^{\left ({\left (x + e^{\left (x e^{\left (-4 \, x\right )} + 4 \, x\right )}\right )} e^{\left (-4 \, x\right )}\right )}\right )} e^{\left (-x e^{\left (-4 \, x\right )}\right )} + 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1620*x-405)*exp(x/exp(4*x))*exp(exp(x/exp(4*x)))+exp(4*x))*exp(-405*exp(exp(x/exp(4*x)))+x)/exp(4*
x),x, algorithm="fricas")

[Out]

e^(-3*(x*e^(x*e^(-4*x)) + 135*e^((x + e^(x*e^(-4*x) + 4*x))*e^(-4*x)))*e^(-x*e^(-4*x)) + 4*x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (405 \, {\left (4 \, x - 1\right )} e^{\left (x e^{\left (-4 \, x\right )} + e^{\left (x e^{\left (-4 \, x\right )}\right )}\right )} + e^{\left (4 \, x\right )}\right )} e^{\left (-3 \, x - 405 \, e^{\left (e^{\left (x e^{\left (-4 \, x\right )}\right )}\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1620*x-405)*exp(x/exp(4*x))*exp(exp(x/exp(4*x)))+exp(4*x))*exp(-405*exp(exp(x/exp(4*x)))+x)/exp(4*
x),x, algorithm="giac")

[Out]

integrate((405*(4*x - 1)*e^(x*e^(-4*x) + e^(x*e^(-4*x))) + e^(4*x))*e^(-3*x - 405*e^(e^(x*e^(-4*x)))), x)

________________________________________________________________________________________

maple [A]  time = 0.09, size = 14, normalized size = 0.82

method	result	size
risch	\({\mathrm e}^{-405 \,{\mathrm e}^{{\mathrm e}^{x \,{\mathrm e}^{-4 x}}}+x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1620*x-405)*exp(x/exp(4*x))*exp(exp(x/exp(4*x)))+exp(4*x))*exp(-405*exp(exp(x/exp(4*x)))+x)/exp(4*x),x,m
ethod=_RETURNVERBOSE)

[Out]

exp(-405*exp(exp(x*exp(-4*x)))+x)

________________________________________________________________________________________

maxima [A]  time = 0.78, size = 13, normalized size = 0.76 \begin {gather*} e^{\left (x - 405 \, e^{\left (e^{\left (x e^{\left (-4 \, x\right )}\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1620*x-405)*exp(x/exp(4*x))*exp(exp(x/exp(4*x)))+exp(4*x))*exp(-405*exp(exp(x/exp(4*x)))+x)/exp(4*
x),x, algorithm="maxima")

[Out]

e^(x - 405*e^(e^(x*e^(-4*x))))

________________________________________________________________________________________

mupad [B]  time = 1.57, size = 14, normalized size = 0.82 \begin {gather*} {\mathrm {e}}^{-405\,{\mathrm {e}}^{{\mathrm {e}}^{x\,{\mathrm {e}}^{-4\,x}}}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4*x)*exp(x - 405*exp(exp(x*exp(-4*x))))*(exp(4*x) + exp(x*exp(-4*x))*exp(exp(x*exp(-4*x)))*(1620*x -
405)),x)

[Out]

exp(-405*exp(exp(x*exp(-4*x))))*exp(x)

________________________________________________________________________________________

sympy [A]  time = 1.51, size = 14, normalized size = 0.82 \begin {gather*} e^{x - 405 e^{e^{x e^{- 4 x}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1620*x-405)*exp(x/exp(4*x))*exp(exp(x/exp(4*x)))+exp(4*x))*exp(-405*exp(exp(x/exp(4*x)))+x)/exp(4*
x),x)

[Out]

exp(x - 405*exp(exp(x*exp(-4*x))))

________________________________________________________________________________________