3.52.20 \(\int (4+e^{64-16 e^{2 e^{4 x}-x}+2 e^{4 x}-x} (16-128 e^{4 x})) \, dx\)

Optimal. Leaf size=26 \[ -4+e^{16 \left (4-e^{2 e^{4 x}-x}\right )}+4 x \]

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Rubi [F]  time = 0.54, 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 \left (4+e^{64-16 e^{2 e^{4 x}-x}+2 e^{4 x}-x} \left (16-128 e^{4 x}\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[4 + E^(64 - 16*E^(2*E^(4*x) - x) + 2*E^(4*x) - x)*(16 - 128*E^(4*x)),x]

[Out]

4*x + 16*Defer[Subst][Defer[Int][E^(2*(32 - (8*E^(2*x^4))/x + x^4))/x^2, x], x, E^x] - 128*Defer[Subst][Defer[
Int][E^(2*(32 - (8*E^(2*x^4))/x + x^4))*x^2, x], x, E^x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 x+\int e^{64-16 e^{2 e^{4 x}-x}+2 e^{4 x}-x} \left (16-128 e^{4 x}\right ) \, dx\\ &=4 x+\operatorname {Subst}\left (\int \frac {16 e^{2 \left (32-\frac {8 e^{2 x^4}}{x}+x^4\right )} \left (1-8 x^4\right )}{x^2} \, dx,x,e^x\right )\\ &=4 x+16 \operatorname {Subst}\left (\int \frac {e^{2 \left (32-\frac {8 e^{2 x^4}}{x}+x^4\right )} \left (1-8 x^4\right )}{x^2} \, dx,x,e^x\right )\\ &=4 x+16 \operatorname {Subst}\left (\int \left (\frac {e^{2 \left (32-\frac {8 e^{2 x^4}}{x}+x^4\right )}}{x^2}-8 e^{2 \left (32-\frac {8 e^{2 x^4}}{x}+x^4\right )} x^2\right ) \, dx,x,e^x\right )\\ &=4 x+16 \operatorname {Subst}\left (\int \frac {e^{2 \left (32-\frac {8 e^{2 x^4}}{x}+x^4\right )}}{x^2} \, dx,x,e^x\right )-128 \operatorname {Subst}\left (\int e^{2 \left (32-\frac {8 e^{2 x^4}}{x}+x^4\right )} x^2 \, dx,x,e^x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 23, normalized size = 0.88 \begin {gather*} e^{64-16 e^{2 e^{4 x}-x}}+4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4 + E^(64 - 16*E^(2*E^(4*x) - x) + 2*E^(4*x) - x)*(16 - 128*E^(4*x)),x]

[Out]

E^(64 - 16*E^(2*E^(4*x) - x)) + 4*x

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fricas [B]  time = 0.62, size = 50, normalized size = 1.92 \begin {gather*} {\left (4 \, x e^{\left (-x + 2 \, e^{\left (4 \, x\right )}\right )} + e^{\left (-x + 2 \, e^{\left (4 \, x\right )} - 16 \, e^{\left (-x + 2 \, e^{\left (4 \, x\right )}\right )} + 64\right )}\right )} e^{\left (x - 2 \, e^{\left (4 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-128*exp(4*x)+16)*exp(2*exp(4*x)-x)*exp(-16*exp(2*exp(4*x)-x)+64)+4,x, algorithm="fricas")

[Out]

(4*x*e^(-x + 2*e^(4*x)) + e^(-x + 2*e^(4*x) - 16*e^(-x + 2*e^(4*x)) + 64))*e^(x - 2*e^(4*x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -16 \, {\left (8 \, e^{\left (4 \, x\right )} - 1\right )} e^{\left (-x + 2 \, e^{\left (4 \, x\right )} - 16 \, e^{\left (-x + 2 \, e^{\left (4 \, x\right )}\right )} + 64\right )} + 4\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-128*exp(4*x)+16)*exp(2*exp(4*x)-x)*exp(-16*exp(2*exp(4*x)-x)+64)+4,x, algorithm="giac")

[Out]

integrate(-16*(8*e^(4*x) - 1)*e^(-x + 2*e^(4*x) - 16*e^(-x + 2*e^(4*x)) + 64) + 4, x)

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maple [A]  time = 0.11, size = 21, normalized size = 0.81

method	result	size
default	\(4 x +{\mathrm e}^{-16 \,{\mathrm e}^{2 \,{\mathrm e}^{4 x}-x}+64}\)	\(21\)
norman	\(4 x +{\mathrm e}^{-16 \,{\mathrm e}^{2 \,{\mathrm e}^{4 x}-x}+64}\)	\(21\)
risch	\(4 x +{\mathrm e}^{-16 \,{\mathrm e}^{2 \,{\mathrm e}^{4 x}-x}+64}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-128*exp(4*x)+16)*exp(2*exp(4*x)-x)*exp(-16*exp(2*exp(4*x)-x)+64)+4,x,method=_RETURNVERBOSE)

[Out]

4*x+exp(-16*exp(2*exp(4*x)-x)+64)

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maxima [A]  time = 0.56, size = 20, normalized size = 0.77 \begin {gather*} 4 \, x + e^{\left (-16 \, e^{\left (-x + 2 \, e^{\left (4 \, x\right )}\right )} + 64\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-128*exp(4*x)+16)*exp(2*exp(4*x)-x)*exp(-16*exp(2*exp(4*x)-x)+64)+4,x, algorithm="maxima")

[Out]

4*x + e^(-16*e^(-x + 2*e^(4*x)) + 64)

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mupad [B]  time = 0.11, size = 21, normalized size = 0.81 \begin {gather*} 4\,x+{\mathrm {e}}^{-16\,{\mathrm {e}}^{2\,{\mathrm {e}}^{4\,x}}\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4 - exp(2*exp(4*x) - x)*exp(64 - 16*exp(2*exp(4*x) - x))*(128*exp(4*x) - 16),x)

[Out]

4*x + exp(-16*exp(2*exp(4*x))*exp(-x))*exp(64)

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sympy [A]  time = 0.27, size = 17, normalized size = 0.65 \begin {gather*} 4 x + e^{64 - 16 e^{- x + 2 e^{4 x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-128*exp(4*x)+16)*exp(2*exp(4*x)-x)*exp(-16*exp(2*exp(4*x)-x)+64)+4,x)

[Out]

4*x + exp(64 - 16*exp(-x + 2*exp(4*x)))

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