3.38.37 \(\int \frac {e^{4-100 e^{4-2 x}-x} (e^{-4+2 x} (-1+x)+200 x)}{x^2} \, dx\)

Optimal. Leaf size=19 \[ 5+\frac {e^{-100 e^{4-2 x}+x}}{x} \]

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Rubi [F]  time = 0.74, 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 {e^{4-100 e^{4-2 x}-x} \left (e^{-4+2 x} (-1+x)+200 x\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(4 - 100*E^(4 - 2*x) - x)*(E^(-4 + 2*x)*(-1 + x) + 200*x))/x^2,x]

[Out]

-Defer[Int][E^(-100*E^(4 - 2*x) + x)/x^2, x] + 200*Defer[Int][E^(4 - 100*E^(4 - 2*x) - x)/x, x] + Defer[Int][E
^(-100*E^(4 - 2*x) + x)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^{-100 e^{4-2 x}+x} (-1+x)}{x^2}+\frac {200 e^{4-100 e^{4-2 x}-x}}{x}\right ) \, dx\\ &=200 \int \frac {e^{4-100 e^{4-2 x}-x}}{x} \, dx+\int \frac {e^{-100 e^{4-2 x}+x} (-1+x)}{x^2} \, dx\\ &=200 \int \frac {e^{4-100 e^{4-2 x}-x}}{x} \, dx+\int \left (-\frac {e^{-100 e^{4-2 x}+x}}{x^2}+\frac {e^{-100 e^{4-2 x}+x}}{x}\right ) \, dx\\ &=200 \int \frac {e^{4-100 e^{4-2 x}-x}}{x} \, dx-\int \frac {e^{-100 e^{4-2 x}+x}}{x^2} \, dx+\int \frac {e^{-100 e^{4-2 x}+x}}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 17, normalized size = 0.89 \begin {gather*} \frac {e^{-100 e^{4-2 x}+x}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4 - 100*E^(4 - 2*x) - x)*(E^(-4 + 2*x)*(-1 + x) + 200*x))/x^2,x]

[Out]

E^(-100*E^(4 - 2*x) + x)/x

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fricas [A]  time = 0.94, size = 30, normalized size = 1.58 \begin {gather*} \frac {e^{\left (-{\left ({\left (x - 4\right )} e^{\left (2 \, x - 4\right )} + 100\right )} e^{\left (-2 \, x + 4\right )} + 2 \, x - 4\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-1)*exp(x-2)^2+200*x)*exp(x)/x^2/exp(x-2)^2/exp(100/exp(x-2)^2),x, algorithm="fricas")

[Out]

e^(-((x - 4)*e^(2*x - 4) + 100)*e^(-2*x + 4) + 2*x - 4)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (x - 1\right )} e^{\left (2 \, x - 4\right )} + 200 \, x\right )} e^{\left (-x - 100 \, e^{\left (-2 \, x + 4\right )} + 4\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-1)*exp(x-2)^2+200*x)*exp(x)/x^2/exp(x-2)^2/exp(100/exp(x-2)^2),x, algorithm="giac")

[Out]

integrate(((x - 1)*e^(2*x - 4) + 200*x)*e^(-x - 100*e^(-2*x + 4) + 4)/x^2, x)

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maple [A]  time = 0.09, size = 16, normalized size = 0.84

method	result	size
risch	\(\frac {{\mathrm e}^{x -100 \,{\mathrm e}^{4-2 x}}}{x}\)	\(16\)
norman	\(\frac {{\mathrm e}^{-4} {\mathrm e}^{4} {\mathrm e}^{x} {\mathrm e}^{-100 \,{\mathrm e}^{4-2 x}}}{x}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x-1)*exp(x-2)^2+200*x)*exp(x)/x^2/exp(x-2)^2/exp(100/exp(x-2)^2),x,method=_RETURNVERBOSE)

[Out]

1/x*exp(x-100*exp(4-2*x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (x - 1\right )} e^{\left (2 \, x - 4\right )} + 200 \, x\right )} e^{\left (-x - 100 \, e^{\left (-2 \, x + 4\right )} + 4\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-1)*exp(x-2)^2+200*x)*exp(x)/x^2/exp(x-2)^2/exp(100/exp(x-2)^2),x, algorithm="maxima")

[Out]

integrate(((x - 1)*e^(2*x - 4) + 200*x)*e^(-x - 100*e^(-2*x + 4) + 4)/x^2, x)

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mupad [B]  time = 2.19, size = 15, normalized size = 0.79 \begin {gather*} \frac {{\mathrm {e}}^{-100\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^4}\,{\mathrm {e}}^x}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-100*exp(4 - 2*x))*exp(4 - 2*x)*exp(x)*(200*x + exp(2*x - 4)*(x - 1)))/x^2,x)

[Out]

(exp(-100*exp(-2*x)*exp(4))*exp(x))/x

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sympy [A]  time = 0.18, size = 15, normalized size = 0.79 \begin {gather*} \frac {e^{x} e^{- 100 e^{4} e^{- 2 x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-1)*exp(x-2)**2+200*x)*exp(x)/x**2/exp(x-2)**2/exp(100/exp(x-2)**2),x)

[Out]

exp(x)*exp(-100*exp(4)*exp(-2*x))/x

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