3.103.45 \(\int \frac {e^{-2 x+\frac {e^{-2 x} (9-10000 e^{2 x} x^2)}{10000 x^2}} (-9-9 x)}{5000 x^3} \, dx\)

Optimal. Leaf size=16 \[ e^{-1+\frac {9 e^{-2 x}}{10000 x^2}} \]

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Rubi [F]  time = 0.53, 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 {\exp \left (-2 x+\frac {e^{-2 x} \left (9-10000 e^{2 x} x^2\right )}{10000 x^2}\right ) (-9-9 x)}{5000 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-2*x + (9 - 10000*E^(2*x)*x^2)/(10000*E^(2*x)*x^2))*(-9 - 9*x))/(5000*x^3),x]

[Out]

(-9*Defer[Int][E^(-1 + 9/(10000*E^(2*x)*x^2) - 2*x)/x^3, x])/5000 - (9*Defer[Int][E^(-1 + 9/(10000*E^(2*x)*x^2
) - 2*x)/x^2, x])/5000

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {\exp \left (-2 x+\frac {e^{-2 x} \left (9-10000 e^{2 x} x^2\right )}{10000 x^2}\right ) (-9-9 x)}{x^3} \, dx}{5000}\\ &=\frac {\int \frac {9 e^{-1+\frac {9 e^{-2 x}}{10000 x^2}-2 x} (-1-x)}{x^3} \, dx}{5000}\\ &=\frac {9 \int \frac {e^{-1+\frac {9 e^{-2 x}}{10000 x^2}-2 x} (-1-x)}{x^3} \, dx}{5000}\\ &=\frac {9 \int \left (-\frac {e^{-1+\frac {9 e^{-2 x}}{10000 x^2}-2 x}}{x^3}-\frac {e^{-1+\frac {9 e^{-2 x}}{10000 x^2}-2 x}}{x^2}\right ) \, dx}{5000}\\ &=-\frac {9 \int \frac {e^{-1+\frac {9 e^{-2 x}}{10000 x^2}-2 x}}{x^3} \, dx}{5000}-\frac {9 \int \frac {e^{-1+\frac {9 e^{-2 x}}{10000 x^2}-2 x}}{x^2} \, dx}{5000}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 16, normalized size = 1.00 \begin {gather*} e^{-1+\frac {9 e^{-2 x}}{10000 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-2*x + (9 - 10000*E^(2*x)*x^2)/(10000*E^(2*x)*x^2))*(-9 - 9*x))/(5000*x^3),x]

[Out]

E^(-1 + 9/(10000*E^(2*x)*x^2))

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fricas [B]  time = 1.26, size = 31, normalized size = 1.94 \begin {gather*} e^{\left (2 \, x - \frac {{\left (10000 \, {\left (2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} - 9\right )} e^{\left (-2 \, x\right )}}{10000 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*(-9*x-9)*exp(1/10000*(-10000*exp(x)^2*x^2+9)/exp(x)^2/x^2)/exp(x)^2/x^3,x, algorithm="fricas"
)

[Out]

e^(2*x - 1/10000*(10000*(2*x^3 + x^2)*e^(2*x) - 9)*e^(-2*x)/x^2)

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giac [B]  time = 0.17, size = 27, normalized size = 1.69 \begin {gather*} e^{\left (2 \, x - \frac {20000 \, x^{3} + 10000 \, x^{2} - 9 \, e^{\left (-2 \, x\right )}}{10000 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*(-9*x-9)*exp(1/10000*(-10000*exp(x)^2*x^2+9)/exp(x)^2/x^2)/exp(x)^2/x^3,x, algorithm="giac")

[Out]

e^(2*x - 1/10000*(20000*x^3 + 10000*x^2 - 9*e^(-2*x))/x^2)

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maple [A]  time = 0.09, size = 22, normalized size = 1.38

method	result	size
norman	\({\mathrm e}^{\frac {\left (-10000 \,{\mathrm e}^{2 x} x^{2}+9\right ) {\mathrm e}^{-2 x}}{10000 x^{2}}}\)	\(22\)
risch	\({\mathrm e}^{-\frac {\left (10000 \,{\mathrm e}^{2 x} x^{2}-9\right ) {\mathrm e}^{-2 x}}{10000 x^{2}}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5000*(-9*x-9)*exp(1/10000*(-10000*exp(x)^2*x^2+9)/exp(x)^2/x^2)/exp(x)^2/x^3,x,method=_RETURNVERBOSE)

[Out]

exp(1/10000*(-10000*exp(x)^2*x^2+9)/exp(x)^2/x^2)

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maxima [A]  time = 0.50, size = 12, normalized size = 0.75 \begin {gather*} e^{\left (\frac {9 \, e^{\left (-2 \, x\right )}}{10000 \, x^{2}} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*(-9*x-9)*exp(1/10000*(-10000*exp(x)^2*x^2+9)/exp(x)^2/x^2)/exp(x)^2/x^3,x, algorithm="maxima"
)

[Out]

e^(9/10000*e^(-2*x)/x^2 - 1)

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mupad [B]  time = 6.30, size = 13, normalized size = 0.81 \begin {gather*} {\mathrm {e}}^{-1}\,{\mathrm {e}}^{\frac {9\,{\mathrm {e}}^{-2\,x}}{10000\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-2*x)*exp(-(exp(-2*x)*(x^2*exp(2*x) - 9/10000))/x^2)*(9*x + 9))/(5000*x^3),x)

[Out]

exp(-1)*exp((9*exp(-2*x))/(10000*x^2))

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sympy [A]  time = 0.29, size = 20, normalized size = 1.25 \begin {gather*} e^{\frac {\left (- x^{2} e^{2 x} + \frac {9}{10000}\right ) e^{- 2 x}}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*(-9*x-9)*exp(1/10000*(-10000*exp(x)**2*x**2+9)/exp(x)**2/x**2)/exp(x)**2/x**3,x)

[Out]

exp((-x**2*exp(2*x) + 9/10000)*exp(-2*x)/x**2)

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