3.28.70 \(\int \frac {e^x+e^x (-2+x) \log (\frac {1}{x})}{x^3 \log ^2(\frac {1}{x})} \, dx\)

Optimal. Leaf size=13 \[ \frac {e^x}{x^2 \log \left (\frac {1}{x}\right )} \]

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Rubi [F]  time = 0.42, 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^x+e^x (-2+x) \log \left (\frac {1}{x}\right )}{x^3 \log ^2\left (\frac {1}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x + E^x*(-2 + x)*Log[x^(-1)])/(x^3*Log[x^(-1)]^2),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 13, normalized size = 1.00 \begin {gather*} \frac {e^x}{x^2 \log \left (\frac {1}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x + E^x*(-2 + x)*Log[x^(-1)])/(x^3*Log[x^(-1)]^2),x]

[Out]

E^x/(x^2*Log[x^(-1)])

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fricas [A]  time = 0.66, size = 12, normalized size = 0.92 \begin {gather*} \frac {e^{x}}{x^{2} \log \left (\frac {1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^x/(x^2*log(1/x))

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giac [A]  time = 0.17, size = 11, normalized size = 0.85 \begin {gather*} -\frac {e^{x}}{x^{2} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^x/(x^2*log(x))

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maple [A]  time = 0.04, size = 12, normalized size = 0.92

method	result	size
risch	\(-\frac {{\mathrm e}^{x}}{\ln \relax (x ) x^{2}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(x)/ln(x)/x^2

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maxima [A]  time = 0.44, size = 11, normalized size = 0.85 \begin {gather*} -\frac {e^{x}}{x^{2} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^x/(x^2*log(x))

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mupad [B]  time = 1.96, size = 12, normalized size = 0.92 \begin {gather*} \frac {{\mathrm {e}}^x}{x^2\,\ln \left (\frac {1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x) + log(1/x)*exp(x)*(x - 2))/(x^3*log(1/x)^2),x)

[Out]

exp(x)/(x^2*log(1/x))

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sympy [A]  time = 0.24, size = 10, normalized size = 0.77 \begin {gather*} \frac {e^{x}}{x^{2} \log {\left (\frac {1}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-2)*exp(x)*ln(1/x)+exp(x))/x**3/ln(1/x)**2,x)

[Out]

exp(x)/(x**2*log(1/x))

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