3.71.20 \(\int \frac {(2-e^x x) \log (x)+\log (-3 e^{-e^x} x^2)}{x} \, dx\) [7020]

Optimal. Leaf size=21 \[ \log (x) \log \left (3 e^{-e^x} (x-x (1+x))\right ) \]

[Out]

ln(x)*ln(3*(x-(1+x)*x)/exp(exp(x)))

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Rubi [F]
time = 0.06, 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 {\left (2-e^x x\right ) \log (x)+\log \left (-3 e^{-e^x} x^2\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 16, normalized size = 0.76 \begin {gather*} \log (x) \log \left (-3 e^{-e^x} x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x]*Log[(-3*x^2)/E^E^x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(19)=38\).
time = 0.85, size = 50, normalized size = 2.38

method	result	size
default	\(-{\mathrm e}^{x} \ln \left (x \right )-\left (\ln \left ({\mathrm e}^{{\mathrm e}^{x}}\right )-{\mathrm e}^{x}\right ) \ln \left (x \right )+\left (\ln \left (-3 \,{\mathrm e}^{-{\mathrm e}^{x}} x^{2}\right )-2 \ln \left (x \right )+\ln \left ({\mathrm e}^{{\mathrm e}^{x}}\right )\right ) \ln \left (x \right )+2 \ln \left (x \right )^{2}\)	\(50\)
risch	\(-\ln \left (x \right ) \ln \left ({\mathrm e}^{{\mathrm e}^{x}}\right )+2 \ln \left (x \right )^{2}-\frac {i \ln \left (x \right ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{2}+i \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\frac {i \ln \left (x \right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}-\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-{\mathrm e}^{x}}\right )}{2}+\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-{\mathrm e}^{x}}\right )^{2}}{2}+\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{x}}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-{\mathrm e}^{x}}\right )^{2}}{2}+\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-{\mathrm e}^{x}}\right )^{3}}{2}-i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{-{\mathrm e}^{x}}\right )^{2}+i \pi \ln \left (x \right )+\ln \left (3\right ) \ln \left (x \right )\)	\(212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(x)*ln(x)-(ln(exp(exp(x)))-exp(x))*ln(x)+(ln(-3*x^2/exp(exp(x)))-2*ln(x)+ln(exp(exp(x))))*ln(x)+2*ln(x)^2

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3)*log(x) + 2*log(x)^2 - log(x)*log(-e^(e^x)) + Ei(x) - integrate(e^x/x, x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.48, size = 20, normalized size = 0.95 \begin {gather*} {\left (i \, \pi - e^{x} + \log \left (3\right )\right )} \log \left (x\right ) + 2 \, \log \left (x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(I*pi - e^x + log(3))*log(x) + 2*log(x)^2

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Sympy [C] Result contains complex when optimal does not.
time = 0.18, size = 22, normalized size = 1.05 \begin {gather*} - e^{x} \log {\left (x \right )} + 2 \log {\left (x \right )}^{2} + \left (\log {\left (3 \right )} + i \pi \right ) \log {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(x)*log(x) + 2*log(x)**2 + (log(3) + I*pi)*log(x)

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Giac [A]
time = 0.41, size = 33, normalized size = 1.57 \begin {gather*} \frac {3}{2} \, \pi ^{2} \mathrm {sgn}\left (x\right ) - \frac {3}{2} \, \pi ^{2} - e^{x} \log \left ({\left | x \right |}\right ) + \log \left (3\right ) \log \left ({\left | x \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*pi^2*sgn(x) - 3/2*pi^2 - e^x*log(abs(x)) + log(3)*log(abs(x)) + 2*log(abs(x))^2

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Mupad [B]
time = 4.35, size = 18, normalized size = 0.86 \begin {gather*} \ln \left (x\right )\,\left (\ln \left (3\,x^2\right )-{\mathrm {e}}^x+\pi \,1{}\mathrm {i}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)*(pi*1i + log(3*x^2) - exp(x))

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