3.172 \(\int e^x x \log (x) \, dx\)

Optimal. Leaf size=22 \[ \text{ExpIntegralEi}(x)-e^x-e^x \log (x)+e^x x \log (x) \]

[Out]

-E^x + ExpIntegralEi[x] - E^x*Log[x] + E^x*x*Log[x]

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Rubi [A]  time = 0.0760769, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714 \[ \text{ExpIntegralEi}(x)-e^x-e^x \log (x)+e^x x \log (x) \]

Antiderivative was successfully verified.

[In]  Int[E^x*x*Log[x],x]

[Out]

-E^x + ExpIntegralEi[x] - E^x*Log[x] + E^x*x*Log[x]

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Rubi in Sympy [A]  time = 4.45879, size = 20, normalized size = 0.91 \[ x e^{x} \log{\left (x \right )} - e^{x} \log{\left (x \right )} - e^{x} + \operatorname{Ei}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)*x*ln(x),x)

[Out]

x*exp(x)*log(x) - exp(x)*log(x) - exp(x) + Ei(x)

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Mathematica [A]  time = 0.0109994, size = 17, normalized size = 0.77 \[ \text{ExpIntegralEi}(x)-e^x+e^x (x-1) \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[E^x*x*Log[x],x]

[Out]

-E^x + ExpIntegralEi[x] + E^x*(-1 + x)*Log[x]

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Maple [A]  time = 0.009, size = 21, normalized size = 1. \[ \left ( -1+x \right ){{\rm e}^{x}}\ln \left ( x \right ) -{\it Ei} \left ( 1,-x \right ) -{{\rm e}^{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)*x*ln(x),x)

[Out]

(-1+x)*exp(x)*ln(x)-Ei(1,-x)-exp(x)

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Maxima [A]  time = 1.41295, size = 20, normalized size = 0.91 \[{\left (x - 1\right )} e^{x} \log \left (x\right ) +{\rm Ei}\left (x\right ) - e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^x*log(x),x, algorithm="maxima")

[Out]

(x - 1)*e^x*log(x) + Ei(x) - e^x

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Fricas [A]  time = 0.220526, size = 20, normalized size = 0.91 \[{\left (x - 1\right )} e^{x} \log \left (x\right ) +{\rm Ei}\left (x\right ) - e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^x*log(x),x, algorithm="fricas")

[Out]

(x - 1)*e^x*log(x) + Ei(x) - e^x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x e^{x} \log{\left (x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(x)*x*ln(x),x)

[Out]

Integral(x*exp(x)*log(x), x)

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GIAC/XCAS [A]  time = 0.198027, size = 20, normalized size = 0.91 \[{\left (x - 1\right )} e^{x}{\rm ln}\left (x\right ) +{\rm Ei}\left (x\right ) - e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^x*log(x),x, algorithm="giac")

[Out]

(x - 1)*e^x*ln(x) + Ei(x) - e^x