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

Optimal. Leaf size=14 \[ -e^{x+(x-\log (x))^2} \]

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Rubi [A]  time = 0.29, antiderivative size = 18, normalized size of antiderivative = 1.29, number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6706} \begin {gather*} x^{-2 x} \left (-e^{x^2+x+\log ^2(x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(x + x^2 + Log[x]^2)/x^(2*x))

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-e^{x+x^2+\log ^2(x)} x^{-2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 18, normalized size = 1.29 \begin {gather*} -e^{x+x^2+\log ^2(x)} x^{-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(x + x^2 + Log[x]^2)/x^(2*x))

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fricas [A]  time = 0.86, size = 17, normalized size = 1.21 \begin {gather*} -e^{\left (x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(x^2 - 2*x*log(x) + log(x)^2 + x)

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giac [A]  time = 0.30, size = 17, normalized size = 1.21 \begin {gather*} -e^{\left (x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(x^2 - 2*x*log(x) + log(x)^2 + x)

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maple [A]  time = 0.03, size = 18, normalized size = 1.29

method	result	size
norman	\(-{\mathrm e}^{\ln \relax (x )^{2}-2 x \ln \relax (x )+x^{2}+x}\)	\(18\)
risch	\(-x^{-2 x} {\mathrm e}^{\ln \relax (x )^{2}+x^{2}+x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.56, size = 17, normalized size = 1.21 \begin {gather*} -e^{\left (x^{2} - 2 \, x \log \relax (x) + \log \relax (x)^{2} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(x^2 - 2*x*log(x) + log(x)^2 + x)

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mupad [B]  time = 0.72, size = 20, normalized size = 1.43 \begin {gather*} -\frac {{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\ln \relax (x)}^2}\,{\mathrm {e}}^x}{x^{2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.29, size = 19, normalized size = 1.36 \begin {gather*} - e^{x^{2} - 2 x \log {\relax (x )} + x + \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(x**2 - 2*x*log(x) + x + log(x)**2)

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