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

Optimal. Leaf size=15 \[ \frac {e^{3+x+\frac {x}{\log (x)}}}{x^2} \]

________________________________________________________________________________________

Rubi [F]  time = 1.26, 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^{\frac {x+(3+x) \log (x)}{\log (x)}} \left (-x+x \log (x)+(-2+x) \log ^2(x)\right )}{x^3 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 15, normalized size = 1.00 \begin {gather*} \frac {e^{3+x+\frac {x}{\log (x)}}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(3 + x + x/Log[x])/x^2

________________________________________________________________________________________

fricas [A]  time = 0.90, size = 18, normalized size = 1.20 \begin {gather*} \frac {e^{\left (\frac {{\left (x + 3\right )} \log \relax (x) + x}{\log \relax (x)}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(((x + 3)*log(x) + x)/log(x))/x^2

________________________________________________________________________________________

giac [A]  time = 0.13, size = 20, normalized size = 1.33 \begin {gather*} \frac {e^{\left (\frac {x \log \relax (x) + x + 3 \, \log \relax (x)}{\log \relax (x)}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^((x*log(x) + x + 3*log(x))/log(x))/x^2

________________________________________________________________________________________

maple [A]  time = 0.02, size = 21, normalized size = 1.40

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {x \ln \relax (x )+3 \ln \relax (x )+x}{\ln \relax (x )}}}{x^{2}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 14, normalized size = 0.93 \begin {gather*} \frac {e^{\left (x + \frac {x}{\log \relax (x)} + 3\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x + x/log(x) + 3)/x^2

________________________________________________________________________________________

mupad [B]  time = 4.44, size = 15, normalized size = 1.00 \begin {gather*} \frac {{\mathrm {e}}^3\,{\mathrm {e}}^{\frac {x}{\ln \relax (x)}}\,{\mathrm {e}}^x}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(3)*exp(x/log(x))*exp(x))/x^2

________________________________________________________________________________________

sympy [A]  time = 0.30, size = 15, normalized size = 1.00 \begin {gather*} \frac {e^{\frac {x + \left (x + 3\right ) \log {\relax (x )}}{\log {\relax (x )}}}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp((x + (x + 3)*log(x))/log(x))/x**2

________________________________________________________________________________________