\(\int \frac {e^{\frac {x-x^3 \log (x)}{\log ^2(x)}} (2+(-1-x^2) \log (x)+3 x^2 \log ^2(x))}{\log ^3(x)} \, dx\) [7742]

Optimal result

Integrand size = 42, antiderivative size = 32 \[ \int \frac {e^{\frac {x-x^3 \log (x)}{\log ^2(x)}} \left (2+\left (-1-x^2\right ) \log (x)+3 x^2 \log ^2(x)\right )}{\log ^3(x)} \, dx=9+e^{4-e^4}-e^{\frac {-x^3+\frac {x}{\log (x)}}{\log (x)}} \]

[Out]

exp(4-exp(4))+9-exp((x/ln(x)-x^3)/ln(x))

Rubi [F]

\[ \int \frac {e^{\frac {x-x^3 \log (x)}{\log ^2(x)}} \left (2+\left (-1-x^2\right ) \log (x)+3 x^2 \log ^2(x)\right )}{\log ^3(x)} \, dx=\int \frac {e^{\frac {x-x^3 \log (x)}{\log ^2(x)}} \left (2+\left (-1-x^2\right ) \log (x)+3 x^2 \log ^2(x)\right )}{\log ^3(x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {e^{\frac {x-x^3 \log (x)}{\log ^2(x)}} \left (2+\left (-1-x^2\right ) \log (x)+3 x^2 \log ^2(x)\right )}{\log ^3(x)} \, dx=-e^{\frac {x}{\log ^2(x)}-\frac {x^3}{\log (x)}} \]

[In]

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

[Out]

-E^(x/Log[x]^2 - x^3/Log[x])

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.56

method	result	size
parallelrisch	\(-{\mathrm e}^{\frac {-x^{3} \ln \left (x \right )+x}{\ln \left (x \right )^{2}}}\)	\(18\)
risch	\(-{\mathrm e}^{-\frac {x \left (x^{2} \ln \left (x \right )-1\right )}{\ln \left (x \right )^{2}}}\)	\(19\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59 \[ \int \frac {e^{\frac {x-x^3 \log (x)}{\log ^2(x)}} \left (2+\left (-1-x^2\right ) \log (x)+3 x^2 \log ^2(x)\right )}{\log ^3(x)} \, dx=-e^{\left (-\frac {x^{3} \log \left (x\right ) - x}{\log \left (x\right )^{2}}\right )} \]

[In]

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

[Out]

-e^(-(x^3*log(x) - x)/log(x)^2)

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {x-x^3 \log (x)}{\log ^2(x)}} \left (2+\left (-1-x^2\right ) \log (x)+3 x^2 \log ^2(x)\right )}{\log ^3(x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {x-x^3 \log (x)}{\log ^2(x)}} \left (2+\left (-1-x^2\right ) \log (x)+3 x^2 \log ^2(x)\right )}{\log ^3(x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59 \[ \int \frac {e^{\frac {x-x^3 \log (x)}{\log ^2(x)}} \left (2+\left (-1-x^2\right ) \log (x)+3 x^2 \log ^2(x)\right )}{\log ^3(x)} \, dx=-e^{\left (-\frac {x^{3}}{\log \left (x\right )} + \frac {x}{\log \left (x\right )^{2}}\right )} \]

[In]

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

[Out]

-e^(-x^3/log(x) + x/log(x)^2)

Mupad [B] (verification not implemented)

Time = 12.35 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\frac {x-x^3 \log (x)}{\log ^2(x)}} \left (2+\left (-1-x^2\right ) \log (x)+3 x^2 \log ^2(x)\right )}{\log ^3(x)} \, dx=-{\mathrm {e}}^{\frac {x-x^3\,\ln \left (x\right )}{{\ln \left (x\right )}^2}} \]

[In]

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

[Out]

-exp((x - x^3*log(x))/log(x)^2)