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

Optimal. Leaf size=32 \[ 9+e^{4-e^4}-e^{\frac {-x^3+\frac {x}{\log (x)}}{\log (x)}} \]

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Rubi [F]  time = 0.78, 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-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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 20, normalized size = 0.62 \begin {gather*} -e^{\frac {x}{\log ^2(x)}-\frac {x^3}{\log (x)}} \end {gather*}

Antiderivative was successfully verified.

[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])

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fricas [A]  time = 0.76, size = 19, normalized size = 0.59 \begin {gather*} -e^{\left (-\frac {x^{3} \log \relax (x) - x}{\log \relax (x)^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.19, size = 19, normalized size = 0.59 \begin {gather*} -e^{\left (-\frac {x^{3}}{\log \relax (x)} + \frac {x}{\log \relax (x)^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.02, size = 19, normalized size = 0.59

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

Verification of antiderivative is not currently implemented for this CAS.

[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*(x^2*ln(x)-1)/ln(x)^2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 5.62, size = 17, normalized size = 0.53 \begin {gather*} -{\mathrm {e}}^{\frac {x-x^3\,\ln \relax (x)}{{\ln \relax (x)}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.43, size = 15, normalized size = 0.47 \begin {gather*} - e^{\frac {- x^{3} \log {\relax (x )} + x}{\log {\relax (x )}^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

-exp((-x**3*log(x) + x)/log(x)**2)

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