3.15.38 \(\int e^{-3 x^2+x^3} (-1+e^{3 x^2-x^3} (1-2 x)-6 x+3 x^2+(-1+6 x^2-3 x^3) \log (x)) \, dx\)

Optimal. Leaf size=27 \[ x-x^2+e^{-\left ((3-x) x^2\right )} (1-x \log (x)) \]

________________________________________________________________________________________

Rubi [F]  time = 1.08, 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 e^{-3 x^2+x^3} \left (-1+e^{3 x^2-x^3} (1-2 x)-6 x+3 x^2+\left (-1+6 x^2-3 x^3\right ) \log (x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-3*x^2 + x^3)*(-1 + E^(3*x^2 - x^3)*(1 - 2*x) - 6*x + 3*x^2 + (-1 + 6*x^2 - 3*x^3)*Log[x]),x]

[Out]

x - x^2 - (E^(-3*x^2 + x^3)*(2*x^2 - x^3)*Log[x])/(2*x - x^2) - 6*Defer[Int][E^(-3*x^2 + x^3)*x, x] + 3*Defer[
Int][E^(-3*x^2 + x^3)*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-e^{-3 x^2+x^3}-2 x-6 e^{-3 x^2+x^3} x+3 e^{-3 x^2+x^3} x^2-e^{-3 x^2+x^3} \left (1-6 x^2+3 x^3\right ) \log (x)\right ) \, dx\\ &=x-x^2+3 \int e^{-3 x^2+x^3} x^2 \, dx-6 \int e^{-3 x^2+x^3} x \, dx-\int e^{-3 x^2+x^3} \, dx-\int e^{-3 x^2+x^3} \left (1-6 x^2+3 x^3\right ) \log (x) \, dx\\ &=x-x^2-\frac {e^{-3 x^2+x^3} \left (2 x^2-x^3\right ) \log (x)}{2 x-x^2}+3 \int e^{-3 x^2+x^3} x^2 \, dx-6 \int e^{-3 x^2+x^3} x \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 1.38, size = 30, normalized size = 1.11 \begin {gather*} e^{(-3+x) x^2}+x-x^2-e^{(-3+x) x^2} x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-3*x^2 + x^3)*(-1 + E^(3*x^2 - x^3)*(1 - 2*x) - 6*x + 3*x^2 + (-1 + 6*x^2 - 3*x^3)*Log[x]),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.94, size = 32, normalized size = 1.19 \begin {gather*} -x e^{\left (x^{3} - 3 \, x^{2}\right )} \log \relax (x) - x^{2} + x + e^{\left (x^{3} - 3 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^3+6*x^2-1)*log(x)+(1-2*x)*exp(-x^3+3*x^2)+3*x^2-6*x-1)/exp(-x^3+3*x^2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.34, size = 32, normalized size = 1.19 \begin {gather*} -x e^{\left (x^{3} - 3 \, x^{2}\right )} \log \relax (x) - x^{2} + x + e^{\left (x^{3} - 3 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^3+6*x^2-1)*log(x)+(1-2*x)*exp(-x^3+3*x^2)+3*x^2-6*x-1)/exp(-x^3+3*x^2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 30, normalized size = 1.11

method	result	size
default	\(x +\left (-x \ln \relax (x )+1\right ) {\mathrm e}^{x^{3}-3 x^{2}}-x^{2}\)	\(30\)
risch	\(-x \,{\mathrm e}^{x^{2} \left (x -3\right )} \ln \relax (x )-\left (x^{2} {\mathrm e}^{-x^{2} \left (x -3\right )}-x \,{\mathrm e}^{-x^{2} \left (x -3\right )}-1\right ) {\mathrm e}^{x^{2} \left (x -3\right )}\)	\(52\)
norman	\(\left (1+x \,{\mathrm e}^{-x^{3}+3 x^{2}}-x \ln \relax (x )-x^{2} {\mathrm e}^{-x^{3}+3 x^{2}}\right ) {\mathrm e}^{x^{3}-3 x^{2}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^3+6*x^2-1)*ln(x)+(1-2*x)*exp(-x^3+3*x^2)+3*x^2-6*x-1)/exp(-x^3+3*x^2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.67, size = 25, normalized size = 0.93 \begin {gather*} -x^{2} - {\left (x \log \relax (x) - 1\right )} e^{\left (x^{3} - 3 \, x^{2}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^3+6*x^2-1)*log(x)+(1-2*x)*exp(-x^3+3*x^2)+3*x^2-6*x-1)/exp(-x^3+3*x^2),x, algorithm="maxima")

[Out]

-x^2 - (x*log(x) - 1)*e^(x^3 - 3*x^2) + x

________________________________________________________________________________________

mupad [B]  time = 1.08, size = 32, normalized size = 1.19 \begin {gather*} x+{\mathrm {e}}^{x^3-3\,x^2}-x^2-x\,{\mathrm {e}}^{x^3-3\,x^2}\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x^3 - 3*x^2)*(6*x + log(x)*(3*x^3 - 6*x^2 + 1) + exp(3*x^2 - x^3)*(2*x - 1) - 3*x^2 + 1),x)

[Out]

x + exp(x^3 - 3*x^2) - x^2 - x*exp(x^3 - 3*x^2)*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.36, size = 20, normalized size = 0.74 \begin {gather*} - x^{2} + x + \left (- x \log {\relax (x )} + 1\right ) e^{x^{3} - 3 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**3+6*x**2-1)*ln(x)+(1-2*x)*exp(-x**3+3*x**2)+3*x**2-6*x-1)/exp(-x**3+3*x**2),x)

[Out]

-x**2 + x + (-x*log(x) + 1)*exp(x**3 - 3*x**2)

________________________________________________________________________________________