\(\int \frac {-x+e^{\frac {e^{x^2}}{x}+x^2} (3-6 x^2)}{3 x^2} \, dx\) [5054]

Optimal result

Integrand size = 34, antiderivative size = 23 \[ \int \frac {-x+e^{\frac {e^{x^2}}{x}+x^2} \left (3-6 x^2\right )}{3 x^2} \, dx=3-e^{\frac {e^{x^2}}{x}}+\log (4)-\frac {\log (x)}{3} \]

[Out]

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

Rubi [F]

\[ \int \frac {-x+e^{\frac {e^{x^2}}{x}+x^2} \left (3-6 x^2\right )}{3 x^2} \, dx=\int \frac {-x+e^{\frac {e^{x^2}}{x}+x^2} \left (3-6 x^2\right )}{3 x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-x+e^{\frac {e^{x^2}}{x}+x^2} \left (3-6 x^2\right )}{3 x^2} \, dx=\frac {1}{3} \left (-3 e^{\frac {e^{x^2}}{x}}-\log (x)\right ) \]

[In]

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

[Out]

(-3*E^(E^x^2/x) - Log[x])/3

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74

method	result	size
norman	\(-{\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{x}}-\frac {\ln \left (x \right )}{3}\)	\(17\)
risch	\(-{\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{x}}-\frac {\ln \left (x \right )}{3}\)	\(17\)
parallelrisch	\(-{\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{x}}-\frac {\ln \left (x \right )}{3}\)	\(17\)
parts	\(-{\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{x}}-\frac {\ln \left (x \right )}{3}\)	\(17\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {-x+e^{\frac {e^{x^2}}{x}+x^2} \left (3-6 x^2\right )}{3 x^2} \, dx=-\frac {1}{3} \, {\left (e^{\left (x^{2}\right )} \log \left (x\right ) + 3 \, e^{\left (\frac {x^{3} + e^{\left (x^{2}\right )}}{x}\right )}\right )} e^{\left (-x^{2}\right )} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {-x+e^{\frac {e^{x^2}}{x}+x^2} \left (3-6 x^2\right )}{3 x^2} \, dx=- e^{\frac {e^{x^{2}}}{x}} - \frac {\log {\left (x \right )}}{3} \]

[In]

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

[Out]

-exp(exp(x**2)/x) - log(x)/3

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {-x+e^{\frac {e^{x^2}}{x}+x^2} \left (3-6 x^2\right )}{3 x^2} \, dx=-e^{\left (\frac {e^{\left (x^{2}\right )}}{x}\right )} - \frac {1}{3} \, \log \left (x\right ) \]

[In]

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

[Out]

-e^(e^(x^2)/x) - 1/3*log(x)

Giac [F]

\[ \int \frac {-x+e^{\frac {e^{x^2}}{x}+x^2} \left (3-6 x^2\right )}{3 x^2} \, dx=\int { -\frac {3 \, {\left (2 \, x^{2} - 1\right )} e^{\left (x^{2} + \frac {e^{\left (x^{2}\right )}}{x}\right )} + x}{3 \, x^{2}} \,d x } \]

[In]

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

[Out]

integrate(-1/3*(3*(2*x^2 - 1)*e^(x^2 + e^(x^2)/x) + x)/x^2, x)

Mupad [B] (verification not implemented)

Time = 11.68 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {-x+e^{\frac {e^{x^2}}{x}+x^2} \left (3-6 x^2\right )}{3 x^2} \, dx=-{\mathrm {e}}^{\frac {{\mathrm {e}}^{x^2}}{x}}-\frac {\ln \left (x\right )}{3} \]

[In]

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

[Out]

- exp(exp(x^2)/x) - log(x)/3