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

Optimal. Leaf size=23 \[ 3-e^{\frac {e^{x^2}}{x}}+\log (4)-\frac {\log (x)}{3} \]

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Rubi [F]  time = 0.28, 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 {-x+e^{\frac {e^{x^2}}{x}+x^2} \left (3-6 x^2\right )}{3 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

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

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Mathematica [A]  time = 0.10, size = 22, normalized size = 0.96 \begin {gather*} \frac {1}{3} \left (-3 e^{\frac {e^{x^2}}{x}}-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.75, size = 31, normalized size = 1.35 \begin {gather*} -\frac {1}{3} \, {\left (e^{\left (x^{2}\right )} \log \relax (x) + 3 \, e^{\left (\frac {x^{3} + e^{\left (x^{2}\right )}}{x}\right )}\right )} e^{\left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.04, size = 17, normalized size = 0.74

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.41, size = 16, normalized size = 0.70 \begin {gather*} -e^{\left (\frac {e^{\left (x^{2}\right )}}{x}\right )} - \frac {1}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 3.54, size = 16, normalized size = 0.70 \begin {gather*} -{\mathrm {e}}^{\frac {{\mathrm {e}}^{x^2}}{x}}-\frac {\ln \relax (x)}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.29, size = 14, normalized size = 0.61 \begin {gather*} - e^{\frac {e^{x^{2}}}{x}} - \frac {\log {\relax (x )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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