3.69.12 \(\int \frac {e^{\frac {3 e^{x^2}+2 x-3 \log (x)}{6 x}} (-1+e^{x^2} (-1+2 x^2)+\log (x))}{8 x^2} \, dx\)

Optimal. Leaf size=28 \[ \frac {1}{4} e^{\frac {e^{x^2}+\frac {2 x}{3}-\log (x)}{2 x}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.54, antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {12, 6706} \begin {gather*} \frac {1}{4} e^{\frac {3 e^{x^2}+2 x}{6 x}} x^{\left .-\frac {1}{2}\right /x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((3*E^x^2 + 2*x - 3*Log[x])/(6*x))*(-1 + E^x^2*(-1 + 2*x^2) + Log[x]))/(8*x^2),x]

[Out]

E^((3*E^x^2 + 2*x)/(6*x))/(4*x^(1/(2*x)))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \frac {e^{\frac {3 e^{x^2}+2 x-3 \log (x)}{6 x}} \left (-1+e^{x^2} \left (-1+2 x^2\right )+\log (x)\right )}{x^2} \, dx\\ &=\frac {1}{4} e^{\frac {3 e^{x^2}+2 x}{6 x}} x^{\left .-\frac {1}{2}\right /x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.62, size = 31, normalized size = 1.11 \begin {gather*} \frac {1}{4} e^{\frac {1}{3}+\frac {e^{x^2}}{2 x}} x^{\left .-\frac {1}{2}\right /x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((3*E^x^2 + 2*x - 3*Log[x])/(6*x))*(-1 + E^x^2*(-1 + 2*x^2) + Log[x]))/(8*x^2),x]

[Out]

E^(1/3 + E^x^2/(2*x))/(4*x^(1/(2*x)))

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{4} \, e^{\left (\frac {2 \, x + 3 \, e^{\left (x^{2}\right )} - 3 \, \log \relax (x)}{6 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(log(x)+(2*x^2-1)*exp(x^2)-1)*exp(1/6*(-3*log(x)+3*exp(x^2)+2*x)/x)/x^2,x, algorithm="fricas")

[Out]

1/4*e^(1/6*(2*x + 3*e^(x^2) - 3*log(x))/x)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 21, normalized size = 0.75 \begin {gather*} \frac {1}{4} \, e^{\left (\frac {e^{\left (x^{2}\right )}}{2 \, x} - \frac {\log \relax (x)}{2 \, x} + \frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(log(x)+(2*x^2-1)*exp(x^2)-1)*exp(1/6*(-3*log(x)+3*exp(x^2)+2*x)/x)/x^2,x, algorithm="giac")

[Out]

1/4*e^(1/2*e^(x^2)/x - 1/2*log(x)/x + 1/3)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 23, normalized size = 0.82

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {3 \ln \relax (x )-3 \,{\mathrm e}^{x^{2}}-2 x}{6 x}}}{4}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*(ln(x)+(2*x^2-1)*exp(x^2)-1)*exp(1/6*(-3*ln(x)+3*exp(x^2)+2*x)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/4*exp(-1/6*(3*ln(x)-3*exp(x^2)-2*x)/x)

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 21, normalized size = 0.75 \begin {gather*} \frac {1}{4} \, e^{\left (\frac {e^{\left (x^{2}\right )}}{2 \, x} - \frac {\log \relax (x)}{2 \, x} + \frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(log(x)+(2*x^2-1)*exp(x^2)-1)*exp(1/6*(-3*log(x)+3*exp(x^2)+2*x)/x)/x^2,x, algorithm="maxima")

[Out]

1/4*e^(1/2*e^(x^2)/x - 1/2*log(x)/x + 1/3)

________________________________________________________________________________________

mupad [B]  time = 4.13, size = 23, normalized size = 0.82 \begin {gather*} \frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^{x^2}}{2\,x}+\frac {1}{3}}}{4\,x^{\frac {1}{2\,x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x/3 + exp(x^2)/2 - log(x)/2)/x)*(log(x) + exp(x^2)*(2*x^2 - 1) - 1))/(8*x^2),x)

[Out]

exp(exp(x^2)/(2*x) + 1/3)/(4*x^(1/(2*x)))

________________________________________________________________________________________

sympy [A]  time = 0.51, size = 19, normalized size = 0.68 \begin {gather*} \frac {e^{\frac {\frac {x}{3} + \frac {e^{x^{2}}}{2} - \frac {\log {\relax (x )}}{2}}{x}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(ln(x)+(2*x**2-1)*exp(x**2)-1)*exp(1/6*(-3*ln(x)+3*exp(x**2)+2*x)/x)/x**2,x)

[Out]

exp((x/3 + exp(x**2)/2 - log(x)/2)/x)/4

________________________________________________________________________________________