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

Optimal. Leaf size=17 \[ \frac {e^{x+\frac {\log (3)}{x}}}{2 x^2} \]

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Rubi [A]  time = 0.24, antiderivative size = 15, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6688, 12, 2288} \begin {gather*} \frac {3^{\frac {1}{x}} e^x}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3^x^(-1)*E^x)/(2*x^2)

Rule 12

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 15, normalized size = 0.88 \begin {gather*} \frac {3^{\frac {1}{x}} e^x}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3^x^(-1)*E^x)/(2*x^2)

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fricas [A]  time = 0.55, size = 19, normalized size = 1.12 \begin {gather*} e^{\left (\frac {x^{2} + x \log \left (\frac {1}{2 \, x^{2}}\right ) + \log \relax (3)}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^((x^2 + x*log(1/2/x^2) + log(3))/x)

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giac [A]  time = 0.19, size = 17, normalized size = 1.00 \begin {gather*} e^{\left (x + \frac {\log \relax (3)}{x} - \log \left (2 \, x^{2}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(x + log(3)/x - log(2*x^2))

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maple [A]  time = 0.18, size = 13, normalized size = 0.76

method	result	size
risch	\(\frac {3^{\frac {1}{x}} {\mathrm e}^{x}}{2 x^{2}}\)	\(13\)
gosper	\({\mathrm e}^{\frac {x \ln \left (\frac {1}{2 x^{2}}\right )+\ln \relax (3)+x^{2}}{x}}\)	\(20\)
default	\({\mathrm e}^{\frac {x \ln \left (\frac {1}{2 x^{2}}\right )+\ln \relax (3)+x^{2}}{x}}\)	\(20\)
norman	\({\mathrm e}^{\frac {x \ln \left (\frac {1}{2 x^{2}}\right )+\ln \relax (3)+x^{2}}{x}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 14, normalized size = 0.82 \begin {gather*} \frac {e^{\left (x + \frac {\log \relax (3)}{x}\right )}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^(x + log(3)/x)/x^2

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mupad [B]  time = 3.58, size = 12, normalized size = 0.71 \begin {gather*} \frac {3^{1/x}\,{\mathrm {e}}^x}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.33, size = 19, normalized size = 1.12 \begin {gather*} e^{\frac {x^{2} + x \log {\left (\frac {1}{2 x^{2}} \right )} + \log {\relax (3 )}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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