3.64.38 \(\int \frac {-6 x+e^x (12+3 \log (3))+e^{2 e^{-x}+x} (24-2 x^2+(6-x^2) \log (3))+(-3 e^x+e^{2 e^{-x}+x} (-6+x^2)) \log (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} (-6+x^2)})}{-3 e^x x^2+e^{2 e^{-x}+x} (-6 x^2+x^4)} \, dx\)

Optimal. Leaf size=33 \[ \frac {3+\log (3)-\log \left (\frac {x}{6+3 e^{-2 e^{-x}}-x^2}\right )}{x} \]

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Rubi [A]  time = 6.35, antiderivative size = 52, normalized size of antiderivative = 1.58, number of steps used = 24, number of rules used = 7, integrand size = 132, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6742, 6688, 6725, 453, 206, 2551, 207} \begin {gather*} -\frac {\log \left (\frac {e^{2 e^{-x}} x}{e^{2 e^{-x}} \left (6-x^2\right )+3}\right )}{x}-\frac {1}{x}+\frac {4+\log (3)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6*x + E^x*(12 + 3*Log[3]) + E^(2/E^x + x)*(24 - 2*x^2 + (6 - x^2)*Log[3]) + (-3*E^x + E^(2/E^x + x)*(-6
+ x^2))*Log[-((E^(2/E^x)*x)/(-3 + E^(2/E^x)*(-6 + x^2)))])/(-3*E^x*x^2 + E^(2/E^x + x)*(-6*x^2 + x^4)),x]

[Out]

-x^(-1) + (4 + Log[3])/x - Log[(E^(2/E^x)*x)/(3 + E^(2/E^x)*(6 - x^2))]/x

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 2551

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 6688

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

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {6 e^{-x}}{x \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}+\frac {-12 \left (1+\frac {\log (3)}{4}\right )-24 e^{2 e^{-x}} \left (1+\frac {\log (3)}{4}\right )+2 e^{2 e^{-x}} x^2 \left (1+\frac {\log (3)}{2}\right )+3 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )+6 e^{2 e^{-x}} \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )-e^{2 e^{-x}} x^2 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (3+6 e^{2 e^{-x}}-e^{2 e^{-x}} x^2\right )}\right ) \, dx\\ &=-\left (6 \int \frac {e^{-x}}{x \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx\right )+\int \frac {-12 \left (1+\frac {\log (3)}{4}\right )-24 e^{2 e^{-x}} \left (1+\frac {\log (3)}{4}\right )+2 e^{2 e^{-x}} x^2 \left (1+\frac {\log (3)}{2}\right )+3 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )+6 e^{2 e^{-x}} \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )-e^{2 e^{-x}} x^2 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (3+6 e^{2 e^{-x}}-e^{2 e^{-x}} x^2\right )} \, dx\\ &=-\left (6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\right )+\int \frac {-12 \left (1+\frac {\log (3)}{4}\right )-e^{2 e^{-x}} \left (24-x^2 (2+\log (3))+\log (729)\right )-\left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (3-e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\\ &=-\left (6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\right )+\int \left (\frac {6}{\left (-6+x^2\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}+\frac {-24 \left (1+\frac {\log (3)}{4}\right )+2 x^2 \left (1+\frac {\log (3)}{2}\right )+6 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )-x^2 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (6-x^2\right )}\right ) \, dx\\ &=6 \int \frac {1}{\left (-6+x^2\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx+\int \frac {-24 \left (1+\frac {\log (3)}{4}\right )+2 x^2 \left (1+\frac {\log (3)}{2}\right )+6 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )-x^2 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (6-x^2\right )} \, dx\\ &=-\left (6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\right )+6 \int \left (-\frac {1}{2 \sqrt {6} \left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}-\frac {1}{2 \sqrt {6} \left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}\right ) \, dx+\int \frac {x^2 (2+\log (3))-6 (4+\log (3))-\left (-6+x^2\right ) \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2 \left (6-x^2\right )} \, dx\\ &=-\left (6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\right )-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx+\int \left (\frac {x^2 (2+\log (3))-6 (4+\log (3))}{x^2 \left (6-x^2\right )}+\frac {\log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2}\right ) \, dx\\ &=-\left (6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\right )-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx+\int \frac {x^2 (2+\log (3))-6 (4+\log (3))}{x^2 \left (6-x^2\right )} \, dx+\int \frac {\log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{x^2} \, dx\\ &=\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-2 \int \frac {1}{6-x^2} \, dx-6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx+\int \frac {e^{-x} \left (3 e^x-6 x+e^{2 e^{-x}+x} \left (6+x^2\right )\right )}{x^2 \left (3-e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\\ &=-\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx+\int \left (\frac {6 e^{-x}}{x \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}-\frac {3+6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2}{x^2 \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}\right ) \, dx\\ &=-\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}+6 \int \frac {e^{-x}}{x \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-6 \int \frac {e^{-x}}{x \left (-3+e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\int \frac {3+6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2}{x^2 \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx\\ &=-\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\int \frac {-3-e^{2 e^{-x}} \left (6+x^2\right )}{x^2 \left (3-e^{2 e^{-x}} \left (-6+x^2\right )\right )} \, dx\\ &=-\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\int \left (\frac {6+x^2}{x^2 \left (-6+x^2\right )}+\frac {6}{\left (-6+x^2\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}\right ) \, dx\\ &=-\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-6 \int \frac {1}{\left (-6+x^2\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\int \frac {6+x^2}{x^2 \left (-6+x^2\right )} \, dx\\ &=-\frac {1}{x}-\sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}-2 \int \frac {1}{-6+x^2} \, dx-6 \int \left (-\frac {1}{2 \sqrt {6} \left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}-\frac {1}{2 \sqrt {6} \left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )}\right ) \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx-\sqrt {\frac {3}{2}} \int \frac {1}{\left (\sqrt {6}+x\right ) \left (-3-6 e^{2 e^{-x}}+e^{2 e^{-x}} x^2\right )} \, dx\\ &=-\frac {1}{x}+\frac {4+\log (3)}{x}-\frac {\log \left (\frac {e^{2 e^{-x}} x}{3+e^{2 e^{-x}} \left (6-x^2\right )}\right )}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 45, normalized size = 1.36 \begin {gather*} \frac {18+\log (729)-6 \log \left (-\frac {e^{2 e^{-x}} x}{-3+e^{2 e^{-x}} \left (-6+x^2\right )}\right )}{6 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*x + E^x*(12 + 3*Log[3]) + E^(2/E^x + x)*(24 - 2*x^2 + (6 - x^2)*Log[3]) + (-3*E^x + E^(2/E^x + x
)*(-6 + x^2))*Log[-((E^(2/E^x)*x)/(-3 + E^(2/E^x)*(-6 + x^2)))])/(-3*E^x*x^2 + E^(2/E^x + x)*(-6*x^2 + x^4)),x
]

[Out]

(18 + Log[729] - 6*Log[-((E^(2/E^x)*x)/(-3 + E^(2/E^x)*(-6 + x^2)))])/(6*x)

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fricas [A]  time = 0.65, size = 51, normalized size = 1.55 \begin {gather*} \frac {\log \relax (3) - \log \left (-\frac {x e^{\left ({\left (x e^{x} + 2\right )} e^{\left (-x\right )}\right )}}{{\left (x^{2} - 6\right )} e^{\left ({\left (x e^{x} + 2\right )} e^{\left (-x\right )}\right )} - 3 \, e^{x}}\right ) + 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-6)*exp(x)*exp(2/exp(x))-3*exp(x))*log(-x*exp(2/exp(x))/((x^2-6)*exp(2/exp(x))-3))+((-x^2+6)*l
og(3)-2*x^2+24)*exp(x)*exp(2/exp(x))+(3*log(3)+12)*exp(x)-6*x)/((x^4-6*x^2)*exp(x)*exp(2/exp(x))-3*exp(x)*x^2)
,x, algorithm="fricas")

[Out]

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

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giac [C]  time = 0.20, size = 60, normalized size = 1.82 \begin {gather*} -\frac {2 \, x {\rm Ei}\left (-x\right ) + 2 \, e^{\left (-x\right )} - \log \relax (3) - \log \left (x^{2} e^{\left (2 \, e^{\left (-x\right )}\right )} - 6 \, e^{\left (2 \, e^{\left (-x\right )}\right )} - 3\right ) + \log \left (-x\right ) - 3}{x} + 2 \, {\rm Ei}\left (-x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-6)*exp(x)*exp(2/exp(x))-3*exp(x))*log(-x*exp(2/exp(x))/((x^2-6)*exp(2/exp(x))-3))+((-x^2+6)*l
og(3)-2*x^2+24)*exp(x)*exp(2/exp(x))+(3*log(3)+12)*exp(x)-6*x)/((x^4-6*x^2)*exp(x)*exp(2/exp(x))-3*exp(x)*x^2)
,x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.38, size = 587, normalized size = 17.79

method	result	size
risch	\(-\frac {\ln \left ({\mathrm e}^{2 \,{\mathrm e}^{-x}}\right )}{x}+\frac {-i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{-x}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )-i \pi \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{3}+2 i \pi \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}-i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{-x}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )^{3}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right ) \mathrm {csgn}\left (\frac {i x \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}}{{\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3}\right )-2 i \pi +6+2 \ln \relax (3)-2 \ln \relax (x )+2 \ln \left ({\mathrm e}^{2 \,{\mathrm e}^{-x}} x^{2}-6 \,{\mathrm e}^{2 \,{\mathrm e}^{-x}}-3\right )}{2 x}\)	\(587\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-6)*exp(x)*exp(2/exp(x))-3*exp(x))*ln(-x*exp(2/exp(x))/((x^2-6)*exp(2/exp(x))-3))+((-x^2+6)*ln(3)-2*
x^2+24)*exp(x)*exp(2/exp(x))+(3*ln(3)+12)*exp(x)-6*x)/((x^4-6*x^2)*exp(x)*exp(2/exp(x))-3*exp(x)*x^2),x,method
=_RETURNVERBOSE)

[Out]

-1/x*ln(exp(2*exp(-x)))+1/2*(-I*Pi*csgn(I*exp(2*exp(-x))/(exp(2*exp(-x))*x^2-6*exp(2*exp(-x))-3))*csgn(I*x/(ex
p(2*exp(-x))*x^2-6*exp(2*exp(-x))-3)*exp(2*exp(-x)))^2+I*Pi*csgn(I/(exp(2*exp(-x))*x^2-6*exp(2*exp(-x))-3))*cs
gn(I*exp(2*exp(-x)))*csgn(I*exp(2*exp(-x))/(exp(2*exp(-x))*x^2-6*exp(2*exp(-x))-3))-I*Pi*csgn(I*x/(exp(2*exp(-
x))*x^2-6*exp(2*exp(-x))-3)*exp(2*exp(-x)))^3+2*I*Pi*csgn(I*x/(exp(2*exp(-x))*x^2-6*exp(2*exp(-x))-3)*exp(2*ex
p(-x)))^2-I*Pi*csgn(I*x)*csgn(I*x/(exp(2*exp(-x))*x^2-6*exp(2*exp(-x))-3)*exp(2*exp(-x)))^2-I*Pi*csgn(I/(exp(2
*exp(-x))*x^2-6*exp(2*exp(-x))-3))*csgn(I*exp(2*exp(-x))/(exp(2*exp(-x))*x^2-6*exp(2*exp(-x))-3))^2-I*Pi*csgn(
I*exp(2*exp(-x)))*csgn(I*exp(2*exp(-x))/(exp(2*exp(-x))*x^2-6*exp(2*exp(-x))-3))^2+I*Pi*csgn(I*exp(2*exp(-x))/
(exp(2*exp(-x))*x^2-6*exp(2*exp(-x))-3))^3+I*Pi*csgn(I*x)*csgn(I*exp(2*exp(-x))/(exp(2*exp(-x))*x^2-6*exp(2*ex
p(-x))-3))*csgn(I*x/(exp(2*exp(-x))*x^2-6*exp(2*exp(-x))-3)*exp(2*exp(-x)))-2*I*Pi+6+2*ln(3)-2*ln(x)+2*ln(exp(
2*exp(-x))*x^2-6*exp(2*exp(-x))-3))/x

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maxima [A]  time = 0.50, size = 38, normalized size = 1.15 \begin {gather*} -\frac {2 \, e^{\left (-x\right )} - \log \relax (3) - \log \left (-{\left (x^{2} - 6\right )} e^{\left (2 \, e^{\left (-x\right )}\right )} + 3\right ) + \log \relax (x) - 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-6)*exp(x)*exp(2/exp(x))-3*exp(x))*log(-x*exp(2/exp(x))/((x^2-6)*exp(2/exp(x))-3))+((-x^2+6)*l
og(3)-2*x^2+24)*exp(x)*exp(2/exp(x))+(3*log(3)+12)*exp(x)-6*x)/((x^4-6*x^2)*exp(x)*exp(2/exp(x))-3*exp(x)*x^2)
,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.52, size = 36, normalized size = 1.09 \begin {gather*} -\frac {\ln \left (-\frac {x\,{\mathrm {e}}^{2\,{\mathrm {e}}^{-x}}}{3\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^{-x}}\,\left (x^2-6\right )-3\right )}\right )-3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + log(-(x*exp(2*exp(-x)))/(exp(2*exp(-x))*(x^2 - 6) - 3))*(3*exp(x) - exp(2*exp(-x))*exp(x)*(x^2 - 6)
) - exp(x)*(3*log(3) + 12) + exp(2*exp(-x))*exp(x)*(log(3)*(x^2 - 6) + 2*x^2 - 24))/(3*x^2*exp(x) + exp(2*exp(
-x))*exp(x)*(6*x^2 - x^4)),x)

[Out]

-(log(-(x*exp(2*exp(-x)))/(3*(exp(2*exp(-x))*(x^2 - 6) - 3))) - 3)/x

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sympy [A]  time = 0.94, size = 36, normalized size = 1.09 \begin {gather*} - \frac {\log {\left (- \frac {x e^{2 e^{- x}}}{\left (x^{2} - 6\right ) e^{2 e^{- x}} - 3} \right )}}{x} - \frac {-3 - \log {\relax (3 )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-6)*exp(x)*exp(2/exp(x))-3*exp(x))*ln(-x*exp(2/exp(x))/((x**2-6)*exp(2/exp(x))-3))+((-x**2+6)
*ln(3)-2*x**2+24)*exp(x)*exp(2/exp(x))+(3*ln(3)+12)*exp(x)-6*x)/((x**4-6*x**2)*exp(x)*exp(2/exp(x))-3*exp(x)*x
**2),x)

[Out]

-log(-x*exp(2*exp(-x))/((x**2 - 6)*exp(2*exp(-x)) - 3))/x - (-3 - log(3))/x

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