\(\int \frac {9-6 x^2+x^4+e^{-\frac {3 x}{-3+x^2}} (9 x+3 x^3)}{e^{-\frac {3 x}{-3+x^2}} (9 x-6 x^3+x^5)+(9 x-6 x^3+x^5) \log (x)} \, dx\) [2167]

Optimal result

Integrand size = 76, antiderivative size = 18 \[ \int \frac {9-6 x^2+x^4+e^{-\frac {3 x}{-3+x^2}} \left (9 x+3 x^3\right )}{e^{-\frac {3 x}{-3+x^2}} \left (9 x-6 x^3+x^5\right )+\left (9 x-6 x^3+x^5\right ) \log (x)} \, dx=\log \left (e^{\frac {3 x}{3-x^2}}+\log (x)\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {9-6 x^2+x^4+e^{-\frac {3 x}{-3+x^2}} \left (9 x+3 x^3\right )}{e^{-\frac {3 x}{-3+x^2}} \left (9 x-6 x^3+x^5\right )+\left (9 x-6 x^3+x^5\right ) \log (x)} \, dx=\int \frac {9-6 x^2+x^4+e^{-\frac {3 x}{-3+x^2}} \left (9 x+3 x^3\right )}{e^{-\frac {3 x}{-3+x^2}} \left (9 x-6 x^3+x^5\right )+\left (9 x-6 x^3+x^5\right ) \log (x)} \, dx \]

[In]

Int[(9 - 6*x^2 + x^4 + (9*x + 3*x^3)/E^((3*x)/(-3 + x^2)))/((9*x - 6*x^3 + x^5)/E^((3*x)/(-3 + x^2)) + (9*x -
6*x^3 + x^5)*Log[x]),x]

[Out]

Log[Log[x]] - (Sqrt[3]*Defer[Int][1/((Sqrt[3] - x)*(1 + E^((3*x)/(-3 + x^2))*Log[x])), x])/2 - (Sqrt[3]*Defer[
Int][1/((Sqrt[3] + x)*(1 + E^((3*x)/(-3 + x^2))*Log[x])), x])/2 + 18*Defer[Int][1/((-3 + x^2)^2*(1 + E^((3*x)/
(-3 + x^2))*Log[x])), x] - Defer[Int][1/(x*Log[x]*(1 + E^((3*x)/(-3 + x^2))*Log[x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\frac {3 x}{-3+x^2}} \left (9-6 x^2+x^4+e^{-\frac {3 x}{-3+x^2}} \left (9 x+3 x^3\right )\right )}{x \left (3-x^2\right )^2 \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx \\ & = \int \frac {e^{\frac {3 x}{-3+x^2}} \left (-3+x^2\right )^2+3 x \left (3+x^2\right )}{\left (3-x^2\right )^2 \left (x+e^{\frac {3 x}{-3+x^2}} x \log (x)\right )} \, dx \\ & = \int \left (\frac {1}{x \log (x)}-\frac {9-6 x^2+x^4-9 x \log (x)-3 x^3 \log (x)}{x \left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx \\ & = \int \frac {1}{x \log (x)} \, dx-\int \frac {9-6 x^2+x^4-9 x \log (x)-3 x^3 \log (x)}{x \left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx \\ & = -\int \left (\frac {9-6 x^2+x^4-9 x \log (x)-3 x^3 \log (x)}{9 x \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {x \left (9-6 x^2+x^4-9 x \log (x)-3 x^3 \log (x)\right )}{3 \left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}-\frac {x \left (9-6 x^2+x^4-9 x \log (x)-3 x^3 \log (x)\right )}{9 \left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = \log (\log (x))-\frac {1}{9} \int \frac {9-6 x^2+x^4-9 x \log (x)-3 x^3 \log (x)}{x \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\frac {1}{9} \int \frac {x \left (9-6 x^2+x^4-9 x \log (x)-3 x^3 \log (x)\right )}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-\frac {1}{3} \int \frac {x \left (9-6 x^2+x^4-9 x \log (x)-3 x^3 \log (x)\right )}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx \\ & = \log (\log (x))-\frac {1}{9} \int \left (-\frac {9}{1+e^{\frac {3 x}{-3+x^2}} \log (x)}-\frac {3 x^2}{1+e^{\frac {3 x}{-3+x^2}} \log (x)}+\frac {9}{x \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}-\frac {6 x}{\log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {x^3}{\log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx+\frac {1}{9} \int \left (-\frac {9 x^2}{\left (-3+x^2\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}-\frac {3 x^4}{\left (-3+x^2\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {9 x}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}-\frac {6 x^3}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {x^5}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx-\frac {1}{3} \int \left (-\frac {9 x^2}{\left (-3+x^2\right )^2 \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}-\frac {3 x^4}{\left (-3+x^2\right )^2 \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {9 x}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}-\frac {6 x^3}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {x^5}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx \\ & = \log (\log (x))-\frac {1}{9} \int \frac {x^3}{\log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\frac {1}{9} \int \frac {x^5}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\frac {1}{3} \int \frac {x^2}{1+e^{\frac {3 x}{-3+x^2}} \log (x)} \, dx-\frac {1}{3} \int \frac {x^4}{\left (-3+x^2\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-\frac {1}{3} \int \frac {x^5}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\frac {2}{3} \int \frac {x}{\log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-\frac {2}{3} \int \frac {x^3}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+2 \int \frac {x^3}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+3 \int \frac {x^2}{\left (-3+x^2\right )^2 \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-3 \int \frac {x}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\int \frac {1}{1+e^{\frac {3 x}{-3+x^2}} \log (x)} \, dx+\int \frac {x^4}{\left (-3+x^2\right )^2 \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-\int \frac {x^2}{\left (-3+x^2\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-\int \frac {1}{x \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\int \frac {x}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx \\ & = \log (\log (x))-\frac {1}{9} \int \frac {x^3}{\log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\frac {1}{9} \int \left (\frac {3 x}{\log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {x^3}{\log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {9 x}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx+\frac {1}{3} \int \frac {x^2}{1+e^{\frac {3 x}{-3+x^2}} \log (x)} \, dx-\frac {1}{3} \int \left (\frac {3}{1+e^{\frac {3 x}{-3+x^2}} \log (x)}+\frac {x^2}{1+e^{\frac {3 x}{-3+x^2}} \log (x)}+\frac {9}{\left (-3+x^2\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx-\frac {1}{3} \int \left (\frac {x}{\log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {9 x}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {6 x}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx+\frac {2}{3} \int \frac {x}{\log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-\frac {2}{3} \int \left (\frac {x}{\log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {3 x}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx+2 \int \left (\frac {3 x}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {x}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx-3 \int \frac {x}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+3 \int \left (\frac {3}{\left (-3+x^2\right )^2 \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {1}{\left (-3+x^2\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx+\int \frac {1}{1+e^{\frac {3 x}{-3+x^2}} \log (x)} \, dx-\int \frac {1}{x \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-\int \left (\frac {1}{1+e^{\frac {3 x}{-3+x^2}} \log (x)}+\frac {3}{\left (-3+x^2\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx+\int \left (\frac {1}{1+e^{\frac {3 x}{-3+x^2}} \log (x)}+\frac {9}{\left (-3+x^2\right )^2 \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {6}{\left (-3+x^2\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx+\int \left (-\frac {1}{2 \left (\sqrt {3}-x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {1}{2 \left (\sqrt {3}+x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx \\ & = \log (\log (x))-\frac {1}{2} \int \frac {1}{\left (\sqrt {3}-x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\frac {1}{2} \int \frac {1}{\left (\sqrt {3}+x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-2 \int \frac {x}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-3 \int \frac {1}{\left (-3+x^2\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-2 \left (3 \int \frac {x}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx\right )+6 \int \frac {1}{\left (-3+x^2\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+6 \int \frac {x}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+2 \left (9 \int \frac {1}{\left (-3+x^2\right )^2 \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx\right )-\int \frac {1}{x \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\int \frac {x}{\left (-3+x^2\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx \\ & = \log (\log (x))-\frac {1}{2} \int \frac {1}{\left (\sqrt {3}-x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\frac {1}{2} \int \frac {1}{\left (\sqrt {3}+x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-2 \int \left (-\frac {1}{2 \left (\sqrt {3}-x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {1}{2 \left (\sqrt {3}+x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx-2 \left (3 \int \frac {x}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx\right )-3 \int \left (-\frac {1}{2 \sqrt {3} \left (\sqrt {3}-x\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}-\frac {1}{2 \sqrt {3} \left (\sqrt {3}+x\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx+6 \int \frac {x}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+6 \int \left (-\frac {1}{2 \sqrt {3} \left (\sqrt {3}-x\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}-\frac {1}{2 \sqrt {3} \left (\sqrt {3}+x\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx+2 \left (9 \int \frac {1}{\left (-3+x^2\right )^2 \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx\right )-\int \frac {1}{x \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\int \left (-\frac {1}{2 \left (\sqrt {3}-x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}+\frac {1}{2 \left (\sqrt {3}+x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )}\right ) \, dx \\ & = \log (\log (x))-2 \left (\frac {1}{2} \int \frac {1}{\left (\sqrt {3}-x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx\right )+2 \left (\frac {1}{2} \int \frac {1}{\left (\sqrt {3}+x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx\right )-2 \left (3 \int \frac {x}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx\right )+6 \int \frac {x}{\left (-3+x^2\right )^2 \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+2 \left (9 \int \frac {1}{\left (-3+x^2\right )^2 \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx\right )+\frac {1}{2} \sqrt {3} \int \frac {1}{\left (\sqrt {3}-x\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\frac {1}{2} \sqrt {3} \int \frac {1}{\left (\sqrt {3}+x\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-\sqrt {3} \int \frac {1}{\left (\sqrt {3}-x\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-\sqrt {3} \int \frac {1}{\left (\sqrt {3}+x\right ) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx+\int \frac {1}{\left (\sqrt {3}-x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-\int \frac {1}{x \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx-\int \frac {1}{\left (\sqrt {3}+x\right ) \log (x) \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right )} \, dx \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(18)=36\).

Time = 0.46 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.06 \[ \int \frac {9-6 x^2+x^4+e^{-\frac {3 x}{-3+x^2}} \left (9 x+3 x^3\right )}{e^{-\frac {3 x}{-3+x^2}} \left (9 x-6 x^3+x^5\right )+\left (9 x-6 x^3+x^5\right ) \log (x)} \, dx=\frac {1}{2} \sqrt {3} \log \left (\sqrt {3}-x\right )-\frac {1}{2} \sqrt {3} \log \left (\sqrt {3}+x\right )+\log \left (1+e^{\frac {3 x}{-3+x^2}} \log (x)\right ) \]

[In]

Integrate[(9 - 6*x^2 + x^4 + (9*x + 3*x^3)/E^((3*x)/(-3 + x^2)))/((9*x - 6*x^3 + x^5)/E^((3*x)/(-3 + x^2)) + (
9*x - 6*x^3 + x^5)*Log[x]),x]

[Out]

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

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

method	result	size
risch	\(\ln \left (\ln \left (x \right )+{\mathrm e}^{-\frac {3 x}{x^{2}-3}}\right )\)	\(16\)
parallelrisch	\(\ln \left (\ln \left (x \right )+{\mathrm e}^{-\frac {3 x}{x^{2}-3}}\right )\)	\(16\)

[In]

int(((3*x^3+9*x)*exp(-3*x/(x^2-3))+x^4-6*x^2+9)/((x^5-6*x^3+9*x)*ln(x)+(x^5-6*x^3+9*x)*exp(-3*x/(x^2-3))),x,me
thod=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {9-6 x^2+x^4+e^{-\frac {3 x}{-3+x^2}} \left (9 x+3 x^3\right )}{e^{-\frac {3 x}{-3+x^2}} \left (9 x-6 x^3+x^5\right )+\left (9 x-6 x^3+x^5\right ) \log (x)} \, dx=\log \left (e^{\left (-\frac {3 \, x}{x^{2} - 3}\right )} + \log \left (x\right )\right ) \]

[In]

integrate(((3*x^3+9*x)*exp(-3*x/(x^2-3))+x^4-6*x^2+9)/((x^5-6*x^3+9*x)*log(x)+(x^5-6*x^3+9*x)*exp(-3*x/(x^2-3)
)),x, algorithm="fricas")

[Out]

log(e^(-3*x/(x^2 - 3)) + log(x))

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {9-6 x^2+x^4+e^{-\frac {3 x}{-3+x^2}} \left (9 x+3 x^3\right )}{e^{-\frac {3 x}{-3+x^2}} \left (9 x-6 x^3+x^5\right )+\left (9 x-6 x^3+x^5\right ) \log (x)} \, dx=\log {\left (\log {\left (x \right )} + e^{- \frac {3 x}{x^{2} - 3}} \right )} \]

[In]

integrate(((3*x**3+9*x)*exp(-3*x/(x**2-3))+x**4-6*x**2+9)/((x**5-6*x**3+9*x)*ln(x)+(x**5-6*x**3+9*x)*exp(-3*x/
(x**2-3))),x)

[Out]

log(log(x) + exp(-3*x/(x**2 - 3)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (15) = 30\).

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.00 \[ \int \frac {9-6 x^2+x^4+e^{-\frac {3 x}{-3+x^2}} \left (9 x+3 x^3\right )}{e^{-\frac {3 x}{-3+x^2}} \left (9 x-6 x^3+x^5\right )+\left (9 x-6 x^3+x^5\right ) \log (x)} \, dx=-\frac {3 \, x}{x^{2} - 3} + \log \left (\frac {e^{\left (\frac {3 \, x}{x^{2} - 3}\right )} \log \left (x\right ) + 1}{\log \left (x\right )}\right ) + \log \left (\log \left (x\right )\right ) \]

[In]

integrate(((3*x^3+9*x)*exp(-3*x/(x^2-3))+x^4-6*x^2+9)/((x^5-6*x^3+9*x)*log(x)+(x^5-6*x^3+9*x)*exp(-3*x/(x^2-3)
)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {9-6 x^2+x^4+e^{-\frac {3 x}{-3+x^2}} \left (9 x+3 x^3\right )}{e^{-\frac {3 x}{-3+x^2}} \left (9 x-6 x^3+x^5\right )+\left (9 x-6 x^3+x^5\right ) \log (x)} \, dx=\log \left (e^{\left (-\frac {3 \, x}{x^{2} - 3}\right )} + \log \left (x\right )\right ) \]

[In]

integrate(((3*x^3+9*x)*exp(-3*x/(x^2-3))+x^4-6*x^2+9)/((x^5-6*x^3+9*x)*log(x)+(x^5-6*x^3+9*x)*exp(-3*x/(x^2-3)
)),x, algorithm="giac")

[Out]

log(e^(-3*x/(x^2 - 3)) + log(x))

Mupad [B] (verification not implemented)

Time = 9.52 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {9-6 x^2+x^4+e^{-\frac {3 x}{-3+x^2}} \left (9 x+3 x^3\right )}{e^{-\frac {3 x}{-3+x^2}} \left (9 x-6 x^3+x^5\right )+\left (9 x-6 x^3+x^5\right ) \log (x)} \, dx=\ln \left ({\mathrm {e}}^{-\frac {3\,x}{x^2-3}}+\ln \left (x\right )\right ) \]

[In]

int((exp(-(3*x)/(x^2 - 3))*(9*x + 3*x^3) - 6*x^2 + x^4 + 9)/(exp(-(3*x)/(x^2 - 3))*(9*x - 6*x^3 + x^5) + log(x
)*(9*x - 6*x^3 + x^5)),x)

[Out]

log(exp(-(3*x)/(x^2 - 3)) + log(x))