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

Optimal. Leaf size=18 \[ \log \left (e^{\frac {3 x}{3-x^2}}+\log (x)\right ) \]

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Rubi [F]  time = 17.29, 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 {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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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+\operatorname {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\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 1.55, size = 55, normalized size = 3.06 \begin {gather*} \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 ) \end {gather*}

Warning: Unable to verify antiderivative.

[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]]

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fricas [A]  time = 1.00, size = 15, normalized size = 0.83 \begin {gather*} \log \left (e^{\left (-\frac {3 \, x}{x^{2} - 3}\right )} + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.23, size = 15, normalized size = 0.83 \begin {gather*} \log \left (e^{\left (-\frac {3 \, x}{x^{2} - 3}\right )} + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.03, size = 16, normalized size = 0.89

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

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maxima [B]  time = 0.49, size = 36, normalized size = 2.00 \begin {gather*} -\frac {3 \, x}{x^{2} - 3} + \log \left (\frac {e^{\left (\frac {3 \, x}{x^{2} - 3}\right )} \log \relax (x) + 1}{\log \relax (x)}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 1.46, size = 15, normalized size = 0.83 \begin {gather*} \ln \left ({\mathrm {e}}^{-\frac {3\,x}{x^2-3}}+\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.56, size = 15, normalized size = 0.83 \begin {gather*} \log {\left (\log {\relax (x )} + e^{- \frac {3 x}{x^{2} - 3}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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