3.86.11 \(\int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} (2 x-x^2+(-2 x+2 x^2) \log (x)+(30+3 x) \log ^2(x))}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{x-e^{\frac {1}{30} x \left (3+\frac {-2+x}{\log (x)}\right )} x} \]

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Rubi [F]  time = 10.06, 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 {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-30*Log[x]^2 + E^((-2*x + x^2 + 3*x*Log[x])/(30*Log[x]))*(2*x - x^2 + (-2*x + 2*x^2)*Log[x] + (30 + 3*x)*
Log[x]^2))/(30*x^2*Log[x]^2 - 60*E^((-2*x + x^2 + 3*x*Log[x])/(30*Log[x]))*x^2*Log[x]^2 + 30*E^((-2*x + x^2 +
3*x*Log[x])/(15*Log[x]))*x^2*Log[x]^2),x]

[Out]

-Defer[Int][E^((2*x)/(15*Log[x]))/((E^(x/(15*Log[x])) - E^((x*(x + 3*Log[x]))/(30*Log[x])))^2*x^2), x] + Defer
[Int][E^((x*(2 + x + 3*Log[x]))/(30*Log[x]))/((E^(x/(15*Log[x])) - E^((x*(x + 3*Log[x]))/(30*Log[x])))^2*x^2),
x] + Defer[Int][E^((x*(2 + x + 3*Log[x]))/(30*Log[x]))/((E^(x/(15*Log[x])) - E^((x*(x + 3*Log[x]))/(30*Log[x]
)))^2*x), x]/10 - Defer[Int][E^((x*(2 + x + 3*Log[x]))/(30*Log[x]))/((E^(x/(15*Log[x])) - E^((x*(x + 3*Log[x])
)/(30*Log[x])))^2*Log[x]^2), x]/30 + Defer[Int][E^((x*(2 + x + 3*Log[x]))/(30*Log[x]))/((E^(x/(15*Log[x])) - E
^((x*(x + 3*Log[x]))/(30*Log[x])))^2*x*Log[x]^2), x]/15 + Defer[Int][E^((x*(2 + x + 3*Log[x]))/(30*Log[x]))/((
E^(x/(15*Log[x])) - E^((x*(x + 3*Log[x]))/(30*Log[x])))^2*Log[x]), x]/15 - Defer[Int][E^((x*(2 + x + 3*Log[x])
)/(30*Log[x]))/((E^(x/(15*Log[x])) - E^((x*(x + 3*Log[x]))/(30*Log[x])))^2*x*Log[x]), x]/15

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {2 x}{15 \log (x)}} \left (-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )\right )}{30 \left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x^2 \log ^2(x)} \, dx\\ &=\frac {1}{30} \int \frac {e^{\frac {2 x}{15 \log (x)}} \left (-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x^2 \log ^2(x)} \, dx\\ &=\frac {1}{30} \int \left (-\frac {30 e^{\frac {2 x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x^2}+\frac {30 \exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x^2}+\frac {3 \exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x}-\frac {\exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 \log ^2(x)}+\frac {2 \exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x \log ^2(x)}+\frac {2 \exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 \log (x)}-\frac {2 \exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x \log (x)}\right ) \, dx\\ &=-\left (\frac {1}{30} \int \frac {\exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 \log ^2(x)} \, dx\right )+\frac {1}{15} \int \frac {\exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x \log ^2(x)} \, dx+\frac {1}{15} \int \frac {\exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 \log (x)} \, dx-\frac {1}{15} \int \frac {\exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x \log (x)} \, dx+\frac {1}{10} \int \frac {\exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x} \, dx-\int \frac {e^{\frac {2 x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x^2} \, dx+\int \frac {\exp \left (\frac {2 x}{15 \log (x)}+\frac {x (-2+x+3 \log (x))}{30 \log (x)}\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x^2} \, dx\\ &=-\left (\frac {1}{30} \int \frac {e^{\frac {x (2+x+3 \log (x))}{30 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 \log ^2(x)} \, dx\right )+\frac {1}{15} \int \frac {e^{\frac {x (2+x+3 \log (x))}{30 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x \log ^2(x)} \, dx+\frac {1}{15} \int \frac {e^{\frac {x (2+x+3 \log (x))}{30 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 \log (x)} \, dx-\frac {1}{15} \int \frac {e^{\frac {x (2+x+3 \log (x))}{30 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x \log (x)} \, dx+\frac {1}{10} \int \frac {e^{\frac {x (2+x+3 \log (x))}{30 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x} \, dx-\int \frac {e^{\frac {2 x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x^2} \, dx+\int \frac {e^{\frac {x (2+x+3 \log (x))}{30 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-30*Log[x]^2 + E^((-2*x + x^2 + 3*x*Log[x])/(30*Log[x]))*(2*x - x^2 + (-2*x + 2*x^2)*Log[x] + (30 +
3*x)*Log[x]^2))/(30*x^2*Log[x]^2 - 60*E^((-2*x + x^2 + 3*x*Log[x])/(30*Log[x]))*x^2*Log[x]^2 + 30*E^((-2*x +
x^2 + 3*x*Log[x])/(15*Log[x]))*x^2*Log[x]^2),x]

[Out]

$Aborted

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fricas [A]  time = 0.85, size = 29, normalized size = 1.21 \begin {gather*} -\frac {1}{x e^{\left (\frac {x^{2} + 3 \, x \log \relax (x) - 2 \, x}{30 \, \log \relax (x)}\right )} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x+30)*log(x)^2+(2*x^2-2*x)*log(x)-x^2+2*x)*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))-30*log(x)^2)/(
30*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))^2-60*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))+
30*x^2*log(x)^2),x, algorithm="fricas")

[Out]

-1/(x*e^(1/30*(x^2 + 3*x*log(x) - 2*x)/log(x)) - x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x+30)*log(x)^2+(2*x^2-2*x)*log(x)-x^2+2*x)*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))-30*log(x)^2)/(
30*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))^2-60*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))+
30*x^2*log(x)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{50625,[0,19]%%%}+%%%{-607500,[0,18]%%%}+%%%{3037500,[0,1
7]%%%}+%%%{

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maple [A]  time = 0.04, size = 25, normalized size = 1.04

method	result	size
risch	\(-\frac {1}{x \left ({\mathrm e}^{\frac {x \left (3 \ln \relax (x )+x -2\right )}{30 \ln \relax (x )}}-1\right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*x+30)*ln(x)^2+(2*x^2-2*x)*ln(x)-x^2+2*x)*exp(1/30*(3*x*ln(x)+x^2-2*x)/ln(x))-30*ln(x)^2)/(30*x^2*ln(x
)^2*exp(1/30*(3*x*ln(x)+x^2-2*x)/ln(x))^2-60*x^2*ln(x)^2*exp(1/30*(3*x*ln(x)+x^2-2*x)/ln(x))+30*x^2*ln(x)^2),x
,method=_RETURNVERBOSE)

[Out]

-1/x/(exp(1/30*x*(3*ln(x)+x-2)/ln(x))-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{30} \, \int \frac {{\left (3 \, {\left (x + 10\right )} \log \relax (x)^{2} - x^{2} + 2 \, {\left (x^{2} - x\right )} \log \relax (x) + 2 \, x\right )} e^{\left (\frac {x^{2} + 3 \, x \log \relax (x) - 2 \, x}{30 \, \log \relax (x)}\right )} - 30 \, \log \relax (x)^{2}}{x^{2} e^{\left (\frac {x^{2} + 3 \, x \log \relax (x) - 2 \, x}{15 \, \log \relax (x)}\right )} \log \relax (x)^{2} - 2 \, x^{2} e^{\left (\frac {x^{2} + 3 \, x \log \relax (x) - 2 \, x}{30 \, \log \relax (x)}\right )} \log \relax (x)^{2} + x^{2} \log \relax (x)^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x+30)*log(x)^2+(2*x^2-2*x)*log(x)-x^2+2*x)*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))-30*log(x)^2)/(
30*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))^2-60*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))+
30*x^2*log(x)^2),x, algorithm="maxima")

[Out]

1/30*integrate(((3*(x + 10)*log(x)^2 - x^2 + 2*(x^2 - x)*log(x) + 2*x)*e^(1/30*(x^2 + 3*x*log(x) - 2*x)/log(x)
) - 30*log(x)^2)/(x^2*e^(1/15*(x^2 + 3*x*log(x) - 2*x)/log(x))*log(x)^2 - 2*x^2*e^(1/30*(x^2 + 3*x*log(x) - 2*
x)/log(x))*log(x)^2 + x^2*log(x)^2), x)

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mupad [B]  time = 6.16, size = 77, normalized size = 3.21 \begin {gather*} -\frac {x\,\left (3\,{\ln \relax (x)}^2-2\,\ln \relax (x)+2\right )+x^2\,\left (2\,\ln \relax (x)-1\right )}{x^2\,\left ({\mathrm {e}}^{\frac {x}{10}-\frac {x}{15\,\ln \relax (x)}+\frac {x^2}{30\,\ln \relax (x)}}-1\right )\,\left (3\,{\ln \relax (x)}^2-2\,\ln \relax (x)-x+2\,x\,\ln \relax (x)+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(30*log(x)^2 - exp(((x*log(x))/10 - x/15 + x^2/30)/log(x))*(2*x - log(x)*(2*x - 2*x^2) - x^2 + log(x)^2*(
3*x + 30)))/(30*x^2*log(x)^2 - 60*x^2*exp(((x*log(x))/10 - x/15 + x^2/30)/log(x))*log(x)^2 + 30*x^2*exp((2*((x
*log(x))/10 - x/15 + x^2/30))/log(x))*log(x)^2),x)

[Out]

-(x*(3*log(x)^2 - 2*log(x) + 2) + x^2*(2*log(x) - 1))/(x^2*(exp(x/10 - x/(15*log(x)) + x^2/(30*log(x))) - 1)*(
3*log(x)^2 - 2*log(x) - x + 2*x*log(x) + 2))

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sympy [A]  time = 0.42, size = 26, normalized size = 1.08 \begin {gather*} - \frac {1}{x e^{\frac {\frac {x^{2}}{30} + \frac {x \log {\relax (x )}}{10} - \frac {x}{15}}{\log {\relax (x )}}} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x+30)*ln(x)**2+(2*x**2-2*x)*ln(x)-x**2+2*x)*exp(1/30*(3*x*ln(x)+x**2-2*x)/ln(x))-30*ln(x)**2)/(
30*x**2*ln(x)**2*exp(1/30*(3*x*ln(x)+x**2-2*x)/ln(x))**2-60*x**2*ln(x)**2*exp(1/30*(3*x*ln(x)+x**2-2*x)/ln(x))
+30*x**2*ln(x)**2),x)

[Out]

-1/(x*exp((x**2/30 + x*log(x)/10 - x/15)/log(x)) - x)

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