3.16.26 \(\int \frac {45 e^{-6+2 x} x+e^{-6+2 x} (90 x-90 x^2) \log (x) \log (\log (x))+((-15 x+9 x^2) \log (x)+30 x \log ^2(x)) \log ^2(\log (x))}{225 e^{-12+4 x} \log (x)+(90 e^{-6+2 x} x \log (x)+150 e^{-6+2 x} \log ^2(x)) \log (\log (x))+(9 x^2 \log (x)+30 x \log ^2(x)+25 \log ^3(x)) \log ^2(\log (x))} \, dx\)

Optimal. Leaf size=40 \[ \frac {x}{5 \left (\frac {\frac {x}{5}+\frac {\log (x)}{3}}{x}+\frac {e^{-6+2 x}}{x \log (\log (x))}\right )} \]

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Rubi [F]  time = 7.59, 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 {45 e^{-6+2 x} x+e^{-6+2 x} \left (90 x-90 x^2\right ) \log (x) \log (\log (x))+\left (\left (-15 x+9 x^2\right ) \log (x)+30 x \log ^2(x)\right ) \log ^2(\log (x))}{225 e^{-12+4 x} \log (x)+\left (90 e^{-6+2 x} x \log (x)+150 e^{-6+2 x} \log ^2(x)\right ) \log (\log (x))+\left (9 x^2 \log (x)+30 x \log ^2(x)+25 \log ^3(x)\right ) \log ^2(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(45*E^(-6 + 2*x)*x + E^(-6 + 2*x)*(90*x - 90*x^2)*Log[x]*Log[Log[x]] + ((-15*x + 9*x^2)*Log[x] + 30*x*Log[
x]^2)*Log[Log[x]]^2)/(225*E^(-12 + 4*x)*Log[x] + (90*E^(-6 + 2*x)*x*Log[x] + 150*E^(-6 + 2*x)*Log[x]^2)*Log[Lo
g[x]] + (9*x^2*Log[x] + 30*x*Log[x]^2 + 25*Log[x]^3)*Log[Log[x]]^2),x]

[Out]

-15*E^12*Defer[Int][(x*Log[Log[x]])/(15*E^(2*x) + 3*E^6*x*Log[Log[x]] + 5*E^6*Log[x]*Log[Log[x]])^2, x] - 9*E^
12*Defer[Int][(x^2*Log[Log[x]])/(Log[x]*(15*E^(2*x) + 3*E^6*x*Log[Log[x]] + 5*E^6*Log[x]*Log[Log[x]])^2), x] -
15*E^12*Defer[Int][(x*Log[Log[x]]^2)/(15*E^(2*x) + 3*E^6*x*Log[Log[x]] + 5*E^6*Log[x]*Log[Log[x]])^2, x] - 9*
E^12*Defer[Int][(x^2*Log[Log[x]]^2)/(15*E^(2*x) + 3*E^6*x*Log[Log[x]] + 5*E^6*Log[x]*Log[Log[x]])^2, x] + 18*E
^12*Defer[Int][(x^3*Log[Log[x]]^2)/(15*E^(2*x) + 3*E^6*x*Log[Log[x]] + 5*E^6*Log[x]*Log[Log[x]])^2, x] + 30*E^
12*Defer[Int][(x^2*Log[x]*Log[Log[x]]^2)/(15*E^(2*x) + 3*E^6*x*Log[Log[x]] + 5*E^6*Log[x]*Log[Log[x]])^2, x] +
3*E^6*Defer[Int][x/(Log[x]*(15*E^(2*x) + 3*E^6*x*Log[Log[x]] + 5*E^6*Log[x]*Log[Log[x]])), x] + 6*E^6*Defer[I
nt][(x*Log[Log[x]])/(15*E^(2*x) + 3*E^6*x*Log[Log[x]] + 5*E^6*Log[x]*Log[Log[x]]), x] - 6*E^6*Defer[Int][(x^2*
Log[Log[x]])/(15*E^(2*x) + 3*E^6*x*Log[Log[x]] + 5*E^6*Log[x]*Log[Log[x]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^6 x \left (15 e^{2 x}+10 e^6 \log ^2(x) \log ^2(\log (x))+\log (x) \log (\log (x)) \left (-30 e^{2 x} (-1+x)+e^6 (-5+3 x) \log (\log (x))\right )\right )}{\log (x) \left (15 e^{2 x}+e^6 (3 x+5 \log (x)) \log (\log (x))\right )^2} \, dx\\ &=\left (3 e^6\right ) \int \frac {x \left (15 e^{2 x}+10 e^6 \log ^2(x) \log ^2(\log (x))+\log (x) \log (\log (x)) \left (-30 e^{2 x} (-1+x)+e^6 (-5+3 x) \log (\log (x))\right )\right )}{\log (x) \left (15 e^{2 x}+e^6 (3 x+5 \log (x)) \log (\log (x))\right )^2} \, dx\\ &=\left (3 e^6\right ) \int \left (-\frac {x (-1-2 \log (x) \log (\log (x))+2 x \log (x) \log (\log (x)))}{\log (x) \left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )}+\frac {e^6 x \log (\log (x)) \left (-3 x-5 \log (x)-5 \log (x) \log (\log (x))-3 x \log (x) \log (\log (x))+6 x^2 \log (x) \log (\log (x))+10 x \log ^2(x) \log (\log (x))\right )}{\log (x) \left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2}\right ) \, dx\\ &=-\left (\left (3 e^6\right ) \int \frac {x (-1-2 \log (x) \log (\log (x))+2 x \log (x) \log (\log (x)))}{\log (x) \left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )} \, dx\right )+\left (3 e^{12}\right ) \int \frac {x \log (\log (x)) \left (-3 x-5 \log (x)-5 \log (x) \log (\log (x))-3 x \log (x) \log (\log (x))+6 x^2 \log (x) \log (\log (x))+10 x \log ^2(x) \log (\log (x))\right )}{\log (x) \left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2} \, dx\\ &=-\left (\left (3 e^6\right ) \int \left (-\frac {x}{\log (x) \left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )}-\frac {2 x \log (\log (x))}{15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))}+\frac {2 x^2 \log (\log (x))}{15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))}\right ) \, dx\right )+\left (3 e^{12}\right ) \int \left (-\frac {5 x \log (\log (x))}{\left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2}-\frac {3 x^2 \log (\log (x))}{\log (x) \left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2}-\frac {5 x \log ^2(\log (x))}{\left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2}-\frac {3 x^2 \log ^2(\log (x))}{\left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2}+\frac {6 x^3 \log ^2(\log (x))}{\left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2}+\frac {10 x^2 \log (x) \log ^2(\log (x))}{\left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2}\right ) \, dx\\ &=\left (3 e^6\right ) \int \frac {x}{\log (x) \left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )} \, dx+\left (6 e^6\right ) \int \frac {x \log (\log (x))}{15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))} \, dx-\left (6 e^6\right ) \int \frac {x^2 \log (\log (x))}{15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))} \, dx-\left (9 e^{12}\right ) \int \frac {x^2 \log (\log (x))}{\log (x) \left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2} \, dx-\left (9 e^{12}\right ) \int \frac {x^2 \log ^2(\log (x))}{\left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2} \, dx-\left (15 e^{12}\right ) \int \frac {x \log (\log (x))}{\left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2} \, dx-\left (15 e^{12}\right ) \int \frac {x \log ^2(\log (x))}{\left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2} \, dx+\left (18 e^{12}\right ) \int \frac {x^3 \log ^2(\log (x))}{\left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2} \, dx+\left (30 e^{12}\right ) \int \frac {x^2 \log (x) \log ^2(\log (x))}{\left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 36, normalized size = 0.90 \begin {gather*} \frac {3 e^6 x^2 \log (\log (x))}{15 e^{2 x}+e^6 (3 x+5 \log (x)) \log (\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(45*E^(-6 + 2*x)*x + E^(-6 + 2*x)*(90*x - 90*x^2)*Log[x]*Log[Log[x]] + ((-15*x + 9*x^2)*Log[x] + 30*
x*Log[x]^2)*Log[Log[x]]^2)/(225*E^(-12 + 4*x)*Log[x] + (90*E^(-6 + 2*x)*x*Log[x] + 150*E^(-6 + 2*x)*Log[x]^2)*
Log[Log[x]] + (9*x^2*Log[x] + 30*x*Log[x]^2 + 25*Log[x]^3)*Log[Log[x]]^2),x]

[Out]

(3*E^6*x^2*Log[Log[x]])/(15*E^(2*x) + E^6*(3*x + 5*Log[x])*Log[Log[x]])

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fricas [A]  time = 0.86, size = 31, normalized size = 0.78 \begin {gather*} \frac {3 \, x^{2} \log \left (\log \relax (x)\right )}{{\left (3 \, x + 5 \, \log \relax (x)\right )} \log \left (\log \relax (x)\right ) + 15 \, e^{\left (2 \, x - 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x*log(x)^2+(9*x^2-15*x)*log(x))*log(log(x))^2+(-90*x^2+90*x)*exp(x-3)^2*log(x)*log(log(x))+45*x
*exp(x-3)^2)/((25*log(x)^3+30*x*log(x)^2+9*x^2*log(x))*log(log(x))^2+(150*exp(x-3)^2*log(x)^2+90*x*exp(x-3)^2*
log(x))*log(log(x))+225*exp(x-3)^4*log(x)),x, algorithm="fricas")

[Out]

3*x^2*log(log(x))/((3*x + 5*log(x))*log(log(x)) + 15*e^(2*x - 6))

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giac [A]  time = 0.31, size = 36, normalized size = 0.90 \begin {gather*} \frac {3 \, x^{2} e^{6} \log \left (\log \relax (x)\right )}{3 \, x e^{6} \log \left (\log \relax (x)\right ) + 5 \, e^{6} \log \relax (x) \log \left (\log \relax (x)\right ) + 15 \, e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x*log(x)^2+(9*x^2-15*x)*log(x))*log(log(x))^2+(-90*x^2+90*x)*exp(x-3)^2*log(x)*log(log(x))+45*x
*exp(x-3)^2)/((25*log(x)^3+30*x*log(x)^2+9*x^2*log(x))*log(log(x))^2+(150*exp(x-3)^2*log(x)^2+90*x*exp(x-3)^2*
log(x))*log(log(x))+225*exp(x-3)^4*log(x)),x, algorithm="giac")

[Out]

3*x^2*e^6*log(log(x))/(3*x*e^6*log(log(x)) + 5*e^6*log(x)*log(log(x)) + 15*e^(2*x))

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maple [A]  time = 0.05, size = 62, normalized size = 1.55

method	result	size
risch	\(\frac {3 x^{2}}{3 x +5 \ln \relax (x )}-\frac {45 x^{2} {\mathrm e}^{2 x -6}}{\left (3 x +5 \ln \relax (x )\right ) \left (5 \ln \relax (x ) \ln \left (\ln \relax (x )\right )+3 x \ln \left (\ln \relax (x )\right )+15 \,{\mathrm e}^{2 x -6}\right )}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((30*x*ln(x)^2+(9*x^2-15*x)*ln(x))*ln(ln(x))^2+(-90*x^2+90*x)*exp(x-3)^2*ln(x)*ln(ln(x))+45*x*exp(x-3)^2)/
((25*ln(x)^3+30*x*ln(x)^2+9*x^2*ln(x))*ln(ln(x))^2+(150*exp(x-3)^2*ln(x)^2+90*x*exp(x-3)^2*ln(x))*ln(ln(x))+22
5*exp(x-3)^4*ln(x)),x,method=_RETURNVERBOSE)

[Out]

3*x^2/(3*x+5*ln(x))-45*x^2*exp(2*x-6)/(3*x+5*ln(x))/(5*ln(x)*ln(ln(x))+3*x*ln(ln(x))+15*exp(2*x-6))

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maxima [A]  time = 0.62, size = 35, normalized size = 0.88 \begin {gather*} \frac {3 \, x^{2} e^{6} \log \left (\log \relax (x)\right )}{{\left (3 \, x e^{6} + 5 \, e^{6} \log \relax (x)\right )} \log \left (\log \relax (x)\right ) + 15 \, e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x*log(x)^2+(9*x^2-15*x)*log(x))*log(log(x))^2+(-90*x^2+90*x)*exp(x-3)^2*log(x)*log(log(x))+45*x
*exp(x-3)^2)/((25*log(x)^3+30*x*log(x)^2+9*x^2*log(x))*log(log(x))^2+(150*exp(x-3)^2*log(x)^2+90*x*exp(x-3)^2*
log(x))*log(log(x))+225*exp(x-3)^4*log(x)),x, algorithm="maxima")

[Out]

3*x^2*e^6*log(log(x))/((3*x*e^6 + 5*e^6*log(x))*log(log(x)) + 15*e^(2*x))

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mupad [B]  time = 1.29, size = 280, normalized size = 7.00 \begin {gather*} 2\,x+\frac {50}{9\,\left (x+\frac {5}{3}\right )}-\frac {\frac {3\,x^2\,\left (3\,x-5\right )}{3\,x+5}+\frac {30\,x^2\,\ln \relax (x)}{3\,x+5}}{3\,x+5\,\ln \relax (x)}-\frac {45\,\left (25\,x^3\,{\mathrm {e}}^{2\,x-6}\,{\ln \relax (x)}^3+30\,x^4\,{\mathrm {e}}^{2\,x-6}\,{\ln \relax (x)}^2-75\,x^3\,{\mathrm {e}}^{4\,x-12}\,{\ln \relax (x)}^2-45\,x^4\,{\mathrm {e}}^{4\,x-12}\,{\ln \relax (x)}^2+150\,x^4\,{\mathrm {e}}^{4\,x-12}\,{\ln \relax (x)}^3+90\,x^5\,{\mathrm {e}}^{4\,x-12}\,{\ln \relax (x)}^2+9\,x^5\,{\mathrm {e}}^{2\,x-6}\,\ln \relax (x)\right )}{\left (3\,x+5\,\ln \relax (x)\right )\,\left (15\,{\mathrm {e}}^{2\,x-6}+\ln \left (\ln \relax (x)\right )\,\left (3\,x+5\,\ln \relax (x)\right )\right )\,\left (25\,x\,{\ln \relax (x)}^3+9\,x^3\,\ln \relax (x)+30\,x^2\,{\ln \relax (x)}^2-45\,x^2\,{\mathrm {e}}^{2\,x-6}\,{\ln \relax (x)}^2+150\,x^2\,{\mathrm {e}}^{2\,x-6}\,{\ln \relax (x)}^3+90\,x^3\,{\mathrm {e}}^{2\,x-6}\,{\ln \relax (x)}^2-75\,x\,{\mathrm {e}}^{2\,x-6}\,{\ln \relax (x)}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(x))^2*(30*x*log(x)^2 - log(x)*(15*x - 9*x^2)) + 45*x*exp(2*x - 6) + log(log(x))*exp(2*x - 6)*log(
x)*(90*x - 90*x^2))/(log(log(x))^2*(30*x*log(x)^2 + 9*x^2*log(x) + 25*log(x)^3) + log(log(x))*(150*exp(2*x - 6
)*log(x)^2 + 90*x*exp(2*x - 6)*log(x)) + 225*exp(4*x - 12)*log(x)),x)

[Out]

2*x + 50/(9*(x + 5/3)) - ((3*x^2*(3*x - 5))/(3*x + 5) + (30*x^2*log(x))/(3*x + 5))/(3*x + 5*log(x)) - (45*(25*
x^3*exp(2*x - 6)*log(x)^3 + 30*x^4*exp(2*x - 6)*log(x)^2 - 75*x^3*exp(4*x - 12)*log(x)^2 - 45*x^4*exp(4*x - 12
)*log(x)^2 + 150*x^4*exp(4*x - 12)*log(x)^3 + 90*x^5*exp(4*x - 12)*log(x)^2 + 9*x^5*exp(2*x - 6)*log(x)))/((3*
x + 5*log(x))*(15*exp(2*x - 6) + log(log(x))*(3*x + 5*log(x)))*(25*x*log(x)^3 + 9*x^3*log(x) + 30*x^2*log(x)^2
- 45*x^2*exp(2*x - 6)*log(x)^2 + 150*x^2*exp(2*x - 6)*log(x)^3 + 90*x^3*exp(2*x - 6)*log(x)^2 - 75*x*exp(2*x
- 6)*log(x)^2))

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sympy [A]  time = 0.38, size = 36, normalized size = 0.90 \begin {gather*} \frac {3 x^{2} \log {\left (\log {\relax (x )} \right )}}{3 x \log {\left (\log {\relax (x )} \right )} + 15 e^{2 x - 6} + 5 \log {\relax (x )} \log {\left (\log {\relax (x )} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((30*x*ln(x)**2+(9*x**2-15*x)*ln(x))*ln(ln(x))**2+(-90*x**2+90*x)*exp(x-3)**2*ln(x)*ln(ln(x))+45*x*e
xp(x-3)**2)/((25*ln(x)**3+30*x*ln(x)**2+9*x**2*ln(x))*ln(ln(x))**2+(150*exp(x-3)**2*ln(x)**2+90*x*exp(x-3)**2*
ln(x))*ln(ln(x))+225*exp(x-3)**4*ln(x)),x)

[Out]

3*x**2*log(log(x))/(3*x*log(log(x)) + 15*exp(2*x - 6) + 5*log(x)*log(log(x)))

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