3.24.88 \(\int \frac {e^{-4 x/9} (e^{4 x/9} (e^{25} x-5 e^{20} x^2+10 e^{15} x^3-10 e^{10} x^4+5 e^5 x^5-x^6) \log (x)+(324 e^5-324 x) \log ^3(\log (x))+(324 x-36 e^5 x+36 x^2) \log (x) \log ^4(\log (x)))}{(e^{25} x-5 e^{20} x^2+10 e^{15} x^3-10 e^{10} x^4+5 e^5 x^5-x^6) \log (x)} \, dx\)

Optimal. Leaf size=26 \[ 3+x+\frac {81 e^{-4 x/9} \log ^4(\log (x))}{\left (-e^5+x\right )^4} \]

________________________________________________________________________________________

Rubi [F]  time = 7.03, 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 {e^{-4 x/9} \left (e^{4 x/9} \left (e^{25} x-5 e^{20} x^2+10 e^{15} x^3-10 e^{10} x^4+5 e^5 x^5-x^6\right ) \log (x)+\left (324 e^5-324 x\right ) \log ^3(\log (x))+\left (324 x-36 e^5 x+36 x^2\right ) \log (x) \log ^4(\log (x))\right )}{\left (e^{25} x-5 e^{20} x^2+10 e^{15} x^3-10 e^{10} x^4+5 e^5 x^5-x^6\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4*x)/9)*(E^25*x - 5*E^20*x^2 + 10*E^15*x^3 - 10*E^10*x^4 + 5*E^5*x^5 - x^6)*Log[x] + (324*E^5 - 324*x
)*Log[Log[x]]^3 + (324*x - 36*E^5*x + 36*x^2)*Log[x]*Log[Log[x]]^4)/(E^((4*x)/9)*(E^25*x - 5*E^20*x^2 + 10*E^1
5*x^3 - 10*E^10*x^4 + 5*E^5*x^5 - x^6)*Log[x]),x]

[Out]

x + 324*Defer[Int][(E^(-5 - (4*x)/9)*Log[Log[x]]^3)/((E^5 - x)^4*Log[x]), x] + 324*Defer[Int][(E^(-10 - (4*x)/
9)*Log[Log[x]]^3)/((E^5 - x)^3*Log[x]), x] + 324*Defer[Int][(E^(-15 - (4*x)/9)*Log[Log[x]]^3)/((E^5 - x)^2*Log
[x]), x] + 324*Defer[Int][(E^(-20 - (4*x)/9)*Log[Log[x]]^3)/((E^5 - x)*Log[x]), x] + 324*Defer[Int][(E^(-20 -
(4*x)/9)*Log[Log[x]]^3)/(x*Log[x]), x] + 324*Defer[Int][Log[Log[x]]^4/(E^((4*x)/9)*(E^5 - x)^5), x] - 36*Defer
[Int][Log[Log[x]]^4/(E^((4*x)/9)*(E^5 - x)^4), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-4 x/9} \left (324 \left (e^5-x\right ) \log ^3(\log (x))+x \log (x) \left (e^{4 x/9} \left (e^5-x\right )^5-36 \left (-9+e^5-x\right ) \log ^4(\log (x))\right )\right )}{\left (e^5-x\right )^5 x \log (x)} \, dx\\ &=\int \left (1+\frac {36 e^{-4 x/9} \log ^3(\log (x)) \left (9 e^5-9 x+9 \left (1-\frac {e^5}{9}\right ) x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\left (e^5-x\right )^5 x \log (x)}\right ) \, dx\\ &=x+36 \int \frac {e^{-4 x/9} \log ^3(\log (x)) \left (9 e^5-9 x+9 \left (1-\frac {e^5}{9}\right ) x \log (x) \log (\log (x))+x^2 \log (x) \log (\log (x))\right )}{\left (e^5-x\right )^5 x \log (x)} \, dx\\ &=x+36 \int \frac {e^{-4 x/9} \log ^3(\log (x)) \left (9 e^5-9 x-\left (-9+e^5-x\right ) x \log (x) \log (\log (x))\right )}{\left (e^5-x\right )^5 x \log (x)} \, dx\\ &=x+36 \int \left (\frac {9 e^{-4 x/9} \log ^3(\log (x))}{\left (e^5-x\right )^4 x \log (x)}-\frac {e^{-4 x/9} \left (-9+e^5-x\right ) \log ^4(\log (x))}{\left (e^5-x\right )^5}\right ) \, dx\\ &=x-36 \int \frac {e^{-4 x/9} \left (-9+e^5-x\right ) \log ^4(\log (x))}{\left (e^5-x\right )^5} \, dx+324 \int \frac {e^{-4 x/9} \log ^3(\log (x))}{\left (e^5-x\right )^4 x \log (x)} \, dx\\ &=x-36 \int \left (-\frac {9 e^{-4 x/9} \log ^4(\log (x))}{\left (e^5-x\right )^5}+\frac {e^{-4 x/9} \log ^4(\log (x))}{\left (e^5-x\right )^4}\right ) \, dx+324 \int \left (\frac {e^{-5-\frac {4 x}{9}} \log ^3(\log (x))}{\left (e^5-x\right )^4 \log (x)}+\frac {e^{-10-\frac {4 x}{9}} \log ^3(\log (x))}{\left (e^5-x\right )^3 \log (x)}+\frac {e^{-15-\frac {4 x}{9}} \log ^3(\log (x))}{\left (e^5-x\right )^2 \log (x)}+\frac {e^{-20-\frac {4 x}{9}} \log ^3(\log (x))}{\left (e^5-x\right ) \log (x)}+\frac {e^{-20-\frac {4 x}{9}} \log ^3(\log (x))}{x \log (x)}\right ) \, dx\\ &=x-36 \int \frac {e^{-4 x/9} \log ^4(\log (x))}{\left (e^5-x\right )^4} \, dx+324 \int \frac {e^{-5-\frac {4 x}{9}} \log ^3(\log (x))}{\left (e^5-x\right )^4 \log (x)} \, dx+324 \int \frac {e^{-10-\frac {4 x}{9}} \log ^3(\log (x))}{\left (e^5-x\right )^3 \log (x)} \, dx+324 \int \frac {e^{-15-\frac {4 x}{9}} \log ^3(\log (x))}{\left (e^5-x\right )^2 \log (x)} \, dx+324 \int \frac {e^{-20-\frac {4 x}{9}} \log ^3(\log (x))}{\left (e^5-x\right ) \log (x)} \, dx+324 \int \frac {e^{-20-\frac {4 x}{9}} \log ^3(\log (x))}{x \log (x)} \, dx+324 \int \frac {e^{-4 x/9} \log ^4(\log (x))}{\left (e^5-x\right )^5} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.71, size = 25, normalized size = 0.96 \begin {gather*} x+\frac {81 e^{-4 x/9} \log ^4(\log (x))}{\left (e^5-x\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4*x)/9)*(E^25*x - 5*E^20*x^2 + 10*E^15*x^3 - 10*E^10*x^4 + 5*E^5*x^5 - x^6)*Log[x] + (324*E^5 -
324*x)*Log[Log[x]]^3 + (324*x - 36*E^5*x + 36*x^2)*Log[x]*Log[Log[x]]^4)/(E^((4*x)/9)*(E^25*x - 5*E^20*x^2 +
10*E^15*x^3 - 10*E^10*x^4 + 5*E^5*x^5 - x^6)*Log[x]),x]

[Out]

x + (81*Log[Log[x]]^4)/(E^((4*x)/9)*(E^5 - x)^4)

________________________________________________________________________________________

fricas [B]  time = 0.50, size = 74, normalized size = 2.85 \begin {gather*} \frac {{\left (81 \, \log \left (\log \relax (x)\right )^{4} + {\left (x^{5} - 4 \, x^{4} e^{5} + 6 \, x^{3} e^{10} - 4 \, x^{2} e^{15} + x e^{20}\right )} e^{\left (\frac {4}{9} \, x\right )}\right )} e^{\left (-\frac {4}{9} \, x\right )}}{x^{4} - 4 \, x^{3} e^{5} + 6 \, x^{2} e^{10} - 4 \, x e^{15} + e^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-36*x*exp(5)+36*x^2+324*x)*log(x)*log(log(x))^4+(324*exp(5)-324*x)*log(log(x))^3+(x*exp(5)^5-5*x^2
*exp(5)^4+10*x^3*exp(5)^3-10*x^4*exp(5)^2+5*x^5*exp(5)-x^6)*exp(1/9*x)^4*log(x))/(x*exp(5)^5-5*x^2*exp(5)^4+10
*x^3*exp(5)^3-10*x^4*exp(5)^2+5*x^5*exp(5)-x^6)/exp(1/9*x)^4/log(x),x, algorithm="fricas")

[Out]

(81*log(log(x))^4 + (x^5 - 4*x^4*e^5 + 6*x^3*e^10 - 4*x^2*e^15 + x*e^20)*e^(4/9*x))*e^(-4/9*x)/(x^4 - 4*x^3*e^
5 + 6*x^2*e^10 - 4*x*e^15 + e^20)

________________________________________________________________________________________

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(((-36*x*exp(5)+36*x^2+324*x)*log(x)*log(log(x))^4+(324*exp(5)-324*x)*log(log(x))^3+(x*exp(5)^5-5*x^2
*exp(5)^4+10*x^3*exp(5)^3-10*x^4*exp(5)^2+5*x^5*exp(5)-x^6)*exp(1/9*x)^4*log(x))/(x*exp(5)^5-5*x^2*exp(5)^4+10
*x^3*exp(5)^3-10*x^4*exp(5)^2+5*x^5*exp(5)-x^6)/exp(1/9*x)^4/log(x),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 8.53Polynomial exponent overflow. Error: Bad Argument Value

________________________________________________________________________________________

maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-36 x \,{\mathrm e}^{5}+36 x^{2}+324 x \right ) \ln \relax (x ) \ln \left (\ln \relax (x )\right )^{4}+\left (324 \,{\mathrm e}^{5}-324 x \right ) \ln \left (\ln \relax (x )\right )^{3}+\left (x \,{\mathrm e}^{25}-5 x^{2} {\mathrm e}^{20}+10 \,{\mathrm e}^{15} x^{3}-10 x^{4} {\mathrm e}^{10}+5 x^{5} {\mathrm e}^{5}-x^{6}\right ) {\mathrm e}^{\frac {4 x}{9}} \ln \relax (x )\right ) {\mathrm e}^{-\frac {4 x}{9}}}{\left (x \,{\mathrm e}^{25}-5 x^{2} {\mathrm e}^{20}+10 \,{\mathrm e}^{15} x^{3}-10 x^{4} {\mathrm e}^{10}+5 x^{5} {\mathrm e}^{5}-x^{6}\right ) \ln \relax (x )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-36*x*exp(5)+36*x^2+324*x)*ln(x)*ln(ln(x))^4+(324*exp(5)-324*x)*ln(ln(x))^3+(x*exp(5)^5-5*x^2*exp(5)^4+1
0*x^3*exp(5)^3-10*x^4*exp(5)^2+5*x^5*exp(5)-x^6)*exp(1/9*x)^4*ln(x))/(x*exp(5)^5-5*x^2*exp(5)^4+10*x^3*exp(5)^
3-10*x^4*exp(5)^2+5*x^5*exp(5)-x^6)/exp(1/9*x)^4/ln(x),x)

[Out]

int(((-36*x*exp(5)+36*x^2+324*x)*ln(x)*ln(ln(x))^4+(324*exp(5)-324*x)*ln(ln(x))^3+(x*exp(5)^5-5*x^2*exp(5)^4+1
0*x^3*exp(5)^3-10*x^4*exp(5)^2+5*x^5*exp(5)-x^6)*exp(1/9*x)^4*ln(x))/(x*exp(5)^5-5*x^2*exp(5)^4+10*x^3*exp(5)^
3-10*x^4*exp(5)^2+5*x^5*exp(5)-x^6)/exp(1/9*x)^4/ln(x),x)

________________________________________________________________________________________

maxima [B]  time = 0.87, size = 324, normalized size = 12.46 \begin {gather*} \frac {81 \, e^{\left (-\frac {4}{9} \, x\right )} \log \left (\log \relax (x)\right )^{4}}{x^{4} - 4 \, x^{3} e^{5} + 6 \, x^{2} e^{10} - 4 \, x e^{15} + e^{20}} + \frac {12 \, x^{5} - 48 \, x^{4} e^{5} - 48 \, x^{3} e^{10} + 252 \, x^{2} e^{15} - 248 \, x e^{20} + 77 \, e^{25}}{12 \, {\left (x^{4} - 4 \, x^{3} e^{5} + 6 \, x^{2} e^{10} - 4 \, x e^{15} + e^{20}\right )}} + \frac {5 \, {\left (48 \, x^{3} e^{10} - 108 \, x^{2} e^{15} + 88 \, x e^{20} - 25 \, e^{25}\right )}}{12 \, {\left (x^{4} - 4 \, x^{3} e^{5} + 6 \, x^{2} e^{10} - 4 \, x e^{15} + e^{20}\right )}} - \frac {5 \, {\left (4 \, x^{3} e^{10} - 6 \, x^{2} e^{15} + 4 \, x e^{20} - e^{25}\right )}}{2 \, {\left (x^{4} - 4 \, x^{3} e^{5} + 6 \, x^{2} e^{10} - 4 \, x e^{15} + e^{20}\right )}} + \frac {5 \, {\left (6 \, x^{2} e^{15} - 4 \, x e^{20} + e^{25}\right )}}{6 \, {\left (x^{4} - 4 \, x^{3} e^{5} + 6 \, x^{2} e^{10} - 4 \, x e^{15} + e^{20}\right )}} - \frac {5 \, {\left (4 \, x e^{20} - e^{25}\right )}}{12 \, {\left (x^{4} - 4 \, x^{3} e^{5} + 6 \, x^{2} e^{10} - 4 \, x e^{15} + e^{20}\right )}} + \frac {e^{25}}{4 \, {\left (x^{4} - 4 \, x^{3} e^{5} + 6 \, x^{2} e^{10} - 4 \, x e^{15} + e^{20}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-36*x*exp(5)+36*x^2+324*x)*log(x)*log(log(x))^4+(324*exp(5)-324*x)*log(log(x))^3+(x*exp(5)^5-5*x^2
*exp(5)^4+10*x^3*exp(5)^3-10*x^4*exp(5)^2+5*x^5*exp(5)-x^6)*exp(1/9*x)^4*log(x))/(x*exp(5)^5-5*x^2*exp(5)^4+10
*x^3*exp(5)^3-10*x^4*exp(5)^2+5*x^5*exp(5)-x^6)/exp(1/9*x)^4/log(x),x, algorithm="maxima")

[Out]

81*e^(-4/9*x)*log(log(x))^4/(x^4 - 4*x^3*e^5 + 6*x^2*e^10 - 4*x*e^15 + e^20) + 1/12*(12*x^5 - 48*x^4*e^5 - 48*
x^3*e^10 + 252*x^2*e^15 - 248*x*e^20 + 77*e^25)/(x^4 - 4*x^3*e^5 + 6*x^2*e^10 - 4*x*e^15 + e^20) + 5/12*(48*x^
3*e^10 - 108*x^2*e^15 + 88*x*e^20 - 25*e^25)/(x^4 - 4*x^3*e^5 + 6*x^2*e^10 - 4*x*e^15 + e^20) - 5/2*(4*x^3*e^1
0 - 6*x^2*e^15 + 4*x*e^20 - e^25)/(x^4 - 4*x^3*e^5 + 6*x^2*e^10 - 4*x*e^15 + e^20) + 5/6*(6*x^2*e^15 - 4*x*e^2
0 + e^25)/(x^4 - 4*x^3*e^5 + 6*x^2*e^10 - 4*x*e^15 + e^20) - 5/12*(4*x*e^20 - e^25)/(x^4 - 4*x^3*e^5 + 6*x^2*e
^10 - 4*x*e^15 + e^20) + 1/4*e^25/(x^4 - 4*x^3*e^5 + 6*x^2*e^10 - 4*x*e^15 + e^20)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{-\frac {4\,x}{9}}\,\left (\ln \relax (x)\,\left (324\,x-36\,x\,{\mathrm {e}}^5+36\,x^2\right )\,{\ln \left (\ln \relax (x)\right )}^4+\left (324\,{\mathrm {e}}^5-324\,x\right )\,{\ln \left (\ln \relax (x)\right )}^3+{\mathrm {e}}^{\frac {4\,x}{9}}\,\ln \relax (x)\,\left (-x^6+5\,{\mathrm {e}}^5\,x^5-10\,{\mathrm {e}}^{10}\,x^4+10\,{\mathrm {e}}^{15}\,x^3-5\,{\mathrm {e}}^{20}\,x^2+{\mathrm {e}}^{25}\,x\right )\right )}{\ln \relax (x)\,\left (-x^6+5\,{\mathrm {e}}^5\,x^5-10\,{\mathrm {e}}^{10}\,x^4+10\,{\mathrm {e}}^{15}\,x^3-5\,{\mathrm {e}}^{20}\,x^2+{\mathrm {e}}^{25}\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(4*x)/9)*(log(log(x))^4*log(x)*(324*x - 36*x*exp(5) + 36*x^2) - log(log(x))^3*(324*x - 324*exp(5)) +
exp((4*x)/9)*log(x)*(x*exp(25) + 5*x^5*exp(5) - 10*x^4*exp(10) + 10*x^3*exp(15) - 5*x^2*exp(20) - x^6)))/(log
(x)*(x*exp(25) + 5*x^5*exp(5) - 10*x^4*exp(10) + 10*x^3*exp(15) - 5*x^2*exp(20) - x^6)),x)

[Out]

int((exp(-(4*x)/9)*(log(log(x))^4*log(x)*(324*x - 36*x*exp(5) + 36*x^2) - log(log(x))^3*(324*x - 324*exp(5)) +
exp((4*x)/9)*log(x)*(x*exp(25) + 5*x^5*exp(5) - 10*x^4*exp(10) + 10*x^3*exp(15) - 5*x^2*exp(20) - x^6)))/(log
(x)*(x*exp(25) + 5*x^5*exp(5) - 10*x^4*exp(10) + 10*x^3*exp(15) - 5*x^2*exp(20) - x^6)), x)

________________________________________________________________________________________

sympy [A]  time = 0.58, size = 46, normalized size = 1.77 \begin {gather*} x + \frac {81 e^{- \frac {4 x}{9}} \log {\left (\log {\relax (x )} \right )}^{4}}{x^{4} - 4 x^{3} e^{5} + 6 x^{2} e^{10} - 4 x e^{15} + e^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-36*x*exp(5)+36*x**2+324*x)*ln(x)*ln(ln(x))**4+(324*exp(5)-324*x)*ln(ln(x))**3+(x*exp(5)**5-5*x**2
*exp(5)**4+10*x**3*exp(5)**3-10*x**4*exp(5)**2+5*x**5*exp(5)-x**6)*exp(1/9*x)**4*ln(x))/(x*exp(5)**5-5*x**2*ex
p(5)**4+10*x**3*exp(5)**3-10*x**4*exp(5)**2+5*x**5*exp(5)-x**6)/exp(1/9*x)**4/ln(x),x)

[Out]

x + 81*exp(-4*x/9)*log(log(x))**4/(x**4 - 4*x**3*exp(5) + 6*x**2*exp(10) - 4*x*exp(15) + exp(20))

________________________________________________________________________________________