3.77.79 \(\int \frac {-9+e^{e^x} (-3-3 e^x x-6 \log (5))+e^{2 e^x} (-\log (5)-\log ^2(5))}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} (5 x \log (5)+(-10+5 x) \log ^2(5))} \, dx\)

Optimal. Leaf size=26 \[ 2-\frac {1}{5} \log \left (-2+x+\frac {x}{3 e^{-e^x}+\log (5)}\right ) \]

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Rubi [F]  time = 9.57, 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+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-9 + E^E^x*(-3 - 3*E^x*x - 6*Log[5]) + E^(2*E^x)*(-Log[5] - Log[5]^2))/(-90 + 45*x + E^E^x*(15*x + (-60 +
30*x)*Log[5]) + E^(2*E^x)*(5*x*Log[5] + (-10 + 5*x)*Log[5]^2)),x]

[Out]

Log[3 + E^E^x*Log[5]]/5 + (Log[25]*Defer[Int][E^(E^x + x)/(-6 + 3*x - 2*E^E^x*Log[5] + E^E^x*x*(1 + Log[5])),
x])/5 + (3*Log[25]*Defer[Int][1/(x*(-6 + 3*x - 2*E^E^x*Log[5] + E^E^x*x*(1 + Log[5]))), x])/5 + (Log[25]*(1 +
Log[25])*Defer[Int][E^E^x/(x*(-6 + 3*x - 2*E^E^x*Log[5] + E^E^x*x*(1 + Log[5]))), x])/5 + (Log[5]*(1 + Log[5])
*Log[25]*Defer[Int][E^(2*E^x)/(x*(-6 + 3*x - 2*E^E^x*Log[5] + E^E^x*x*(1 + Log[5]))), x])/15 + (3*(1 + Log[5])
*Defer[Int][(6 - 3*x - E^E^x*x*(1 + Log[5]) + E^E^x*Log[25])^(-1), x])/5 + ((1 + Log[5])*(1 + Log[25])*Defer[I
nt][E^E^x/(6 - 3*x - E^E^x*x*(1 + Log[5]) + E^E^x*Log[25]), x])/5 + (Log[5]*(1 + Log[5])^2*Defer[Int][E^(2*E^x
)/(6 - 3*x - E^E^x*x*(1 + Log[5]) + E^E^x*Log[25]), x])/15 + ((1 + Log[5])*Defer[Int][(E^(E^x + x)*x)/(6 - 3*x
- E^E^x*x*(1 + Log[5]) + E^E^x*Log[25]), x])/5 + (9*Log[5]*Defer[Int][1/(x*(9 + E^E^x*Log[125])), x])/5 + (3*
Log[5]*(1 + Log[25])*Defer[Int][E^E^x/(x*(9 + E^E^x*Log[125])), x])/5 + (Log[5]^2*(1 + Log[5])*Defer[Int][E^(2
*E^x)/(x*(9 + E^E^x*Log[125])), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9-e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )-e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{5 \left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx\\ &=\frac {1}{5} \int \frac {9-e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )-e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx\\ &=\frac {1}{5} \int \left (\frac {9}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {3 e^{e^x+x} x}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {e^{2 e^x} \log (5) (1+\log (5))}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {3 e^{e^x} (1+\log (25))}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}\right ) \, dx\\ &=\frac {3}{5} \int \frac {e^{e^x+x} x}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {9}{5} \int \frac {1}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} (\log (5) (1+\log (5))) \int \frac {e^{2 e^x}}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} (3 (1+\log (25))) \int \frac {e^{e^x}}{\left (3+e^{e^x} \log (5)\right ) \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx\\ &=\frac {3}{5} \int \left (\frac {e^{e^x+x} \log (5)}{3 \left (3+e^{e^x} \log (5)\right )}+\frac {e^{e^x+x} (x (1+\log (5))-\log (25))}{3 \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}\right ) \, dx+\frac {9}{5} \int \left (\frac {x (1+\log (5))-\log (25)}{3 x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {\log (5)}{x \left (9+e^{e^x} \log (125)\right )}\right ) \, dx+\frac {1}{5} (\log (5) (1+\log (5))) \int \left (\frac {e^{2 e^x} (x (1+\log (5))-\log (25))}{3 x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {e^{2 e^x} \log (5)}{x \left (9+e^{e^x} \log (125)\right )}\right ) \, dx+\frac {1}{5} (3 (1+\log (25))) \int \left (\frac {e^{e^x} (x (1+\log (5))-\log (25))}{3 x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )}+\frac {e^{e^x} \log (5)}{x \left (9+e^{e^x} \log (125)\right )}\right ) \, dx\\ &=\frac {1}{5} \int \frac {e^{e^x+x} (x (1+\log (5))-\log (25))}{6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))} \, dx+\frac {3}{5} \int \frac {x (1+\log (5))-\log (25)}{x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} \log (5) \int \frac {e^{e^x+x}}{3+e^{e^x} \log (5)} \, dx+\frac {1}{5} (9 \log (5)) \int \frac {1}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{15} (\log (5) (1+\log (5))) \int \frac {e^{2 e^x} (x (1+\log (5))-\log (25))}{x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} \left (\log ^2(5) (1+\log (5))\right ) \int \frac {e^{2 e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} (1+\log (25)) \int \frac {e^{e^x} (x (1+\log (5))-\log (25))}{x \left (6-3 x+2 e^{e^x} \log (5)-e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} (3 \log (5) (1+\log (25))) \int \frac {e^{e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx\\ &=\frac {1}{5} \int \left (\frac {e^{e^x+x} \log (25)}{-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))}+\frac {e^{e^x+x} x (1+\log (5))}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)}\right ) \, dx+\frac {3}{5} \int \left (\frac {\log (25)}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )}+\frac {1+\log (5)}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)}\right ) \, dx+\frac {1}{5} \log (5) \operatorname {Subst}\left (\int \frac {e^x}{3+e^x \log (5)} \, dx,x,e^x\right )+\frac {1}{5} (9 \log (5)) \int \frac {1}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{15} (\log (5) (1+\log (5))) \int \left (\frac {e^{2 e^x} \log (25)}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )}+\frac {e^{2 e^x} (1+\log (5))}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)}\right ) \, dx+\frac {1}{5} \left (\log ^2(5) (1+\log (5))\right ) \int \frac {e^{2 e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} (1+\log (25)) \int \left (\frac {e^{e^x} \log (25)}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )}+\frac {e^{e^x} (1+\log (5))}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)}\right ) \, dx+\frac {1}{5} (3 \log (5) (1+\log (25))) \int \frac {e^{e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx\\ &=\frac {1}{5} \log (5) \operatorname {Subst}\left (\int \frac {1}{3+x \log (5)} \, dx,x,e^{e^x}\right )+\frac {1}{5} (9 \log (5)) \int \frac {1}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} (1+\log (5)) \int \frac {e^{e^x+x} x}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} (3 (1+\log (5))) \int \frac {1}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} \left (\log ^2(5) (1+\log (5))\right ) \int \frac {e^{2 e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{15} \left (\log (5) (1+\log (5))^2\right ) \int \frac {e^{2 e^x}}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} \log (25) \int \frac {e^{e^x+x}}{-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))} \, dx+\frac {1}{5} (3 \log (25)) \int \frac {1}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{15} (\log (5) (1+\log (5)) \log (25)) \int \frac {e^{2 e^x}}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} (3 \log (5) (1+\log (25))) \int \frac {e^{e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} ((1+\log (5)) (1+\log (25))) \int \frac {e^{e^x}}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} (\log (25) (1+\log (25))) \int \frac {e^{e^x}}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx\\ &=\frac {1}{5} \log \left (3+e^{e^x} \log (5)\right )+\frac {1}{5} (9 \log (5)) \int \frac {1}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} (1+\log (5)) \int \frac {e^{e^x+x} x}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} (3 (1+\log (5))) \int \frac {1}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} \left (\log ^2(5) (1+\log (5))\right ) \int \frac {e^{2 e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{15} \left (\log (5) (1+\log (5))^2\right ) \int \frac {e^{2 e^x}}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} \log (25) \int \frac {e^{e^x+x}}{-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))} \, dx+\frac {1}{5} (3 \log (25)) \int \frac {1}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{15} (\log (5) (1+\log (5)) \log (25)) \int \frac {e^{2 e^x}}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx+\frac {1}{5} (3 \log (5) (1+\log (25))) \int \frac {e^{e^x}}{x \left (9+e^{e^x} \log (125)\right )} \, dx+\frac {1}{5} ((1+\log (5)) (1+\log (25))) \int \frac {e^{e^x}}{6-3 x-e^{e^x} x (1+\log (5))+e^{e^x} \log (25)} \, dx+\frac {1}{5} (\log (25) (1+\log (25))) \int \frac {e^{e^x}}{x \left (-6+3 x-2 e^{e^x} \log (5)+e^{e^x} x (1+\log (5))\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 3.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-9+e^{e^x} \left (-3-3 e^x x-6 \log (5)\right )+e^{2 e^x} \left (-\log (5)-\log ^2(5)\right )}{-90+45 x+e^{e^x} (15 x+(-60+30 x) \log (5))+e^{2 e^x} \left (5 x \log (5)+(-10+5 x) \log ^2(5)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-9 + E^E^x*(-3 - 3*E^x*x - 6*Log[5]) + E^(2*E^x)*(-Log[5] - Log[5]^2))/(-90 + 45*x + E^E^x*(15*x +
(-60 + 30*x)*Log[5]) + E^(2*E^x)*(5*x*Log[5] + (-10 + 5*x)*Log[5]^2)),x]

[Out]

Integrate[(-9 + E^E^x*(-3 - 3*E^x*x - 6*Log[5]) + E^(2*E^x)*(-Log[5] - Log[5]^2))/(-90 + 45*x + E^E^x*(15*x +
(-60 + 30*x)*Log[5]) + E^(2*E^x)*(5*x*Log[5] + (-10 + 5*x)*Log[5]^2)), x]

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fricas [B]  time = 0.77, size = 54, normalized size = 2.08 \begin {gather*} -\frac {1}{5} \, \log \left ({\left (x - 2\right )} \log \relax (5) + x\right ) + \frac {1}{5} \, \log \left (e^{\left (e^{x}\right )} \log \relax (5) + 3\right ) - \frac {1}{5} \, \log \left (\frac {{\left ({\left (x - 2\right )} \log \relax (5) + x\right )} e^{\left (e^{x}\right )} + 3 \, x - 6}{{\left (x - 2\right )} \log \relax (5) + x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(5)^2-log(5))*exp(exp(x))^2+(-3*exp(x)*x-6*log(5)-3)*exp(exp(x))-9)/(((5*x-10)*log(5)^2+5*x*lo
g(5))*exp(exp(x))^2+((30*x-60)*log(5)+15*x)*exp(exp(x))+45*x-90),x, algorithm="fricas")

[Out]

-1/5*log((x - 2)*log(5) + x) + 1/5*log(e^(e^x)*log(5) + 3) - 1/5*log((((x - 2)*log(5) + x)*e^(e^x) + 3*x - 6)/
((x - 2)*log(5) + x))

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giac [A]  time = 0.65, size = 43, normalized size = 1.65 \begin {gather*} -\frac {1}{5} \, e^{x} - \frac {1}{5} \, \log \left (x e^{\left (e^{x}\right )} \log \relax (5) + x e^{\left (e^{x}\right )} - 2 \, e^{\left (e^{x}\right )} \log \relax (5) + 3 \, x - 6\right ) + \frac {1}{5} \, \log \left (e^{\left (e^{x}\right )} \log \relax (5) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(5)^2-log(5))*exp(exp(x))^2+(-3*exp(x)*x-6*log(5)-3)*exp(exp(x))-9)/(((5*x-10)*log(5)^2+5*x*lo
g(5))*exp(exp(x))^2+((30*x-60)*log(5)+15*x)*exp(exp(x))+45*x-90),x, algorithm="giac")

[Out]

-1/5*e^x - 1/5*log(x*e^(e^x)*log(5) + x*e^(e^x) - 2*e^(e^x)*log(5) + 3*x - 6) + 1/5*log(e^(e^x)*log(5) + 3)

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maple [A]  time = 0.30, size = 40, normalized size = 1.54

method	result	size
norman	\(\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{x}} \ln \relax (5)+3\right )}{5}-\frac {\ln \left (\ln \relax (5) {\mathrm e}^{{\mathrm e}^{x}} x -2 \,{\mathrm e}^{{\mathrm e}^{x}} \ln \relax (5)+x \,{\mathrm e}^{{\mathrm e}^{x}}+3 x -6\right )}{5}\)	\(40\)
risch	\(-\frac {\ln \left (\left (\ln \relax (5)+1\right ) x -2 \ln \relax (5)\right )}{5}+\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{x}}+\frac {3}{\ln \relax (5)}\right )}{5}-\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{x}}+\frac {3 x -6}{x \ln \relax (5)-2 \ln \relax (5)+x}\right )}{5}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-ln(5)^2-ln(5))*exp(exp(x))^2+(-3*exp(x)*x-6*ln(5)-3)*exp(exp(x))-9)/(((5*x-10)*ln(5)^2+5*x*ln(5))*exp(e
xp(x))^2+((30*x-60)*ln(5)+15*x)*exp(exp(x))+45*x-90),x,method=_RETURNVERBOSE)

[Out]

1/5*ln(exp(exp(x))*ln(5)+3)-1/5*ln(ln(5)*exp(exp(x))*x-2*exp(exp(x))*ln(5)+x*exp(exp(x))+3*x-6)

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maxima [B]  time = 0.51, size = 68, normalized size = 2.62 \begin {gather*} -\frac {1}{5} \, \log \left (x {\left (\log \relax (5) + 1\right )} - 2 \, \log \relax (5)\right ) - \frac {1}{5} \, \log \left (\frac {{\left (x {\left (\log \relax (5) + 1\right )} - 2 \, \log \relax (5)\right )} e^{\left (e^{x}\right )} + 3 \, x - 6}{x {\left (\log \relax (5) + 1\right )} - 2 \, \log \relax (5)}\right ) + \frac {1}{5} \, \log \left (\frac {e^{\left (e^{x}\right )} \log \relax (5) + 3}{\log \relax (5)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(5)^2-log(5))*exp(exp(x))^2+(-3*exp(x)*x-6*log(5)-3)*exp(exp(x))-9)/(((5*x-10)*log(5)^2+5*x*lo
g(5))*exp(exp(x))^2+((30*x-60)*log(5)+15*x)*exp(exp(x))+45*x-90),x, algorithm="maxima")

[Out]

-1/5*log(x*(log(5) + 1) - 2*log(5)) - 1/5*log(((x*(log(5) + 1) - 2*log(5))*e^(e^x) + 3*x - 6)/(x*(log(5) + 1)
- 2*log(5))) + 1/5*log((e^(e^x)*log(5) + 3)/log(5))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (6\,\ln \relax (5)+3\,x\,{\mathrm {e}}^x+3\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (\ln \relax (5)+{\ln \relax (5)}^2\right )+9}{45\,x+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (5\,x\,\ln \relax (5)+{\ln \relax (5)}^2\,\left (5\,x-10\right )\right )+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (15\,x+\ln \relax (5)\,\left (30\,x-60\right )\right )-90} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(x))*(6*log(5) + 3*x*exp(x) + 3) + exp(2*exp(x))*(log(5) + log(5)^2) + 9)/(45*x + exp(2*exp(x))*(
5*x*log(5) + log(5)^2*(5*x - 10)) + exp(exp(x))*(15*x + log(5)*(30*x - 60)) - 90),x)

[Out]

int(-(exp(exp(x))*(6*log(5) + 3*x*exp(x) + 3) + exp(2*exp(x))*(log(5) + log(5)^2) + 9)/(45*x + exp(2*exp(x))*(
5*x*log(5) + log(5)^2*(5*x - 10)) + exp(exp(x))*(15*x + log(5)*(30*x - 60)) - 90), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-ln(5)**2-ln(5))*exp(exp(x))**2+(-3*exp(x)*x-6*ln(5)-3)*exp(exp(x))-9)/(((5*x-10)*ln(5)**2+5*x*ln(
5))*exp(exp(x))**2+((30*x-60)*ln(5)+15*x)*exp(exp(x))+45*x-90),x)

[Out]

Exception raised: PolynomialError

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