3.81.66 \(\int \frac {x^6+5 x^5 \log (x)+10 x^4 \log ^2(x)+10 x^3 \log ^3(x)+5 x^2 \log ^4(x)+x \log ^5(x)+e^{e^{-1+e^{e^x}}} (-16-16 x+e^{-1+e^{e^x}+e^x} (4 e^x x^2+4 e^x x \log (x)))}{x^6+5 x^5 \log (x)+10 x^4 \log ^2(x)+10 x^3 \log ^3(x)+5 x^2 \log ^4(x)+x \log ^5(x)} \, dx\)

Optimal. Leaf size=22 \[ 1+x+\frac {4 e^{e^{-1+e^{e^x}}}}{(x+\log (x))^4} \]

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Rubi [F]  time = 3.97, 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 {x^6+5 x^5 \log (x)+10 x^4 \log ^2(x)+10 x^3 \log ^3(x)+5 x^2 \log ^4(x)+x \log ^5(x)+e^{e^{-1+e^{e^x}}} \left (-16-16 x+e^{-1+e^{e^x}+e^x} \left (4 e^x x^2+4 e^x x \log (x)\right )\right )}{x^6+5 x^5 \log (x)+10 x^4 \log ^2(x)+10 x^3 \log ^3(x)+5 x^2 \log ^4(x)+x \log ^5(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^6 + 5*x^5*Log[x] + 10*x^4*Log[x]^2 + 10*x^3*Log[x]^3 + 5*x^2*Log[x]^4 + x*Log[x]^5 + E^E^(-1 + E^E^x)*(
-16 - 16*x + E^(-1 + E^E^x + E^x)*(4*E^x*x^2 + 4*E^x*x*Log[x])))/(x^6 + 5*x^5*Log[x] + 10*x^4*Log[x]^2 + 10*x^
3*Log[x]^3 + 5*x^2*Log[x]^4 + x*Log[x]^5),x]

[Out]

x - 16*Defer[Int][E^E^(-1 + E^E^x)/(x + Log[x])^5, x] - 16*Defer[Int][E^E^(-1 + E^E^x)/(x*(x + Log[x])^5), x]
+ 4*Defer[Int][E^(-1 + E^E^x + E^(-1 + E^E^x) + E^x + x)/(x + Log[x])^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^6+5 x^5 \log (x)+10 x^4 \log ^2(x)+10 x^3 \log ^3(x)+5 x^2 \log ^4(x)+x \log ^5(x)+e^{e^{-1+e^{e^x}}} \left (-16-16 x+e^{-1+e^{e^x}+e^x} \left (4 e^x x^2+4 e^x x \log (x)\right )\right )}{x (x+\log (x))^5} \, dx\\ &=\int \left (-\frac {16 e^{e^{-1+e^{e^x}}}}{(x+\log (x))^5}-\frac {16 e^{e^{-1+e^{e^x}}}}{x (x+\log (x))^5}+\frac {x^5}{(x+\log (x))^5}+\frac {5 x^4 \log (x)}{(x+\log (x))^5}+\frac {10 x^3 \log ^2(x)}{(x+\log (x))^5}+\frac {10 x^2 \log ^3(x)}{(x+\log (x))^5}+\frac {5 x \log ^4(x)}{(x+\log (x))^5}+\frac {\log ^5(x)}{(x+\log (x))^5}+\frac {4 e^{-1+e^{e^x}+e^{-1+e^{e^x}}+e^x+x}}{(x+\log (x))^4}\right ) \, dx\\ &=4 \int \frac {e^{-1+e^{e^x}+e^{-1+e^{e^x}}+e^x+x}}{(x+\log (x))^4} \, dx+5 \int \frac {x^4 \log (x)}{(x+\log (x))^5} \, dx+5 \int \frac {x \log ^4(x)}{(x+\log (x))^5} \, dx+10 \int \frac {x^3 \log ^2(x)}{(x+\log (x))^5} \, dx+10 \int \frac {x^2 \log ^3(x)}{(x+\log (x))^5} \, dx-16 \int \frac {e^{e^{-1+e^{e^x}}}}{(x+\log (x))^5} \, dx-16 \int \frac {e^{e^{-1+e^{e^x}}}}{x (x+\log (x))^5} \, dx+\int \frac {x^5}{(x+\log (x))^5} \, dx+\int \frac {\log ^5(x)}{(x+\log (x))^5} \, dx\\ &=4 \int \frac {e^{-1+e^{e^x}+e^{-1+e^{e^x}}+e^x+x}}{(x+\log (x))^4} \, dx+5 \int \left (-\frac {x^5}{(x+\log (x))^5}+\frac {x^4}{(x+\log (x))^4}\right ) \, dx+5 \int \left (\frac {x^5}{(x+\log (x))^5}-\frac {4 x^4}{(x+\log (x))^4}+\frac {6 x^3}{(x+\log (x))^3}-\frac {4 x^2}{(x+\log (x))^2}+\frac {x}{x+\log (x)}\right ) \, dx+10 \int \left (\frac {x^5}{(x+\log (x))^5}-\frac {2 x^4}{(x+\log (x))^4}+\frac {x^3}{(x+\log (x))^3}\right ) \, dx+10 \int \left (-\frac {x^5}{(x+\log (x))^5}+\frac {3 x^4}{(x+\log (x))^4}-\frac {3 x^3}{(x+\log (x))^3}+\frac {x^2}{(x+\log (x))^2}\right ) \, dx-16 \int \frac {e^{e^{-1+e^{e^x}}}}{(x+\log (x))^5} \, dx-16 \int \frac {e^{e^{-1+e^{e^x}}}}{x (x+\log (x))^5} \, dx+\int \frac {x^5}{(x+\log (x))^5} \, dx+\int \left (1-\frac {x^5}{(x+\log (x))^5}+\frac {5 x^4}{(x+\log (x))^4}-\frac {10 x^3}{(x+\log (x))^3}+\frac {10 x^2}{(x+\log (x))^2}-\frac {5 x}{x+\log (x)}\right ) \, dx\\ &=x+4 \int \frac {e^{-1+e^{e^x}+e^{-1+e^{e^x}}+e^x+x}}{(x+\log (x))^4} \, dx+2 \left (5 \int \frac {x^4}{(x+\log (x))^4} \, dx\right )+2 \left (10 \int \frac {x^2}{(x+\log (x))^2} \, dx\right )-16 \int \frac {e^{e^{-1+e^{e^x}}}}{(x+\log (x))^5} \, dx-16 \int \frac {e^{e^{-1+e^{e^x}}}}{x (x+\log (x))^5} \, dx-2 \left (20 \int \frac {x^4}{(x+\log (x))^4} \, dx\right )-20 \int \frac {x^2}{(x+\log (x))^2} \, dx+30 \int \frac {x^4}{(x+\log (x))^4} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 29, normalized size = 1.32 \begin {gather*} \frac {e x+\frac {4 e^{1+e^{-1+e^{e^x}}}}{(x+\log (x))^4}}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^6 + 5*x^5*Log[x] + 10*x^4*Log[x]^2 + 10*x^3*Log[x]^3 + 5*x^2*Log[x]^4 + x*Log[x]^5 + E^E^(-1 + E^
E^x)*(-16 - 16*x + E^(-1 + E^E^x + E^x)*(4*E^x*x^2 + 4*E^x*x*Log[x])))/(x^6 + 5*x^5*Log[x] + 10*x^4*Log[x]^2 +
10*x^3*Log[x]^3 + 5*x^2*Log[x]^4 + x*Log[x]^5),x]

[Out]

(E*x + (4*E^(1 + E^(-1 + E^E^x)))/(x + Log[x])^4)/E

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fricas [B]  time = 0.83, size = 78, normalized size = 3.55 \begin {gather*} \frac {x^{5} + 4 \, x^{4} \log \relax (x) + 6 \, x^{3} \log \relax (x)^{2} + 4 \, x^{2} \log \relax (x)^{3} + x \log \relax (x)^{4} + 4 \, e^{\left (e^{\left (e^{\left (e^{x}\right )} - 1\right )}\right )}}{x^{4} + 4 \, x^{3} \log \relax (x) + 6 \, x^{2} \log \relax (x)^{2} + 4 \, x \log \relax (x)^{3} + \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x*exp(x)*log(x)+4*exp(x)*x^2)*exp(exp(x))*exp(exp(exp(x))-1)-16*x-16)*exp(exp(exp(exp(x))-1))+x
*log(x)^5+5*x^2*log(x)^4+10*x^3*log(x)^3+10*x^4*log(x)^2+5*x^5*log(x)+x^6)/(x*log(x)^5+5*x^2*log(x)^4+10*x^3*l
og(x)^3+10*x^4*log(x)^2+5*x^5*log(x)+x^6),x, algorithm="fricas")

[Out]

(x^5 + 4*x^4*log(x) + 6*x^3*log(x)^2 + 4*x^2*log(x)^3 + x*log(x)^4 + 4*e^(e^(e^(e^x) - 1)))/(x^4 + 4*x^3*log(x
) + 6*x^2*log(x)^2 + 4*x*log(x)^3 + log(x)^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + 5 \, x^{5} \log \relax (x) + 10 \, x^{4} \log \relax (x)^{2} + 10 \, x^{3} \log \relax (x)^{3} + 5 \, x^{2} \log \relax (x)^{4} + x \log \relax (x)^{5} + 4 \, {\left ({\left (x^{2} e^{x} + x e^{x} \log \relax (x)\right )} e^{\left (e^{x} + e^{\left (e^{x}\right )} - 1\right )} - 4 \, x - 4\right )} e^{\left (e^{\left (e^{\left (e^{x}\right )} - 1\right )}\right )}}{x^{6} + 5 \, x^{5} \log \relax (x) + 10 \, x^{4} \log \relax (x)^{2} + 10 \, x^{3} \log \relax (x)^{3} + 5 \, x^{2} \log \relax (x)^{4} + x \log \relax (x)^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x*exp(x)*log(x)+4*exp(x)*x^2)*exp(exp(x))*exp(exp(exp(x))-1)-16*x-16)*exp(exp(exp(exp(x))-1))+x
*log(x)^5+5*x^2*log(x)^4+10*x^3*log(x)^3+10*x^4*log(x)^2+5*x^5*log(x)+x^6)/(x*log(x)^5+5*x^2*log(x)^4+10*x^3*l
og(x)^3+10*x^4*log(x)^2+5*x^5*log(x)+x^6),x, algorithm="giac")

[Out]

integrate((x^6 + 5*x^5*log(x) + 10*x^4*log(x)^2 + 10*x^3*log(x)^3 + 5*x^2*log(x)^4 + x*log(x)^5 + 4*((x^2*e^x
+ x*e^x*log(x))*e^(e^x + e^(e^x) - 1) - 4*x - 4)*e^(e^(e^(e^x) - 1)))/(x^6 + 5*x^5*log(x) + 10*x^4*log(x)^2 +
10*x^3*log(x)^3 + 5*x^2*log(x)^4 + x*log(x)^5), x)

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maple [B]  time = 0.06, size = 45, normalized size = 2.05

method	result	size
risch	\(x +\frac {4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}-1}}}{x^{4}+4 x^{3} \ln \relax (x )+6 x^{2} \ln \relax (x )^{2}+4 x \ln \relax (x )^{3}+\ln \relax (x )^{4}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x*exp(x)*ln(x)+4*exp(x)*x^2)*exp(exp(x))*exp(exp(exp(x))-1)-16*x-16)*exp(exp(exp(exp(x))-1))+x*ln(x)^
5+5*x^2*ln(x)^4+10*x^3*ln(x)^3+10*x^4*ln(x)^2+5*x^5*ln(x)+x^6)/(x*ln(x)^5+5*x^2*ln(x)^4+10*x^3*ln(x)^3+10*x^4*
ln(x)^2+5*x^5*ln(x)+x^6),x,method=_RETURNVERBOSE)

[Out]

x+4/(x^4+4*x^3*ln(x)+6*x^2*ln(x)^2+4*x*ln(x)^3+ln(x)^4)*exp(exp(exp(exp(x))-1))

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maxima [B]  time = 0.41, size = 78, normalized size = 3.55 \begin {gather*} \frac {x^{5} + 4 \, x^{4} \log \relax (x) + 6 \, x^{3} \log \relax (x)^{2} + 4 \, x^{2} \log \relax (x)^{3} + x \log \relax (x)^{4} + 4 \, e^{\left (e^{\left (e^{\left (e^{x}\right )} - 1\right )}\right )}}{x^{4} + 4 \, x^{3} \log \relax (x) + 6 \, x^{2} \log \relax (x)^{2} + 4 \, x \log \relax (x)^{3} + \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x*exp(x)*log(x)+4*exp(x)*x^2)*exp(exp(x))*exp(exp(exp(x))-1)-16*x-16)*exp(exp(exp(exp(x))-1))+x
*log(x)^5+5*x^2*log(x)^4+10*x^3*log(x)^3+10*x^4*log(x)^2+5*x^5*log(x)+x^6)/(x*log(x)^5+5*x^2*log(x)^4+10*x^3*l
og(x)^3+10*x^4*log(x)^2+5*x^5*log(x)+x^6),x, algorithm="maxima")

[Out]

(x^5 + 4*x^4*log(x) + 6*x^3*log(x)^2 + 4*x^2*log(x)^3 + x*log(x)^4 + 4*e^(e^(e^(e^x) - 1)))/(x^4 + 4*x^3*log(x
) + 6*x^2*log(x)^2 + 4*x*log(x)^3 + log(x)^4)

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mupad [B]  time = 7.41, size = 18, normalized size = 0.82 \begin {gather*} x+\frac {4\,{\mathrm {e}}^{{\mathrm {e}}^{-1}\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}}{{\left (x+\ln \relax (x)\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*log(x)^5 + 5*x^5*log(x) + 5*x^2*log(x)^4 + 10*x^3*log(x)^3 + 10*x^4*log(x)^2 - exp(exp(exp(exp(x)) - 1)
)*(16*x - exp(exp(exp(x)) - 1)*exp(exp(x))*(4*x^2*exp(x) + 4*x*exp(x)*log(x)) + 16) + x^6)/(x*log(x)^5 + 5*x^5
*log(x) + 5*x^2*log(x)^4 + 10*x^3*log(x)^3 + 10*x^4*log(x)^2 + x^6),x)

[Out]

x + (4*exp(exp(-1)*exp(exp(exp(x)))))/(x + log(x))^4

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sympy [B]  time = 1.62, size = 48, normalized size = 2.18 \begin {gather*} x + \frac {4 e^{e^{e^{e^{x}} - 1}}}{x^{4} + 4 x^{3} \log {\relax (x )} + 6 x^{2} \log {\relax (x )}^{2} + 4 x \log {\relax (x )}^{3} + \log {\relax (x )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x*exp(x)*ln(x)+4*exp(x)*x**2)*exp(exp(x))*exp(exp(exp(x))-1)-16*x-16)*exp(exp(exp(exp(x))-1))+x
*ln(x)**5+5*x**2*ln(x)**4+10*x**3*ln(x)**3+10*x**4*ln(x)**2+5*x**5*ln(x)+x**6)/(x*ln(x)**5+5*x**2*ln(x)**4+10*
x**3*ln(x)**3+10*x**4*ln(x)**2+5*x**5*ln(x)+x**6),x)

[Out]

x + 4*exp(exp(exp(exp(x)) - 1))/(x**4 + 4*x**3*log(x) + 6*x**2*log(x)**2 + 4*x*log(x)**3 + log(x)**4)

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