3.29.85 \(\int \frac {-5+4 e^{1+x} x^3+(40+e^{1+x} (-128 x^3-32 x^4)+(5+e^{1+x} (-16 x^3-4 x^4)) \log (x)) \log (8+\log (x))}{(8+\log (x)) \log ^2(8+\log (x))} \, dx\)

Optimal. Leaf size=21 \[ \frac {x \left (5-4 e^{1+x} x^3\right )}{\log (8+\log (x))} \]

________________________________________________________________________________________

Rubi [F]  time = 1.06, 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 {-5+4 e^{1+x} x^3+\left (40+e^{1+x} \left (-128 x^3-32 x^4\right )+\left (5+e^{1+x} \left (-16 x^3-4 x^4\right )\right ) \log (x)\right ) \log (8+\log (x))}{(8+\log (x)) \log ^2(8+\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5 + 4*E^(1 + x)*x^3 + (40 + E^(1 + x)*(-128*x^3 - 32*x^4) + (5 + E^(1 + x)*(-16*x^3 - 4*x^4))*Log[x])*Lo
g[8 + Log[x]])/((8 + Log[x])*Log[8 + Log[x]]^2),x]

[Out]

(-4*E^(1 + x)*x^3*(8*x*Log[8 + Log[x]] + x*Log[x]*Log[8 + Log[x]]))/((8 + Log[x])*Log[8 + Log[x]]^2) - 5*Defer
[Int][1/((8 + Log[x])*Log[8 + Log[x]]^2), x] + 5*Defer[Int][Log[8 + Log[x]]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5+4 e^{1+x} x^3-\left (-5+4 e^{1+x} x^3 (4+x)\right ) (8+\log (x)) \log (8+\log (x))}{(8+\log (x)) \log ^2(8+\log (x))} \, dx\\ &=\int \left (\frac {5 (-1+8 \log (8+\log (x))+\log (x) \log (8+\log (x)))}{(8+\log (x)) \log ^2(8+\log (x))}-\frac {4 e^{1+x} x^3 (-1+32 \log (8+\log (x))+8 x \log (8+\log (x))+4 \log (x) \log (8+\log (x))+x \log (x) \log (8+\log (x)))}{(8+\log (x)) \log ^2(8+\log (x))}\right ) \, dx\\ &=-\left (4 \int \frac {e^{1+x} x^3 (-1+32 \log (8+\log (x))+8 x \log (8+\log (x))+4 \log (x) \log (8+\log (x))+x \log (x) \log (8+\log (x)))}{(8+\log (x)) \log ^2(8+\log (x))} \, dx\right )+5 \int \frac {-1+8 \log (8+\log (x))+\log (x) \log (8+\log (x))}{(8+\log (x)) \log ^2(8+\log (x))} \, dx\\ &=-\frac {4 e^{1+x} x^3 (8 x \log (8+\log (x))+x \log (x) \log (8+\log (x)))}{(8+\log (x)) \log ^2(8+\log (x))}+5 \int \frac {-\frac {1}{8+\log (x)}+\log (8+\log (x))}{\log ^2(8+\log (x))} \, dx\\ &=-\frac {4 e^{1+x} x^3 (8 x \log (8+\log (x))+x \log (x) \log (8+\log (x)))}{(8+\log (x)) \log ^2(8+\log (x))}+5 \int \left (-\frac {1}{(8+\log (x)) \log ^2(8+\log (x))}+\frac {1}{\log (8+\log (x))}\right ) \, dx\\ &=-\frac {4 e^{1+x} x^3 (8 x \log (8+\log (x))+x \log (x) \log (8+\log (x)))}{(8+\log (x)) \log ^2(8+\log (x))}-5 \int \frac {1}{(8+\log (x)) \log ^2(8+\log (x))} \, dx+5 \int \frac {1}{\log (8+\log (x))} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.23, size = 22, normalized size = 1.05 \begin {gather*} -\frac {x \left (-5+4 e^{1+x} x^3\right )}{\log (8+\log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + 4*E^(1 + x)*x^3 + (40 + E^(1 + x)*(-128*x^3 - 32*x^4) + (5 + E^(1 + x)*(-16*x^3 - 4*x^4))*Log[
x])*Log[8 + Log[x]])/((8 + Log[x])*Log[8 + Log[x]]^2),x]

[Out]

-((x*(-5 + 4*E^(1 + x)*x^3))/Log[8 + Log[x]])

________________________________________________________________________________________

fricas [A]  time = 1.00, size = 22, normalized size = 1.05 \begin {gather*} -\frac {4 \, x^{4} e^{\left (x + 1\right )} - 5 \, x}{\log \left (\log \relax (x) + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x^4-16*x^3)*exp(x+1)+5)*log(x)+(-32*x^4-128*x^3)*exp(x+1)+40)*log(log(x)+8)+4*x^3*exp(x+1)-5)
/(log(x)+8)/log(log(x)+8)^2,x, algorithm="fricas")

[Out]

-(4*x^4*e^(x + 1) - 5*x)/log(log(x) + 8)

________________________________________________________________________________________

giac [A]  time = 0.33, size = 22, normalized size = 1.05 \begin {gather*} -\frac {4 \, x^{4} e^{\left (x + 1\right )} - 5 \, x}{\log \left (\log \relax (x) + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x^4-16*x^3)*exp(x+1)+5)*log(x)+(-32*x^4-128*x^3)*exp(x+1)+40)*log(log(x)+8)+4*x^3*exp(x+1)-5)
/(log(x)+8)/log(log(x)+8)^2,x, algorithm="giac")

[Out]

-(4*x^4*e^(x + 1) - 5*x)/log(log(x) + 8)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 22, normalized size = 1.05

method	result	size
risch	\(-\frac {x \left (4 x^{3} {\mathrm e}^{x +1}-5\right )}{\ln \left (\ln \relax (x )+8\right )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((-4*x^4-16*x^3)*exp(x+1)+5)*ln(x)+(-32*x^4-128*x^3)*exp(x+1)+40)*ln(ln(x)+8)+4*x^3*exp(x+1)-5)/(ln(x)+8
)/ln(ln(x)+8)^2,x,method=_RETURNVERBOSE)

[Out]

-x*(4*x^3*exp(x+1)-5)/ln(ln(x)+8)

________________________________________________________________________________________

maxima [A]  time = 0.91, size = 22, normalized size = 1.05 \begin {gather*} -\frac {4 \, x^{4} e^{\left (x + 1\right )} - 5 \, x}{\log \left (\log \relax (x) + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x^4-16*x^3)*exp(x+1)+5)*log(x)+(-32*x^4-128*x^3)*exp(x+1)+40)*log(log(x)+8)+4*x^3*exp(x+1)-5)
/(log(x)+8)/log(log(x)+8)^2,x, algorithm="maxima")

[Out]

-(4*x^4*e^(x + 1) - 5*x)/log(log(x) + 8)

________________________________________________________________________________________

mupad [B]  time = 1.82, size = 21, normalized size = 1.00 \begin {gather*} -\frac {x\,\left (4\,x^3\,{\mathrm {e}}^{x+1}-5\right )}{\ln \left (\ln \relax (x)+8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(log(x) + 8)*(log(x)*(exp(x + 1)*(16*x^3 + 4*x^4) - 5) + exp(x + 1)*(128*x^3 + 32*x^4) - 40) - 4*x^3*
exp(x + 1) + 5)/(log(log(x) + 8)^2*(log(x) + 8)),x)

[Out]

-(x*(4*x^3*exp(x + 1) - 5))/log(log(x) + 8)

________________________________________________________________________________________

sympy [A]  time = 0.39, size = 26, normalized size = 1.24 \begin {gather*} - \frac {4 x^{4} e^{x + 1}}{\log {\left (\log {\relax (x )} + 8 \right )}} + \frac {5 x}{\log {\left (\log {\relax (x )} + 8 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-4*x**4-16*x**3)*exp(x+1)+5)*ln(x)+(-32*x**4-128*x**3)*exp(x+1)+40)*ln(ln(x)+8)+4*x**3*exp(x+1)-
5)/(ln(x)+8)/ln(ln(x)+8)**2,x)

[Out]

-4*x**4*exp(x + 1)/log(log(x) + 8) + 5*x/log(log(x) + 8)

________________________________________________________________________________________