3.8.38 \(\int \frac {256 x-4 e^{4+4 x} x+(e^{4+4 x} (1-x)-256 x+256 x^2) \log (\frac {1}{256} (e^{4+4 x}-256 x))}{(e^{4+5 x}-256 e^x x) \log ^2(\frac {1}{256} (e^{4+4 x}-256 x))} \, dx\)

Optimal. Leaf size=25 \[ \frac {e^{-x} x}{\log \left (\frac {1}{256} e^{4+4 x}-x\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 4.12, 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 {256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}{\left (e^{4+5 x}-256 e^x x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(256*x - 4*E^(4 + 4*x)*x + (E^(4 + 4*x)*(1 - x) - 256*x + 256*x^2)*Log[(E^(4 + 4*x) - 256*x)/256])/((E^(4
+ 5*x) - 256*E^x*x)*Log[(E^(4 + 4*x) - 256*x)/256]^2),x]

[Out]

-4*Defer[Int][x/(E^x*Log[(E^(4 + 4*x) - 256*x)/256]^2), x] + 256*Defer[Int][x/(E^x*(E^(4 + 4*x) - 256*x)*Log[(
E^(4 + 4*x) - 256*x)/256]^2), x] + 1024*Defer[Int][x^2/(E^x*(-E^(4 + 4*x) + 256*x)*Log[(E^(4 + 4*x) - 256*x)/2
56]^2), x] + Defer[Int][1/(E^x*Log[(E^(4 + 4*x) - 256*x)/256]), x] - Defer[Int][x/(E^x*Log[(E^(4 + 4*x) - 256*
x)/256]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )\right )}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ &=\int \frac {e^{-x} \left (-\frac {4 \left (-64+e^{4+4 x}\right ) x}{e^{4+4 x}-256 x}-(-1+x) \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ &=\int \left (\frac {256 e^{-x} (1-4 x) x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}+\frac {e^{-x} \left (-4 x+\log \left (\frac {1}{256} e^{4+4 x}-x\right )-x \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx\\ &=256 \int \frac {e^{-x} (1-4 x) x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \frac {e^{-x} \left (-4 x+\log \left (\frac {1}{256} e^{4+4 x}-x\right )-x \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ &=256 \int \left (\frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}+\frac {4 e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx+\int \frac {e^{-x} \left (-4 x-(-1+x) \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ &=256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \left (-\frac {4 e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}+\frac {e^{-x} (1-x)}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\right )+256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \frac {e^{-x} (1-x)}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ &=-\left (4 \int \frac {e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\right )+256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \left (\frac {e^{-x}}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}-\frac {e^{-x} x}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\right )+256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \frac {e^{-x}}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx-\int \frac {e^{-x} x}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.95, size = 25, normalized size = 1.00 \begin {gather*} \frac {e^{-x} x}{\log \left (\frac {1}{256} e^{4+4 x}-x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(256*x - 4*E^(4 + 4*x)*x + (E^(4 + 4*x)*(1 - x) - 256*x + 256*x^2)*Log[(E^(4 + 4*x) - 256*x)/256])/(
(E^(4 + 5*x) - 256*E^x*x)*Log[(E^(4 + 4*x) - 256*x)/256]^2),x]

[Out]

x/(E^x*Log[E^(4 + 4*x)/256 - x])

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 21, normalized size = 0.84 \begin {gather*} \frac {x e^{\left (-x\right )}}{\log \left (-x + \frac {1}{256} \, e^{\left (4 \, x + 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+1)*exp(x+1)^4+256*x^2-256*x)*log(1/256*exp(x+1)^4-x)-4*x*exp(x+1)^4+256*x)/(exp(x)*exp(x+1)^4-
256*exp(x)*x)/log(1/256*exp(x+1)^4-x)^2,x, algorithm="fricas")

[Out]

x*e^(-x)/log(-x + 1/256*e^(4*x + 4))

________________________________________________________________________________________

giac [A]  time = 0.40, size = 27, normalized size = 1.08 \begin {gather*} -\frac {x}{8 \, e^{x} \log \relax (2) - e^{x} \log \left (-256 \, x + e^{\left (4 \, x + 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+1)*exp(x+1)^4+256*x^2-256*x)*log(1/256*exp(x+1)^4-x)-4*x*exp(x+1)^4+256*x)/(exp(x)*exp(x+1)^4-
256*exp(x)*x)/log(1/256*exp(x+1)^4-x)^2,x, algorithm="giac")

[Out]

-x/(8*e^x*log(2) - e^x*log(-256*x + e^(4*x + 4)))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 22, normalized size = 0.88

method	result	size
risch	\(\frac {x \,{\mathrm e}^{-x}}{\ln \left (\frac {{\mathrm e}^{4 x +4}}{256}-x \right )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((1-x)*exp(x+1)^4+256*x^2-256*x)*ln(1/256*exp(x+1)^4-x)-4*x*exp(x+1)^4+256*x)/(exp(x)*exp(x+1)^4-256*exp(
x)*x)/ln(1/256*exp(x+1)^4-x)^2,x,method=_RETURNVERBOSE)

[Out]

x*exp(-x)/ln(1/256*exp(4*x+4)-x)

________________________________________________________________________________________

maxima [A]  time = 0.74, size = 27, normalized size = 1.08 \begin {gather*} -\frac {x}{8 \, e^{x} \log \relax (2) - e^{x} \log \left (-256 \, x + e^{\left (4 \, x + 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+1)*exp(x+1)^4+256*x^2-256*x)*log(1/256*exp(x+1)^4-x)-4*x*exp(x+1)^4+256*x)/(exp(x)*exp(x+1)^4-
256*exp(x)*x)/log(1/256*exp(x+1)^4-x)^2,x, algorithm="maxima")

[Out]

-x/(8*e^x*log(2) - e^x*log(-256*x + e^(4*x + 4)))

________________________________________________________________________________________

mupad [B]  time = 0.92, size = 106, normalized size = 4.24 \begin {gather*} \frac {x\,{\mathrm {e}}^{-x}-\frac {{\mathrm {e}}^{-x}\,\ln \left (\frac {{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^4}{256}-x\right )\,\left (256\,x-{\mathrm {e}}^{4\,x+4}\right )\,\left (x-1\right )}{4\,\left ({\mathrm {e}}^{4\,x+4}-64\right )}}{\ln \left (\frac {{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^4}{256}-x\right )}+{\mathrm {e}}^{-x}\,\left (x-x^2\right )+\frac {{\mathrm {e}}^{3\,x}\,\left (x^2-\frac {5\,x}{4}+\frac {1}{4}\right )}{{\mathrm {e}}^{4\,x}-64\,{\mathrm {e}}^{-4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(exp(4*x + 4)/256 - x)*(256*x + exp(4*x + 4)*(x - 1) - 256*x^2) - 256*x + 4*x*exp(4*x + 4))/(log(exp(4
*x + 4)/256 - x)^2*(256*x*exp(x) - exp(4*x + 4)*exp(x))),x)

[Out]

(x*exp(-x) - (exp(-x)*log((exp(4*x)*exp(4))/256 - x)*(256*x - exp(4*x + 4))*(x - 1))/(4*(exp(4*x + 4) - 64)))/
log((exp(4*x)*exp(4))/256 - x) + exp(-x)*(x - x^2) + (exp(3*x)*(x^2 - (5*x)/4 + 1/4))/(exp(4*x) - 64*exp(-4))

________________________________________________________________________________________

sympy [A]  time = 0.21, size = 17, normalized size = 0.68 \begin {gather*} \frac {x e^{- x}}{\log {\left (- x + \frac {e^{4} e^{4 x}}{256} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+1)*exp(x+1)**4+256*x**2-256*x)*ln(1/256*exp(x+1)**4-x)-4*x*exp(x+1)**4+256*x)/(exp(x)*exp(x+1)
**4-256*exp(x)*x)/ln(1/256*exp(x+1)**4-x)**2,x)

[Out]

x*exp(-x)/log(-x + exp(4)*exp(4*x)/256)

________________________________________________________________________________________