3.100.89 \(\int \frac {e^{-x} (2048 x-128 x^2+(1920 x^2-128 x^3) \log (x)+(8192+7168 x-256 x^2-288 x^3+33 x^4-x^5) \log ^2(x))}{(256 x^2-32 x^3+x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=31 \[ e^{-x} \left (x+\frac {8 \left (4+\frac {x}{\log (x)}\right )}{-x+\frac {x^2}{16}}\right ) \]

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Rubi [F]  time = 1.88, 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^{-x} \left (2048 x-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right ) \log ^2(x)\right )}{\left (256 x^2-32 x^3+x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2048*x - 128*x^2 + (1920*x^2 - 128*x^3)*Log[x] + (8192 + 7168*x - 256*x^2 - 288*x^3 + 33*x^4 - x^5)*Log[x
]^2)/(E^x*(256*x^2 - 32*x^3 + x^4)*Log[x]^2),x]

[Out]

-32/(E^x*(16 - x)) - 32/(E^x*x) + x/E^x - 8*Defer[Int][1/(E^x*(-16 + x)*Log[x]^2), x] + 8*Defer[Int][1/(E^x*x*
Log[x]^2), x] - 128*Defer[Int][1/(E^x*(-16 + x)^2*Log[x]), x] - 128*Defer[Int][1/(E^x*(-16 + x)*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (2048 x-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right ) \log ^2(x)\right )}{x^2 \left (256-32 x+x^2\right ) \log ^2(x)} \, dx\\ &=\int \frac {e^{-x} \left (2048 x-128 x^2+\left (1920 x^2-128 x^3\right ) \log (x)+\left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right ) \log ^2(x)\right )}{(-16+x)^2 x^2 \log ^2(x)} \, dx\\ &=\int \left (\frac {e^{-x} \left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right )}{(-16+x)^2 x^2}-\frac {128 e^{-x}}{(-16+x) x \log ^2(x)}-\frac {128 e^{-x} (-15+x)}{(-16+x)^2 \log (x)}\right ) \, dx\\ &=-\left (128 \int \frac {e^{-x}}{(-16+x) x \log ^2(x)} \, dx\right )-128 \int \frac {e^{-x} (-15+x)}{(-16+x)^2 \log (x)} \, dx+\int \frac {e^{-x} \left (8192+7168 x-256 x^2-288 x^3+33 x^4-x^5\right )}{(-16+x)^2 x^2} \, dx\\ &=-\left (128 \int \left (\frac {e^{-x}}{16 (-16+x) \log ^2(x)}-\frac {e^{-x}}{16 x \log ^2(x)}\right ) \, dx\right )-128 \int \left (\frac {e^{-x}}{(-16+x)^2 \log (x)}+\frac {e^{-x}}{(-16+x) \log (x)}\right ) \, dx+\int \left (e^{-x}-\frac {32 e^{-x}}{(-16+x)^2}-\frac {32 e^{-x}}{-16+x}+\frac {32 e^{-x}}{x^2}+\frac {32 e^{-x}}{x}-e^{-x} x\right ) \, dx\\ &=-\left (8 \int \frac {e^{-x}}{(-16+x) \log ^2(x)} \, dx\right )+8 \int \frac {e^{-x}}{x \log ^2(x)} \, dx-32 \int \frac {e^{-x}}{(-16+x)^2} \, dx-32 \int \frac {e^{-x}}{-16+x} \, dx+32 \int \frac {e^{-x}}{x^2} \, dx+32 \int \frac {e^{-x}}{x} \, dx-128 \int \frac {e^{-x}}{(-16+x)^2 \log (x)} \, dx-128 \int \frac {e^{-x}}{(-16+x) \log (x)} \, dx+\int e^{-x} \, dx-\int e^{-x} x \, dx\\ &=-e^{-x}-\frac {32 e^{-x}}{16-x}-\frac {32 e^{-x}}{x}+e^{-x} x-\frac {32 \text {Ei}(16-x)}{e^{16}}+32 \text {Ei}(-x)-8 \int \frac {e^{-x}}{(-16+x) \log ^2(x)} \, dx+8 \int \frac {e^{-x}}{x \log ^2(x)} \, dx+32 \int \frac {e^{-x}}{-16+x} \, dx-32 \int \frac {e^{-x}}{x} \, dx-128 \int \frac {e^{-x}}{(-16+x)^2 \log (x)} \, dx-128 \int \frac {e^{-x}}{(-16+x) \log (x)} \, dx-\int e^{-x} \, dx\\ &=-\frac {32 e^{-x}}{16-x}-\frac {32 e^{-x}}{x}+e^{-x} x-8 \int \frac {e^{-x}}{(-16+x) \log ^2(x)} \, dx+8 \int \frac {e^{-x}}{x \log ^2(x)} \, dx-128 \int \frac {e^{-x}}{(-16+x)^2 \log (x)} \, dx-128 \int \frac {e^{-x}}{(-16+x) \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 37, normalized size = 1.19 \begin {gather*} e^{-x} \left (\frac {32}{-16+x}-\frac {32}{x}+x\right )+\frac {128 e^{-x}}{(-16+x) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2048*x - 128*x^2 + (1920*x^2 - 128*x^3)*Log[x] + (8192 + 7168*x - 256*x^2 - 288*x^3 + 33*x^4 - x^5)
*Log[x]^2)/(E^x*(256*x^2 - 32*x^3 + x^4)*Log[x]^2),x]

[Out]

(32/(-16 + x) - 32/x + x)/E^x + 128/(E^x*(-16 + x)*Log[x])

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fricas [A]  time = 2.56, size = 39, normalized size = 1.26 \begin {gather*} \frac {{\left (x^{3} - 16 \, x^{2} + 512\right )} e^{\left (-x\right )} \log \relax (x) + 128 \, x e^{\left (-x\right )}}{{\left (x^{2} - 16 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5+33*x^4-288*x^3-256*x^2+7168*x+8192)*log(x)^2+(-128*x^3+1920*x^2)*log(x)-128*x^2+2048*x)/(x^4-
32*x^3+256*x^2)/exp(x)/log(x)^2,x, algorithm="fricas")

[Out]

((x^3 - 16*x^2 + 512)*e^(-x)*log(x) + 128*x*e^(-x))/((x^2 - 16*x)*log(x))

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giac [A]  time = 0.16, size = 52, normalized size = 1.68 \begin {gather*} \frac {x^{3} e^{\left (-x\right )} \log \relax (x) - 16 \, x^{2} e^{\left (-x\right )} \log \relax (x) + 128 \, x e^{\left (-x\right )} + 512 \, e^{\left (-x\right )} \log \relax (x)}{x^{2} \log \relax (x) - 16 \, x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5+33*x^4-288*x^3-256*x^2+7168*x+8192)*log(x)^2+(-128*x^3+1920*x^2)*log(x)-128*x^2+2048*x)/(x^4-
32*x^3+256*x^2)/exp(x)/log(x)^2,x, algorithm="giac")

[Out]

(x^3*e^(-x)*log(x) - 16*x^2*e^(-x)*log(x) + 128*x*e^(-x) + 512*e^(-x)*log(x))/(x^2*log(x) - 16*x*log(x))

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maple [A]  time = 0.03, size = 40, normalized size = 1.29

method	result	size
risch	\(\frac {\left (x^{3}-16 x^{2}+512\right ) {\mathrm e}^{-x}}{x \left (x -16\right )}+\frac {128 \,{\mathrm e}^{-x}}{\left (x -16\right ) \ln \relax (x )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^5+33*x^4-288*x^3-256*x^2+7168*x+8192)*ln(x)^2+(-128*x^3+1920*x^2)*ln(x)-128*x^2+2048*x)/(x^4-32*x^3+2
56*x^2)/exp(x)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

(x^3-16*x^2+512)/x/(x-16)*exp(-x)+128*exp(-x)/(x-16)/ln(x)

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maxima [A]  time = 0.42, size = 35, normalized size = 1.13 \begin {gather*} \frac {{\left ({\left (x^{3} - 16 \, x^{2} + 512\right )} \log \relax (x) + 128 \, x\right )} e^{\left (-x\right )}}{{\left (x^{2} - 16 \, x\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5+33*x^4-288*x^3-256*x^2+7168*x+8192)*log(x)^2+(-128*x^3+1920*x^2)*log(x)-128*x^2+2048*x)/(x^4-
32*x^3+256*x^2)/exp(x)/log(x)^2,x, algorithm="maxima")

[Out]

((x^3 - 16*x^2 + 512)*log(x) + 128*x)*e^(-x)/((x^2 - 16*x)*log(x))

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mupad [B]  time = 7.97, size = 36, normalized size = 1.16 \begin {gather*} x\,{\mathrm {e}}^{-x}+\frac {128\,x\,{\mathrm {e}}^{-x}+512\,{\mathrm {e}}^{-x}\,\ln \relax (x)}{x\,\ln \relax (x)\,\left (x-16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*(2048*x + log(x)*(1920*x^2 - 128*x^3) + log(x)^2*(7168*x - 256*x^2 - 288*x^3 + 33*x^4 - x^5 + 819
2) - 128*x^2))/(log(x)^2*(256*x^2 - 32*x^3 + x^4)),x)

[Out]

x*exp(-x) + (128*x*exp(-x) + 512*exp(-x)*log(x))/(x*log(x)*(x - 16))

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sympy [B]  time = 0.36, size = 39, normalized size = 1.26 \begin {gather*} \frac {\left (x^{3} \log {\relax (x )} - 16 x^{2} \log {\relax (x )} + 128 x + 512 \log {\relax (x )}\right ) e^{- x}}{x^{2} \log {\relax (x )} - 16 x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**5+33*x**4-288*x**3-256*x**2+7168*x+8192)*ln(x)**2+(-128*x**3+1920*x**2)*ln(x)-128*x**2+2048*x)
/(x**4-32*x**3+256*x**2)/exp(x)/ln(x)**2,x)

[Out]

(x**3*log(x) - 16*x**2*log(x) + 128*x + 512*log(x))*exp(-x)/(x**2*log(x) - 16*x*log(x))

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