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