Optimal. Leaf size=24 \[ \left (4-\frac {2 e^{-4/x} \log ^2(x)}{5 (4+x)}\right )^2 \]
[Out]
________________________________________________________________________________________
Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(24)=48\).
time = 6.45, antiderivative size = 57, normalized size of antiderivative = 2.38, number of steps
used = 6, number of rules used = 7, integrand size = 107, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6820, 12,
6874, 2260, 2209, 2326, 2634} \begin {gather*} \frac {4 e^{-8/x} \log ^3(x) (x \log (x)+4 \log (x))}{25 (x+4)^3}-\frac {16 e^{-4/x} \log (x) (x \log (x)+4 \log (x))}{5 (x+4)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2209
Rule 2260
Rule 2326
Rule 2634
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^{-8/x} \log (x) \left (2 x (4+x)-\left (-16-4 x+x^2\right ) \log (x)\right ) \left (-10 e^{4/x} (4+x)+\log ^2(x)\right )}{25 x^2 (4+x)^3} \, dx\\ &=\frac {8}{25} \int \frac {e^{-8/x} \log (x) \left (2 x (4+x)-\left (-16-4 x+x^2\right ) \log (x)\right ) \left (-10 e^{4/x} (4+x)+\log ^2(x)\right )}{x^2 (4+x)^3} \, dx\\ &=\frac {8}{25} \int \left (\frac {10 e^{-4/x} \log (x) \left (-8 x-2 x^2-16 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{x^2 (4+x)^2}-\frac {e^{-8/x} \log ^3(x) \left (-8 x-2 x^2-16 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{x^2 (4+x)^3}\right ) \, dx\\ &=-\left (\frac {8}{25} \int \frac {e^{-8/x} \log ^3(x) \left (-8 x-2 x^2-16 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{x^2 (4+x)^3} \, dx\right )+\frac {16}{5} \int \frac {e^{-4/x} \log (x) \left (-8 x-2 x^2-16 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{x^2 (4+x)^2} \, dx\\ &=-\frac {16 e^{-4/x} \log (x) (4 \log (x)+x \log (x))}{5 (4+x)^2}+\frac {4 e^{-8/x} \log ^3(x) (4 \log (x)+x \log (x))}{25 (4+x)^3}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 43, normalized size = 1.79 \begin {gather*} -\frac {8}{25} \left (\frac {10 e^{-4/x} \log ^2(x)}{4+x}-\frac {e^{-8/x} \log ^4(x)}{2 (4+x)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.13, size = 36, normalized size = 1.50
method | result | size |
risch | \(\frac {4 \,{\mathrm e}^{-\frac {8}{x}} \ln \left (x \right )^{4}}{25 \left (4+x \right )^{2}}-\frac {16 \,{\mathrm e}^{-\frac {4}{x}} \ln \left (x \right )^{2}}{5 \left (4+x \right )}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.37, size = 39, normalized size = 1.62 \begin {gather*} \frac {4 \, {\left (e^{\left (-\frac {8}{x}\right )} \log \left (x\right )^{4} - 20 \, {\left (x + 4\right )} e^{\left (-\frac {4}{x}\right )} \log \left (x\right )^{2}\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 40, normalized size = 1.67 \begin {gather*} -\frac {4 \, {\left (20 \, {\left (x + 4\right )} e^{\frac {4}{x}} \log \left (x\right )^{2} - \log \left (x\right )^{4}\right )} e^{\left (-\frac {8}{x}\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (19) = 38\).
time = 0.21, size = 66, normalized size = 2.75 \begin {gather*} \frac {\left (20 x \log {\left (x \right )}^{4} + 80 \log {\left (x \right )}^{4}\right ) e^{- \frac {8}{x}} + \left (- 400 x^{2} \log {\left (x \right )}^{2} - 3200 x \log {\left (x \right )}^{2} - 6400 \log {\left (x \right )}^{2}\right ) e^{- \frac {4}{x}}}{125 x^{3} + 1500 x^{2} + 6000 x + 8000} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{-\frac {8}{x}}\,\left (\left (-8\,x^2+32\,x+128\right )\,{\ln \left (x\right )}^4+\left (16\,x^2+64\,x\right )\,{\ln \left (x\right )}^3-{\mathrm {e}}^{4/x}\,\left (-80\,x^3+2560\,x+5120\right )\,{\ln \left (x\right )}^2-{\mathrm {e}}^{4/x}\,\left (160\,x^3+1280\,x^2+2560\,x\right )\,\ln \left (x\right )\right )}{25\,x^5+300\,x^4+1200\,x^3+1600\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________