Optimal. Leaf size=34 \[ e^{\frac {-\frac {1}{x}+x}{x}}+(2+x) \left (2+\frac {2 \log (\log (5))}{-e^x+x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 1.81, 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 {2 e^{\frac {-1+x^2}{x^2}} x^2+2 x^5+e^{2 x} \left (2 e^{\frac {-1+x^2}{x^2}}+2 x^3\right )+e^x \left (-4 e^{\frac {-1+x^2}{x^2}} x-4 x^4\right )+2 \left (-2 x^3+e^x \left (x^3+x^4\right )\right ) \log (\log (5))}{e^{2 x} x^3-2 e^x x^4+x^5} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{\frac {-1+x^2}{x^2}} x^2+2 x^5+e^{2 x} \left (2 e^{\frac {-1+x^2}{x^2}}+2 x^3\right )+e^x \left (-4 e^{\frac {-1+x^2}{x^2}} x-4 x^4\right )+2 \left (-2 x^3+e^x \left (x^3+x^4\right )\right ) \log (\log (5))}{\left (e^x-x\right )^2 x^3} \, dx\\ &=\int \left (\frac {2 e^{-\frac {1}{x^2}} \left (e+e^{\frac {1}{x^2}} x^3\right )}{x^3}+\frac {2 (1+x) \log (\log (5))}{e^x-x}+\frac {2 \left (-2+x+x^2\right ) \log (\log (5))}{\left (e^x-x\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{-\frac {1}{x^2}} \left (e+e^{\frac {1}{x^2}} x^3\right )}{x^3} \, dx+(2 \log (\log (5))) \int \frac {1+x}{e^x-x} \, dx+(2 \log (\log (5))) \int \frac {-2+x+x^2}{\left (e^x-x\right )^2} \, dx\\ &=2 \int \left (1+\frac {e^{1-\frac {1}{x^2}}}{x^3}\right ) \, dx+(2 \log (\log (5))) \int \left (\frac {1}{e^x-x}+\frac {x}{e^x-x}\right ) \, dx+(2 \log (\log (5))) \int \left (-\frac {2}{\left (e^x-x\right )^2}+\frac {x}{\left (e^x-x\right )^2}+\frac {x^2}{\left (e^x-x\right )^2}\right ) \, dx\\ &=2 x+2 \int \frac {e^{1-\frac {1}{x^2}}}{x^3} \, dx+(2 \log (\log (5))) \int \frac {1}{e^x-x} \, dx+(2 \log (\log (5))) \int \frac {x}{\left (e^x-x\right )^2} \, dx+(2 \log (\log (5))) \int \frac {x}{e^x-x} \, dx+(2 \log (\log (5))) \int \frac {x^2}{\left (e^x-x\right )^2} \, dx-(4 \log (\log (5))) \int \frac {1}{\left (e^x-x\right )^2} \, dx\\ &=e^{1-\frac {1}{x^2}}+2 x+(2 \log (\log (5))) \int \frac {1}{e^x-x} \, dx+(2 \log (\log (5))) \int \frac {x}{\left (e^x-x\right )^2} \, dx+(2 \log (\log (5))) \int \frac {x}{e^x-x} \, dx+(2 \log (\log (5))) \int \frac {x^2}{\left (e^x-x\right )^2} \, dx-(4 \log (\log (5))) \int \frac {1}{\left (e^x-x\right )^2} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 34, normalized size = 1.00 \begin {gather*} 2 \left (\frac {1}{2} e^{1-\frac {1}{x^2}}+x-\frac {(2+x) \log (\log (5))}{e^x-x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.30, size = 32, normalized size = 0.94
method | result | size |
risch | \(2 x +\frac {2 \left (2+x \right ) \ln \left (\ln \left (5\right )\right )}{x -{\mathrm e}^{x}}+{\mathrm e}^{\frac {\left (x -1\right ) \left (x +1\right )}{x^{2}}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.53, size = 51, normalized size = 1.50 \begin {gather*} \frac {{\left (x e + 2 \, {\left (x^{2} - x e^{x} + x \log \left (\log \left (5\right )\right ) + 2 \, \log \left (\log \left (5\right )\right )\right )} e^{\left (\frac {1}{x^{2}}\right )} - e^{\left (x + 1\right )}\right )} e^{\left (-\frac {1}{x^{2}}\right )}}{x - e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 53, normalized size = 1.56 \begin {gather*} \frac {2 \, x^{2} - {\left (2 \, x + e^{\left (\frac {x^{2} - 1}{x^{2}}\right )}\right )} e^{x} + x e^{\left (\frac {x^{2} - 1}{x^{2}}\right )} + 2 \, {\left (x + 2\right )} \log \left (\log \left (5\right )\right )}{x - e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.15, size = 34, normalized size = 1.00 \begin {gather*} 2 x + e^{\frac {x^{2} - 1}{x^{2}}} + \frac {- 2 x \log {\left (\log {\left (5 \right )} \right )} - 4 \log {\left (\log {\left (5 \right )} \right )}}{- x + e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.43, size = 38, normalized size = 1.12 \begin {gather*} \frac {2 \, {\left (x^{2} - x e^{x} + x \log \left (\log \left (5\right )\right ) + 2 \, \log \left (\log \left (5\right )\right )\right )}}{x - e^{x}} + e^{\left (-\frac {1}{x^{2}} + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.67, size = 45, normalized size = 1.32 \begin {gather*} 2\,x+{\mathrm {e}}^{1-\frac {1}{x^2}}+\frac {2\,\left (\ln \left (\ln \left (5\right )\right )\,x^2+\ln \left (\ln \left (5\right )\right )\,x-2\,\ln \left (\ln \left (5\right )\right )\right )}{\left (x-{\mathrm {e}}^x\right )\,\left (x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________