Optimal. Leaf size=32 \[ \frac {x}{5+e^{\frac {e^2}{x \left (-\frac {2}{x}+x\right )}}}+\frac {1+x}{2} \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 2.62, 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 {140-140 x^2+35 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (4-4 x^2+x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (48-48 x^2+4 e^2 x^2+12 x^4\right )}{200-200 x^2+50 x^4+e^{\frac {2 e^2}{-2+x^2}} \left (8-8 x^2+2 x^4\right )+e^{\frac {e^2}{-2+x^2}} \left (80-80 x^2+20 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^{2+\frac {e^2}{-2+x^2}} x^2+35 \left (-2+x^2\right )^2+12 e^{\frac {e^2}{-2+x^2}} \left (-2+x^2\right )^2+e^{\frac {2 e^2}{-2+x^2}} \left (-2+x^2\right )^2}{2 \left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (2-x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {4 e^{2+\frac {e^2}{-2+x^2}} x^2+35 \left (-2+x^2\right )^2+12 e^{\frac {e^2}{-2+x^2}} \left (-2+x^2\right )^2+e^{\frac {2 e^2}{-2+x^2}} \left (-2+x^2\right )^2}{\left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (2-x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \left (1-\frac {20 e^2 x^2}{\left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (-2+x^2\right )^2}+\frac {2 \left (4-2 \left (2-e^2\right ) x^2+x^4\right )}{\left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (2-x^2\right )^2}\right ) \, dx\\ &=\frac {x}{2}-\left (10 e^2\right ) \int \frac {x^2}{\left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (-2+x^2\right )^2} \, dx+\int \frac {4-2 \left (2-e^2\right ) x^2+x^4}{\left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (2-x^2\right )^2} \, dx\\ &=\frac {x}{2}-\left (10 e^2\right ) \int \left (\frac {2}{\left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (-2+x^2\right )^2}+\frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (-2+x^2\right )}\right ) \, dx+\int \left (\frac {1}{5+e^{\frac {e^2}{-2+x^2}}}+\frac {4 e^2}{\left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (-2+x^2\right )^2}+\frac {2 e^2}{\left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (-2+x^2\right )}\right ) \, dx\\ &=\frac {x}{2}+\left (2 e^2\right ) \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (-2+x^2\right )} \, dx+\left (4 e^2\right ) \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (-2+x^2\right )^2} \, dx-\left (10 e^2\right ) \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (-2+x^2\right )} \, dx-\left (20 e^2\right ) \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (-2+x^2\right )^2} \, dx+\int \frac {1}{5+e^{\frac {e^2}{-2+x^2}}} \, dx\\ &=\frac {x}{2}+\left (2 e^2\right ) \int \left (-\frac {1}{2 \sqrt {2} \left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (\sqrt {2}-x\right )}-\frac {1}{2 \sqrt {2} \left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (\sqrt {2}+x\right )}\right ) \, dx+\left (4 e^2\right ) \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (-2+x^2\right )^2} \, dx-\left (10 e^2\right ) \int \left (-\frac {1}{2 \sqrt {2} \left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (\sqrt {2}-x\right )}-\frac {1}{2 \sqrt {2} \left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (\sqrt {2}+x\right )}\right ) \, dx-\left (20 e^2\right ) \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (-2+x^2\right )^2} \, dx+\int \frac {1}{5+e^{\frac {e^2}{-2+x^2}}} \, dx\\ &=\frac {x}{2}+\left (4 e^2\right ) \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (-2+x^2\right )^2} \, dx-\left (20 e^2\right ) \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (-2+x^2\right )^2} \, dx-\frac {e^2 \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (\sqrt {2}-x\right )} \, dx}{\sqrt {2}}-\frac {e^2 \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right ) \left (\sqrt {2}+x\right )} \, dx}{\sqrt {2}}+\frac {\left (5 e^2\right ) \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (\sqrt {2}-x\right )} \, dx}{\sqrt {2}}+\frac {\left (5 e^2\right ) \int \frac {1}{\left (5+e^{\frac {e^2}{-2+x^2}}\right )^2 \left (\sqrt {2}+x\right )} \, dx}{\sqrt {2}}+\int \frac {1}{5+e^{\frac {e^2}{-2+x^2}}} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 26, normalized size = 0.81 \begin {gather*} \frac {1}{2} \left (x+\frac {2 x}{5+e^{\frac {e^2}{-2+x^2}}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.05, size = 22, normalized size = 0.69
method | result | size |
risch | \(\frac {x}{2}+\frac {x}{{\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{2}-2}}+5}\) | \(22\) |
norman | \(\frac {-7 x +\frac {7 x^{3}}{2}-{\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{2}-2}} x +\frac {{\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{2}-2}} x^{3}}{2}}{\left ({\mathrm e}^{\frac {{\mathrm e}^{2}}{x^{2}-2}}+5\right ) \left (x^{2}-2\right )}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.33, size = 34, normalized size = 1.06 \begin {gather*} \frac {x e^{\left (\frac {e^{2}}{x^{2} - 2}\right )} + 7 \, x}{2 \, {\left (e^{\left (\frac {e^{2}}{x^{2} - 2}\right )} + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 34, normalized size = 1.06 \begin {gather*} \frac {x e^{\left (\frac {e^{2}}{x^{2} - 2}\right )} + 7 \, x}{2 \, {\left (e^{\left (\frac {e^{2}}{x^{2} - 2}\right )} + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.07, size = 15, normalized size = 0.47 \begin {gather*} \frac {x}{2} + \frac {x}{e^{\frac {e^{2}}{x^{2} - 2}} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.48, size = 21, normalized size = 0.66 \begin {gather*} \frac {x}{2}+\frac {x}{{\mathrm {e}}^{\frac {{\mathrm {e}}^2}{x^2-2}}+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________