Optimal. Leaf size=24 \[ \frac {x+e^{-\frac {25 x^4}{4}} x}{2 \left (e^6+x\right )} \]
________________________________________________________________________________________
Rubi [B] time = 0.34, antiderivative size = 49, normalized size of antiderivative = 2.04, number of steps used = 5, number of rules used = 4, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {27, 12, 6742, 2288} \begin {gather*} \frac {e^{-\frac {25 x^4}{4}} \left (x^5+e^6 x^4\right )}{2 x^3 \left (x+e^6\right )^2}-\frac {e^6}{2 \left (x+e^6\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 27
Rule 2288
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {25 x^4}{4}} \left (e^{6+\frac {25 x^4}{4}}-25 x^5+e^6 \left (1-25 x^4\right )\right )}{2 \left (e^6+x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{-\frac {25 x^4}{4}} \left (e^{6+\frac {25 x^4}{4}}-25 x^5+e^6 \left (1-25 x^4\right )\right )}{\left (e^6+x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {e^6}{\left (e^6+x\right )^2}+\frac {e^{-\frac {25 x^4}{4}} \left (e^6-25 e^6 x^4-25 x^5\right )}{\left (e^6+x\right )^2}\right ) \, dx\\ &=-\frac {e^6}{2 \left (e^6+x\right )}+\frac {1}{2} \int \frac {e^{-\frac {25 x^4}{4}} \left (e^6-25 e^6 x^4-25 x^5\right )}{\left (e^6+x\right )^2} \, dx\\ &=-\frac {e^6}{2 \left (e^6+x\right )}+\frac {e^{-\frac {25 x^4}{4}} \left (e^6 x^4+x^5\right )}{2 x^3 \left (e^6+x\right )^2}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 28, normalized size = 1.17 \begin {gather*} \frac {-e^6+e^{-\frac {25 x^4}{4}} x}{2 \left (e^6+x\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 31, normalized size = 1.29 \begin {gather*} \frac {{\left (x e^{6} - e^{\left (\frac {25}{4} \, x^{4} + 12\right )}\right )} e^{\left (-\frac {25}{4} \, x^{4} - 6\right )}}{2 \, {\left (x + e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 33, normalized size = 1.38 \begin {gather*} \frac {x - e^{\left (\frac {25}{4} \, x^{4} + 6\right )}}{2 \, {\left (x e^{\left (\frac {25}{4} \, x^{4}\right )} + e^{\left (\frac {25}{4} \, x^{4} + 6\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.17, size = 27, normalized size = 1.12
method | result | size |
risch | \(-\frac {{\mathrm e}^{6}}{2 \left ({\mathrm e}^{6}+x \right )}+\frac {x \,{\mathrm e}^{-\frac {25 x^{4}}{4}}}{2 \,{\mathrm e}^{6}+2 x}\) | \(27\) |
norman | \(\frac {\left (-\frac {{\mathrm e}^{6} {\mathrm e}^{\frac {25 x^{4}}{4}}}{2}+\frac {x}{2}\right ) {\mathrm e}^{-\frac {25 x^{4}}{4}}}{{\mathrm e}^{6}+x}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 21, normalized size = 0.88 \begin {gather*} \frac {x e^{\left (-\frac {25}{4} \, x^{4}\right )} - e^{6}}{2 \, {\left (x + e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.18, size = 21, normalized size = 0.88 \begin {gather*} \frac {x+x\,{\mathrm {e}}^{-\frac {25\,x^4}{4}}}{2\,x+2\,{\mathrm {e}}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.23, size = 29, normalized size = 1.21 \begin {gather*} \frac {x e^{- \frac {25 x^{4}}{4}}}{2 x + 2 e^{6}} - \frac {e^{6}}{2 x + 2 e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________