Optimal. Leaf size=20 \[ 4 e^{-x} \sqrt {e^{2+x} x^2} \]
________________________________________________________________________________________
Rubi [B] time = 0.33, antiderivative size = 52, normalized size of antiderivative = 2.60, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {12, 6719, 2281, 2187, 2176, 2194} \begin {gather*} \frac {8 e^{-x} \sqrt {e^{x+2} x^2}}{x}-\frac {4 e^{-x} (2-x) \sqrt {e^{x+2} x^2}}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2176
Rule 2187
Rule 2194
Rule 2281
Rule 6719
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=2 \int \frac {e^{-x} (2-x) \sqrt {e^{2+x} x^2}}{x} \, dx\\ &=\frac {\left (2 \sqrt {e^{2+x} x^2}\right ) \int e^{-x} \sqrt {e^{2+x}} (2-x) \, dx}{\sqrt {e^{2+x}} x}\\ &=\frac {\left (2 e^{\frac {1}{2} (-2-x)} \sqrt {e^{2+x} x^2}\right ) \int e^{-x+\frac {2+x}{2}} (2-x) \, dx}{x}\\ &=\frac {\left (2 e^{\frac {1}{2} (-2-x)} \sqrt {e^{2+x} x^2}\right ) \int e^{1-\frac {x}{2}} (2-x) \, dx}{x}\\ &=-\frac {4 e^{-x} (2-x) \sqrt {e^{2+x} x^2}}{x}-\frac {\left (4 e^{\frac {1}{2} (-2-x)} \sqrt {e^{2+x} x^2}\right ) \int e^{1-\frac {x}{2}} \, dx}{x}\\ &=\frac {8 e^{-x} \sqrt {e^{2+x} x^2}}{x}-\frac {4 e^{-x} (2-x) \sqrt {e^{2+x} x^2}}{x}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 21, normalized size = 1.05 \begin {gather*} \frac {4 e^2 x^2}{\sqrt {e^{2+x} x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 17, normalized size = 0.85 \begin {gather*} 4 \, \sqrt {x^{2}} e^{\left (-\frac {1}{2} \, \sqrt {x^{2}} + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 11, normalized size = 0.55 \begin {gather*} 4 \, x e^{\left (-\frac {1}{2} \, x + 1\right )} \mathrm {sgn}\relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 17, normalized size = 0.85
method | result | size |
gosper | \(4 \sqrt {x^{2} {\mathrm e}^{2+x}}\, {\mathrm e}^{-x}\) | \(17\) |
risch | \(4 \sqrt {x^{2} {\mathrm e}^{2+x}}\, {\mathrm e}^{-x}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.37, size = 24, normalized size = 1.20 \begin {gather*} 4 \, {\left (x e + 2 \, e\right )} e^{\left (-\frac {1}{2} \, x\right )} - 8 \, e^{\left (-\frac {1}{2} \, x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.34, size = 13, normalized size = 0.65 \begin {gather*} 4\,{\mathrm {e}}^{1-\frac {x}{2}}\,\sqrt {x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 67.17, size = 20, normalized size = 1.00 \begin {gather*} 4 e \sqrt {x^{2}} e^{- x} \sqrt {e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________