Optimal. Leaf size=32 \[ -\frac {1}{2} e^{-2 x} x^2-\frac {1}{2} e^{-2 x} x-\frac {e^{-2 x}}{4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2176, 2194} \[ -\frac {1}{2} e^{-2 x} x^2-\frac {1}{2} e^{-2 x} x-\frac {e^{-2 x}}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2176
Rule 2194
Rubi steps
\begin {align*} \int e^{-2 x} x^2 \, dx &=-\frac {1}{2} e^{-2 x} x^2+\int e^{-2 x} x \, dx\\ &=-\frac {1}{2} e^{-2 x} x-\frac {1}{2} e^{-2 x} x^2+\frac {1}{2} \int e^{-2 x} \, dx\\ &=-\frac {1}{4} e^{-2 x}-\frac {1}{2} e^{-2 x} x-\frac {1}{2} e^{-2 x} x^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 19, normalized size = 0.59 \[ -\frac {1}{4} e^{-2 x} \left (2 x^2+2 x+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.38, size = 16, normalized size = 0.50 \[ -\frac {1}{4} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.01, size = 16, normalized size = 0.50 \[ -\frac {1}{4} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 19, normalized size = 0.59 \[ -\frac {\left (2 x^{2}+2 x +1\right ) {\mathrm e}^{-2 x}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.65, size = 16, normalized size = 0.50 \[ -\frac {1}{4} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.08, size = 16, normalized size = 0.50 \[ -\frac {{\mathrm {e}}^{-2\,x}\,\left (4\,x^2+4\,x+2\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 17, normalized size = 0.53 \[ \frac {\left (- 2 x^{2} - 2 x - 1\right ) e^{- 2 x}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________