Optimal. Leaf size=20 \[ -3+e^{4 e^{-2 x} x^2}-3 x+x^2 \]
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Rubi [F]
time = 0.24, 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 {-3 x+2 x^2+4 e^{-2 x+4 e^{-2 x} x^2} (2-2 x) x^2}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-3+2 x-8 e^{2 x \left (-1+2 e^{-2 x} x\right )} (-1+x) x\right ) \, dx\\ &=-3 x+x^2-8 \int e^{2 x \left (-1+2 e^{-2 x} x\right )} (-1+x) x \, dx\\ &=-3 x+x^2-8 \int \left (-e^{2 x \left (-1+2 e^{-2 x} x\right )} x+e^{2 x \left (-1+2 e^{-2 x} x\right )} x^2\right ) \, dx\\ &=-3 x+x^2+8 \int e^{2 x \left (-1+2 e^{-2 x} x\right )} x \, dx-8 \int e^{2 x \left (-1+2 e^{-2 x} x\right )} x^2 \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 19, normalized size = 0.95 \begin {gather*} e^{4 e^{-2 x} x^2}-3 x+x^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 20, normalized size = 1.00
method | result | size |
risch | \(x^{2}-3 x +{\mathrm e}^{4 x^{2} {\mathrm e}^{-2 x}}\) | \(18\) |
default | \(x^{2}-3 x +{\mathrm e}^{{\mathrm e}^{\ln \left (4 x^{2}\right )-2 x}}\) | \(20\) |
norman | \(x^{2}-3 x +{\mathrm e}^{{\mathrm e}^{\ln \left (4 x^{2}\right )-2 x}}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 17, normalized size = 0.85 \begin {gather*} x^{2} - 3 \, x + e^{\left (4 \, x^{2} e^{\left (-2 \, x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (20) = 40\).
time = 0.36, size = 56, normalized size = 2.80 \begin {gather*} {\left ({\left (x^{2} - 3 \, x\right )} e^{\left (-2 \, x + \log \left (4 \, x^{2}\right )\right )} + e^{\left (-2 \, x + e^{\left (-2 \, x + \log \left (4 \, x^{2}\right )\right )} + \log \left (4 \, x^{2}\right )\right )}\right )} e^{\left (2 \, x - \log \left (4 \, x^{2}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 17, normalized size = 0.85 \begin {gather*} x^{2} - 3 x + e^{4 x^{2} e^{- 2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [B]
time = 0.90, size = 17, normalized size = 0.85 \begin {gather*} {\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{-2\,x}}-3\,x+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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