Optimal. Leaf size=26 \[ x+\left (-e^{4 \left (-1+e^2-4 e^{2 x}\right )}+x^2\right )^2 \]
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Rubi [A]
time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.85, number of steps
used = 4, number of rules used = 3, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2320, 2225,
2326} \begin {gather*} x^4-2 e^{-16 e^{2 x}-4 \left (1-e^2\right )} x^2+x+e^{-32 e^{2 x}-8 \left (1-e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 2320
Rule 2326
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=x+x^4-64 \int e^{-8+8 e^2-32 e^{2 x}+2 x} \, dx+\int e^{-4+4 e^2-16 e^{2 x}} \left (-4 x+64 e^{2 x} x^2\right ) \, dx\\ &=x-2 e^{-16 e^{2 x}-4 \left (1-e^2\right )} x^2+x^4-32 \text {Subst}\left (\int e^{-8 \left (1-e^2\right )-32 x} \, dx,x,e^{2 x}\right )\\ &=e^{-32 e^{2 x}-8 \left (1-e^2\right )}+x-2 e^{-16 e^{2 x}-4 \left (1-e^2\right )} x^2+x^4\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 43, normalized size = 1.65 \begin {gather*} e^{-32 e^{2 x}+8 \left (-1+e^2\right )}+x-2 e^{4 \left (-1+e^2-4 e^{2 x}\right )} x^2+x^4 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 41, normalized size = 1.58
method | result | size |
risch | \(x -2 x^{2} {\mathrm e}^{-16 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{2}-4}+x^{4}+{\mathrm e}^{-32 \,{\mathrm e}^{2 x}+8 \,{\mathrm e}^{2}-8}\) | \(37\) |
default | \(x -2 x^{2} {\mathrm e}^{-16 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{2}-4}+x^{4}+{\mathrm e}^{-32 \,{\mathrm e}^{2 x}+8 \,{\mathrm e}^{2}-8}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 36, normalized size = 1.38 \begin {gather*} x^{4} - 2 \, x^{2} e^{\left (4 \, e^{2} - 16 \, e^{\left (2 \, x\right )} - 4\right )} + x + e^{\left (8 \, e^{2} - 32 \, e^{\left (2 \, x\right )} - 8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 36, normalized size = 1.38 \begin {gather*} x^{4} - 2 \, x^{2} e^{\left (4 \, e^{2} - 16 \, e^{\left (2 \, x\right )} - 4\right )} + x + e^{\left (8 \, e^{2} - 32 \, e^{\left (2 \, x\right )} - 8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 39, normalized size = 1.50 \begin {gather*} x^{4} - 2 x^{2} e^{- 16 e^{2 x} - 4 + 4 e^{2}} + x + e^{- 32 e^{2 x} - 8 + 8 e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 36, normalized size = 1.38 \begin {gather*} x^{4} - 2 \, x^{2} e^{\left (4 \, e^{2} - 16 \, e^{\left (2 \, x\right )} - 4\right )} + x + e^{\left (8 \, e^{2} - 32 \, e^{\left (2 \, x\right )} - 8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.00, size = 36, normalized size = 1.38 \begin {gather*} x+{\mathrm {e}}^{8\,{\mathrm {e}}^2-32\,{\mathrm {e}}^{2\,x}-8}-2\,x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^2-16\,{\mathrm {e}}^{2\,x}-4}+x^4 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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