Optimal. Leaf size=31 \[ \frac {e^{-2 x}}{2}+\frac {e^{2 x}}{2}+\frac {e^{4 x}}{4}-x \]
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Rubi [A]
time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2320, 14}
\begin {gather*} -x+\frac {e^{-2 x}}{2}+\frac {e^{2 x}}{2}+\frac {e^{4 x}}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2320
Rubi steps
\begin {align*} \int e^x \left (-e^{-x}+e^x\right ) \left (e^{-x}+e^x\right )^2 \, dx &=\text {Subst}\left (\int \frac {-1-\frac {1}{x^2}+x^2+x^4}{x} \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \left (-\frac {1}{x^3}-\frac {1}{x}+x+x^3\right ) \, dx,x,e^x\right )\\ &=\frac {e^{-2 x}}{2}+\frac {e^{2 x}}{2}+\frac {e^{4 x}}{4}-x\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} \frac {e^{-2 x}}{2}+\frac {e^{2 x}}{2}+\frac {e^{4 x}}{4}-x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 23, normalized size = 0.74
method | result | size |
default | \(-x +\frac {{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{4 x}}{4}+\frac {{\mathrm e}^{-2 x}}{2}\) | \(23\) |
risch | \(-x +\frac {{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{4 x}}{4}+\frac {{\mathrm e}^{-2 x}}{2}\) | \(23\) |
norman | \(\left (\frac {{\mathrm e}^{5 x}}{2}+\frac {{\mathrm e}^{7 x}}{4}-{\mathrm e}^{3 x} x +\frac {{\mathrm e}^{x}}{2}\right ) {\mathrm e}^{-3 x}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 24, normalized size = 0.77 \begin {gather*} \frac {1}{4} \, {\left (2 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (4 \, x\right )} - x + \frac {1}{2} \, e^{\left (-2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 27, normalized size = 0.87 \begin {gather*} -\frac {1}{4} \, {\left (4 \, x e^{\left (2 \, x\right )} - e^{\left (6 \, x\right )} - 2 \, e^{\left (4 \, x\right )} - 2\right )} e^{\left (-2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 22, normalized size = 0.71 \begin {gather*} - x + \frac {e^{4 x}}{4} + \frac {e^{2 x}}{2} + \frac {e^{- 2 x}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.04, size = 28, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, {\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} - x + \frac {1}{4} \, e^{\left (4 \, x\right )} + \frac {1}{2} \, e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.39, size = 22, normalized size = 0.71 \begin {gather*} \frac {{\mathrm {e}}^{-2\,x}}{2}-x+\frac {{\mathrm {e}}^{2\,x}}{2}+\frac {{\mathrm {e}}^{4\,x}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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