Optimal. Leaf size=36 \[ -\frac {1}{4} e^{-4 x}+2 e^{-2 x}-2 e^{2 x}+\frac {e^{4 x}}{4}+6 x \]
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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2320, 272, 45}
\begin {gather*} 6 x-\frac {e^{-4 x}}{4}+2 e^{-2 x}-2 e^{2 x}+\frac {e^{4 x}}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 2320
Rubi steps
\begin {align*} \int \left (-e^{-x}+e^x\right )^4 \, dx &=\text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^5} \, dx,x,e^x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(1-x)^4}{x^3} \, dx,x,e^{2 x}\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-4+\frac {1}{x^3}-\frac {4}{x^2}+\frac {6}{x}+x\right ) \, dx,x,e^{2 x}\right )\\ &=-\frac {1}{4} e^{-4 x}+2 e^{-2 x}-2 e^{2 x}+\frac {e^{4 x}}{4}+6 x\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 37, normalized size = 1.03 \begin {gather*} \frac {1}{4} e^{-4 x} \left (-1+8 e^{2 x}-8 e^{6 x}+e^{8 x}\right )+6 \log \left (e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 31, normalized size = 0.86
method | result | size |
risch | \(6 x +\frac {{\mathrm e}^{4 x}}{4}-2 \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{-2 x}-\frac {{\mathrm e}^{-4 x}}{4}\) | \(29\) |
derivativedivides | \(\frac {{\mathrm e}^{4 x}}{4}-2 \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{-2 x}-\frac {{\mathrm e}^{-4 x}}{4}+6 \ln \left ({\mathrm e}^{x}\right )\) | \(31\) |
default | \(\frac {{\mathrm e}^{4 x}}{4}-2 \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{-2 x}-\frac {{\mathrm e}^{-4 x}}{4}+6 \ln \left ({\mathrm e}^{x}\right )\) | \(31\) |
norman | \(\left (-\frac {1}{4}+2 \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{6 x}+\frac {{\mathrm e}^{8 x}}{4}+6 x \,{\mathrm e}^{4 x}\right ) {\mathrm e}^{-4 x}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.73, size = 28, normalized size = 0.78 \begin {gather*} 6 \, x + \frac {1}{4} \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 2 \, e^{\left (-2 \, x\right )} - \frac {1}{4} \, e^{\left (-4 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.71, size = 31, normalized size = 0.86 \begin {gather*} \frac {1}{4} \, {\left (24 \, x e^{\left (4 \, x\right )} + e^{\left (8 \, x\right )} - 8 \, e^{\left (6 \, x\right )} + 8 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-4 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 31, normalized size = 0.86 \begin {gather*} 6 x + \frac {e^{4 x}}{4} - 2 e^{2 x} + 2 e^{- 2 x} - \frac {e^{- 4 x}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 36, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \, {\left (18 \, e^{\left (4 \, x\right )} - 8 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-4 \, x\right )} + 6 \, x + \frac {1}{4} \, e^{\left (4 \, x\right )} - 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 = 0.34, size = 28, normalized size = 0.78 \begin {gather*} 6\,x+2\,{\mathrm {e}}^{-2\,x}-2\,{\mathrm {e}}^{2\,x}-\frac {{\mathrm {e}}^{-4\,x}}{4}+\frac {{\mathrm {e}}^{4\,x}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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