Integrand size = 13, antiderivative size = 31 \[ \int \left (-e^{-x}+e^x\right )^3 \, dx=\frac {e^{-3 x}}{3}-3 e^{-x}-3 e^x+\frac {e^{3 x}}{3} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2320, 276} \[ \int \left (-e^{-x}+e^x\right )^3 \, dx=\frac {e^{-3 x}}{3}-3 e^{-x}-3 e^x+\frac {e^{3 x}}{3} \]
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Rule 276
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (-1+x^2\right )^3}{x^4} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \left (-3-\frac {1}{x^4}+\frac {3}{x^2}+x^2\right ) \, dx,x,e^x\right ) \\ & = \frac {e^{-3 x}}{3}-3 e^{-x}-3 e^x+\frac {e^{3 x}}{3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \left (-e^{-x}+e^x\right )^3 \, dx=\frac {1}{3} e^{-3 x} \left (1-9 e^{2 x}-9 e^{4 x}+e^{6 x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {{\mathrm e}^{3 x}}{3}-3 \,{\mathrm e}^{x}-3 \,{\mathrm e}^{-x}+\frac {{\mathrm e}^{-3 x}}{3}\) | \(24\) |
default | \(\frac {{\mathrm e}^{3 x}}{3}-3 \,{\mathrm e}^{x}-3 \,{\mathrm e}^{-x}+\frac {{\mathrm e}^{-3 x}}{3}\) | \(24\) |
risch | \(\frac {{\mathrm e}^{3 x}}{3}-3 \,{\mathrm e}^{x}-3 \,{\mathrm e}^{-x}+\frac {{\mathrm e}^{-3 x}}{3}\) | \(24\) |
parts | \(\frac {{\mathrm e}^{3 x}}{3}-3 \,{\mathrm e}^{x}-3 \,{\mathrm e}^{-x}+\frac {{\mathrm e}^{-3 x}}{3}\) | \(24\) |
meijerg | \(\frac {16}{3}+\frac {{\mathrm e}^{-3 x}}{3}-3 \,{\mathrm e}^{-x}-3 \,{\mathrm e}^{x}+\frac {{\mathrm e}^{3 x}}{3}\) | \(25\) |
norman | \(\left (\frac {1}{3}-3 \,{\mathrm e}^{2 x}-3 \,{\mathrm e}^{4 x}+\frac {{\mathrm e}^{6 x}}{3}\right ) {\mathrm e}^{-3 x}\) | \(26\) |
parallelrisch | \(-\frac {\left (-{\mathrm e}^{6 x}-1+9 \,{\mathrm e}^{4 x}+9 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-3 x}}{3}\) | \(27\) |
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Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \left (-e^{-x}+e^x\right )^3 \, dx=\frac {1}{3} \, {\left (e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} \]
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Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \left (-e^{-x}+e^x\right )^3 \, dx=\frac {e^{3 x}}{3} - 3 e^{x} - 3 e^{- x} + \frac {e^{- 3 x}}{3} \]
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Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \left (-e^{-x}+e^x\right )^3 \, dx=\frac {1}{3} \, e^{\left (3 \, x\right )} - 3 \, e^{\left (-x\right )} + \frac {1}{3} \, e^{\left (-3 \, x\right )} - 3 \, e^{x} \]
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \left (-e^{-x}+e^x\right )^3 \, dx=-\frac {1}{3} \, {\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-3 \, x\right )} + \frac {1}{3} \, e^{\left (3 \, x\right )} - 3 \, e^{x} \]
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Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \left (-e^{-x}+e^x\right )^3 \, dx=\frac {{\mathrm {e}}^{-3\,x}}{3}-3\,{\mathrm {e}}^{-x}+\frac {{\mathrm {e}}^{3\,x}}{3}-3\,{\mathrm {e}}^x \]
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