Integrand size = 13, antiderivative size = 22 \[ \int \left (-e^{-x}+e^x\right )^2 \, dx=-\frac {1}{2} e^{-2 x}+\frac {e^{2 x}}{2}-2 x \]
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Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2320, 272, 45} \[ \int \left (-e^{-x}+e^x\right )^2 \, dx=-2 x-\frac {e^{-2 x}}{2}+\frac {e^{2 x}}{2} \]
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Rule 45
Rule 272
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^3} \, dx,x,e^x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {(1-x)^2}{x^2} \, dx,x,e^{2 x}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (1+\frac {1}{x^2}-\frac {2}{x}\right ) \, dx,x,e^{2 x}\right ) \\ & = -\frac {1}{2} e^{-2 x}+\frac {e^{2 x}}{2}-2 x \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \left (-e^{-x}+e^x\right )^2 \, dx=-\frac {1}{2} e^{-2 x}+\frac {e^{2 x}}{2}-2 x \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77
method | result | size |
risch | \(-2 x +\frac {{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{2}\) | \(17\) |
parts | \(-2 x +\frac {{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{2}\) | \(17\) |
derivativedivides | \(\frac {{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{2}-2 \ln \left ({\mathrm e}^{x}\right )\) | \(19\) |
default | \(\frac {{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{-2 x}}{2}-2 \ln \left ({\mathrm e}^{x}\right )\) | \(19\) |
parallelrisch | \(\frac {\left (-1+{\mathrm e}^{4 x}-4 \,{\mathrm e}^{2 x} x \right ) {\mathrm e}^{-2 x}}{2}\) | \(20\) |
norman | \(\left (-\frac {1}{2}+\frac {{\mathrm e}^{4 x}}{2}-2 \,{\mathrm e}^{2 x} x \right ) {\mathrm e}^{-2 x}\) | \(21\) |
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \left (-e^{-x}+e^x\right )^2 \, dx=-\frac {1}{2} \, {\left (4 \, x e^{\left (2 \, x\right )} - e^{\left (4 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \left (-e^{-x}+e^x\right )^2 \, dx=- 2 x + \frac {e^{2 x}}{2} - \frac {e^{- 2 x}}{2} \]
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Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \left (-e^{-x}+e^x\right )^2 \, dx=-2 \, x + \frac {1}{2} \, e^{\left (2 \, x\right )} - \frac {1}{2} \, e^{\left (-2 \, x\right )} \]
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (-e^{-x}+e^x\right )^2 \, dx=\frac {1}{2} \, {\left (2 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} - 2 \, x + \frac {1}{2} \, e^{\left (2 \, x\right )} \]
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Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.36 \[ \int \left (-e^{-x}+e^x\right )^2 \, dx=\mathrm {sinh}\left (2\,x\right )-2\,x \]
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