Optimal. Leaf size=22 \[ 2 x-\frac {e^{-2 x}}{2}+\frac {e^{2 x}}{2} \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2282, 266, 43} \[ 2 x-\frac {e^{-2 x}}{2}+\frac {e^{2 x}}{2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2282
Rubi steps
\begin {align*} \int \left (e^{-x}+e^x\right )^2 \, dx &=\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^3} \, dx,x,e^x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(1+x)^2}{x^2} \, dx,x,e^{2 x}\right )\\ &=\frac {1}{2} \operatorname {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*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.91 \[ \frac {1}{2} \left (4 x-e^{-2 x}+e^{2 x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 19, normalized size = 0.86 \[ \frac {1}{2} \, {\left (4 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 24, normalized size = 1.09 \[ -\frac {1}{2} \, {\left (2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} + 2 \, x + \frac {1}{2} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 17, normalized size = 0.77 \[ 2 x -\frac {{\mathrm e}^{-2 x}}{2}+\frac {{\mathrm e}^{2 x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 16, normalized size = 0.73 \[ 2 \, x + \frac {1}{2} \, e^{\left (2 \, x\right )} - \frac {1}{2} \, e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.58, size = 8, normalized size = 0.36 \[ 2\,x+\mathrm {sinh}\left (2\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 17, normalized size = 0.77 \[ 2 x + \frac {e^{2 x}}{2} - \frac {e^{- 2 x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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