Optimal. Leaf size=20 \[ -e^{-1+\frac {e^3+x}{4 x}}+x \]
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Rubi [A] time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {12, 14, 2209} \begin {gather*} x-e^{\frac {e^3}{4 x}-\frac {3}{4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2209
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {e^{3+\frac {e^3-3 x}{4 x}}+4 x^2}{x^2} \, dx\\ &=\frac {1}{4} \int \left (4+\frac {e^{\frac {9}{4}+\frac {e^3}{4 x}}}{x^2}\right ) \, dx\\ &=x+\frac {1}{4} \int \frac {e^{\frac {9}{4}+\frac {e^3}{4 x}}}{x^2} \, dx\\ &=-e^{-\frac {3}{4}+\frac {e^3}{4 x}}+x\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 20, normalized size = 1.00 \begin {gather*} -e^{-\frac {3}{4}+\frac {e^3}{4 x}}+x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 22, normalized size = 1.10 \begin {gather*} {\left (x e^{3} - e^{\left (\frac {9 \, x + e^{3}}{4 \, x}\right )}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 59, normalized size = 2.95 \begin {gather*} -\frac {{\left (\frac {{\left (9 \, x + e^{3}\right )} e^{\left (\frac {9 \, x + e^{3}}{4 \, x}\right )}}{x} - e^{6} - 9 \, e^{\left (\frac {9 \, x + e^{3}}{4 \, x}\right )}\right )} e^{\left (-3\right )}}{\frac {9 \, x + e^{3}}{x} - 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 15, normalized size = 0.75
method | result | size |
derivativedivides | \(x -{\mathrm e}^{-\frac {3}{4}+\frac {{\mathrm e}^{3}}{4 x}}\) | \(15\) |
default | \(x -{\mathrm e}^{-\frac {3}{4}+\frac {{\mathrm e}^{3}}{4 x}}\) | \(15\) |
risch | \(x -{\mathrm e}^{\frac {{\mathrm e}^{3}-3 x}{4 x}}\) | \(17\) |
norman | \(\frac {x^{2}-x \,{\mathrm e}^{\frac {{\mathrm e}^{3}-3 x}{4 x}}}{x}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 14, normalized size = 0.70 \begin {gather*} x - e^{\left (\frac {e^{3}}{4 \, x} - \frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.51, size = 14, normalized size = 0.70 \begin {gather*} x-{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{4\,x}-\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 14, normalized size = 0.70 \begin {gather*} x - e^{\frac {- \frac {3 x}{4} + \frac {e^{3}}{4}}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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