Optimal. Leaf size=28 \[ 1+\frac {e^3 \left (e^{e+\frac {x^2}{4}}+\frac {x}{21+x}\right )}{x} \]
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Rubi [A] time = 0.38, antiderivative size = 26, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {1594, 27, 12, 6688, 2288} \begin {gather*} \frac {e^{\frac {x^2}{4}+e+3}}{x}+\frac {e^3}{x+21} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 1594
Rule 2288
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{x^2 \left (882+84 x+2 x^2\right )} \, dx\\ &=\int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{2 x^2 (21+x)^2} \, dx\\ &=\frac {1}{2} \int \frac {-2 e^3 x^2+e^{3+\frac {1}{4} \left (4 e+x^2\right )} \left (-882-84 x+439 x^2+42 x^3+x^4\right )}{x^2 (21+x)^2} \, dx\\ &=\frac {1}{2} \int \left (e^{3+e+\frac {x^2}{4}} \left (1-\frac {2}{x^2}\right )-\frac {2 e^3}{(21+x)^2}\right ) \, dx\\ &=\frac {e^3}{21+x}+\frac {1}{2} \int e^{3+e+\frac {x^2}{4}} \left (1-\frac {2}{x^2}\right ) \, dx\\ &=\frac {e^{3+e+\frac {x^2}{4}}}{x}+\frac {e^3}{21+x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 26, normalized size = 0.93 \begin {gather*} \frac {e^{3+e+\frac {x^2}{4}}}{x}+\frac {e^3}{21+x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 29, normalized size = 1.04 \begin {gather*} \frac {x e^{3} + {\left (x + 21\right )} e^{\left (\frac {1}{4} \, x^{2} + e + 3\right )}}{x^{2} + 21 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 39, normalized size = 1.39 \begin {gather*} \frac {x e^{3} + x e^{\left (\frac {1}{4} \, x^{2} + e + 3\right )} + 21 \, e^{\left (\frac {1}{4} \, x^{2} + e + 3\right )}}{x^{2} + 21 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 24, normalized size = 0.86
method | result | size |
risch | \(\frac {{\mathrm e}^{3}}{x +21}+\frac {{\mathrm e}^{3+{\mathrm e}+\frac {x^{2}}{4}}}{x}\) | \(24\) |
default | \(\frac {{\mathrm e}^{3+{\mathrm e}} {\mathrm e}^{\frac {x^{2}}{4}}}{x}+\frac {{\mathrm e}^{3}}{x +21}\) | \(25\) |
norman | \(\frac {x \,{\mathrm e}^{3}+x \,{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}+\frac {x^{2}}{4}}+21 \,{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}+\frac {x^{2}}{4}}}{x \left (x +21\right )}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 23, normalized size = 0.82 \begin {gather*} \frac {e^{3}}{x + 21} + \frac {e^{\left (\frac {1}{4} \, x^{2} + e + 3\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 37, normalized size = 1.32 \begin {gather*} \frac {21\,{\mathrm {e}}^{\frac {x^2}{4}+\mathrm {e}+3}+x\,\left ({\mathrm {e}}^{\frac {x^2}{4}+\mathrm {e}+3}+{\mathrm {e}}^3\right )}{x\,\left (x+21\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 20, normalized size = 0.71 \begin {gather*} \frac {e^{3}}{x + 21} + \frac {e^{3} e^{\frac {x^{2}}{4} + e}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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