Optimal. Leaf size=28 \[ \frac {x}{e^{\frac {1+\frac {1}{e^5}}{\left (3-e^{e^4}\right ) x}}+x} \]
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Rubi [A]
time = 6.89, antiderivative size = 31, normalized size of antiderivative = 1.11, number of steps
used = 4, number of rules used = 4, integrand size = 167, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6, 6820, 12,
6819} \begin {gather*} \frac {x}{x+e^{\frac {1+e^5}{e^5 \left (3-e^{e^4}\right ) x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 6819
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-1+e^5 (-1-3 x)+e^{5+e^4} x\right )}{\left (-3 e^5+e^{5+e^4}\right ) x^3+e^{\frac {2 \left (-1-e^5\right )}{-3 e^5 x+e^{5+e^4} x}} \left (-3 e^5 x+e^{5+e^4} x\right )+e^{\frac {-1-e^5}{-3 e^5 x+e^{5+e^4} x}} \left (-6 e^5 x^2+2 e^{5+e^4} x^2\right )} \, dx\\ &=\int \frac {\exp \left (\frac {1+e^5-5 e^5 \left (3-e^{e^4}\right ) x}{e^5 \left (3-e^{e^4}\right ) x}\right ) \left (1+e^5+e^5 \left (3-e^{e^4}\right ) x\right )}{\left (3-e^{e^4}\right ) x \left (e^{\frac {1+e^5}{3 e^5 x-e^{5+e^4} x}}+x\right )^2} \, dx\\ &=\frac {\int \frac {\exp \left (\frac {1+e^5-5 e^5 \left (3-e^{e^4}\right ) x}{e^5 \left (3-e^{e^4}\right ) x}\right ) \left (1+e^5+e^5 \left (3-e^{e^4}\right ) x\right )}{x \left (e^{\frac {1+e^5}{3 e^5 x-e^{5+e^4} x}}+x\right )^2} \, dx}{3-e^{e^4}}\\ &=\frac {x}{e^{\frac {1+e^5}{e^5 \left (3-e^{e^4}\right ) x}}+x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.28, size = 33, normalized size = 1.18 \begin {gather*} \frac {x}{e^{\frac {1+e^5}{3 e^5 x-e^{5+e^4} x}}+x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.69, size = 30, normalized size = 1.07
method | result | size |
risch | \(\frac {x}{x +{\mathrm e}^{\frac {{\mathrm e}^{5}+1}{x \left (3 \,{\mathrm e}^{5}-{\mathrm e}^{5+{\mathrm e}^{4}}\right )}}}\) | \(30\) |
norman | \(-\frac {{\mathrm e}^{\frac {-{\mathrm e}^{5}-1}{x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-3 x \,{\mathrm e}^{5}}}}{x +{\mathrm e}^{\frac {-{\mathrm e}^{5}-1}{x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-3 x \,{\mathrm e}^{5}}}}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.61, size = 40, normalized size = 1.43 \begin {gather*} -\frac {1}{x e^{\left (-\frac {1}{x {\left (3 \, e^{5} - e^{\left (e^{4} + 5\right )}\right )}} + \frac {1}{x {\left (e^{\left (e^{4}\right )} - 3\right )}}\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 28, normalized size = 1.00 \begin {gather*} \frac {x}{x + e^{\left (\frac {e^{5} + 1}{3 \, x e^{5} - x e^{\left (e^{4} + 5\right )}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 27, normalized size = 0.96 \begin {gather*} \frac {x}{x + e^{\frac {- e^{5} - 1}{- 3 x e^{5} + x e^{5} e^{e^{4}}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.12, size = 44, normalized size = 1.57 \begin {gather*} \frac {x}{x+{\mathrm {e}}^{\frac {{\mathrm {e}}^5}{3\,x\,{\mathrm {e}}^5-x\,{\mathrm {e}}^5\,{\mathrm {e}}^{{\mathrm {e}}^4}}}\,{\mathrm {e}}^{\frac {1}{3\,x\,{\mathrm {e}}^5-x\,{\mathrm {e}}^5\,{\mathrm {e}}^{{\mathrm {e}}^4}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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