Optimal. Leaf size=24 \[ \frac {x}{1-\frac {5 \left (e^2-2 x\right )}{2 (-4+2 x)}} \]
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Rubi [A] time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.42, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1984, 27, 683} \begin {gather*} \frac {2 x}{7}+\frac {5 \left (32+12 e^2-5 e^4\right )}{49 \left (-14 x+5 e^2+8\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 683
Rule 1984
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 \left (8+5 e^2\right )-8 \left (8+5 e^2\right ) x+56 x^2}{\left (8+5 e^2\right )^2-28 \left (8+5 e^2\right ) x+196 x^2} \, dx\\ &=\int \frac {8 \left (8+5 e^2\right )-8 \left (8+5 e^2\right ) x+56 x^2}{\left (8+5 e^2-14 x\right )^2} \, dx\\ &=\int \left (\frac {2}{7}-\frac {10 \left (-32-12 e^2+5 e^4\right )}{7 \left (8+5 e^2-14 x\right )^2}\right ) \, dx\\ &=\frac {5 \left (32+12 e^2-5 e^4\right )}{49 \left (8+5 e^2-14 x\right )}+\frac {2 x}{7}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 1.67 \begin {gather*} -\frac {2 \left (-48+25 e^4+e^2 (10-70 x)-112 x+98 x^2\right )}{49 \left (8+5 e^2-14 x\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 36, normalized size = 1.50 \begin {gather*} \frac {196 \, x^{2} - 10 \, {\left (7 \, x + 6\right )} e^{2} - 112 \, x + 25 \, e^{4} - 160}{49 \, {\left (14 \, x - 5 \, e^{2} - 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 11.85, size = 24, normalized size = 1.00
method | result | size |
norman | \(\frac {-4 x^{2}+\frac {20 \,{\mathrm e}^{2}}{7}+\frac {32}{7}}{5 \,{\mathrm e}^{2}-14 x +8}\) | \(24\) |
gosper | \(\frac {-4 x^{2}+\frac {20 \,{\mathrm e}^{2}}{7}+\frac {32}{7}}{5 \,{\mathrm e}^{2}-14 x +8}\) | \(25\) |
risch | \(\frac {2 x}{7}-\frac {5 \,{\mathrm e}^{4}}{49 \left ({\mathrm e}^{2}-\frac {14 x}{5}+\frac {8}{5}\right )}+\frac {12 \,{\mathrm e}^{2}}{49 \left ({\mathrm e}^{2}-\frac {14 x}{5}+\frac {8}{5}\right )}+\frac {32}{49 \left ({\mathrm e}^{2}-\frac {14 x}{5}+\frac {8}{5}\right )}\) | \(42\) |
meijerg | \(-\frac {32 x}{7 \left (-\frac {5 \,{\mathrm e}^{2}}{14}-\frac {4}{7}\right ) \left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right ) \left (5 \,{\mathrm e}^{2}+8\right )}+\frac {\left (-40 \,{\mathrm e}^{2}-64\right ) \left (\frac {14 x}{\left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right ) \left (5 \,{\mathrm e}^{2}+8\right )}+\ln \left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right )\right )}{196}+\frac {\left (5 \,{\mathrm e}^{2}+8\right )^{2} \left (-\frac {14 x \left (-\frac {42 x}{5 \,{\mathrm e}^{2}+8}+6\right )}{3 \left (5 \,{\mathrm e}^{2}+8\right ) \left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right )}-2 \ln \left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right )\right )}{-245 \,{\mathrm e}^{2}-392}-\frac {20 \,{\mathrm e}^{2} x}{7 \left (-\frac {5 \,{\mathrm e}^{2}}{14}-\frac {4}{7}\right ) \left (1-\frac {14 x}{5 \,{\mathrm e}^{2}+8}\right ) \left (5 \,{\mathrm e}^{2}+8\right )}\) | \(195\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 27, normalized size = 1.12 \begin {gather*} \frac {2}{7} \, x + \frac {5 \, {\left (5 \, e^{4} - 12 \, e^{2} - 32\right )}}{49 \, {\left (14 \, x - 5 \, e^{2} - 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.49, size = 26, normalized size = 1.08 \begin {gather*} \frac {2\,x}{7}+\frac {\frac {60\,{\mathrm {e}}^2}{49}-\frac {25\,{\mathrm {e}}^4}{49}+\frac {160}{49}}{5\,{\mathrm {e}}^2-14\,x+8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 26, normalized size = 1.08 \begin {gather*} \frac {2 x}{7} + \frac {- 60 e^{2} - 160 + 25 e^{4}}{686 x - 245 e^{2} - 392} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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