Optimal. Leaf size=19 \[ \frac {5 x}{-4+e^4+e^{1+\frac {2}{x}}} \]
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Rubi [A] time = 0.24, antiderivative size = 23, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 4, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6, 6688, 12, 6687} \begin {gather*} -\frac {5 x}{-e^{\frac {2}{x}+1}+4-e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 6687
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-20+5 e^4\right ) x+e^{\frac {2+x}{x}} (10+5 x)}{16 x-8 e^4 x+e^8 x+e^{\frac {2 (2+x)}{x}} x+e^{\frac {2+x}{x}} \left (-8 x+2 e^4 x\right )} \, dx\\ &=\int \frac {\left (-20+5 e^4\right ) x+e^{\frac {2+x}{x}} (10+5 x)}{e^8 x+e^{\frac {2 (2+x)}{x}} x+\left (16-8 e^4\right ) x+e^{\frac {2+x}{x}} \left (-8 x+2 e^4 x\right )} \, dx\\ &=\int \frac {\left (-20+5 e^4\right ) x+e^{\frac {2+x}{x}} (10+5 x)}{e^{\frac {2 (2+x)}{x}} x+\left (16-8 e^4+e^8\right ) x+e^{\frac {2+x}{x}} \left (-8 x+2 e^4 x\right )} \, dx\\ &=\int \frac {5 \left (-4 \left (1-\frac {e^4}{4}\right ) x+e^{1+\frac {2}{x}} (2+x)\right )}{\left (e^{1+\frac {2}{x}}-4 \left (1-\frac {e^4}{4}\right )\right )^2 x} \, dx\\ &=5 \int \frac {-4 \left (1-\frac {e^4}{4}\right ) x+e^{1+\frac {2}{x}} (2+x)}{\left (e^{1+\frac {2}{x}}-4 \left (1-\frac {e^4}{4}\right )\right )^2 x} \, dx\\ &=-\frac {5 x}{4-e^4-e^{1+\frac {2}{x}}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.21, size = 19, normalized size = 1.00 \begin {gather*} \frac {5 x}{-4+e^4+e^{1+\frac {2}{x}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.47, size = 17, normalized size = 0.89 \begin {gather*} \frac {5 \, x}{e^{4} + e^{\left (\frac {x + 2}{x}\right )} - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 28, normalized size = 1.47 \begin {gather*} \frac {5}{\frac {e^{4}}{x} + \frac {e^{\left (\frac {2}{x} + 1\right )}}{x} - \frac {4}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.84, size = 18, normalized size = 0.95
method | result | size |
norman | \(\frac {5 x}{-4+{\mathrm e}^{4}+{\mathrm e}^{\frac {2+x}{x}}}\) | \(18\) |
risch | \(\frac {5 x}{-4+{\mathrm e}^{4}+{\mathrm e}^{\frac {2+x}{x}}}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 17, normalized size = 0.89 \begin {gather*} \frac {5 \, x}{e^{4} + e^{\left (\frac {2}{x} + 1\right )} - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.91, size = 17, normalized size = 0.89 \begin {gather*} \frac {5\,x}{{\mathrm {e}}^4+{\mathrm {e}}^{\frac {2}{x}+1}-4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 14, normalized size = 0.74 \begin {gather*} \frac {5 x}{e^{\frac {x + 2}{x}} - 4 + e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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