Optimal. Leaf size=31 \[ \frac {4-x+4 \log \left (1+e^{-\frac {1}{4}-\frac {5}{x}-x+x^2}\right )}{x} \]
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Rubi [F] time = 3.97, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-4 x+e^{\frac {-20-x-4 x^2+4 x^3}{4 x}} \left (20-4 x-4 x^2+8 x^3\right )+\left (-4 x-4 e^{\frac {-20-x-4 x^2+4 x^3}{4 x}} x\right ) \log \left (1+e^{\frac {-20-x-4 x^2+4 x^3}{4 x}}\right )}{x^3+e^{\frac {-20-x-4 x^2+4 x^3}{4 x}} x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 e^{\frac {1}{4}+\frac {5}{x}+x} x+4 e^{x^2} \left (5-x-x^2+2 x^3\right )-4 \left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x \log \left (1+e^{-\frac {1}{4}-\frac {5}{x}-x+x^2}\right )}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x^3} \, dx\\ &=\int \left (-\frac {4 e^{\frac {1}{4}+\frac {5}{x}+x} \left (5-x^2+2 x^3\right )}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x^3}+\frac {4 \left (5-x-x^2+2 x^3-x \log \left (1+e^{-\frac {1}{4}-\frac {5}{x}-x+x^2}\right )\right )}{x^3}\right ) \, dx\\ &=-\left (4 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x} \left (5-x^2+2 x^3\right )}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x^3} \, dx\right )+4 \int \frac {5-x-x^2+2 x^3-x \log \left (1+e^{-\frac {1}{4}-\frac {5}{x}-x+x^2}\right )}{x^3} \, dx\\ &=-\left (4 \int \left (\frac {2 e^{\frac {1}{4}+\frac {5}{x}+x}}{e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}}+\frac {5 e^{\frac {1}{4}+\frac {5}{x}+x}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x^3}-\frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x}\right ) \, dx\right )+4 \int \left (\frac {5-x-x^2+2 x^3}{x^3}-\frac {\log \left (1+e^{-\frac {1}{4}-\frac {5}{x}-x+x^2}\right )}{x^2}\right ) \, dx\\ &=4 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x} \, dx+4 \int \frac {5-x-x^2+2 x^3}{x^3} \, dx-4 \int \frac {\log \left (1+e^{-\frac {1}{4}-\frac {5}{x}-x+x^2}\right )}{x^2} \, dx-8 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}} \, dx-20 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x^3} \, dx\\ &=\frac {4 \log \left (1+e^{-\frac {1}{4}-\frac {5}{x}-x+x^2}\right )}{x}+4 \int \left (2+\frac {5}{x^3}-\frac {1}{x^2}-\frac {1}{x}\right ) \, dx+4 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x} \, dx-4 \int \frac {e^{x^2} \left (5-x^2+2 x^3\right )}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x^3} \, dx-8 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}} \, dx-20 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x^3} \, dx\\ &=-\frac {10}{x^2}+\frac {4}{x}+8 x+\frac {4 \log \left (1+e^{-\frac {1}{4}-\frac {5}{x}-x+x^2}\right )}{x}-4 \log (x)-4 \int \left (\frac {2 e^{x^2}}{e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}}+\frac {5 e^{x^2}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x^3}-\frac {e^{x^2}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x}\right ) \, dx+4 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x} \, dx-8 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}} \, dx-20 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x^3} \, dx\\ &=-\frac {10}{x^2}+\frac {4}{x}+8 x+\frac {4 \log \left (1+e^{-\frac {1}{4}-\frac {5}{x}-x+x^2}\right )}{x}-4 \log (x)+4 \int \frac {e^{x^2}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x} \, dx+4 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x} \, dx-8 \int \frac {e^{x^2}}{e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}} \, dx-8 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}} \, dx-20 \int \frac {e^{x^2}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x^3} \, dx-20 \int \frac {e^{\frac {1}{4}+\frac {5}{x}+x}}{\left (e^{x^2}+e^{\frac {1}{4}+\frac {5}{x}+x}\right ) x^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 28, normalized size = 0.90 \begin {gather*} \frac {4 \left (1+x+\log \left (1+e^{-\frac {1}{4}-\frac {5}{x}-x+x^2}\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 31, normalized size = 1.00 \begin {gather*} \frac {4 \, {\left (\log \left (e^{\left (\frac {4 \, x^{3} - 4 \, x^{2} - x - 20}{4 \, x}\right )} + 1\right ) + 1\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.84, size = 31, normalized size = 1.00 \begin {gather*} \frac {4 \, {\left (\log \left (e^{\left (\frac {4 \, x^{3} - 4 \, x^{2} - x - 20}{4 \, x}\right )} + 1\right ) + 1\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 1.16
method | result | size |
norman | \(\frac {4 x +4 x \ln \left ({\mathrm e}^{\frac {4 x^{3}-4 x^{2}-x -20}{4 x}}+1\right )}{x^{2}}\) | \(36\) |
risch | \(\frac {4 \ln \left ({\mathrm e}^{\frac {4 x^{3}-4 x^{2}-x -20}{4 x}}+1\right )}{x}+\frac {4}{x}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 27, normalized size = 0.87 \begin {gather*} \frac {4 \, x \log \left (e^{\left (x^{2}\right )} + e^{\left (x + \frac {5}{x} + \frac {1}{4}\right )}\right ) + 3 \, x - 20}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 27, normalized size = 0.87 \begin {gather*} \frac {4\,\left (\ln \left ({\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-\frac {1}{4}}\,{\mathrm {e}}^{-\frac {5}{x}}+1\right )+1\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 24, normalized size = 0.77 \begin {gather*} \frac {4 \log {\left (e^{\frac {x^{3} - x^{2} - \frac {x}{4} - 5}{x}} + 1 \right )}}{x} + \frac {4}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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