Optimal. Leaf size=33 \[ (4+5 x) \left (2+\frac {e^x}{x}+x-\frac {\log \left (\frac {3}{4}+\frac {x^2}{4}\right )}{x}\right ) \]
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Rubi [A] time = 0.90, antiderivative size = 47, normalized size of antiderivative = 1.42, number of steps used = 18, number of rules used = 12, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1593, 6725, 2199, 2194, 2177, 2178, 6688, 1810, 635, 203, 260, 2455} \begin {gather*} 5 x^2-5 \log \left (x^2+3\right )-\frac {4 \log \left (\frac {x^2}{4}+\frac {3}{4}\right )}{x}+14 x+5 e^x+\frac {4 e^x}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1593
Rule 1810
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rule 2455
Rule 6688
Rule 6725
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {34 x^2+20 x^3+14 x^4+10 x^5+e^x \left (-12+12 x+11 x^2+4 x^3+5 x^4\right )+\left (12+4 x^2\right ) \log \left (\frac {1}{4} \left (3+x^2\right )\right )}{x^2 \left (3+x^2\right )} \, dx\\ &=\int \left (\frac {e^x \left (-4+4 x+5 x^2\right )}{x^2}+\frac {2 \left (17 x^2+10 x^3+7 x^4+5 x^5+6 \log \left (\frac {1}{4} \left (3+x^2\right )\right )+2 x^2 \log \left (\frac {1}{4} \left (3+x^2\right )\right )\right )}{x^2 \left (3+x^2\right )}\right ) \, dx\\ &=2 \int \frac {17 x^2+10 x^3+7 x^4+5 x^5+6 \log \left (\frac {1}{4} \left (3+x^2\right )\right )+2 x^2 \log \left (\frac {1}{4} \left (3+x^2\right )\right )}{x^2 \left (3+x^2\right )} \, dx+\int \frac {e^x \left (-4+4 x+5 x^2\right )}{x^2} \, dx\\ &=2 \int \left (\frac {17+10 x+7 x^2+5 x^3}{3+x^2}+\frac {2 \log \left (\frac {3}{4}+\frac {x^2}{4}\right )}{x^2}\right ) \, dx+\int \left (5 e^x-\frac {4 e^x}{x^2}+\frac {4 e^x}{x}\right ) \, dx\\ &=2 \int \frac {17+10 x+7 x^2+5 x^3}{3+x^2} \, dx-4 \int \frac {e^x}{x^2} \, dx+4 \int \frac {e^x}{x} \, dx+4 \int \frac {\log \left (\frac {3}{4}+\frac {x^2}{4}\right )}{x^2} \, dx+5 \int e^x \, dx\\ &=5 e^x+\frac {4 e^x}{x}+4 \text {Ei}(x)-\frac {4 \log \left (\frac {3}{4}+\frac {x^2}{4}\right )}{x}+2 \int \frac {1}{\frac {3}{4}+\frac {x^2}{4}} \, dx+2 \int \left (7+5 x-\frac {4+5 x}{3+x^2}\right ) \, dx-4 \int \frac {e^x}{x} \, dx\\ &=5 e^x+\frac {4 e^x}{x}+14 x+5 x^2+\frac {8 \tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {4 \log \left (\frac {3}{4}+\frac {x^2}{4}\right )}{x}-2 \int \frac {4+5 x}{3+x^2} \, dx\\ &=5 e^x+\frac {4 e^x}{x}+14 x+5 x^2+\frac {8 \tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {4 \log \left (\frac {3}{4}+\frac {x^2}{4}\right )}{x}-8 \int \frac {1}{3+x^2} \, dx-10 \int \frac {x}{3+x^2} \, dx\\ &=5 e^x+\frac {4 e^x}{x}+14 x+5 x^2-\frac {4 \log \left (\frac {3}{4}+\frac {x^2}{4}\right )}{x}-5 \log \left (3+x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [C] time = 0.27, size = 107, normalized size = 3.24 \begin {gather*} 14 x+5 x^2+\frac {e^x (4+5 x)}{x}+\frac {8 \tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} i \left (15 i+4 \sqrt {3}\right ) \log \left (\sqrt {3}+i x\right )-\frac {1}{3} \left (15+4 i \sqrt {3}\right ) \log \left (i \sqrt {3}+x\right )-\frac {4 \log \left (\frac {1}{4} \left (3+x^2\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 38, normalized size = 1.15 \begin {gather*} \frac {5 \, x^{3} + 14 \, x^{2} + {\left (5 \, x + 4\right )} e^{x} - {\left (5 \, x + 4\right )} \log \left (\frac {1}{4} \, x^{2} + \frac {3}{4}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 43, normalized size = 1.30 \begin {gather*} \frac {5 \, x^{3} + 14 \, x^{2} + 5 \, x e^{x} - 5 \, x \log \left (x^{2} + 3\right ) + 4 \, e^{x} - 4 \, \log \left (\frac {1}{4} \, x^{2} + \frac {3}{4}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 47, normalized size = 1.42
method | result | size |
default | \(\frac {4 \,{\mathrm e}^{x}}{x}+5 \,{\mathrm e}^{x}-\frac {4 \ln \left (x^{2}+3\right )}{x}+\frac {8 \ln \relax (2)}{x}+5 x^{2}+14 x -5 \ln \left (x^{2}+3\right )\) | \(47\) |
risch | \(-\frac {4 \ln \left (\frac {3}{4}+\frac {x^{2}}{4}\right )}{x}-\frac {-5 x^{3}+5 \ln \left (x^{2}+3\right ) x -14 x^{2}-5 \,{\mathrm e}^{x} x -4 \,{\mathrm e}^{x}}{x}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 42, normalized size = 1.27 \begin {gather*} 5 \, x^{2} + 14 \, x + \frac {{\left (5 \, x + 4\right )} e^{x} + 8 \, \log \relax (2) - 4 \, \log \left (x^{2} + 3\right )}{x} - 5 \, \log \left (x^{2} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.98, size = 41, normalized size = 1.24 \begin {gather*} 14\,x-5\,\ln \left (x^2+3\right )-\frac {4\,\ln \left (\frac {x^2}{4}+\frac {3}{4}\right )}{x}+5\,x^2+\frac {{\mathrm {e}}^x\,\left (5\,x+4\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 39, normalized size = 1.18 \begin {gather*} 5 x^{2} + 14 x - 5 \log {\left (x^{2} + 3 \right )} + \frac {\left (5 x + 4\right ) e^{x}}{x} - \frac {4 \log {\left (\frac {x^{2}}{4} + \frac {3}{4} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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