Optimal. Leaf size=24 \[ \frac {\left (e^{-4+x}+\log (x)\right ) \left (5+\log \left (\frac {1}{5 (3+x)}\right )\right )}{x} \]
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Rubi [B] time = 2.81, antiderivative size = 136, normalized size of antiderivative = 5.67, number of steps used = 37, number of rules used = 18, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1593, 6742, 6688, 2177, 2178, 2197, 2554, 2357, 2304, 2301, 2317, 2391, 2395, 36, 29, 31, 2376, 2392} \begin {gather*} \frac {5 e^{x-4}}{x}-\frac {1}{3} \log ^2(x)+\frac {1}{3} \log \left (\frac {x}{3}+1\right ) \log (x)-\frac {1}{3} (1-\log (x)) \log (x)+\frac {5 \log (x)}{x}+\frac {1}{3} \log (3) \log (x)+\frac {\log (x)}{3}+\frac {1}{3} (1-\log (x)) \log (x+3)-\frac {1}{3} \log (x+3)-\frac {(1-\log (x)) \log \left (\frac {1}{5 x+15}\right )}{x}+\frac {e^{x-4} \log \left (\frac {1}{5 x+15}\right )}{x}+\frac {\log \left (\frac {1}{5 x+15}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 1593
Rule 2177
Rule 2178
Rule 2197
Rule 2301
Rule 2304
Rule 2317
Rule 2357
Rule 2376
Rule 2391
Rule 2392
Rule 2395
Rule 2554
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {15+5 x+e^{-4+x} \left (-15+9 x+5 x^2\right )+\left (3+x+e^{-4+x} \left (-3+2 x+x^2\right )\right ) \log \left (\frac {1}{15+5 x}\right )+\log (x) \left (-15-6 x+(-3-x) \log \left (\frac {1}{15+5 x}\right )\right )}{x^2 (3+x)} \, dx\\ &=\int \left (\frac {e^{-4+x} \left (-15+9 x+5 x^2-3 \log \left (\frac {1}{15+5 x}\right )+2 x \log \left (\frac {1}{15+5 x}\right )+x^2 \log \left (\frac {1}{15+5 x}\right )\right )}{x^2 (3+x)}+\frac {15+5 x-15 \log (x)-6 x \log (x)+3 \log \left (\frac {1}{15+5 x}\right )+x \log \left (\frac {1}{15+5 x}\right )-3 \log (x) \log \left (\frac {1}{15+5 x}\right )-x \log (x) \log \left (\frac {1}{15+5 x}\right )}{x^2 (3+x)}\right ) \, dx\\ &=\int \frac {e^{-4+x} \left (-15+9 x+5 x^2-3 \log \left (\frac {1}{15+5 x}\right )+2 x \log \left (\frac {1}{15+5 x}\right )+x^2 \log \left (\frac {1}{15+5 x}\right )\right )}{x^2 (3+x)} \, dx+\int \frac {15+5 x-15 \log (x)-6 x \log (x)+3 \log \left (\frac {1}{15+5 x}\right )+x \log \left (\frac {1}{15+5 x}\right )-3 \log (x) \log \left (\frac {1}{15+5 x}\right )-x \log (x) \log \left (\frac {1}{15+5 x}\right )}{x^2 (3+x)} \, dx\\ &=\int \frac {e^{-4+x} \left (-15+9 x+5 x^2+\left (-3+2 x+x^2\right ) \log \left (\frac {1}{15+5 x}\right )\right )}{x^2 (3+x)} \, dx+\int \frac {(3+x) \left (5+\log \left (\frac {1}{15+5 x}\right )\right )-\log (x) \left (15+6 x+(3+x) \log \left (\frac {1}{15+5 x}\right )\right )}{x^2 (3+x)} \, dx\\ &=\int \left (\frac {e^{-4+x} \left (-15+9 x+5 x^2\right )}{x^2 (3+x)}+\frac {e^{-4+x} (-1+x) \log \left (\frac {1}{15+5 x}\right )}{x^2}\right ) \, dx+\int \left (\frac {15+5 x-15 \log (x)-6 x \log (x)}{x^2 (3+x)}-\frac {(-1+\log (x)) \log \left (\frac {1}{15+5 x}\right )}{x^2}\right ) \, dx\\ &=\int \frac {e^{-4+x} \left (-15+9 x+5 x^2\right )}{x^2 (3+x)} \, dx+\int \frac {15+5 x-15 \log (x)-6 x \log (x)}{x^2 (3+x)} \, dx+\int \frac {e^{-4+x} (-1+x) \log \left (\frac {1}{15+5 x}\right )}{x^2} \, dx-\int \frac {(-1+\log (x)) \log \left (\frac {1}{15+5 x}\right )}{x^2} \, dx\\ &=-\frac {1}{3} (1-\log (x)) \log (x)+\frac {1}{3} (1-\log (x)) \log (3+x)+\frac {e^{-4+x} \log \left (\frac {1}{15+5 x}\right )}{x}-\frac {(1-\log (x)) \log \left (\frac {1}{15+5 x}\right )}{x}-\int \frac {e^{-4+x}}{(-3-x) x} \, dx+\int \left (-\frac {5 e^{-4+x}}{x^2}+\frac {14 e^{-4+x}}{3 x}+\frac {e^{-4+x}}{3 (3+x)}\right ) \, dx+\int \left (\frac {5}{x^2}-\frac {3 (5+2 x) \log (x)}{x^2 (3+x)}\right ) \, dx+\int \left (-\frac {\log (x)}{3 x}+\frac {\log (3+x)}{3 x}-\frac {\log \left (\frac {1}{15+5 x}\right )}{x^2}\right ) \, dx\\ &=-\frac {5}{x}-\frac {1}{3} (1-\log (x)) \log (x)+\frac {1}{3} (1-\log (x)) \log (3+x)+\frac {e^{-4+x} \log \left (\frac {1}{15+5 x}\right )}{x}-\frac {(1-\log (x)) \log \left (\frac {1}{15+5 x}\right )}{x}+\frac {1}{3} \int \frac {e^{-4+x}}{3+x} \, dx-\frac {1}{3} \int \frac {\log (x)}{x} \, dx+\frac {1}{3} \int \frac {\log (3+x)}{x} \, dx-3 \int \frac {(5+2 x) \log (x)}{x^2 (3+x)} \, dx+\frac {14}{3} \int \frac {e^{-4+x}}{x} \, dx-5 \int \frac {e^{-4+x}}{x^2} \, dx-\int \left (-\frac {e^{-4+x}}{3 x}+\frac {e^{-4+x}}{3 (3+x)}\right ) \, dx-\int \frac {\log \left (\frac {1}{15+5 x}\right )}{x^2} \, dx\\ &=-\frac {5}{x}+\frac {5 e^{-4+x}}{x}+\frac {14 \text {Ei}(x)}{3 e^4}+\frac {\text {Ei}(3+x)}{3 e^7}+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} (1-\log (x)) \log (x)-\frac {\log ^2(x)}{6}+\frac {1}{3} (1-\log (x)) \log (3+x)+\frac {\log \left (\frac {1}{15+5 x}\right )}{x}+\frac {e^{-4+x} \log \left (\frac {1}{15+5 x}\right )}{x}-\frac {(1-\log (x)) \log \left (\frac {1}{15+5 x}\right )}{x}+\frac {1}{3} \int \frac {e^{-4+x}}{x} \, dx-\frac {1}{3} \int \frac {e^{-4+x}}{3+x} \, dx+\frac {1}{3} \int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx-3 \int \left (\frac {5 \log (x)}{3 x^2}+\frac {\log (x)}{9 x}-\frac {\log (x)}{9 (3+x)}\right ) \, dx-5 \int \frac {e^{-4+x}}{x} \, dx+5 \int \frac {1}{x (15+5 x)} \, dx\\ &=-\frac {5}{x}+\frac {5 e^{-4+x}}{x}+\frac {1}{3} \log (3) \log (x)-\frac {1}{3} (1-\log (x)) \log (x)-\frac {\log ^2(x)}{6}+\frac {1}{3} (1-\log (x)) \log (3+x)+\frac {\log \left (\frac {1}{15+5 x}\right )}{x}+\frac {e^{-4+x} \log \left (\frac {1}{15+5 x}\right )}{x}-\frac {(1-\log (x)) \log \left (\frac {1}{15+5 x}\right )}{x}-\frac {\text {Li}_2\left (-\frac {x}{3}\right )}{3}+\frac {1}{3} \int \frac {1}{x} \, dx-\frac {1}{3} \int \frac {\log (x)}{x} \, dx+\frac {1}{3} \int \frac {\log (x)}{3+x} \, dx-\frac {5}{3} \int \frac {1}{15+5 x} \, dx-5 \int \frac {\log (x)}{x^2} \, dx\\ &=\frac {5 e^{-4+x}}{x}+\frac {\log (x)}{3}+\frac {5 \log (x)}{x}+\frac {1}{3} \log (3) \log (x)+\frac {1}{3} \log \left (1+\frac {x}{3}\right ) \log (x)-\frac {1}{3} (1-\log (x)) \log (x)-\frac {\log ^2(x)}{3}-\frac {1}{3} \log (3+x)+\frac {1}{3} (1-\log (x)) \log (3+x)+\frac {\log \left (\frac {1}{15+5 x}\right )}{x}+\frac {e^{-4+x} \log \left (\frac {1}{15+5 x}\right )}{x}-\frac {(1-\log (x)) \log \left (\frac {1}{15+5 x}\right )}{x}-\frac {\text {Li}_2\left (-\frac {x}{3}\right )}{3}-\frac {1}{3} \int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx\\ &=\frac {5 e^{-4+x}}{x}+\frac {\log (x)}{3}+\frac {5 \log (x)}{x}+\frac {1}{3} \log (3) \log (x)+\frac {1}{3} \log \left (1+\frac {x}{3}\right ) \log (x)-\frac {1}{3} (1-\log (x)) \log (x)-\frac {\log ^2(x)}{3}-\frac {1}{3} \log (3+x)+\frac {1}{3} (1-\log (x)) \log (3+x)+\frac {\log \left (\frac {1}{15+5 x}\right )}{x}+\frac {e^{-4+x} \log \left (\frac {1}{15+5 x}\right )}{x}-\frac {(1-\log (x)) \log \left (\frac {1}{15+5 x}\right )}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.69, size = 27, normalized size = 1.12 \begin {gather*} \frac {\left (e^x+e^4 \log (x)\right ) \left (5+\log \left (\frac {1}{15+5 x}\right )\right )}{e^4 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 37, normalized size = 1.54 \begin {gather*} \frac {{\left (\log \left (\frac {1}{5 \, {\left (x + 3\right )}}\right ) + 5\right )} \log \relax (x) + e^{\left (x - 4\right )} \log \left (\frac {1}{5 \, {\left (x + 3\right )}}\right ) + 5 \, e^{\left (x - 4\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 38, normalized size = 1.58 \begin {gather*} -\frac {{\left (e^{4} \log \left (5 \, x + 15\right ) \log \relax (x) + e^{x} \log \left (5 \, x + 15\right ) - 5 \, e^{4} \log \relax (x) - 5 \, e^{x}\right )} e^{\left (-4\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 36, normalized size = 1.50
method | result | size |
risch | \(-\frac {\left (\ln \relax (x )+{\mathrm e}^{x -4}\right ) \ln \left (3+x \right )}{x}-\frac {\left (-10+2 \ln \relax (5)\right ) \left (\ln \relax (x )+{\mathrm e}^{x -4}\right )}{2 x}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 47, normalized size = 1.96 \begin {gather*} -\frac {{\left ({\left (\log \relax (5) - 5\right )} e^{4} \log \relax (x) + {\left (\log \relax (5) - 5\right )} e^{x} + {\left (e^{4} \log \relax (x) + e^{x}\right )} \log \left (x + 3\right ) - 5 \, e^{4}\right )} e^{\left (-4\right )}}{x} - \frac {5}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {5\,x+{\mathrm {e}}^{x-4}\,\left (5\,x^2+9\,x-15\right )-\ln \relax (x)\,\left (6\,x+\ln \left (\frac {1}{5\,x+15}\right )\,\left (x+3\right )+15\right )+\ln \left (\frac {1}{5\,x+15}\right )\,\left (x+{\mathrm {e}}^{x-4}\,\left (x^2+2\,x-3\right )+3\right )+15}{x^3+3\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.89, size = 36, normalized size = 1.50 \begin {gather*} \frac {\left (\log {\left (\frac {1}{5 x + 15} \right )} + 5\right ) e^{x - 4}}{x} + \frac {\log {\relax (x )} \log {\left (\frac {1}{5 x + 15} \right )}}{x} + \frac {5 \log {\relax (x )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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