Optimal. Leaf size=17 \[ (5+x) \left (1+\frac {\log \left (\log \left (-\frac {3 x}{4}\right )\right )}{x}\right ) \]
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Rubi [A]
time = 0.48, antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps
used = 13, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {6874, 2395,
2346, 2209, 2339, 29, 2602} \begin {gather*} x+\log \left (\log \left (-\frac {3 x}{4}\right )\right )+\frac {5 \log \left (\log \left (-\frac {3 x}{4}\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2209
Rule 2339
Rule 2346
Rule 2395
Rule 2602
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {5+x+x^2 \log \left (-\frac {3 x}{4}\right )}{x^2 \log \left (-\frac {3 x}{4}\right )}-\frac {5 \log \left (\log \left (-\frac {3 x}{4}\right )\right )}{x^2}\right ) \, dx\\ &=-\left (5 \int \frac {\log \left (\log \left (-\frac {3 x}{4}\right )\right )}{x^2} \, dx\right )+\int \frac {5+x+x^2 \log \left (-\frac {3 x}{4}\right )}{x^2 \log \left (-\frac {3 x}{4}\right )} \, dx\\ &=\frac {5 \log \left (\log \left (-\frac {3 x}{4}\right )\right )}{x}-5 \int \frac {1}{x^2 \log \left (-\frac {3 x}{4}\right )} \, dx+\int \left (1+\frac {5+x}{x^2 \log \left (-\frac {3 x}{4}\right )}\right ) \, dx\\ &=x+\frac {5 \log \left (\log \left (-\frac {3 x}{4}\right )\right )}{x}+\frac {15}{4} \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (-\frac {3 x}{4}\right )\right )+\int \frac {5+x}{x^2 \log \left (-\frac {3 x}{4}\right )} \, dx\\ &=x+\frac {15}{4} \text {Ei}\left (-\log \left (-\frac {3 x}{4}\right )\right )+\frac {5 \log \left (\log \left (-\frac {3 x}{4}\right )\right )}{x}+\int \left (\frac {5}{x^2 \log \left (-\frac {3 x}{4}\right )}+\frac {1}{x \log \left (-\frac {3 x}{4}\right )}\right ) \, dx\\ &=x+\frac {15}{4} \text {Ei}\left (-\log \left (-\frac {3 x}{4}\right )\right )+\frac {5 \log \left (\log \left (-\frac {3 x}{4}\right )\right )}{x}+5 \int \frac {1}{x^2 \log \left (-\frac {3 x}{4}\right )} \, dx+\int \frac {1}{x \log \left (-\frac {3 x}{4}\right )} \, dx\\ &=x+\frac {15}{4} \text {Ei}\left (-\log \left (-\frac {3 x}{4}\right )\right )+\frac {5 \log \left (\log \left (-\frac {3 x}{4}\right )\right )}{x}-\frac {15}{4} \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (-\frac {3 x}{4}\right )\right )+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (-\frac {3 x}{4}\right )\right )\\ &=x+\log \left (\log \left (-\frac {3 x}{4}\right )\right )+\frac {5 \log \left (\log \left (-\frac {3 x}{4}\right )\right )}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.20, size = 21, normalized size = 1.24 \begin {gather*} x+\log \left (\log \left (-\frac {3 x}{4}\right )\right )+\frac {5 \log \left (\log \left (-\frac {3 x}{4}\right )\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 2.31, size = 70, normalized size = 4.12
method | result | size |
risch | \(\frac {5 \ln \left (\ln \left (-\frac {3 x}{4}\right )\right )}{x}+x +\ln \left (\ln \left (-\frac {3 x}{4}\right )\right )\) | \(18\) |
norman | \(\frac {x^{2}+x \ln \left (\ln \left (-\frac {3 x}{4}\right )\right )+5 \ln \left (\ln \left (-\frac {3 x}{4}\right )\right )}{x}\) | \(23\) |
default | \(x +\ln \left (\ln \left (3\right )-2 \ln \left (2\right )+\ln \left (-x \right )\right )+5 \,{\mathrm e}^{\ln \left (3\right )-2 \ln \left (2\right )} \expIntegral \left (1, \ln \left (3\right )-2 \ln \left (2\right )+\ln \left (-x \right )\right )+\frac {5 \ln \left (\ln \left (3\right )-2 \ln \left (2\right )+\ln \left (-x \right )\right )}{x}-\frac {15 \expIntegral \left (1, \ln \left (3\right )-2 \ln \left (2\right )+\ln \left (-x \right )\right )}{4}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 17, normalized size = 1.00 \begin {gather*} x + \frac {5 \, \log \left (\log \left (-\frac {3}{4} \, x\right )\right )}{x} + \log \left (\log \left (-\frac {3}{4} \, x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 17, normalized size = 1.00 \begin {gather*} \frac {x^{2} + {\left (x + 5\right )} \log \left (\log \left (-\frac {3}{4} \, x\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 24, normalized size = 1.41 \begin {gather*} x + \log {\left (\log {\left (- \frac {3 x}{4} \right )} \right )} + \frac {5 \log {\left (\log {\left (- \frac {3 x}{4} \right )} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 22, normalized size = 1.29 \begin {gather*} x + \frac {5 \, \log \left (\log \left (-\frac {3}{4} \, x\right )\right )}{x} + \log \left (-2 \, \log \left (2\right ) + \log \left (-3 \, x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.44, size = 17, normalized size = 1.00 \begin {gather*} x+\ln \left (\ln \left (-\frac {3\,x}{4}\right )\right )+\frac {5\,\ln \left (\ln \left (-\frac {3\,x}{4}\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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