Optimal. Leaf size=13 \[ x+\log \left (\log \left (16 \left (-\frac {3}{4}-2 x\right )\right )\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 11, normalized size of antiderivative = 0.85, number of steps
used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6820, 2437, 12,
2339, 29} \begin {gather*} x+\log (\log (-4 (8 x+3))) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 2339
Rule 2437
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {8}{(3+8 x) \log (-12-32 x)}\right ) \, dx\\ &=x+8 \int \frac {1}{(3+8 x) \log (-12-32 x)} \, dx\\ &=x-\frac {1}{4} \text {Subst}\left (\int -\frac {4}{x \log (x)} \, dx,x,-12-32 x\right )\\ &=x+\text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,-12-32 x\right )\\ &=x+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (-4 (3+8 x))\right )\\ &=x+\log (\log (-4 (3+8 x)))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 11, normalized size = 0.85 \begin {gather*} x+\log (\log (-4 (3+8 x))) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 11, normalized size = 0.85
method | result | size |
norman | \(\ln \left (\ln \left (-32 x -12\right )\right )+x\) | \(10\) |
risch | \(\ln \left (\ln \left (-32 x -12\right )\right )+x\) | \(10\) |
derivativedivides | \(x +\frac {3}{8}+\ln \left (\ln \left (-32 x -12\right )\right )\) | \(11\) |
default | \(x +\frac {3}{8}+\ln \left (\ln \left (-32 x -12\right )\right )\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.52, size = 71, normalized size = 5.46 \begin {gather*} -\frac {3}{8} \, {\left (i \, \pi + 2 \, \log \left (2\right ) + \log \left (8 \, x + 3\right )\right )} \log \left (i \, \pi + 2 \, \log \left (2\right ) + \log \left (8 \, x + 3\right )\right ) + \frac {3}{8} \, \log \left (i \, \pi + 2 \, \log \left (2\right ) + \log \left (8 \, x + 3\right )\right ) \log \left (-32 \, x - 12\right ) + x + \log \left (i \, \pi + 2 \, \log \left (2\right ) + \log \left (8 \, x + 3\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 9, normalized size = 0.69 \begin {gather*} x + \log \left (\log \left (-32 \, x - 12\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 10, normalized size = 0.77 \begin {gather*} x + \log {\left (\log {\left (- 32 x - 12 \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 9, normalized size = 0.69 \begin {gather*} x + \log \left (\log \left (-32 \, x - 12\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 9, normalized size = 0.69 \begin {gather*} x+\ln \left (\ln \left (-32\,x-12\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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