Optimal. Leaf size=35 \[ -\log \left (b+\frac {a}{x}\right ) \log \left (-\frac {a}{b x}\right )-\text {Li}_2\left (1+\frac {a}{b x}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2511, 2504,
2441, 2352} \begin {gather*} -\text {PolyLog}\left (2,\frac {a}{b x}+1\right )-\log \left (\frac {a}{x}+b\right ) \log \left (-\frac {a}{b x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2441
Rule 2504
Rule 2511
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {a+b x}{x}\right )}{x} \, dx &=\int \frac {\log \left (b+\frac {a}{x}\right )}{x} \, dx\\ &=-\text {Subst}\left (\int \frac {\log (b+a x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\log \left (b+\frac {a}{x}\right ) \log \left (-\frac {a}{b x}\right )+a \text {Subst}\left (\int \frac {\log \left (-\frac {a x}{b}\right )}{b+a x} \, dx,x,\frac {1}{x}\right )\\ &=-\log \left (b+\frac {a}{x}\right ) \log \left (-\frac {a}{b x}\right )-\text {Li}_2\left (1+\frac {a}{b x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 36, normalized size = 1.03 \begin {gather*} -\log \left (b+\frac {a}{x}\right ) \log \left (-\frac {a}{b x}\right )-\text {Li}_2\left (\frac {b+\frac {a}{x}}{b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 34, normalized size = 0.97
method | result | size |
derivativedivides | \(-\dilog \left (-\frac {a}{b x}\right )-\ln \left (b +\frac {a}{x}\right ) \ln \left (-\frac {a}{b x}\right )\) | \(34\) |
default | \(-\dilog \left (-\frac {a}{b x}\right )-\ln \left (b +\frac {a}{x}\right ) \ln \left (-\frac {a}{b x}\right )\) | \(34\) |
risch | \(-\dilog \left (-\frac {a}{b x}\right )-\ln \left (b +\frac {a}{x}\right ) \ln \left (-\frac {a}{b x}\right )\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 67, normalized size = 1.91 \begin {gather*} -{\left (\log \left (b x + a\right ) - \log \left (x\right )\right )} \log \left (x\right ) + \log \left (b x + a\right ) \log \left (x\right ) - \log \left (\frac {b x}{a} + 1\right ) \log \left (x\right ) - \frac {1}{2} \, \log \left (x\right )^{2} + \log \left (x\right ) \log \left (\frac {b x + a}{x}\right ) - {\rm Li}_2\left (-\frac {b x}{a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (\frac {a}{x} + b \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 204 vs.
\(2 (34) = 68\).
time = 4.34, size = 204, normalized size = 5.83 \begin {gather*} \frac {a^{3} {\left (\frac {\log \left (\frac {{\left | b x + a \right |}}{{\left | x \right |}}\right )}{b^{2}} - \frac {\log \left ({\left | -b + \frac {b x + a}{x} \right |}\right )}{b^{2}} + \frac {1}{{\left (b - \frac {b x + a}{x}\right )} b}\right )} - \frac {a^{3} \log \left (-{\left (a - \frac {b}{\frac {{\left (a - \frac {b}{\frac {b}{a} - \frac {b x + a}{a x}}\right )} {\left (\frac {b}{a} - \frac {b x + a}{a x}\right )}}{a} + \frac {b}{a}}\right )} {\left (\frac {{\left (a - \frac {b}{\frac {b}{a} - \frac {b x + a}{a x}}\right )} {\left (\frac {b}{a} - \frac {b x + a}{a x}\right )}}{a} + \frac {b}{a}\right )}\right )}{{\left (b - \frac {b x + a}{x}\right )}^{2}}}{2 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.41, size = 37, normalized size = 1.06 \begin {gather*} -\mathrm {polylog}\left (2,\frac {a}{b\,x}+1\right )-\ln \left (\frac {a+b\,x}{x}\right )\,\ln \left (-\frac {a}{b\,x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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