Optimal. Leaf size=24 \[ \frac {\text {Li}_2\left (1-\frac {2 a}{a+b x}\right )}{2 a b} \]
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Rubi [A]
time = 0.09, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2458, 2378,
2370, 2352} \begin {gather*} \frac {\text {PolyLog}\left (2,1-\frac {2 a}{a+b x}\right )}{2 a b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2370
Rule 2378
Rule 2458
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {2 a}{a+b x}\right )}{(a-b x) (a+b x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\log \left (\frac {2 a}{x}\right )}{(2 a-x) x} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\text {Subst}\left (\int \frac {\log (2 a x)}{\left (2 a-\frac {1}{x}\right ) x} \, dx,x,\frac {1}{a+b x}\right )}{b}\\ &=-\frac {\text {Subst}\left (\int \frac {\log (2 a x)}{-1+2 a x} \, dx,x,\frac {1}{a+b x}\right )}{b}\\ &=\frac {\text {Li}_2\left (1-\frac {2 a}{a+b x}\right )}{2 a b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.12 \begin {gather*} \frac {\text {Li}_2\left (\frac {-a+b x}{a+b x}\right )}{2 a b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.83, size = 20, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\dilog \left (\frac {2 a}{b x +a}\right )}{2 b a}\) | \(20\) |
default | \(\frac {\dilog \left (\frac {2 a}{b x +a}\right )}{2 b a}\) | \(20\) |
risch | \(\frac {\dilog \left (\frac {2 a}{b x +a}\right )}{2 b a}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs.
\(2 (21) = 42\).
time = 0.29, size = 120, normalized size = 5.00 \begin {gather*} \frac {1}{4} \, b {\left (\frac {\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (b x - a\right )}{a b^{2}} + \frac {2 \, {\left (\log \left (b x + a\right ) \log \left (-\frac {b x + a}{2 \, a} + 1\right ) + {\rm Li}_2\left (\frac {b x + a}{2 \, a}\right )\right )}}{a b^{2}}\right )} + \frac {1}{2} \, {\left (\frac {\log \left (b x + a\right )}{a b} - \frac {\log \left (b x - a\right )}{a b}\right )} \log \left (\frac {2 \, a}{b x + a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 21, normalized size = 0.88 \begin {gather*} \frac {{\rm Li}_2\left (-\frac {2 \, a}{b x + a} + 1\right )}{2 \, a b} \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 (2 \right )}}{- a^{2} + b^{2} x^{2}}\, dx - \int \frac {\log {\left (\frac {a}{a + b x} \right )}}{- a^{2} + b^{2} x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [B]
time = 0.20, size = 19, normalized size = 0.79 \begin {gather*} \frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {2\,a}{a+b\,x}\right )}{2\,a\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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