Optimal. Leaf size=86 \[ -\frac {\text {Li}_2\left (1-\frac {2 (a+b x)}{(a+b) (x+1)}\right )}{2 b}+\frac {\tanh ^{-1}(x) \log \left (\frac {2 (a+b x)}{(x+1) (a+b)}\right )}{b}+\frac {\text {Li}_2\left (1-\frac {2}{x+1}\right )}{2 b}-\frac {\log \left (\frac {2}{x+1}\right ) \tanh ^{-1}(x)}{b} \]
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Rubi [A] time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5920, 2402, 2315, 2447} \[ -\frac {\text {PolyLog}\left (2,1-\frac {2 (a+b x)}{(x+1) (a+b)}\right )}{2 b}+\frac {\text {PolyLog}\left (2,1-\frac {2}{x+1}\right )}{2 b}+\frac {\tanh ^{-1}(x) \log \left (\frac {2 (a+b x)}{(x+1) (a+b)}\right )}{b}-\frac {\log \left (\frac {2}{x+1}\right ) \tanh ^{-1}(x)}{b} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 2447
Rule 5920
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(x)}{a+b x} \, dx &=-\frac {\tanh ^{-1}(x) \log \left (\frac {2}{1+x}\right )}{b}+\frac {\tanh ^{-1}(x) \log \left (\frac {2 (a+b x)}{(a+b) (1+x)}\right )}{b}+\frac {\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx}{b}-\frac {\int \frac {\log \left (\frac {2 (a+b x)}{(a+b) (1+x)}\right )}{1-x^2} \, dx}{b}\\ &=-\frac {\tanh ^{-1}(x) \log \left (\frac {2}{1+x}\right )}{b}+\frac {\tanh ^{-1}(x) \log \left (\frac {2 (a+b x)}{(a+b) (1+x)}\right )}{b}-\frac {\text {Li}_2\left (1-\frac {2 (a+b x)}{(a+b) (1+x)}\right )}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+x}\right )}{b}\\ &=-\frac {\tanh ^{-1}(x) \log \left (\frac {2}{1+x}\right )}{b}+\frac {\tanh ^{-1}(x) \log \left (\frac {2 (a+b x)}{(a+b) (1+x)}\right )}{b}+\frac {\text {Li}_2\left (1-\frac {2}{1+x}\right )}{2 b}-\frac {\text {Li}_2\left (1-\frac {2 (a+b x)}{(a+b) (1+x)}\right )}{2 b}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 260, normalized size = 3.02 \[ \frac {-4 \text {Li}_2\left (e^{-2 \left (\tanh ^{-1}\left (\frac {a}{b}\right )+\tanh ^{-1}(x)\right )}\right )+8 \tanh ^{-1}(x) \tanh ^{-1}\left (\frac {a}{b}\right )+8 \tanh ^{-1}\left (\frac {a}{b}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {a}{b}\right )+\tanh ^{-1}(x)\right )}\right )+8 \tanh ^{-1}(x) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {a}{b}\right )+\tanh ^{-1}(x)\right )}\right )-8 \tanh ^{-1}\left (\frac {a}{b}\right ) \log \left (2 i \sinh \left (\tanh ^{-1}\left (\frac {a}{b}\right )+\tanh ^{-1}(x)\right )\right )+8 \tanh ^{-1}(x) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {a}{b}\right )+\tanh ^{-1}(x)\right )\right )-8 \tanh ^{-1}(x) \log \left (2 i \sinh \left (\tanh ^{-1}\left (\frac {a}{b}\right )+\tanh ^{-1}(x)\right )\right )+4 \tanh ^{-1}\left (\frac {a}{b}\right )^2-4 \text {Li}_2\left (-e^{2 \tanh ^{-1}(x)}\right )+4 i \pi \log \left (\frac {2}{\sqrt {1-x^2}}\right )+8 \log \left (\frac {2}{\sqrt {1-x^2}}\right ) \tanh ^{-1}(x)+4 \log \left (1-x^2\right ) \tanh ^{-1}(x)+8 \tanh ^{-1}(x)^2+4 i \pi \tanh ^{-1}(x)-8 \tanh ^{-1}(x) \log \left (e^{2 \tanh ^{-1}(x)}+1\right )-4 i \pi \log \left (e^{2 \tanh ^{-1}(x)}+1\right )-\pi ^2}{8 b} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\relax (x)}{b x + a}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\relax (x)}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 110, normalized size = 1.28 \[ \frac {\ln \left (b x +a \right ) \arctanh \relax (x )}{b}-\frac {\ln \left (b x +a \right ) \ln \left (\frac {b x +b}{-a +b}\right )}{2 b}-\frac {\dilog \left (\frac {b x +b}{-a +b}\right )}{2 b}+\frac {\ln \left (b x +a \right ) \ln \left (\frac {b x -b}{-a -b}\right )}{2 b}+\frac {\dilog \left (\frac {b x -b}{-a -b}\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 119, normalized size = 1.38 \[ -\frac {{\left (\log \left (x + 1\right ) - \log \left (x - 1\right )\right )} \log \left (b x + a\right )}{2 \, b} + \frac {\operatorname {artanh}\relax (x) \log \left (b x + a\right )}{b} - \frac {\log \left (x - 1\right ) \log \left (\frac {b x - b}{a + b} + 1\right ) + {\rm Li}_2\left (-\frac {b x - b}{a + b}\right )}{2 \, b} + \frac {\log \left (x + 1\right ) \log \left (\frac {b x + b}{a - b} + 1\right ) + {\rm Li}_2\left (-\frac {b x + b}{a - b}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {atanh}\relax (x)}{a+b\,x} \,d x \]
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 \[ \int \frac {\operatorname {atanh}{\relax (x )}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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