Optimal. Leaf size=35 \[ \frac {\text {Li}_2\left (-\frac {1}{a+b x}\right )}{2 b}-\frac {\text {Li}_2\left (\frac {1}{a+b x}\right )}{2 b} \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6108, 5913} \[ \frac {\text {PolyLog}\left (2,-\frac {1}{a+b x}\right )}{2 b}-\frac {\text {PolyLog}\left (2,\frac {1}{a+b x}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 5913
Rule 6108
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a+b x)}{a+b x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\coth ^{-1}(x)}{x} \, dx,x,a+b x\right )}{b}\\ &=\frac {\text {Li}_2\left (-\frac {1}{a+b x}\right )}{2 b}-\frac {\text {Li}_2\left (\frac {1}{a+b x}\right )}{2 b}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 286, normalized size = 8.17 \[ -\frac {\text {Li}_2(-a-b x)}{2 b}+\frac {\text {Li}_2(a+b x)}{2 b}-\frac {\log ^2\left (\frac {a b-(a-1) b}{b (a+b x)}\right )}{4 b}+\frac {\log ^2\left (\frac {a b-(a+1) b}{b (a+b x)}\right )}{4 b}-\frac {\log \left (\frac {b (a+b x-1)}{(a-1) b-a b}\right ) \log \left (\frac {a b-(a-1) b}{b (a+b x)}\right )}{2 b}+\frac {\log \left (\frac {a+b x-1}{a+b x}\right ) \log \left (\frac {a b-(a-1) b}{b (a+b x)}\right )}{2 b}+\frac {\log \left (\frac {b (-a-b x-1)}{(-a-1) b+a b}\right ) \log \left (\frac {a b-(a+1) b}{b (a+b x)}\right )}{2 b}-\frac {\log \left (\frac {a b-(a+1) b}{b (a+b x)}\right ) \log \left (\frac {a+b x+1}{a+b x}\right )}{2 b} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (b x + a\right )}{b x + a}, x\right ) \]
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 \[ \int \frac {\operatorname {arcoth}\left (b x + a\right )}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 59, normalized size = 1.69 \[ \frac {\ln \left (b x +a \right ) \mathrm {arccoth}\left (b x +a \right )}{b}-\frac {\dilog \left (b x +a \right )}{2 b}-\frac {\dilog \left (b x +a +1\right )}{2 b}-\frac {\ln \left (b x +a \right ) \ln \left (b x +a +1\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 112, normalized size = 3.20 \[ -\frac {1}{2} \, b {\left (\frac {\log \left (b x + a\right ) \log \left (b x + a - 1\right ) + {\rm Li}_2\left (-b x - a + 1\right )}{b^{2}} - \frac {\log \left (b x + a + 1\right ) \log \left (-b x - a\right ) + {\rm Li}_2\left (b x + a + 1\right )}{b^{2}}\right )} - \frac {1}{2} \, {\left (\frac {\log \left (b x + a + 1\right )}{b} - \frac {\log \left (b x + a - 1\right )}{b}\right )} \log \left (b x + a\right ) + \frac {\operatorname {arcoth}\left (b x + a\right ) \log \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acoth}\left (a+b\,x\right )}{a+b\,x} \,d x \]
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 \[ \int \frac {\operatorname {acoth}{\left (a + b x \right )}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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