Optimal. Leaf size=35 \[ \frac {\text {Li}_2\left (-\frac {1}{a+b x}\right )}{2 d}-\frac {\text {Li}_2\left (\frac {1}{a+b x}\right )}{2 d} \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6108, 12, 5913} \[ \frac {\text {PolyLog}\left (2,-\frac {1}{a+b x}\right )}{2 d}-\frac {\text {PolyLog}\left (2,\frac {1}{a+b x}\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 5913
Rule 6108
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b \coth ^{-1}(x)}{d x} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\coth ^{-1}(x)}{x} \, dx,x,a+b x\right )}{d}\\ &=\frac {\text {Li}_2\left (-\frac {1}{a+b x}\right )}{2 d}-\frac {\text {Li}_2\left (\frac {1}{a+b x}\right )}{2 d}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 312, normalized size = 8.91 \[ b \left (-\frac {\text {Li}_2(-a-b x)}{2 b d}+\frac {\text {Li}_2(a+b x)}{2 b d}-\frac {\log ^2\left (\frac {a b-(a-1) b}{b (a+b x)}\right )}{4 b d}+\frac {\log ^2\left (\frac {a b-(a+1) b}{b (a+b x)}\right )}{4 b d}-\frac {\log \left (\frac {b (a+b x-1)}{(a-1) b-a b}\right ) \log \left (\frac {a b-(a-1) b}{b (a+b x)}\right )}{2 b d}+\frac {\log \left (\frac {a+b x-1}{a+b x}\right ) \log \left (\frac {a b-(a-1) b}{b (a+b x)}\right )}{2 b d}+\frac {\log \left (\frac {b (-a-b x-1)}{(-a-1) b+a b}\right ) \log \left (\frac {a b-(a+1) b}{b (a+b x)}\right )}{2 b d}-\frac {\log \left (\frac {a b-(a+1) b}{b (a+b x)}\right ) \log \left (\frac {a+b x+1}{a+b x}\right )}{2 b d}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcoth}\left (b x + a\right )}{b d x + a d}, x\right ) \]
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 \[ \int \frac {\operatorname {arcoth}\left (b x + a\right )}{d x + \frac {a d}{b}}\,{d x} \]
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 \[ \frac {\ln \left (b x +a \right ) \mathrm {arccoth}\left (b x +a \right )}{d}-\frac {\dilog \left (b x +a \right )}{2 d}-\frac {\dilog \left (b x +a +1\right )}{2 d}-\frac {\ln \left (b x +a \right ) \ln \left (b x +a +1\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 132, normalized size = 3.77 \[ -\frac {1}{2} \, b {\left (\frac {\log \left (b x + a\right ) \log \left (b x + a - 1\right ) + {\rm Li}_2\left (-b x - a + 1\right )}{b d} - \frac {\log \left (b x + a + 1\right ) \log \left (-b x - a\right ) + {\rm Li}_2\left (b x + a + 1\right )}{b d}\right )} - \frac {b {\left (\frac {\log \left (b x + a + 1\right )}{b} - \frac {\log \left (b x + a - 1\right )}{b}\right )} \log \left (d x + \frac {a d}{b}\right )}{2 \, d} + \frac {\operatorname {arcoth}\left (b x + a\right ) \log \left (d x + \frac {a d}{b}\right )}{d} \]
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 \[ \int \frac {\mathrm {acoth}\left (a+b\,x\right )}{d\,x+\frac {a\,d}{b}} \,d x \]
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 \[ \frac {b \int \frac {\operatorname {acoth}{\left (a + b x \right )}}{a + b x}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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