Optimal. Leaf size=120 \[ -\frac {\text {Li}_2\left (1-\frac {2 b (c+d x)}{(b c-a d+d) (a+b x+1)}\right )}{2 d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(a+b x+1) (-a d+b c+d)}\right )}{d}+\frac {\text {Li}_2\left (1-\frac {2}{a+b x+1}\right )}{2 d}-\frac {\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)}{d} \]
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Rubi [A] time = 0.13, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6112, 5921, 2402, 2315, 2447} \[ -\frac {\text {PolyLog}\left (2,1-\frac {2 b (c+d x)}{(a+b x+1) (-a d+b c+d)}\right )}{2 d}+\frac {\text {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )}{2 d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(a+b x+1) (-a d+b c+d)}\right )}{d}-\frac {\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)}{d} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 2447
Rule 5921
Rule 6112
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a+b x)}{c+d x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\coth ^{-1}(x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,a+b x\right )}{d}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (\frac {2 \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{\left (\frac {d}{b}+\frac {b c-a d}{b}\right ) (1+x)}\right )}{1-x^2} \, dx,x,a+b x\right )}{d}\\ &=-\frac {\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}-\frac {\text {Li}_2\left (1-\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{2 d}+\frac {\operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+a+b x}\right )}{d}\\ &=-\frac {\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{d}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{d}+\frac {\text {Li}_2\left (1-\frac {2}{1+a+b x}\right )}{2 d}-\frac {\text {Li}_2\left (1-\frac {2 b (c+d x)}{(b c+d-a d) (1+a+b x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 185, normalized size = 1.54 \[ -\frac {\text {Li}_2\left (\frac {b (c+d x)}{b c-a d-d}\right )}{2 d}+\frac {\text {Li}_2\left (\frac {b (c+d x)}{b c-a d+d}\right )}{2 d}+\frac {\log (c+d x) \log \left (\frac {d (-a-b x+1)}{-a d+b c+d}\right )}{2 d}-\frac {\log \left (\frac {a+b x-1}{a+b x}\right ) \log (c+d x)}{2 d}-\frac {\log (c+d x) \log \left (-\frac {d (a+b x+1)}{-a d+b c-d}\right )}{2 d}+\frac {\log \left (\frac {a+b x+1}{a+b x}\right ) \log (c+d x)}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (b x + a\right )}{d x + c}, x\right ) \]
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 \[ \int \frac {\operatorname {arcoth}\left (b x + a\right )}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 176, normalized size = 1.47 \[ \frac {\ln \left (d \left (b x +a \right )-a d +b c \right ) \mathrm {arccoth}\left (b x +a \right )}{d}+\frac {\ln \left (d \left (b x +a \right )-a d +b c \right ) \ln \left (\frac {d \left (b x +a \right )-d}{a d -b c -d}\right )}{2 d}+\frac {\dilog \left (\frac {d \left (b x +a \right )-d}{a d -b c -d}\right )}{2 d}-\frac {\ln \left (\frac {d \left (b x +a \right )+d}{a d -b c +d}\right ) \ln \left (d \left (b x +a \right )-a d +b c \right )}{2 d}-\frac {\dilog \left (\frac {d \left (b x +a \right )+d}{a d -b c +d}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 192, normalized size = 1.60 \[ -\frac {1}{2} \, b {\left (\frac {\log \left (b x + a - 1\right ) \log \left (\frac {b d x + a d - d}{b c - a d + d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d - d}{b c - a d + d}\right )}{b d} - \frac {\log \left (b x + a + 1\right ) \log \left (\frac {b d x + a d + d}{b c - a d - d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d + d}{b c - a d - d}\right )}{b d}\right )} - \frac {b {\left (\frac {\log \left (b x + a + 1\right )}{b} - \frac {\log \left (b x + a - 1\right )}{b}\right )} \log \left (d x + c\right )}{2 \, d} + \frac {\operatorname {arcoth}\left (b x + a\right ) \log \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acoth}\left (a+b\,x\right )}{c+d\,x} \,d x \]
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 \[ \int \frac {\operatorname {acoth}{\left (a + b x \right )}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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