Optimal. Leaf size=148 \[ \frac {1}{2} \text {Li}_3\left (1-\frac {2}{a+b x+1}\right )-\frac {1}{2} \text {Li}_3\left (1-\frac {2 b x}{(1-a) (a+b x+1)}\right )+\text {Li}_2\left (1-\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)-\text {Li}_2\left (1-\frac {2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x)-\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)^2+\log \left (\frac {2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x)^2 \]
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Rubi [A] time = 0.09, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6112, 5923} \[ \frac {1}{2} \text {PolyLog}\left (3,1-\frac {2}{a+b x+1}\right )-\frac {1}{2} \text {PolyLog}\left (3,1-\frac {2 b x}{(1-a) (a+b x+1)}\right )+\coth ^{-1}(a+b x) \text {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )-\coth ^{-1}(a+b x) \text {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (a+b x+1)}\right )-\log \left (\frac {2}{a+b x+1}\right ) \coth ^{-1}(a+b x)^2+\log \left (\frac {2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x)^2 \]
Antiderivative was successfully verified.
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Rule 5923
Rule 6112
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a+b x)^2}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\coth ^{-1}(x)^2}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\coth ^{-1}(a+b x)^2 \log \left (\frac {2}{1+a+b x}\right )+\coth ^{-1}(a+b x)^2 \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )+\coth ^{-1}(a+b x) \text {Li}_2\left (1-\frac {2}{1+a+b x}\right )-\coth ^{-1}(a+b x) \text {Li}_2\left (1-\frac {2 b x}{(1-a) (1+a+b x)}\right )+\frac {1}{2} \text {Li}_3\left (1-\frac {2}{1+a+b x}\right )-\frac {1}{2} \text {Li}_3\left (1-\frac {2 b x}{(1-a) (1+a+b x)}\right )\\ \end {align*}
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Mathematica [C] time = 2.87, size = 547, normalized size = 3.70 \[ \frac {2}{3} \sqrt {1-\frac {1}{a^2}} a e^{\tanh ^{-1}\left (\frac {1}{a}\right )} \coth ^{-1}(a+b x)^3+2 \coth ^{-1}(a+b x) \text {Li}_2\left (-\sqrt {\frac {a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right )+2 \coth ^{-1}(a+b x) \text {Li}_2\left (\sqrt {\frac {a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right )-\coth ^{-1}(a+b x) \text {Li}_2\left (e^{2 \coth ^{-1}(a+b x)}\right )-2 \text {Li}_3\left (-\sqrt {\frac {a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right )-2 \text {Li}_3\left (\sqrt {\frac {a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right )+\frac {1}{2} \text {Li}_3\left (e^{2 \coth ^{-1}(a+b x)}\right )-\frac {2}{3} a \coth ^{-1}(a+b x)^3-\frac {2}{3} \coth ^{-1}(a+b x)^3+\coth ^{-1}(a+b x)^2 \log \left (1-\sqrt {\frac {a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right )+\coth ^{-1}(a+b x)^2 \log \left (\sqrt {\frac {a-1}{a+1}} e^{\coth ^{-1}(a+b x)}+1\right )-\coth ^{-1}(a+b x)^2 \log \left (1-e^{2 \coth ^{-1}(a+b x)}\right )+\coth ^{-1}(a+b x)^2 \log \left (\frac {1}{2} e^{-\coth ^{-1}(a+b x)} \left (a \left (e^{2 \coth ^{-1}(a+b x)}-1\right )-e^{2 \coth ^{-1}(a+b x)}-1\right )\right )-\log \left (-\frac {b x}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right ) \coth ^{-1}(a+b x)^2-i \pi \coth ^{-1}(a+b x) \log \left (\frac {1}{2} \left (e^{-\coth ^{-1}(a+b x)}+e^{\coth ^{-1}(a+b x)}\right )\right )+i \pi \log \left (\frac {1}{\sqrt {1-\frac {1}{(a+b x)^2}}}\right ) \coth ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac {1}{a}\right ) \coth ^{-1}(a+b x) \log \left (\frac {1}{2} i \left (e^{\coth ^{-1}(a+b x)-\tanh ^{-1}\left (\frac {1}{a}\right )}-e^{\tanh ^{-1}\left (\frac {1}{a}\right )-\coth ^{-1}(a+b x)}\right )\right )+2 \tanh ^{-1}\left (\frac {1}{a}\right ) \coth ^{-1}(a+b x) \log \left (i \sinh \left (\coth ^{-1}(a+b x)-\tanh ^{-1}\left (\frac {1}{a}\right )\right )\right )-\frac {i \pi ^3}{24} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.15, size = 985, normalized size = 6.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (b x + a\right )^{2}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {acoth}\left (a+b\,x\right )}^2}{x} \,d x \]
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 \[ \int \frac {\operatorname {acoth}^{2}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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