Optimal. Leaf size=92 \[ \frac {1}{2} \text {Li}_2\left (1-\frac {2}{a+b x+1}\right )-\frac {1}{2} \text {Li}_2\left (1-\frac {2 b x}{(1-a) (a+b x+1)}\right )+\log \left (\frac {2}{a+b x+1}\right ) \left (-\coth ^{-1}(a+b x)\right )+\log \left (\frac {2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6112, 5921, 2402, 2315, 2447} \[ \frac {1}{2} \text {PolyLog}\left (2,1-\frac {2}{a+b x+1}\right )-\frac {1}{2} \text {PolyLog}\left (2,1-\frac {2 b x}{(1-a) (a+b x+1)}\right )+\log \left (\frac {2}{a+b x+1}\right ) \left (-\coth ^{-1}(a+b x)\right )+\log \left (\frac {2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2315
Rule 2402
Rule 2447
Rule 5921
Rule 6112
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a+b x)}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\coth ^{-1}(x)}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )+\coth ^{-1}(a+b x) \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )+\operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+x}\right )}{1-x^2} \, dx,x,a+b x\right )-\operatorname {Subst}\left (\int \frac {\log \left (\frac {2 \left (-\frac {a}{b}+\frac {x}{b}\right )}{\left (\frac {1}{b}-\frac {a}{b}\right ) (1+x)}\right )}{1-x^2} \, dx,x,a+b x\right )\\ &=-\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )+\coth ^{-1}(a+b x) \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )-\frac {1}{2} \text {Li}_2\left (1-\frac {2 b x}{(1-a) (1+a+b x)}\right )+\operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+a+b x}\right )\\ &=-\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )+\coth ^{-1}(a+b x) \log \left (\frac {2 b x}{(1-a) (1+a+b x)}\right )+\frac {1}{2} \text {Li}_2\left (1-\frac {2}{1+a+b x}\right )-\frac {1}{2} \text {Li}_2\left (1-\frac {2 b x}{(1-a) (1+a+b x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.17, size = 259, normalized size = 2.82 \[ \frac {1}{8} \left (-4 \text {Li}_2\left (e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right )-4 \text {Li}_2\left (-e^{2 \tanh ^{-1}(a+b x)}\right )+4 \left (\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right )^2-\left (\pi -2 i \tanh ^{-1}(a+b x)\right )^2-8 \left (\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right ) \log \left (1-e^{2 \tanh ^{-1}(a)-2 \tanh ^{-1}(a+b x)}\right )-4 i \left (\pi -2 i \tanh ^{-1}(a+b x)\right ) \log \left (e^{2 \tanh ^{-1}(a+b x)}+1\right )+4 \log \left (\frac {2}{\sqrt {1-(a+b x)^2}}\right ) \left (2 \tanh ^{-1}(a+b x)+i \pi \right )+8 \left (\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right ) \log \left (-2 i \sinh \left (\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right )\right )\right )+\tanh ^{-1}(a+b x) \left (-\log \left (\frac {1}{\sqrt {1-(a+b x)^2}}\right )+\log \left (-i \sinh \left (\tanh ^{-1}(a)-\tanh ^{-1}(a+b x)\right )\right )\right )+\log (x) \left (\coth ^{-1}(a+b x)-\tanh ^{-1}(a+b x)\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (b x + a\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (b x + a\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 81, normalized size = 0.88 \[ \ln \left (b x \right ) \mathrm {arccoth}\left (b x +a \right )-\frac {\dilog \left (\frac {b x +a +1}{1+a}\right )}{2}-\frac {\ln \left (b x \right ) \ln \left (\frac {b x +a +1}{1+a}\right )}{2}+\frac {\dilog \left (\frac {b x +a -1}{a -1}\right )}{2}+\frac {\ln \left (b x \right ) \ln \left (\frac {b x +a -1}{a -1}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 128, normalized size = 1.39 \[ -\frac {1}{2} \, b {\left (\frac {\log \left (b x + a + 1\right )}{b} - \frac {\log \left (b x + a - 1\right )}{b}\right )} \log \relax (x) + \frac {1}{2} \, b {\left (\frac {\log \left (b x + a + 1\right ) \log \left (-\frac {b x + a + 1}{a + 1} + 1\right ) + {\rm Li}_2\left (\frac {b x + a + 1}{a + 1}\right )}{b} - \frac {\log \left (b x + a - 1\right ) \log \left (-\frac {b x + a - 1}{a - 1} + 1\right ) + {\rm Li}_2\left (\frac {b x + a - 1}{a - 1}\right )}{b}\right )} + \operatorname {arcoth}\left (b x + a\right ) \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acoth}\left (a+b\,x\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acoth}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________