Optimal. Leaf size=56 \[ \frac {1}{2} \text {Li}_3\left (1+\frac {2}{b x}\right )-\text {Li}_2\left (1+\frac {2}{b x}\right ) \tanh ^{-1}(b x+1)-\log \left (-\frac {2}{b x}\right ) \tanh ^{-1}(b x+1)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6111, 5918, 5948, 6058, 6610} \[ \frac {1}{2} \text {PolyLog}\left (3,\frac {2}{b x}+1\right )-\tanh ^{-1}(b x+1) \text {PolyLog}\left (2,\frac {2}{b x}+1\right )-\log \left (-\frac {2}{b x}\right ) \tanh ^{-1}(b x+1)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5918
Rule 5948
Rule 6058
Rule 6111
Rule 6610
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(1+b x)^2}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\tanh ^{-1}(x)^2}{-\frac {1}{b}+\frac {x}{b}} \, dx,x,1+b x\right )}{b}\\ &=-\tanh ^{-1}(1+b x)^2 \log \left (-\frac {2}{b x}\right )+2 \operatorname {Subst}\left (\int \frac {\tanh ^{-1}(x) \log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,1+b x\right )\\ &=-\tanh ^{-1}(1+b x)^2 \log \left (-\frac {2}{b x}\right )-\tanh ^{-1}(1+b x) \text {Li}_2\left (1+\frac {2}{b x}\right )+\operatorname {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1-x}\right )}{1-x^2} \, dx,x,1+b x\right )\\ &=-\tanh ^{-1}(1+b x)^2 \log \left (-\frac {2}{b x}\right )-\tanh ^{-1}(1+b x) \text {Li}_2\left (1+\frac {2}{b x}\right )+\frac {1}{2} \text {Li}_3\left (1+\frac {2}{b x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 75, normalized size = 1.34 \[ \tanh ^{-1}(b x+1) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(b x+1)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{-2 \tanh ^{-1}(b x+1)}\right )-\frac {2}{3} \tanh ^{-1}(b x+1)^3-\tanh ^{-1}(b x+1)^2 \log \left (e^{-2 \tanh ^{-1}(b x+1)}+1\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (b x + 1\right )^{2}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (b x + 1\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.45, size = 160, normalized size = 2.86 \[ \ln \left (b x \right ) \arctanh \left (b x +1\right )^{2}-\arctanh \left (b x +1\right ) \polylog \left (2, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )+\frac {\polylog \left (3, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )}{2}-\left (i \pi \mathrm {csgn}\left (\frac {i}{1+\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i}{1+\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}}\right )^{2}+i \pi +\ln \relax (2)\right ) \arctanh \left (b x +1\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{12} \, \log \left (-b x\right )^{3} + \frac {1}{4} \, \log \left (b x + 2\right )^{2} \log \left (-x\right ) - \frac {1}{4} \, \int \frac {2 \, {\left (b x \log \relax (b) + 2 \, {\left (b x + 1\right )} \log \left (-x\right ) + 2 \, \log \relax (b)\right )} \log \left (b x + 2\right )}{b x^{2} + 2 \, x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {atanh}\left (b\,x+1\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}^{2}{\left (b x + 1 \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________