Optimal. Leaf size=156 \[ \frac{5}{3} a \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )-a \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac{1}{3} a^3 x^2 \tanh ^{-1}(a x)+\frac{a^2 x}{3}-2 a^2 x \tanh ^{-1}(a x)^2-\frac{2}{3} a \tanh ^{-1}(a x)^2-\frac{1}{3} a \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{10}{3} a \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)+2 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.418459, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {6012, 5910, 5984, 5918, 2402, 2315, 5916, 5988, 5932, 2447, 5980, 321, 206} \[ \frac{5}{3} a \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )-a \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )+\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac{1}{3} a^3 x^2 \tanh ^{-1}(a x)+\frac{a^2 x}{3}-2 a^2 x \tanh ^{-1}(a x)^2-\frac{2}{3} a \tanh ^{-1}(a x)^2-\frac{1}{3} a \tanh ^{-1}(a x)-\frac{\tanh ^{-1}(a x)^2}{x}+\frac{10}{3} a \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)+2 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6012
Rule 5910
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 5916
Rule 5988
Rule 5932
Rule 2447
Rule 5980
Rule 321
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{x^2} \, dx &=\int \left (-2 a^2 \tanh ^{-1}(a x)^2+\frac{\tanh ^{-1}(a x)^2}{x^2}+a^4 x^2 \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int \tanh ^{-1}(a x)^2 \, dx\right )+a^4 \int x^2 \tanh ^{-1}(a x)^2 \, dx+\int \frac{\tanh ^{-1}(a x)^2}{x^2} \, dx\\ &=-\frac{\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+(2 a) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx+\left (4 a^3\right ) \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac{1}{3} \left (2 a^5\right ) \int \frac{x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=-a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+(2 a) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\left (4 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx+\frac{1}{3} \left (2 a^3\right ) \int x \tanh ^{-1}(a x) \, dx-\frac{1}{3} \left (2 a^3\right ) \int \frac{x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac{1}{3} a^3 x^2 \tanh ^{-1}(a x)-\frac{2}{3} a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+4 a \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )-\frac{1}{3} \left (2 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{1-a x} \, dx-\left (2 a^2\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx-\left (4 a^2\right ) \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx-\frac{1}{3} a^4 \int \frac{x^2}{1-a^2 x^2} \, dx\\ &=\frac{a^2 x}{3}+\frac{1}{3} a^3 x^2 \tanh ^{-1}(a x)-\frac{2}{3} a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac{10}{3} a \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )-a \text{Li}_2\left (-1+\frac{2}{1+a x}\right )+(4 a) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )-\frac{1}{3} a^2 \int \frac{1}{1-a^2 x^2} \, dx+\frac{1}{3} \left (2 a^2\right ) \int \frac{\log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac{a^2 x}{3}-\frac{1}{3} a \tanh ^{-1}(a x)+\frac{1}{3} a^3 x^2 \tanh ^{-1}(a x)-\frac{2}{3} a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac{10}{3} a \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )+2 a \text{Li}_2\left (1-\frac{2}{1-a x}\right )-a \text{Li}_2\left (-1+\frac{2}{1+a x}\right )-\frac{1}{3} (2 a) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-a x}\right )\\ &=\frac{a^2 x}{3}-\frac{1}{3} a \tanh ^{-1}(a x)+\frac{1}{3} a^3 x^2 \tanh ^{-1}(a x)-\frac{2}{3} a \tanh ^{-1}(a x)^2-\frac{\tanh ^{-1}(a x)^2}{x}-2 a^2 x \tanh ^{-1}(a x)^2+\frac{1}{3} a^4 x^3 \tanh ^{-1}(a x)^2+\frac{10}{3} a \tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )+2 a \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )+\frac{5}{3} a \text{Li}_2\left (1-\frac{2}{1-a x}\right )-a \text{Li}_2\left (-1+\frac{2}{1+a x}\right )\\ \end{align*}
Mathematica [A] time = 0.056467, size = 182, normalized size = 1.17 \[ \frac{1}{3} a \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )-\left (1-a^2 x^2\right ) \left (a x \tanh ^{-1}(a x)+1\right ) \tanh ^{-1}(a x)+a x+a x \tanh ^{-1}(a x)^2-\tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x) \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )\right )-2 a \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+a \left (\tanh ^{-1}(a x) \left (-\frac{\tanh ^{-1}(a x)}{a x}+\tanh ^{-1}(a x)+2 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )-\text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )\right )-2 a \tanh ^{-1}(a x) \left (a x \tanh ^{-1}(a x)-\tanh ^{-1}(a x)-2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.056, size = 222, normalized size = 1.4 \begin{align*}{\frac{{a}^{4}{x}^{3} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{3}}-2\,{a}^{2}x \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{x}}+{\frac{{a}^{3}{x}^{2}{\it Artanh} \left ( ax \right ) }{3}}-{\frac{8\,a{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{3}}+2\,a{\it Artanh} \left ( ax \right ) \ln \left ( ax \right ) -{\frac{8\,a{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{3}}+{\frac{{a}^{2}x}{3}}+{\frac{a\ln \left ( ax-1 \right ) }{6}}-{\frac{a\ln \left ( ax+1 \right ) }{6}}-a{\it dilog} \left ( ax \right ) -a{\it dilog} \left ( ax+1 \right ) -a\ln \left ( ax \right ) \ln \left ( ax+1 \right ) -{\frac{2\,a \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{3}}+{\frac{8\,a}{3}{\it dilog} \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{4\,a\ln \left ( ax-1 \right ) }{3}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{4\,a}{3}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{4\,a\ln \left ( ax+1 \right ) }{3}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }+{\frac{2\,a \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.976506, size = 270, normalized size = 1.73 \begin{align*} \frac{1}{6} \, a^{2}{\left (\frac{2 \,{\left (a x + 2 \, \log \left (a x + 1\right )^{2} - 4 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 2 \, \log \left (a x - 1\right )^{2}\right )}}{a} + \frac{16 \,{\left (\log \left (a x - 1\right ) \log \left (\frac{1}{2} \, a x + \frac{1}{2}\right ) +{\rm Li}_2\left (-\frac{1}{2} \, a x + \frac{1}{2}\right )\right )}}{a} - \frac{6 \,{\left (\log \left (a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x\right )\right )}}{a} + \frac{6 \,{\left (\log \left (-a x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x\right )\right )}}{a} - \frac{\log \left (a x + 1\right )}{a} + \frac{\log \left (a x - 1\right )}{a}\right )} + \frac{1}{3} \,{\left (a^{2} x^{2} - 8 \, \log \left (a x + 1\right ) - 8 \, \log \left (a x - 1\right ) + 6 \, \log \left (x\right )\right )} a \operatorname{artanh}\left (a x\right ) + \frac{1}{3} \,{\left (a^{4} x^{3} - 6 \, a^{2} x - \frac{3}{x}\right )} \operatorname{artanh}\left (a x\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname{artanh}\left (a x\right )^{2}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname{atanh}^{2}{\left (a x \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname{artanh}\left (a x\right )^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]