Optimal. Leaf size=152 \[ -a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+a^2 \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-a^2 \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \left (-\tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )\right ) \tanh ^{-1}(a x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.415348, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6026, 6008, 266, 63, 208, 6020, 4182, 2531, 2282, 6589} \[ -a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+a^2 \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-a^2 \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+a^2 \left (-\tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )\right ) \tanh ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6026
Rule 6008
Rule 266
Rule 63
Rule 208
Rule 6020
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^2}{x^3 \sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+a \int \frac{\tanh ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx+\frac{1}{2} a^2 \int \frac{\tanh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int x^2 \text{csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )+a^2 \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )-a^2 \operatorname{Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+a^2 \operatorname{Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+a^2 \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+a^2 \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-a^2 \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+a^2 \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+a^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )-a^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )\\ &=-\frac{a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+a^2 \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+a^2 \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-a^2 \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 1.19724, size = 188, normalized size = 1.24 \[ \frac{1}{8} a^2 \left (8 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-8 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )+8 \text{PolyLog}\left (3,-e^{-\tanh ^{-1}(a x)}\right )-8 \text{PolyLog}\left (3,e^{-\tanh ^{-1}(a x)}\right )+4 \tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right ) \tanh ^{-1}(a x)+4 \tanh ^{-1}(a x)^2 \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-4 \tanh ^{-1}(a x)^2 \log \left (e^{-\tanh ^{-1}(a x)}+1\right )+8 \log \left (\tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )-4 \tanh ^{-1}(a x) \coth \left (\frac{1}{2} \tanh ^{-1}(a x)\right )+\tanh ^{-1}(a x)^2 \left (-\text{csch}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )-\tanh ^{-1}(a x)^2 \text{sech}^2\left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.283, size = 231, normalized size = 1.5 \begin{align*} -{\frac{{\it Artanh} \left ( ax \right ) \left ( 2\,ax+{\it Artanh} \left ( ax \right ) \right ) }{2\,{x}^{2}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2}\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{a}^{2}{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +{a}^{2}{\it polylog} \left ( 3,-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +{\frac{{a}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{2}\ln \left ( 1-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{a}^{2}{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) -{a}^{2}{\it polylog} \left ( 3,{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) -2\,{a}^{2}{\it Artanh} \left ({\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1} x^{3}}\,{d x} \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{\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )^{2}}{a^{2} x^{5} - x^{3}}, 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{\operatorname{atanh}^{2}{\left (a x \right )}}{x^{3} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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{\operatorname{artanh}\left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]