Optimal. Leaf size=68 \[ -2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.141289, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {6020, 4182, 2531, 2282, 6589} \[ -2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{\tanh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-e^{\tanh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,e^{\tanh ^{-1}(a x)}\right )-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6020
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx &=\operatorname{Subst}\left (\int x^2 \text{csch}(x) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-2 \operatorname{Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+2 \operatorname{Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+2 \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-2 \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )-2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )\\ &=-2 \tanh ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x) \text{Li}_2\left (-e^{\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \text{Li}_2\left (e^{\tanh ^{-1}(a x)}\right )+2 \text{Li}_3\left (-e^{\tanh ^{-1}(a x)}\right )-2 \text{Li}_3\left (e^{\tanh ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.0838509, size = 100, normalized size = 1.47 \[ 2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-e^{-\tanh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,e^{-\tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x)^2 \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x)^2 \log \left (e^{-\tanh ^{-1}(a x)}+1\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.251, size = 158, normalized size = 2.3 \begin{align*} - \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) -2\,{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) +2\,{\it polylog} \left ( 3,-{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) + \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) +2\,{\it Artanh} \left ( ax \right ){\it polylog} \left ( 2,{\frac{ax+1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) -2\,{\it polylog} \left ( 3,{\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}\,{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^{3} - x}, 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 \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]