Optimal. Leaf size=144 \[ \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{1}{6} a x \sqrt{1-a^2 x^2}+\frac{1}{3} \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)+\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{7}{6} \sin ^{-1}(a x)-2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23155, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6014, 6010, 6018, 216, 5994, 195} \[ \text{PolyLog}\left (2,-\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right )-\frac{1}{6} a x \sqrt{1-a^2 x^2}+\frac{1}{3} \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)+\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)-\frac{7}{6} \sin ^{-1}(a x)-2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6014
Rule 6010
Rule 6018
Rule 216
Rule 5994
Rule 195
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{x} \, dx &=-\left (a^2 \int x \sqrt{1-a^2 x^2} \tanh ^{-1}(a x) \, dx\right )+\int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x} \, dx\\ &=\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{1}{3} \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)-\frac{1}{3} a \int \sqrt{1-a^2 x^2} \, dx-a \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx+\int \frac{\tanh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{1}{6} a x \sqrt{1-a^2 x^2}-\sin ^{-1}(a x)+\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{1}{3} \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)-2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+\text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\frac{1}{6} a \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{1}{6} a x \sqrt{1-a^2 x^2}-\frac{7}{6} \sin ^{-1}(a x)+\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+\frac{1}{3} \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)-2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )+\text{Li}_2\left (-\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )-\text{Li}_2\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right )\\ \end{align*}
Mathematica [A] time = 0.211454, size = 143, normalized size = 0.99 \[ \frac{1}{6} \left (6 \text{PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-6 \text{PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )-a x \sqrt{1-a^2 x^2}-2 a^2 x^2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+8 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)+6 \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right )-14 \tan ^{-1}\left (\tanh \left (\frac{1}{2} \tanh ^{-1}(a x)\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.2, size = 132, normalized size = 0.9 \begin{align*} -{\frac{2\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) +ax-8\,{\it Artanh} \left ( ax \right ) }{6}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}-{\frac{7}{3}\arctan \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\it dilog} \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) -{\it dilog} \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) -{\it Artanh} \left ( ax \right ) \ln \left ( 1+{(ax+1){\frac{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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \operatorname{artanh}\left (a x\right )}{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{{\left (a^{2} x^{2} - 1\right )} \sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )}{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{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \operatorname{atanh}{\left (a x \right )}}{x}\, 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 )}^{\frac{3}{2}} \operatorname{artanh}\left (a x\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]