Optimal. Leaf size=70 \[ -\frac{a \sqrt{1-a^2 x^2}}{6 x^2}+\frac{1}{6} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0816485, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6008, 266, 47, 63, 208} \[ -\frac{a \sqrt{1-a^2 x^2}}{6 x^2}+\frac{1}{6} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6008
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}{x^4} \, dx &=-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}+\frac{1}{3} a \int \frac{\sqrt{1-a^2 x^2}}{x^3} \, dx\\ &=-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}+\frac{1}{6} a \operatorname{Subst}\left (\int \frac{\sqrt{1-a^2 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{a \sqrt{1-a^2 x^2}}{6 x^2}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-\frac{1}{12} a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{a \sqrt{1-a^2 x^2}}{6 x^2}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}+\frac{1}{6} a \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}}{6 x^2}-\frac{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}+\frac{1}{6} a^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0719546, size = 79, normalized size = 1.13 \[ -\frac{a x \sqrt{1-a^2 x^2}+a^3 x^3 \log (x)-a^3 x^3 \log \left (\sqrt{1-a^2 x^2}+1\right )+2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{6 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.236, size = 96, normalized size = 1.4 \begin{align*}{\frac{2\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) -ax-2\,{\it Artanh} \left ( ax \right ) }{6\,{x}^{3}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{{a}^{3}}{6}\ln \left ( 1+{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{{a}^{3}}{6}\ln \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.43899, size = 122, normalized size = 1.74 \begin{align*} \frac{1}{6} \,{\left (a^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \sqrt{-a^{2} x^{2} + 1} a^{2} - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{x^{2}}\right )} a - \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \operatorname{artanh}\left (a x\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.04219, size = 163, normalized size = 2.33 \begin{align*} -\frac{a^{3} x^{3} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt{-a^{2} x^{2} + 1}{\left (a x -{\left (a^{2} x^{2} - 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )\right )}}{6 \, x^{3}} \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{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname{atanh}{\left (a x \right )}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.27278, size = 263, normalized size = 3.76 \begin{align*} \frac{1}{12} \, a^{3} \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) - \frac{1}{12} \, a^{3} \log \left (-\sqrt{-a^{2} x^{2} + 1} + 1\right ) + \frac{1}{48} \,{\left (\frac{{\left (a^{4} - \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{x^{2}}\right )} a^{6} x^{3}}{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left | a \right |}} + \frac{\frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4}}{x} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{x^{3}}}{a^{2}{\left | a \right |}}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - \frac{\sqrt{-a^{2} x^{2} + 1} a}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]