Optimal. Leaf size=63 \[ \frac{a \sqrt{1-a^2 x^2}}{x}-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{1}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0604718, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6124, 835, 807, 266, 63, 208} \[ \frac{a \sqrt{1-a^2 x^2}}{x}-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{1}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6124
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac{1-a x}{x^3 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{2 x^2}-\frac{1}{2} \int \frac{2 a-a^2 x}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{2 x^2}+\frac{a \sqrt{1-a^2 x^2}}{x}+\frac{1}{2} a^2 \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{2 x^2}+\frac{a \sqrt{1-a^2 x^2}}{x}+\frac{1}{4} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1-a^2 x^2}}{2 x^2}+\frac{a \sqrt{1-a^2 x^2}}{x}-\frac{1}{2} \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{\sqrt{1-a^2 x^2}}{2 x^2}+\frac{a \sqrt{1-a^2 x^2}}{x}-\frac{1}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0419927, size = 57, normalized size = 0.9 \[ \frac{1}{2} \left (\frac{(2 a x-1) \sqrt{1-a^2 x^2}}{x^2}-a^2 \log \left (\sqrt{1-a^2 x^2}+1\right )+a^2 \log (x)\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.043, size = 186, normalized size = 3. \begin{align*}{\frac{a}{x} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{a}^{3}x\sqrt{-{a}^{2}{x}^{2}+1}+{{a}^{3}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{{a}^{2}}{2}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{a}^{2}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }-{{a}^{3}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{2\,{x}^{2}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \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{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.93545, size = 113, normalized size = 1.79 \begin{align*} \frac{a^{2} x^{2} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x - 1\right )}}{2 \, x^{2}} \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 )}}{x^{3} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.21876, size = 215, normalized size = 3.41 \begin{align*} \frac{{\left (a^{3} - \frac{4 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a}{x}\right )} a^{4} x^{2}}{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left | a \right |}} - \frac{a^{3} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} + \frac{\frac{4 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a{\left | a \right |}}{x} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]