Optimal. Leaf size=39 \[ -\frac{\sqrt{9-4 x^2}}{18 x^2}-\frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0159081, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 206} \[ -\frac{\sqrt{9-4 x^2}}{18 x^2}-\frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{9-4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{9-4 x} x^2} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{9-4 x^2}}{18 x^2}+\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt{9-4 x} x} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{9-4 x^2}}{18 x^2}-\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{\frac{9}{4}-\frac{x^2}{4}} \, dx,x,\sqrt{9-4 x^2}\right )\\ &=-\frac{\sqrt{9-4 x^2}}{18 x^2}-\frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0097624, size = 37, normalized size = 0.95 \[ \frac{1}{54} \left (-\frac{3 \sqrt{9-4 x^2}}{x^2}-4 \tanh ^{-1}\left (\sqrt{1-\frac{4 x^2}{9}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 30, normalized size = 0.8 \begin{align*} -{\frac{1}{18\,{x}^{2}}\sqrt{-4\,{x}^{2}+9}}-{\frac{2}{27}{\it Artanh} \left ( 3\,{\frac{1}{\sqrt{-4\,{x}^{2}+9}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 3.38938, size = 54, normalized size = 1.38 \begin{align*} -\frac{\sqrt{-4 \, x^{2} + 9}}{18 \, x^{2}} - \frac{2}{27} \, \log \left (\frac{6 \, \sqrt{-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.27832, size = 93, normalized size = 2.38 \begin{align*} \frac{4 \, x^{2} \log \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{x}\right ) - 3 \, \sqrt{-4 \, x^{2} + 9}}{54 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.24583, size = 99, normalized size = 2.54 \begin{align*} \begin{cases} - \frac{2 \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{27} + \frac{1}{9 x \sqrt{-1 + \frac{9}{4 x^{2}}}} - \frac{1}{4 x^{3} \sqrt{-1 + \frac{9}{4 x^{2}}}} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\\frac{2 i \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{27} - \frac{i}{9 x \sqrt{1 - \frac{9}{4 x^{2}}}} + \frac{i}{4 x^{3} \sqrt{1 - \frac{9}{4 x^{2}}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.71776, size = 61, normalized size = 1.56 \begin{align*} -\frac{\sqrt{-4 \, x^{2} + 9}}{18 \, x^{2}} - \frac{1}{27} \, \log \left (\sqrt{-4 \, x^{2} + 9} + 3\right ) + \frac{1}{27} \, \log \left (-\sqrt{-4 \, x^{2} + 9} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]