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 = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 51
Rule 63
Rule 206
Rule 266
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 = 0.01, 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]
________________________________________________________________________________________
fricas [A] time = 0.91, size = 38, normalized size = 0.97 \[ \frac {4 \, x^{2} \log \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{x}\right ) - 3 \, \sqrt {-4 \, x^{2} + 9}}{54 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.21, size = 45, normalized size = 1.15 \[ -\frac {\sqrt {-4 \, x^{2} + 9}}{18 \, x^{2}} - \frac {1}{27} \, \log \left (\sqrt {-4 \, x^{2} + 9} + 3\right ) + \frac {1}{27} \, \log \left (-\sqrt {-4 \, x^{2} + 9} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 30, normalized size = 0.77 \[ -\frac {2 \arctanh \left (\frac {3}{\sqrt {-4 x^{2}+9}}\right )}{27}-\frac {\sqrt {-4 x^{2}+9}}{18 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.93, size = 40, normalized size = 1.03 \[ -\frac {\sqrt {-4 \, x^{2} + 9}}{18 \, x^{2}} - \frac {2}{27} \, \log \left (\frac {6 \, \sqrt {-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 35, normalized size = 0.90 \[ \frac {2\,\ln \left (\sqrt {\frac {9}{4\,x^2}-1}-\frac {3\,\sqrt {\frac {1}{x^2}}}{2}\right )}{27}-\frac {\sqrt {\frac {9}{4}-x^2}}{9\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.17, size = 99, normalized size = 2.54 \[ \begin {cases} - \frac {2 \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{27} + \frac {1}{9 x \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {1}{4 x^{3} \sqrt {-1 + \frac {9}{4 x^{2}}}} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\\frac {2 i \operatorname {asin}{\left (\frac {3}{2 x} \right )}}{27} - \frac {i}{9 x \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {i}{4 x^{3} \sqrt {1 - \frac {9}{4 x^{2}}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________