Optimal. Leaf size=39 \[ \frac {\sqrt {4 x^2-9}}{18 x^2}+\frac {2}{27} \tan ^{-1}\left (\frac {1}{3} \sqrt {4 x^2-9}\right ) \]
[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, 203} \[ \frac {\sqrt {4 x^2-9}}{18 x^2}+\frac {2}{27} \tan ^{-1}\left (\frac {1}{3} \sqrt {4 x^2-9}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 203
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}{x^2 \sqrt {-9+4 x}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {-9+4 x^2}}{18 x^2}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-9+4 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} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9+4 x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 54, normalized size = 1.38 \[ \frac {4}{81} \sqrt {4 x^2-9} \left (\frac {9}{8 x^2}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {4 x^2}{9}}\right )}{2 \sqrt {1-\frac {4 x^2}{9}}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 38, normalized size = 0.97 \[ \frac {8 \, x^{2} \arctan \left (-\frac {2}{3} \, x + \frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) + 3 \, \sqrt {4 \, x^{2} - 9}}{54 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.10, size = 29, normalized size = 0.74 \[ \frac {\sqrt {4 \, x^{2} - 9}}{18 \, x^{2}} + \frac {2}{27} \, \arctan \left (\frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 30, normalized size = 0.77 \[ -\frac {2 \arctan \left (\frac {3}{\sqrt {4 x^{2}-9}}\right )}{27}+\frac {\sqrt {4 x^{2}-9}}{18 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.93, size = 24, normalized size = 0.62 \[ \frac {\sqrt {4 \, x^{2} - 9}}{18 \, x^{2}} - \frac {2}{27} \, \arcsin \left (\frac {3}{2 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.88, size = 29, normalized size = 0.74 \[ \frac {2\,\mathrm {atan}\left (\frac {\sqrt {4\,x^2-9}}{3}\right )}{27}+\frac {\sqrt {4\,x^2-9}}{18\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.13, size = 99, normalized size = 2.54 \[ \begin {cases} \frac {2 i \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{27} - \frac {i}{9 x \sqrt {-1 + \frac {9}{4 x^{2}}}} + \frac {i}{4 x^{3} \sqrt {-1 + \frac {9}{4 x^{2}}}} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\- \frac {2 \operatorname {asin}{\left (\frac {3}{2 x} \right )}}{27} + \frac {1}{9 x \sqrt {1 - \frac {9}{4 x^{2}}}} - \frac {1}{4 x^{3} \sqrt {1 - \frac {9}{4 x^{2}}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________