Optimal. Leaf size=30 \[ \sqrt {9-x^2}-3 \tanh ^{-1}\left (\frac {\sqrt {9-x^2}}{3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 30, 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 = {272, 52, 65,
212} \begin {gather*} \sqrt {9-x^2}-3 \tanh ^{-1}\left (\frac {\sqrt {9-x^2}}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 212
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt {9-x^2}}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {9-x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {9-x^2}+\frac {9}{2} \text {Subst}\left (\int \frac {1}{\sqrt {9-x} x} \, dx,x,x^2\right )\\ &=\sqrt {9-x^2}-9 \text {Subst}\left (\int \frac {1}{9-x^2} \, dx,x,\sqrt {9-x^2}\right )\\ &=\sqrt {9-x^2}-3 \tanh ^{-1}\left (\frac {\sqrt {9-x^2}}{3}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 30, normalized size = 1.00 \begin {gather*} \sqrt {9-x^2}-3 \tanh ^{-1}\left (\frac {\sqrt {9-x^2}}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 25, normalized size = 0.83
method | result | size |
default | \(\sqrt {-x^{2}+9}-3 \arctanh \left (\frac {3}{\sqrt {-x^{2}+9}}\right )\) | \(25\) |
trager | \(\sqrt {-x^{2}+9}+3 \ln \left (\frac {\sqrt {-x^{2}+9}-3}{x}\right )\) | \(29\) |
meijerg | \(-\frac {3 \left (-2 \left (2-2 \ln \left (2\right )+2 \ln \left (x \right )-2 \ln \left (3\right )+i \pi \right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {-\frac {x^{2}}{9}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-\frac {x^{2}}{9}+1}}{2}\right )\right )}{4 \sqrt {\pi }}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 3.50, size = 35, normalized size = 1.17 \begin {gather*} \sqrt {-x^{2} + 9} - 3 \, \log \left (\frac {6 \, \sqrt {-x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.59, size = 28, normalized size = 0.93 \begin {gather*} \sqrt {-x^{2} + 9} + 3 \, \log \left (\frac {\sqrt {-x^{2} + 9} - 3}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.65, size = 66, normalized size = 2.20 \begin {gather*} \begin {cases} i \sqrt {x^{2} - 9} - 3 \log {\left (x \right )} + \frac {3 \log {\left (x^{2} \right )}}{2} + 3 i \operatorname {asin}{\left (\frac {3}{x} \right )} & \text {for}\: \left |{x^{2}}\right | > 9 \\\sqrt {9 - x^{2}} + \frac {3 \log {\left (x^{2} \right )}}{2} - 3 \log {\left (\sqrt {1 - \frac {x^{2}}{9}} + 1 \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.42, size = 40, normalized size = 1.33 \begin {gather*} \sqrt {-x^{2} + 9} - \frac {3}{2} \, \log \left (\sqrt {-x^{2} + 9} + 3\right ) + \frac {3}{2} \, \log \left (-\sqrt {-x^{2} + 9} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 30, normalized size = 1.00 \begin {gather*} 3\,\ln \left (\sqrt {\frac {9}{x^2}-1}-3\,\sqrt {\frac {1}{x^2}}\right )+\sqrt {9-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________