Optimal. Leaf size=34 \[ -\frac {\sqrt {-1+9 x^2}}{x}+3 \tanh ^{-1}\left (\frac {3 x}{\sqrt {-1+9 x^2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {283, 223, 212}
\begin {gather*} 3 \tanh ^{-1}\left (\frac {3 x}{\sqrt {9 x^2-1}}\right )-\frac {\sqrt {9 x^2-1}}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 223
Rule 283
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+9 x^2}}{x^2} \, dx &=-\frac {\sqrt {-1+9 x^2}}{x}+9 \int \frac {1}{\sqrt {-1+9 x^2}} \, dx\\ &=-\frac {\sqrt {-1+9 x^2}}{x}+9 \text {Subst}\left (\int \frac {1}{1-9 x^2} \, dx,x,\frac {x}{\sqrt {-1+9 x^2}}\right )\\ &=-\frac {\sqrt {-1+9 x^2}}{x}+3 \tanh ^{-1}\left (\frac {3 x}{\sqrt {-1+9 x^2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 35, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {-1+9 x^2}}{x}-3 \log \left (-3 x+\sqrt {-1+9 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [A]
time = 1.81, size = 21, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {-1+9 x^2}}{x}+3 \text {ArcCosh}\left [3 x\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 47, normalized size = 1.38
method | result | size |
trager | \(-\frac {\sqrt {9 x^{2}-1}}{x}+3 \ln \left (3 x +\sqrt {9 x^{2}-1}\right )\) | \(32\) |
risch | \(-\frac {\sqrt {9 x^{2}-1}}{x}+\ln \left (\sqrt {9}\, x +\sqrt {9 x^{2}-1}\right ) \sqrt {9}\) | \(36\) |
default | \(\frac {\left (9 x^{2}-1\right )^{\frac {3}{2}}}{x}-9 x \sqrt {9 x^{2}-1}+\ln \left (\sqrt {9}\, x +\sqrt {9 x^{2}-1}\right ) \sqrt {9}\) | \(47\) |
meijerg | \(-\frac {3 i \sqrt {\mathrm {signum}\left (9 x^{2}-1\right )}\, \left (-\frac {4 i \sqrt {\pi }\, \sqrt {-9 x^{2}+1}}{3 x}-4 i \sqrt {\pi }\, \arcsin \left (3 x \right )\right )}{4 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (9 x^{2}-1\right )}}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.35, size = 33, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {9 \, x^{2} - 1}}{x} + 3 \, \log \left (18 \, x + 6 \, \sqrt {9 \, x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.33, size = 35, normalized size = 1.03 \begin {gather*} -\frac {3 \, x \log \left (-3 \, x + \sqrt {9 \, x^{2} - 1}\right ) + 3 \, x + \sqrt {9 \, x^{2} - 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.11, size = 17, normalized size = 0.50 \begin {gather*} 3 \operatorname {acosh}{\left (3 x \right )} - \frac {\sqrt {9 x^{2} - 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 50, normalized size = 1.47 \begin {gather*} 2 \left (-\frac {3}{\left (\sqrt {9 x^{2}-1}-3 x\right )^{2}+1}-\frac {3}{4} \ln \left (\left (\sqrt {9 x^{2}-1}-3 x\right )^{2}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.52, size = 32, normalized size = 0.94 \begin {gather*} -\frac {\left (\frac {3\,x\,\mathrm {asin}\left (3\,x\right )}{\sqrt {1-9\,x^2}}+1\right )\,\sqrt {9\,x^2-1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________