Optimal. Leaf size=23 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+3}}{\sqrt{3}}\right )}{\sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0110963, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 63, 207} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+3}}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{3+x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3+x}} \, dx,x,x^2\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{-3+x^2} \, dx,x,\sqrt{3+x^2}\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{3+x^2}}{\sqrt{3}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0042559, size = 23, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+3}}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 18, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\sqrt{3}{\frac{1}{\sqrt{{x}^{2}+3}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.40106, size = 19, normalized size = 0.83 \begin{align*} -\frac{1}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{\sqrt{3}}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.2903, size = 63, normalized size = 2.74 \begin{align*} \frac{1}{3} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - \sqrt{x^{2} + 3}}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.987104, size = 15, normalized size = 0.65 \begin{align*} - \frac{\sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{3}}{x} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.05048, size = 50, normalized size = 2.17 \begin{align*} -\frac{1}{6} \, \sqrt{3} \log \left (\sqrt{3} + \sqrt{x^{2} + 3}\right ) + \frac{1}{6} \, \sqrt{3} \log \left (-\sqrt{3} + \sqrt{x^{2} + 3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]