Optimal. Leaf size=38 \[ \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-\sqrt{a+\frac{b}{x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0227823, antiderivative size = 38, normalized size of antiderivative = 1., 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, 50, 63, 208} \[ \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-\sqrt{a+\frac{b}{x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+\frac{b}{x^2}}}{x} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\sqrt{a+\frac{b}{x^2}}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\sqrt{a+\frac{b}{x^2}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^2}}\right )}{b}\\ &=-\sqrt{a+\frac{b}{x^2}}+\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0620198, size = 69, normalized size = 1.82 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (-\sqrt{a} \sqrt{b} x \sqrt{\frac{a x^2}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )+a x^2+b\right )}{a x^2+b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.005, size = 81, normalized size = 2.1 \begin{align*}{\frac{1}{b}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ({a}^{{\frac{3}{2}}}\sqrt{a{x}^{2}+b}{x}^{2}- \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}\sqrt{a}+\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ) xab \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.54115, size = 252, normalized size = 6.63 \begin{align*} \left [\frac{1}{2} \, \sqrt{a} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) - \sqrt{\frac{a x^{2} + b}{x^{2}}}, -\sqrt{-a} \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) - \sqrt{\frac{a x^{2} + b}{x^{2}}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.42436, size = 56, normalized size = 1.47 \begin{align*} \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )} - \frac{a x}{\sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{\sqrt{b}}{x \sqrt{\frac{a x^{2}}{b} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18872, size = 82, normalized size = 2.16 \begin{align*} -\frac{1}{2} \, \sqrt{a} \log \left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2}\right ) \mathrm{sgn}\left (x\right ) + \frac{2 \, \sqrt{a} b \mathrm{sgn}\left (x\right )}{{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]