Optimal. Leaf size=17 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0025019, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {216} \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{9-b x^2}} \, dx &=\frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0072171, size = 17, normalized size = 1. \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 21, normalized size = 1.2 \begin{align*}{\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{-b{x}^{2}+9}}}} \right ){\frac{1}{\sqrt{b}}}} \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 [B] time = 1.27607, size = 146, normalized size = 8.59 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-\sqrt{-b} x - \sqrt{-b x^{2} + 9}\right )}{b}, -\frac{2 \, \arctan \left (\frac{\sqrt{-b x^{2} + 9} - 3}{\sqrt{b} x}\right )}{\sqrt{b}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.00037, size = 39, normalized size = 2.29 \begin{align*} \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{b} x}{3} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x^{2}}\right |}{9} > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{3} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 2.40067, size = 36, normalized size = 2.12 \begin{align*} -\frac{\log \left (-\sqrt{-b} x + \sqrt{-b x^{2} + 9}\right )}{\sqrt{-b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]