Optimal. Leaf size=26 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2-9}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0045402, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {217, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2-9}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-9-b x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{x}{\sqrt{-9-b x^2}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-9-b x^2}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0060969, size = 26, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2-9}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.8 \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.
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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.
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Fricas [A] time = 1.29094, size = 173, normalized size = 6.65 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{-b x^{2} - 9} \sqrt{-b} x - 9\right )}{2 \, b}, -\frac{\arctan \left (\frac{\sqrt{-b x^{2} - 9} \sqrt{b} x}{b x^{2} + 9}\right )}{\sqrt{b}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.947742, size = 17, normalized size = 0.65 \begin{align*} - \frac{i \operatorname{asinh}{\left (\frac{\sqrt{b} x}{3} \right )}}{\sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38766, size = 38, normalized size = 1.46 \begin{align*} -\frac{\log \left ({\left | -\sqrt{-b} x + \sqrt{-b x^{2} - 9} \right |}\right )}{\sqrt{-b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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