Optimal. Leaf size=28 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-a-b x^2}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.004854, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {217, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-a-b x^2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-a-b x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{x}{\sqrt{-a-b x^2}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-a-b x^2}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0058657, size = 28, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-a-b x^2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, normalized size = 0.8 \begin{align*}{\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{-b{x}^{2}-a}}}} \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 = 1.36821, size = 173, normalized size = 6.18 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{-b x^{2} - a} \sqrt{-b} x - a\right )}{2 \, b}, -\frac{\arctan \left (\frac{\sqrt{-b x^{2} - a} \sqrt{b} x}{b x^{2} + a}\right )}{\sqrt{b}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.00626, size = 20, normalized size = 0.71 \begin{align*} - \frac{i \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.67519, size = 41, normalized size = 1.46 \begin{align*} -\frac{\log \left ({\left | -\sqrt{-b} x + \sqrt{-b x^{2} - a} \right |}\right )}{\sqrt{-b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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