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 = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 203
Rule 217
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*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {-a-b x^2}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 74, normalized size = 2.64 \[ \left [-\frac {\sqrt {-b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {-b x^{2} - a} \sqrt {-b} x - a\right )}{2 \, b}, -\frac {\arctan \left (\frac {\sqrt {-b x^{2} - a} \sqrt {b} x}{b x^{2} + a}\right )}{\sqrt {b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 30, normalized size = 1.07 \[ -\frac {\log \left ({\left | -\sqrt {-b} x + \sqrt {-b x^{2} - a} \right |}\right )}{\sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.82 \[ \frac {\arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}-a}}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.32, size = 14, normalized size = 0.50 \[ -\frac {i \, \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.13, size = 27, normalized size = 0.96 \[ \frac {\ln \left (\sqrt {-b\,x^2-a}+\sqrt {-b}\,x\right )}{\sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.02, size = 20, normalized size = 0.71 \[ - \frac {i \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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