Optimal. Leaf size=25 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2-9}}\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {217, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2-9}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
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 {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {-9+b x^2}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2-9}}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 57, normalized size = 2.28 \[ \left [\frac {\log \left (2 \, b x^{2} + 2 \, \sqrt {b x^{2} - 9} \sqrt {b} x - 9\right )}{2 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} - 9}}\right )}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 23, normalized size = 0.92 \[ -\frac {\log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} - 9} \right |}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.84 \[ \frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}-9}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 24, normalized size = 0.96 \[ \frac {\log \left (2 \, b x + 2 \, \sqrt {b x^{2} - 9} \sqrt {b}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 20, normalized size = 0.80 \[ \frac {\ln \left (\sqrt {b\,x^2-9}+\sqrt {b}\,x\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 39, normalized size = 1.56 \[ \begin {cases} \frac {\operatorname {acosh}{\left (\frac {\sqrt {b} x}{3} \right )}}{\sqrt {b}} & \text {for}\: \frac {\left |{b x^{2}}\right |}{9} > 1 \\- \frac {i \operatorname {asin}{\left (\frac {\sqrt {b} x}{3} \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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