Optimal. Leaf size=24 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 208
Rubi steps
\begin {align*} \int \frac {1}{a-b x^2} \, dx &=\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 68, normalized size = 2.83 \[ \left [\frac {\sqrt {a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {a b} x + a}{b x^{2} - a}\right )}{2 \, a b}, -\frac {\sqrt {-a b} \arctan \left (\frac {\sqrt {-a b} x}{a}\right )}{a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 18, normalized size = 0.75 \[ -\frac {\arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{\sqrt {-a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.67 \[ \frac {\arctanh \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 31, normalized size = 1.29 \[ -\frac {\log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{2 \, \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 16, normalized size = 0.67 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.14, size = 46, normalized size = 1.92 \[ - \frac {\sqrt {\frac {1}{a b}} \log {\left (- a \sqrt {\frac {1}{a b}} + x \right )}}{2} + \frac {\sqrt {\frac {1}{a b}} \log {\left (a \sqrt {\frac {1}{a b}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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