Optimal. Leaf size=34 \[ \frac {\tanh ^{-1}\left (\frac {x \sqrt {b-a c}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b-a c}} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {208} \[ \frac {\tanh ^{-1}\left (\frac {x \sqrt {b-a c}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b-a c}} \]
Antiderivative was successfully verified.
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Rule 208
Rubi steps
\begin {align*} \int \frac {1}{a-(b-a c) x^2} \, dx &=\frac {\tanh ^{-1}\left (\frac {\sqrt {b-a c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b-a c}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 1.06 \[ \frac {\tan ^{-1}\left (\frac {x \sqrt {a c-b}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {a c-b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 105, normalized size = 3.09 \[ \left [-\frac {\sqrt {-a^{2} c + a b} \log \left (\frac {{\left (a c - b\right )} x^{2} - 2 \, \sqrt {-a^{2} c + a b} x - a}{{\left (a c - b\right )} x^{2} + a}\right )}{2 \, {\left (a^{2} c - a b\right )}}, \frac {\arctan \left (\frac {\sqrt {a^{2} c - a b} x}{a}\right )}{\sqrt {a^{2} c - a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 36, normalized size = 1.06 \[ \frac {\arctan \left (\frac {a c x - b x}{\sqrt {a^{2} c - a b}}\right )}{\sqrt {a^{2} c - a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 1.00 \[ \frac {\arctan \left (\frac {\left (a c -b \right ) x}{\sqrt {\left (a c -b \right ) a}}\right )}{\sqrt {\left (a c -b \right ) a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.64, size = 38, normalized size = 1.12 \[ -\frac {\mathrm {atan}\left (\frac {x\,\left (2\,b-2\,a\,c\right )}{2\,\sqrt {a^2\,c-a\,b}}\right )}{\sqrt {a^2\,c-a\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.23, size = 66, normalized size = 1.94 \[ - \frac {\sqrt {- \frac {1}{a \left (a c - b\right )}} \log {\left (- a \sqrt {- \frac {1}{a \left (a c - b\right )}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a \left (a c - b\right )}} \log {\left (a \sqrt {- \frac {1}{a \left (a c - b\right )}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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