Optimal. Leaf size=30 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {1-a}}\right )}{\sqrt {(1-a) a}} \]
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Rubi [A] time = 0.03, antiderivative size = 30, 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} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {1-a}}\right )}{\sqrt {(1-a) a}} \]
Antiderivative was successfully verified.
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Rule 208
Rubi steps
\begin {align*} \int \frac {1}{-1+a+a x^2} \, dx &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {1-a}}\right )}{\sqrt {(1-a) a}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.93 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a-1}}\right )}{\sqrt {a-1} \sqrt {a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 82, normalized size = 2.73 \[ \left [-\frac {\sqrt {-a^{2} + a} \log \left (\frac {a x^{2} - 2 \, \sqrt {-a^{2} + a} x - a + 1}{a x^{2} + a - 1}\right )}{2 \, {\left (a^{2} - a\right )}}, \frac {\arctan \left (\frac {\sqrt {a^{2} - a} x}{a - 1}\right )}{\sqrt {a^{2} - a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 23, normalized size = 0.77 \[ \frac {\arctan \left (\frac {a x}{\sqrt {a^{2} - a}}\right )}{\sqrt {a^{2} - a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 0.67 \[ \frac {\arctan \left (\frac {a x}{\sqrt {\left (a -1\right ) a}}\right )}{\sqrt {\left (a -1\right ) a}} \]
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 = 0.17, size = 23, normalized size = 0.77 \[ \frac {\mathrm {atan}\left (\frac {a\,x}{\sqrt {a^2-a}}\right )}{\sqrt {a^2-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 83, normalized size = 2.77 \[ - \frac {\sqrt {- \frac {1}{a \left (a - 1\right )}} \log {\left (- a \sqrt {- \frac {1}{a \left (a - 1\right )}} + x + \sqrt {- \frac {1}{a \left (a - 1\right )}} \right )}}{2} + \frac {\sqrt {- \frac {1}{a \left (a - 1\right )}} \log {\left (a \sqrt {- \frac {1}{a \left (a - 1\right )}} + x - \sqrt {- \frac {1}{a \left (a - 1\right )}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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