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 = 0.0267875, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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*}
Mathematica [A] time = 0.0101971, 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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Maple [A] time = 0.005, size = 20, normalized size = 0.7 \begin{align*}{\arctan \left ({ax{\frac{1}{\sqrt{ \left ( a-1 \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a-1 \right ) a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18032, size = 189, normalized size = 6.3 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.149289, size = 83, normalized size = 2.77 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.22977, size = 31, normalized size = 1.03 \begin{align*} \frac{\arctan \left (\frac{a x}{\sqrt{a^{2} - a}}\right )}{\sqrt{a^{2} - a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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