Optimal. Leaf size=24 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}} \]
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Rubi [A] time = 0.0050706, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 205
Rubi steps
\begin{align*} \int \frac{1}{a+c x^2} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0046259, size = 24, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 16, normalized size = 0.7 \begin{align*}{\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 2.11155, size = 151, normalized size = 6.29 \begin{align*} \left [-\frac{\sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{2 \, a c}, \frac{\sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{a c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.153677, size = 53, normalized size = 2.21 \begin{align*} - \frac{\sqrt{- \frac{1}{a c}} \log{\left (- a \sqrt{- \frac{1}{a c}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a c}} \log{\left (a \sqrt{- \frac{1}{a c}} + x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23682, size = 20, normalized size = 0.83 \begin{align*} \frac{\arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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