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 = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 67, normalized size = 2.79 \[ \left [-\frac {\sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{2 \, a c}, \frac {\sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 15, normalized size = 0.62 \[ \frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 16, normalized size = 0.67 \[ \frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.90, size = 15, normalized size = 0.62 \[ \frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 16, normalized size = 0.67 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.14, size = 53, normalized size = 2.21 \[ - \frac {\sqrt {- \frac {1}{a c}} \log {\left (- a \sqrt {- \frac {1}{a c}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a c}} \log {\left (a \sqrt {- \frac {1}{a c}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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