Optimal. Leaf size=34 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b-a c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b-a c}} \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {x \sqrt {b-a c}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b-a c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rubi steps
\begin {align*} \int \frac {1}{a+(b-a c) x^2} \, dx &=\frac {\tan ^{-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 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {-b+a c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {-b+a c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 34, normalized size = 1.00
method | result | size |
default | \(\frac {\arctanh \left (\frac {\left (a c -b \right ) x}{\sqrt {a \left (a c -b \right )}}\right )}{\sqrt {a \left (a c -b \right )}}\) | \(34\) |
risch | \(\frac {\ln \left (\left (-a c +b \right ) x -\sqrt {a \left (a c -b \right )}\right )}{2 \sqrt {a \left (a c -b \right )}}-\frac {\ln \left (\left (a c -b \right ) x -\sqrt {a \left (a c -b \right )}\right )}{2 \sqrt {a \left (a c -b \right )}}\) | \(75\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.45, size = 106, normalized size = 3.12 \begin {gather*} \left [\frac {\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 \, \sqrt {a^{2} c - a b}}, -\frac {\sqrt {-a^{2} c + a b} \arctan \left (\frac {\sqrt {-a^{2} c + a b} x}{a}\right )}{a^{2} c - a b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (29) = 58\).
time = 0.11, size = 60, normalized size = 1.76 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.43, size = 37, normalized size = 1.09 \begin {gather*} -\frac {\arctan \left (\frac {a c x - b x}{\sqrt {-a^{2} c + a b}}\right )}{\sqrt {-a^{2} c + a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.14, size = 38, normalized size = 1.12 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {x\,\left (2\,b-2\,a\,c\right )}{2\,\sqrt {a}\,\sqrt {a\,c-b}}\right )}{\sqrt {a}\,\sqrt {a\,c-b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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