Optimal. Leaf size=25 \[ \frac {\arctan \left (\sqrt {\frac {b}{a}} x^2\right )}{2 \sqrt {a b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.16, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {281, 211}
\begin {gather*} \frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 281
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )\\ &=\frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.00, size = 29, normalized size = 1.16 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 19, normalized size = 0.76
method | result | size |
default | \(\frac {\arctan \left (\frac {b \,x^{2}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\) | \(19\) |
risch | \(-\frac {\ln \left (x^{2} \sqrt {-a b}-a \right )}{4 \sqrt {-a b}}+\frac {\ln \left (x^{2} \sqrt {-a b}+a \right )}{4 \sqrt {-a b}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.43, size = 18, normalized size = 0.72 \begin {gather*} \frac {\arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.62, size = 72, normalized size = 2.88 \begin {gather*} \left [-\frac {\sqrt {-a b} \log \left (\frac {b x^{4} - 2 \, \sqrt {-a b} x^{2} - a}{b x^{4} + a}\right )}{4 \, a b}, -\frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b x^{2}}\right )}{2 \, 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. 56 vs.
\(2 (19) = 38\).
time = 0.07, size = 56, normalized size = 2.24 \begin {gather*} - \frac {\sqrt {- \frac {1}{a b}} \log {\left (- a \sqrt {- \frac {1}{a b}} + x^{2} \right )}}{4} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (a \sqrt {- \frac {1}{a b}} + x^{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 18, normalized size = 0.72 \begin {gather*} \frac {\arctan \left (\frac {b x^{2}}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 19, normalized size = 0.76 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x^2}{\sqrt {a}}\right )}{2\,\sqrt {a}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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