Optimal. Leaf size=45 \[ \frac {x}{2 a \left (a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {28, 205, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {x}{2 a \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 205
Rule 211
Rubi steps
\begin {align*} \int \frac {1}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac {1}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac {x}{2 a \left (a+b x^2\right )}+\frac {b \int \frac {1}{a b+b^2 x^2} \, dx}{2 a}\\ &=\frac {x}{2 a \left (a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} \frac {x}{2 a \left (a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 36, normalized size = 0.80
method | result | size |
default | \(\frac {x}{2 a \left (b \,x^{2}+a \right )}+\frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\) | \(36\) |
risch | \(\frac {x}{2 a \left (b \,x^{2}+a \right )}-\frac {\ln \left (b x +\sqrt {-a b}\right )}{4 \sqrt {-a b}\, a}+\frac {\ln \left (-b x +\sqrt {-a b}\right )}{4 \sqrt {-a b}\, a}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 35, normalized size = 0.78 \begin {gather*} \frac {x}{2 \, {\left (a b x^{2} + a^{2}\right )}} + \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 120, normalized size = 2.67 \begin {gather*} \left [\frac {2 \, a b x - {\left (b x^{2} + a\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{4 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}, \frac {a b x + {\left (b x^{2} + a\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{2 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}\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. 78 vs.
\(2 (36) = 72\).
time = 0.09, size = 78, normalized size = 1.73 \begin {gather*} \frac {x}{2 a^{2} + 2 a b x^{2}} - \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.12, size = 35, normalized size = 0.78 \begin {gather*} \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} + \frac {x}{2 \, {\left (b x^{2} + a\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 33, normalized size = 0.73 \begin {gather*} \frac {x}{2\,a\,\left (b\,x^2+a\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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