Optimal. Leaf size=50 \[ -\frac {1}{a \sqrt {d x^2}}-\frac {\sqrt {b} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {d x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {15, 331, 211}
\begin {gather*} -\frac {\sqrt {b} x \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {d x^2}}-\frac {1}{a \sqrt {d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 211
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {d x^2} \left (a+b x^2\right )} \, dx &=\frac {x \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{\sqrt {d x^2}}\\ &=-\frac {1}{a \sqrt {d x^2}}-\frac {(b x) \int \frac {1}{a+b x^2} \, dx}{a \sqrt {d x^2}}\\ &=-\frac {1}{a \sqrt {d x^2}}-\frac {\sqrt {b} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 46, normalized size = 0.92 \begin {gather*} -\frac {d x^2 \left (\sqrt {a}+\sqrt {b} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )}{a^{3/2} \left (d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 36, normalized size = 0.72
method | result | size |
default | \(-\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right ) x +\sqrt {a b}}{\sqrt {d \,x^{2}}\, a \sqrt {a b}}\) | \(36\) |
risch | \(-\frac {1}{a \sqrt {d \,x^{2}}}+\frac {x \left (\munderset {\textit {\_R} =\RootOf \left (a^{3} \textit {\_Z}^{2}+b \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{3}+2 b \right ) x +a^{2} \textit {\_R} \right )\right )}{2 \sqrt {d \,x^{2}}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 35, normalized size = 0.70 \begin {gather*} -\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a \sqrt {d}} - \frac {1}{a \sqrt {d} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.25, size = 132, normalized size = 2.64 \begin {gather*} \left [\frac {d x^{2} \sqrt {-\frac {b}{a d}} \log \left (\frac {b x^{2} - 2 \, \sqrt {d x^{2}} a \sqrt {-\frac {b}{a d}} - a}{b x^{2} + a}\right ) - 2 \, \sqrt {d x^{2}}}{2 \, a d x^{2}}, -\frac {d x^{2} \sqrt {\frac {b}{a d}} \arctan \left (\sqrt {d x^{2}} \sqrt {\frac {b}{a d}}\right ) + \sqrt {d x^{2}}}{a d x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {d x^{2}} \left (a + b x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 43, normalized size = 0.86 \begin {gather*} -\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a \sqrt {d} \mathrm {sgn}\left (x\right )} - \frac {1}{a \sqrt {d} x \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 38, normalized size = 0.76 \begin {gather*} -\frac {1}{a\,\sqrt {d}\,\sqrt {x^2}}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x^2}}{\sqrt {a}}\right )}{a^{3/2}\,\sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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