Optimal. Leaf size=21 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2+x^2}}{a}\right )}{a} \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 65, 213}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2+x^2}}{a}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a^2+x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {a^2+x}} \, dx,x,x^2\right )\\ &=\text {Subst}\left (\int \frac {1}{-a^2+x^2} \, dx,x,\sqrt {a^2+x^2}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2+x^2}}{a}\right )}{a}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(21)=42\).
time = 0.03, size = 49, normalized size = 2.33 \begin {gather*} -\frac {\log \left (a+\sqrt {a^2+x^2}\right )}{2 a}+\frac {\log \left (-a^2+a \sqrt {a^2+x^2}\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 35, normalized size = 1.67
method | result | size |
default | \(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}+x^{2}}}{x}\right )}{\sqrt {a^{2}}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.96, size = 12, normalized size = 0.57 \begin {gather*} -\frac {\operatorname {arsinh}\left (\frac {a}{{\left | x \right |}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs.
\(2 (19) = 38\).
time = 0.74, size = 40, normalized size = 1.90 \begin {gather*} -\frac {\log \left (a - x + \sqrt {a^{2} + x^{2}}\right ) - \log \left (-a - x + \sqrt {a^{2} + x^{2}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.43, size = 7, normalized size = 0.33 \begin {gather*} - \frac {\operatorname {asinh}{\left (\frac {a}{x} \right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.88, size = 37, normalized size = 1.76 \begin {gather*} -\frac {\log \left (a + \sqrt {a^{2} + x^{2}}\right )}{2 \, a} + \frac {\log \left (-a + \sqrt {a^{2} + x^{2}}\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 26, normalized size = 1.24 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {a^2+x^2}}{\sqrt {-a^2}}\right )}{\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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