Optimal. Leaf size=14 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {a^2+x^2}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = {223, 212}
\begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {a^2+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a^2+x^2}} \, dx &=\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {a^2+x^2}}\right )\\ &=\tanh ^{-1}\left (\frac {x}{\sqrt {a^2+x^2}}\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(14)=28\).
time = 0.00, size = 42, normalized size = 3.00 \begin {gather*} -\frac {1}{2} \log \left (1-\frac {x}{\sqrt {a^2+x^2}}\right )+\frac {1}{2} \log \left (1+\frac {x}{\sqrt {a^2+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 13, normalized size = 0.93
method | result | size |
default | \(\ln \left (x +\sqrt {a^{2}+x^{2}}\right )\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.16, size = 6, normalized size = 0.43 \begin {gather*} \operatorname {arsinh}\left (\frac {x}{a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 16, normalized size = 1.14 \begin {gather*} -\log \left (-x + \sqrt {a^{2} + x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 3, normalized size = 0.21 \begin {gather*} \operatorname {asinh}{\left (\frac {x}{a} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs.
\(2 (12) = 24\).
time = 0.47, size = 32, normalized size = 2.29 \begin {gather*} -\frac {1}{2} \, a^{2} \log \left (-x + \sqrt {a^{2} + x^{2}}\right ) + \frac {1}{2} \, \sqrt {a^{2} + x^{2}} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 12, normalized size = 0.86 \begin {gather*} \ln \left (x+\sqrt {a^2+x^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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