Optimal. Leaf size=16 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {-a^2+x^2}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {223, 212}
\begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-a^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. \(46\) vs. \(2(16)=32\).
time = 0.00, size = 46, normalized size = 2.88 \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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.30, size = 27, normalized size = 1.69 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {ArcCosh}\left [\frac {x}{a}\right ],\text {Abs}\left [\frac {x^2}{a^2}\right ]>1\right \}\right \},-I \text {ArcSin}\left [\frac {x}{a}\right ]\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.04, size = 15, normalized size = 0.94
method | result | size |
default | \(\ln \left (x +\sqrt {-a^{2}+x^{2}}\right )\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 18, normalized size = 1.12 \begin {gather*} \log \left (2 \, x + 2 \, \sqrt {-a^{2} + x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 18, normalized size = 1.12 \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 = 0.47, size = 19, normalized size = 1.19 \begin {gather*} \begin {cases} \operatorname {acosh}{\left (\frac {x}{a} \right )} & \text {for}\: \left |{\frac {x^{2}}{a^{2}}}\right | > 1 \\- i \operatorname {asin}{\left (\frac {x}{a} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 18, normalized size = 1.12 \begin {gather*} -\ln \left |\sqrt {-a^{2}+x^{2}}-x\right | \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 14, normalized size = 0.88 \begin {gather*} \ln \left (x+\sqrt {x^2-a^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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