Optimal. Leaf size=10 \[ \arctan \left (\sqrt {-1+x^2}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 65, 209}
\begin {gather*} \arctan \left (\sqrt {x^2-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 272
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^2\right )\\ &=\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^2}\right )\\ &=\arctan \left (\sqrt {-1+x^2}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 10, normalized size = 1.00 \begin {gather*} \arctan \left (\sqrt {-1+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 11, normalized size = 1.10
method | result | size |
pseudoelliptic | \(\arctan \left (\sqrt {x^{2}-1}\right )\) | \(9\) |
default | \(-\arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )\) | \(11\) |
trager | \(\mathit {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\mathit {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{2}-1}}{x}\right )\) | \(27\) |
meijerg | \(\frac {\sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \left (\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{2}+1}}{2}\right )\right )}{2 \sqrt {\pi }\, \sqrt {\mathrm {signum}\left (x^{2}-1\right )}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.43, size = 7, normalized size = 0.70 \begin {gather*} -\arcsin \left (\frac {1}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.58, size = 14, normalized size = 1.40 \begin {gather*} 2 \, \arctan \left (-x + \sqrt {x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.66, size = 17, normalized size = 1.70 \begin {gather*} \begin {cases} i \operatorname {acosh}{\left (\frac {1}{x} \right )} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\- \operatorname {asin}{\left (\frac {1}{x} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 8, normalized size = 0.80 \begin {gather*} \arctan \left (\sqrt {x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 8, normalized size = 0.80 \begin {gather*} \mathrm {atan}\left (\sqrt {x^2-1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 9, normalized size = 0.90 \begin {gather*} -\frac {1}{2 \left (x^{2}-1\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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