Optimal. Leaf size=46 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt {\alpha ^2+\epsilon ^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {272, 65, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt {\alpha ^2+\epsilon ^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 211
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2}} \, dr &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r}} \, dr,r,r^2\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{-\frac {-\alpha ^2-\epsilon ^2}{2 h}+\frac {r^2}{2 h}} \, dr,r,\sqrt {-\alpha ^2-\epsilon ^2+2 h r^2}\right )}{2 h}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt {\alpha ^2+\epsilon ^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 46, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt {\alpha ^2+\epsilon ^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 66, normalized size = 1.43
method | result | size |
default | \(-\frac {\ln \left (\frac {-2 \alpha ^{2}-2 \epsilon ^{2}+2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}\, \sqrt {2 h \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{r}\right )}{\sqrt {-\alpha ^{2}-\epsilon ^{2}}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.05, size = 59, normalized size = 1.28 \begin {gather*} -\frac {\arcsin \left (\frac {\sqrt {2} \alpha ^{2}}{2 \, \sqrt {{\left (\alpha ^{2} + \epsilon ^{2}\right )} h} {\left | r \right |}} + \frac {\sqrt {2} \epsilon ^{2}}{2 \, \sqrt {{\left (\alpha ^{2} + \epsilon ^{2}\right )} h} {\left | r \right |}}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.72, size = 41, normalized size = 0.89 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {\alpha ^{2} + \epsilon ^{2}}}{\sqrt {2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2}}}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.54, size = 42, normalized size = 0.91 \begin {gather*} - \frac {\operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {\operatorname {polar\_lift}{\left (- \alpha ^{2} - \epsilon ^{2} \right )}}}{2 \sqrt {h} r} \right )}}{\sqrt {\operatorname {polar\_lift}{\left (- \alpha ^{2} - \epsilon ^{2} \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 40, normalized size = 0.87 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2}}}{\sqrt {\alpha ^{2} + \epsilon ^{2}}}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.66, size = 40, normalized size = 0.87 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {-\alpha ^2-\epsilon ^2+2\,h\,r^2}}{\sqrt {\alpha ^2+\epsilon ^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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