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 = 0.03, 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 = {266, 63, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt {\alpha ^2+\epsilon ^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 205
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2}} \, dr &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r}} \, dr,r,r^2\right )\\ &=\frac {\operatorname {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 = 0.01, size = 46, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {-\alpha ^2-\epsilon ^2+2 h r^2}}{\sqrt {\alpha ^2+\epsilon ^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 41, normalized size = 0.89 \[ -\frac {\arctan \left (\frac {\sqrt {\alpha ^{2} + \epsilon ^{2}}}{\sqrt {2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2}}}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 38, normalized size = 0.83 \[ \frac {1.00000000000000 \times 10^{12} \, \arctan \left (\frac {1.00000000000000 \times 10^{12} \, \sqrt {2.00000000000000 \, h r^{2} - 1.00000000000000 \, \alpha ^{2} - 1.00000000000000 \times 10^{-24}}}{\sqrt {1.00000000000000 \times 10^{24} \, \alpha ^{2} + 1.00000000000000}}\right )}{\sqrt {1.00000000000000 \times 10^{24} \, \alpha ^{2} + 1.00000000000000}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 1.43 \[ -\frac {\ln \left (\frac {-2 \alpha ^{2}-2 \epsilon ^{2}+2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}\, \sqrt {2 h \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{r}\right )}{\sqrt {-\alpha ^{2}-\epsilon ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 57, normalized size = 1.24 \[ -\frac {\arcsin \left (\frac {\sqrt {2} \alpha ^{2}}{2 \, \sqrt {{\left (\alpha ^{2} + \epsilon ^{2}\right )} h} r} + \frac {\sqrt {2} \epsilon ^{2}}{2 \, \sqrt {{\left (\alpha ^{2} + \epsilon ^{2}\right )} h} r}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \]
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 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {-\alpha ^2-\epsilon ^2+2\,h\,r^2}}{\sqrt {\alpha ^2+\epsilon ^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.25, size = 42, normalized size = 0.91 \[ - \frac {\operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {\operatorname {polar\_lift}{\left (- \alpha ^{2} - \epsilon ^{2} \right )}}}{2 \sqrt {h} r} \right )}}{\sqrt {\operatorname {polar\_lift}{\left (- \alpha ^{2} - \epsilon ^{2} \right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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