Optimal. Leaf size=61 \[ -\frac {\tan ^{-1}\left (\frac {\alpha ^2+\epsilon ^2+k r}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {724, 204} \[ -\frac {\tan ^{-1}\left (\frac {\alpha ^2+\epsilon ^2+k r}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 724
Rubi steps
\begin {align*} \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{4 \left (-\alpha ^2-\epsilon ^2\right )-r^2} \, dr,r,\frac {2 \left (-\alpha ^2-\epsilon ^2\right )-2 k r}{\sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}}\right )\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\alpha ^2+\epsilon ^2+k r}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 1.07 \[ \frac {\tan ^{-1}\left (\frac {-\alpha ^2-\epsilon ^2-k r}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2+2 r (h r-k)}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 97, normalized size = 1.59 \[ -\frac {\arctan \left (-\frac {\sqrt {2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2} - 2 \, k r} {\left (\alpha ^{2} + \epsilon ^{2} + k r\right )} \sqrt {\alpha ^{2} + \epsilon ^{2}}}{\alpha ^{4} + 2 \, \alpha ^{2} \epsilon ^{2} + \epsilon ^{4} - 2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} h r^{2} + 2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} k r}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.74, size = 51, normalized size = 0.84 \[ \frac {2.00000000000000 \times 10^{12} \, \arctan \left (\frac {\left (6.55360000000000 \times 10^{-8}\right ) \, {\left (-2.15791864375777 \times 10^{19} \, \sqrt {h} r + 1.52587890625000 \times 10^{19} \, \sqrt {2.00000000000000 \, h r^{2} - \alpha ^{2} - 2.00000000000000 \, k r - 1.00000000000000 \times 10^{-24}}\right )}}{\sqrt {1.00000000000000 \times 10^{24} \, \alpha ^{2} + 1.00000000000000}}\right )}{\sqrt {1.00000000000000 \times 10^{24} \, \alpha ^{2} + 1.00000000000000}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 1.21 \[ -\frac {\ln \left (\frac {-2 \alpha ^{2}-2 \epsilon ^{2}-2 k r +2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}\, \sqrt {2 h \,r^{2}-\alpha ^{2}-\epsilon ^{2}-2 k r}}{r}\right )}{\sqrt {-\alpha ^{2}-\epsilon ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 77, normalized size = 1.26 \[ -\frac {\arcsin \left (\frac {k}{\sqrt {2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} h + k^{2}}} + \frac {\alpha ^{2}}{\sqrt {2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} h + k^{2}} r} + \frac {\epsilon ^{2}}{\sqrt {2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} h + k^{2}} r}\right )}{\sqrt {\alpha ^{2} + \epsilon ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 72, normalized size = 1.18 \[ -\frac {\ln \left (\frac {\sqrt {-\alpha ^2-\epsilon ^2}\,\sqrt {-\alpha ^2-\epsilon ^2+2\,h\,r^2-2\,k\,r}}{r}-\frac {\alpha ^2+\epsilon ^2}{r}-k\right )}{\sqrt {-\alpha ^2-\epsilon ^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{r \sqrt {- \alpha ^{2} - \epsilon ^{2} + 2 h r^{2} - 2 k r}}\, dr \]
Verification of antiderivative is not currently implemented for this CAS.
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