Optimal. Leaf size=61 \[ -\frac {\tan ^{-1}\left (\frac {\alpha ^2+\epsilon ^2+k r}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}}\right )}{\sqrt {\alpha ^2+\epsilon ^2}} \]
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Rubi [A]
time = 0.01, 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 = {738, 210}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 738
Rubi steps
\begin {align*} \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2-2 k r+2 h r^2}} \, dr &=-\left (2 \text {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 = 0.11, size = 66, normalized size = 1.08 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {h} r-\sqrt {-\alpha ^2-\epsilon ^2-2 k r+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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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 74, normalized size = 1.21
method | result | size |
default | \(-\frac {\ln \left (\frac {-2 \alpha ^{2}-2 \epsilon ^{2}-2 k r +2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}\, \sqrt {2 h \,r^{2}-\alpha ^{2}-\epsilon ^{2}-2 k r}}{r}\right )}{\sqrt {-\alpha ^{2}-\epsilon ^{2}}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 77, normalized size = 1.26 \begin {gather*} -\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 97, normalized size = 1.59 \begin {gather*} -\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}}} \end {gather*}
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 \begin {gather*} \int \frac {1}{r \sqrt {- \alpha ^{2} - \epsilon ^{2} + 2 h r^{2} - 2 k r}}\, dr \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 54, normalized size = 0.89 \begin {gather*} \frac {\left (2.0\times 10^{12}\right ) \arctan \left (\frac {\sqrt {-\left (1.0\times 10^{24}\right ) \alpha ^{2}+\left (2.0\times 10^{24}\right ) h r^{2}-\left (2.0\times 10^{24}\right ) k r-1.0}-\sqrt {\left (2.0\times 10^{24}\right ) h} r}{\sqrt {\left (1.0\times 10^{24}\right ) \alpha ^{2}+1.0}}\right )}{\sqrt {\left (1.0\times 10^{24}\right ) \alpha ^{2}+1.0}} \end {gather*}
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 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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