Optimal. Leaf size=68 \[ -\frac {\tan ^{-1}\left (\frac {\alpha ^2+\epsilon ^2-h r^2}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}}\right )}{2 \sqrt {\alpha ^2+\epsilon ^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1114, 724, 204} \[ -\frac {\tan ^{-1}\left (\frac {\alpha ^2+\epsilon ^2-h r^2}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}}\right )}{2 \sqrt {\alpha ^2+\epsilon ^2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 724
Rule 1114
Rubi steps
\begin {align*} \int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}} \, dr &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{r \sqrt {-\alpha ^2-\epsilon ^2+2 h r-2 k r^2}} \, dr,r,r^2\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{4 \left (-\alpha ^2-\epsilon ^2\right )-r^2} \, dr,r,\frac {2 \left (-\alpha ^2-\epsilon ^2+h r^2\right )}{\sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {-\alpha ^2-\epsilon ^2+h r^2}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}}\right )}{2 \sqrt {\alpha ^2+\epsilon ^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 71, normalized size = 1.04 \[ \frac {\tan ^{-1}\left (\frac {-\alpha ^2-\epsilon ^2+h r^2}{\sqrt {\alpha ^2+\epsilon ^2} \sqrt {-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}}\right )}{2 \sqrt {\alpha ^2+\epsilon ^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 106, normalized size = 1.56 \[ -\frac {\arctan \left (\frac {\sqrt {-2 \, k r^{4} + 2 \, h r^{2} - \alpha ^{2} - \epsilon ^{2}} {\left (h r^{2} - \alpha ^{2} - \epsilon ^{2}\right )} \sqrt {\alpha ^{2} + \epsilon ^{2}}}{2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} k r^{4} + \alpha ^{4} + 2 \, \alpha ^{2} \epsilon ^{2} + \epsilon ^{4} - 2 \, {\left (\alpha ^{2} + \epsilon ^{2}\right )} h r^{2}}\right )}{2 \, \sqrt {\alpha ^{2} + \epsilon ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 45, normalized size = 0.66 \[ \frac {\arctan \left (-\frac {\sqrt {2} \sqrt {-k} r^{2} - \sqrt {-2 \, k r^{4} + 2 \, h r^{2} - \alpha ^{2}}}{\alpha }\right )}{\alpha } \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 78, normalized size = 1.15 \[ -\frac {\ln \left (\frac {2 h \,r^{2}-2 \alpha ^{2}-2 \epsilon ^{2}+2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}\, \sqrt {-2 k \,r^{4}+2 h \,r^{2}-\alpha ^{2}-\epsilon ^{2}}}{r^{2}}\right )}{2 \sqrt {-\alpha ^{2}-\epsilon ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 72, normalized size = 1.06 \[ -\frac {\ln \left (h-\frac {\alpha ^2+\epsilon ^2}{r^2}+\frac {\sqrt {-\alpha ^2-\epsilon ^2}\,\sqrt {-\alpha ^2-\epsilon ^2-2\,k\,r^4+2\,h\,r^2}}{r^2}\right )}{2\,\sqrt {-\alpha ^2-\epsilon ^2}} \]
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 \[ \int \frac {1}{r \sqrt {- \alpha ^{2} - \epsilon ^{2} + 2 h r^{2} - 2 k r^{4}}}\, dr \]
Verification of antiderivative is not currently implemented for this CAS.
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