Optimal. Leaf size=37 \[ -\frac {\tan ^{-1}\left (\frac {\alpha ^2+k r}{\alpha \sqrt {-\alpha ^2+2 h r^2-2 k r}}\right )}{\alpha } \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {724, 204} \[ -\frac {\tan ^{-1}\left (\frac {\alpha ^2+k r}{\alpha \sqrt {-\alpha ^2+2 h r^2-2 k r}}\right )}{\alpha } \]
Antiderivative was successfully verified.
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Rule 204
Rule 724
Rubi steps
\begin {align*} \int \frac {1}{r \sqrt {-\alpha ^2-2 k r+2 h r^2}} \, dr &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-4 \alpha ^2-r^2} \, dr,r,\frac {-2 \alpha ^2-2 k r}{\sqrt {-\alpha ^2-2 k r+2 h r^2}}\right )\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\alpha ^2+k r}{\alpha \sqrt {-\alpha ^2-2 k r+2 h r^2}}\right )}{\alpha }\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 1.05 \[ \frac {\tan ^{-1}\left (\frac {-\alpha ^2-k r}{\alpha \sqrt {2 r (h r-k)-\alpha ^2}}\right )}{\alpha } \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 52, normalized size = 1.41 \[ -\frac {\arctan \left (\frac {\sqrt {2 \, h r^{2} - \alpha ^{2} - 2 \, k r} {\left (\alpha ^{2} + k r\right )}}{2 \, \alpha h r^{2} - \alpha ^{3} - 2 \, \alpha k r}\right )}{\alpha } \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 40, normalized size = 1.08 \[ \frac {2 \, \arctan \left (-\frac {\sqrt {2} \sqrt {h} r - \sqrt {2 \, h r^{2} - \alpha ^{2} - 2 \, k r}}{\alpha }\right )}{\alpha } \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 1.41 \[ -\frac {\ln \left (\frac {-2 \alpha ^{2}-2 k r +2 \sqrt {-\alpha ^{2}}\, \sqrt {2 h \,r^{2}-\alpha ^{2}-2 k r}}{r}\right )}{\sqrt {-\alpha ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 40, normalized size = 1.08 \[ -\frac {\arcsin \left (\frac {k}{\sqrt {2 \, \alpha ^{2} h + k^{2}}} + \frac {\alpha ^{2}}{\sqrt {2 \, \alpha ^{2} h + k^{2}} r}\right )}{\alpha } \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 51, normalized size = 1.38 \[ -\frac {\ln \left (\frac {\sqrt {-\alpha ^2}\,\sqrt {-\alpha ^2+2\,h\,r^2-2\,k\,r}}{r}-\frac {\alpha ^2}{r}-k\right )}{\sqrt {-\alpha ^2}} \]
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 \[ \int \frac {1}{r \sqrt {- \alpha ^{2} + 2 h r^{2} - 2 k r}}\, dr \]
Verification of antiderivative is not currently implemented for this CAS.
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