Optimal. Leaf size=37 \[ -\frac {\tan ^{-1}\left (\frac {\alpha ^2+k r}{\alpha \sqrt {-\alpha ^2-2 k r+2 h r^2}}\right )}{\alpha } \]
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Rubi [A]
time = 0.01, 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 = {738, 210}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\alpha ^2+k r}{\alpha \sqrt {-\alpha ^2+2 h r^2-2 k r}}\right )}{\alpha } \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 738
Rubi steps
\begin {align*} \int \frac {1}{r \sqrt {-\alpha ^2-2 k r+2 h r^2}} \, dr &=-\left (2 \text {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 = 0.08, size = 45, normalized size = 1.22 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {h} r-\sqrt {-\alpha ^2-2 k r+2 h r^2}}{\alpha }\right )}{\alpha } \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 52, normalized size = 1.41
method | result | size |
default | \(-\frac {\ln \left (\frac {-2 \alpha ^{2}-2 k r +2 \sqrt {-\alpha ^{2}}\, \sqrt {2 h \,r^{2}-\alpha ^{2}-2 k r}}{r}\right )}{\sqrt {-\alpha ^{2}}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.14, size = 46, normalized size = 1.24 \begin {gather*} -\frac {\arcsin \left (\frac {\alpha ^{2}}{\sqrt {2 \, \alpha ^{2} h + k^{2}} {\left | r \right |}} + \frac {k r}{\sqrt {2 \, \alpha ^{2} h + k^{2}} {\left | r \right |}}\right )}{\alpha } \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.75, size = 52, normalized size = 1.41 \begin {gather*} -\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 } \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} + 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 = 0.56, size = 40, normalized size = 1.08 \begin {gather*} \frac {2 \, \arctan \left (-\frac {\sqrt {2} \sqrt {h} r - \sqrt {2 \, h r^{2} - \alpha ^{2} - 2 \, k r}}{\alpha }\right )}{\alpha } \end {gather*}
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 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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