Optimal. Leaf size=56 \[ -\frac {\tan ^{-1}\left (\frac {e-2 k r^2}{\sqrt {2} \sqrt {k} \sqrt {-\alpha ^2+2 e r^2-2 k r^4}}\right )}{2 \sqrt {2} \sqrt {k}} \]
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Rubi [A]
time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1121, 635, 210}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {e-2 k r^2}{\sqrt {2} \sqrt {k} \sqrt {-\alpha ^2+2 e r^2-2 k r^4}}\right )}{2 \sqrt {2} \sqrt {k}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 635
Rule 1121
Rubi steps
\begin {align*} \int \frac {r}{\sqrt {-\alpha ^2+2 e r^2-2 k r^4}} \, dr &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-\alpha ^2+2 e r-2 k r^2}} \, dr,r,r^2\right )\\ &=\text {Subst}\left (\int \frac {1}{-8 k-r^2} \, dr,r,\frac {2 \left (e-2 k r^2\right )}{\sqrt {-\alpha ^2+2 e r^2-2 k r^4}}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {e-2 k r^2}{\sqrt {2} \sqrt {k} \sqrt {-\alpha ^2+2 e r^2-2 k r^4}}\right )}{2 \sqrt {2} \sqrt {k}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(150\) vs. \(2(56)=112\).
time = 0.18, size = 150, normalized size = 2.68 \begin {gather*} -\frac {2 \sqrt {-k} \tan ^{-1}\left (\frac {\sqrt {k} \left (2 \sqrt {-k} r^2-\sqrt {2} \sqrt {-\alpha ^2+2 e r^2-2 k r^4}\right )}{e}\right )+\sqrt {k} \log \left (e^2+4 e k r^2-2 k \left (\alpha ^2+4 k r^4+2 \sqrt {2} \sqrt {-k} r^2 \sqrt {-\alpha ^2+2 e r^2-2 k r^4}\right )\right )}{4 \sqrt {2} \sqrt {-k^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 47, normalized size = 0.84
method | result | size |
default | \(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {k}\, \left (r^{2}-\frac {e}{2 k}\right )}{\sqrt {-2 k \,r^{4}+2 e \,r^{2}-\alpha ^{2}}}\right )}{4 \sqrt {k}}\) | \(47\) |
elliptic | \(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {k}\, \left (r^{2}-\frac {e}{2 k}\right )}{\sqrt {-2 k \,r^{4}+2 e \,r^{2}-\alpha ^{2}}}\right )}{4 \sqrt {k}}\) | \(47\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.69, size = 155, normalized size = 2.77 \begin {gather*} \left [-\frac {\sqrt {2} \sqrt {-k} \log \left (8 \, k^{2} r^{4} - 8 \, k r^{2} e + 2 \, \alpha ^{2} k - 2 \, \sqrt {2} \sqrt {-2 \, k r^{4} + 2 \, r^{2} e - \alpha ^{2}} {\left (2 \, k r^{2} - e\right )} \sqrt {-k} + e^{2}\right )}{8 \, k}, -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-2 \, k r^{4} + 2 \, r^{2} e - \alpha ^{2}} {\left (2 \, k r^{2} - e\right )} \sqrt {k}}{2 \, {\left (2 \, k^{2} r^{4} - 2 \, k r^{2} e + \alpha ^{2} k\right )}}\right )}{4 \, \sqrt {k}}\right ] \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 {r}{\sqrt {- \alpha ^{2} + 2 e r^{2} - 2 k r^{4}}}\, dr \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 60, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {2} \log \left ({\left | \sqrt {2} {\left (\sqrt {2} \sqrt {-k} r^{2} - \sqrt {-2 \, k r^{4} + 2 \, r^{2} e - \alpha ^{2}}\right )} \sqrt {-k} + e \right |}\right )}{4 \, \sqrt {-k}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.98, size = 50, normalized size = 0.89 \begin {gather*} \frac {\sqrt {2}\,\ln \left (\sqrt {-\alpha ^2-2\,k\,r^4+2\,e\,r^2}+\frac {\sqrt {2}\,\left (e-2\,k\,r^2\right )}{2\,\sqrt {-k}}\right )}{4\,\sqrt {-k}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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