Optimal. Leaf size=68 \[ -\frac{\tan ^{-1}\left (\frac{\alpha ^2+\epsilon ^2-h r^2}{\sqrt{\alpha ^2+\epsilon ^2} \sqrt{-\alpha ^2-\epsilon ^2+2 h r^2-2 k r^4}}\right )}{2 \sqrt{\alpha ^2+\epsilon ^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0488941, antiderivative size = 68, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 1114
Rule 724
Rule 204
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*}
Mathematica [A] time = 0.025217, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 78, normalized size = 1.2 \begin{align*} -{\frac{1}{2}\ln \left ({\frac{1}{{r}^{2}} \left ( -2\,{\alpha }^{2}-2\,{\epsilon }^{2}+2\,h{r}^{2}+2\,\sqrt{-{\alpha }^{2}-{\epsilon }^{2}}\sqrt{-2\,k{r}^{4}+2\,h{r}^{2}-{\alpha }^{2}-{\epsilon }^{2}} \right ) } \right ){\frac{1}{\sqrt{-{\alpha }^{2}-{\epsilon }^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.89919, size = 339, normalized size = 4.99 \begin{align*} -\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{r \sqrt{- \alpha ^{2} - \epsilon ^{2} + 2 h r^{2} - 2 k r^{4}}}\, dr \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25976, size = 42, normalized size = 0.62 \begin{align*} -\frac{\arcsin \left (-\frac{h - \frac{\alpha ^{2}}{r^{2}}}{\sqrt{-2 \, \alpha ^{2} k + h^{2}}}\right )}{2 \,{\left | \alpha \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]