Optimal. Leaf size=40 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {h} r}{\sqrt {2 h r^2-\alpha ^2}}\right )}{\sqrt {2} \sqrt {h}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {217, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {h} r}{\sqrt {2 h r^2-\alpha ^2}}\right )}{\sqrt {2} \sqrt {h}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-\alpha ^2+2 h r^2}} \, dr &=\operatorname {Subst}\left (\int \frac {1}{1-2 h r^2} \, dr,r,\frac {r}{\sqrt {-\alpha ^2+2 h r^2}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {h} r}{\sqrt {-\alpha ^2+2 h r^2}}\right )}{\sqrt {2} \sqrt {h}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {h} r}{\sqrt {2 h r^2-\alpha ^2}}\right )}{\sqrt {2} \sqrt {h}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 85, normalized size = 2.12 \[ \left [\frac {\sqrt {2} \log \left (4 \, h r^{2} + 2 \, \sqrt {2} \sqrt {2 \, h r^{2} - \alpha ^{2}} \sqrt {h} r - \alpha ^{2}\right )}{4 \, \sqrt {h}}, -\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{h}} \arctan \left (\frac {\sqrt {2} h r \sqrt {-\frac {1}{h}}}{\sqrt {2 \, h r^{2} - \alpha ^{2}}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.15, size = 34, normalized size = 0.85 \[ -\frac {\sqrt {2} \log \left ({\left | -\sqrt {2} \sqrt {h} r + \sqrt {2 \, h r^{2} - \alpha ^{2}} \right |}\right )}{2 \, \sqrt {h}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 33, normalized size = 0.82 \[ \frac {\sqrt {2}\, \ln \left (\sqrt {2}\, \sqrt {h}\, r +\sqrt {2 h \,r^{2}-\alpha ^{2}}\right )}{2 \sqrt {h}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 36, normalized size = 0.90 \[ \frac {\sqrt {2} \log \left (4 \, h r + 2 \, \sqrt {2} \sqrt {2 \, h r^{2} - \alpha ^{2}} \sqrt {h}\right )}{2 \, \sqrt {h}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.48, size = 32, normalized size = 0.80 \[ \frac {\sqrt {2}\,\ln \left (\sqrt {2\,h\,r^2-\alpha ^2}+\sqrt {2}\,\sqrt {h}\,r\right )}{2\,\sqrt {h}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.14, size = 66, normalized size = 1.65 \[ \begin {cases} \frac {\sqrt {2} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {h} r}{\alpha } \right )}}{2 \sqrt {h}} & \text {for}\: 2 \left |{\frac {h r^{2}}{\alpha ^{2}}}\right | > 1 \\- \frac {\sqrt {2} i \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {h} r}{\alpha } \right )}}{2 \sqrt {h}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________