Integrand size = 17, antiderivative size = 22 \[ \int \frac {1}{x \sqrt {-a^2+x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {-a^2+x^2}}{a}\right )}{a} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {272, 65, 209} \[ \int \frac {1}{x \sqrt {-a^2+x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {x^2-a^2}}{a}\right )}{a} \]
[In]
[Out]
Rule 65
Rule 209
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {-a^2+x}} \, dx,x,x^2\right ) \\ & = \text {Subst}\left (\int \frac {1}{a^2+x^2} \, dx,x,\sqrt {-a^2+x^2}\right ) \\ & = \frac {\arctan \left (\frac {\sqrt {-a^2+x^2}}{a}\right )}{a} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {-a^2+x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {-a^2+x^2}}{a}\right )}{a} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(\frac {\arctan \left (\frac {\sqrt {-a^{2}+x^{2}}}{a}\right )}{a}\) | \(21\) |
default | \(-\frac {\ln \left (\frac {-2 a^{2}+2 \sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}}{x}\right )}{\sqrt {-a^{2}}}\) | \(41\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x \sqrt {-a^2+x^2}} \, dx=\frac {2 \, \arctan \left (-\frac {x - \sqrt {-a^{2} + x^{2}}}{a}\right )}{a} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {-a^2+x^2}} \, dx=\begin {cases} \frac {i \operatorname {acosh}{\left (\frac {a}{x} \right )}}{a} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\- \frac {\operatorname {asin}{\left (\frac {a}{x} \right )}}{a} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x \sqrt {-a^2+x^2}} \, dx=-\frac {\arcsin \left (\frac {a}{{\left | x \right |}}\right )}{a} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x \sqrt {-a^2+x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {-a^{2} + x^{2}}}{a}\right )}{a} \]
[In]
[Out]
Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \sqrt {-a^2+x^2}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {x^2-a^2}}{\sqrt {a^2}}\right )}{\sqrt {a^2}} \]
[In]
[Out]