Optimal. Leaf size=16 \[ \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {223, 209}
\begin {gather*} \text {ArcTan}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 223
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a^2-x^2}} \, dx &=\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {a^2-x^2}}\right )\\ &=\tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 15, normalized size = 0.94
method | result | size |
default | \(\arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 3.26, size = 6, normalized size = 0.38 \begin {gather*} \arcsin \left (\frac {x}{a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.99, size = 23, normalized size = 1.44 \begin {gather*} -2 \, \arctan \left (-\frac {a - \sqrt {a^{2} - x^{2}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.44, size = 19, normalized size = 1.19 \begin {gather*} \begin {cases} - i \operatorname {acosh}{\left (\frac {x}{a} \right )} & \text {for}\: \left |{\frac {x^{2}}{a^{2}}}\right | > 1 \\\operatorname {asin}{\left (\frac {x}{a} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.48, size = 28, normalized size = 1.75 \begin {gather*} \frac {1}{2} \, a^{2} \arcsin \left (\frac {x}{a}\right ) \mathrm {sgn}\left (a\right ) + \frac {1}{2} \, \sqrt {a^{2} - x^{2}} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.16, size = 14, normalized size = 0.88 \begin {gather*} \mathrm {atan}\left (\frac {x}{\sqrt {a^2-x^2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________