Optimal. Leaf size=24 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rubi steps
\begin {align*} \int \frac {1}{a+b x^2} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 16, normalized size = 0.67
method | result | size |
default | \(\frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\) | \(16\) |
risch | \(-\frac {\ln \left (b x +\sqrt {-a b}\right )}{2 \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right )}{2 \sqrt {-a b}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 1.57, size = 15, normalized size = 0.62 \begin {gather*} \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.72, size = 67, normalized size = 2.79 \begin {gather*} \left [-\frac {\sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b}, \frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{a b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (22) = 44\).
time = 0.05, size = 53, normalized size = 2.21 \begin {gather*} - \frac {\sqrt {- \frac {1}{a b}} \log {\left (- a \sqrt {- \frac {1}{a b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (a \sqrt {- \frac {1}{a b}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.43, size = 15, normalized size = 0.62 \begin {gather*} \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.10, size = 16, normalized size = 0.67 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________