Optimal. Leaf size=31 \[ \frac {x}{a}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {193, 321, 205} \[ \frac {x}{a}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 193
Rule 205
Rule 321
Rubi steps
\begin {align*} \int \frac {1}{a+\frac {b}{x^2}} \, dx &=\int \frac {x^2}{b+a x^2} \, dx\\ &=\frac {x}{a}-\frac {b \int \frac {1}{b+a x^2} \, dx}{a}\\ &=\frac {x}{a}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ \frac {x}{a}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 82, normalized size = 2.65 \[ \left [\frac {\sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right ) + 2 \, x}{2 \, a}, -\frac {\sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right ) - x}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 26, normalized size = 0.84 \[ -\frac {b \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 27, normalized size = 0.87 \[ -\frac {b \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.93, size = 26, normalized size = 0.84 \[ -\frac {b \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.09, size = 23, normalized size = 0.74 \[ \frac {x}{a}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{a^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.16, size = 56, normalized size = 1.81 \[ \frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (- a \sqrt {- \frac {b}{a^{3}}} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (a \sqrt {- \frac {b}{a^{3}}} + x \right )}}{2} + \frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________