Optimal. Leaf size=33 \[ \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {325, 208} \[ \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a-b x^2\right )} \, dx &=-\frac {1}{a x}+\frac {b \int \frac {1}{a-b x^2} \, dx}{a}\\ &=-\frac {1}{a x}+\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.84, size = 82, normalized size = 2.48 \[ \left [\frac {x \sqrt {\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {\frac {b}{a}} + a}{b x^{2} - a}\right ) - 2}{2 \, a x}, -\frac {x \sqrt {-\frac {b}{a}} \arctan \left (x \sqrt {-\frac {b}{a}}\right ) + 1}{a x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.62, size = 31, normalized size = 0.94 \[ -\frac {b \arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{\sqrt {-a b} a} - \frac {1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 29, normalized size = 0.88 \[ \frac {b \arctanh \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}-\frac {1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.89, size = 44, normalized size = 1.33 \[ -\frac {b \log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{2 \, \sqrt {a b} a} - \frac {1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.61, size = 25, normalized size = 0.76 \[ \frac {\sqrt {b}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.19, size = 58, normalized size = 1.76 \[ - \frac {\sqrt {\frac {b}{a^{3}}} \log {\left (- \frac {a^{2} \sqrt {\frac {b}{a^{3}}}}{b} + x \right )}}{2} + \frac {\sqrt {\frac {b}{a^{3}}} \log {\left (\frac {a^{2} \sqrt {\frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {1}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________