Optimal. Leaf size=33 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x} \]
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Rubi [A] time = 0.0121497, antiderivative size = 33, normalized size of antiderivative = 1., 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.
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Rule 325
Rule 208
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 = 0.0109404, size = 33, normalized size = 1. \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}-\frac{1}{a x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 29, normalized size = 0.9 \begin{align*}{\frac{b}{a}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{ax}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2156, size = 173, normalized size = 5.24 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.329602, size = 58, normalized size = 1.76 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.93039, size = 42, normalized size = 1.27 \begin{align*} -\frac{b \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{\sqrt{-a b} a} - \frac{1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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