Optimal. Leaf size=48 \[ \frac{x \tanh ^{-1}(a x)}{c \sqrt{c-a^2 c x^2}}-\frac{1}{a c \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.0289554, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {5958} \[ \frac{x \tanh ^{-1}(a x)}{c \sqrt{c-a^2 c x^2}}-\frac{1}{a c \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 5958
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=-\frac{1}{a c \sqrt{c-a^2 c x^2}}+\frac{x \tanh ^{-1}(a x)}{c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0553198, size = 43, normalized size = 0.9 \[ \frac{\sqrt{c-a^2 c x^2} \left (1-a x \tanh ^{-1}(a x)\right )}{a c^2 \left (a^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.25, size = 74, normalized size = 1.5 \begin{align*} -{\frac{{\it Artanh} \left ( ax \right ) -1}{2\,a \left ( ax-1 \right ){c}^{2}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}}-{\frac{{\it Artanh} \left ( ax \right ) +1}{2\,a \left ( ax+1 \right ){c}^{2}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}} \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 = 1.6681, size = 115, normalized size = 2.4 \begin{align*} -\frac{\sqrt{-a^{2} c x^{2} + c}{\left (a x \log \left (-\frac{a x + 1}{a x - 1}\right ) - 2\right )}}{2 \,{\left (a^{3} c^{2} x^{2} - a c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27115, size = 95, normalized size = 1.98 \begin{align*} -\frac{\sqrt{-a^{2} c x^{2} + c} x \log \left (-\frac{a x + 1}{a x - 1}\right )}{2 \,{\left (a^{2} c x^{2} - c\right )} c} - \frac{1}{\sqrt{-a^{2} c x^{2} + c} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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