Optimal. Leaf size=37 \[ \frac{x \coth ^{-1}(x)}{a \sqrt{a-a x^2}}-\frac{1}{a \sqrt{a-a x^2}} \]
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Rubi [A] time = 0.0254153, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {5959} \[ \frac{x \coth ^{-1}(x)}{a \sqrt{a-a x^2}}-\frac{1}{a \sqrt{a-a x^2}} \]
Antiderivative was successfully verified.
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Rule 5959
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(x)}{\left (a-a x^2\right )^{3/2}} \, dx &=-\frac{1}{a \sqrt{a-a x^2}}+\frac{x \coth ^{-1}(x)}{a \sqrt{a-a x^2}}\\ \end{align*}
Mathematica [A] time = 0.0438355, size = 30, normalized size = 0.81 \[ \frac{\sqrt{a-a x^2} \left (1-x \coth ^{-1}(x)\right )}{a^2 \left (x^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.226, size = 52, normalized size = 1.4 \begin{align*} -{\frac{{\rm arccoth} \left (x\right )-1}{ \left ( -2+2\,x \right ){a}^{2}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}-{\frac{{\rm arccoth} \left (x\right )+1}{ \left ( 2+2\,x \right ){a}^{2}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}} \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 = 1.5362, size = 92, normalized size = 2.49 \begin{align*} -\frac{\sqrt{-a x^{2} + a}{\left (x \log \left (\frac{x + 1}{x - 1}\right ) - 2\right )}}{2 \,{\left (a^{2} x^{2} - a^{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{acoth}{\left (x \right )}}{\left (- a \left (x - 1\right ) \left (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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (x\right )}{{\left (-a x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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