Optimal. Leaf size=83 \[ -\frac{2}{3 a^2 \sqrt{a-a x^2}}+\frac{2 x \coth ^{-1}(x)}{3 a^2 \sqrt{a-a x^2}}-\frac{1}{9 a \left (a-a x^2\right )^{3/2}}+\frac{x \coth ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0534298, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {5961, 5959} \[ -\frac{2}{3 a^2 \sqrt{a-a x^2}}+\frac{2 x \coth ^{-1}(x)}{3 a^2 \sqrt{a-a x^2}}-\frac{1}{9 a \left (a-a x^2\right )^{3/2}}+\frac{x \coth ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5961
Rule 5959
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx &=-\frac{1}{9 a \left (a-a x^2\right )^{3/2}}+\frac{x \coth ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}}+\frac{2 \int \frac{\coth ^{-1}(x)}{\left (a-a x^2\right )^{3/2}} \, dx}{3 a}\\ &=-\frac{1}{9 a \left (a-a x^2\right )^{3/2}}-\frac{2}{3 a^2 \sqrt{a-a x^2}}+\frac{x \coth ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}}+\frac{2 x \coth ^{-1}(x)}{3 a^2 \sqrt{a-a x^2}}\\ \end{align*}
Mathematica [A] time = 0.0499682, size = 45, normalized size = 0.54 \[ -\frac{\sqrt{a-a x^2} \left (-6 x^2+\left (6 x^3-9 x\right ) \coth ^{-1}(x)+7\right )}{9 a^3 \left (x^2-1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.238, size = 112, normalized size = 1.4 \begin{align*}{\frac{ \left ( 1+x \right ) \left ( -1+3\,{\rm arccoth} \left (x\right ) \right ) }{72\, \left ( -1+x \right ) ^{2}{a}^{3}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}-{\frac{3\,{\rm arccoth} \left (x\right )-3}{ \left ( -8+8\,x \right ){a}^{3}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}-{\frac{3\,{\rm arccoth} \left (x\right )+3}{ \left ( 8+8\,x \right ){a}^{3}}\sqrt{- \left ( -1+x \right ) \left ( 1+x \right ) a}}+{\frac{ \left ( 1+3\,{\rm arccoth} \left (x\right ) \right ) \left ( -1+x \right ) }{72\, \left ( 1+x \right ) ^{2}{a}^{3}}\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 [A] time = 1.01709, size = 90, normalized size = 1.08 \begin{align*} \frac{1}{3} \,{\left (\frac{2 \, x}{\sqrt{-a x^{2} + a} a^{2}} + \frac{x}{{\left (-a x^{2} + a\right )}^{\frac{3}{2}} a}\right )} \operatorname{arcoth}\left (x\right ) - \frac{2}{3 \, \sqrt{-a x^{2} + a} a^{2}} - \frac{1}{9 \,{\left (-a x^{2} + a\right )}^{\frac{3}{2}} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58402, size = 140, normalized size = 1.69 \begin{align*} \frac{\sqrt{-a x^{2} + a}{\left (12 \, x^{2} - 3 \,{\left (2 \, x^{3} - 3 \, x\right )} \log \left (\frac{x + 1}{x - 1}\right ) - 14\right )}}{18 \,{\left (a^{3} x^{4} - 2 \, a^{3} x^{2} + a^{3}\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{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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