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 = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, 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 5959
Rule 5961
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*}
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Mathematica [A] time = 0.05, size = 45, normalized size = 0.54 \[ -\frac {\sqrt {a-a x^2} \left (\left (6 x^3-9 x\right ) \coth ^{-1}(x)-6 x^2+7\right )}{9 a^3 \left (x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 61, normalized size = 0.73 \[ \frac {\sqrt {-a x^{2} + a} {\left (12 \, x^{2} - 3 \, {\left (2 \, x^{3} - 3 \, x\right )} \log \left (\frac {x + 1}{x - 1}\right ) - 14\right )}}{18 \, {\left (a^{3} x^{4} - 2 \, a^{3} x^{2} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\relax (x)}{{\left (-a x^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 112, normalized size = 1.35 \[ \frac {\left (1+x \right ) \left (-1+3 \,\mathrm {arccoth}\relax (x )\right ) \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{72 \left (-1+x \right )^{2} a^{3}}-\frac {3 \left (\mathrm {arccoth}\relax (x )-1\right ) \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{8 \left (-1+x \right ) a^{3}}-\frac {3 \left (\mathrm {arccoth}\relax (x )+1\right ) \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{8 \left (1+x \right ) a^{3}}+\frac {\left (1+3 \,\mathrm {arccoth}\relax (x )\right ) \left (-1+x \right ) \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{72 \left (1+x \right )^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 67, normalized size = 0.81 \[ \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-a x^{2} + a} a^{2}} + \frac {x}{{\left (-a x^{2} + a\right )}^{\frac {3}{2}} a}\right )} \operatorname {arcoth}\relax (x) - \frac {2}{3 \, \sqrt {-a x^{2} + a} a^{2}} - \frac {1}{9 \, {\left (-a x^{2} + a\right )}^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acoth}\relax (x)}{{\left (a-a\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acoth}{\relax (x )}}{\left (- a \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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