Integrand size = 15, antiderivative size = 83 \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=-\frac {1}{9 a \left (a-a x^2\right )^{3/2}}-\frac {2}{3 a^2 \sqrt {a-a x^2}}+\frac {x \coth ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}}+\frac {2 x \coth ^{-1}(x)}{3 a^2 \sqrt {a-a x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6108, 6106} \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=-\frac {2}{3 a^2 \sqrt {a-a x^2}}+\frac {2 x \coth ^{-1}(x)}{3 a^2 \sqrt {a-a x^2}}-\frac {1}{9 a \left (a-a x^2\right )^{3/2}}+\frac {x \coth ^{-1}(x)}{3 a \left (a-a x^2\right )^{3/2}} \]
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Rule 6106
Rule 6108
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.54 \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=-\frac {\sqrt {a-a x^2} \left (7-6 x^2+\left (-9 x+6 x^3\right ) \coth ^{-1}(x)\right )}{9 a^3 \left (-1+x^2\right )^2} \]
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Time = 0.60 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.98
method | result | size |
risch | \(-\frac {x \left (2 x^{2}-3\right ) \ln \left (x -1\right )}{6 a^{2} \left (x^{2}-1\right ) \sqrt {-a \left (x^{2}-1\right )}}+\frac {6 x^{3} \ln \left (1+x \right )-12 x^{2}-9 \ln \left (1+x \right ) x +14}{18 a^{2} \left (x^{2}-1\right ) \sqrt {-a \left (x^{2}-1\right )}}\) | \(81\) |
default | \(\frac {\left (1+x \right ) \left (-1+3 \,\operatorname {arccoth}\left (x \right )\right ) \sqrt {-\left (1+x \right ) \left (x -1\right ) a}}{72 \left (x -1\right )^{2} a^{3}}-\frac {3 \left (-1+\operatorname {arccoth}\left (x \right )\right ) \sqrt {-\left (1+x \right ) \left (x -1\right ) a}}{8 a^{3} \left (x -1\right )}-\frac {3 \left (1+\operatorname {arccoth}\left (x \right )\right ) \sqrt {-\left (1+x \right ) \left (x -1\right ) a}}{8 \left (1+x \right ) a^{3}}+\frac {\left (1+3 \,\operatorname {arccoth}\left (x \right )\right ) \left (x -1\right ) \sqrt {-\left (1+x \right ) \left (x -1\right ) a}}{72 \left (1+x \right )^{2} a^{3}}\) | \(112\) |
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Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=\frac {\sqrt {-a x^{2} + a} {\left (12 \, x^{2} - 3 \, {\left (2 \, x^{3} - 3 \, x\right )} \log \left (\frac {x + 1}{x - 1}\right ) - 14\right )}}{18 \, {\left (a^{3} x^{4} - 2 \, a^{3} x^{2} + a^{3}\right )}} \]
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\[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acoth}{\left (x \right )}}{\left (- a \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81 \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=\frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-a x^{2} + a} a^{2}} + \frac {x}{{\left (-a x^{2} + a\right )}^{\frac {3}{2}} a}\right )} \operatorname {arcoth}\left (x\right ) - \frac {2}{3 \, \sqrt {-a x^{2} + a} a^{2}} - \frac {1}{9 \, {\left (-a x^{2} + a\right )}^{\frac {3}{2}} a} \]
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Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.08 \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=-\frac {\sqrt {-a x^{2} + a} x {\left (\frac {2 \, x^{2}}{a} - \frac {3}{a}\right )} \log \left (-\frac {\frac {1}{x} + 1}{\frac {1}{x} - 1}\right )}{6 \, {\left (a x^{2} - a\right )}^{2}} - \frac {6 \, a x^{2} - 7 \, a}{9 \, {\left (a x^{2} - a\right )} \sqrt {-a x^{2} + a} a^{2}} \]
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Timed out. \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {acoth}\left (x\right )}{{\left (a-a\,x^2\right )}^{5/2}} \,d x \]
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