Integrand size = 15, antiderivative size = 124 \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{7/2}} \, dx=-\frac {1}{25 a \left (a-a x^2\right )^{5/2}}-\frac {4}{45 a^2 \left (a-a x^2\right )^{3/2}}-\frac {8}{15 a^3 \sqrt {a-a x^2}}+\frac {x \coth ^{-1}(x)}{5 a \left (a-a x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(x)}{15 a^2 \left (a-a x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(x)}{15 a^3 \sqrt {a-a x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6108, 6106} \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{7/2}} \, dx=-\frac {8}{15 a^3 \sqrt {a-a x^2}}+\frac {8 x \coth ^{-1}(x)}{15 a^3 \sqrt {a-a x^2}}-\frac {4}{45 a^2 \left (a-a x^2\right )^{3/2}}+\frac {4 x \coth ^{-1}(x)}{15 a^2 \left (a-a x^2\right )^{3/2}}-\frac {1}{25 a \left (a-a x^2\right )^{5/2}}+\frac {x \coth ^{-1}(x)}{5 a \left (a-a x^2\right )^{5/2}} \]
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Rule 6106
Rule 6108
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{25 a \left (a-a x^2\right )^{5/2}}+\frac {x \coth ^{-1}(x)}{5 a \left (a-a x^2\right )^{5/2}}+\frac {4 \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{5/2}} \, dx}{5 a} \\ & = -\frac {1}{25 a \left (a-a x^2\right )^{5/2}}-\frac {4}{45 a^2 \left (a-a x^2\right )^{3/2}}+\frac {x \coth ^{-1}(x)}{5 a \left (a-a x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(x)}{15 a^2 \left (a-a x^2\right )^{3/2}}+\frac {8 \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{3/2}} \, dx}{15 a^2} \\ & = -\frac {1}{25 a \left (a-a x^2\right )^{5/2}}-\frac {4}{45 a^2 \left (a-a x^2\right )^{3/2}}-\frac {8}{15 a^3 \sqrt {a-a x^2}}+\frac {x \coth ^{-1}(x)}{5 a \left (a-a x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(x)}{15 a^2 \left (a-a x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(x)}{15 a^3 \sqrt {a-a x^2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.44 \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{7/2}} \, dx=\frac {\sqrt {a-a x^2} \left (149-260 x^2+120 x^4-15 x \left (15-20 x^2+8 x^4\right ) \coth ^{-1}(x)\right )}{225 a^4 \left (-1+x^2\right )^3} \]
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Time = 0.58 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.81
method | result | size |
risch | \(-\frac {x \left (8 x^{4}-20 x^{2}+15\right ) \ln \left (x -1\right )}{30 a^{3} \left (x^{2}-1\right )^{2} \sqrt {-a \left (x^{2}-1\right )}}+\frac {120 x^{5} \ln \left (1+x \right )-240 x^{4}-300 x^{3} \ln \left (1+x \right )+520 x^{2}+225 \ln \left (1+x \right ) x -298}{450 a^{3} \left (x^{2}-1\right )^{2} \sqrt {-a \left (x^{2}-1\right )}}\) | \(100\) |
default | \(-\frac {\left (1+x \right )^{2} \left (-1+5 \,\operatorname {arccoth}\left (x \right )\right ) \sqrt {-\left (1+x \right ) \left (x -1\right ) a}}{800 \left (x -1\right )^{3} a^{4}}+\frac {5 \left (1+x \right ) \left (-1+3 \,\operatorname {arccoth}\left (x \right )\right ) \sqrt {-\left (1+x \right ) \left (x -1\right ) a}}{288 a^{4} \left (x -1\right )^{2}}-\frac {5 \left (-1+\operatorname {arccoth}\left (x \right )\right ) \sqrt {-\left (1+x \right ) \left (x -1\right ) a}}{16 \left (x -1\right ) a^{4}}-\frac {5 \left (1+\operatorname {arccoth}\left (x \right )\right ) \sqrt {-\left (1+x \right ) \left (x -1\right ) a}}{16 \left (1+x \right ) a^{4}}+\frac {5 \left (1+3 \,\operatorname {arccoth}\left (x \right )\right ) \left (x -1\right ) \sqrt {-\left (1+x \right ) \left (x -1\right ) a}}{288 \left (1+x \right )^{2} a^{4}}-\frac {\left (1+5 \,\operatorname {arccoth}\left (x \right )\right ) \left (x -1\right )^{2} \sqrt {-\left (1+x \right ) \left (x -1\right ) a}}{800 \left (1+x \right )^{3} a^{4}}\) | \(176\) |
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Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.65 \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{7/2}} \, dx=\frac {{\left (240 \, x^{4} - 520 \, x^{2} - 15 \, {\left (8 \, x^{5} - 20 \, x^{3} + 15 \, x\right )} \log \left (\frac {x + 1}{x - 1}\right ) + 298\right )} \sqrt {-a x^{2} + a}}{450 \, {\left (a^{4} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{4} x^{2} - a^{4}\right )}} \]
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\[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {acoth}{\left (x \right )}}{\left (- a \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {7}{2}}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.80 \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{7/2}} \, dx=\frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {-a x^{2} + a} a^{3}} + \frac {4 \, x}{{\left (-a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} + \frac {3 \, x}{{\left (-a x^{2} + a\right )}^{\frac {5}{2}} a}\right )} \operatorname {arcoth}\left (x\right ) - \frac {8}{15 \, \sqrt {-a x^{2} + a} a^{3}} - \frac {4}{45 \, {\left (-a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {1}{25 \, {\left (-a x^{2} + a\right )}^{\frac {5}{2}} a} \]
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Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98 \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{7/2}} \, dx=-\frac {\sqrt {-a x^{2} + a} {\left (4 \, x^{2} {\left (\frac {2 \, x^{2}}{a} - \frac {5}{a}\right )} + \frac {15}{a}\right )} x \log \left (-\frac {\frac {1}{x} + 1}{\frac {1}{x} - 1}\right )}{30 \, {\left (a x^{2} - a\right )}^{3}} - \frac {120 \, {\left (a x^{2} - a\right )}^{2} - 20 \, {\left (a x^{2} - a\right )} a + 9 \, a^{2}}{225 \, {\left (a x^{2} - a\right )}^{2} \sqrt {-a x^{2} + a} a^{3}} \]
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Timed out. \[ \int \frac {\coth ^{-1}(x)}{\left (a-a x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {acoth}\left (x\right )}{{\left (a-a\,x^2\right )}^{7/2}} \,d x \]
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