Optimal. Leaf size=124 \[ -\frac {1}{25 a \left (a-a x^2\right )^{5/2}}-\frac {4}{45 a^2 \left (a-a x^2\right )^{3/2}}-\frac {8}{15 a^3 \sqrt {a-a x^2}}+\frac {x \coth ^{-1}(x)}{5 a \left (a-a x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(x)}{15 a^2 \left (a-a x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(x)}{15 a^3 \sqrt {a-a x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6108, 6106}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6106
Rule 6108
Rubi steps
\begin {align*} \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 {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*}
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Mathematica [A]
time = 0.04, size = 55, normalized size = 0.44 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 176, normalized size = 1.42
method | result | size |
risch | \(\frac {x \left (8 x^{4}-20 x^{2}+15\right ) \ln \left (1+x \right )}{30 a^{3} \left (x^{2}-1\right )^{2} \sqrt {-a \left (x^{2}-1\right )}}-\frac {120 x^{5} \ln \left (-1+x \right )+240 x^{4}-300 x^{3} \ln \left (-1+x \right )-520 x^{2}+225 \ln \left (-1+x \right ) x +298}{450 a^{3} \left (x^{2}-1\right )^{2} \sqrt {-a \left (x^{2}-1\right )}}\) | \(100\) |
default | \(-\frac {\left (1+x \right )^{2} \left (-1+5 \,\mathrm {arccoth}\left (x \right )\right ) \sqrt {-a \left (1+x \right ) \left (-1+x \right )}}{800 \left (-1+x \right )^{3} a^{4}}+\frac {5 \left (1+x \right ) \left (-1+3 \,\mathrm {arccoth}\left (x \right )\right ) \sqrt {-a \left (1+x \right ) \left (-1+x \right )}}{288 \left (-1+x \right )^{2} a^{4}}-\frac {5 \left (-1+\mathrm {arccoth}\left (x \right )\right ) \sqrt {-a \left (1+x \right ) \left (-1+x \right )}}{16 \left (-1+x \right ) a^{4}}-\frac {5 \left (1+\mathrm {arccoth}\left (x \right )\right ) \sqrt {-a \left (1+x \right ) \left (-1+x \right )}}{16 \left (1+x \right ) a^{4}}+\frac {5 \left (1+3 \,\mathrm {arccoth}\left (x \right )\right ) \left (-1+x \right ) \sqrt {-a \left (1+x \right ) \left (-1+x \right )}}{288 \left (1+x \right )^{2} a^{4}}-\frac {\left (1+5 \,\mathrm {arccoth}\left (x \right )\right ) \left (-1+x \right )^{2} \sqrt {-a \left (1+x \right ) \left (-1+x \right )}}{800 \left (1+x \right )^{3} a^{4}}\) | \(176\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 99, normalized size = 0.80 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 81, normalized size = 0.65 \begin {gather*} \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 )}} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {acoth}{\left (x \right )}}{\left (- a \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 122, normalized size = 0.98 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {acoth}\left (x\right )}{{\left (a-a\,x^2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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