Optimal. Leaf size=67 \[ -\frac {3}{16 \left (1-x^2\right )}-\frac {1}{16 \left (1-x^2\right )^2}+\frac {3 x \coth ^{-1}(x)}{8 \left (1-x^2\right )}+\frac {x \coth ^{-1}(x)}{4 \left (1-x^2\right )^2}+\frac {3}{16} \coth ^{-1}(x)^2 \]
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Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5961, 5957, 261} \[ -\frac {3}{16 \left (1-x^2\right )}-\frac {1}{16 \left (1-x^2\right )^2}+\frac {3 x \coth ^{-1}(x)}{8 \left (1-x^2\right )}+\frac {x \coth ^{-1}(x)}{4 \left (1-x^2\right )^2}+\frac {3}{16} \coth ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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Rule 261
Rule 5957
Rule 5961
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(x)}{\left (1-x^2\right )^3} \, dx &=-\frac {1}{16 \left (1-x^2\right )^2}+\frac {x \coth ^{-1}(x)}{4 \left (1-x^2\right )^2}+\frac {3}{4} \int \frac {\coth ^{-1}(x)}{\left (1-x^2\right )^2} \, dx\\ &=-\frac {1}{16 \left (1-x^2\right )^2}+\frac {x \coth ^{-1}(x)}{4 \left (1-x^2\right )^2}+\frac {3 x \coth ^{-1}(x)}{8 \left (1-x^2\right )}+\frac {3}{16} \coth ^{-1}(x)^2-\frac {3}{8} \int \frac {x}{\left (1-x^2\right )^2} \, dx\\ &=-\frac {1}{16 \left (1-x^2\right )^2}-\frac {3}{16 \left (1-x^2\right )}+\frac {x \coth ^{-1}(x)}{4 \left (1-x^2\right )^2}+\frac {3 x \coth ^{-1}(x)}{8 \left (1-x^2\right )}+\frac {3}{16} \coth ^{-1}(x)^2\\ \end {align*}
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Mathematica [A] time = 0.06, size = 43, normalized size = 0.64 \[ -\frac {-3 x^2+2 \left (3 x^2-5\right ) x \coth ^{-1}(x)-3 \left (x^2-1\right )^2 \coth ^{-1}(x)^2+4}{16 \left (x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 66, normalized size = 0.99 \[ \frac {3 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (\frac {x + 1}{x - 1}\right )^{2} + 12 \, x^{2} - 4 \, {\left (3 \, x^{3} - 5 \, x\right )} \log \left (\frac {x + 1}{x - 1}\right ) - 16}{64 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \]
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 \[ \int -\frac {\operatorname {arcoth}\relax (x)}{{\left (x^{2} - 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 131, normalized size = 1.96 \[ \frac {\mathrm {arccoth}\relax (x )}{16 \left (-1+x \right )^{2}}-\frac {3 \,\mathrm {arccoth}\relax (x )}{16 \left (-1+x \right )}-\frac {3 \,\mathrm {arccoth}\relax (x ) \ln \left (-1+x \right )}{16}-\frac {\mathrm {arccoth}\relax (x )}{16 \left (1+x \right )^{2}}-\frac {3 \,\mathrm {arccoth}\relax (x )}{16 \left (1+x \right )}+\frac {3 \,\mathrm {arccoth}\relax (x ) \ln \left (1+x \right )}{16}-\frac {3 \ln \left (-1+x \right )^{2}}{64}+\frac {3 \ln \left (-1+x \right ) \ln \left (\frac {1}{2}+\frac {x}{2}\right )}{32}-\frac {3 \ln \left (1+x \right )^{2}}{64}+\frac {3 \left (\ln \left (1+x \right )-\ln \left (\frac {1}{2}+\frac {x}{2}\right )\right ) \ln \left (-\frac {x}{2}+\frac {1}{2}\right )}{32}-\frac {1}{64 \left (-1+x \right )^{2}}+\frac {7}{64 \left (-1+x \right )}-\frac {1}{64 \left (1+x \right )^{2}}-\frac {7}{64 \left (1+x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 118, normalized size = 1.76 \[ -\frac {1}{16} \, {\left (\frac {2 \, {\left (3 \, x^{3} - 5 \, x\right )}}{x^{4} - 2 \, x^{2} + 1} - 3 \, \log \left (x + 1\right ) + 3 \, \log \left (x - 1\right )\right )} \operatorname {arcoth}\relax (x) - \frac {3 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x + 1\right )^{2} - 6 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x + 1\right ) \log \left (x - 1\right ) + 3 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x - 1\right )^{2} - 12 \, x^{2} + 16}{64 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 112, normalized size = 1.67 \[ \frac {3\,{\ln \left (\frac {1}{x}+1\right )}^2}{64}-\ln \left (1-\frac {1}{x}\right )\,\left (\frac {3\,\ln \left (\frac {1}{x}+1\right )}{32}+\frac {\frac {5\,x}{16}-\frac {3\,x^3}{16}}{x^4-2\,x^2+1}\right )+\frac {3\,{\ln \left (1-\frac {1}{x}\right )}^2}{64}+\frac {\frac {3\,x^2}{16}-\frac {1}{4}}{x^4-2\,x^2+1}+\frac {\ln \left (\frac {1}{x}+1\right )\,\left (\frac {5\,x}{16}-\frac {3\,x^3}{16}\right )}{x^4-2\,x^2+1} \]
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 \[ - \int \frac {\operatorname {acoth}{\relax (x )}}{x^{6} - 3 x^{4} + 3 x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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