Optimal. Leaf size=50 \[ -\frac {3 x}{32 \left (1-x^2\right )}-\frac {x}{16 \left (1-x^2\right )^2}+\frac {\coth ^{-1}(x)}{4 \left (1-x^2\right )^2}-\frac {3}{32} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5995, 199, 206} \[ -\frac {3 x}{32 \left (1-x^2\right )}-\frac {x}{16 \left (1-x^2\right )^2}+\frac {\coth ^{-1}(x)}{4 \left (1-x^2\right )^2}-\frac {3}{32} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 5995
Rubi steps
\begin {align*} \int \frac {x \coth ^{-1}(x)}{\left (1-x^2\right )^3} \, dx &=\frac {\coth ^{-1}(x)}{4 \left (1-x^2\right )^2}-\frac {1}{4} \int \frac {1}{\left (1-x^2\right )^3} \, dx\\ &=-\frac {x}{16 \left (1-x^2\right )^2}+\frac {\coth ^{-1}(x)}{4 \left (1-x^2\right )^2}-\frac {3}{16} \int \frac {1}{\left (1-x^2\right )^2} \, dx\\ &=-\frac {x}{16 \left (1-x^2\right )^2}-\frac {3 x}{32 \left (1-x^2\right )}+\frac {\coth ^{-1}(x)}{4 \left (1-x^2\right )^2}-\frac {3}{32} \int \frac {1}{1-x^2} \, dx\\ &=-\frac {x}{16 \left (1-x^2\right )^2}-\frac {3 x}{32 \left (1-x^2\right )}+\frac {\coth ^{-1}(x)}{4 \left (1-x^2\right )^2}-\frac {3}{32} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 50, normalized size = 1.00 \[ \frac {1}{64} \left (\frac {6 x}{x^2-1}-\frac {4 x}{\left (x^2-1\right )^2}+\frac {16 \coth ^{-1}(x)}{\left (x^2-1\right )^2}+3 \log (1-x)-3 \log (x+1)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 47, normalized size = 0.94 \[ \frac {6 \, x^{3} - {\left (3 \, x^{4} - 6 \, x^{2} - 5\right )} \log \left (\frac {x + 1}{x - 1}\right ) - 10 \, x}{64 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \]
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 \[ \int -\frac {x \operatorname {arcoth}\relax (x)}{{\left (x^{2} - 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 53, normalized size = 1.06 \[ \frac {\mathrm {arccoth}\relax (x )}{4 \left (x^{2}-1\right )^{2}}-\frac {1}{64 \left (-1+x \right )^{2}}+\frac {3}{64 \left (-1+x \right )}+\frac {3 \ln \left (-1+x \right )}{64}+\frac {1}{64 \left (1+x \right )^{2}}+\frac {3}{64 \left (1+x \right )}-\frac {3 \ln \left (1+x \right )}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 47, normalized size = 0.94 \[ \frac {3 \, x^{3} - 5 \, x}{32 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} + \frac {\operatorname {arcoth}\relax (x)}{4 \, {\left (x^{2} - 1\right )}^{2}} - \frac {3}{64} \, \log \left (x + 1\right ) + \frac {3}{64} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 34, normalized size = 0.68 \[ \frac {3\,\ln \left (x-1\right )}{64}-\frac {3\,\ln \left (x+1\right )}{64}+\frac {\frac {\mathrm {acoth}\relax (x)}{4}-\frac {5\,x}{32}+\frac {3\,x^3}{32}}{{\left (x^2-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.91, size = 88, normalized size = 1.76 \[ - \frac {3 x^{4} \operatorname {acoth}{\relax (x )}}{32 x^{4} - 64 x^{2} + 32} + \frac {3 x^{3}}{32 x^{4} - 64 x^{2} + 32} + \frac {6 x^{2} \operatorname {acoth}{\relax (x )}}{32 x^{4} - 64 x^{2} + 32} - \frac {5 x}{32 x^{4} - 64 x^{2} + 32} + \frac {5 \operatorname {acoth}{\relax (x )}}{32 x^{4} - 64 x^{2} + 32} \]
Verification of antiderivative is not currently implemented for this CAS.
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