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 = 0.0341809, antiderivative size = 50, normalized size of antiderivative = 1., 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 5995
Rule 199
Rule 206
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*}
Mathematica [A] time = 0.0505672, size = 50, normalized size = 1. \[ \frac{1}{64} \left (\frac{6 x}{x^2-1}-\frac{4 x}{\left (x^2-1\right )^2}+\frac{16 \coth ^{-1}(x)}{\left (x^2-1\right )^2}+3 \log (1-x)-3 \log (x+1)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 53, normalized size = 1.1 \begin{align*}{\frac{{\rm arccoth} \left (x\right )}{4\, \left ({x}^{2}-1 \right ) ^{2}}}-{\frac{1}{64\, \left ( -1+x \right ) ^{2}}}+{\frac{3}{-64+64\,x}}+{\frac{3\,\ln \left ( -1+x \right ) }{64}}+{\frac{1}{64\, \left ( 1+x \right ) ^{2}}}+{\frac{3}{64+64\,x}}-{\frac{3\,\ln \left ( 1+x \right ) }{64}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949001, size = 63, normalized size = 1.26 \begin{align*} \frac{3 \, x^{3} - 5 \, x}{32 \,{\left (x^{4} - 2 \, x^{2} + 1\right )}} + \frac{\operatorname{arcoth}\left (x\right )}{4 \,{\left (x^{2} - 1\right )}^{2}} - \frac{3}{64} \, \log \left (x + 1\right ) + \frac{3}{64} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59255, size = 111, normalized size = 2.22 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.32896, size = 88, normalized size = 1.76 \begin{align*} - \frac{3 x^{4} \operatorname{acoth}{\left (x \right )}}{32 x^{4} - 64 x^{2} + 32} + \frac{3 x^{3}}{32 x^{4} - 64 x^{2} + 32} + \frac{6 x^{2} \operatorname{acoth}{\left (x \right )}}{32 x^{4} - 64 x^{2} + 32} - \frac{5 x}{32 x^{4} - 64 x^{2} + 32} + \frac{5 \operatorname{acoth}{\left (x \right )}}{32 x^{4} - 64 x^{2} + 32} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x \operatorname{arcoth}\left (x\right )}{{\left (x^{2} - 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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