Optimal. Leaf size=62 \[ \frac {x}{4 \left (1-x^2\right )}-\frac {\coth ^{-1}(x)}{2 \left (1-x^2\right )}+\frac {x \coth ^{-1}(x)^2}{2 \left (1-x^2\right )}+\frac {1}{6} \coth ^{-1}(x)^3+\frac {1}{4} \tanh ^{-1}(x) \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6104, 6142,
205, 212} \begin {gather*} \frac {x}{4 \left (1-x^2\right )}+\frac {x \coth ^{-1}(x)^2}{2 \left (1-x^2\right )}-\frac {\coth ^{-1}(x)}{2 \left (1-x^2\right )}+\frac {1}{4} \tanh ^{-1}(x)+\frac {1}{6} \coth ^{-1}(x)^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 212
Rule 6104
Rule 6142
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(x)^2}{\left (1-x^2\right )^2} \, dx &=\frac {x \coth ^{-1}(x)^2}{2 \left (1-x^2\right )}+\frac {1}{6} \coth ^{-1}(x)^3-\int \frac {x \coth ^{-1}(x)}{\left (1-x^2\right )^2} \, dx\\ &=-\frac {\coth ^{-1}(x)}{2 \left (1-x^2\right )}+\frac {x \coth ^{-1}(x)^2}{2 \left (1-x^2\right )}+\frac {1}{6} \coth ^{-1}(x)^3+\frac {1}{2} \int \frac {1}{\left (1-x^2\right )^2} \, dx\\ &=\frac {x}{4 \left (1-x^2\right )}-\frac {\coth ^{-1}(x)}{2 \left (1-x^2\right )}+\frac {x \coth ^{-1}(x)^2}{2 \left (1-x^2\right )}+\frac {1}{6} \coth ^{-1}(x)^3+\frac {1}{4} \int \frac {1}{1-x^2} \, dx\\ &=\frac {x}{4 \left (1-x^2\right )}-\frac {\coth ^{-1}(x)}{2 \left (1-x^2\right )}+\frac {x \coth ^{-1}(x)^2}{2 \left (1-x^2\right )}+\frac {1}{6} \coth ^{-1}(x)^3+\frac {1}{4} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 61, normalized size = 0.98 \begin {gather*} \frac {-6 x+12 \coth ^{-1}(x)-12 x \coth ^{-1}(x)^2+4 \left (-1+x^2\right ) \coth ^{-1}(x)^3-3 \left (-1+x^2\right ) \log (1-x)+3 \left (-1+x^2\right ) \log (1+x)}{24 \left (-1+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.24, size = 707, normalized size = 11.40
method | result | size |
risch | \(\frac {\ln \left (1+x \right )^{3}}{48}-\frac {\left (\ln \left (-1+x \right ) x^{2}+2 x -\ln \left (-1+x \right )\right ) \ln \left (1+x \right )^{2}}{16 \left (x^{2}-1\right )}+\frac {\left (x^{2} \ln \left (-1+x \right )^{2}+4 \ln \left (-1+x \right ) x -\ln \left (-1+x \right )^{2}+4\right ) \ln \left (1+x \right )}{16 \left (-1+x \right ) \left (1+x \right )}+\frac {-x^{2} \ln \left (-1+x \right )^{3}+6 \ln \left (1+x \right ) x^{2}-6 \ln \left (-1+x \right ) x^{2}-6 x \ln \left (-1+x \right )^{2}+\ln \left (-1+x \right )^{3}-6 \ln \left (1+x \right )-6 \ln \left (-1+x \right )-12 x}{48 \left (-1+x \right ) \left (1+x \right )}\) | \(158\) |
default | \(-\frac {\mathrm {arccoth}\left (x \right )^{2}}{4 \left (1+x \right )}+\frac {\mathrm {arccoth}\left (x \right )^{2} \ln \left (1+x \right )}{4}-\frac {\mathrm {arccoth}\left (x \right )^{2}}{4 \left (-1+x \right )}-\frac {\mathrm {arccoth}\left (x \right )^{2} \ln \left (-1+x \right )}{4}+\frac {\mathrm {arccoth}\left (x \right )^{2} \ln \left (\frac {-1+x}{1+x}\right )}{4}+\frac {3 i \mathrm {arccoth}\left (x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (1+x \right )}{-1+x}\right )^{3} x^{2}-3 i \mathrm {csgn}\left (\frac {i \left (1+x \right )}{-1+x}\right ) \mathrm {csgn}\left (\frac {i}{\sqrt {\frac {-1+x}{1+x}}}\right )^{2} \mathrm {arccoth}\left (x \right )^{2} \pi -3 i \mathrm {csgn}\left (\frac {i \left (1+x \right )}{\left (-1+x \right ) \left (\frac {1+x}{-1+x}-1\right )}\right ) \mathrm {csgn}\left (\frac {i \left (1+x \right )}{-1+x}\right ) \mathrm {csgn}\left (\frac {i}{\frac {1+x}{-1+x}-1}\right ) \mathrm {arccoth}\left (x \right )^{2} \pi +6 i \mathrm {csgn}\left (\frac {i \left (1+x \right )}{-1+x}\right )^{2} \mathrm {csgn}\left (\frac {i}{\sqrt {\frac {-1+x}{1+x}}}\right ) \mathrm {arccoth}\left (x \right )^{2} \pi +3 i \mathrm {arccoth}\left (x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (1+x \right )}{\left (-1+x \right ) \left (\frac {1+x}{-1+x}-1\right )}\right )^{3} x^{2}-3 i \mathrm {csgn}\left (\frac {i \left (1+x \right )}{-1+x}\right )^{3} \mathrm {arccoth}\left (x \right )^{2} \pi +3 i \mathrm {csgn}\left (\frac {i \left (1+x \right )}{\left (-1+x \right ) \left (\frac {1+x}{-1+x}-1\right )}\right )^{2} \mathrm {csgn}\left (\frac {i \left (1+x \right )}{-1+x}\right ) \mathrm {arccoth}\left (x \right )^{2} \pi -6 i \mathrm {arccoth}\left (x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (1+x \right )}{-1+x}\right )^{2} \mathrm {csgn}\left (\frac {i}{\sqrt {\frac {-1+x}{1+x}}}\right ) x^{2}+3 i \mathrm {csgn}\left (\frac {i \left (1+x \right )}{\left (-1+x \right ) \left (\frac {1+x}{-1+x}-1\right )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\frac {1+x}{-1+x}-1}\right ) \mathrm {arccoth}\left (x \right )^{2} \pi +3 i \mathrm {arccoth}\left (x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (1+x \right )}{-1+x}\right ) \mathrm {csgn}\left (\frac {i}{\sqrt {\frac {-1+x}{1+x}}}\right )^{2} x^{2}-3 i \mathrm {arccoth}\left (x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (1+x \right )}{-1+x}\right ) \mathrm {csgn}\left (\frac {i \left (1+x \right )}{\left (-1+x \right ) \left (\frac {1+x}{-1+x}-1\right )}\right )^{2} x^{2}-3 i \mathrm {csgn}\left (\frac {i \left (1+x \right )}{\left (-1+x \right ) \left (\frac {1+x}{-1+x}-1\right )}\right )^{3} \mathrm {arccoth}\left (x \right )^{2} \pi -3 i \mathrm {arccoth}\left (x \right )^{2} \pi \mathrm {csgn}\left (\frac {i \left (1+x \right )}{\left (-1+x \right ) \left (\frac {1+x}{-1+x}-1\right )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\frac {1+x}{-1+x}-1}\right ) x^{2}+3 i \mathrm {arccoth}\left (x \right )^{2} \pi \,\mathrm {csgn}\left (\frac {i \left (1+x \right )}{-1+x}\right ) \mathrm {csgn}\left (\frac {i \left (1+x \right )}{\left (-1+x \right ) \left (\frac {1+x}{-1+x}-1\right )}\right ) \mathrm {csgn}\left (\frac {i}{\frac {1+x}{-1+x}-1}\right ) x^{2}+4 \mathrm {arccoth}\left (x \right )^{3} x^{2}-4 \mathrm {arccoth}\left (x \right )^{3}+6 \,\mathrm {arccoth}\left (x \right ) x^{2}+6 \,\mathrm {arccoth}\left (x \right )-6 x}{24 \left (-1+x \right ) \left (1+x \right )}\) | \(707\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs.
\(2 (46) = 92\).
time = 0.25, size = 171, normalized size = 2.76 \begin {gather*} -\frac {1}{4} \, {\left (\frac {2 \, x}{x^{2} - 1} - \log \left (x + 1\right ) + \log \left (x - 1\right )\right )} \operatorname {arcoth}\left (x\right )^{2} - \frac {{\left ({\left (x^{2} - 1\right )} \log \left (x + 1\right )^{2} - 2 \, {\left (x^{2} - 1\right )} \log \left (x + 1\right ) \log \left (x - 1\right ) + {\left (x^{2} - 1\right )} \log \left (x - 1\right )^{2} - 4\right )} \operatorname {arcoth}\left (x\right )}{8 \, {\left (x^{2} - 1\right )}} + \frac {{\left (x^{2} - 1\right )} \log \left (x + 1\right )^{3} - 3 \, {\left (x^{2} - 1\right )} \log \left (x + 1\right )^{2} \log \left (x - 1\right ) - {\left (x^{2} - 1\right )} \log \left (x - 1\right )^{3} + 3 \, {\left ({\left (x^{2} - 1\right )} \log \left (x - 1\right )^{2} + 2 \, x^{2} - 2\right )} \log \left (x + 1\right ) - 6 \, {\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 12 \, x}{48 \, {\left (x^{2} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 63, normalized size = 1.02 \begin {gather*} \frac {{\left (x^{2} - 1\right )} \log \left (\frac {x + 1}{x - 1}\right )^{3} - 6 \, x \log \left (\frac {x + 1}{x - 1}\right )^{2} + 6 \, {\left (x^{2} + 1\right )} \log \left (\frac {x + 1}{x - 1}\right ) - 12 \, x}{48 \, {\left (x^{2} - 1\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}^{2}{\left (x \right )}}{\left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 53, normalized size = 0.85 \begin {gather*} -\frac {{\left (x - 1\right )} \log \left (\frac {x + 1}{x - 1}\right )^{2}}{16 \, {\left (x + 1\right )}} - \frac {{\left (x - 1\right )} \log \left (\frac {x + 1}{x - 1}\right )}{8 \, {\left (x + 1\right )}} - \frac {x - 1}{8 \, {\left (x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.68, size = 201, normalized size = 3.24 \begin {gather*} \frac {{\ln \left (\frac {1}{x}+1\right )}^3}{48}-\frac {{\ln \left (1-\frac {1}{x}\right )}^3}{48}-\frac {x}{4\,\left (x^2-1\right )}+\ln \left (1-\frac {1}{x}\right )\,\left (\frac {\frac {3\,x}{32}-\frac {1}{8}}{x^2-1}-\frac {\frac {x}{8}+\frac {1}{8}}{x^2-1}-\frac {{\ln \left (\frac {1}{x}+1\right )}^2}{16}+\frac {x}{32\,\left (x^2-1\right )}+\ln \left (\frac {1}{x}+1\right )\,\left (\frac {\frac {x}{4}+\frac {1}{16}}{x^2-1}-\frac {1}{16\,\left (x^2-1\right )}\right )\right )+{\ln \left (1-\frac {1}{x}\right )}^2\,\left (\frac {\ln \left (\frac {1}{x}+1\right )}{16}-\frac {x}{8\,\left (x^2-1\right )}\right )+\frac {\ln \left (\frac {1}{x}+1\right )}{4\,\left (x^2-1\right )}-\frac {x\,{\ln \left (\frac {1}{x}+1\right )}^2}{8\,\left (x^2-1\right )}-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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