Integrand size = 8, antiderivative size = 40 \[ \int x^2 \coth ^{-1}(a x) \, dx=\frac {x^2}{6 a}+\frac {1}{3} x^3 \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{6 a^3} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6038, 272, 45} \[ \int x^2 \coth ^{-1}(a x) \, dx=\frac {\log \left (1-a^2 x^2\right )}{6 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)+\frac {x^2}{6 a} \]
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Rule 45
Rule 272
Rule 6038
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{3} a \int \frac {x^3}{1-a^2 x^2} \, dx \\ & = \frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \text {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right ) \\ & = \frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \text {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {x^2}{6 a}+\frac {1}{3} x^3 \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{6 a^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int x^2 \coth ^{-1}(a x) \, dx=\frac {x^2}{6 a}+\frac {1}{3} x^3 \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{6 a^3} \]
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Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95
method | result | size |
parts | \(\frac {x^{3} \operatorname {arccoth}\left (a x \right )}{3}+\frac {a \left (\frac {x^{2}}{2 a^{2}}+\frac {\ln \left (a^{2} x^{2}-1\right )}{2 a^{4}}\right )}{3}\) | \(38\) |
parallelrisch | \(-\frac {-2 a^{3} x^{3} \operatorname {arccoth}\left (a x \right )-a^{2} x^{2}-2 \ln \left (a x -1\right )-2 \,\operatorname {arccoth}\left (a x \right )}{6 a^{3}}\) | \(41\) |
derivativedivides | \(\frac {\frac {a^{3} x^{3} \operatorname {arccoth}\left (a x \right )}{3}+\frac {a^{2} x^{2}}{6}+\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}}{a^{3}}\) | \(42\) |
default | \(\frac {\frac {a^{3} x^{3} \operatorname {arccoth}\left (a x \right )}{3}+\frac {a^{2} x^{2}}{6}+\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}}{a^{3}}\) | \(42\) |
risch | \(\frac {x^{3} \ln \left (a x +1\right )}{6}-\frac {\ln \left (a x -1\right ) x^{3}}{6}+\frac {x^{2}}{6 a}+\frac {\ln \left (a^{2} x^{2}-1\right )}{6 a^{3}}\) | \(47\) |
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Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.10 \[ \int x^2 \coth ^{-1}(a x) \, dx=\frac {a^{3} x^{3} \log \left (\frac {a x + 1}{a x - 1}\right ) + a^{2} x^{2} + \log \left (a^{2} x^{2} - 1\right )}{6 \, a^{3}} \]
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Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int x^2 \coth ^{-1}(a x) \, dx=\begin {cases} \frac {x^{3} \operatorname {acoth}{\left (a x \right )}}{3} + \frac {x^{2}}{6 a} + \frac {\log {\left (a x + 1 \right )}}{3 a^{3}} - \frac {\operatorname {acoth}{\left (a x \right )}}{3 a^{3}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{3}}{6} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int x^2 \coth ^{-1}(a x) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcoth}\left (a x\right ) + \frac {1}{6} \, a {\left (\frac {x^{2}}{a^{2}} + \frac {\log \left (a^{2} x^{2} - 1\right )}{a^{4}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (34) = 68\).
Time = 0.28 (sec) , antiderivative size = 206, normalized size of antiderivative = 5.15 \[ \int x^2 \coth ^{-1}(a x) \, dx=\frac {1}{3} \, a {\left (\frac {\log \left (\frac {{\left | a x + 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{4}} - \frac {\log \left ({\left | \frac {a x + 1}{a x - 1} - 1 \right |}\right )}{a^{4}} + \frac {{\left (\frac {3 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + 1\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{a^{4} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{3}} + \frac {2 \, {\left (a x + 1\right )}}{{\left (a x - 1\right )} a^{4} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{2}}\right )} \]
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Time = 4.40 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int x^2 \coth ^{-1}(a x) \, dx=\frac {\frac {\ln \left (a^2\,x^2-1\right )}{6}+\frac {a^2\,x^2}{6}}{a^3}+\frac {x^3\,\mathrm {acoth}\left (a\,x\right )}{3} \]
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