Integrand size = 8, antiderivative size = 50 \[ \int x^4 \coth ^{-1}(a x) \, dx=\frac {x^2}{10 a^3}+\frac {x^4}{20 a}+\frac {1}{5} x^5 \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{10 a^5} \]
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Time = 0.03 (sec) , antiderivative size = 50, 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} \[ \int x^4 \coth ^{-1}(a x) \, dx=\frac {x^2}{10 a^3}+\frac {\log \left (1-a^2 x^2\right )}{10 a^5}+\frac {1}{5} x^5 \coth ^{-1}(a x)+\frac {x^4}{20 a} \]
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Rule 45
Rule 272
Rule 6038
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{5} a \int \frac {x^5}{1-a^2 x^2} \, dx \\ & = \frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \text {Subst}\left (\int \frac {x^2}{1-a^2 x} \, dx,x,x^2\right ) \\ & = \frac {1}{5} x^5 \coth ^{-1}(a x)-\frac {1}{10} a \text {Subst}\left (\int \left (-\frac {1}{a^4}-\frac {x}{a^2}-\frac {1}{a^4 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {x^2}{10 a^3}+\frac {x^4}{20 a}+\frac {1}{5} x^5 \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{10 a^5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int x^4 \coth ^{-1}(a x) \, dx=\frac {x^2}{10 a^3}+\frac {x^4}{20 a}+\frac {1}{5} x^5 \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{10 a^5} \]
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Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94
method | result | size |
parts | \(\frac {x^{5} \operatorname {arccoth}\left (a x \right )}{5}+\frac {a \left (\frac {\frac {1}{2} x^{4} a^{2}+x^{2}}{2 a^{4}}+\frac {\ln \left (a^{2} x^{2}-1\right )}{2 a^{6}}\right )}{5}\) | \(47\) |
derivativedivides | \(\frac {\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )}{5}+\frac {a^{4} x^{4}}{20}+\frac {a^{2} x^{2}}{10}+\frac {\ln \left (a x -1\right )}{10}+\frac {\ln \left (a x +1\right )}{10}}{a^{5}}\) | \(50\) |
default | \(\frac {\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )}{5}+\frac {a^{4} x^{4}}{20}+\frac {a^{2} x^{2}}{10}+\frac {\ln \left (a x -1\right )}{10}+\frac {\ln \left (a x +1\right )}{10}}{a^{5}}\) | \(50\) |
parallelrisch | \(-\frac {-4 a^{5} x^{5} \operatorname {arccoth}\left (a x \right )-a^{4} x^{4}-2-2 a^{2} x^{2}-4 \ln \left (a x -1\right )-4 \,\operatorname {arccoth}\left (a x \right )}{20 a^{5}}\) | \(50\) |
risch | \(\frac {x^{5} \ln \left (a x +1\right )}{10}-\frac {\ln \left (a x -1\right ) x^{5}}{10}+\frac {x^{4}}{20 a}+\frac {x^{2}}{10 a^{3}}+\frac {\ln \left (a^{2} x^{2}-1\right )}{10 a^{5}}+\frac {1}{20 a^{5}}\) | \(60\) |
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Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.10 \[ \int x^4 \coth ^{-1}(a x) \, dx=\frac {2 \, a^{5} x^{5} \log \left (\frac {a x + 1}{a x - 1}\right ) + a^{4} x^{4} + 2 \, a^{2} x^{2} + 2 \, \log \left (a^{2} x^{2} - 1\right )}{20 \, a^{5}} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.08 \[ \int x^4 \coth ^{-1}(a x) \, dx=\begin {cases} \frac {x^{5} \operatorname {acoth}{\left (a x \right )}}{5} + \frac {x^{4}}{20 a} + \frac {x^{2}}{10 a^{3}} + \frac {\log {\left (a x + 1 \right )}}{5 a^{5}} - \frac {\operatorname {acoth}{\left (a x \right )}}{5 a^{5}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{5}}{10} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92 \[ \int x^4 \coth ^{-1}(a x) \, dx=\frac {1}{5} \, x^{5} \operatorname {arcoth}\left (a x\right ) + \frac {1}{20} \, a {\left (\frac {a^{2} x^{4} + 2 \, x^{2}}{a^{4}} + \frac {2 \, \log \left (a^{2} x^{2} - 1\right )}{a^{6}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (42) = 84\).
Time = 0.28 (sec) , antiderivative size = 255, normalized size of antiderivative = 5.10 \[ \int x^4 \coth ^{-1}(a x) \, dx=\frac {1}{5} \, a {\left (\frac {\log \left (\frac {{\left | a x + 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{6}} - \frac {\log \left ({\left | \frac {a x + 1}{a x - 1} - 1 \right |}\right )}{a^{6}} + \frac {4 \, {\left (\frac {{\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} - \frac {{\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {a x + 1}{a x - 1}\right )}}{a^{6} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{4}} + \frac {{\left (\frac {5 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {10 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + 1\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{a^{6} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{5}}\right )} \]
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Time = 4.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86 \[ \int x^4 \coth ^{-1}(a x) \, dx=\frac {\frac {\ln \left (a^2\,x^2-1\right )}{10}+\frac {a^2\,x^2}{10}+\frac {a^4\,x^4}{20}}{a^5}+\frac {x^5\,\mathrm {acoth}\left (a\,x\right )}{5} \]
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