Integrand size = 10, antiderivative size = 149 \[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\frac {x \coth ^{-1}(a x)}{a^2}-\frac {\coth ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \coth ^{-1}(a x)^2}{2 a}+\frac {\coth ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {\log \left (1-a^2 x^2\right )}{2 a^3}-\frac {\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^3} \]
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Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6038, 6128, 6022, 266, 6096, 6132, 6056, 6206, 6745} \[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^3}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a^3}+\frac {\coth ^{-1}(a x)^3}{3 a^3}-\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a^3}+\frac {x \coth ^{-1}(a x)}{a^2}+\frac {\log \left (1-a^2 x^2\right )}{2 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3+\frac {x^2 \coth ^{-1}(a x)^2}{2 a} \]
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Rule 266
Rule 6022
Rule 6038
Rule 6056
Rule 6096
Rule 6128
Rule 6132
Rule 6206
Rule 6745
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \coth ^{-1}(a x)^3-a \int \frac {x^3 \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx \\ & = \frac {1}{3} x^3 \coth ^{-1}(a x)^3+\frac {\int x \coth ^{-1}(a x)^2 \, dx}{a}-\frac {\int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{a} \\ & = \frac {x^2 \coth ^{-1}(a x)^2}{2 a}+\frac {\coth ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3-\frac {\int \frac {\coth ^{-1}(a x)^2}{1-a x} \, dx}{a^2}-\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx \\ & = \frac {x^2 \coth ^{-1}(a x)^2}{2 a}+\frac {\coth ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {\int \coth ^{-1}(a x) \, dx}{a^2}-\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{a^2}+\frac {2 \int \frac {\coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2} \\ & = \frac {x \coth ^{-1}(a x)}{a^2}-\frac {\coth ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \coth ^{-1}(a x)^2}{2 a}+\frac {\coth ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}-\frac {\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3}+\frac {\int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2}-\frac {\int \frac {x}{1-a^2 x^2} \, dx}{a} \\ & = \frac {x \coth ^{-1}(a x)}{a^2}-\frac {\coth ^{-1}(a x)^2}{2 a^3}+\frac {x^2 \coth ^{-1}(a x)^2}{2 a}+\frac {\coth ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3}+\frac {\log \left (1-a^2 x^2\right )}{2 a^3}-\frac {\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3}+\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.96 \[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\frac {-i \pi ^3+24 a x \coth ^{-1}(a x)-12 \coth ^{-1}(a x)^2+12 a^2 x^2 \coth ^{-1}(a x)^2+8 \coth ^{-1}(a x)^3+8 a^3 x^3 \coth ^{-1}(a x)^3-24 \coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-24 \log \left (\frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )-24 \log \left (\frac {1}{a x}\right )-24 \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right )}{24 a^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.88 (sec) , antiderivative size = 681, normalized size of antiderivative = 4.57
method | result | size |
parts | \(\frac {x^{3} \operatorname {arccoth}\left (a x \right )^{3}}{3}+\frac {\frac {a^{2} x^{2} \operatorname {arccoth}\left (a x \right )^{2}}{2}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}+\operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x +1}{a x -1}-1\right )+\frac {\operatorname {arccoth}\left (a x \right ) \left (3 i \operatorname {arccoth}\left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}-6 i \operatorname {arccoth}\left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2} \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )-3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}+3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2}-3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}+3 i \operatorname {arccoth}\left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{3}+4 \operatorname {arccoth}\left (a x \right )^{2}-12 \,\operatorname {arccoth}\left (a x \right ) \ln \left (2\right )-6 \,\operatorname {arccoth}\left (a x \right )+12 a x +12\right )}{12}-\ln \left (\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}-1\right )-\ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\operatorname {arccoth}\left (a x \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\operatorname {arccoth}\left (a x \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a^{3}}\) | \(681\) |
derivativedivides | \(\frac {\frac {\operatorname {arccoth}\left (a x \right )^{3} a^{3} x^{3}}{3}+\frac {a^{2} x^{2} \operatorname {arccoth}\left (a x \right )^{2}}{2}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}+\operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x +1}{a x -1}-1\right )+\frac {\operatorname {arccoth}\left (a x \right ) \left (3 i \operatorname {arccoth}\left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}-6 i \operatorname {arccoth}\left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2} \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )-3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}+3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2}-3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}+3 i \operatorname {arccoth}\left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{3}+4 \operatorname {arccoth}\left (a x \right )^{2}-12 \,\operatorname {arccoth}\left (a x \right ) \ln \left (2\right )-6 \,\operatorname {arccoth}\left (a x \right )+12 a x +12\right )}{12}-\ln \left (\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}-1\right )-\ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\operatorname {arccoth}\left (a x \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\operatorname {arccoth}\left (a x \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a^{3}}\) | \(683\) |
default | \(\frac {\frac {\operatorname {arccoth}\left (a x \right )^{3} a^{3} x^{3}}{3}+\frac {a^{2} x^{2} \operatorname {arccoth}\left (a x \right )^{2}}{2}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}+\operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x +1}{a x -1}-1\right )+\frac {\operatorname {arccoth}\left (a x \right ) \left (3 i \operatorname {arccoth}\left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}-6 i \operatorname {arccoth}\left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2} \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )-3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}+3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2}-3 i \operatorname {arccoth}\left (a x \right ) \pi \,\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}+3 i \operatorname {arccoth}\left (a x \right ) \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{\left (a x -1\right ) \left (\frac {a x +1}{a x -1}-1\right )}\right )^{3}+4 \operatorname {arccoth}\left (a x \right )^{2}-12 \,\operatorname {arccoth}\left (a x \right ) \ln \left (2\right )-6 \,\operatorname {arccoth}\left (a x \right )+12 a x +12\right )}{12}-\ln \left (\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}-1\right )-\ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\operatorname {arccoth}\left (a x \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\operatorname {arccoth}\left (a x \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a^{3}}\) | \(683\) |
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\[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\int { x^{2} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]
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\[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\int x^{2} \operatorname {acoth}^{3}{\left (a x \right )}\, dx \]
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\[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\int { x^{2} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]
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\[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\int { x^{2} \operatorname {arcoth}\left (a x\right )^{3} \,d x } \]
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Timed out. \[ \int x^2 \coth ^{-1}(a x)^3 \, dx=\int x^2\,{\mathrm {acoth}\left (a\,x\right )}^3 \,d x \]
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