Integrand size = 10, antiderivative size = 150 \[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )+\frac {3}{2} \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+a x}\right )-\frac {3}{2} \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 a x}{1+a x}\right )+\frac {3}{2} \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+a x}\right )-\frac {3}{2} \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 a x}{1+a x}\right )+\frac {3}{4} \operatorname {PolyLog}\left (4,1-\frac {2}{1+a x}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,1-\frac {2 a x}{1+a x}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6034, 6200, 6096, 6204, 6208, 6745} \[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\frac {3}{4} \operatorname {PolyLog}\left (4,1-\frac {2}{a x+1}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,1-\frac {2 a x}{a x+1}\right )+\frac {3}{2} \operatorname {PolyLog}\left (2,1-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2-\frac {3}{2} \operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+1}\right ) \coth ^{-1}(a x)^2+\frac {3}{2} \operatorname {PolyLog}\left (3,1-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\frac {3}{2} \operatorname {PolyLog}\left (3,1-\frac {2 a x}{a x+1}\right ) \coth ^{-1}(a x)+2 \coth ^{-1}\left (1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^3 \]
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Rule 6034
Rule 6096
Rule 6200
Rule 6204
Rule 6208
Rule 6745
Rubi steps \begin{align*} \text {integral}& = 2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )-(6 a) \int \frac {\coth ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx \\ & = 2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )+(3 a) \int \frac {\coth ^{-1}(a x)^2 \log \left (\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx-(3 a) \int \frac {\coth ^{-1}(a x)^2 \log \left (\frac {2 a x}{1+a x}\right )}{1-a^2 x^2} \, dx \\ & = 2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )+\frac {3}{2} \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+a x}\right )-\frac {3}{2} \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 a x}{1+a x}\right )-(3 a) \int \frac {\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx+(3 a) \int \frac {\coth ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2 a x}{1+a x}\right )}{1-a^2 x^2} \, dx \\ & = 2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )+\frac {3}{2} \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+a x}\right )-\frac {3}{2} \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 a x}{1+a x}\right )+\frac {3}{2} \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+a x}\right )-\frac {3}{2} \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 a x}{1+a x}\right )-\frac {1}{2} (3 a) \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx+\frac {1}{2} (3 a) \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2 a x}{1+a x}\right )}{1-a^2 x^2} \, dx \\ & = 2 \coth ^{-1}(a x)^3 \coth ^{-1}\left (1-\frac {2}{1-a x}\right )+\frac {3}{2} \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+a x}\right )-\frac {3}{2} \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2 a x}{1+a x}\right )+\frac {3}{2} \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+a x}\right )-\frac {3}{2} \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,1-\frac {2 a x}{1+a x}\right )+\frac {3}{4} \operatorname {PolyLog}\left (4,1-\frac {2}{1+a x}\right )-\frac {3}{4} \operatorname {PolyLog}\left (4,1-\frac {2 a x}{1+a x}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.04 \[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\frac {1}{64} \left (-\pi ^4+32 \coth ^{-1}(a x)^4+64 \coth ^{-1}(a x)^3 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )-64 \coth ^{-1}(a x)^3 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )-96 \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-96 \coth ^{-1}(a x)^2 \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(a x)}\right )-96 \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right )+96 \coth ^{-1}(a x) \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(a x)}\right )-48 \operatorname {PolyLog}\left (4,-e^{-2 \coth ^{-1}(a x)}\right )-48 \operatorname {PolyLog}\left (4,e^{2 \coth ^{-1}(a x)}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.67 (sec) , antiderivative size = 536, normalized size of antiderivative = 3.57
method | result | size |
derivativedivides | \(\ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )^{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \left (\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right )-\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right )+\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (a x \right )^{3}}{2}+\operatorname {arccoth}\left (a x \right )^{3} \ln \left (\frac {a x +1}{a x -1}-1\right )-\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \operatorname {polylog}\left (4, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \operatorname {polylog}\left (4, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{a x -1}\right )}{2}-\frac {3 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{a x -1}\right )}{2}+\frac {3 \operatorname {polylog}\left (4, -\frac {a x +1}{a x -1}\right )}{4}\) | \(536\) |
default | \(\ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )^{3}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \left (\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right )-\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )-\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right )+\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2}\right ) \operatorname {arccoth}\left (a x \right )^{3}}{2}+\operatorname {arccoth}\left (a x \right )^{3} \ln \left (\frac {a x +1}{a x -1}-1\right )-\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \operatorname {polylog}\left (4, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+6 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-6 \operatorname {polylog}\left (4, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )+\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{a x -1}\right )}{2}-\frac {3 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{a x -1}\right )}{2}+\frac {3 \operatorname {polylog}\left (4, -\frac {a x +1}{a x -1}\right )}{4}\) | \(536\) |
parts | \(\ln \left (x \right ) \operatorname {arccoth}\left (a x \right )^{3}+3 a \left (\frac {\left (i \pi \,\operatorname {csgn}\left (\frac {i}{a}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{a \left (\frac {a x +1}{a x -1}-1\right )}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{a}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{a \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )-i \pi \,\operatorname {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2}-i \pi \,\operatorname {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{a \left (\frac {a x +1}{a x -1}-1\right )}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{a \left (\frac {a x +1}{a x -1}-1\right )}\right )^{3}+2 \ln \left (a \right )\right ) \operatorname {arccoth}\left (a x \right )^{3}}{6 a}+\frac {\operatorname {arccoth}\left (a x \right )^{3} \ln \left (\frac {a x +1}{a x -1}-1\right )}{3 a}-\frac {\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{3 a}-\frac {\operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}+\frac {2 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {2 \operatorname {polylog}\left (4, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {\operatorname {arccoth}\left (a x \right )^{3} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{3 a}-\frac {\operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}+\frac {2 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {2 \operatorname {polylog}\left (4, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {a x +1}{a x -1}\right )}{2 a}-\frac {\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (3, -\frac {a x +1}{a x -1}\right )}{2 a}+\frac {\operatorname {polylog}\left (4, -\frac {a x +1}{a x -1}\right )}{4 a}\right )\) | \(859\) |
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\[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x}\, dx \]
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\[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(a x)^3}{x} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x} \,d x \]
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