Integrand size = 10, antiderivative size = 79 \[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6038, 6136, 6080, 6096, 6204, 6745} \[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=-\frac {3}{2} a \operatorname {PolyLog}\left (3,\frac {2}{a x+1}-1\right )-3 a \operatorname {PolyLog}\left (2,\frac {2}{a x+1}-1\right ) \coth ^{-1}(a x)+a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)^2 \]
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Rule 6038
Rule 6080
Rule 6096
Rule 6136
Rule 6204
Rule 6745
Rubi steps \begin{align*} \text {integral}& = -\frac {\coth ^{-1}(a x)^3}{x}+(3 a) \int \frac {\coth ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx \\ & = a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+(3 a) \int \frac {\coth ^{-1}(a x)^2}{x (1+a x)} \, dx \\ & = a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-\left (6 a^2\right ) \int \frac {\coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx \\ & = a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )+\left (3 a^2\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx \\ & = a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.91 \[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=\frac {(-1+a x) \coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )-3 a \coth ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-\frac {3}{2} a \operatorname {PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.61 (sec) , antiderivative size = 718, normalized size of antiderivative = 9.09
method | result | size |
parts | \(-\frac {\operatorname {arccoth}\left (a x \right )^{3}}{x}-3 a \left (\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}-\ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )^{2}+\frac {\operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}+\frac {\operatorname {arccoth}\left (a x \right )^{3}}{3}-\frac {\left (-i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}-i \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}+i \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}-i \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 )}{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 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 )-2 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}+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}{\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}-i \pi \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -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}+2 i \pi \,\operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2}+4 \ln \left (2\right )\right ) \operatorname {arccoth}\left (a x \right )^{2}}{4}-\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{a x -1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {a x +1}{a x -1}\right )}{2}\right )\) | \(718\) |
derivativedivides | \(a \left (-\frac {\operatorname {arccoth}\left (a x \right )^{3}}{a x}-\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}-\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}+3 \ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )^{2}-\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}-\operatorname {arccoth}\left (a x \right )^{3}+\frac {3 \left (-i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}-i \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}+i \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}-i \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 )}{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 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 )-2 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}+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}{\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}-i \pi \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -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}+2 i \pi \,\operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2}+4 \ln \left (2\right )\right ) \operatorname {arccoth}\left (a x \right )^{2}}{4}+3 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{a x -1}\right )-\frac {3 \operatorname {polylog}\left (3, -\frac {a x +1}{a x -1}\right )}{2}\right )\) | \(719\) |
default | \(a \left (-\frac {\operatorname {arccoth}\left (a x \right )^{3}}{a x}-\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}-\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}+3 \ln \left (a x \right ) \operatorname {arccoth}\left (a x \right )^{2}-\frac {3 \operatorname {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}-\operatorname {arccoth}\left (a x \right )^{3}+\frac {3 \left (-i \pi \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}-i \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}+i \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}-i \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 )}{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 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 )-2 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}+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}{\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}-i \pi \operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2} \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -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}+2 i \pi \,\operatorname {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right ) \operatorname {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2}+4 \ln \left (2\right )\right ) \operatorname {arccoth}\left (a x \right )^{2}}{4}+3 \,\operatorname {arccoth}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +1}{a x -1}\right )-\frac {3 \operatorname {polylog}\left (3, -\frac {a x +1}{a x -1}\right )}{2}\right )\) | \(719\) |
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\[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{2}} \,d x } \]
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\[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=\int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{2}} \,d x } \]
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\[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=\int { \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx=\int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^2} \,d x \]
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