Optimal. Leaf size=141 \[ -a^4 \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {1}{4} a^4 \tanh ^{-1}(a x)+\frac {1}{4} a^4 \coth ^{-1}(a x)^3+a^4 \coth ^{-1}(a x)^2+2 a^4 \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\frac {a^3}{4 x}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}-\frac {a^2 \coth ^{-1}(a x)}{4 x^2}-\frac {\coth ^{-1}(a x)^3}{4 x^4}-\frac {a \coth ^{-1}(a x)^2}{4 x^3} \]
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Rubi [A] time = 0.46, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5917, 5983, 325, 206, 5989, 5933, 2447, 5949} \[ -a^4 \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )-\frac {a^2 \coth ^{-1}(a x)}{4 x^2}-\frac {a^3}{4 x}+\frac {1}{4} a^4 \tanh ^{-1}(a x)+\frac {1}{4} a^4 \coth ^{-1}(a x)^3+a^4 \coth ^{-1}(a x)^2-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}+2 a^4 \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {\coth ^{-1}(a x)^3}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 206
Rule 325
Rule 2447
Rule 5917
Rule 5933
Rule 5949
Rule 5983
Rule 5989
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)^3}{x^5} \, dx &=-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} (3 a) \int \frac {\coth ^{-1}(a x)^2}{x^4 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} (3 a) \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx+\frac {1}{4} \left (3 a^3\right ) \int \frac {\coth ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{2} a^2 \int \frac {\coth ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx+\frac {1}{4} \left (3 a^3\right ) \int \frac {\coth ^{-1}(a x)^2}{x^2} \, dx+\frac {1}{4} \left (3 a^5\right ) \int \frac {\coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{2} a^2 \int \frac {\coth ^{-1}(a x)}{x^3} \, dx+\frac {1}{2} a^4 \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} \left (3 a^4\right ) \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a^2 \coth ^{-1}(a x)}{4 x^2}+a^4 \coth ^{-1}(a x)^2-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} a^3 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} a^4 \int \frac {\coth ^{-1}(a x)}{x (1+a x)} \, dx+\frac {1}{2} \left (3 a^4\right ) \int \frac {\coth ^{-1}(a x)}{x (1+a x)} \, dx\\ &=-\frac {a^3}{4 x}-\frac {a^2 \coth ^{-1}(a x)}{4 x^2}+a^4 \coth ^{-1}(a x)^2-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{4 x^4}+2 a^4 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{4} a^5 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{2} a^5 \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx-\frac {1}{2} \left (3 a^5\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^3}{4 x}-\frac {a^2 \coth ^{-1}(a x)}{4 x^2}+a^4 \coth ^{-1}(a x)^2-\frac {a \coth ^{-1}(a x)^2}{4 x^3}-\frac {3 a^3 \coth ^{-1}(a x)^2}{4 x}+\frac {1}{4} a^4 \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} a^4 \tanh ^{-1}(a x)+2 a^4 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-a^4 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.24, size = 118, normalized size = 0.84 \[ \frac {-4 a^4 x^4 \text {Li}_2\left (-e^{-2 \coth ^{-1}(a x)}\right )+\left (a^4 x^4-1\right ) \coth ^{-1}(a x)^3-a^3 x^3+a^2 x^2 \coth ^{-1}(a x) \left (a^2 x^2+8 a^2 x^2 \log \left (e^{-2 \coth ^{-1}(a x)}+1\right )-1\right )+a x \left (4 a^3 x^3-3 a^2 x^2-1\right ) \coth ^{-1}(a x)^2}{4 x^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (a x\right )^{3}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.50, size = 661, normalized size = 4.69 \[ \frac {a^{4} \mathrm {arccoth}\left (a x \right )}{4}+\frac {a^{4}}{4}-a^{4} \mathrm {arccoth}\left (a x \right )^{2}+\frac {a^{4} \mathrm {arccoth}\left (a x \right )^{3}}{4}-\frac {3 a^{4} \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{8}+\frac {3 a^{4} \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{8}+2 a^{4} \mathrm {arccoth}\left (a x \right ) \ln \left (1+\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 a^{4} \mathrm {arccoth}\left (a x \right ) \ln \left (1-\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+\frac {3 a^{4} \mathrm {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{8}-\frac {\mathrm {arccoth}\left (a x \right )^{3}}{4 x^{4}}-\frac {a \mathrm {arccoth}\left (a x \right )^{2}}{4 x^{3}}-\frac {3 a^{3} \mathrm {arccoth}\left (a x \right )^{2}}{4 x}-\frac {a^{3}}{4 x}-\frac {3 i a^{4} \pi \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {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} \mathrm {arccoth}\left (a x \right )^{2}}{16}-\frac {a^{2} \mathrm {arccoth}\left (a x \right )}{4 x^{2}}+2 a^{4} \dilog \left (1+\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+2 a^{4} \dilog \left (1-\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )+\frac {3 i a^{4} \pi \mathrm {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right )^{2} \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \mathrm {arccoth}\left (a x \right )^{2}}{16}-\frac {3 i a^{4} \pi \,\mathrm {csgn}\left (\frac {i}{\sqrt {\frac {a x -1}{a x +1}}}\right ) \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{2} \mathrm {arccoth}\left (a x \right )^{2}}{8}+\frac {3 i a^{4} \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3} \mathrm {arccoth}\left (a x \right )^{2}}{16}+\frac {3 i a^{4} \pi \mathrm {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} \mathrm {arccoth}\left (a x \right )^{2}}{16}+\frac {3 i a^{4} \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {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 ) \mathrm {arccoth}\left (a x \right )^{2}}{16}-\frac {3 i a^{4} \pi \,\mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right ) \mathrm {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} \mathrm {arccoth}\left (a x \right )^{2}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 342, normalized size = 2.43 \[ \frac {1}{8} \, {\left (3 \, a^{3} \log \left (a x + 1\right ) - 3 \, a^{3} \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} + 1\right )}}{x^{3}}\right )} a \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{32} \, {\left ({\left (32 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )} a - 32 \, {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )} a + 32 \, {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )} a + 4 \, a \log \left (a x + 1\right ) - 4 \, a \log \left (a x - 1\right ) + \frac {a x \log \left (a x + 1\right )^{3} - a x \log \left (a x - 1\right )^{3} - 8 \, a x \log \left (a x - 1\right )^{2} - {\left (3 \, a x \log \left (a x - 1\right ) - 8 \, a x\right )} \log \left (a x + 1\right )^{2} + {\left (3 \, a x \log \left (a x - 1\right )^{2} - 16 \, a x \log \left (a x - 1\right )\right )} \log \left (a x + 1\right ) - 8}{x}\right )} a^{2} + 2 \, {\left (32 \, a^{2} \log \relax (x) - \frac {3 \, a^{2} x^{2} \log \left (a x + 1\right )^{2} + 3 \, a^{2} x^{2} \log \left (a x - 1\right )^{2} + 16 \, a^{2} x^{2} \log \left (a x - 1\right ) - 2 \, {\left (3 \, a^{2} x^{2} \log \left (a x - 1\right ) - 8 \, a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 4}{x^{2}}\right )} a \operatorname {arcoth}\left (a x\right )\right )} a - \frac {\operatorname {arcoth}\left (a x\right )^{3}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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