Optimal. Leaf size=79 \[ a \coth ^{-1}(a x)^3-\frac {\coth ^{-1}(a x)^3}{x}+3 a \coth ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \coth ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \]
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Rubi [A]
time = 0.14, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6038, 6136,
6080, 6096, 6204, 6745} \begin {gather*} -\frac {3}{2} a \text {Li}_3\left (\frac {2}{a x+1}-1\right )-3 a \text {Li}_2\left (\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 \end {gather*}
Antiderivative was successfully verified.
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Rule 6038
Rule 6080
Rule 6096
Rule 6136
Rule 6204
Rule 6745
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)^3}{x^2} \, dx &=-\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) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\left (3 a^2\right ) \int \frac {\text {Li}_2\left (-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) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \text {Li}_3\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 72, normalized size = 0.91 \begin {gather*} \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) \text {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )-\frac {3}{2} a \text {PolyLog}\left (3,-e^{-2 \coth ^{-1}(a x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 3.21, size = 719, normalized size = 9.10
method | result | size |
derivativedivides | \(a \left (-\frac {\mathrm {arccoth}\left (a x \right )^{3}}{a x}-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}+3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x \right )-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}-\mathrm {arccoth}\left (a x \right )^{3}+\frac {3 \left (2 i \pi \mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{3}-2 i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {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 \,\mathrm {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \mathrm {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 \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \mathrm {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 \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \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 )+i \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}-i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}+2 i \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}+i \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}-i \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 )-i \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}+4 \ln \left (2\right )\right ) \mathrm {arccoth}\left (a x \right )^{2}}{4}+3 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, -\frac {a x +1}{a x -1}\right )-\frac {3 \polylog \left (3, -\frac {a x +1}{a x -1}\right )}{2}\right )\) | \(719\) |
default | \(a \left (-\frac {\mathrm {arccoth}\left (a x \right )^{3}}{a x}-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x -1\right )}{2}+3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x \right )-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (a x +1\right )}{2}-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (\frac {a x -1}{a x +1}\right )}{2}-\mathrm {arccoth}\left (a x \right )^{3}+\frac {3 \left (2 i \pi \mathrm {csgn}\left (\frac {i \left (1+\frac {a x +1}{a x -1}\right )}{\frac {a x +1}{a x -1}-1}\right )^{3}-2 i \pi \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {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 \,\mathrm {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \mathrm {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 \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \mathrm {csgn}\left (i \left (1+\frac {a x +1}{a x -1}\right )\right ) \mathrm {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 \,\mathrm {csgn}\left (\frac {i}{\frac {a x +1}{a x -1}-1}\right ) \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 )+i \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}-i \pi \mathrm {csgn}\left (\frac {i \left (a x +1\right )}{a x -1}\right )^{3}+2 i \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}+i \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}-i \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 )-i \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}+4 \ln \left (2\right )\right ) \mathrm {arccoth}\left (a x \right )^{2}}{4}+3 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, -\frac {a x +1}{a x -1}\right )-\frac {3 \polylog \left (3, -\frac {a x +1}{a x -1}\right )}{2}\right )\) | \(719\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {acoth}^{3}{\left (a x \right )}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {acoth}\left (a\,x\right )}^3}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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