Optimal. Leaf size=31 \[ \frac {1}{2} a^2 \tanh ^{-1}(a x)-\frac {\coth ^{-1}(a x)}{2 x^2}-\frac {a}{2 x} \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5917, 325, 206} \[ \frac {1}{2} a^2 \tanh ^{-1}(a x)-\frac {\coth ^{-1}(a x)}{2 x^2}-\frac {a}{2 x} \]
Antiderivative was successfully verified.
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Rule 206
Rule 325
Rule 5917
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)}{x^3} \, dx &=-\frac {\coth ^{-1}(a x)}{2 x^2}+\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a}{2 x}-\frac {\coth ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^3 \int \frac {1}{1-a^2 x^2} \, dx\\ &=-\frac {a}{2 x}-\frac {\coth ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \tanh ^{-1}(a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 47, normalized size = 1.52 \[ -\frac {1}{4} a^2 \log (1-a x)+\frac {1}{4} a^2 \log (a x+1)-\frac {\coth ^{-1}(a x)}{2 x^2}-\frac {a}{2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 35, normalized size = 1.13 \[ -\frac {2 \, a x - {\left (a^{2} x^{2} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{4 \, x^{2}} \]
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 \[ \int \frac {\operatorname {arcoth}\left (a x\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 39, normalized size = 1.26 \[ -\frac {\mathrm {arccoth}\left (a x \right )}{2 x^{2}}-\frac {a}{2 x}-\frac {a^{2} \ln \left (a x -1\right )}{4}+\frac {a^{2} \ln \left (a x +1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 36, normalized size = 1.16 \[ \frac {1}{4} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} a - \frac {\operatorname {arcoth}\left (a x\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 40, normalized size = 1.29 \[ \frac {a\,\mathrm {atan}\left (\frac {a^2\,x}{\sqrt {-a^2}}\right )\,\sqrt {-a^2}}{2}-\frac {\frac {\mathrm {acoth}\left (a\,x\right )}{2}+\frac {a\,x}{2}}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 24, normalized size = 0.77 \[ \frac {a^{2} \operatorname {acoth}{\left (a x \right )}}{2} - \frac {a}{2 x} - \frac {\operatorname {acoth}{\left (a x \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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