Optimal. Leaf size=47 \[ \frac {1}{3} a^3 \log (x)-\frac {1}{6} a^3 \log \left (1-a^2 x^2\right )-\frac {\coth ^{-1}(a x)}{3 x^3}-\frac {a}{6 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5917, 266, 44} \[ -\frac {1}{6} a^3 \log \left (1-a^2 x^2\right )+\frac {1}{3} a^3 \log (x)-\frac {a}{6 x^2}-\frac {\coth ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 5917
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)}{x^4} \, dx &=-\frac {\coth ^{-1}(a x)}{3 x^3}+\frac {1}{3} a \int \frac {1}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\coth ^{-1}(a x)}{3 x^3}+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {\coth ^{-1}(a x)}{3 x^3}+\frac {1}{6} a \operatorname {Subst}\left (\int \left (\frac {1}{x^2}+\frac {a^2}{x}-\frac {a^4}{-1+a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a}{6 x^2}-\frac {\coth ^{-1}(a x)}{3 x^3}+\frac {1}{3} a^3 \log (x)-\frac {1}{6} a^3 \log \left (1-a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 47, normalized size = 1.00 \[ \frac {1}{3} a^3 \log (x)-\frac {1}{6} a^3 \log \left (1-a^2 x^2\right )-\frac {\coth ^{-1}(a x)}{3 x^3}-\frac {a}{6 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 50, normalized size = 1.06 \[ -\frac {a^{3} x^{3} \log \left (a^{2} x^{2} - 1\right ) - 2 \, a^{3} x^{3} \log \relax (x) + a x + \log \left (\frac {a x + 1}{a x - 1}\right )}{6 \, x^{3}} \]
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 \[ \int \frac {\operatorname {arcoth}\left (a x\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 48, normalized size = 1.02 \[ -\frac {\mathrm {arccoth}\left (a x \right )}{3 x^{3}}-\frac {a}{6 x^{2}}+\frac {a^{3} \ln \left (a x \right )}{3}-\frac {a^{3} \ln \left (a x -1\right )}{6}-\frac {a^{3} \ln \left (a x +1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 40, normalized size = 0.85 \[ -\frac {1}{6} \, {\left (a^{2} \log \left (a^{2} x^{2} - 1\right ) - a^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} a - \frac {\operatorname {arcoth}\left (a x\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 39, normalized size = 0.83 \[ \frac {a^3\,\ln \relax (x)}{3}-\frac {\frac {\mathrm {acoth}\left (a\,x\right )}{3}+\frac {a\,x}{6}}{x^3}-\frac {a^3\,\ln \left (a^2\,x^2-1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 46, normalized size = 0.98 \[ \frac {a^{3} \log {\relax (x )}}{3} - \frac {a^{3} \log {\left (a x + 1 \right )}}{3} + \frac {a^{3} \operatorname {acoth}{\left (a x \right )}}{3} - \frac {a}{6 x^{2}} - \frac {\operatorname {acoth}{\left (a x \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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