Optimal. Leaf size=41 \[ \frac {1}{4} a^4 \tanh ^{-1}(a x)-\frac {a^3}{4 x}-\frac {\coth ^{-1}(a x)}{4 x^4}-\frac {a}{12 x^3} \]
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Rubi [A] time = 0.02, antiderivative size = 41, 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, 325, 206} \[ -\frac {a^3}{4 x}+\frac {1}{4} a^4 \tanh ^{-1}(a x)-\frac {a}{12 x^3}-\frac {\coth ^{-1}(a x)}{4 x^4} \]
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^5} \, dx &=-\frac {\coth ^{-1}(a x)}{4 x^4}+\frac {1}{4} a \int \frac {1}{x^4 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a}{12 x^3}-\frac {\coth ^{-1}(a x)}{4 x^4}+\frac {1}{4} a^3 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a}{12 x^3}-\frac {a^3}{4 x}-\frac {\coth ^{-1}(a x)}{4 x^4}+\frac {1}{4} a^5 \int \frac {1}{1-a^2 x^2} \, dx\\ &=-\frac {a}{12 x^3}-\frac {a^3}{4 x}-\frac {\coth ^{-1}(a x)}{4 x^4}+\frac {1}{4} a^4 \tanh ^{-1}(a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 57, normalized size = 1.39 \[ -\frac {1}{8} a^4 \log (1-a x)+\frac {1}{8} a^4 \log (a x+1)-\frac {a^3}{4 x}-\frac {\coth ^{-1}(a x)}{4 x^4}-\frac {a}{12 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 43, normalized size = 1.05 \[ -\frac {6 \, a^{3} x^{3} + 2 \, a x - 3 \, {\left (a^{4} x^{4} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{24 \, x^{4}} \]
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 \[ \int \frac {\operatorname {arcoth}\left (a x\right )}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 47, normalized size = 1.15 \[ -\frac {\mathrm {arccoth}\left (a x \right )}{4 x^{4}}-\frac {a}{12 x^{3}}-\frac {a^{3}}{4 x}-\frac {a^{4} \ln \left (a x -1\right )}{8}+\frac {a^{4} \ln \left (a x +1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 51, normalized size = 1.24 \[ \frac {1}{24} \, {\left (3 \, a^{3} \log \left (a x + 1\right ) - 3 \, a^{3} \log \left (a x - 1\right ) - \frac {2 \, {\left (3 \, a^{2} x^{2} + 1\right )}}{x^{3}}\right )} a - \frac {\operatorname {arcoth}\left (a x\right )}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.61, size = 60, normalized size = 1.46 \[ \frac {\ln \left (1-\frac {1}{a\,x}\right )}{8\,x^4}-\frac {\ln \left (\frac {1}{a\,x}+1\right )}{8\,x^4}-\frac {a^3\,x^2+\frac {a}{3}}{4\,x^3}-\frac {a^4\,\mathrm {atan}\left (a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.16, size = 32, normalized size = 0.78 \[ \frac {a^{4} \operatorname {acoth}{\left (a x \right )}}{4} - \frac {a^{3}}{4 x} - \frac {a}{12 x^{3}} - \frac {\operatorname {acoth}{\left (a x \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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