Optimal. Leaf size=103 \[ -\frac {a^2}{3 x}-\frac {a \coth ^{-1}(a x)}{3 x^2}+\frac {1}{3} a^3 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^3 \tanh ^{-1}(a x)+\frac {2}{3} a^3 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{3} a^3 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6038, 6130,
331, 212, 6136, 6080, 2497} \begin {gather*} -\frac {1}{3} a^3 \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {1}{3} a^3 \tanh ^{-1}(a x)+\frac {1}{3} a^3 \coth ^{-1}(a x)^2+\frac {2}{3} a^3 \log \left (2-\frac {2}{a x+1}\right ) \coth ^{-1}(a x)-\frac {a^2}{3 x}-\frac {\coth ^{-1}(a x)^2}{3 x^3}-\frac {a \coth ^{-1}(a x)}{3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 331
Rule 2497
Rule 6038
Rule 6080
Rule 6130
Rule 6136
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)^2}{x^4} \, dx &=-\frac {\coth ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} (2 a) \int \frac {\coth ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {\coth ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} (2 a) \int \frac {\coth ^{-1}(a x)}{x^3} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\coth ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {a \coth ^{-1}(a x)}{3 x^2}+\frac {1}{3} a^3 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^2 \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {\coth ^{-1}(a x)}{x (1+a x)} \, dx\\ &=-\frac {a^2}{3 x}-\frac {a \coth ^{-1}(a x)}{3 x^2}+\frac {1}{3} a^3 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{3} a^4 \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{3} \left (2 a^4\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a^2}{3 x}-\frac {a \coth ^{-1}(a x)}{3 x^2}+\frac {1}{3} a^3 \coth ^{-1}(a x)^2-\frac {\coth ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} a^3 \tanh ^{-1}(a x)+\frac {2}{3} a^3 \coth ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{3} a^3 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 87, normalized size = 0.84 \begin {gather*} \frac {-a^2 x^2+\left (-1+a^3 x^3\right ) \coth ^{-1}(a x)^2+a x \coth ^{-1}(a x) \left (-1+a^2 x^2+2 a^2 x^2 \log \left (1+e^{-2 \coth ^{-1}(a x)}\right )\right )-a^3 x^3 \text {PolyLog}\left (2,-e^{-2 \coth ^{-1}(a x)}\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs.
\(2(89)=178\).
time = 0.08, size = 185, normalized size = 1.80
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {\mathrm {arccoth}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\mathrm {arccoth}\left (a x \right )}{3 a^{2} x^{2}}+\frac {2 \ln \left (a x \right ) \mathrm {arccoth}\left (a x \right )}{3}-\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{3}-\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {1}{3 a x}-\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{6}-\frac {\ln \left (a x -1\right )^{2}}{12}+\frac {\ln \left (a x +1\right )^{2}}{12}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{6}-\frac {\dilog \left (a x +1\right )}{3}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {\dilog \left (a x \right )}{3}\right )\) | \(185\) |
default | \(a^{3} \left (-\frac {\mathrm {arccoth}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\mathrm {arccoth}\left (a x \right )}{3 a^{2} x^{2}}+\frac {2 \ln \left (a x \right ) \mathrm {arccoth}\left (a x \right )}{3}-\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{3}-\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {1}{3 a x}-\frac {\ln \left (a x -1\right )}{6}+\frac {\ln \left (a x +1\right )}{6}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{6}-\frac {\ln \left (a x -1\right )^{2}}{12}+\frac {\ln \left (a x +1\right )^{2}}{12}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{6}-\frac {\dilog \left (a x +1\right )}{3}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{3}-\frac {\dilog \left (a x \right )}{3}\right )\) | \(185\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 176, normalized size = 1.71 \begin {gather*} \frac {1}{12} \, {\left (4 \, {\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 - 4 \, {\left (\log \left (a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x\right )\right )} a + 4 \, {\left (\log \left (-a x + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x\right )\right )} a + 2 \, a \log \left (a x + 1\right ) - 2 \, a \log \left (a x - 1\right ) + \frac {a x \log \left (a x + 1\right )^{2} - 2 \, a x \log \left (a x + 1\right ) \log \left (a x - 1\right ) - a x \log \left (a x - 1\right )^{2} - 4}{x}\right )} a^{2} - \frac {1}{3} \, {\left (a^{2} \log \left (a^{2} x^{2} - 1\right ) - a^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} a \operatorname {arcoth}\left (a x\right ) - \frac {\operatorname {arcoth}\left (a x\right )^{2}}{3 \, x^{3}} \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}^{2}{\left (a x \right )}}{x^{4}}\, 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 )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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