Optimal. Leaf size=84 \[ -\frac {a}{2 x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6129, 6037,
331, 212, 6135, 6079, 2497} \begin {gather*} -\frac {1}{2} a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right )+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2+\frac {1}{2} a^2 \tanh ^{-1}(a x)+a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{2 x^2}-\frac {a}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 331
Rule 2497
Rule 6037
Rule 6079
Rule 6129
Rule 6135
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx+\int \frac {\tanh ^{-1}(a x)}{x^3} \, dx\\ &=-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2+\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )} \, dx+a^2 \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx\\ &=-\frac {a}{2 x}-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{2} a^3 \int \frac {1}{1-a^2 x^2} \, dx-a^3 \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {a}{2 x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )-\frac {1}{2} a^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 60, normalized size = 0.71 \begin {gather*} -\frac {1}{2} a^2 \left (\frac {1}{a x}-\tanh ^{-1}(a x) \left (1-\frac {1}{a^2 x^2}+\tanh ^{-1}(a x)+2 \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )\right )+\text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )\right ) \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. \(169\) vs.
\(2(74)=148\).
time = 0.38, size = 170, normalized size = 2.02
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\arctanh \left (a x \right )}{2 a^{2} x^{2}}+\arctanh \left (a x \right ) \ln \left (a x \right )-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {1}{2 a x}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\dilog \left (a x \right )}{2}-\frac {\dilog \left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}\right )\) | \(170\) |
default | \(a^{2} \left (-\frac {\arctanh \left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\arctanh \left (a x \right )}{2 a^{2} x^{2}}+\arctanh \left (a x \right ) \ln \left (a x \right )-\frac {\arctanh \left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\ln \left (a x -1\right )}{4}+\frac {\ln \left (a x +1\right )}{4}-\frac {1}{2 a x}+\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (a x -1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\dilog \left (a x \right )}{2}-\frac {\dilog \left (a x +1\right )}{2}-\frac {\ln \left (a x \right ) \ln \left (a x +1\right )}{2}\right )\) | \(170\) |
risch | \(-\frac {a}{2 x}-\frac {a^{2} \ln \left (a x \right )}{4}+\frac {a^{2} \ln \left (a x +1\right )}{4}-\frac {\ln \left (a x +1\right )}{4 x^{2}}-\frac {a^{2} \ln \left (a x +1\right )^{2}}{8}-\frac {a^{2} \dilog \left (a x +1\right )}{2}-\frac {a^{2} \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{4}+\frac {a^{2} \dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {a^{2} \ln \left (-a x \right )}{4}-\frac {a^{2} \ln \left (-a x +1\right )}{4}+\frac {\ln \left (-a x +1\right )}{4 x^{2}}+\frac {a^{2} \ln \left (-a x +1\right )^{2}}{8}+\frac {a^{2} \dilog \left (-a x +1\right )}{2}+\frac {a^{2} \ln \left (\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (-a x +1\right )}{4}-\frac {a^{2} \dilog \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs.
\(2 (73) = 146\).
time = 0.27, size = 162, normalized size = 1.93 \begin {gather*} \frac {1}{8} \, {\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 - \frac {1}{2} \, {\left (a^{2} \log \left (a^{2} x^{2} - 1\right ) - a^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} \operatorname {artanh}\left (a x\right ) \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 {atanh}{\left (a x \right )}}{a^{2} x^{5} - x^{3}}\, 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 {atanh}\left (a\,x\right )}{x^3\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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