Optimal. Leaf size=200 \[ \frac {3}{2} a^2 \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^4+3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )-\frac {3}{4} a^2 \text {PolyLog}\left (4,-1+\frac {2}{1+a x}\right ) \]
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Rubi [A]
time = 0.34, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6129, 6037,
6135, 6079, 2497, 6095, 6203, 6207, 6745} \begin {gather*} -\frac {3}{2} a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right )-\frac {3}{4} a^2 \text {Li}_4\left (\frac {2}{a x+1}-1\right )-\frac {3}{2} a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)^2-\frac {3}{2} a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {1}{4} a^2 \tanh ^{-1}(a x)^4+\frac {1}{2} a^2 \tanh ^{-1}(a x)^3+\frac {3}{2} a^2 \tanh ^{-1}(a x)^2+a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3+3 a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)^3}{2 x^2}-\frac {3 a \tanh ^{-1}(a x)^2}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2497
Rule 6037
Rule 6079
Rule 6095
Rule 6129
Rule 6135
Rule 6203
Rule 6207
Rule 6745
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{x^3 \left (1-a^2 x^2\right )} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x^3} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^4+\frac {1}{2} (3 a) \int \frac {\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx+a^2 \int \frac {\tanh ^{-1}(a x)^3}{x (1+a x)} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )+\frac {1}{2} (3 a) \int \frac {\tanh ^{-1}(a x)^2}{x^2} \, dx+\frac {1}{2} \left (3 a^3\right ) \int \frac {\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx-\left (3 a^3\right ) \int \frac {\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {3 a \tanh ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx+\left (3 a^3\right ) \int \frac {\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {3}{2} a^2 \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )+\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx+\frac {1}{2} \left (3 a^3\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {3}{2} a^2 \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^4+3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )-\frac {3}{4} a^2 \text {Li}_4\left (-1+\frac {2}{1+a x}\right )-\left (3 a^3\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {3}{2} a^2 \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{2 x}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^4+3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )-\frac {3}{4} a^2 \text {Li}_4\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 165, normalized size = 0.82 \begin {gather*} -\frac {1}{64} a^2 \left (-\pi ^4-96 \tanh ^{-1}(a x)^2+\frac {96 \tanh ^{-1}(a x)^2}{a x}+\frac {32 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}{a^2 x^2}+16 \tanh ^{-1}(a x)^4-192 \tanh ^{-1}(a x) \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )-64 \tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+96 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )-96 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+96 \tanh ^{-1}(a x) \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )-48 \text {PolyLog}\left (4,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. \(368\) vs.
\(2(182)=364\).
time = 85.05, size = 369, normalized size = 1.84
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\arctanh \left (a x \right )^{4}}{4}+\frac {\arctanh \left (a x \right )^{2} \left (a x \arctanh \left (a x \right )+\arctanh \left (a x \right )+3 a x \right ) \left (a x -1\right )}{2 a^{2} x^{2}}+\arctanh \left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \arctanh \left (a x \right ) \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\arctanh \left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right )^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \arctanh \left (a x \right ) \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 \arctanh \left (a x \right )^{2}+3 \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(369\) |
default | \(a^{2} \left (-\frac {\arctanh \left (a x \right )^{4}}{4}+\frac {\arctanh \left (a x \right )^{2} \left (a x \arctanh \left (a x \right )+\arctanh \left (a x \right )+3 a x \right ) \left (a x -1\right )}{2 a^{2} x^{2}}+\arctanh \left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \arctanh \left (a x \right ) \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\arctanh \left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right )^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \arctanh \left (a x \right ) \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 \arctanh \left (a x \right )^{2}+3 \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(369\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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}^{3}{\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 = -0.00 \begin {gather*} -\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{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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