Optimal. Leaf size=302 \[ -\frac {3 a^3 x}{8 \left (1-a^2 x^2\right )}-\frac {3}{8} a^2 \tanh ^{-1}(a x)+\frac {3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{2} a^2 \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{2 x}-\frac {3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^4+3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+2 a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-3 a^2 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-3 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a^2 \text {PolyLog}\left (4,-1+\frac {2}{1+a x}\right ) \]
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Rubi [A]
time = 0.66, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 14, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used =
{6177, 6129, 6037, 6135, 6079, 2497, 6095, 6203, 6207, 6745, 6141, 6103, 205, 212}
\begin {gather*} -\frac {3}{2} a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right )-\frac {3}{2} a^2 \text {Li}_4\left (\frac {2}{a x+1}-1\right )-3 a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)^2-3 a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {1}{2} a^2 \tanh ^{-1}(a x)^4+\frac {1}{4} a^2 \tanh ^{-1}(a x)^3+\frac {3}{2} a^2 \tanh ^{-1}(a x)^2-\frac {3}{8} 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 {3 a^3 x}{8 \left (1-a^2 x^2\right )}-\frac {3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\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 205
Rule 212
Rule 2497
Rule 6037
Rule 6079
Rule 6095
Rule 6103
Rule 6129
Rule 6135
Rule 6141
Rule 6177
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 )^2} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x^3 \left (1-a^2 x^2\right )} \, dx\\ &=2 \left (a^2 \int \frac {\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx\right )+a^4 \int \frac {x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x^3} \, dx\\ &=-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{2} (3 a) \int \frac {\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx+2 \left (\frac {1}{4} a^2 \tanh ^{-1}(a x)^4+a^2 \int \frac {\tanh ^{-1}(a x)^3}{x (1+a x)} \, dx\right )-\frac {1}{2} \left (3 a^3\right ) \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} a^2 \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\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+2 \left (\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 )-\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\right )+\frac {1}{2} \left (3 a^4\right ) \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3 a \tanh ^{-1}(a x)^2}{2 x}-\frac {3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx-\frac {1}{4} \left (3 a^3\right ) \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx+2 \left (\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^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\right )\\ &=-\frac {3 a^3 x}{8 \left (1-a^2 x^2\right )}+\frac {3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{2} a^2 \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{2 x}-\frac {3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx-\frac {1}{8} \left (3 a^3\right ) \int \frac {1}{1-a^2 x^2} \, dx+2 \left (\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 )+\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\right )\\ &=-\frac {3 a^3 x}{8 \left (1-a^2 x^2\right )}-\frac {3}{8} a^2 \tanh ^{-1}(a x)+\frac {3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{2} a^2 \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{2 x}-\frac {3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )+2 \left (\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 )-\frac {3}{4} a^2 \text {Li}_4\left (-1+\frac {2}{1+a x}\right )\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 a^3 x}{8 \left (1-a^2 x^2\right )}-\frac {3}{8} a^2 \tanh ^{-1}(a x)+\frac {3 a^2 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{2} a^2 \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{2 x}-\frac {3 a^3 x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {1}{4} a^2 \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{2 x^2}+\frac {a^2 \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+3 a^2 \tanh ^{-1}(a x) \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 )+2 \left (\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 )-\frac {3}{4} a^2 \text {Li}_4\left (-1+\frac {2}{1+a x}\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 215, normalized size = 0.71 \begin {gather*} \frac {1}{32} a^2 \left (\pi ^4+48 \tanh ^{-1}(a x)^2-\frac {48 \tanh ^{-1}(a x)^2}{a x}-\frac {16 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}{a^2 x^2}-16 \tanh ^{-1}(a x)^4+12 \tanh ^{-1}(a x) \cosh \left (2 \tanh ^{-1}(a x)\right )+8 \tanh ^{-1}(a x)^3 \cosh \left (2 \tanh ^{-1}(a x)\right )+96 \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 )-48 \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 )-6 \sinh \left (2 \tanh ^{-1}(a x)\right )-12 \tanh ^{-1}(a x)^2 \sinh \left (2 \tanh ^{-1}(a x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 306.89, size = 447, normalized size = 1.48
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\arctanh \left (a x \right )^{4}}{2}-\frac {\left (a x +1\right ) \left (4 \arctanh \left (a x \right )^{3}-6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )-3\right )}{32 \left (a x -1\right )}-\frac {\left (4 \arctanh \left (a x \right )^{3}+6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )+3\right ) \left (a x -1\right )}{32 \left (a x +1\right )}+\frac {\arctanh \left (a x \right )^{2} \left (a x \arctanh \left (a x \right )+\arctanh \left (a x \right )+3 a x \right ) \left (a x -1\right )}{2 a^{2} x^{2}}-3 \arctanh \left (a x \right )^{2}+3 \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \arctanh \left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \arctanh \left (a x \right ) \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+12 \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \arctanh \left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right )^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \arctanh \left (a x \right ) \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+12 \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(447\) |
default | \(a^{2} \left (-\frac {\arctanh \left (a x \right )^{4}}{2}-\frac {\left (a x +1\right ) \left (4 \arctanh \left (a x \right )^{3}-6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )-3\right )}{32 \left (a x -1\right )}-\frac {\left (4 \arctanh \left (a x \right )^{3}+6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )+3\right ) \left (a x -1\right )}{32 \left (a x +1\right )}+\frac {\arctanh \left (a x \right )^{2} \left (a x \arctanh \left (a x \right )+\arctanh \left (a x \right )+3 a x \right ) \left (a x -1\right )}{2 a^{2} x^{2}}-3 \arctanh \left (a x \right )^{2}+3 \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \arctanh \left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \arctanh \left (a x \right ) \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+12 \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 \arctanh \left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right )^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-12 \arctanh \left (a x \right ) \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+12 \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(447\) |
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 )}}{x^{3} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, 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 )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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