Optimal. Leaf size=281 \[ -\frac {3 a}{128 \left (1-a^2 x^2\right )^2}-\frac {93 a}{128 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {93}{128} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right ) \]
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Rubi [A]
time = 0.50, antiderivative size = 281, normalized size of antiderivative = 1.00, number
of steps used = 21, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules
used = {6177, 6129, 6037, 6135, 6079, 6095, 6203, 6745, 6103, 6141, 267, 6111, 6107}
\begin {gather*} -\frac {93 a}{128 \left (1-a^2 x^2\right )}-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3}{2} a \text {Li}_3\left (\frac {2}{a x+1}-1\right )-3 a \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {15}{32} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {93}{128} a \tanh ^{-1}(a x)^2+3 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 6037
Rule 6079
Rule 6095
Rule 6103
Rule 6107
Rule 6111
Rule 6129
Rule 6135
Rule 6141
Rule 6177
Rule 6203
Rule 6745
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )^3} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+\frac {1}{4} \left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx+a^2 \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {7}{32} a \tanh ^{-1}(a x)^4+\frac {1}{32} \left (9 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+a^2 \int \frac {\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx-\frac {1}{8} \left (9 a^3\right ) \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{2} \left (3 a^3\right ) \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x^2} \, dx\\ &=-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {9 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {9}{128} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} a \tanh ^{-1}(a x)^4+(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx+\frac {1}{8} \left (9 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+\frac {1}{2} \left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{64} \left (9 a^3\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}-\frac {9 a}{128 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {93}{128} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} a \tanh ^{-1}(a x)^4+(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx-\frac {1}{16} \left (9 a^3\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{4} \left (3 a^3\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}-\frac {93 a}{128 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {93}{128} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-\left (6 a^2\right ) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}-\frac {93 a}{128 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {93}{128} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\left (3 a^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}-\frac {93 a}{128 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {93}{128} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \text {Li}_3\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.43, size = 218, normalized size = 0.78 \begin {gather*} -a \left (-\frac {i \pi ^3}{8}+\tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{a x}-\frac {a x \tanh ^{-1}(a x)^3}{1-a^2 x^2}-\frac {15}{32} \tanh ^{-1}(a x)^4+\frac {3}{8} \cosh \left (2 \tanh ^{-1}(a x)\right )+\frac {3}{4} \tanh ^{-1}(a x)^2 \cosh \left (2 \tanh ^{-1}(a x)\right )+\frac {3 \cosh \left (4 \tanh ^{-1}(a x)\right )}{1024}+\frac {3}{128} \tanh ^{-1}(a x)^2 \cosh \left (4 \tanh ^{-1}(a x)\right )-3 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-3 \tanh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+\frac {3}{2} \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )-\frac {3}{4} \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )-\frac {3}{256} \tanh ^{-1}(a x) \sinh \left (4 \tanh ^{-1}(a x)\right )-\frac {1}{32} \tanh ^{-1}(a x)^3 \sinh \left (4 \tanh ^{-1}(a x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 17.69, size = 351, normalized size = 1.25
method | result | size |
derivativedivides | \(a \left (\frac {15 \arctanh \left (a x \right )^{4}}{32}+\frac {\left (32 \arctanh \left (a x \right )^{3}-24 \arctanh \left (a x \right )^{2}+12 \arctanh \left (a x \right )-3\right ) \left (a x +1\right )^{2}}{2048 \left (a x -1\right )^{2}}-\frac {\left (a x +1\right ) \left (4 \arctanh \left (a x \right )^{3}-6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )-3\right )}{16 \left (a x -1\right )}+\frac {\left (4 \arctanh \left (a x \right )^{3}+6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )+3\right ) \left (a x -1\right )}{16 a x +16}-\frac {\left (32 \arctanh \left (a x \right )^{3}+24 \arctanh \left (a x \right )^{2}+12 \arctanh \left (a x \right )+3\right ) \left (a x -1\right )^{2}}{2048 \left (a x +1\right )^{2}}+\frac {\arctanh \left (a x \right )^{3} \left (a x -1\right )}{a x}-2 \arctanh \left (a x \right )^{3}+3 \arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(351\) |
default | \(a \left (\frac {15 \arctanh \left (a x \right )^{4}}{32}+\frac {\left (32 \arctanh \left (a x \right )^{3}-24 \arctanh \left (a x \right )^{2}+12 \arctanh \left (a x \right )-3\right ) \left (a x +1\right )^{2}}{2048 \left (a x -1\right )^{2}}-\frac {\left (a x +1\right ) \left (4 \arctanh \left (a x \right )^{3}-6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )-3\right )}{16 \left (a x -1\right )}+\frac {\left (4 \arctanh \left (a x \right )^{3}+6 \arctanh \left (a x \right )^{2}+6 \arctanh \left (a x \right )+3\right ) \left (a x -1\right )}{16 a x +16}-\frac {\left (32 \arctanh \left (a x \right )^{3}+24 \arctanh \left (a x \right )^{2}+12 \arctanh \left (a x \right )+3\right ) \left (a x -1\right )^{2}}{2048 \left (a x +1\right )^{2}}+\frac {\arctanh \left (a x \right )^{3} \left (a x -1\right )}{a x}-2 \arctanh \left (a x \right )^{3}+3 \arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 \arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(351\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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^{6} x^{8} - 3 a^{4} x^{6} + 3 a^{2} x^{4} - x^{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^2\,{\left (a^2\,x^2-1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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