Optimal. Leaf size=305 \[ \frac {3 a^2 \tanh ^{-1}(a x)^2}{2 c}-\frac {3 a \tanh ^{-1}(a x)^2}{2 c x}-\frac {a^2 \tanh ^{-1}(a x)^3}{2 c}-\frac {\tanh ^{-1}(a x)^3}{2 c x^2}+\frac {a \tanh ^{-1}(a x)^3}{c x}+\frac {3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a^2 \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a^2 \tanh ^{-1}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a^2 \text {PolyLog}\left (4,-1+\frac {2}{1+a x}\right )}{4 c} \]
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Rubi [A]
time = 0.52, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6081, 6037,
6129, 6135, 6079, 2497, 6095, 6203, 6745, 6207} \begin {gather*} -\frac {3 a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right )}{2 c}+\frac {3 a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right )}{2 c}-\frac {3 a^2 \text {Li}_4\left (\frac {2}{a x+1}-1\right )}{4 c}-\frac {3 a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)^2}{2 c}+\frac {3 a^2 \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)}{c}-\frac {3 a^2 \text {Li}_3\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)}{2 c}-\frac {a^2 \tanh ^{-1}(a x)^3}{2 c}+\frac {3 a^2 \tanh ^{-1}(a x)^2}{2 c}+\frac {a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{c}-\frac {3 a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2}{c}+\frac {3 a^2 \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)}{c}-\frac {\tanh ^{-1}(a x)^3}{2 c x^2}+\frac {a \tanh ^{-1}(a x)^3}{c x}-\frac {3 a \tanh ^{-1}(a x)^2}{2 c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2497
Rule 6037
Rule 6079
Rule 6081
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 (c+a c x)} \, dx &=-\left (a \int \frac {\tanh ^{-1}(a x)^3}{x^2 (c+a c x)} \, dx\right )+\frac {\int \frac {\tanh ^{-1}(a x)^3}{x^3} \, dx}{c}\\ &=-\frac {\tanh ^{-1}(a x)^3}{2 c x^2}+a^2 \int \frac {\tanh ^{-1}(a x)^3}{x (c+a c x)} \, dx-\frac {a \int \frac {\tanh ^{-1}(a x)^3}{x^2} \, dx}{c}+\frac {(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx}{2 c}\\ &=-\frac {\tanh ^{-1}(a x)^3}{2 c x^2}+\frac {a \tanh ^{-1}(a x)^3}{c x}+\frac {a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x^2} \, dx}{2 c}-\frac {\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx}{c}+\frac {\left (3 a^3\right ) \int \frac {\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 c}-\frac {\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}{c}\\ &=-\frac {3 a \tanh ^{-1}(a x)^2}{2 c x}-\frac {a^2 \tanh ^{-1}(a x)^3}{2 c}-\frac {\tanh ^{-1}(a x)^3}{2 c x^2}+\frac {a \tanh ^{-1}(a x)^3}{c x}+\frac {a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx}{c}-\frac {\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx}{c}+\frac {\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}{c}\\ &=\frac {3 a^2 \tanh ^{-1}(a x)^2}{2 c}-\frac {3 a \tanh ^{-1}(a x)^2}{2 c x}-\frac {a^2 \tanh ^{-1}(a x)^3}{2 c}-\frac {\tanh ^{-1}(a x)^3}{2 c x^2}+\frac {a \tanh ^{-1}(a x)^3}{c x}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a^2 \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {\left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{x (1+a x)} \, dx}{c}+\frac {\left (3 a^3\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{2 c}+\frac {\left (6 a^3\right ) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {3 a^2 \tanh ^{-1}(a x)^2}{2 c}-\frac {3 a \tanh ^{-1}(a x)^2}{2 c x}-\frac {a^2 \tanh ^{-1}(a x)^3}{2 c}-\frac {\tanh ^{-1}(a x)^3}{2 c x^2}+\frac {a \tanh ^{-1}(a x)^3}{c x}+\frac {3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {3 a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a^2 \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a^2 \text {Li}_4\left (-1+\frac {2}{1+a x}\right )}{4 c}-\frac {\left (3 a^3\right ) \int \frac {\log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}-\frac {\left (3 a^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac {3 a^2 \tanh ^{-1}(a x)^2}{2 c}-\frac {3 a \tanh ^{-1}(a x)^2}{2 c x}-\frac {a^2 \tanh ^{-1}(a x)^3}{2 c}-\frac {\tanh ^{-1}(a x)^3}{2 c x^2}+\frac {a \tanh ^{-1}(a x)^3}{c x}+\frac {3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{c}+\frac {a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a^2 \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{c}-\frac {3 a^2 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{2 c}+\frac {3 a^2 \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a^2 \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{2 c}-\frac {3 a^2 \text {Li}_4\left (-1+\frac {2}{1+a x}\right )}{4 c}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.39, size = 222, normalized size = 0.73 \begin {gather*} \frac {a^2 \left (-8 i \pi ^3+\pi ^4+96 \tanh ^{-1}(a x)^2-\frac {96 \tanh ^{-1}(a x)^2}{a x}+96 \tanh ^{-1}(a x)^3-\frac {32 \tanh ^{-1}(a x)^3}{a^2 x^2}+\frac {64 \tanh ^{-1}(a x)^3}{a x}-32 \tanh ^{-1}(a x)^4+192 \tanh ^{-1}(a x) \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )-192 \tanh ^{-1}(a x)^2 \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 \left (-2+\tanh ^{-1}(a x)\right ) \tanh ^{-1}(a x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+96 \text {PolyLog}\left (3,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 )}{64 c} \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. \(601\) vs.
\(2(287)=574\).
time = 32.82, size = 602, normalized size = 1.97
method | result | size |
derivativedivides | \(a^{2} \left (-\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 c \,a^{2} x^{2}}-\frac {\arctanh \left (a x \right )^{4}}{2 c}+\frac {\arctanh \left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctanh \left (a x \right ) \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {\arctanh \left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \arctanh \left (a x \right )^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctanh \left (a x \right ) \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {3 \arctanh \left (a x \right )^{2}}{c}+\frac {3 \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {2 \arctanh \left (a x \right )^{3}}{c}-\frac {3 \arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {3 \arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}\right )\) | \(602\) |
default | \(a^{2} \left (-\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 c \,a^{2} x^{2}}-\frac {\arctanh \left (a x \right )^{4}}{2 c}+\frac {\arctanh \left (a x \right )^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctanh \left (a x \right ) \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \polylog \left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {\arctanh \left (a x \right )^{3} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \arctanh \left (a x \right )^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctanh \left (a x \right ) \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \polylog \left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {3 \arctanh \left (a x \right )^{2}}{c}+\frac {3 \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {3 \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {2 \arctanh \left (a x \right )^{3}}{c}-\frac {3 \arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {3 \arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}-\frac {6 \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}+\frac {6 \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}\right )\) | \(602\) |
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*} \frac {\int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a x^{4} + x^{3}}\, dx}{c} \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 (c+a\,c\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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